# Boundary Effects on Ideal Fluid Forces and Kelvin's Minimum Energy Theorem

Kyle I. McKee1<EMAIL_ADDRESS>1Department of Mathematics, Massachusetts
Institute of Technology, Cambridge, MA 02139, USA

###### Abstract

The electrostatic force on a charge above a neutral conductor is generally
attractive. Surprisingly, that force becomes repulsive in certain geometries
Levin & Johnson (2011), a result that follows from an energy theorem in
electrostatics. Based on the analogous minimum energy theorem of Kelvin
(1849), valid in the theory of ideal fluids, we show corresponding effects on
steady and unsteady fluid forces in the presence of boundaries. Two main
results are presented regarding the unsteady force. First, the added mass is
proven to always increase in the presence of boundaries. Second, in a model of
a body approaching a boundary, where the unsteady force is typically repulsive
(Lamb, 1975, §137), we present a geometry where the force can be attractive.
As for the steady force, there is one main result: in a model of a Bernoulli
suction gripper, for which the steady force is typically attractive, we show
that force becomes repulsive in some geometries. Both the unsteady and steady
forces are shown to reverse sign when boundaries approximate flow streamlines,
at energy minima predicted by Kelvin’s theorem.

##  1 Introduction

It is generally stated that the electrostatic force on a charge above a
conductor is attractive (Griffiths, 2013, pp. 99). However, Levin & Johnson
(2011) showed that the force may be repulsive in certain geometries (see
figure 1C). Repulsion occurs when the electric field energy depends non-
monotonically on the charge-conductor separation distance. Non-monotonicity
occurs when conductor geometries resemble natural contours of electrical
potential, as follows from the energy theorem of Thomson (Lord Kelvin)
(Jackson, 1999, pp. 53). An analogous energy theorem due to Kelvin (1849)
holds in an ideal fluid. Whereas repulsion occurs in electrostatics when
conductors resemble equipotential lines, we show surprising effects on fluid
forces when boundaries resemble streamlines. In this paper, we analyse ideal
fluid forces in such geometries, through consideration of both unsteady and
steady components of force (Sedov, 1965, pp. 17).

The unsteady force on a submerged body is relevant during transient motions.
It is often characterized by an effective added mass that limits large
accelerations (Newman, 2018; McKee & Czarnecki, 2019) and alters the natural
vibration frequencies of submerged structures (Valentín et al., 2014; Newman,
2018). Moreover, unsteady forces play a critical role in swimming mechanisms
(Saffman, 1967; Childress, 1981; Weymouth & Triantafyllou, 2013; Steele et
al., 2017) of cephalopods and biomimetic robots (Bujard et al., 2021).

The steady force is relevant even when there is no acceleration, and vanishes
(along the direction of motion) in an infinite fluid according to d’Alembert’s
paradox (Batchelor, 2000, 404-405). The steady force is generally non-zero in
higher connectivities; for example, there is a non-zero steady force between
two interacting circular cylinders (Wang, 2004; Tchieu et al., 2010). Notably,
the steady force is exploited for industrial applications in non-contact
Bernoulli grippers (Giesen et al., 2013; Davis et al., 2008), in which a fluid
source creates a region of high velocity and low pressure, resulting in an
attractive lift force that can be used to manipulate objects (see Ralph (2014)
for a video).

In the presence of boundaries, ideal fluid forces have been studied in various
contexts. Basset (1888, pp. 215) gave the exact added mass of a cylinder in a
concentric confining cylinder. Lamb (1975, §93) solved the spherical version
of that problem, which was verified experimentally by Ackermann & Arbhabhirama
(1964). Confinement effects were studied further in the context of nuclear
reactor core oscillations at the Argonne National Laboratory(Chung & Chen,
1984; Wambsganss et al., 1974). Brennen (1982) then provided an extensive
review of added mass and analysed boundary effects in various geometries.
Later, Wang (2004) presented analytical formulae for the fluid forces in the
two-cylinder problem. More recently, Tchieu et al. (2010) formulated the two-
body problem using a conformal map approach and revisited the two-cylinder
problem.

