# On the Hinch-Kim dualism between singularity and Faxén operators in the
hydromechanics of arbitrary bodies in Stokes flows

Giuseppe Procopio  Massimiliano Giona<EMAIL_ADDRESS>Dipartimento di Ingegneria Chimica Materiali Ambiente, Sapienza Università di
Roma, via Eudossiana 18, Rome 00184, Italy

###### Abstract

We generalize the multipole expansion and the structure of the Faxén operator
in Stokes flows obtained for bodies with no-slip to generic boundary
conditions, addressing the assumptions under which this generalization is
conceivable. We show that a disturbance field generated by a body immersed in
an ambient flow can be expressed as a multipole expansion the coefficients of
which are the moments of the volume forces, independently on the boundary
conditions. We find that the dualism between the operator giving the
disturbance field of an $n$-th order ambient flow and the $n$-th order Faxén
operator, referred to as the Hinch-Kim dualism, holds only if the boundary
conditions satisfy a property that we call Boundary-Condition reciprocity (BC-
reciprocity). If this property is fulfilled, the Faxén operators can be
expressed in terms of the $(m,n)$-th order geometrical moments of the volume
forces (defined in the article). In addition, it is shown that in these cases,
the hydromechanics of the fluid-body system is completely determined by the
entire set of the Faxén operators. Finally, classical boundary conditions of
hydrodynamic practice are investigated in the light of this property: boundary
conditions for rigid bodies, Newtonian drops at the mechanical equilibrium,
porous bodies modeled by the Brinkman equations are BC-reciprocal, while
deforming linear elastic bodies, deforming Newtonian drops, non-Newtonian
drops and porous bodies modeled by the Darcy equations do not have this
property. For Navier-slip boundary conditions on a rigid body, we find the
analytical expression for low order Faxén operators.

##  I Introduction

The detailed description of the low-Reynolds number hydromechanics of
particles immersed in a fluid is highly significant in rheology and colloidal
science [1, 2, 3], fundamental statistical physics [4, 5, 6, 7], biological
locomotion and life science [10, 8, 9], microfluidics [12, 11, 13], etc.

One of the most powerful mathematical tools to investigate particle-fluid
interactions is, when available, the so called Faxén operator, which is the
operator that once applied to a generic ambient flow (defined as the flow of
the fluid without the disturbance due to the particle inclusion), furnishes
forces, torques, stresses and higher-order moments on the particle, without
the need of solving the specific hydrodynamic problem. The introduction of
this operator is originally due to Hilding Faxén (hence the name, see the book
by Oseen [14] or by Happel and Brenner [15]), who found that the force acting
on a sphere with no-slip boundary conditions immersed in a generic ambient
flow can be expressed in a simple way in terms of the value of the ambient
field and of its Laplacian at the center of the sphere. Moreover, the torque
is proportional to the vorticity of the ambient field at the center of the
sphere. The Faxén laws for the no-slip sphere are, essentially, an application
to the solutions of the Stokes equations [16] of the mean value theorem for
biharmonic (the velocity solution of the Stokes equations) and harmonic (the
vorticity) functions, yielding, respectively, the $0$-th and the asymmetric
$1$-st order moments of surface traction on the surface of the sphere.

Thereafter, many authors have obtained Faxén operators for several
combinations of surface moments, shapes of immersed bodies, boundary
conditions, fluid regimes. In the case of the stationary Stokes regime,
literature results include the analytic expressions for the Faxén operators of
lower orders, specifically: the symmetric $1$-st order operator for a sphere
with no-slip boundary conditions [17], $0$-th and $1$-st order for spheroids
[18, 19] and, more generally for ellipsoids [20, 21] with no-slip boundary
conditions, $0$-th and the Faxén operator for the Stresslet contribution for a
spherical Newtonian drop [22, 23], $0$-th, asymmetric $1$-st order operators
and the Faxén operator for the Stresslet contribution for a sphere with
Navier-slip boundary conditions [24, 25], $0$-th and asymmetric $1$-st order
operators for porous spheres using the Darcy model [26], and $0$-th and $1$-st
order using the Brinkman model [28, 27]. Faxén operators for spheres has been
obtained in other flow regimes: $0$-th and asymmetric $1$-st order operators
for a sphere with no-slip boundary conditions in unsteady Stokes flow [29, 30,
31] and for the linearized compressible Navier-Stokes flow [32], $0$-th order
operator for a spherical Newtonian drop [33], $0$-th and asymmetric $1$-st
operators for a sphere with Navier-slip boundary conditions in unsteady Stokes
flows [34, 35], $0$-th and $1$-st order operators for porous spheres using the
Brinkman model [36], $0$-th order Faxén counterpart for a sphere in a
potential flow [37].

By definition, a Faxén operator is independent of the ambient flow and of the
viscosity of the fluids, and depends solely on the geometrical structure of
the body and on the parameters specifying the boundary conditions. An explicit
expression of the Faxén operator for forces and torques has been given by
Brenner [20] for bodies with no-slip conditions and arbitrary shape in terms
of an infinite series of differential operators with polidiadic coefficients.
The Brenner coefficients depend only on the geometry of the body and
correspond to: i) the moments of the surface traction associated with the
solution of the Stokes problem for the translating body in the unbounded fluid
(in the case of the Faxén operator for the force), and, ii) the moments of the
surface traction related the solution of the Stokes problem for the rotating
body in the unbounded fluid (in the case of the Faxén operator for the
torque).

Hinch [38] observed that the operator applied to the pole of the unbounded
Green function of the Stokes flow (usually referred to as the Stokeslet [39])
and returning the disturbance field generated by a no-slip sphere in a
symmetric linear flow is exactly the $1$-order symmetric Faxén operator found
by Batchelor and Green [17], thus intuitively concluding that this is not a
simple coincidence but the consequence of the Lorentz reciprocal theorem for
the Stokes flows. The dualism between the singularity representation of the
flow generated by an arbitrary body immersed in a fluid and the Faxén
operators of the body has been proved in a conclusive way by Kim [19] by means
of the Lorentz reciprocal theorem. More precisely, in the case no-slip
boundary conditions at the body surface are assumed, the Faxén operator for
the force of a body with arbitrary shape coincides (up to a multiplicative
constant $8\pi\mu$, where $\mu$ is the viscosity of the fluid) with the
operator that, applied to the pole of the Stokeslet, yields the velocity field
of the fluid due to the translations of the body. The extension of the dualism
between higher order Faxén operators (torques, Stresslet, etc.) and higher
order singularity operators (giving the field for rotations, strains, etc.) is
a straightforward consequence of the Kim’s proof. We refer to this
correspondence as the Hinch-Kim dualism.

Hinch-Kim dualism implies several important consequences of theoretical and
practical interest: i) by solving a single hydrodynamic problem (either
analytically or numerically) it is possible to obtain a Faxén operator even
for particles with complex shape (at least for the leading order terms), ii)
the flow generated by an immersed particle can be represented in a compact way
by its Faxén operator, the leading-order terms of which can be evaluated using
the Brenner polidiadic expansion even for particles with complex shape, iii)
long range particle-particle and particle-channel interactions can be
investigated taking advantage of this symmetry in order to obtain hydrodynamic
properties of complex systems of particles [17, 40, 38, 41, 42], of active
microswimmers near walls [43, 44, 45], of microfluidic flow and separation
devices [46, 47], either applying theoretical approaches or by means of
numerical methods, such as the Stokesian dynamics [48].

The last two decades have seen a growing interest in generalizing the nature
of the boundary conditions, going beyond the no-slip case, and in
investigating the interactions with ambient flows more complex then purely
constant and linear fields. This is mainly due to: i) the rise of
microfluidics [49], where surfaces are chemically treated and the properties
of the resulting solid-liquid interfaces exploited, hinging for a more
detailed hydrodynamic description [50], ii) the development of the
hydromechanics of biological particles [8], where the assumption of rigid
translating and rotating particles equipped with no-slip boundary conditions
is evidently too simplified and limiting, and where it has been verified that
the inclusion of only lower order moments, such as forces, torques and
Stresslets, is not sufficient to explain many interesting hydrodynamics
behaviors of biological particles [51].

The scope of this article is to generalize and extend the results obtained for
the singularity and Faxén operators and their mutual relationships enforcing
no-slip at the solid boundaries to generic boundary conditions and to ambient
flows of any order. This extension yields several novel results related to: i)
the analytic expression for the singularity operator in terms of volume
moments, ii) the definition of an analytic criterion upon which the Hinch-Kim
duality holds, iii) the application of this criterion to a broad class of
boundary conditions of hydrodynamic interest.

The main technical tool in the present theory is the bitensorial
distributional analysis developed in Procopio and Giona[52], in which the
moments with respect to the volume forces acting on the body - instead of the
moments associated with the surface tractions considered in the literature -
are introduced and applied in order to express the singularity expansion of a
disturbance flow. These two hierarchies of moments coincide in the no-slip
case. The advantage of this approach is that it makes it possible to obtain a
general expression for the singularity operator of a disturbance flow in terms
of an infinite series of differential operators with the moments of the volume
forces as coefficients, independently of the boundary conditions assumed at
the fluid-body interface.

The article is organized as follows. Section II briefly reviews the
bitensorial distributional theory of hydrodynamic singularities introduced by
Procopio and Giona [52]. In Section III we define the $(n,m)$-th order
geometrical moments as the $m$-th order moments on the body immersed in an
$n$-th order ambient field, and we show that the $n$-th order singularity
operator of an arbitrary body can be expressed in series of differential
operators with the $(n,m)$-th order geometrical moments as coefficients. In
Section IV, we introduce a parity constraint, referred to as boundary
condition reciprocity, for the boundary conditions assumed at the body-fluid
interface. In Section V, we investigate the Hinch-Kim dualism between $n$-th
order singularity operators and $n$-th order Faxén operator for an arbitrary
body. We show that the dualism is not a general property deriving from the
Lorentz reciprocity theorem, as it applies solely to a subclass of boundary
conditions assumed at the surface of the body satisfying the boundary
condition reciprocity introduced in Section IV. We show that, whenever this
dualism holds (hence reciprocal boundary conditions are considered), the
Brenner expression for the Faxén operators can be generalized by considering
the moments of the volume forces. In addition, it is possible to generalize
also the property found in the article by Procopio and Giona[52], namely that
the hydromechanics of a body in Stokes flows is completely described by the
entire set of its Faxén operators (or geometrical moments). This is a
fundamental result for the development of a theory describing the hydrodynamic
interaction between many bodies based on the knowledge of the hydrodynamics of
each individual component[53], hence represented by their Faxén operators.

A similar investigation of the Hinch-Kim dualism has been carried out by
Dolata and Zia [54] following a method, completely different from the present
approach, based on energetic considerations and expressing the reciprocity
between operators instead of fields. Although their main result (the
conditions under which the dualism hold) can be mapped into the present
theory, these authors have reached some misleading conclusions, such as the
validity of the dualism for porous particles modeled by the Darcy law. In this
article we show that this is not case. In fact, in the second part of this
work (from Section VI to Section VIII) we analyze a broad class of typical
hydrodynamic boundary conditions, determining, case by case, whether the
dualism holds or not. In Section VI, we investigate the boundary conditions at
the solid-fluid interface, finding that the dualism holds for rigid bodies
with Navier-slip boundary conditions (even with a non uniform slip length
along the surface), but not for linear elastic bodies in deformation. In
Section VII, we analyze the dualism for fluid-fluid boundary conditions
finding that it is verified solely for Newtonian drops at the mechanical
equilibrium. In Section VIII, we consider the case of porous bodies, finding
that the dualism applies in the Brinkman model for porous media, but not for
the Darcy model. Finally, we use the analytical approach developed in the
previous Sections to derive a systematic method for obtaining higher-order
Faxén operators for a sphere. By means of this method, we provide a closed-
form expression for the $0$-th (already available in literature [24]), $1$-st
and $2$-nd (to the best of our knowledge, not yet present in the literature)
order Faxén operators for a sphere with Navier-slip boundary conditions,
obtaining, hence, anaclitic expressions for the associated high order flows
around the sphere.

##  II Formulation of the problem

Consider a body immersed in a unbounded Stokes fluid. The domain of the body
is $D_{b}\subset\mathbb{R}^{3}$ with boundaries $\partial D_{b}$ and the
domain of the fluid is $D_{f}\equiv\mathbb{R}^{3}/D_{b}$ with boundaries
$\partial D_{f}\equiv\partial D_{b}\cup\partial D_{\infty}$, where $\partial
D_{\infty}$ is an ideal surface at infinity. The ambient flow of the fluid
(i.e. the flow of the fluid without the body inclusion) is
${\boldsymbol{u}}({\boldsymbol{x}})$ with associated pressure
${p}({\boldsymbol{x}})$ and stress tensor
${\boldsymbol{\pi}}({\boldsymbol{x}})$, solution of the Stokes equations

$\begin{cases}-\nabla\cdot{\boldsymbol{\pi}}({\boldsymbol{x}})=\mu\Delta{\boldsymbol{u}}({\boldsymbol{x}})-\nabla
p({\boldsymbol{x}})=0\\\
\nabla\cdot{\boldsymbol{u}}({\boldsymbol{x}})=0\qquad{\boldsymbol{x}}\in\mathbb{R}^{3}\end{cases}$
(1)

The presence of the body generates a disturbance flow at the boundaries
$\partial D_{b}$ of the body, that we indicate as
${\boldsymbol{w}}^{S}({\boldsymbol{x}})$, and thus, a disturbance flow
${\boldsymbol{w}}({\boldsymbol{x}})$ in the whole domain of the fluid with
associated pressure $q({\boldsymbol{x}})$ and stress tensor
${\boldsymbol{\tau}}({\boldsymbol{x}})$ that are solution of the Stokes
equations

$\begin{cases}-\nabla\cdot{\boldsymbol{\tau}}({\boldsymbol{x}})=\mu\Delta{\boldsymbol{w}}({\boldsymbol{x}})-\nabla
q({\boldsymbol{x}})=0\\\
\nabla\cdot{\boldsymbol{w}}({\boldsymbol{x}})=0\qquad{\boldsymbol{x}}\in
D_{f}\\\
{\boldsymbol{w}}({\boldsymbol{x}})={\boldsymbol{w}}^{S}({\boldsymbol{x}}),\
{\boldsymbol{\tau}}({\boldsymbol{x}})={\boldsymbol{\tau}}^{S}({\boldsymbol{x}})\qquad{\boldsymbol{x}}\in\partial
D_{b}\end{cases}$ (2)

where ${\boldsymbol{\tau}}^{S}({\boldsymbol{x}})$ is the stress tensor of the
disturbance flow at the surface of the body. The only assumption on
${\boldsymbol{w}}^{S}({\boldsymbol{x}})$ is that

$\int_{\partial
D_{b}}{\boldsymbol{w}}^{(s)}({\boldsymbol{x}})\cdot{\boldsymbol{n}}({\boldsymbol{x}})dS=0$
(3)

which is a necessary condition to have a single layer potential expression for
${\boldsymbol{w}}({\boldsymbol{x}})$ [55] and, hence, a multipole expansion.
The total field
$({\boldsymbol{v}}({\boldsymbol{x}}),{\boldsymbol{\sigma}}({\boldsymbol{x}}))=({\boldsymbol{u}}({\boldsymbol{x}}),{\boldsymbol{\pi}}({\boldsymbol{x}}))+({\boldsymbol{w}}({\boldsymbol{x}}),{\boldsymbol{\tau}}({\boldsymbol{x}}))$
is the solution of the Stokes equations

$\begin{cases}-\nabla\cdot{\boldsymbol{\sigma}}({\boldsymbol{x}})=\mu\Delta{\boldsymbol{v}}({\boldsymbol{x}})-\nabla
s({\boldsymbol{x}})=0\\\
\nabla\cdot{\boldsymbol{v}}({\boldsymbol{x}})=0\qquad{\boldsymbol{x}}\in
D_{f}\\\
{\boldsymbol{v}}({\boldsymbol{x}})={\boldsymbol{v}}^{S}({\boldsymbol{x}}),\
{\boldsymbol{\sigma}}({\boldsymbol{x}})={\boldsymbol{\sigma}}^{S}({\boldsymbol{x}})\qquad{\boldsymbol{x}}\in\partial
D_{b}\\\
{\boldsymbol{v}}({\boldsymbol{x}})={\boldsymbol{u}}({\boldsymbol{x}})\qquad{\boldsymbol{x}}\rightarrow\infty\end{cases}$
(4)

$s({\boldsymbol{x}})=p({\boldsymbol{x}})+q({\boldsymbol{x}})$ being the total
pressure field,
${\boldsymbol{v}}^{S}({\boldsymbol{x}})={\boldsymbol{w}}^{S}({\boldsymbol{x}})+{\boldsymbol{u}}({\boldsymbol{x}})$
and
${\boldsymbol{\sigma}}^{S}({\boldsymbol{x}})={\boldsymbol{\tau}}^{S}({\boldsymbol{x}})+{\boldsymbol{\pi}}({\boldsymbol{x}})$
the total velocity field and stress tensor, respectively, at the surface of
the body.

Making use of the Ladyzhenskaya volume potential [55], it is possible to
express the disturbance flow as the solution of the non-homogeneous Stokes
equations defined in the whole domain $\mathbb{R}^{3}$

$\begin{cases}-\nabla\cdot{\boldsymbol{\tau}}({\boldsymbol{x}})=\mu\Delta{\boldsymbol{w}}({\boldsymbol{x}})-\nabla
q({\boldsymbol{x}})=-{\boldsymbol{\psi}}({\boldsymbol{x}})\\\
\nabla\cdot{\boldsymbol{w}}({\boldsymbol{x}})=0\qquad{{\boldsymbol{x}}\in\mathbb{R}^{3}}\end{cases}$
(5)

${\boldsymbol{\psi}}({\boldsymbol{x}})$ being any force field distribution,
with compact support in the domain of the body $D_{b}$, satisfying the
relation

$\dfrac{1}{8\pi\mu}\int_{D_{b}}\psi_{\alpha}({\boldsymbol{\xi}})S_{a\,\alpha}({\boldsymbol{x}},{\boldsymbol{\xi}})dV({\boldsymbol{\xi}})={w}_{a}^{S}({\boldsymbol{x}})\qquad{\boldsymbol{x}}\in\partial
D_{b}$ (6)

where $dV({\boldsymbol{\xi}})$ is the volume element at ${\boldsymbol{\xi}}$
and $S_{a\,\alpha}({\boldsymbol{x}},{\boldsymbol{\xi}})$ are the entries of
the Oseen bitensor or Stokeslet. Since
${\boldsymbol{\psi}}({\boldsymbol{\xi}})$ admits compact support localized on
$D_{b}$, the integration in eq. (6) can be performed equivalently either on
$D_{b}$ or on $\mathbb{R}^{3}$. The Stokeslet
$S_{a\,\alpha}({\boldsymbol{x}},{\boldsymbol{\xi}})$ is the Green function of
the Stokes equations in the unbounded domain, thus it is solution of the
equations [52]

$\begin{cases}-\nabla_{b}\Sigma_{ab\alpha}({\boldsymbol{x}},{\boldsymbol{\xi}})=\Delta
S_{a\,\alpha}({\boldsymbol{x}},{\boldsymbol{\xi}})-\nabla_{a}P_{\alpha}({\boldsymbol{x}},{\boldsymbol{\xi}})=-8\pi\delta_{a\alpha}\delta({\boldsymbol{x}}-{\boldsymbol{\xi}})\\\
\nabla_{a}S_{a\,\alpha}({\boldsymbol{x}},{\boldsymbol{\xi}})=0\\\ \end{cases}$
(7)

with $\Sigma_{ab\,\alpha}({\boldsymbol{x}},{\boldsymbol{\xi}})$ and
$P_{\alpha}({\boldsymbol{x}},{\boldsymbol{\xi}})$ the associated stress tensor
and pressure. The explicit solution of eqs. (7) is [16, 39, 55, 56]

$\displaystyle S_{a\alpha}({\boldsymbol{x}},{\boldsymbol{\xi}})$
$\displaystyle=$
$\displaystyle\dfrac{\delta_{a\alpha}}{r}+\dfrac{({\boldsymbol{x}}-{\boldsymbol{\xi}})_{a}({\boldsymbol{x}}-{\boldsymbol{\xi}})_{\alpha}}{r^{3}}$
$\displaystyle P_{\alpha}({\boldsymbol{x}},{\boldsymbol{\xi}})$
$\displaystyle=$ $\displaystyle
2\dfrac{({\boldsymbol{x}}-{\boldsymbol{\xi}})_{\alpha}}{r^{3}}$ (8)
$\displaystyle\Sigma_{ab\alpha}({\boldsymbol{x}},{\boldsymbol{\xi}})$
$\displaystyle=$ $\displaystyle
6\dfrac{({\boldsymbol{x}}-{\boldsymbol{\xi}})_{a}({\boldsymbol{x}}-{\boldsymbol{\xi}})_{b}({\boldsymbol{x}}-{\boldsymbol{\xi}})_{\alpha}}{r^{5}}$

In eqs. (6)-(8) and throughout this article the Einstein summation convention
for repeated indexes is adopted, as well as the distinction between indexes
according to the bitensorial formalism proposed by Procopio and Giona[52].
Therefore, Greek letters ($\alpha,\beta,...=1,2,3$) are used for indexes
referred to field entries at the source point ${\boldsymbol{\xi}}$, while
Latin letters ($a,b,...=1,2,3$) apply for indexes referring to field entries
at the field point ${\boldsymbol{x}}$. Consistently, the operator $\nabla_{a}$
represents the entries of the gradient with respect to the field-point
coordinate ${\boldsymbol{x}}$, while the operator $\nabla_{\alpha}$ indicates
the entries of the gradient with respect to the source point
${\boldsymbol{\xi}}$. Since we are analyzing bodies immersed in a unbounded
fluid and since, for the sake of simplicity, we consider the poles of
singularities located at a single point, it is always possible to express both
source and field points in the same Cartesian coordinate system, and thus the
parallel propagator [52, 57] is simply
$g_{a\alpha}({\boldsymbol{x}},{\boldsymbol{\xi}})=\delta_{a\alpha}$. However,
behind the formal correctness in distinguishing entries at different points,
the use of the bitensorial convention provides, apart from a higher notational
clearness, some practical advantages. Specifically, i) it provides a direct
extension of the results obtained in simple systems to more complex
geometries; ii) in many cases it is useful to consider poles lying on a
manifold belonging to the domain of the body, such as for ellipsoids [39] in
order to avoid infinite series in the singularity expansion. In these
situations bitensorial formalism is necessary in order to distinguish the
coordinate system describing the pole manifold from the coordinates associated
with the field point; iii) the points where the operators are applied are
naturally specified; iv) different properties and symmetries between the
entries are clearly highlighted.