In all these studies, boundaries were found to increase the added mass,
relative to the boundary-free problem. One might wonder if there exists a
boundary geometry in which the added mass of a body can actually be decreased.
If this were possible, strategic boundary placement could give large
accelerations during transient motions. We prove that no such boundary
geometry exists, a result that follows from the minimum energy theorem of
Kelvin (1849). We then use Kelvin’s minimum energy theorem to make a direct
analogy with the electrostatic results of Levin & Johnson (2011). This analogy
is discussed in detail in §2.2 and illustrated in figure 1.

Specifically, the unexpected repulsive forces found by Levin & Johnson (2011)
stem from a non-monotonic dependence of the electric field energy on the
conductor-charge distance. If the energy is non-monotonic, then the force must
take both positive and negative values, since force is related to the gradient
of energy. Using an energy theorem of electrostatics, Levin & Johnson (2011)
showed that a non-monotonic energy results when conductors resemble contours
of potential. By analogy, we pose the question: _Does the ideal fluid kinetic
energy of a system always depend monotonically on its separation distance to a
boundary?_ We answer this question in the negative and analyse the effect of
energy non-monotonicity on ideal fluid forces.

The remainder of this paper is arranged as follows. In §2.1, we state Kelvin’s
minimum energy theorem along with an important corollary regarding boundaries.
In §2.2, we use the results of §2.1 to demonstrate that the fluid kinetic
energy can be non-monotonic in the boundary separation distance, if boundaries
approximate streamlines. §3 is dedicated to unsteady force effects. In §3.1,
we show that boundaries cannot decrease the added mass. In §3.2, we analyse
the unsteady force by revisiting a calculation of Lamb (1975, §137), but with
a streamline-approximating boundary. In §4, we analyse the steady force
through two models of a Bernoulli gripper, which we solve exactly using the
framework of Crowdy (2020). In both §3.2 and §4, forces reverse sign near
energy minima predicted by Kelvin’s minimum energy theorem, when boundaries
approximate streamlines.

##  2 Ideal Fluid Energy and Kelvin’s Theorem

###  2.1 Kelvin’s Minimum Energy Theorem

We begin by stating the minimum energy theorem of Kelvin (1849), which will be
generalized thereafter. The statement is as follows.

###### Theorem 1

Consider an ideal fluid domain $D\subseteq\mathbb{R}^{2}$, with velocity field
$\boldsymbol{v}(\boldsymbol{x})=\bnabla\phi(\boldsymbol{x})$ satisfying
$\nabla^{2}\phi=0$ with no-flux boundary conditions on $\partial D$, where
$\phi$ is single-valued. Further assume the volume of each boundary does not
change in time and the velocity vanishes at infinity. Then any other
incompressible flow $\boldsymbol{q}(\boldsymbol{x})$, satisfying the boundary
conditions on $\partial D$, will possess kinetic energy greater than or equal
to that of $\boldsymbol{v}$,

$\frac{1}{2}\rho\int_{D}||\boldsymbol{v}||^{2}dV\leq\frac{1}{2}\rho\int_{D}||\boldsymbol{q}||^{2}dV,$
(1)

when $||\boldsymbol{v}||^{2}$ contains no non-integrable singularities.

When there exist source singularities in $D$, the integrals in (1) are not
well-defined. This point is important since our model in §4 involves point
sources. We reserve a discussion of this technicality for §4.1 and Appendix
LABEL:reg_proc. In these analyses, we show that all results of §2 still hold
if the flow possesses a source singularity, as long as the energy is defined
appropriately. Until §4, we will continue to assume that there are no non-
integrable singularities in the flow. We now present the proof of Kelvin’s
minimum energy theorem as stated in Theorem 1, when the flow possesses no
singularities.

######  Proof 2.2.

We now prove Theorem 1. We choose to decompose $\boldsymbol{q}$ according to
$\boldsymbol{q}=\boldsymbol{v}+\boldsymbol{w}$, where $\boldsymbol{w}$ is
finite in $D$. Since $\boldsymbol{v}$ already satisfies the appropriate
boundary conditions, $\boldsymbol{w}$ must satisfy the no-flux condition
$\boldsymbol{w}\cdot\boldsymbol{n}=0$ on the boundary,
$\boldsymbol{x}\in\partial D$. Noting that
$(||\boldsymbol{q}||^{2}-||\boldsymbol{v}||^{2})=2\bnabla\phi\cdot\boldsymbol{w}+||\boldsymbol{w}||^{2}$,
we find that

$\frac{1}{2}\rho\int_{D}\left(||\boldsymbol{q}||^{2}-||\boldsymbol{v}||^{2}\right)dV=\rho\left(\int_{C}\phi\boldsymbol{w\cdot
n}dA-\int_{D}\phi\bnabla\cdot\boldsymbol{w}dV\right)+\frac{1}{2}\rho\int_{D}||\boldsymbol{w}||^{2}dV,$
(2)

where $C$ is equal to $\partial D$, since we took $\phi$ to be single valued.
Then the first integral on the right hand side vanishes by the no-flux
condition on $\boldsymbol{w}$. The second integral on the right hand side
vanishes by the incompressibility of $\boldsymbol{q}$. Meanwhile, the last
term is non-negative.

In general, there is no unique solution to the Laplace problem with Neumann
boundary conditions: solutions are only determined up to an unspecified amount
of circulation around each boundary. In the above proof, assuming that $\phi$
was a single-valued function specified a particular solution to the Laplace
problem. This solution is defined by the property that the flow $\nabla\phi$
possesses no circulation in the sense that
$\int_{\gamma}\nabla\phi\cdot\boldsymbol{dl}=0$ around any closed loop
$\gamma$. Hence, Theorem 1 is applicable to the solution which possesses no
circulation. As was noted by Gonzalez & Taha (2022), the correct conclusion is
that the minimal kinetic energy solution is the one possessing no circulation;
in their paper, it is argued that the true solution is the one minimizing
curvature. However, in the present paper, we focus on the zero circulation
solution.

In Appendix B, we present a generalization of Theorem 1 to the case where
arbitrary circulations may exist around each flow boundary. When circulation
is present, the above proof is modified because the contour $C$ cannot be
taken equal to $\partial D$; instead, $C$ must be chosen not to cross branch
cuts associated with a multivalued $\phi$. It turns out that Theorem 1 is
still applicable even when there is circulation, provided that a particular
flux condition is satisfied across branch cuts (see Appendix B).

We emphasize again that in the present paper, we restrict our attention to the
behaviour of the zero circulation solutions, to which Theorem 1 applies. We
proceed by showing that an added stationary boundary never decreases the
energy. This corollary will be important in establishing the result of §2.2,
regarding energy non-monotonicity.

######  Corollary 2.3.

Suppose a stationary impenetrable body $C$, with boundary $\partial C\subset
D$, is added to the previous flow. Then a potential flow solution exists with
velocity $\bnabla\phi_{C}$ defined over $D\backslash C$, where
$\nabla^{2}\phi_{C}=0$. Furthermore, the kinetic energy of that flow is
greater than the boundary-less potential flow solution,

$\frac{1}{2}\rho\int_{D\backslash
C}||\bnabla\phi_{C}||^{2}dV\geq\frac{1}{2}\rho\int_{D}||\boldsymbol{v}||^{2}dV.$
(3)

######  Proof 2.4.

The ideal flow, in the region $D\backslash C$, has a solution by standard
existence theorems (Thomson, 1848). That solution has no flux into the region
$C$ and a perfectly valid choice of an incompressible flow $\boldsymbol{q}$
defined over all $D$, as defined in Theorem LABEL:thm2, is

$\boldsymbol{q}(\boldsymbol{x})=\left\\{\begin{array}[]{ll}\bnabla\phi_{C}&\boldsymbol{x}\in
D\backslash C\\\ \boldsymbol{0}&\boldsymbol{x}\in C.\end{array}\right.$ (4)

This flow has precisely the kinetic energy of the potential flow solution in
the presence of the boundary. However, by Theorem 1, the flow possesses more
energy than the boundary-less flow. Note that $\partial C$ has measure zero,
$\boldsymbol{v}$ is defined over $D$, and $\bnabla\phi_{C}$ over $D\backslash
C$.