Let us define the $n$-th order moments

$M_{\alpha\,{\boldsymbol{\alpha}}_{n}}({\boldsymbol{\xi}})=\int_{{D_{b}}}({\boldsymbol{x}}-{\boldsymbol{\xi}})_{{\boldsymbol{\alpha}}_{n}}\psi_{\alpha}({\boldsymbol{x}})dV({\boldsymbol{x}})$
(9)

where, formally, $\psi_{\alpha}({\boldsymbol{x}})=\delta_{\alpha
a}\psi_{a}({\boldsymbol{x}})$ are the entries of the force field distribution
at the point ${\boldsymbol{x}}$ expressed in the coordinate system of the
point ${\boldsymbol{\xi}}$ and where $dV({\boldsymbol{x}})$ is the volume
element at the point ${\boldsymbol{x}}$. As shown in the article by Procopio
and Giona [52], the classical multipole expansion of a disturbance field for a
body with no-slip boundary conditions (expressed in terms of surface moments
[39]), can be reformulated in terms of volume moments defined by eq. (9). Eq.
(4.3) in the article by Procopio and Giona[52] has been referred to a body
with no-slip boundary conditions. In point of fact, it is sufficient to
exchange
$-{\boldsymbol{u}}({\boldsymbol{\xi}})\rightarrow{\boldsymbol{w}}^{s}({\boldsymbol{\xi}})$
to see that the distributional analysis developed by Procopio and Giona [52]
applies for generic boundary conditions on the surface of the immersed body.
Hence, given the set of volume moments
$M_{\alpha\,{\boldsymbol{\alpha}}_{n}}({\boldsymbol{\xi}})$ for a generic
body, the disturbance field can be expressed as

$w_{a}({\boldsymbol{x}})=\dfrac{1}{8\pi\mu}\sum_{n=0}^{\infty}\dfrac{M_{\alpha\,{\boldsymbol{\alpha}}_{n}}({\boldsymbol{\xi}})}{n!}\nabla_{{\boldsymbol{\alpha}}_{n}}S_{a\,\alpha}({\boldsymbol{x}},{\boldsymbol{\xi}})$
(10)

and similarly for the pressure

$q({\boldsymbol{x}})=\dfrac{1}{8\pi}\sum_{n=0}^{\infty}\dfrac{M_{\alpha\,{\boldsymbol{\alpha}}_{n}}({\boldsymbol{\xi}})}{n!}\nabla_{{\boldsymbol{\alpha}}_{n}}P_{\alpha}({\boldsymbol{x}},{\boldsymbol{\xi}})$
(11)

and for the stress field

$\tau_{ab}({\boldsymbol{x}})=\dfrac{1}{8\pi}\sum_{n=0}^{\infty}\dfrac{M_{\alpha\,{\boldsymbol{\alpha}}_{n}}({\boldsymbol{\xi}})}{n!}\nabla_{{\boldsymbol{\alpha}}_{n}}\Sigma_{a\,b\,\alpha}({\boldsymbol{x}},{\boldsymbol{\xi}})$
(12)

where ${\boldsymbol{\alpha}}_{n}=\alpha_{1}...\alpha_{n}$ is a multi-index,
$({\boldsymbol{x}}-{\boldsymbol{\xi}})_{{\boldsymbol{\alpha}}_{n}}=({\boldsymbol{x}}-{\boldsymbol{\xi}})_{\alpha_{1}}...({\boldsymbol{x}}-{\boldsymbol{\xi}})_{\alpha_{n}}$
and
$\nabla_{{\boldsymbol{\alpha}}_{n}}=\nabla_{\alpha_{1}}...\nabla_{\alpha_{n}}$.

This is a key point for the general analysis developed below. In fact, while
classical expressions for the coefficients of the multipole expansion in terms
of surface integrals are referred to specific boundary conditions (for
example, the coefficients of multipole expansions reported by many authors
[58, 59] are referred to a rigid body with no-slip boundary conditions, while
the coefficients of multipole expansion reported in the book by Kim and
Karrila [39, pp. 48-49] are referred to rigid particles and drops), the volume
moments in eq. (9) giving the multipole expansion in eqs. (10)-(12) are valid
regardless of the boundary conditions assumed on the surface of the body, once
the integral condition eq. (3), deriving by eq. (6) and by the
incompressibility of the Stokeslet, is satisfied. As a consequence, eqs.
(10)-(12) allow us to express disturbance fields around a generic body by
means of a differential operator applied at the pole point of the Stokeslet.
In the the article by Procopio and Giona[52, Appendix B], it is shown that the
moments defined by eq. (9) reduce to the surface moments defined by several
authors [58, 59, 39] in the case no-slip boundary conditions are imposed at
the surface of the body $\partial D_{b}$. The evaluation of the volume moments
directly from their definition eq. (9) is not straightforward as the
distribution ${\boldsymbol{\psi}}({\boldsymbol{x}})$ is in principle not
unique. This issue can be overcome since these moments can be expressed as
surface integrals of the associated total Stokes flows as addressed in the
next Section.

##  III Generalized geometrical moment expansion

Consider a $n$-th order unbounded polynomial ambient Stokes flow, centered at
a point ${\boldsymbol{\xi}}\in D_{b}$ (see the schematic representation in
Fig. 1)

$u^{(n)}_{a}({\boldsymbol{x}},{\boldsymbol{\xi}})=A_{{a}{\boldsymbol{a}}_{n}}({\boldsymbol{x}}-{\boldsymbol{\xi}})_{{\boldsymbol{a}}_{n}}$
(13)

where hereafter the superscript $(n)$ indicates any quantity referred to a
$n$-th order ambient flow and $A_{a\,{\boldsymbol{a}}_{n}}$ is an intensity
constant tensor with $N_{E}=3^{n+1}$ entries.

Figure 1:  Schematic representation of a body immersed in an $0$-th, $1$-st
and $2$-nd order polynomial ambient flow centered at the point
${\boldsymbol{\xi}}^{\prime}$ (blue arrows) as defined by eqs. (13)-(15).
Black arrows represent the position vector with respect the point
${\boldsymbol{\xi}}$ of the force field
${\boldsymbol{\psi}}({\boldsymbol{x}})$ (red arrows) by which geometrical
moments, defined in eq. (27), are evaluated.

To ensure that the ambient flow in eq. (13) satisfies the Stokes eqs. (1), the
entries of $A_{a{\boldsymbol{a}}_{n}}$ should fulfill the incompressibility
constraint

$A_{a{\boldsymbol{a}}_{n}}\nabla_{a}({\boldsymbol{x}}-{\boldsymbol{\xi}})_{{\boldsymbol{a}}_{n}}=0$
(14)

and, for $n\geq 3$, also the harmonicity constraint for the vorticity

$\varepsilon_{abc}\,A_{b\,{\boldsymbol{a}}_{n}}\Delta\nabla_{c}({\boldsymbol{x}}-{\boldsymbol{\xi}})_{{\boldsymbol{a}}_{n}}=0$
(15)

$\varepsilon_{abc}$ being the Ricci-Levi Civita symbol. It can be shown by
means of elementary combinatorial analysis [60], that eqs. (14) and (15)
constitute a system of

$N_{C}=\begin{cases}\dfrac{(n+2)!}{n!\,2}\qquad\qquad\qquad\quad\;n\leq
2\vspace{0.5cm}\\\ \dfrac{(n+2)!}{n!\,2}+\dfrac{n!\,3}{(n-2)!\,2}\qquad n\geq
3\end{cases}$

constraints for the $N_{E}=3^{n+1}$ entries of $A_{a{\boldsymbol{a}}_{n}}$.
Therefore, since $N_{E}\geq N_{C}\quad(\forall n\in\mathbb{N})$, the ambient
flow defined by eqs. (13)-(15) exists for all $n\in\mathbb{N}$. Specifically,
the flows defined by eqs. (13)-(15), represent a more general class of the
external singularities (singular at infinity) defined by Chwang and Wu [61].
For example, the fundamental external singularity defined in their work [61],
referred to as the Stokeson, can be obtained by choosing
$A_{aa_{1}a_{2}}=f_{\alpha}(\delta_{a\alpha}\delta_{a_{1}a_{2}}-\delta_{a_{1}\alpha}\delta_{aa_{2}})$,
$f_{\alpha}$ being the arbitrary intensities.

The constraint eq. (15) ensures, by the Schwarz’s theorem, that there exists a
scalar field $p^{(n)}({\boldsymbol{x}})$ (the $n$-th order ambient pressure)
such that

$\nabla_{a}p^{(n)}({\boldsymbol{x}},{\boldsymbol{\xi}})=\mu
A_{a\,{\boldsymbol{a}}_{n}}\Delta({\boldsymbol{x}}-{\boldsymbol{\xi}})_{{\boldsymbol{a}}_{n}}$
(16)

from which, letting ${\boldsymbol{x}}_{0}$ be an arbitrary point at which the
the pressure vanishes, we have

$p^{(n)}({\boldsymbol{x}},{\boldsymbol{\xi}})=\mu
A_{a\,{\boldsymbol{a}}_{n}}\int_{\Gamma({\boldsymbol{x}}_{0},{\boldsymbol{x}})}\Delta({\boldsymbol{y}}-{\boldsymbol{\xi}})_{{\boldsymbol{a}}_{n}}d{\boldsymbol{y}}_{a}$
(17)

where $\Gamma({\boldsymbol{x}}_{0},{\boldsymbol{x}})$ is any Lipshitz curve,
the endpoint of which are ${\boldsymbol{x}}_{0}$ and ${\boldsymbol{x}}$.

By defining

$p_{a{\boldsymbol{a}}_{n}}({\boldsymbol{x}},{\boldsymbol{\xi}})=\int_{\Gamma({\boldsymbol{x}}_{0},{\boldsymbol{x}})}\Delta({\boldsymbol{y}}-{\boldsymbol{\xi}})_{{\boldsymbol{a}}_{n}}d{\boldsymbol{y}}_{a}$
(18)

the gradient of which is obviously

$\nabla_{b}\,p_{a{\boldsymbol{a}}_{n}}({\boldsymbol{x}},{\boldsymbol{\xi}})=\delta_{ab}\Delta({\boldsymbol{x}}-{\boldsymbol{\xi}})_{{\boldsymbol{a}}_{n}}$
(19)

and

$\pi_{bca{\boldsymbol{a}}_{n}}({\boldsymbol{x}},{\boldsymbol{\xi}})=\delta_{bc}p_{a{\boldsymbol{a}}_{n}}({\boldsymbol{x}},{\boldsymbol{\xi}})-\left(\delta_{ac}\nabla_{b}({\boldsymbol{x}}-{\boldsymbol{\xi}})_{{\boldsymbol{a}}_{n}}+\delta_{ab}\nabla_{c}({\boldsymbol{x}}-{\boldsymbol{\xi}})_{{\boldsymbol{a}}_{n}}\right)$
(20)

it is possible to exploit the linearity of the Stokes problem in order to
express the pressure $p^{(n)}({\boldsymbol{x}},{\boldsymbol{\xi}})$ and the
stress tensor $\pi^{(n)}_{bc}({\boldsymbol{x}},{\boldsymbol{\xi}})$ of the
$n$-th order ambient flows as

$\displaystyle p^{(n)}({\boldsymbol{x}},{\boldsymbol{\xi}})=\mu
A_{a{\boldsymbol{a}}_{n}}\,p_{a{\boldsymbol{a}}_{n}}({\boldsymbol{x}},{\boldsymbol{\xi}}),\qquad\pi^{(n)}_{bc}({\boldsymbol{x}},{\boldsymbol{\xi}})=\mu
A_{a{\boldsymbol{a}}_{n}}\pi_{bca{\boldsymbol{a}}_{n}}({\boldsymbol{x}},{\boldsymbol{\xi}})$
(21)

where the dependence on the viscosity $\mu$, the intensity
$A_{a{\boldsymbol{a}}_{n}}$ and the geometric fields
$p_{a{\boldsymbol{a}}_{n}}({\boldsymbol{x}},{\boldsymbol{\xi}})$ and
$\pi_{bca{\boldsymbol{a}}_{n}}({\boldsymbol{x}},{\boldsymbol{\xi}})$ has been
factorized. By using this factorization it is possible to express the volume
moments in term of surface integral moments.

To this aim, consider a generic total velocity field for a body immersed in an
ambient field
${\boldsymbol{v}}({\boldsymbol{x}})={\boldsymbol{u}}({\boldsymbol{x}})+{\boldsymbol{w}}({\boldsymbol{x}})$
in the non-homogeneous form defined in $\mathbb{R}^{3}$

$\begin{cases}-\nabla\cdot{\boldsymbol{\sigma}}({\boldsymbol{x}})=\mu\Delta{\boldsymbol{v}}({\boldsymbol{x}})-\nabla
s({\boldsymbol{x}})=-{\boldsymbol{\psi}}({\boldsymbol{x}})\\\
\nabla\cdot{\boldsymbol{v}}({\boldsymbol{x}})=0\qquad{\boldsymbol{x}}\in\mathbb{R}^{3}\\\
{\boldsymbol{v}}({\boldsymbol{x}})={\boldsymbol{u}}({\boldsymbol{x}})\qquad{\boldsymbol{x}}\rightarrow\infty\end{cases}$
(22)

The application of the Lorentz reciprocal theorem in the differential form
[16] to the fields
$({\boldsymbol{u}}^{(n)}({\boldsymbol{x}},{\boldsymbol{\xi}}),{\boldsymbol{\pi}}^{(n)}({\boldsymbol{x}},{\boldsymbol{\xi}}))$
and
$({\boldsymbol{v}}({\boldsymbol{x}}),{\boldsymbol{\sigma}}({\boldsymbol{x}}))$
provides

$u^{(n)}_{a}({\boldsymbol{x}},{\boldsymbol{\xi}})\nabla_{b}\sigma_{ab}({\boldsymbol{x}})-v_{a}({\boldsymbol{x}})\nabla_{b}\pi^{(n)}_{ab}({\boldsymbol{x}},{\boldsymbol{\xi}})=\nabla_{b}\left(u^{(n)}_{a}({\boldsymbol{x}},{\boldsymbol{\xi}})\sigma_{ab}({\boldsymbol{x}})-v_{a}({\boldsymbol{x}})\pi^{(n)}_{ab}({\boldsymbol{x}},{\boldsymbol{\xi}})\right)$
(23)

From eq. (23), considering that
$\nabla_{b}\pi^{(n)}_{bc}({\boldsymbol{x}},{\boldsymbol{\xi}})=0$ and making
use of (13)-(22), it follows that

$\psi_{a}({\boldsymbol{x}})({\boldsymbol{x}}-{\boldsymbol{\xi}})_{{\boldsymbol{a}}_{n}}=\nabla_{b}\left(\sigma_{ab}({\boldsymbol{x}})({\boldsymbol{x}}-{\boldsymbol{\xi}})_{{\boldsymbol{a}}_{n}}-\mu
v_{c}({\boldsymbol{x}})\,\pi_{bca{\boldsymbol{a}}_{n}}({\boldsymbol{x}},{\boldsymbol{\xi}})\right)$
(24)

Integrating the latter equation over the volume of the body, using the Gauss
theorem for the r.h.s of the resulting equation, and enforcing the definition
eq. (9), the moments on volume forces
$M_{\alpha{\boldsymbol{\alpha}}_{n}}({\boldsymbol{\xi}})$ can be expressed as
the surface integrals

$M_{\alpha{\boldsymbol{\alpha}}_{n}}({\boldsymbol{\xi}})=\int_{\partial
D_{b}}\left(\sigma_{\alpha
b}({\boldsymbol{x}})({\boldsymbol{x}}-{\boldsymbol{\xi}})_{{\boldsymbol{\alpha}}_{n}}-\mu
v_{c}({\boldsymbol{x}})\,\pi_{bc\alpha{\boldsymbol{\alpha}}_{n}}({\boldsymbol{x}},{\boldsymbol{\xi}})\right)n_{b}({\boldsymbol{x}})dS({\boldsymbol{x}})$
(25)

where $n_{b}({\boldsymbol{x}})$ are the entries of the outwardly oriented
normal unit vector at point ${\boldsymbol{x}}$ of $\partial D_{b}$ and
$\sigma_{\alpha b}({\boldsymbol{x}})=\delta_{\alpha
a}\sigma_{ab}({\boldsymbol{x}})$. Therefore, by using the expression eq. (21)
for $\pi_{bca{\boldsymbol{a}}_{n}}({\boldsymbol{x}},{\boldsymbol{\xi}})$, the
functional relation connecting the moments to the values of
${\boldsymbol{v}}({\boldsymbol{x}})$ and
${\boldsymbol{\sigma}}({\boldsymbol{x}})$ assigned at the boundary of the body
follows

$\displaystyle M_{\alpha{\boldsymbol{\alpha}}_{n}}({\boldsymbol{\xi}})$
$\displaystyle=$ $\displaystyle\int_{\partial
D_{b}}({\boldsymbol{x}}-{\boldsymbol{\xi}})_{{\boldsymbol{\alpha}}_{n}}\sigma_{\alpha
b}({\boldsymbol{x}})n_{b}({\boldsymbol{x}})dS({\boldsymbol{x}})-\mu\int_{\partial
D_{b}}p_{\alpha{\boldsymbol{\alpha}}_{n}}({\boldsymbol{x}},{\boldsymbol{\xi}})v_{b}({\boldsymbol{x}})n_{b}({\boldsymbol{x}})dS({\boldsymbol{x}})$
(26) $\displaystyle+$ $\displaystyle\mu\int_{\partial
D_{b}}\,\left[v_{\alpha}({\boldsymbol{x}})n_{b}({\boldsymbol{x}})\nabla_{b}({\boldsymbol{x}}-{\boldsymbol{\xi}})_{{\boldsymbol{\alpha}}_{n}}+n_{\alpha}({\boldsymbol{x}})v_{c}({\boldsymbol{x}})\nabla_{c}({\boldsymbol{x}}-{\boldsymbol{\xi}})_{{\boldsymbol{\alpha}}_{n}}\right]dS({\boldsymbol{x}})$

We can introduce the geometrical moments
$m_{\alpha{\boldsymbol{\alpha}}_{m}\beta^{\prime}{\boldsymbol{\beta}}^{\prime}_{n}}({\boldsymbol{\xi}},{\boldsymbol{\xi}}^{\prime})$
as defined in the article by Procopio and Giona[52] by the relation

$M^{(n)}_{\alpha{\boldsymbol{\alpha}}_{m}}({\boldsymbol{\xi}},{\boldsymbol{\xi}}^{\prime})=8\pi\mu
A_{\beta^{\prime}{\boldsymbol{\beta}}_{n}^{\prime}}m_{\alpha{\boldsymbol{\alpha}}_{m}\beta^{\prime}{\boldsymbol{\beta}}^{\prime}_{n}}({\boldsymbol{\xi}},{\boldsymbol{\xi}}^{\prime})$
(27)

where
$M^{(n)}_{\alpha{\boldsymbol{\alpha}}_{m}}({\boldsymbol{\xi}},{\boldsymbol{\xi}}^{\prime})$
is the $m$-th order moments on the body with respect to the point
${\boldsymbol{\xi}}$ immersed in the $n$-th order ambient flow centered at the
point ${\boldsymbol{\xi}}^{\prime}$ and the index
$\beta^{\prime}{\boldsymbol{\beta}}^{\prime}_{n}$ refers to the entries at the
point ${\boldsymbol{\xi}}^{\prime}$. In eq. (27) we explicit the functional
dependence of the moments on the pole ${\boldsymbol{\xi}}^{\prime}$ of the
ambient flow and on its order $n$.

A schematic representation of these special hydrodynamic systems is reported
in Fig. 1, where a generic body is immersed in $0$-th, $1$-st and $2$-nd order
ambient flows. The relation between the geometrical moments and the entries of
the grand-resistance matrix [62, 63, 64, 39] is addressed in Appendix C.

The $n$-th order disturbance field
$({\boldsymbol{w}}^{(n)}({\boldsymbol{x}},{\boldsymbol{\xi}}^{\prime}),{\boldsymbol{\tau}}^{(n)}({\boldsymbol{x}},{\boldsymbol{\xi}}^{\prime}))$
of the ambient field
$({\boldsymbol{u}}^{(n)}({\boldsymbol{x}},{\boldsymbol{\xi}}^{\prime}),{\boldsymbol{\pi}}^{(n)}({\boldsymbol{x}},{\boldsymbol{\xi}}^{\prime}))$
centered at the point ${\boldsymbol{\xi}}^{\prime}\in D_{b}$ is the solution
of the Stokes equations

$\begin{cases}-\nabla\cdot{\boldsymbol{\tau}}^{(n)}({\boldsymbol{x}},{\boldsymbol{\xi}}^{\prime})=\mu\Delta{\boldsymbol{w}}^{(n)}({\boldsymbol{x}},{\boldsymbol{\xi}}^{\prime})-\nabla
q^{(n)}({\boldsymbol{x}},{\boldsymbol{\xi}}^{\prime})=0\\\
\nabla\cdot{\boldsymbol{w}}^{(n)}({\boldsymbol{x}},{\boldsymbol{\xi}}^{\prime})=0\qquad{\boldsymbol{x}}\in
D_{f}\\\
{\boldsymbol{w}}^{(n)}({\boldsymbol{x}},{\boldsymbol{\xi}}^{\prime})={\boldsymbol{w}}^{(S,n)}({\boldsymbol{x}},{\boldsymbol{\xi}}^{\prime}),\
{\boldsymbol{\tau}}^{(n)}({\boldsymbol{x}},{\boldsymbol{\xi}}^{\prime})={\boldsymbol{\tau}}^{(S,n)}({\boldsymbol{x}},{\boldsymbol{\xi}}^{\prime})\qquad{\boldsymbol{x}}\in\partial
D_{b}\end{cases}$ (28)

where ${\boldsymbol{w}}^{(S,n)}({\boldsymbol{x}},{\boldsymbol{\xi}}^{\prime})$
and
${\boldsymbol{\tau}}^{(S,n)}({\boldsymbol{x}},{\boldsymbol{\xi}}^{\prime})$
are the $n$-th order disturbance velocity field and stress tensor at the
surface of the body depending on the assigned boundary conditions. Introducing
the singularity operator
$\mathcal{F}_{\alpha\beta^{\prime}{\boldsymbol{\beta}}^{\prime}_{n}}$ defined
starting from the hierarchy of the geometrical moments
$m_{\alpha{\boldsymbol{\alpha}}_{m}\beta^{\prime}{\boldsymbol{\beta}}^{\prime}_{n}}({\boldsymbol{\xi}},{\boldsymbol{\xi}}^{\prime})$

$\mathcal{F}_{\alpha\beta^{\prime}{\boldsymbol{\beta}}^{\prime}_{n}}=\sum_{m=0}^{\infty}\dfrac{m_{\alpha{\boldsymbol{\alpha}}_{m}\beta^{\prime}{\boldsymbol{\beta}}^{\prime}_{n}}({\boldsymbol{\xi}},{\boldsymbol{\xi}}^{\prime})}{m!}\nabla_{{\boldsymbol{\alpha}}_{m}}$
(29)

it is possible to express the $n$-th order disturbance field
$({\boldsymbol{w}}^{(n)}({\boldsymbol{x}},{\boldsymbol{\xi}}^{\prime}),{\boldsymbol{\tau}}^{(n)}({\boldsymbol{x}},{\boldsymbol{\xi}}^{\prime}))$
in the form

$w^{(n)}_{a}({\boldsymbol{x}},{\boldsymbol{\xi}}^{\prime})=A_{\beta^{\prime}{\boldsymbol{\beta}}^{\prime}_{n}}{\mathcal{F}_{\alpha\beta^{\prime}{\boldsymbol{\beta}}^{\prime}_{n}}S_{a\alpha}({\boldsymbol{x}},{\boldsymbol{\xi}})}$
(30)
$q^{(n)}({\boldsymbol{x}},{\boldsymbol{\xi}}^{\prime})=A_{\beta^{\prime}{\boldsymbol{\beta}}^{\prime}_{n}}{\mathcal{F}_{\alpha\beta^{\prime}{\boldsymbol{\beta}}^{\prime}_{n}}P_{\alpha}({\boldsymbol{x}},{\boldsymbol{\xi}})}$
(31)
$\tau^{(n)}_{ab}({\boldsymbol{x}},{\boldsymbol{\xi}}^{\prime})=A_{\beta^{\prime}{\boldsymbol{\beta}}^{\prime}_{n}}{\mathcal{F}_{\alpha\beta^{\prime}{\boldsymbol{\beta}}^{\prime}_{n}}\Sigma_{ab\alpha}({\boldsymbol{x}},{\boldsymbol{\xi}})}$
(32)

The $n$-th order total velocity field is
$({\boldsymbol{v}}^{(n)}({\boldsymbol{x}},{\boldsymbol{\xi}}^{\prime}),{\boldsymbol{\sigma}}^{(n)}({\boldsymbol{x}},{\boldsymbol{\xi}}^{\prime}))=({\boldsymbol{u}}^{(n)}({\boldsymbol{x}},{\boldsymbol{\xi}}^{\prime}),{\boldsymbol{\pi}}^{(n)}({\boldsymbol{x}},{\boldsymbol{\xi}}^{\prime}))+({\boldsymbol{w}}^{(n)}({\boldsymbol{x}},{\boldsymbol{\xi}}^{\prime}),{\boldsymbol{\tau}}^{(n)}({\boldsymbol{x}},{\boldsymbol{\xi}}^{\prime}))$
and its entries can be expressed by enforcing the linearity of the Stokes flow
as
$(v^{(n)}_{a}({\boldsymbol{x}},{\boldsymbol{\xi}}^{\prime}),\sigma^{(n)}_{ab}({\boldsymbol{x}},{\boldsymbol{\xi}}^{\prime}))=A_{\beta^{\prime}{\boldsymbol{\beta}}^{\prime}_{n}}(v_{a\beta^{\prime}{\boldsymbol{\beta}}^{\prime}_{n}}({\boldsymbol{x}},{\boldsymbol{\xi}}^{\prime}),\mu\sigma_{ab\beta^{\prime}{\boldsymbol{\beta}}^{\prime}_{n}}({\boldsymbol{x}},{\boldsymbol{\xi}}^{\prime}))$.
Therefore, from eqs. (26) and (27), the geometrical moments can be evaluated
as the following surface integrals

$\displaystyle
8\pi\,m_{\alpha{\boldsymbol{\alpha}}_{m}\beta^{\prime}{\boldsymbol{\beta}}^{\prime}_{n}}({\boldsymbol{\xi}},{\boldsymbol{\xi}}^{\prime})=$
$\displaystyle\int_{\partial
D_{b}}({\boldsymbol{x}}-{\boldsymbol{\xi}})_{{\boldsymbol{\alpha}}_{m}}\sigma_{\alpha
b\beta^{\prime}{\boldsymbol{\beta}}^{\prime}_{n}}({\boldsymbol{x}},{\boldsymbol{\xi}}^{\prime})n_{b}({\boldsymbol{x}})dS({\boldsymbol{x}})-\int_{\partial
D_{b}}p_{\alpha{\boldsymbol{\alpha}}_{m}}({\boldsymbol{x}},{\boldsymbol{\xi}})v_{b\beta^{\prime}{\boldsymbol{\beta}}^{\prime}_{n}}({\boldsymbol{x}},{\boldsymbol{\xi}}^{\prime})n_{b}({\boldsymbol{x}})dS({\boldsymbol{x}})$
$\displaystyle+\int_{\partial
D_{b}}\,\left(v_{\alpha\beta^{\prime}{\boldsymbol{\beta}}^{\prime}_{n}}({\boldsymbol{x}},{\boldsymbol{\xi}}^{\prime})n_{b}({\boldsymbol{x}})\nabla_{b}({\boldsymbol{x}}-{\boldsymbol{\xi}})_{{\boldsymbol{\alpha}}_{m}}+n_{\alpha}({\boldsymbol{x}})v_{c\beta^{\prime}{\boldsymbol{\beta}}^{\prime}_{n}}({\boldsymbol{x}},{\boldsymbol{\xi}}^{\prime})\nabla_{c}({\boldsymbol{x}}-{\boldsymbol{\xi}})_{{\boldsymbol{\alpha}}_{m}}\right)dS({\boldsymbol{x}})$
(33)

Without loss of generality, we can always consider
${\boldsymbol{\xi}}={\boldsymbol{\xi}}^{\prime}$ in all the case addressed in
the remainder since the distinction between these points is unnecessary. For
example, as regards the geometrical moments
$m_{\alpha{\boldsymbol{\alpha}}_{m}\beta{\boldsymbol{\beta}}_{n}}({\boldsymbol{\xi}},{\boldsymbol{\xi}})=m_{\alpha{\boldsymbol{\alpha}}_{m}\beta^{\prime}{\boldsymbol{\beta}}^{\prime}_{n}}({\boldsymbol{\xi}},{\boldsymbol{\xi}}^{\prime})|_{{\boldsymbol{\xi}}^{\prime}={\boldsymbol{\xi}}}$.

For the sake of notational simplicity, we are assuming an infinite series of
singularities centered at a single pole for describing the disturbance flows,
from which eq. (29) for $\mathcal{F}_{\alpha\beta{\boldsymbol{\beta}}_{n}}$
follows. There are cases, e.g. associated with ellipsoidal bodies [21], where
it is possible to avoid such an infinite series by letting the poles lie on an
manifold $\Omega\subset D_{b}$ belonging to the domain of the body and
following its symmetries. In these cases, the definition of
$\mathcal{F}_{\alpha\beta{\boldsymbol{\beta}}_{n}}$ can be easily extended to
a continuous set of singularities in the form of an integro-differential
linear operator

$\mathcal{F}_{\alpha\beta^{\prime}{\boldsymbol{\beta}}^{\prime}_{n}}=\dfrac{1}{\text{meas}({\Omega})}\int
d\Omega({\boldsymbol{\xi}})\sum_{m=0}^{\infty}\dfrac{m_{\alpha{\boldsymbol{\alpha}}_{m}\beta^{\prime}{\boldsymbol{\beta}}^{\prime}_{n}}({\boldsymbol{\xi}},{\boldsymbol{\xi}}^{\prime})}{m!}\nabla_{{\boldsymbol{\alpha}}_{m}}$
(34)

where $d\Omega({\boldsymbol{\xi}})$ is the measure element at the point
${\boldsymbol{\xi}}$ and

$\text{meas}({\Omega})=\int_{\omega}d\Omega({\boldsymbol{\xi}})$ (35)

is the Lebesgue measure of the manifold $\Omega$. This extension is addressed
in the article by Procopio and Giona [52], and the reader is referred to this
article for the mathematical details.

All the results obtained in the remainder hold regardless of the exact form of
$\mathcal{F}_{\alpha\beta{\boldsymbol{\beta}}_{n}}$, and thus can be extended
to the integro-differential representation eq. (34).

##  IV Reciprocal boundary conditions

Before investigating the relations between the singularity operators
$\mathcal{F}_{\alpha\beta{\boldsymbol{\beta}}_{n}}$ and the $n$-th order Faxén
operators, it is convenient to introduce the concept of BC-reciprocity, which
as shown in the next Section, is crucial in order to establish a dualism
between the two operators.

For the sake of notational conciseness, we indicate with the symbol
$[\cdot,\cdot]$ the bilinear operator acting on two generic Stokes flows ${\bf
v}({\boldsymbol{x}})$ and ${\bf v}^{\prime}({\boldsymbol{x}})$ at the field
point ${\boldsymbol{x}}$, corresponding to the surface integral on the body

$[{\bf v},{\bf v}^{\prime}]=\int_{\partial
D_{b}}\left({\boldsymbol{\sigma}}\\{{\bf
v}^{\prime}({\boldsymbol{x}})\\}\cdot{\bf
v}({\boldsymbol{x}})-{\boldsymbol{\sigma}}\\{{\bf
v}({\boldsymbol{x}})\\}\cdot{\bf
v}^{\prime}({\boldsymbol{x}})\right)\cdot{\boldsymbol{n}}({\boldsymbol{x}})dS({\boldsymbol{x}})$
(36)

where ${\boldsymbol{\sigma}}\\{{\bf v}({\boldsymbol{x}})\\}$ is the stress
tensor related to the field ${\bf v}({\boldsymbol{x}})$.

It is easy to verify that

$[{\bf v},{\bf v}^{\prime}]=-[{\bf v}^{\prime},{\bf v}]$ (37)

and therefore the operator $[\cdot,\cdot]$ admits odd parity. Since the
ambient fields are regular homogeneous solutions of the Stokes equations in
the domain of the body, given two ambient fields
${\boldsymbol{u}}({\boldsymbol{x}})$ and
${\boldsymbol{u}}^{\prime}({\boldsymbol{x}})$, we can simply apply the Gauss
theorem to have

$[{\boldsymbol{u}},{\boldsymbol{u}}^{\prime}]=0$ (38)

Eq. (38) is indeed a direct consequence of the Lorentz reciprocal theorem
applied to the Stokes fluid in the domain $D_{b}$ bounded by $\partial D_{b}$.
In the case we consider two disturbance fields
${\boldsymbol{w}}({\boldsymbol{x}})$ and
${\boldsymbol{w}}^{\prime}({\boldsymbol{x}})$ (not defined in $D_{b}$),
applying the Lorentz reciprocal theorem on the surface $\partial
D_{f}\equiv\partial D_{b}\cup\partial D_{\infty}$ and considering that the
disturbance velocity fields vanish at $\partial D_{\infty}$ as a Stokeslet
${\boldsymbol{w}}({\boldsymbol{x}})=O(1/|{\boldsymbol{x}}|)$ and the stress
tensor disturbance field as a Stresslet, i.e. $O(1/|{\boldsymbol{x}}|^{2})$
[16, 39], we obtain

$[{\boldsymbol{w}},{\boldsymbol{w}}^{\prime}]=\int_{\partial
D_{\infty}}\left({\boldsymbol{\sigma}}\\{{\boldsymbol{w}}^{\prime}({\boldsymbol{x}})\\}\cdot{\boldsymbol{w}}({\boldsymbol{x}})-{\boldsymbol{\sigma}}\\{{\boldsymbol{w}}({\boldsymbol{x}})\\}\cdot{\boldsymbol{w}}^{\prime}({\boldsymbol{x}})\right)\cdot{\boldsymbol{n}}({\boldsymbol{x}})dS({\boldsymbol{x}})=0$
(39)

On the other hand, given two total fields
${\boldsymbol{v}}({\boldsymbol{x}})={\boldsymbol{u}}({\boldsymbol{x}})+{\boldsymbol{w}}({\boldsymbol{x}})$
and
${\boldsymbol{v}}^{\prime}({\boldsymbol{x}})={\boldsymbol{u}}^{\prime}({\boldsymbol{x}})+{\boldsymbol{w}}^{\prime}({\boldsymbol{x}})$,
the quantity $[{\boldsymbol{v}},{\boldsymbol{v}}^{\prime}]$ does not vanish in
general. This result can be proved as follows. By using the bilinear property
of the operator $[.\,,.]$, we can expand
$[{\boldsymbol{v}},{\boldsymbol{v}}^{\prime}]$, hence

$[{\boldsymbol{v}},{\boldsymbol{v}}^{\prime}]=[{\boldsymbol{u}}+{\boldsymbol{w}},\,{\boldsymbol{u}}^{\prime}+{\boldsymbol{w}}^{\prime}]=[{\boldsymbol{u}},{\boldsymbol{u}}^{\prime}]+[{\boldsymbol{u}},{\boldsymbol{w}}^{\prime}]+[{\boldsymbol{w}},{\boldsymbol{u}}^{\prime}]+[{\boldsymbol{w}},{\boldsymbol{w}}^{\prime}]$
(40)

The first and fourth terms at the r.h.s of eq. (40) vanish because of eqs.
(38) and (39). By using the property expressed by eq. (37), the following
identity holds

$[{\boldsymbol{v}},{\boldsymbol{v}}^{\prime}]=[{\boldsymbol{u}},{\boldsymbol{w}}^{\prime}]-[{\boldsymbol{u}}^{\prime},{\boldsymbol{w}}]$
(41)

By applying the Lorentz reciprocal theorem, it is possible to verify that
$[{\boldsymbol{u}},{\boldsymbol{w}}^{\prime}]-[{\boldsymbol{u}}^{\prime},{\boldsymbol{w}}]$
in general does not vanishes at infinity. For example, consider
$[{\boldsymbol{u}},{\boldsymbol{w}}^{\prime}]$ and apply the Lorentz
reciprocal theorem in the volume of the fluid bounded by $\partial
D_{f}=\partial D_{b}\cup\partial D_{\infty}$

$[{\boldsymbol{u}},{\boldsymbol{w}}^{\prime}]=\int_{\partial
D_{\infty}}\left({\boldsymbol{\sigma}}\\{{\boldsymbol{u}}({\boldsymbol{x}})\\}\cdot{\boldsymbol{w}}^{\prime}({\boldsymbol{x}})-{\boldsymbol{\sigma}}\\{{\boldsymbol{w}}^{\prime}({\boldsymbol{x}})\\}\cdot{\boldsymbol{u}}({\boldsymbol{x}})\right)\cdot{\boldsymbol{n}}({\boldsymbol{x}})dS({\boldsymbol{x}})$
(42)

while for $[{\boldsymbol{u}}^{\prime},{\boldsymbol{w}}]$

$[{\boldsymbol{u}}^{\prime},{\boldsymbol{w}}]=\int_{\partial
D_{\infty}}\left({\boldsymbol{\sigma}}\\{{\boldsymbol{u}}^{\prime}({\boldsymbol{x}})\\}\cdot{\boldsymbol{w}}({\boldsymbol{x}})-{\boldsymbol{\sigma}}\\{{\boldsymbol{w}}({\boldsymbol{x}})\\}\cdot{\boldsymbol{u}}^{\prime}({\boldsymbol{x}})\right)\cdot{\boldsymbol{n}}({\boldsymbol{x}})dS({\boldsymbol{x}})$
(43)

Since ${\boldsymbol{u}}({\boldsymbol{x}})$ and
${\boldsymbol{u}}^{\prime}({\boldsymbol{x}})$ should not necessarily decay at
infinity, the integrals at the r.h.s of eqs. (42) and (43) are in general
different from zero, contrarily to what happens for disturbance flows.

To show this unambiguously, consider the simple case where
${\boldsymbol{u}}({\boldsymbol{x}})={\boldsymbol{u}}=\mbox{const}$ and
${\boldsymbol{u}}^{\prime}({\boldsymbol{x}})={\boldsymbol{A}}\cdot({\boldsymbol{x}}-{\boldsymbol{\xi}})$
a linear ambient flow. Let us start analyzing the integral in eq. (42). If the
body is a sphere in a linear ambient flow, the leading order Stokeslet term in
the multipole expansion of ${\boldsymbol{w}}^{\prime}({\boldsymbol{x}})$ is
vanishing due to the spherical symmetries [16, 39]. Therefore, since
${\boldsymbol{w}}({\boldsymbol{x}})=O(1/|{\boldsymbol{x}}|^{2})$ and the
associated stress
${\boldsymbol{\sigma}}\\{{\boldsymbol{w}}^{\prime}({\boldsymbol{x}})\\}=O(1/|{\boldsymbol{x}}|^{3})$
, the integral in eq. (42) vanishes at infinity and
$[{\boldsymbol{u}},{\boldsymbol{w}}^{\prime}]=0$. Apart from this specific
case possessing spherical symmetry, the Stokeslet term in the disturbance flow
implies ${\boldsymbol{w}}^{\prime}({\boldsymbol{x}})=O(1/|{\boldsymbol{x}}|)$
and
${\boldsymbol{\sigma}}\\{{\boldsymbol{w}}^{\prime}({\boldsymbol{x}})\\}=O(1/|{\boldsymbol{x}}|^{2})$.
Therefore, integrating over a sphere with radius $R_{s}$, enclosing the body,
and letting the radius go to infinity, we have in general

$[{\boldsymbol{u}},{\boldsymbol{w}}^{\prime}]\sim\int_{R_{s}\rightarrow\infty}\dfrac{dr^{2}}{r^{2}}\rightarrow\infty$
(44)

On the other hand, the leading order term in the multipole expansion of a
disturbance field associated with a body in a constant ambient flow is a
Stokeslet independently of the symmetries of the problem. Therefore,
considering $[{\boldsymbol{u}}^{\prime},{\boldsymbol{w}}]$ in eq. (43), where
${\boldsymbol{w}}({\boldsymbol{x}})=O(1/|{\boldsymbol{x}}|)$ and
${\boldsymbol{\sigma}}\\{{\boldsymbol{w}}({\boldsymbol{x}})\\}=O(1/|{\boldsymbol{x}}|^{2})$,
we obtain

$[{\boldsymbol{u}}^{\prime},{\boldsymbol{w}}]\sim\int_{R_{s}\rightarrow\infty}\dfrac{dr^{2}}{r}\rightarrow\infty$
(45)

By the different scaling law between eq. (44) and eq. (45), we can also
conclude that the difference
$[{\boldsymbol{u}},{\boldsymbol{w}}^{\prime}]-[{\boldsymbol{u}}^{\prime},{\boldsymbol{w}}]$
does not vanish in general.

The above example shows that it is not possible to determine whether
$[{\boldsymbol{v}},{\boldsymbol{v}}^{\prime}]=0$ for any pair of total fields
${\boldsymbol{v}}({\boldsymbol{x}})$ and
${\boldsymbol{v}}^{\prime}({\boldsymbol{x}})$ by the asymptotic behavior of
Stokes flows at infinity and, hence, without considering the specific
interactions between the Stokes fluid and the body. Therefore, in order to
establish whether $[{\boldsymbol{v}},{\boldsymbol{v}}^{\prime}]=0$, it is
necessary to take into account the boundary conditions assumed at the surface
of the body, possibly considering the equations governing the internal
behavior of the body (if any), for which the Lorentz reciprocal theorem does
not hold in principle.

We call reciprocal boundary conditions, any boundary condition for which

$[{\boldsymbol{v}},{\boldsymbol{v}}^{\prime}]=0,\qquad\forall\
{\boldsymbol{v}}({\boldsymbol{x}}),{\boldsymbol{v}}^{\prime}({\boldsymbol{x}})$
(46)

Equivalently, eq. (46) is the mathematical definition of the property referred
to as Boundary-Condition reciprocity (BC-reciprocity, for short). The BC-
reciprocity of classical boundary conditions employed to describe the
hydrodynamics of Stokes fluid-body interactions is analyzed in Sections VI-
VIII. In the next Section, we show that, in order to fulfill the Hinch-Kim
dualism, it is necessary to assume reciprocal boundary conditions at the
interface between the body and the Stokes external fluid.