In the proof of Theorem 1, it may have appeared that the reason
$\boldsymbol{q}$ has more energy than $\boldsymbol{v}$ is either that the
former is rotational or possesses non-zero circulation. However, we have
demonstrated a third possibility via corollary 2.3. Namely, if
$\boldsymbol{q}$ is irrotational and possesses zero circulation, but is
subject to additional boundary conditions, it will also possess an energy
greater or equal to that of $\boldsymbol{v}$. This statement is understood
most simply by appealing to a variational argument; see, for example, Whitham
(1974, pp. 434) for a deeper discussion of the variational formulation of
potential flow. The general argument is as follows. A solution to the Laplace
equation $\phi$ can be viewed as a function that extremizes the energy
functional
$S[\phi(\boldsymbol{x})]=(\rho/2)\int{\left(\nabla\phi\cdot\nabla\phi\right)}dV$,
so that $\delta S=0$, subject to a given set of constraints. In other words,
$\phi(\boldsymbol{x})$ can be thought of as the result of a constrained
optimization of $S[\phi(\boldsymbol{x})]$. In potential flow, the
extremization occurs at a minimum of $S[\phi(\boldsymbol{x})]$ and the
constraints are given by the boundary conditions of the flow. It follows that
imposing additional boundary conditions (constraints) on the constrained
optimization problem can never result in a better minimization of
$S[\phi(\boldsymbol{x})]$.

This is in contrast to the electrostatic problem with a neutral conducting
boundary. In the electrostatic problem, surface charge is allowed to
distribute itself on the surface of the conductor, as long as the net charge
is zero. The presence of surface charge affects the relevant energy
functional. Hence, the addition of a conducting boundary presents an
additional degree of freedom in the optimization of the electrostatic energy
functional, and so neutral conducting boundaries decrease (or keep constant)
the electrostatic energy.

###  2.2 Non-Monotonic Kinetic Energy

We now demonstrate that the fluid kinetic energy need not vary monotonically
with the distance between a fluid system and a newly introduced boundary
$\partial C$, a result that follows from corollary 2.3. We define a fluid
system by a collection of boundary conditions on $\partial D$ as in Theorem 1.
A stationary boundary is introduced mathematically as an additional no-flux
boundary $\partial C$, as in corollary 2.3. As usual, slip is allowed in an
ideal fluid. The boundary-separation distance is defined as the distance
between a chosen point attached to $\partial C$ and one in $\partial D$. We
now demonstrate by example, referring to figure 2, as follows.

Consider $D$ to be the exterior to a unit circle that translates vertically at
unit velocity. We choose a boundary $\partial C$ comprising a pair of
infinitely thin arcs resembling streamlines, which are drawn in three distance
configurations, labelled $A,B,C$. The boundary-separation distance, labelled
$x$, is taken as the vertical distance from the centre of the circle to the
bottom of the boundary. We call the kinetic energy of the system $T(x)$. When
$x\rightarrow\infty$, the boundary-less energy is recovered. At other
distances, $T(x)\geq T(\infty)$ by corollary 2.3. However, when $x=x_{B}$, the
boundary does not perturb the cylinder flow, as it coincides precisely with
streamlines, and $T(x_{B})=T(\infty)$. Therefore, $T(x_{B})$ corresponds to a
local minima of the kinetic energy as a function of the boundary-separation
distance. Hence, the kinetic energy dependence need not be monotonic.

Similarly, one can conclude that the kinetic energy of the configuration in
figure 1D is non-monotonic in the boundary separation distance, $y$. Since the
perfect slits do not perturb the source flow when $y=0$, the kinetic energy
$T(y)$ satisfies $T(0)=T(\infty)$. At other distances, $T(y)\geq T(\infty)$.
One can then deduce the schematic plot of the kinetic energy given in
$\ref{fig:compare_electromag}B$. A main goal of our investigation will be to
analyse the physical manifestation of the change in sign of the slope of
$T(y)$, in the highlighted region $y\gtrsim 0$.

The electrostatic energy of the configuration considered by Levin & Johnson
(2011) (see 1C) is plotted schematically in figure 1A. An energy theorem of
electrostatics states that a neutral conducting boundary must decrease (or
keep constant) the potential energy. The curve shape then follows from the
fact that $V(0)=V(\infty)$ and $V(x)\leq V(\infty)$. Thus, the electrostatic
energy theorem predicts the non-monotonic shape of the curve in figure 1A,
where the highlighted window manifests as a repulsive electrostatic force,
$F_{\mathrm{e}}=-dV/dx$.