##  V Generalized $n$-th order Faxén operator and the Hinch-Kim dualism
theorem

It is possible to express the generic ambient field in the domain of the body
by the Ladyzhenskaya boundary integrals [55]

$u_{\alpha}({\boldsymbol{\xi}})=-\dfrac{[{\boldsymbol{S}}_{\alpha}({\boldsymbol{\xi}}),{\boldsymbol{u}}]}{8\pi\mu}=-\int_{\partial
D_{b}}{\bigg{(}}\pi_{ab}({\boldsymbol{x}})\dfrac{S_{a\alpha}({\boldsymbol{x}},{\boldsymbol{\xi}})}{8\pi\mu}-u_{a}({\boldsymbol{x}})\dfrac{\Sigma_{ab\alpha}({\boldsymbol{x}},{\boldsymbol{\xi}})}{8\pi}{\bigg{)}}n_{b}({\boldsymbol{x}})dS({\boldsymbol{x}})$
(47)

Applying the operator
$A_{\beta{\boldsymbol{\beta}}_{n}}\mathcal{F}_{\alpha\beta{\boldsymbol{\beta}}_{n}}$,
acting on the pole point ${\boldsymbol{\xi}}$ (as indicated by using Greek
subscripts), at both sides of eq. (47), and using the relations (30) one
obtains

$A_{\beta{\boldsymbol{\beta}}_{n}}\mathcal{F}_{\alpha\beta{\boldsymbol{\beta}}_{n}}u_{\alpha}({\boldsymbol{\xi}})=-\dfrac{[A_{\beta{\boldsymbol{\beta}}_{n}}\mathcal{F}_{\alpha\beta{\boldsymbol{\beta}}_{n}}{\boldsymbol{S}}_{\alpha}({\boldsymbol{\xi}}),{\boldsymbol{u}}]}{8\pi\mu}=-\dfrac{[{\boldsymbol{w}}^{(n)}({\boldsymbol{\xi}}),{\boldsymbol{u}}]}{8\pi\mu}$
(48)

It is possible to add at the r.h.s. of eq. (48) the vanishing contribution
$[{\boldsymbol{w}}^{(n)}({\boldsymbol{\xi}}),{\boldsymbol{w}}]$ deriving from
two disturbance fields, thus

$A_{\beta{\boldsymbol{\beta}}_{n}}\mathcal{F}_{\alpha\beta{\boldsymbol{\beta}}_{n}}u_{\alpha}({\boldsymbol{\xi}})=-\dfrac{[{\boldsymbol{w}}^{(n)}({\boldsymbol{\xi}}),{\boldsymbol{u}}]+[{\boldsymbol{w}}^{(n)}({\boldsymbol{\xi}}),{\boldsymbol{w}}]}{8\pi\mu}=-\dfrac{[{\boldsymbol{w}}^{(n)}({\boldsymbol{\xi}}),{\boldsymbol{v}}]}{8\pi\mu}$
(49)

that can be expressed, replacing
${\boldsymbol{w}}^{(n)}({\boldsymbol{x}},{\boldsymbol{\xi}})={\boldsymbol{v}}^{(n)}({\boldsymbol{x}},{\boldsymbol{\xi}})-{\boldsymbol{u}}^{(n)}({\boldsymbol{x}},{\boldsymbol{\xi}})$,
in the form

$A_{\beta{\boldsymbol{\beta}}_{n}}\mathcal{F}_{\alpha\beta{\boldsymbol{\beta}}_{n}}u_{\alpha}({\boldsymbol{\xi}})=\dfrac{[{\boldsymbol{u}}^{(n)}({\boldsymbol{\xi}}),{\boldsymbol{v}}]-[{\boldsymbol{v}}^{(n)}({\boldsymbol{\xi}}),{\boldsymbol{v}}]}{8\pi\mu}$
(50)

Applying $A_{\alpha{\boldsymbol{\alpha}}_{n}}$ on both the sides of eq. (25)

$A_{\alpha{\boldsymbol{\alpha}}_{n}}M_{\alpha{\boldsymbol{\alpha}}_{n}}({\boldsymbol{\xi}})=[{\boldsymbol{u}}^{(n)}({\boldsymbol{\xi}}),{\boldsymbol{v}}]$
(51)

Finally, comparing eq. (51) with eq. (50), we obtain

$M_{\beta{\boldsymbol{\beta}}_{n}}({\boldsymbol{\xi}})=8\pi\mu\mathcal{F}_{\alpha\beta{\boldsymbol{\beta}}_{n}}u_{\alpha}({\boldsymbol{\xi}})+[{\boldsymbol{v}}^{(n)}_{\beta{\boldsymbol{\beta}}_{n}}({\boldsymbol{\xi}}),{\boldsymbol{v}}]$
(52)

Eq. (52) is one of the main result of this article, connecting the Hinch-Kim
duality to the condition of BC-reciprocity. From eq. (52) it is possible to
state that the Hinch-Kim dualism, holds whenever reciprocal boundary
conditions are imposed on the surface of the body, i.e. whenever

$[{\boldsymbol{v}}^{(n)}({\boldsymbol{\xi}}),{\boldsymbol{v}}]=\int_{\partial
D_{b}}\big{(}\sigma_{ab}({\boldsymbol{x}})v^{(n)}_{a}({\boldsymbol{x}},{\boldsymbol{\xi}})-v_{a}({\boldsymbol{x}})\sigma^{(n)}_{ab}({\boldsymbol{x}},{\boldsymbol{\xi}})\big{)}n_{b}({\boldsymbol{x}})dS({\boldsymbol{x}})=0,\qquad{{\forall}\,{\boldsymbol{v}}({\boldsymbol{x}})}$
(53)

where ”$\forall\,{\boldsymbol{v}}({\boldsymbol{x}})$”, meaning that
${\boldsymbol{v}}({\boldsymbol{x}})$ can be an arbitrary solution of the
Stokes equations, is a necessary condition for the Faxén operator to be
independent of the ambient flow.

This can be referred to as the Hinch-Kim dualism theorem. In the BC-reciprocal
case, the $n$-th order singularity operator
$\mathcal{F}_{\alpha\beta{\boldsymbol{\beta}}_{n}}$ defined by eq. (29)
furnishes either the $n$-th order disturbance field, if applied to the pole of
the unbounded Green function according eqs. (30)-(32), or the $n$-th order
moment on a particle immersed in an ambient field
${\boldsymbol{u}}({\boldsymbol{x}})$ according to the relation

$M_{\beta{\boldsymbol{\beta}}_{n}}({\boldsymbol{\xi}})=8\pi\mu\mathcal{F}_{\alpha\beta{\boldsymbol{\beta}}_{n}}u_{\alpha}({\boldsymbol{\xi}}),\qquad{\forall\,{\boldsymbol{u}}({\boldsymbol{x}})}$
(54)

Owing to the fact that the operator
$\mathcal{F}_{\alpha\beta{\boldsymbol{\beta}}_{n}}$ returns the $n$-th order
moments on the body if applied to a generic ambient flow,
$\mathcal{F}_{\alpha\beta{\boldsymbol{\beta}}_{n}}$ is sensu-stricto a $n$-th
order generalized Faxén operator.

Let us to show an interesting consequence of reciprocal boundary conditions.
If BC-reciprocity holds, by using eq. (10), the disturbance field due to the
inclusion of the body related to a generic ambient field
${\boldsymbol{u}}({\boldsymbol{x}})$ can be expressed as

$w_{a}({\boldsymbol{x}})=\dfrac{1}{8\pi\mu}\sum_{n=0}^{\infty}\dfrac{\mathcal{F}_{\beta\alpha{\boldsymbol{\alpha}}_{n}}u_{\beta}({\boldsymbol{\xi}})}{n!}\nabla_{{\boldsymbol{\alpha}}_{n}}S_{a\,\alpha}({\boldsymbol{x}},{\boldsymbol{\xi}})$
(55)

and, as shown by Procopio and Giona [52] and briefly reviewed in Appendix A,
due to the following symmetry of the geometric moments

$m_{\alpha{\boldsymbol{\alpha}}_{m}\beta^{\prime}{\boldsymbol{\beta}}^{\prime}_{n}}({\boldsymbol{\xi}},{\boldsymbol{\xi}}^{\prime})=m_{\beta^{\prime}{\boldsymbol{\beta}}^{\prime}_{n}\alpha{\boldsymbol{\alpha}}_{m}}({\boldsymbol{\xi}}^{\prime},{\boldsymbol{\xi}})$
(56)

we obtain an expansion of a generic disturbance field in terms of the Faxén
operators (i.e. in terms of the geometrical moments)

$w_{a}({\boldsymbol{x}})=\sum_{n=0}^{\infty}\dfrac{\nabla_{{\boldsymbol{\beta}}_{n}}u_{\beta}({\boldsymbol{\xi}})}{n!}\mathcal{F}_{\alpha\beta{\boldsymbol{\beta}}_{n}}S_{a\,\alpha}({\boldsymbol{x}},{\boldsymbol{\xi}})$
(57)

Gathering eqs. (54) and (57) a remarkable property follows, namely if BC-
reciprocity holds, the hydromechanics (i.e. the motion of the body due to the
interaction with the fluid and the motion of the fluid due to the interaction
with the body) of a fluid-body system in the Stokes regime can be completely
described by the knowledge of the entire set of $(m,n)$-th order geometrical
moments of the body.

In the next Sections we analize typical hydrodynamic boundary conditions at
the fluid-body interface in order to ascertain in which cases BC-reciprocity
i.e. eq. (46) is fulfilled and
$\mathcal{F}_{\alpha\beta{\boldsymbol{\beta}}_{n}}$ is a Faxén operator.

##  VI Boundary conditions at solid-fluid interfaces

BC-reciprocity, i.e. $[{\boldsymbol{v}},{\boldsymbol{v}}^{\prime}]=0$, is
straightforwardly verified for no-slip boundary conditions. In fact, if
${\boldsymbol{v}}({\boldsymbol{x}})={\boldsymbol{v}}^{\prime}({\boldsymbol{x}})=0$
for ${\boldsymbol{x}}\in\partial D_{b}$ are assumed (as by definition of no-
slip boundary conditions), the integral in eq. (36) vanishes trivially. This
means that the Hinch-Kim dualism holds for any operator
$\mathcal{F}_{\alpha\beta{\boldsymbol{\beta}}_{n}}$ when bodies with no-slip
boundary conditions are considered. A typical example is the case of a sphere
with radius $R_{p}$ and with no-slip boundary conditions immersed in a
constant ambient flow (hence
${\boldsymbol{v}}^{\prime}({\boldsymbol{x}})\equiv{\boldsymbol{v}}^{(0)}({\boldsymbol{x}})$).
In this case, the disturbance flow is [16]

$w^{(0)}_{a}({\boldsymbol{x}})=-A_{\alpha}\left(\dfrac{3}{4}R_{p}+\dfrac{R_{p}^{3}}{8}\Delta_{x}\right)S_{a\,\alpha}({\boldsymbol{x}},{\boldsymbol{\xi}})$
(58)

and, by the well known dualism [39, 16], the force on the same sphere immersed
in a generic ambient flow ${\boldsymbol{u}}({\boldsymbol{x}})$ is given by

$F_{a}=-M_{a}=8\pi\mu\left(\dfrac{3}{4}R_{p}+\dfrac{R_{p}^{3}}{8}\Delta_{x}\right)u_{a}({\boldsymbol{x}})$
(59)

By eq. (58) and (59), it is possible to identify
$\mathcal{F}_{\alpha\beta}=(3R_{p}/4+R_{p}\Delta/8)\delta_{\alpha\beta}$ as
the operator acting on the ambient flow in eq. (54).

Analogously, the BC-reciprocity is straightforward considering complete slip
boundary conditions, for which
${\boldsymbol{\sigma}}\\{{\boldsymbol{v}}({\boldsymbol{x}})\\}\cdot{\boldsymbol{n}}({\boldsymbol{x}})={\boldsymbol{\sigma}}\\{{\boldsymbol{v}}^{\prime}({\boldsymbol{x}})\\}\cdot{\boldsymbol{n}}({\boldsymbol{x}})=0$
for ${\boldsymbol{x}}\in\partial D_{b}$ are assumed in eq. (36).

It is possible to show that BC-reciprocity holds for any linear relation
between velocity and traction at the boundary (of which no-slip and complete
slip boundary conditions are particular cases) although, in this case, neither
${\boldsymbol{v}}({\boldsymbol{x}})$ and
${\boldsymbol{v}}^{\prime}({\boldsymbol{x}})$, nor
${\boldsymbol{\sigma}}\\{{\boldsymbol{v}}({\boldsymbol{x}})\\}\cdot{\boldsymbol{n}}({\boldsymbol{x}})$
and
${\boldsymbol{\sigma}}\\{{\boldsymbol{v}}^{\prime}({\boldsymbol{x}})\\}\cdot{\boldsymbol{n}}({\boldsymbol{x}})$
vanish at the boundary. To this aim, consider the interfacial mobility matrix
${\boldsymbol{\beta}}({\boldsymbol{x}})$ [65] defined by the relation

${\boldsymbol{\beta}}({\boldsymbol{x}})\cdot{\boldsymbol{v}}({\boldsymbol{x}})={\boldsymbol{\sigma}}({\boldsymbol{x}})\cdot{\boldsymbol{n}}({\boldsymbol{x}}),\qquad{\boldsymbol{x}}\in\partial
D_{b}$ (60)

Due to its symmetry,
${\beta}_{ab}({\boldsymbol{x}})={\beta}_{ba}({\boldsymbol{x}})$, we have that

$\displaystyle[{\boldsymbol{v}}^{(n)}({\boldsymbol{\xi}}),{\boldsymbol{v}}]=\int_{\partial
D_{b}}(\sigma_{ab}({\boldsymbol{x}})n_{b}({\boldsymbol{x}})v^{(n)}_{a}({\boldsymbol{x}},{\boldsymbol{\xi}})-v_{a}({\boldsymbol{x}})\sigma^{(n)}_{ab}({\boldsymbol{x}},{\boldsymbol{\xi}})n_{b}({\boldsymbol{x}}))dS({\boldsymbol{x}})=$
$\displaystyle\int_{\partial
D_{b}}\big{(}v_{b}({\boldsymbol{x}}){\beta}_{ab}({\boldsymbol{x}})v^{(n)}_{a}({\boldsymbol{x}},{\boldsymbol{\xi}})-v^{(n)}_{b}({\boldsymbol{x}},{\boldsymbol{\xi}}){\beta}_{ab}({\boldsymbol{x}})v_{a}({\boldsymbol{x}})\big{)}dS({\boldsymbol{x}})=0\qquad{\forall\,{\boldsymbol{v}}^{(n)}({\boldsymbol{x}}),{\boldsymbol{v}}({\boldsymbol{x}})}$
(61)

Consistently, in this case, the Hinch-Kim dualism holds and
$\mathcal{F}_{\alpha\beta{\boldsymbol{\beta}}_{n}}$ is a Faxén operator.

Next, let us focus on the case of Navier-slip boundary conditions. Thus, given
an ambient field ${\boldsymbol{u}}({\boldsymbol{x}})$, the total field
${\boldsymbol{v}}({\boldsymbol{x}})={\boldsymbol{u}}({\boldsymbol{x}})+{\boldsymbol{w}}({\boldsymbol{x}})$
satisfies at the boundaries of the body the relations

$\begin{cases}{\boldsymbol{v}}({\boldsymbol{x}})\cdot{\boldsymbol{n}}({\boldsymbol{x}})=0\\\
{\boldsymbol{v}}({\boldsymbol{x}})\cdot{\boldsymbol{t}}({\boldsymbol{x}})=-\dfrac{\lambda}{\mu}\,{\boldsymbol{h}}({\boldsymbol{x}})\cdot{\boldsymbol{t}}({\boldsymbol{x}}),\qquad{\boldsymbol{x}}\in\partial
D_{b}\end{cases}$ (62)

where $\lambda$ is the slip length of the interface,
${\boldsymbol{t}}({\boldsymbol{x}})={\boldsymbol{I}}-{\boldsymbol{n}}({\boldsymbol{x}})\otimes{\boldsymbol{n}}({\boldsymbol{x}})$
the unit tangent matrix, and
${\boldsymbol{h}}({\boldsymbol{x}})={\boldsymbol{\sigma}}({\boldsymbol{x}})\cdot{\boldsymbol{n}}({\boldsymbol{x}})$
the surface traction of the total velocity field.

Navier-slip boundary conditions eq. (62) represent a particular case where the
relation between velocity and traction at the boundary of the body is linear,
and from what obtained above, BC-reciprocity applies, meaning that the Hinch-
Kim dualism holds. Therefore, given a generic ambient field
${\boldsymbol{u}}({\boldsymbol{x}})$, the moments on the body are given by eq.
(54), and the disturbance field is expressed by eq. (57). The geometrical
moments that are needed to explicit the Faxén operators can be obtained by
substituting the boundary conditions eq. (62) into eq. (33), and by
considering the geometrical surface traction of the body immersed in a $n$-th
order ambient field
$h_{\alpha\beta{\boldsymbol{\beta}}_{n}}({\boldsymbol{x}},{\boldsymbol{\xi}})=\delta_{\alpha
a}\sigma_{ab\beta{\boldsymbol{\beta}}_{n}}({\boldsymbol{x}},{\boldsymbol{\xi}})n_{b}({\boldsymbol{x}})$,
thus

$\displaystyle
m_{\alpha{\boldsymbol{\alpha}}_{m}\beta{\boldsymbol{\beta}}_{n}}({\boldsymbol{\xi}},{\boldsymbol{\xi}})=$
$\displaystyle\int_{\partial
D_{b}}\dfrac{h_{\gamma\beta{\boldsymbol{\beta}}_{n}}({\boldsymbol{x}},{\boldsymbol{\xi}})}{8\pi}\left(\delta_{\alpha\gamma}({\boldsymbol{x}}-{\boldsymbol{\xi}})_{{\boldsymbol{\alpha}}_{m}}-\lambda(t_{\alpha\gamma}({\boldsymbol{x}})n_{b}({\boldsymbol{x}})\nabla_{b}({\boldsymbol{x}}-{\boldsymbol{\xi}})_{{\boldsymbol{\alpha}}_{m}}+n_{\alpha}({\boldsymbol{x}})t_{c\gamma}({\boldsymbol{x}})\nabla_{c}({\boldsymbol{x}}-{\boldsymbol{\xi}})_{{\boldsymbol{\alpha}}_{m}})\right)dS({\boldsymbol{x}})$
(63)

The $n$-th order surface traction
$h_{\alpha\beta{\boldsymbol{\beta}}_{n}}({\boldsymbol{x}},{\boldsymbol{\xi}})$
can be expressed as

$h_{\alpha\beta{\boldsymbol{\beta}}_{n}}({\boldsymbol{x}},{\boldsymbol{\xi}})=f_{\alpha\beta{\boldsymbol{\beta}}_{n}}({\boldsymbol{x}},{\boldsymbol{\xi}})+n_{\alpha}({\boldsymbol{x}})p_{\beta{\boldsymbol{\beta}}_{n}}({\boldsymbol{x}},{\boldsymbol{\xi}})-\left(n_{\beta}({\boldsymbol{x}})\nabla_{\alpha}({\boldsymbol{x}}-{\boldsymbol{\xi}})_{{\boldsymbol{\beta}}_{n}}+\delta_{\beta\alpha}n_{\gamma}({\boldsymbol{x}})\nabla_{\gamma}({\boldsymbol{x}}-{\boldsymbol{\xi}})_{{\boldsymbol{\beta}}_{n}}\right)$
(64)

where
$f_{\alpha\beta{\boldsymbol{\beta}}_{n}}({\boldsymbol{x}},{\boldsymbol{\xi}})=\delta_{\alpha
a}{\tau}_{ab\beta{\boldsymbol{\beta}}_{n}}({\boldsymbol{x}},{\boldsymbol{\xi}})n_{b}({\boldsymbol{x}})$
is the surface traction related to the $n$-th order disturbance field. In
Section IX, the specific case of a sphere with Navier-slip boundary conditions
is addressed in detail and a method, based on this theoretical approach, in
order to evaluate all the $n$-th order Faxén operators for the sphere in a
systematic way is developed.

On the other hand, it is easy to see that
$\mathcal{F}_{\alpha\beta{\boldsymbol{\beta}}_{n}}$ is not a Faxén operator
for a deforming body. In fact, under the assumption that the body is a linear
elastic material solid, the governing equations for the body deformation are
[66]

$\begin{cases}\nabla\cdot{\boldsymbol{\sigma}}^{[s]}({\boldsymbol{x}})=-\rho^{[s]}\ddot{\boldsymbol{u}}^{[s]}({\boldsymbol{x}})\\\
{\sigma}^{[s]}_{ab}({\boldsymbol{x}})=\delta_{ab}\,\lambda^{[s]}\nabla\cdot{\boldsymbol{u}}^{[s]}({\boldsymbol{x}})+\mu^{[s]}(\nabla_{a}u^{[s]}_{b}({\boldsymbol{x}})+\nabla_{b}u^{[s]}_{a}({\boldsymbol{x}})),\qquad{\boldsymbol{x}}\in
D_{b}\end{cases}$ (65)

where ${\boldsymbol{\sigma}}^{[s]}({\boldsymbol{x}})$ is the stress tensor
field in the solid, ${\boldsymbol{u}}^{[s]}({\boldsymbol{x}})$ the
displacement field of the solid, $\rho^{[s]}$ the solid density, and
$\lambda^{[s]}$ and $\mu^{[s]}$ the Lamé coefficients. In eq. (65) any upper
“dot” indicates the derivative operation with respect to time. Enforcing
continuity conditions at the solid-fluid interface [67]

$\begin{cases}{\boldsymbol{\sigma}}({\boldsymbol{x}})\cdot{\boldsymbol{n}}({\boldsymbol{x}})={\boldsymbol{\sigma}}^{[s]}({\boldsymbol{x}})\cdot{\boldsymbol{n}}({\boldsymbol{x}})\\\
{\boldsymbol{v}}({\boldsymbol{x}})=\dot{\boldsymbol{u}}^{[s]}({\boldsymbol{x}})\qquad{\boldsymbol{x}}\in\partial
D_{b}\end{cases}$ (66)

and substituting eqs. (66) in the first integral in eq. (61), from the
Maxwell-Betti theorem [68, 69] it follows that

$\displaystyle[{\boldsymbol{v}}^{(n)}({\boldsymbol{\xi}}),{\boldsymbol{v}}]$
$\displaystyle=$ $\displaystyle\int_{\partial
D_{b}}\big{(}\sigma^{[s]}_{ab}({\boldsymbol{x}})n_{b}({\boldsymbol{x}})\dot{u}^{[s](n)}_{a}({\boldsymbol{x}},{\boldsymbol{\xi}})-\dot{u}^{[s]}_{a}({\boldsymbol{x}})\sigma^{[s](n)}_{ab}({\boldsymbol{x}},{\boldsymbol{\xi}})n_{b}({\boldsymbol{x}})\big{)}dS({\boldsymbol{x}})$
(67) $\displaystyle=$
$\displaystyle-\rho^{[s]}\int_{D_{b}}\big{(}\ddot{u}^{[s]}_{a}({\boldsymbol{x}})\,\dot{u}^{[s](n)}_{a}({\boldsymbol{x}},{\boldsymbol{\xi}})-\dot{u}^{[s]}_{a}({\boldsymbol{x}})\,\ddot{u}^{[s](n)}_{a}({\boldsymbol{x}},{\boldsymbol{\xi}})\big{)}dV({\boldsymbol{x}})$

which does not vanish in general for any flow
${\boldsymbol{v}}({\boldsymbol{x}})$ and
${\boldsymbol{v}}^{(n)}({\boldsymbol{x}},{\boldsymbol{\xi}})$. BC-reciprocity
is ensured only when we can consider the body at the mechanical equilibrium,
i.e. when ${\nabla\cdot{\boldsymbol{\sigma}}^{[s]}({\boldsymbol{x}})=0}$.

##  VII Boundary conditions at fluid-fluid interfaces

In the presence of a fluid body, the most common linear boundary conditions
assumed at the fluid-fluid interface, considered incompressible and
homogeneous, are [70, 71]

$\begin{cases}{\boldsymbol{v}}({\boldsymbol{x}})={\boldsymbol{v}}^{[i]}({\boldsymbol{x}})\\\
{\boldsymbol{v}}({\boldsymbol{x}})\cdot{\boldsymbol{n}}({\boldsymbol{x}})=\dot{r}({\boldsymbol{x}},t)\\\
{\boldsymbol{\sigma}}({\boldsymbol{x}})\cdot{\boldsymbol{n}}({\boldsymbol{x}})={\boldsymbol{\sigma}}^{[i]}({\boldsymbol{x}})\cdot{\boldsymbol{n}}({\boldsymbol{x}})+\gamma\,{\boldsymbol{n}}({\boldsymbol{x}})\,C({\boldsymbol{x}}),\qquad{\boldsymbol{x}}\in\partial
D_{b}\end{cases}$ (68)

where ${\boldsymbol{v}}^{[i]}({\boldsymbol{x}})$ and
${\boldsymbol{\sigma}}^{[i]}({\boldsymbol{x}})$ are the velocity field and the
stress tensor in the disturbing fluid (say a liquid drop or a gas bubble),
$C({\boldsymbol{x}})$ the trace of the curvature tensor of the surface and
$\gamma$ the surface tension.

Applying the reciprocity integral eq. (36) to the fields
${\boldsymbol{v}}({\boldsymbol{x}})$ and
${\boldsymbol{v}}^{(n)}({\boldsymbol{x}})$, it follows that

$\displaystyle[{\boldsymbol{v}}^{(n)}({\boldsymbol{\xi}}),{\boldsymbol{v}}]=\int_{\partial
D_{b}}\big{(}\sigma_{ab}({\boldsymbol{x}})n_{b}({\boldsymbol{x}})v^{(n)}_{a}({\boldsymbol{x}},{\boldsymbol{\xi}})-v_{a}({\boldsymbol{x}})\sigma^{(n)}_{ab}({\boldsymbol{x}},{\boldsymbol{\xi}})n_{b}({\boldsymbol{x}})\big{)}dS({\boldsymbol{x}})$
$\displaystyle=$ $\displaystyle\int_{\partial
D_{b}}\big{(}\sigma^{[i]}_{ab}({\boldsymbol{x}})n_{b}({\boldsymbol{x}})v^{[i](n)}_{a}({\boldsymbol{x}},{\boldsymbol{\xi}})-v^{[i]}_{a}({\boldsymbol{x}})\sigma^{[i](n)}_{ab}({\boldsymbol{x}},{\boldsymbol{\xi}})n_{b}({\boldsymbol{x}})\big{)}dS({\boldsymbol{x}})$
$\displaystyle+$ $\displaystyle\gamma\int_{\partial
D_{b}}(v^{[i](n)}_{a}({\boldsymbol{x}},{\boldsymbol{\xi}})-v^{[i]}_{a}({\boldsymbol{x}}))C({\boldsymbol{x}})n_{a}({\boldsymbol{x}})dS({\boldsymbol{x}})$
(69)

Since the Lorentz reciprocal theorem is a peculiarity of Newtonian fluids
(and, more generally, of continua characterized by linear relations between
fluxes and thermodynamics forces), the first integral at the r.h.s of eq. (69)
does not vanish, at least in principle, in the non-Newtonian case and
consequently $\mathcal{F}_{\alpha\beta{\boldsymbol{\beta}}_{n}}$ cannot be a
Faxén operator. In the case the disturbing fluid is Newtonian, the first
integral at the r.h.s of eq. (69) vanishes due to the Lorentz reciprocal
theorem for Newtonian fluids, but the second integral does not vanish until
the interface shape does not reach the equilibrium state. In fact, the
velocity at the interface
${\boldsymbol{v}}({\boldsymbol{x}})|_{{\boldsymbol{x}}\in\partial D_{b}}$ is
uniquely determined by the Rallison-Acrivos integral equations once the
ambient field is assigned [72, 73]. Therefore, the second integral at the
r.h.s of eq. (69) does not vanishes for any ambient flows, but solely in the
trivial case of
${\boldsymbol{u}}({\boldsymbol{x}})={\boldsymbol{u}}^{(n)}({\boldsymbol{x}},{\boldsymbol{\xi}})$.

If the disturbing fluid is Newtonian and the shape of the body, say a drop or
a bubble, is stationary
(${\boldsymbol{v}}({\boldsymbol{x}})\cdot{\boldsymbol{n}}({\boldsymbol{x}})|_{{\boldsymbol{x}}\in\partial
D_{b}}=0$), the first integral at the r.h.s of eq. (69) vanishes due to
Lorentz’s reciprocity. Furthermore, since the normal velocity is assumed to be
vanishing at the surface of the body, also the second integral at the r.h.s
vanishes, independently of the shape of the body and of the surface tension.
We obtain $[{\boldsymbol{v}}^{(n)}({\boldsymbol{\xi}}),{\boldsymbol{v}}]=0$
and consequently $\mathcal{F}_{\alpha\beta{\boldsymbol{\beta}}_{n}}$ is, in
this case, a Faxén operator. This means that both the velocity field in the
external fluid and the moments on the drop do not depend directly on the
surface tension at the surface, as surface tension has only an indirect
influence related to the geometry of the stationary shape of the drop.

In this case, in order to evaluate the geometrical moments providing the Faxén
operator, the knowledge either of the set of $n$-th surface velocity fields
${v}_{ab{\boldsymbol{b}}_{n}}({\boldsymbol{x}},{\boldsymbol{\xi}})$ or of
$n$-th external surface traction
$h_{ab{\boldsymbol{b}}_{n}}({\boldsymbol{x}},{\boldsymbol{\xi}})=\sigma_{acb{\boldsymbol{b}}_{n}}({\boldsymbol{x}},{\boldsymbol{\xi}})n_{c}({\boldsymbol{x}})$
is required since

$\displaystyle
8\pi\,m_{\alpha{\boldsymbol{\alpha}}_{m}\beta{\boldsymbol{\beta}}_{n}}({\boldsymbol{\xi}},{\boldsymbol{\xi}})=\int_{\partial
D_{b}}({\boldsymbol{x}}-{\boldsymbol{\xi}})_{{\boldsymbol{\alpha}}_{m}}h_{\alpha\beta{\boldsymbol{\beta}}_{n}}({\boldsymbol{x}},{\boldsymbol{\xi}})dS({\boldsymbol{x}})$
$\displaystyle+\int_{\partial
D_{b}}\,\left(v_{\alpha\beta{\boldsymbol{\beta}}_{n}}({\boldsymbol{x}},{\boldsymbol{\xi}})n_{b}({\boldsymbol{x}})\nabla_{b}({\boldsymbol{x}}-{\boldsymbol{\xi}})_{{\boldsymbol{\alpha}}_{m}}+n_{\alpha}({\boldsymbol{x}})v_{c\beta{\boldsymbol{\beta}}_{n}}({\boldsymbol{x}},{\boldsymbol{\xi}})\nabla_{c}({\boldsymbol{x}}-{\boldsymbol{\xi}})_{{\boldsymbol{\alpha}}_{m}}\right)dS({\boldsymbol{x}})$
(70)

##  VIII Boundary conditions at porous body-fluid interfaces

Next, consider the case the inner flow
$({\boldsymbol{v}}^{[i]}({\boldsymbol{x}}),p^{[i]}({\boldsymbol{x}}))$ inside
a porous body is modeled by means of the Darcy equations [74, 75]

$\begin{cases}{\boldsymbol{v}}^{[i]}({\boldsymbol{x}})=-\dfrac{k}{\mu}\nabla
p^{[i]}({\boldsymbol{x}})\\\
\nabla\cdot{\boldsymbol{v}}^{[i]}({\boldsymbol{x}})=0,\qquad{\boldsymbol{x}}\in
D_{b}\end{cases}$ (71)

where $k$ is the permeability of the porous medium. The boundary condition to
be imposed at the interface are the Beavers-Joseph-Saffman boundary conditions
[76, 77], i.e.,

$\begin{cases}({\boldsymbol{v}}({\boldsymbol{x}})-{\boldsymbol{v}}^{[i]}({\boldsymbol{x}}))\cdot({\boldsymbol{I}}-{\boldsymbol{n}}({\boldsymbol{x}})\otimes{\boldsymbol{n}}({\boldsymbol{x}}))=\dfrac{\sqrt{k}}{\alpha}{\boldsymbol{\sigma}}({\boldsymbol{x}})\cdot({\boldsymbol{I}}-{\boldsymbol{n}}({\boldsymbol{x}})\otimes{\boldsymbol{n}}({\boldsymbol{x}}))\\\
{\boldsymbol{v}}({\boldsymbol{x}})\cdot{\boldsymbol{n}}({\boldsymbol{x}})={\boldsymbol{v}}^{[i]}({\boldsymbol{x}})\cdot{\boldsymbol{n}}({\boldsymbol{x}})\\\
p({\boldsymbol{x}})=p^{[i]}({\boldsymbol{x}}),\qquad{\boldsymbol{x}}\in\partial
D_{b}\end{cases}$ (72)

where $\alpha=\alpha_{0}\,\mu$ and $\alpha_{0}$ is a nondimensional constant
depending on the geometry and topology of the pore structure. In this case,
BC-reciprocity is not satisfied because
$[{\boldsymbol{v}},{\boldsymbol{v}}^{\prime}]$ does not vanish in general,
since

$\displaystyle[{\boldsymbol{v}}^{(n)}({\boldsymbol{\boldsymbol{\xi}}}),{\boldsymbol{v}}]=\int_{\partial
D_{b}}\big{(}\sigma_{ab}({\boldsymbol{x}})n_{b}({\boldsymbol{x}})v^{(n)}_{a}({\boldsymbol{x}},{\boldsymbol{\xi}})-v_{a}({\boldsymbol{x}})\sigma^{(n)}_{ab}({\boldsymbol{x}},{\boldsymbol{\xi}})n_{b}({\boldsymbol{x}})\big{)}dS({\boldsymbol{x}})$
$\displaystyle=$ $\displaystyle-2\mu\int_{\partial
D_{b}}\big{(}e_{ab}\\{{\boldsymbol{v}}({\boldsymbol{x}})\\}v_{a}^{[i](n)}({\boldsymbol{x}},{\boldsymbol{\xi}})-e_{ab}\\{{\boldsymbol{v}}^{(n)}({\boldsymbol{x}},{\boldsymbol{\xi}})\\}v_{a}^{[i]}({\boldsymbol{x}})\big{)}n_{b}({\boldsymbol{x}})dS({\boldsymbol{x}})$
$\displaystyle\neq$ $\displaystyle 0$ (73)

where

$e_{ab}\\{{\boldsymbol{v}}({\bf
x})\\}=\dfrac{1}{2}(\nabla_{a}v_{b}({\boldsymbol{x}})+\nabla_{b}v_{a}({\boldsymbol{x}}))$
(74)

It is possible to check this result by identifying the singularities in the
solution provided by Jones [77] for the simpler problem of a porous sphere
with radius $R_{p}$ in a constant flow with components $U_{\beta}$, comparing
the solution with the Faxén theorem obtained by Palaniappan [26] for a Darcy
porous sphere in a generic ambient flow. The disturbance field in the Jones
solution is given by the operator applied at the pole of the Stokeslet
centered at the center of the sphere ${\boldsymbol{\xi}}$

$U_{\beta}\mathcal{F}_{\alpha\beta}=U_{\beta}\left(\dfrac{R_{p}A_{D}}{2}+\dfrac{R_{p}^{3}B_{D}}{2}\Delta_{\xi}\right)\delta_{\alpha\beta}$
(75)

where $\Delta_{\xi}$ is the Laplacian operator acting on the coordinate of the
center of the sphere and where

$\displaystyle A_{D}=-\frac{3R_{p}^{2}\left(2\sqrt{k}+\alpha
R_{p}\right)}{6k^{3/2}+3\alpha kR_{p}+6\sqrt{k}R_{p}^{2}+2\alpha R_{p}^{3}}$
$\displaystyle B_{D}=-\frac{\alpha R_{p}^{3}}{12k^{3/2}+6\alpha
kR_{p}+12\sqrt{k}R_{p}^{2}+4\alpha R_{p}^{3}}$

while the Faxén theorem obtained by Palaniappan [26] states that the force
acting on a sphere in an ambient flow ${\boldsymbol{u}}({\boldsymbol{x}})$ is

$F_{\alpha}=-M_{\alpha}=-8\pi\mu\left(\dfrac{R_{p}A_{D}}{2}+\dfrac{R_{p}^{3}B_{D}^{\prime}}{2}\Delta_{\xi}\right)u_{\alpha}({\boldsymbol{\xi}})$
(76)

where

$B_{D}^{\prime}=B_{D}-\frac{6k^{3/2}+3\alpha kR_{p}}{6k^{3/2}+3\alpha
kR_{p}+6\sqrt{k}R_{p}^{2}+2\alpha R_{p}^{3}}$ (77)

The comparison of eqs. (75) an eq. (76) shows that the terms proportional to
the Laplacian $\Delta_{\xi}$ are different in the two expressions (as
$B_{D}\neq B_{D}^{\prime}$), as it should be if the Hinch-Kim dualism would
not apply. This result, follows almost immediately from the functional
structure of the r.h.s. in eq. (73).

On the other hand, if the flow of the fluid in the porous medium is modeled by
the Brinkman equations [78]

$\begin{cases}\mu\Delta{\boldsymbol{v}}^{[i]}({\boldsymbol{x}})-\nabla
p^{[i]}({\boldsymbol{x}})=\dfrac{\mu}{k}{\boldsymbol{v}}^{[i]}({\boldsymbol{x}})\\\
\nabla\cdot{\boldsymbol{v}}^{[i]}({\boldsymbol{x}})=0,\qquad{\boldsymbol{x}}\in
D_{b}\end{cases}$ (78)

with continuous boundary condition
$({\boldsymbol{v}}({\boldsymbol{x}}),{\boldsymbol{\sigma}}({\boldsymbol{x}}))=({\boldsymbol{v}}^{[i]}({\boldsymbol{x}}),{\boldsymbol{\sigma}}^{[i]}({\boldsymbol{x}}))$
at the interface ${\boldsymbol{x}}\in\partial D_{b}$, the reciprocity of the
boundary conditions is fulfilled, since

$\displaystyle[{\boldsymbol{v}}^{(n)}({\boldsymbol{\xi}}),{\boldsymbol{v}}]=\int_{\partial
D_{b}}\big{(}\sigma_{ab}({\boldsymbol{x}})n_{b}({\boldsymbol{x}})v^{(n)}_{a}({\boldsymbol{x}},{\boldsymbol{\xi}})-v_{a}({\boldsymbol{x}})\sigma^{(n)}_{ab}({\boldsymbol{x}},{\boldsymbol{\xi}})n_{b}({\boldsymbol{x}})\big{)}dS({\boldsymbol{x}})=$
$\displaystyle\int_{\partial
D_{b}}\big{(}\sigma_{ab}^{[i]}({\boldsymbol{x}})n_{b}({\boldsymbol{x}})v^{[i](n)}_{a}({\boldsymbol{x}},{\boldsymbol{\xi}})-v_{a}^{[i]}({\boldsymbol{x}})\sigma^{[i](n)}_{ab}({\boldsymbol{x}},{\boldsymbol{\xi}})n_{b}({\boldsymbol{x}})\big{)}dS({\boldsymbol{x}})=$
$\displaystyle-\dfrac{\mu}{k}\int_{D_{b}}\big{(}v_{a}^{[i]}({\boldsymbol{x}})v^{[i](n)}_{a}({\boldsymbol{x}},{\boldsymbol{\xi}})-v_{a}^{[i]}({\boldsymbol{x}})v^{[i](n)}_{a}({\boldsymbol{x}},{\boldsymbol{\xi}})\big{)}dV({\boldsymbol{x}})=0,\qquad{\forall\,{\boldsymbol{v}}({\boldsymbol{x}})}$
(79)

thus for Brinkman porous bodies
$\mathcal{F}_{\alpha\beta{\boldsymbol{\beta}}_{n}}$ is a Faxén operator.

The $0$-th order Faxén operator for this case can be identified by the
solutions given by Masliyah and al. [79] or by Yu and Kaloni [80] for a
traslating Brinkman porous sphere in the Stokes flow. We observe that, in this
case

$U_{\beta}\mathcal{F}_{\alpha\beta}=U_{\beta}\left(\dfrac{R_{p}A_{B}}{2}+\dfrac{R_{p}^{3}B_{B}}{2}\Delta_{\xi}\right)\delta_{\alpha\beta}$
(80)

where

$\displaystyle
A_{B}=\frac{3R_{p}^{2}\left(R_{p}\cosh\left(\frac{R_{p}}{\sqrt{k}}\right)-\sqrt{k}\sinh\left(\frac{R_{p}}{\sqrt{k}}\right)\right)}{6k^{3/2}\sinh\left(\frac{R_{p}}{\sqrt{k}}\right)-6kR_{p}\cosh\left(\frac{R_{p}}{\sqrt{k}}\right)-4R_{p}^{3}\cosh\left(\frac{R_{p}}{\sqrt{k}}\right)}$
$\displaystyle
B_{B}=\frac{6kR_{p}\cosh\left(\frac{R_{p}}{\sqrt{k}}\right)+R_{p}^{3}\cosh\left(\frac{R_{p}}{\sqrt{k}}\right)-6k^{3/2}\sinh\left(\frac{R_{p}}{\sqrt{k}}\right)-3\sqrt{k}R_{p}^{2}\sinh\left(\frac{R_{p}}{\sqrt{k}}\right)}{12k^{3/2}\sinh\left(\frac{R_{p}}{\sqrt{k}}\right)-12kR_{p}\cosh\left(\frac{R_{p}}{\sqrt{k}}\right)-8R_{p}^{3}\cosh\left(\frac{R_{p}}{\sqrt{k}}\right)}$

The same operator is identifiable in the Faxén theorem found by Padmavathi and
al. [27, 28], according to which the force on a Brinkman porous sphere with
center at ${\boldsymbol{\xi}}$ immersed in a generic ambient flow
${\boldsymbol{u}}({\boldsymbol{x}})$ is, as expected,

$F_{\alpha}=-M_{\alpha}=-8\pi\mu\left(\dfrac{R_{p}A_{B}}{2}+\dfrac{R_{p}^{3}B_{B}}{2}\Delta_{\xi}\right)u_{\alpha}({\boldsymbol{\xi}})$
(81)

In this case, in order to evaluate the geometrical moments we need to
determine the surface traction and the velocity at the boundary, since

$\displaystyle
8\pi\,m_{\alpha{\boldsymbol{\alpha}}_{m}\beta{\boldsymbol{\beta}}_{n}}({\boldsymbol{\xi}},{\boldsymbol{\xi}})=\int_{\partial
D_{b}}({\boldsymbol{x}}-{\boldsymbol{\xi}})_{{\boldsymbol{\alpha}}_{m}}h_{\alpha\beta{\boldsymbol{\beta}}_{n}}({\boldsymbol{x}},{\boldsymbol{\xi}})dS({\boldsymbol{x}})-\int_{\partial
D_{b}}p_{\alpha{\boldsymbol{\alpha}}_{m}}({\boldsymbol{x}},{\boldsymbol{\xi}})v_{b\beta{\boldsymbol{\beta}}_{n}}({\boldsymbol{x}},{\boldsymbol{\xi}})n_{b}({\boldsymbol{x}})dS({\boldsymbol{x}})$
$\displaystyle+\int_{\partial
D_{b}}\,\left(v_{\alpha\beta{\boldsymbol{\beta}}_{n}}({\boldsymbol{x}},{\boldsymbol{\xi}})n_{b}({\boldsymbol{x}})\nabla_{b}({\boldsymbol{x}}-{\boldsymbol{\xi}})_{{\boldsymbol{\alpha}}_{m}}+n_{\alpha}({\boldsymbol{x}})v_{c\beta{\boldsymbol{\beta}}_{n}}({\boldsymbol{x}},{\boldsymbol{\xi}})\nabla_{c}({\boldsymbol{x}}-{\boldsymbol{\xi}})_{{\boldsymbol{\alpha}}_{m}}\right)dS({\boldsymbol{x}})$
(82)

##  IX  Faxén operator for a sphere with Navier-slip boundary conditions

From eqs. (54), (57) and (63) it follows that the hydromechanics of a body in
a Stokes fluid with Navier-slip boundary conditions can be determined if the
complete set of surface traction is known. In this Section we develop an
analytic method, based on the Lorentz reciprocal theorem, for determining the
surface tractions eq. (64) entering eq. (63), assuming Navier-slip boundary
conditions on the surface of a spherical object. The method, here developed
for a sphere with Navier-slip boundary conditions, can be employed
systematically for obtaining analytic expressions of $n$-th order Faxén
operators of spheres with different BC-reciprocal boundary conditions. To this
aim consider, in the remainder, a Cartesian coordinate system for the point
${\boldsymbol{x}}$ with the origin at the center of the sphere. Also the
entries at the source point ${\boldsymbol{\xi}}$ are expressed in the same
Cartesian coordinate system, therefore there is no substantial distinction
between Greek and Latin indexes.