Although these discussions considered infinitely thin boundaries, we show in
§4.2 that the corresponding effects on fluid forces persist for finite
thicknesses.

Figure 1: A) Schematic illustration of the electrostatic energy of the
geometry given in panel C. B) Schematic illustration of the ideal fluid
kinetic energy of the geometry given in panel D. C) The electrostatic geometry
considered by Levin & Johnson (2011) is shown, where an electric point charge
is placed above a semicircular neutral conducting arc. D) An ideal fluid
geometry is given where, when $y=0$, the slits do not perturb the source flow.

##  3 The Unsteady Force

In §3.1, we show that the added mass is always greater than or equal to that
in the boundary-less problem. An alternative derivation is given in Appendix A
using a conformal mapping approach, that does not appeal to Kelvin’s energy
Theorem. In §3.2, we show the unsteady force found by Lamb (1975, §137)
reverses sign when the boundary approximates a streamline. Herein, we ignore
intrinsic body mass.

###  3.1 Added Mass Cannot Decrease

Consider a single translating body in an unbounded ideal fluid. The
Buckingham–Pi theorem (Buckingham, 1914) implies the total fluid kinetic
energy can be written as

$T_{f}=\frac{C_{a}}{2}\rho VU^{2},$ (5)

where $U$ is the body velocity, $C_{a}$ is a dimensionless constant called the
added mass coefficient, and $\rho$ and $V$ are the fluid density and body
volume, respectively. Considering acceleration at a constant rate $a$, (5)
gives the necessary work to move a distance $\delta x$, $\delta W=C_{a}\rho
Va\delta x$. The resulting force is therefore $F=-\delta W/\delta x=-C_{a}\rho
Va$; the reaction is equivalent to a mass augmentation of amount $C_{a}\rho
V$. Corollary 2.3 showed that boundaries may not decrease the total kinetic
energy $T_{f}$. Therefore, when the presence of a boundary is modelled as an
effective change in the added mass, increasing $T_{f}$ implies an increased
$C_{a}$ in (5).

Hence, boundaries cannot be used to decrease the added mass and attain higher
accelerations. The connection between boundary-induced added mass increase and
Kelvin’s theorem was mentioned by Lamb (1975, §93). Although boundaries cannot
reduce the added mass, geometries such as those described in §2.2 can lead to
interesting fluid force effects, to be analysed in the remainder of this
paper. The effects presented in the remainder of the paper rely on energy non-
monotonicity as discussed in §2.2, and are thus in analogy with the
electrostatic results of Levin & Johnson (2011).

Figure 2: A cylinder of unit radius and unit speed is pictured with its
boundary-less streamlines. Configurations of a streamline-approximating
boundary are superimposed and labelled $A,B,C$.

###  3.2 Effect of Non-monotonic Energy on Unsteady Force

When a body translates toward a boundary, according to the added mass
formulation, the energy can be expressed as in (5) with $C_{a}=C_{a}(x)$,
where $x$ is the separation distance to the boundary. Motion can only be taken
as 1-dimensional after either assuming a constraint or by exploiting a
symmetry in the $y$ direction, as is present in figure 2. This assumption is
made to simplify the calculation and illustrate our main argument. Note again
that we will ignore the intrinsic mass of the cylinder. If the cylinder had
mass $m_{c}$, then one should replace every $C_{a}$ with $C_{a}+m_{c}/(\rho
V)$ in the following analysis.

For 1D free motion toward the boundary, the Euler-Lagrange equation is

$2\frac{\partial C_{a}(x)}{\partial
x}\dot{x}^{2}+2C_{a}(x)\ddot{x}-\frac{\partial C_{a}(x)}{\partial
x}\dot{x}^{2}=\frac{1}{\dot{x}}\frac{d}{dt}\left(C_{a}(x)\dot{x}^{2}\right)=0,$
(6)

or $C_{a}(x)\dot{x}^{2}=\mathrm{const}$ since $\dot{x}\neq 0$. Lamb (1975,
§136) justified the Lagrangian formulation, and the validity of (5), in the
context of a body approaching a boundary with such a symmetry. In analysing
the motion of a sphere toward a plane wall, Lamb (1975, §137) deduced from (6)
a repulsive force since $\partial C_{a}/\partial x$ was always negative there.
However, this need not always be the case as discussed in §2.2: if the
boundary coincides, at some finite $x$, with the streamlines of a traveling
body then there is a local minima of $C_{a}(x)$. The equation of motion (6)
implies, setting the constant to 1, that $\partial\dot{x}/\partial
x=-C_{a}^{-\frac{3}{2}}(\partial C_{a}/\partial x)/2$ which switches sign at
the local minima.