###  IX.0.1 $0$-th order Faxén operator

In order to determine the $0$-th order surface traction on a sphere moving in
the unbounded Stokes fluid with velocity $-{\boldsymbol{U}}$, consider the
disturbance field
${\boldsymbol{w}}^{(0)}({\boldsymbol{x}},{\boldsymbol{\xi}})={\boldsymbol{w}}^{(0)}({\boldsymbol{x}})$
due to a sphere with Navier-slip boundary conditions in a constant field
${\boldsymbol{u}}^{(0)}({\boldsymbol{x}},{\boldsymbol{\xi}})={\boldsymbol{U}}$,
which is the solution of the Stokes problem

$\begin{cases}\mu\Delta
w^{(0)}_{a}({\boldsymbol{x}})-\nabla_{a}q^{(0)}({\boldsymbol{x}})=0\\\
\nabla_{a}w^{(0)}_{a}({\boldsymbol{x}})=0,\qquad{\boldsymbol{x}}\in D_{f}\\\
w^{(0)}_{a}(\boldsymbol{x})=-U_{b}\left(\delta_{ab}+{\lambda}h_{bc}({\boldsymbol{x}})t_{ac}({\boldsymbol{x}})\right),\qquad{\boldsymbol{x}}\in\partial
D_{b}\end{cases}$ (83)

and the Stokeslet
$({\boldsymbol{S}}_{\alpha}({\boldsymbol{x}},{\boldsymbol{\xi}}),\mu\,{\boldsymbol{\Sigma}}_{\alpha}({\boldsymbol{x}},{\boldsymbol{\xi}}))$
in eqs. (8). Applying the Lorentz reciprocal theorem to the fields
$({\boldsymbol{w}}^{(0)}({\boldsymbol{x}}),q^{(0)}({\boldsymbol{x}}))$
solution of eqs. (83), and
$({\boldsymbol{S}}_{\alpha}({\boldsymbol{x}},{\boldsymbol{\xi}}),\mu{\boldsymbol{\Sigma}}_{\alpha}({\boldsymbol{x}},{\boldsymbol{\xi}}))$
solution of the eqs. (7) within the domain of the fluid $D_{f}$, bounded by
the surface $\partial D_{b}\cup\partial D_{\infty}$, and considering that both
fields vanish at infinity i.e. on $\partial D_{\infty}$, we have

$[{\boldsymbol{w}}^{(0)},{\boldsymbol{S}}_{\alpha}({\boldsymbol{\xi}})]=0$
(84)

At the surface of the sphere, a Stokeslet with pole at the center of the
sphere (thus for
$r=({\boldsymbol{x}}-{\boldsymbol{\xi}})_{a}({\boldsymbol{x}}-{\boldsymbol{\xi}})_{a}=R_{p}$
and ${\boldsymbol{\xi}}=(0,0,0)$) reads

$\displaystyle{S_{a\alpha}({\boldsymbol{x}},{\boldsymbol{\xi}})}=\dfrac{\delta_{a\alpha}+n_{a\alpha}({\boldsymbol{x}})}{R_{p}}$
$\displaystyle\Sigma_{ab\alpha}({\boldsymbol{x}},{\boldsymbol{\xi}})n_{b}({\boldsymbol{x}})=6\dfrac{\,n_{a\alpha}({\boldsymbol{x}})}{R_{p}^{2}},\qquad
r=R_{p}$ (85)

where
$n_{aa_{1}...a_{n}}({\boldsymbol{x}})=n_{a}({\boldsymbol{x}})n_{a_{1}}({\boldsymbol{x}})...n_{a_{n}}({\boldsymbol{x}})$
and $n_{a}({\boldsymbol{x}})=({\boldsymbol{x}}-{\boldsymbol{\xi}})_{a}/R_{p}$.

Substituting eqs. (85) within eq. (84) and expliciting the
$[\cdot,\cdot]$-operator in eq. (84) according to the definition eq. (36), we
have the following relation between integrals

$-\dfrac{6}{R_{p}}\int_{r=R_{p}}n_{a\alpha}({\boldsymbol{x}})dS({\boldsymbol{x}})=\int_{r=R_{p}}f_{ab}({\boldsymbol{x}})(\delta_{b\alpha}+n_{b\alpha}({\boldsymbol{x}}))dS({\boldsymbol{x}})$
(86)

where $f_{ab}({\boldsymbol{x}})$ is the surface traction related to the
disturbance field introduced in eq. (64). In order to satisfy eq. (86), the
surface traction $f_{ab}({\boldsymbol{x}})$ should have the generic form

$f_{ab}({\boldsymbol{x}})=a\,\delta_{ab}+b\,n_{ab}(\boldsymbol{x})$ (87)

where $a$ and $b$ are constant to be determined. Alternatively, expressed in
terms of normal and tangential components,

$f_{ab}({\boldsymbol{x}})=f^{n}\,n_{ab}(\boldsymbol{x})+f^{t}(\delta_{ab}-n_{ab}(\boldsymbol{x}))$
(88)

where $f^{n}=a+b$ and $f^{t}=a$.

Substituting eq. (88) into eq. (86), a first relation between $f^{n}$ and
$f^{t}$ is obtained

$-\dfrac{6}{R_{p}}\int_{r=R_{p}}n_{a\alpha}({\boldsymbol{x}})dS({\boldsymbol{x}})=2f^{n}\int_{r=R_{p}}n_{a\alpha}(\boldsymbol{x})dS({\boldsymbol{x}})+f^{t}\int_{r=R_{p}}(\delta_{a\alpha}-n_{a\alpha}(\boldsymbol{x}))dS({\boldsymbol{x}})$
(89)

that, solving the surface integrals, attains the simple form

$f^{n}+f^{t}=-\dfrac{3}{R_{p}}$ (90)

To determine $f^{n}$ and $f^{t}$, a further independent relation between
$f^{t}$ and $f^{n}$ is required. To this aim, we can consider the lowest order
potential Stokes singularity, i.e. the so called source doublet
$(-\Delta_{\xi}{\boldsymbol{S}}_{\alpha}({\boldsymbol{x}},{\boldsymbol{\xi}})/2,-\mu\Delta_{\xi}{\boldsymbol{\Sigma}}_{\alpha}({\boldsymbol{x}},{\boldsymbol{\xi}})/2)$
reported in the supplementary material [81]. Also in this case, applying the
Lorentz reciprocal theorem, we obtain

$[{\boldsymbol{w}}^{(0)},\Delta_{\xi}{\boldsymbol{S}}_{\alpha}({\boldsymbol{\xi}})]=0$
(91)

where $\Delta_{\xi}$ is the Laplacian operator acting on the coordinates of
the pole. At the surface of the sphere, the Source Doublet with pole at the
center of the sphere is

$\displaystyle-\dfrac{{\Delta_{\xi}\,S_{a\alpha}({\boldsymbol{x}},{\boldsymbol{\xi}})}}{2}=\dfrac{-\delta_{a\alpha}+3\,n_{a\alpha}({\boldsymbol{x}})}{R_{p}^{3}}$
$\displaystyle-\dfrac{\left(\Delta_{\xi}\,\Sigma_{ab\alpha}({\boldsymbol{x}},{\boldsymbol{\xi}})\right)n_{b}({\boldsymbol{x}})}{2}=6\,\dfrac{-\delta_{a\alpha}+3\,n_{a\alpha}({\boldsymbol{x}})}{R_{p}^{4}},\qquad
r=R_{p}$ (92)

Consequently, the second relation for $f^{n}$ and $f^{t}$ stemming from eq.
(91) is

$\displaystyle\dfrac{6}{R_{p}}\int_{r=R_{p}}(\delta_{a\alpha}-3n_{a\alpha}({\boldsymbol{x}}))dS({\boldsymbol{x}})$
$\displaystyle+$ $\displaystyle\dfrac{6\lambda
f^{t}}{R_{p}}\int_{r=R_{p}}(\delta_{a\alpha}-n_{a\alpha}({\boldsymbol{x}}))dS({\boldsymbol{x}})=$
$\displaystyle
2f^{n}\int_{r=R_{p}}n_{a\alpha}(\boldsymbol{x})dS({\boldsymbol{x}})$
$\displaystyle-$ $\displaystyle
f^{t}\int_{r=R_{p}}(\delta_{ab}-n_{ab}(\boldsymbol{x}))dS({\boldsymbol{x}})$
(93)

that, upon explicit integration, simplifies as

$\left(6\hat{\lambda}+1\right)f^{t}-f^{n}=0,\qquad\hat{\lambda}=\lambda/R_{p}$
(94)

The solution of the linear system eqs. (90) and (94) provides

$\displaystyle
f^{n}=-\dfrac{3(1+6\hat{\lambda})}{2R_{p}(1+3\hat{\lambda})},\qquad
f^{t}=-\dfrac{3}{2R_{p}(1+3\hat{\lambda})}$ (95)

and, thus, the total $0$-th order geometrical surface traction is

$h_{ab}(\boldsymbol{x})=f_{ab}({\boldsymbol{x}})=-\dfrac{3}{2R_{p}}\left(\dfrac{\delta_{ab}+6\hat{\lambda}n_{ab}(\boldsymbol{x})}{1+3\hat{\lambda}}\right)$
(96)

In Appendix B, we determine the analytical expression for the geometrical
moments
$m_{\alpha{\boldsymbol{\alpha}}_{m}\beta{\boldsymbol{\beta}}_{n}}({\boldsymbol{\xi}},{\boldsymbol{\xi}})$
useful to evaluate the Faxén operators of the $0$-th, $1$-st and $2$-nd order,
by means of the surface traction obtained in this Section. To express the
$0$-th order Faxén operator according the eq. (29), we need the geometrical
moments for $n=0$. Geometrical moments for $n=0$ and $m=0,1,2$ are obtained in
eqs. (141)-(144). Due to the symmetry of the sphere, the moments for $n=0$ and
$m=1$ in eq. (142) vanish, and equivalently also the moments for $n=0$ and
$m=3,5,7,...$. It is easy to see that the moments for $n=0$ and $m=4,6,8,...$
contribute to the Faxén operator in eq. (89) by introducing terms proportional
either to the divergence operators $\nabla_{\beta}$ and $\nabla_{\alpha}$ or
to the bilaplacian operator $\Delta_{\xi}\Delta_{\xi}$. Since Stokes fields
are both divergence free and biharmonic, their action is immaterial.
Therefore, in agreement with the result obtained by Premlata and Wei [24], the
$0$-th order Faxén operator for a sphere with Navier-slip boundary conditions
is

$\mathcal{F}_{\beta\alpha}=-\left(\dfrac{1+2\hat{\lambda}}{1+3\hat{\lambda}}\right)\left(\dfrac{3}{4}R_{p}+\dfrac{1}{8}\dfrac{R_{p}^{3}}{(1+2\hat{\lambda})}\Delta_{\xi}\right)\delta_{\alpha\beta}$
(97)

where $\Delta_{\xi}$ is the Laplacian respect to the coordinates of the center
of the sphere ${\boldsymbol{\xi}}$.

Since the force ${F}_{\alpha}$ exerted by the fluid on the particle is the
$0$-th order moment with reverse sign, for a sphere immersed in the ambient
flow ${\boldsymbol{u}}({\boldsymbol{x}})$ we have

$F_{\alpha}=-M_{\alpha}({\boldsymbol{\xi}})=8\pi\mu\left(\dfrac{1+2\hat{\lambda}}{1+3\hat{\lambda}}\right)\left(\dfrac{3}{4}R_{p}+\dfrac{1}{8}\dfrac{R_{p}^{3}}{(1+2\hat{\lambda})}\Delta_{\xi}\right)u_{\alpha}({\boldsymbol{\xi}})$
(98)

###  IX.0.2 $1$-st order Faxén operator

In order to evaluate the surface traction on a sphere immersed in a $1$-st
order ambient field, consider the disturbance field

$\begin{cases}\mu\Delta
w_{a}^{(1)}({\boldsymbol{x}},{\boldsymbol{\xi}})-\nabla_{a}q^{(1)}({\boldsymbol{x}},{\boldsymbol{\xi}})=0\\\
\nabla_{a}w_{a}^{(1)}({\boldsymbol{x}},{\boldsymbol{\xi}})=0,\qquad{\boldsymbol{x}}\in
D_{f}\\\
w_{a}^{(1)}({\boldsymbol{x}},{\boldsymbol{\xi}})=-A_{bb_{1}}\left(({\boldsymbol{x}}-{\boldsymbol{\xi}})_{b_{1}}\delta_{ab}+{\lambda}h_{cbb_{1}}({\boldsymbol{x}},{\boldsymbol{\xi}})t_{ac}({\boldsymbol{x}})\right),\qquad{\boldsymbol{x}}\in\partial
D_{b}\end{cases}$ (99)

with ${\boldsymbol{\xi}}=(0,0,0)$ at the center of the sphere. From eq. (64)
we have

$h_{ab{b}_{1}}({\boldsymbol{x}})=f_{ab{b}_{1}}({\boldsymbol{x}})-\left(n_{b}({\boldsymbol{x}})\,\delta_{ab_{1}}+\delta_{ab}\,n_{b_{1}}({\boldsymbol{x}})\right)$
(100)

The procedure to obtain $f_{ab{b}_{1}}({\boldsymbol{x}})$ (and thus
$h_{ab{b}_{1}}({\boldsymbol{x}})$) is equivalent to that followed in eqs.
(84)-(96) and is reported in detail in the supplementary materials [81]. In
this case, the most general form for $f_{abb_{1}}({\boldsymbol{x}})$ in order
to satisfy reciprocity relations with singularities centered at the center of
the sphere is

$f_{abb_{1}}({\boldsymbol{x}})=a\,\delta_{ab}n_{b_{1}}({\boldsymbol{x}})+b\,\delta_{ab_{1}}n_{b}({\boldsymbol{x}})+c\,n_{abb_{1}}({\boldsymbol{x}})+d\,\delta_{bb_{1}}n_{a1}({\boldsymbol{x}})$
(101)

where $a$, $b$, $c$ and $d$ are constant to be determined. Since, according to
the definition eq. (13), $A_{bb_{1}}=0$ due to the imcompressibility of the
ambient flow, the last term in eq. (101) does not contribute to the surface
traction, and we can set $d=0$. If we apply the Lorentz reciprocal theorem to
the solution
$({\boldsymbol{w}}^{(1)}({\boldsymbol{x}},{\boldsymbol{\xi}}),{\boldsymbol{\tau}}^{(1)}({\boldsymbol{x}},{\boldsymbol{\xi}}))$
of eqs. (99) and to the Stokeslet or the Source doublet, as in the previous
paragraph, we obtain that all the integrals on the surface of the sphere
vanish due to the spherical symmetry. For this reason, it is necessary to
consider the Stokes doublet
$(\nabla_{\beta}{\boldsymbol{S}}_{\alpha}({\boldsymbol{x}},{\boldsymbol{\xi}}),\mu\nabla_{\beta}{\boldsymbol{\Sigma}}_{\alpha}({\boldsymbol{x}},{\boldsymbol{\xi}}))$
with pole at the center of the sphere, and therefore

$[{\boldsymbol{w}}^{(1)}({\boldsymbol{\xi}}),\nabla_{\beta}{\boldsymbol{S}}_{\alpha}({\boldsymbol{\xi}})]=0$
(102)

from which one obtains a system of two linear equations in the coefficients
$a$, $b$, $c$,

$\begin{cases}-(8+6\hat{\lambda})a+(2+24\hat{\lambda})b-3c=24+18\hat{\lambda}\\\
(2+24\hat{\lambda})a-(8+6\hat{\lambda})b-3c=-6+18\hat{\lambda}\end{cases}$
(103)

In order to solve the system we need another linearly independent equation. To
this aim, it is possible to apply the Lorentz reciprocal theorem between the
Source Quadrupole
$(\Delta_{\xi}\nabla_{\beta}{\boldsymbol{S}}_{\alpha}({\boldsymbol{x}},{\boldsymbol{\xi}}),\mu\,\Delta_{\xi}\nabla_{\beta}{\boldsymbol{\Sigma}}_{\alpha}({\boldsymbol{x}},{\boldsymbol{\xi}}))$
and
$({\boldsymbol{w}}^{(1)}({\boldsymbol{x}}),{\boldsymbol{\tau}^{(1)}}({\boldsymbol{x}}))$
. Thus, applying the relation

$[{\boldsymbol{w}}^{(1)}({\boldsymbol{\xi}}),\Delta_{\xi}\nabla_{\beta}{\boldsymbol{S}}_{\alpha}({\boldsymbol{\xi}})]=0$
(104)

one obtains another independent relation amongst $a$, $b$ and $c$,

$c-8\hat{\lambda}(a+b)=-16\hat{\lambda}$ (105)

Solving eqs. (103) and (105) one finally gets

$\displaystyle
a=\frac{-3-7\hat{\lambda}+15\hat{\lambda}^{2}}{(1+5\hat{\lambda})(1+3\hat{\lambda})},\qquad
b=\frac{\hat{\lambda}(8+15\hat{\lambda})}{(1+5\hat{\lambda})(1+3\hat{\lambda})},\qquad
c=-\frac{40\hat{\lambda}}{1+5\hat{\lambda}}$ (106)

The surface tensor in eq. (64) is obtained by substituting the constant
defined by eq. (106) into eq. (101),

$h_{abb_{1}}({\boldsymbol{x}})=-\dfrac{(4+15\hat{\lambda})\,\delta_{ab}n_{b_{1}}({\boldsymbol{x}})+\,\delta_{ab_{1}}n_{b}({\boldsymbol{x}})+40\hat{\lambda}(1+3\hat{\lambda})\,n_{abb_{1}}({\boldsymbol{x}})}{(1+5\hat{\lambda})(1+3\hat{\lambda})}$
(107)

To determine the Faxén operator $\mathcal{F}_{\alpha\beta\beta_{1}}$, the
moments
$m_{\alpha{\boldsymbol{\alpha}}_{m}\beta{\boldsymbol{\beta}}_{n}}({\boldsymbol{\xi}},{\boldsymbol{\xi}})$
at the center of the sphere for $n=1$ should be evaluated using the surface
traction defined by eq. (107). Since geometrical moments with even $m=0,2,...$
vanish due to the symmetry of the sphere, only the geometrical moments for odd
$m=1,3,...$ are needed. Geometrical moments for $m=1$ and $m=3$ are given in
eq. (LABEL:eqB5) and eq. (146) respectively. Higher order geometrical moments
contribute to the Faxén operator by divergence and bilaplacian operators and
thus, they can be neglected as their action is immaterial. Thus, according to
eq. (89), the $1$-st order Faxén operator is given by

$\displaystyle\mathcal{F}_{\alpha\beta\beta_{1}}=-\dfrac{R_{p}^{3}}{6(1+5\hat{\lambda})(1+3\hat{\lambda})}\bigg{\\{}\left[(4+20\hat{\lambda}+15\hat{\lambda}^{2})\delta_{\alpha\beta}\nabla_{\beta_{1}}+(1+5\hat{\lambda}+15\hat{\lambda}^{2})\delta_{\alpha\beta_{1}}\nabla_{\beta}\right]$
$\displaystyle+\dfrac{R_{p}^{2}}{10}\left[(4+12\hat{\lambda}-15\hat{\lambda}^{2})\Delta_{\xi}\nabla_{\beta_{1}}\delta_{\alpha\beta}+(1+3\hat{\lambda}+15\hat{\lambda}^{2})\Delta_{{\xi}}\nabla_{\beta}\delta_{\alpha\beta_{1}}\right]\bigg{\\}}$
(108)

The Faxén operator $\mathcal{T}_{\gamma\alpha}$, yielding the torque
$T_{\alpha}=\varepsilon_{\alpha\beta\beta_{1}}M_{\beta\beta_{1}}({\boldsymbol{\xi}})$
on the body if an ambient flow is applied, or the velocity field due to the
body rotation if applied to the Stokeslet, can be defined by the antisymmetric
part of the first order Faxén operator, i.e.,

$\mathcal{T}_{\gamma\alpha}=\varepsilon_{\alpha\beta\beta_{1}}\mathcal{F}_{\gamma\beta\beta_{1}}$
(109)

Considering that
$\varepsilon_{\gamma\alpha\alpha_{1}}\Delta\nabla_{\alpha_{1}}u_{\alpha}({\boldsymbol{x}})=0$
for any ambient flow, terms containing the third order derivatives in eq.
(108) are immaterial and may be neglected, so we obtain

$\mathcal{T}_{\gamma\alpha}=\dfrac{\varepsilon_{\alpha\gamma\gamma_{1}}R_{p}^{3}\nabla_{\gamma_{1}}}{2(1+3\hat{\lambda})}$
(110)

in agreement with [24].

Thus, the torque on the sphere in the ambient flow
$u_{\alpha}({\boldsymbol{x}})$ is give by

$T_{\alpha}=\dfrac{4\pi\mu\,\varepsilon_{\alpha\gamma\gamma_{1}}R_{p}^{3}\nabla_{\gamma_{1}}u_{\gamma}({\boldsymbol{\xi}})}{(1+3\hat{\lambda})}$
(111)

As shown in Appendix C, the off-diagonal elements of the symmetric part of
$M_{\beta\beta_{1}}({\boldsymbol{\xi}})$ defined by eq. (9) represent, with
reversed sign, the Stresslet
$\mathsf{S}_{\alpha\alpha_{1}}=-\gamma_{\alpha\alpha_{1}\beta\beta_{1}}M_{\beta\beta_{1}}({\boldsymbol{\xi}})$,
where

$\gamma_{\alpha\alpha_{1}\beta\beta_{1}}=\dfrac{\delta_{\alpha\beta}\delta_{\alpha_{1}\beta_{1}}+\delta_{\alpha\beta_{1}}\delta_{\alpha_{1}\beta}}{2}-\dfrac{\delta_{\alpha\alpha_{1}}\delta_{\beta\beta_{1}}}{3}$
(112)

which physically corresponds to the additional contribution of any single
particle to the effective stress of a diluted suspension [40, 39, 25].

Therefore, the associated Faxén operator $\mathcal{E}_{\alpha\beta\beta_{1}}$
providing the Stresslet on the body immersed in an ambient flow
${\boldsymbol{u}}({\boldsymbol{x}})$ according to

$\mathsf{S}_{\beta\beta_{1}}=8\pi\mu\mathcal{E}_{\alpha\beta\beta_{1}}u_{\alpha}({\boldsymbol{\xi}})$
(113)

is generically expressed by

$\mathcal{E}_{\alpha\beta\beta_{1}}=-\dfrac{\mathcal{F}_{\alpha\beta\beta_{1}}+\mathcal{F}_{\alpha\beta_{1}\beta}}{2}+\dfrac{\mathcal{F}_{\alpha\gamma\gamma}\delta_{\beta\beta_{1}}}{3}$
(114)

Since, $\mathcal{F}_{\alpha\beta\beta}u_{\alpha}({\boldsymbol{x}})$ in eq.
(108) vanishes due to the incompressibility, in the case of a spherical body
with Navier-slip boundary conditions, the Faxén operator giving the Stresslet
contribution is simply the symmetric part of the $1$-st order Faxén operator
with opposite sign. Therefore, in agreement with [25], we obtain

$\mathcal{E}_{\alpha\beta\beta_{1}}=-\dfrac{\mathcal{F}_{\alpha\beta\beta_{1}}+\mathcal{F}_{\alpha\beta_{1}\beta}}{2}=\left(\dfrac{5+10\hat{\lambda}}{6+30\hat{\lambda}}+\dfrac{\Delta_{\xi}}{12(1+5\hat{\lambda})}\right)\left(\dfrac{\nabla_{\beta}\delta_{\alpha\beta_{1}}+\nabla_{\beta}\delta_{\alpha\beta_{1}}}{2}\right)$
(115)

According to eq. (30), the Faxén operator in eq. (108) determines the flow
around a sphere in the linear ambient flow
$u_{a}({\boldsymbol{x}})=\delta_{a\,3}\,\delta_{b\,1}\,x_{b}=\delta_{a\,3}\,x_{1}$.
In fact, by choosing
$A_{\beta\beta_{1}}=\delta_{\beta\,3}\delta_{\beta_{1}\,1}$ in eq. (30) the
disturbance field reads

$\displaystyle
w^{(1)}_{a}({\boldsymbol{x}},{\boldsymbol{\xi}})=-\dfrac{R_{p}^{3}}{6(1+5\hat{\lambda})(1+3\hat{\lambda})}\left\\{(4+20\hat{\lambda}+15\hat{\lambda}^{2})S_{a\,3,1}({\boldsymbol{x}},{\boldsymbol{\xi}})+(1+5\hat{\lambda}+15\hat{\lambda}^{2})S_{a\,1,3}({\boldsymbol{x}},{\boldsymbol{\xi}})\right.$
$\displaystyle\left.\dfrac{R_{p}^{2}}{10}\left[(4+12\hat{\lambda}-15\hat{\lambda}^{2})\Delta_{\xi}S_{a\,3,1}({\boldsymbol{x}},{\boldsymbol{\xi}})+(1+3\hat{\lambda}+15\hat{\lambda}^{2})\Delta_{\xi}S_{a\,1,3}({\boldsymbol{x}},{\boldsymbol{\xi}})\right]\right\\}$
(116)

Fig. 2 depicts the streamlines for the total flow in the case of no-slip,
complete slip, and for two different values of $\hat{\lambda}=1,\,10$.

Figure 2: Streamlines on the plane $x_{2}=0$ of the fluid around a sphere in a
linear ambient flow $u_{a}({\boldsymbol{x}})=\delta_{a\,3}\,x_{1}$ for
different values of dimensionless slip length at the surface of the sphere.

###  IX.0.3 $2$-nd order Faxén operator

In order to evaluate the surface traction on a sphere immersed in a $2$-nd
order ambient field, consider the disturbance field

$\begin{cases}\mu\Delta
w_{a}^{(2)}({\boldsymbol{x}},{\boldsymbol{\xi}})-\nabla_{a}q^{(2)}({\boldsymbol{x}},{\boldsymbol{\xi}})=0\\\
\nabla_{a}w_{a}^{(2)}({\boldsymbol{x}},{\boldsymbol{\xi}})=0,\qquad{\boldsymbol{x}}\in
D_{f}\\\
w_{a}^{(2)}({\boldsymbol{x}},{\boldsymbol{\xi}})=-A_{bb_{1}b_{2}}\left(({\boldsymbol{x}}-{\boldsymbol{\xi}})_{b_{1}b_{2}}\delta_{ab}+{\lambda}h_{cbb_{1}b_{2}}({\boldsymbol{x}},{\boldsymbol{\xi}})t_{ac}({\boldsymbol{x}})\right),\qquad{\boldsymbol{x}}\in\partial
D_{b}\end{cases}$ (117)

In this case, the ambient pressure is
$p_{bb_{1}b_{2}}({\boldsymbol{x}},{\boldsymbol{\xi}})=({\boldsymbol{x}}-{\boldsymbol{\xi}})_{b}\Delta({\boldsymbol{x}}-{\boldsymbol{\xi}})_{b_{1}b_{2}}$.
Therefore, from eq. (64) with ${\boldsymbol{\xi}}=(0,0,0)$ we can set

$h_{abb_{1}b_{2}}({\boldsymbol{x}})=f_{ab{b}_{1}b_{2}}({\boldsymbol{x}})-R_{p}\left(n_{bb_{1}}({\boldsymbol{x}})\delta_{ab_{2}}+n_{bb_{2}}({\boldsymbol{x}})\delta_{ab_{1}}+2(\delta_{ab}n_{b_{1}b_{2}}({\boldsymbol{x}})-\delta_{b_{1}b_{2}}n_{ab}({\boldsymbol{x}}))\right)$
(118)

Considering that $f_{ab{b}_{1}b_{2}}({\boldsymbol{x}})$ must be symmetric with
respect to the indexes $b_{1}$ and $b_{2}$, and that $A_{bb}=0$, the most
general form for this traction is

$f_{ab{b}_{1}b_{2}}({\boldsymbol{x}})=a\,\delta_{ab}n_{b_{1}b_{2}}({\boldsymbol{x}})+b\,\delta_{ab}\delta_{b_{1}b_{2}}+c\,(\delta_{ab_{1}}n_{bb_{2}}({\boldsymbol{x}})+\delta_{ab_{2}}n_{bb_{1}}({\boldsymbol{x}}))+d\,\delta_{b_{1}b_{2}}n_{ab}({\boldsymbol{x}})+e\,n_{abb_{1}b_{2}}({\boldsymbol{x}})$
(119)

Following the same procedure used for the geometrical surface tractions of
lower orders, we consider the Stokes Quadrupole
$(\nabla_{\gamma}\nabla_{\beta}{\boldsymbol{S}}_{\alpha}({\boldsymbol{x}},{\boldsymbol{\xi}}),\mu\nabla_{\gamma}\nabla_{\beta}{\boldsymbol{\Sigma}}_{\alpha}({\boldsymbol{x}},{\boldsymbol{\xi}}))$.
Applying the Lorentz reciprocal theorem, we have

$[{\boldsymbol{w}}^{(2)}({\boldsymbol{\xi}}),\nabla_{\gamma}\nabla_{\beta}{\boldsymbol{S}}_{\alpha}({\boldsymbol{\xi}})]=0$
(120)

Solving the integrals (the detailed procedure is reported in the supplementary
material [81]), we obtain four linear equations for the five unknown

$\begin{cases}-2(3+4)a+2(1+20\hat{\lambda})c-2e=(25+24\hat{\lambda})R_{p}\\\
(-5+12\hat{\lambda})a+84\hat{\lambda}b-3(1-8\hat{\lambda})c-14d-4e=3(5+16\hat{\lambda})R_{p}\\\
(1+20\hat{\lambda})a-(5-12\hat{\lambda})c-2e=-(3-52\hat{\lambda})R_{p}\\\
-4(1+6\hat{\lambda})a+42\hat{\lambda}b+6(1-\hat{\lambda})c-7d+e=6(2-9\hat{\lambda})R_{p}\end{cases}$
(121)

out of which only three are linearly independent. In fact, by summing the
fourth equation multiplied by $2$ to the third equation multiplied by $3$, we
obtain the second equation. To obtain a further equation, we consider the
Source Hexapole
$(\Delta_{\xi}\nabla_{\gamma}\nabla_{\beta}{\boldsymbol{S}}_{\alpha}({\boldsymbol{x}},{\boldsymbol{\xi}}),\mu\Delta_{\xi}\nabla_{\gamma}\nabla_{\beta}{\boldsymbol{\Sigma}}_{\alpha}({\boldsymbol{x}},{\boldsymbol{\xi}}))$.
As in the previous cases, the application of the the Lorentz reciprocal
theorem provides

$[{\boldsymbol{w}}^{(2)}({\boldsymbol{\xi}}),\Delta_{\xi}\nabla_{\gamma}\nabla_{\beta}{\boldsymbol{S}}_{\alpha}({\boldsymbol{\xi}})]=0$
(122)

from which it follows that

$-10\hat{\lambda}a-20\hat{\lambda}c+e=-40\hat{\lambda}$ (123)

We need another equation, linearly independent of eqs. (123) and of the three
linearly independent eqs. (121) that can obtained by applying the Lorentz
reciprocal theorem to the Stokeslet

$[{\boldsymbol{w}}^{(2)}({\boldsymbol{\xi}}),{\boldsymbol{S}}_{\alpha}({\boldsymbol{\xi}})]=0$
(124)

resulting the relation

$3a+10b+c+5d+e=-3R_{p}$ (125)

By solving the linear system formed by the first, third, and fourth equations
in eq. (121), eq. (123) and eq. (125), that possesses a non-vanishing
determinant for any $\hat{\lambda}\geq 0$, the constant $a$, $b$, $c$, $d$,
$e$ entering eq. (119) are determined

$\displaystyle\dfrac{a}{R_{p}}=\dfrac{-17-52\hat{\lambda}+244\hat{\lambda}^{2}}{4(1+4\hat{\lambda})(1+7\hat{\lambda})}\qquad$
$\displaystyle\dfrac{b}{R_{p}}=\dfrac{3+17\hat{\lambda}}{4(1+3\hat{\lambda})(1+4\hat{\lambda})(1+7\hat{\lambda})}\qquad\dfrac{e}{R_{p}}=-\dfrac{175\hat{\lambda}}{2+14\hat{\lambda}}$
$\displaystyle\dfrac{c}{R_{p}}=\dfrac{-1+44\hat{\lambda}+112\hat{\lambda}^{2}}{4(1+4\hat{\lambda})(1+7\hat{\lambda})}\qquad$
$\displaystyle\dfrac{d}{R_{p}}=\dfrac{1+28\hat{\lambda}+127\hat{\lambda}^{2}+84\hat{\lambda}^{3}}{2(1+3\hat{\lambda})(1+4\hat{\lambda})(1+7\hat{\lambda})}$
(126)

and therefore, the geometrical surface traction in eq. (118) is given by

$\displaystyle
h_{ab{b}_{1}b_{2}}({\boldsymbol{x}})=\dfrac{R_{p}}{4(1+4\hat{\lambda})(1+7\hat{\lambda})}\left[-5(5+28\hat{\lambda})\delta_{ab}n_{b_{1}b_{2}}({\boldsymbol{x}})\right.$
(127) $\displaystyle+$
$\displaystyle\left.\dfrac{3+17\hat{\lambda}}{1+3\hat{\lambda}}\,\delta_{ab}\delta_{b_{1}b_{2}}-5\,(\delta_{ab_{1}}n_{bb_{2}}({\boldsymbol{x}})+\delta_{ab_{2}}n_{bb_{1}}({\boldsymbol{x}}))\right.$
$\displaystyle+$
$\displaystyle\left.\dfrac{2(5+84\lambda+371\lambda^{2}+420\lambda^{3})}{1+3\hat{\lambda}}\,\delta_{b_{1}b_{2}}n_{ab}({\boldsymbol{x}})-350\hat{\lambda}(1+4\hat{\lambda})\,n_{abb_{1}b_{2}}({\boldsymbol{x}}))\right]$

through which it is possible to obtain the geometrical moments at the center
of the sphere
$m_{\alpha\beta{\boldsymbol{\beta}}_{2}}({\boldsymbol{\xi}},{\boldsymbol{\xi}})$,
$m_{\alpha{\boldsymbol{\alpha}}_{2}\beta{\boldsymbol{\beta}}_{2}}({\boldsymbol{\xi}},{\boldsymbol{\xi}})$,
$m_{\alpha{\boldsymbol{\alpha}}_{4}\beta{\boldsymbol{\beta}}_{2}}({\boldsymbol{\xi}},{\boldsymbol{\xi}})$.
Their analytic expression is reported in Appendix B, eqs. (147), (148), (149).
Using these results it is possible to express analytically the $2$-nd order
Faxén operator

$\displaystyle\mathcal{F}_{\alpha\beta\beta_{1}\beta_{2}}=-\dfrac{R_{p}^{3}}{4(1+4\hat{\lambda})(1+7\hat{\lambda})}\left\\{\dfrac{(1+4\hat{\lambda})(1+7\hat{\lambda})}{1+3\hat{\lambda}}\left[\delta_{\alpha\beta}\delta_{\beta_{1}\beta_{2}}+\hat{\lambda}(\delta_{\beta\beta_{1}}\delta_{\alpha\beta_{2}}+\delta_{\beta\beta_{2}}\delta_{\alpha\beta_{1}})\right]\right.$
$\displaystyle+\dfrac{R_{p}^{2}}{6}\bigg{[}-4\hat{\lambda}^{2}\left(\dfrac{4+21\hat{\lambda}}{1+3\hat{\lambda}}\right)\Delta_{\xi}\delta_{\beta_{1}\beta_{2}}\delta_{\alpha\beta}+5(1+6\hat{\lambda})\nabla_{\beta_{1}\beta_{2}}\delta_{\alpha\beta}+$
$\displaystyle\left.(1+6\hat{\lambda}+28\hat{\lambda}^{2})(\nabla_{\beta\beta_{1}}\delta_{\alpha\beta_{2}}+\nabla_{\beta\beta_{2}}\delta_{\alpha\beta_{1}})+(1+12\hat{\lambda}+56\hat{\lambda}^{2})(\delta_{\alpha\beta_{1}}\delta_{\beta\beta_{2}}+\delta_{\alpha\beta_{2}}\delta_{\beta\beta_{1}})\Delta_{\xi}\bigg{]}\right.$
$\displaystyle\left.+\dfrac{R_{p}^{4}}{84}\left[(5+20\hat{\lambda}-56\hat{\lambda}^{2})\nabla_{\beta_{1}\beta_{2}}\delta_{\alpha\beta}+(1+4\hat{\lambda}+28\hat{\lambda}^{2})(\nabla_{\beta\beta_{1}}\delta_{\alpha\beta_{2}}+\nabla_{\beta\beta_{2}}\delta_{\alpha\beta_{1}})\right]\Delta_{\xi}\right\\}$
(128)

The Faxén operator in eq. (128) can be used to obtain the flow around a sphere
in the unbounded Poiseuille ambient flow

$u^{(2)}_{a}({\boldsymbol{x}})=\delta_{a\,3}(\delta_{a_{1}a_{2}}-\delta_{a_{1}\,3}\delta_{a_{2}\,3})\,({\boldsymbol{x}}-{\boldsymbol{\xi}})_{a_{1}a_{2}}=\delta_{a\,3}\,(({\boldsymbol{x}}-{\boldsymbol{\xi}})_{1}^{2}+({\boldsymbol{x}}-{\boldsymbol{\xi}})_{2}^{2})$

${\boldsymbol{\xi}}$ being the center of the sphere. Choosing
$A_{\beta\beta_{1}\beta_{2}}=\delta_{\beta\,3}(\delta_{\beta_{1}\beta_{2}}-\delta_{\beta_{1}\,3}\delta_{\beta_{2}\,3})$
in eq. (30) the disturbance field reads

$\displaystyle
w^{(2)}_{a}({\boldsymbol{x}},{\boldsymbol{\xi}})=-\dfrac{R_{p}^{3}}{24(1+7\hat{\lambda})}\left\\{12\left(\dfrac{1+7\hat{\lambda}}{1+3\hat{\lambda}}\right)S_{a\,3}({\boldsymbol{x}},{\boldsymbol{\xi}})+\right.$
$\displaystyle\left.{R_{p}^{2}}\left[\left(\dfrac{5+25\hat{\lambda}-42\hat{\lambda}^{2}}{1+3\hat{\lambda}}\right)\Delta_{\xi}S_{a\,3}({\boldsymbol{x}},{\boldsymbol{\xi}})-7(1+2\hat{\lambda})S_{a\,3,3\,3}({\boldsymbol{x}},{\boldsymbol{\xi}})\right]-R_{p}^{4}\dfrac{\Delta_{\xi}S_{a\,3,3\,3}({\boldsymbol{x}},{\boldsymbol{\xi}})}{2}\right\\}$
(129)

The streamlines obtained using eq. (129) in the case of no-slip, complete
slip, Navier-slip with $\hat{\lambda}=1$ and $\hat{\lambda}=10$ are reported
in Fig. 3.

Figure 3: Streamlines on the plane $x_{2}=0$ of the fluid around a sphere in
the unbounded Poiseuille ambient flow
$u_{a}({\boldsymbol{x}})=\delta_{a\,3}\,(x_{1}^{2}+x_{2}^{2})$ for different
values of dimensionless slip length, at the surface of the sphere.

##  X Conclusions

This work has investigated the operators describing the hydromechanics of
bodies immersed in Stokes flow: the singularity operator giving the
disturbance flow due to the body immersed in the fluid and the Faxén operator
providing the moments on the body due to the fluid flowing around it.

By considering moments of the volume forces on the body, we provided a general
expression for the singularity operator. Specifically, a generic $n$-th order
singularity operator can be obtained by defining ($m$, $n$)-th order
geometrical moments as the $m$-th order moments of volume forces on the body
immersed in an $n$-th order ambient flow singular at infinity. Furthermore, it
is found that the same operator is a Faxén operator only if the boundary
conditions assumed at the surface of the body are BC-reciprocal. This shows
that the Hinch-Kim’s dualism is not an intrinsic property of the Stokes flow
due to the reciprocity of its governing equations, but it depends also on the
nature of the interaction with the body. This is not a trivial conclusion,
since other properties following from the reciprocity between thermodynamic
forces and fluxes of the governing equations, such as the symmetry of the
resistance matrix (following from the reciprocity of the Stokes equations and,
firstly, from that of the Onsager relations [83, 84]), are independent on the
nature of the interaction with the body (this is also shown in Appendix A for
the resistance matrix of the Stokes flow).

The results reported in Section V show that bodies for which the Hinch-Kim’s
dualism holds are an interesting class of bodies since their hydromechanic
interaction with the fluid can be completely described by collecting all the
associated geometrical moments. This makes this analysis a fundamental tool
for the development of a theory describing the hydrodynamic interactions
between many bodies in confined geometries starting from the knowledge of the
hydrodynamics of single bodies in unbounded flows and of the confinement Green
function [53]. The strength of this method is that the operators have an
explicit expression that allow us to investigate case by case the dualism, so
we find that Hinch-Kim’s dualism hold only for particular (but of great
interest) boundary conditions (such as for rigid particles, drop at the
mechanical equilibrium or porous bodies modeled by the Birkman equations), but
it does not hold for other systems (elastic deforming body, deforming drop,
non-Newtonian drops, porous bodies modeled by the Darcy law). It is
interesting that in the case where Navier-slip boundary conditions are
considered (for which we found in Section VI that Hinch-Kim’s dualism holds)
the $n$-th order Faxén operator is determined by the $n$-th order surface
traction on the body (i.e. the surface traction on the body immersed in an
$n$-th order ambient field). Therefore, by using a method based on the
Lorentz’s reciprocal theorem, we have obtain the $0$-th, $1$-st and $2$-nd
order surface traction on a sphere with Navier-slip boundary condition and,
then, applying the expressions developed by enforcing the representation in
terms of volume geometrical moments, the $0$-th, $1$-st, $2$-nd Faxén operator
giving both the moments, up to the second order, on the sphere in a generic
ambient flow and the disturbance field due to the sphere in an ambient flow up
to the $2$-nd order, such as the Poiseuille flow.

##  Appendix A Symmetry of the geometrical moments

To show the symmetry in eq. (56), if BC-reciprocity applies, consider the
moments
$M^{(n)}_{\alpha{\boldsymbol{\alpha}}_{m}}({\boldsymbol{\xi}},{\boldsymbol{\xi}}^{\prime})$
and their expression eq. (25). Using the notation developed in Section IV

$A_{\alpha{\boldsymbol{\alpha}}_{m}}M^{(n)}_{\alpha{\boldsymbol{\alpha}}_{m}}({\boldsymbol{\xi}},{\boldsymbol{\xi}}^{\prime})=[{\boldsymbol{u}}^{(m)}({\boldsymbol{\xi}}),{\boldsymbol{v}}^{(n)}({\boldsymbol{\xi}}^{\prime})]$
(130)

and analogously

$A_{\beta^{\prime}{\boldsymbol{\beta}}^{\prime}_{n}}M^{(m)}_{\beta^{\prime}{\boldsymbol{\beta}}^{\prime}_{n}}({\boldsymbol{\xi}}^{\prime},{\boldsymbol{\xi}})=[{\boldsymbol{u}}^{(n)}({\boldsymbol{\xi}}^{\prime}),{\boldsymbol{v}}^{(m)}({\boldsymbol{\xi}})]$
(131)

Therefore, their difference can be expressed as

$\displaystyle
A_{\alpha{\boldsymbol{\alpha}}_{m}}M^{(n)}_{\alpha{\boldsymbol{\alpha}}_{m}}({\boldsymbol{\xi}},{\boldsymbol{\xi}}^{\prime})-A_{\beta^{\prime}{\boldsymbol{\beta}}^{\prime}_{n}}M^{(m)}_{\beta^{\prime}{\boldsymbol{\beta}}^{\prime}_{n}}({\boldsymbol{\xi}}^{\prime},{\boldsymbol{\xi}})=[{\boldsymbol{u}}^{(m)}({\boldsymbol{\xi}}),{\boldsymbol{v}}^{(n)}({\boldsymbol{\xi}}^{\prime})]-[{\boldsymbol{u}}^{(n)}({\boldsymbol{\xi}}^{\prime}),{\boldsymbol{v}}^{(m)}({\boldsymbol{\xi}})]$
$\displaystyle=[{\boldsymbol{u}}^{(m)}({\boldsymbol{\xi}}),{\boldsymbol{u}}^{(n)}({\boldsymbol{\xi}}^{\prime})]+[{\boldsymbol{u}}^{(m)}({\boldsymbol{\xi}}),{\boldsymbol{w}}^{(n)}({\boldsymbol{\xi}}^{\prime})]-[{\boldsymbol{u}}^{(n)}({\boldsymbol{\xi}}^{\prime}),{\boldsymbol{u}}^{(m)}({\boldsymbol{\xi}})]-[{\boldsymbol{u}}^{(n)}({\boldsymbol{\xi}}^{\prime}),{\boldsymbol{w}}^{(m)}({\boldsymbol{\xi}})]$
$\displaystyle=[{\boldsymbol{u}}^{(m)}({\boldsymbol{\xi}}),{\boldsymbol{w}}^{(n)}({\boldsymbol{\xi}}^{\prime})]-[{\boldsymbol{u}}^{(n)}({\boldsymbol{\xi}}^{\prime}),{\boldsymbol{w}}^{(m)}({\boldsymbol{\xi}})]$
(132)

Under the hypothesis of reciprocal boundary conditions, from the identity eq.
(41) it follows that

$[{\boldsymbol{u}}^{(m)}({\boldsymbol{\xi}}),{\boldsymbol{w}}^{(n)}({\boldsymbol{\xi}}^{\prime})]-[{\boldsymbol{u}}^{(n)}({\boldsymbol{\xi}}^{\prime}),{\boldsymbol{w}}^{(m)}({\boldsymbol{\xi}})]=[{\boldsymbol{v}}^{(m)}({\boldsymbol{\xi}}),{\boldsymbol{v}}^{(n)}({\boldsymbol{\xi}}^{\prime})]=0$
(133)

i.e.,

$A_{\alpha{\boldsymbol{\alpha}}_{m}}M^{(n)}_{\alpha{\boldsymbol{\alpha}}_{m}}({\boldsymbol{\xi}},{\boldsymbol{\xi}}^{\prime})=A_{\beta^{\prime}{\boldsymbol{\beta}}^{\prime}_{n}}M^{(m)}_{\beta^{\prime}{\boldsymbol{\beta}}^{\prime}_{n}}({\boldsymbol{\xi}}^{\prime},{\boldsymbol{\xi}})$
(134)

Therefore, from the definition of geometrical moments eq. (27), since
$A_{\beta^{\prime}{\boldsymbol{\beta}}_{n}^{\prime}}$ could be in principle
arbitrary, we have

$m_{\alpha{\boldsymbol{\alpha}}_{m}\beta^{\prime}{\boldsymbol{\beta}}^{\prime}_{n}}({\boldsymbol{\xi}},{\boldsymbol{\xi}}^{\prime})=m_{\beta^{\prime}{\boldsymbol{\beta}}^{\prime}_{n}\alpha{\boldsymbol{\alpha}}_{m}}({\boldsymbol{\xi}}^{\prime},{\boldsymbol{\xi}})$
(135)

corresponding to eq. (56). If the boundary conditions are not reciprocal, the
r.h.s. of eq. (132) possesses the property

$[{\boldsymbol{u}}^{(m)}({\boldsymbol{\xi}}),{\boldsymbol{w}}^{(n)}({\boldsymbol{\xi}}^{\prime})]-[{\boldsymbol{u}}^{(n)}({\boldsymbol{\xi}}^{\prime}),{\boldsymbol{w}}^{(m)}({\boldsymbol{\xi}})]\;\,\begin{cases}=0,\quad
n=m\\\ \neq 0,\quad n\neq m\end{cases}$ (136)

For $n=m=0$ eq. (136) expresses the thermodynamic condition of symmetry of the
resistance matrix, $m_{\alpha\beta}=m_{\beta\alpha}$, independently on the
boundary conditions and on the nature of the immersed body [84]. This result
represents a purely mechanical proof of the symmetry of the resistance matrix
independently of the boundary conditions. The thermodynamic proof has been
given by Landau [83, 84], while the mechanical proof by Brenner uses specific
(no-slip) boundary conditions see [15, p. 166] and the discussion therein.

As seen in Section III, a consequence of the symmetry of the geometrical
moments, in the case the Hinch-Kim dualism holds, is the equivalence between
eq. (10) and eq. (57). This equivalence can be proved by substituting the
Hinch-Kim theorem expressed by eq. (54) in eq. (10), hence

$w_{a}({\boldsymbol{x}})=\dfrac{1}{8\pi\mu}\sum_{m=0}^{\infty}\dfrac{M_{\alpha\,{\boldsymbol{\alpha}}_{m}}({\boldsymbol{\xi}})}{m!}\nabla_{{\boldsymbol{\alpha}}_{m}}S_{a\,\alpha}({\boldsymbol{x}},{\boldsymbol{\xi}})=\sum_{m=0}^{\infty}\dfrac{\mathcal{F}_{\beta^{\prime}\alpha{\boldsymbol{\alpha}}_{m}}u_{\beta^{\prime}}({\boldsymbol{\xi}}^{\prime})}{m!}\nabla_{{\boldsymbol{\alpha}}_{m}}S_{a\,\alpha}({\boldsymbol{x}},{\boldsymbol{\xi}})$
(137)

substituting the expression eq. (29) for the Faxén operator and using the
symmetry eq. (135)

$\displaystyle w_{a}({\boldsymbol{x}})$
$\displaystyle=\sum_{m=0}^{\infty}\sum_{n=0}^{\infty}\dfrac{m_{\beta^{\prime}{\boldsymbol{\beta}}^{\prime}_{n}\alpha{\boldsymbol{\alpha}}_{m}}({\boldsymbol{\xi}}^{\prime},{\boldsymbol{\xi}})\nabla_{{\boldsymbol{\beta}^{\prime}}_{n}}u_{\beta^{\prime}}({\boldsymbol{\xi}}^{\prime})}{m!n!}\nabla_{{\boldsymbol{\alpha}}_{m}}S_{a\,\alpha}({\boldsymbol{x}},{\boldsymbol{\xi}})$
(138)
$\displaystyle=\sum_{m=0}^{\infty}\sum_{n=0}^{\infty}\dfrac{m_{\alpha{\boldsymbol{\alpha}}_{m}\beta^{\prime}{\boldsymbol{\beta}}^{\prime}_{n}}({\boldsymbol{\xi}},{\boldsymbol{\xi}}^{\prime})\nabla_{{\boldsymbol{\beta}^{\prime}}_{n}}u_{\beta^{\prime}}({\boldsymbol{\xi}}^{\prime})}{m!n!}\nabla_{{\boldsymbol{\alpha}}_{m}}S_{a\,\alpha}({\boldsymbol{x}},{\boldsymbol{\xi}})$

in which a new representation of the Faxén operator acting on the point
${\boldsymbol{\xi}}^{\prime}$ is introduced

$\mathcal{F}_{\alpha\beta^{\prime}{\boldsymbol{\beta}}^{\prime}_{n}}=\sum_{m=0}^{\infty}\dfrac{m_{\alpha{\boldsymbol{\alpha}}_{m}\beta^{\prime}{\boldsymbol{\beta}}^{\prime}_{n}}({\boldsymbol{\xi}},{\boldsymbol{\xi}}^{\prime})\nabla_{{\boldsymbol{\alpha}}_{m}}}{m!}$
(139)

from which it follows that

$w_{a}({\boldsymbol{x}})=\sum_{n=0}^{\infty}\dfrac{\nabla_{{\boldsymbol{\beta}}_{n}^{\prime}}u_{\beta^{\prime}}({\boldsymbol{\xi}}^{\prime})}{n!}\mathcal{F}_{\alpha\beta^{\prime}{\boldsymbol{\beta}}_{n}^{\prime}}S_{a\,\alpha}({\boldsymbol{x}},{\boldsymbol{\xi}})$
(140)

that becomes eq. (57) for ${\boldsymbol{\xi}}^{\prime}={\boldsymbol{\xi}}$.

##  Appendix B Geometrical moments for a sphere with Navier-slip conditions

In this Appendix, we provide the analytical expression for the Cartesian
entries
$m_{\alpha{\boldsymbol{\alpha}}_{m}\beta{\boldsymbol{\beta}}_{n}}({\boldsymbol{\xi}},{\boldsymbol{\xi}})$
of the geometrical moments, for a sphere with Navier-slip boundary conditions,
${\boldsymbol{\xi}}$ being the center of the sphere.

###  B.0.1  $(m,n)$-th order geometrical moments with $n=0$

In order to evaluate the geometrical moment with $n=0$, eqs. (63)-(64) can be
applied using for the geometrical surface traction eq. (96).

In the case $m=0$ we obtain the well-known Basset term [82]

$m_{\alpha\beta}=-\dfrac{3}{16\pi
R_{p}}\int_{r=R_{p}}\left[\dfrac{\delta_{\alpha\beta}+6\hat{\lambda}n_{\alpha\beta}(\boldsymbol{x})}{1+3\hat{\lambda}}\right]dS(\boldsymbol{x})=-\dfrac{3}{4}\left[\dfrac{1+2\hat{\lambda}}{1+3\hat{\lambda}}\right]R_{p}\,\delta_{\alpha\beta}$
(141)

For $m=1$ the geometrical moments at the center of the sphere
${\boldsymbol{\xi}}=(0,0,0)$ vanish, in fact

$\displaystyle m_{\alpha{\alpha}_{1}\beta}=m_{\beta\alpha{\alpha}_{1}}=$
$\displaystyle-\dfrac{3}{16\pi}\int_{r=R_{p}}\left[\dfrac{\delta_{\gamma\beta}+6\hat{\lambda}n_{\gamma\beta}(\boldsymbol{x})}{1+3\hat{\lambda}}\right]\left[\delta_{\alpha\gamma}n_{{\alpha}_{1}}({\boldsymbol{x}})-\hat{\lambda}(t_{\alpha\gamma}({\boldsymbol{x}})n_{\alpha_{1}}({\boldsymbol{x}})+n_{\alpha}({\boldsymbol{x}})t_{\alpha_{1}\gamma}({\boldsymbol{x}}))\right]dS({\boldsymbol{x}})=0$
(142)

and, due to the symmetry of the sphere,

$m_{\alpha{\boldsymbol{\alpha}}_{m}\beta}=0,\qquad m=1,3,5\,...$ (143)

for any odd value of $m$. For $m=2$

$\displaystyle
m_{\alpha{\boldsymbol{\alpha}}_{2}\beta}=m_{\beta\alpha{\boldsymbol{\alpha}}_{2}}=-\dfrac{3R_{p}}{16\pi}\int_{r=R_{p}}\left[\dfrac{\delta_{\gamma\beta}+6\hat{\lambda}n_{\gamma\beta}(\boldsymbol{x})}{1+3\hat{\lambda}}\right]\times$
(144) $\displaystyle\times$
$\displaystyle\bigg{[}\delta_{\alpha\gamma}n_{\alpha_{1}\alpha_{2}}({\boldsymbol{x}})-\hat{\lambda}(2t_{\alpha\gamma}n_{\alpha_{1}\alpha_{2}}(\boldsymbol{x})+t_{\alpha_{1}\gamma}n_{\alpha\alpha_{2}}(\boldsymbol{x})+t_{\alpha_{2}\gamma}n_{\alpha\alpha_{1}}(\boldsymbol{x}))\bigg{]}dS({\boldsymbol{x}})=$
$\displaystyle-\dfrac{R_{p}^{3}}{4(1+3\hat{\lambda})}\bigg{[}\delta_{\alpha\beta}\delta_{\alpha_{1}\alpha_{2}}+\hat{\lambda}(\delta_{\alpha\alpha_{1}}\delta_{\beta\alpha_{2}}+\delta_{\alpha\alpha_{2}}\delta_{\beta\alpha_{1}})\bigg{]}$

###  B.0.2 (m,n)-th order geometrical moments with $n=1$

For the geometrical moments with $n=1$, eqs. (63)-(64) can be used with the
geometrical surface traction expressed by eq. (127). By the symmetry expressed
by eq. (142), the moment with $m=0$ vanishes. For $m=1$,

$\displaystyle
m_{\alpha{\alpha}_{1}\beta\beta_{1}}=m_{\beta\beta_{1}\alpha{\alpha}_{1}}=-R_{p}\int_{r=R_{p}}\left[\dfrac{(4+15\hat{\lambda})\,\delta_{\gamma\beta}n_{\beta_{1}}({\boldsymbol{x}})+\,\delta_{\gamma\beta_{1}}n_{\beta}({\boldsymbol{x}})+40\hat{\lambda}(1+3\hat{\lambda})\,n_{\gamma\beta\beta_{1}}({\boldsymbol{x}})}{8\pi(1+5\hat{\lambda})(1+3\hat{\lambda})}\right]$
$\displaystyle\times$
$\displaystyle\bigg{[}\delta_{\alpha\gamma}n_{\alpha_{1}}({\boldsymbol{x}})-\hat{\lambda}(t_{\alpha\gamma}({\boldsymbol{x}})n_{\alpha_{1}}({\boldsymbol{x}})+t_{\alpha_{1}\gamma}({\boldsymbol{x}})n_{\alpha}({\boldsymbol{x}}))\bigg{]}dS({\boldsymbol{x}})=$
$\displaystyle-\dfrac{R_{p}^{3}}{6(1+5\hat{\lambda})(1+3\hat{\lambda})}\bigg{[}\left(4+20\hat{\lambda}+15\hat{\lambda}^{2}\right)\delta_{\alpha\beta}\delta_{\alpha_{1}\beta_{1}}+\left(1+5\hat{\lambda}+15\hat{\lambda}^{2}\right)\delta_{\alpha\beta_{1}}\delta_{\alpha_{1}\beta}+10\hat{\lambda}(1+3\hat{\lambda})\delta_{\alpha\alpha_{1}}\delta_{\beta\beta_{1}}\bigg{]}$

Due to symmetry of the sphere, for odd $n$ and even $m$ the geometrical
moments vanish, therefore
$m_{\alpha{\boldsymbol{\alpha}}_{2}\beta\beta_{1}}=m_{\beta\beta_{1}\alpha{\boldsymbol{\alpha}}_{2}}=0$.
For $m=3$,

$\displaystyle
m_{\alpha{\boldsymbol{\alpha}}_{3}\beta\beta_{1}}=m_{\beta\beta_{1}\alpha{\boldsymbol{\alpha}}_{3}}=$
(146) $\displaystyle-$ $\displaystyle
R_{p}^{3}\int_{r=R_{p}}\left[\dfrac{(4+15\hat{\lambda})\,\delta_{\gamma\beta}n_{\beta_{1}}({\boldsymbol{x}})+\,\delta_{\gamma\beta_{1}}n_{\beta}({\boldsymbol{x}})+40\hat{\lambda}(1+3\hat{\lambda})\,n_{\gamma\beta\beta_{1}}({\boldsymbol{x}})}{8\pi(1+5\hat{\lambda})(1+3\hat{\lambda})}\right]\times$
$\displaystyle\bigg{[}\delta_{\alpha\gamma}n_{\alpha_{1}\alpha_{2}\alpha_{3}}({\boldsymbol{x}})-\hat{\lambda}(3t_{\alpha\gamma}({\boldsymbol{x}})n_{\alpha_{1}\alpha_{2}\alpha_{3}}({\boldsymbol{x}})+t_{\alpha_{1}\gamma}({\boldsymbol{x}})n_{\alpha\alpha_{2}\alpha_{3}}({\boldsymbol{x}})+t_{\alpha_{2}\gamma}({\boldsymbol{x}})n_{\alpha_{1}\alpha\alpha_{3}}({\boldsymbol{x}})+t_{\alpha_{3}\gamma}({\boldsymbol{x}})n_{\alpha_{1}\alpha_{2}\alpha}({\boldsymbol{x}}))\bigg{]}dS({\boldsymbol{x}})$
$\displaystyle=$
$\displaystyle-\dfrac{R_{p}^{5}}{30(1+5\hat{\lambda})(1+3\hat{\lambda})}\bigg{[}\left(4+12\hat{\lambda}-15\hat{\lambda}^{2}\right)\delta_{\alpha\beta}\eta_{\beta_{1}{\boldsymbol{\alpha}}_{3}}+\left(1+3\hat{\lambda}+15\hat{\lambda}^{2}\right)\delta_{\alpha\beta_{1}}\eta_{\beta{\boldsymbol{\alpha}}_{3}}$
$\displaystyle+5\hat{\lambda}(1+3\hat{\lambda})(\delta_{\alpha\alpha_{1}}\eta_{\beta\beta_{1}\alpha_{2}\alpha_{3}}+\delta_{\alpha\alpha_{2}}\eta_{\beta\beta_{1}\alpha_{1}\alpha_{3}}+\delta_{\alpha\alpha_{3}}\eta_{\beta\beta_{1}\alpha_{1}\alpha_{2}})\bigg{]}$

where
$\eta_{\alpha\beta\gamma\delta}=\delta_{\alpha\beta}\delta_{\gamma\delta}+\delta_{\alpha\delta}\delta_{\beta\gamma}+\delta_{\alpha\gamma}\delta_{\beta\delta}$.

###  B.0.3 (m,n)-th order geometrical moments with n=2

It is possible to obtain the geometrical moments for $n=2$ and $m=0$ by
symmetry from eq. (144), thus

$m_{\alpha\beta{\boldsymbol{\beta}}_{2}}=-\dfrac{R_{p}^{3}}{4(1+3\hat{\lambda})}\bigg{[}\delta_{\alpha\beta}\delta_{\beta_{1}\beta_{2}}+\hat{\lambda}(\delta_{\beta\beta_{1}}\delta_{\alpha\beta_{2}}+\delta_{\beta\beta_{2}}\delta_{\alpha\beta_{1}})\bigg{]}$
(147)

For $m=2$

$\displaystyle
m_{\alpha{\boldsymbol{\alpha}}_{2}\beta{\boldsymbol{\beta}}_{2}}=m_{\beta{\boldsymbol{\beta}}_{2}\alpha{\boldsymbol{\alpha}}_{2}}=$
$\displaystyle
R_{p}^{2}\int_{r=R_{p}}\dfrac{h_{\alpha\beta{\boldsymbol{\beta}}_{2}}({\boldsymbol{x}})}{8\pi}\bigg{[}\delta_{\alpha\gamma}n_{\alpha_{1}\alpha_{2}}({\boldsymbol{x}})-\hat{\lambda}(2t_{\alpha\gamma}n_{\alpha_{1}\alpha_{2}}(\boldsymbol{x})+t_{\alpha_{1}\gamma}n_{\alpha\alpha_{2}}(\boldsymbol{x})+t_{\alpha_{2}\gamma}n_{\alpha\alpha_{1}}(\boldsymbol{x}))\bigg{]}dS({\boldsymbol{x}})=$
$\displaystyle-\dfrac{R_{p}^{5}}{24(1+4\hat{\lambda})(1+7\hat{\lambda})}\left\\{\delta_{\alpha\beta}\left[-8\hat{\lambda}^{2}\left(\dfrac{4+21\hat{\lambda}}{1+3\hat{\lambda}}\right)\delta_{\alpha_{1}\alpha_{2}}\delta_{\beta_{1}\beta_{2}}+5(1+6\hat{\lambda})(\delta_{\alpha_{1}\beta_{2}}\delta_{\beta_{1}\alpha_{2}}+\delta_{\alpha_{1}\beta_{1}}\delta_{\alpha_{2}\beta_{2}})\right]\right.$
$\displaystyle+(1+6\hat{\lambda}+28\hat{\lambda}^{2})(\delta_{\alpha\beta_{1}}(\delta_{\alpha_{1}\beta_{2}}\delta_{\alpha_{2}\beta}+\delta_{\alpha_{1}\beta}\delta_{\alpha_{2}\beta_{2}})+\delta_{\alpha\beta_{2}}(\delta_{\alpha_{1}\beta_{1}}\delta_{\alpha_{2}\beta}+\delta_{\alpha_{1}\beta}\delta_{\alpha_{2}\beta_{1}}))$