Consider the geometry of figure 2. A circle translates vertically and
boundary-less streamlines, which are a function of velocity, are drawn for
reference. The circle approaches a boundary that approximates flow
streamlines. If, as the circle evolves according to (6), at some moment the
free streamlines coincide with the boundary (as pictured for distance
$x_{B}$), then the force will change sign when the boundary slits get aligned
with the streamlines, $x=x_{B}$. In a different scenario where the circle is
pushed at constant velocity, the necessary external force to maintain the
velocity reverses sign at $x_{B}$.

##  4 The Steady Force

Here, we analyse the effect of energy non-monotonicity, as described in §2.2,
on the steady fluid force. We consider a source flow with streamline-
approximating boundaries, as illustrated in figure 3. In the simply connected
problem (figure 3A), attractive Bernoulli suction is achieved when the source
is directly above the plate, giving a model of a Bernoulli suction gripper
(Davis et al., 2008; Giesen et al., 2013). However, we show that the sign of
the vertical force is reversed near the point where boundaries lie directly on
streamlines. It is worth noting that in this 2-dimensional model, the geometry
is not the typical Bernoulli gripper with axial symmetry as seen in Ralph
(2014); however the governing principle is the same. Both of the geometries of
figure 3 are solved exactly; the simply connected problem is solved using
conformal maps while the doubly connected problem is solved using the
theoretical framework of Crowdy (2020). We begin by demonstrating that the
analysis of §2, particularly corollary 2.3 and the non-monotonic energy
discussion in §2.2, is still applicable in the presence of a point source.

###  4.1 On the Validity of Kelvin’s Theorem with Source Singularities

In 2 dimensions, the kinetic energy of a source singularity, in an otherwise
unbounded fluid, diverges logarithmically. A source at the origin has velocity
$||\boldsymbol{v}||\propto 1/r$, so that the absolute kinetic energy scales as
$\int_{0}^{2\pi}\int_{\epsilon}^{L}||\boldsymbol{v}||^{2}rdrd\theta\propto\log{\left(L/\epsilon\right)}$.
The integral is divergent both at the source location, as $\epsilon\rightarrow
0$, and at infinity, as $L\rightarrow\infty$. Therefore, the absolute kinetic
energy of a source is not a meaningful quantity. However, forces and physical
observables are related to differences, and in particular gradients, of
energy. Therefore, the energy may be redefined with respect to any reference
system, without altering physical predictions. In what follows, we will
demonstrate a manner in which a finite energy can be defined even in the
presence of sources. It is with respect to this definition of energy that the
arguments of §2 will hold.

We note that a similar issue of energy divergence arises in the theory of
electrostatics. The electromagnetic field energy of a point charge is non-
integrable at the location of a charge in 3 dimensions. The divergence at the
location of the source is eliminated (regularized) by choosing to measure the
energy with respect to that of an isolated point source. Similarly, by
choosing to measure the fluid kinetic energy with respect to that of an
isolated source singularity, the divergence at the location of a fluid source
can be regularized. What is left is to deal with the issue of the integral’s
convergence at infinity. The divergence at infinity is related to the fact
that we are working in 2 dimensions. Sources of the Laplace equation are
integrable at infinity in dimensions $n\geq 3$.

The divergence at infinity may be resolved by two different approaches. The
first approach involves treating the 2-dimensional problem as the limit of a
3-dimensional problem. The 2-dimensional flow due to a point source at $x=y=0$
is equivalent to the flow produced by a uniform continuum (line) of
3-dimensional sources along the entire $z$-axis. If one considers a line of
sources with density $m$, in 3 dimensions, extending from $z=-L$ to $z=L$, the
radial velocity profile in the $z=0$ plane is given by $v_{r}(r)=m/(2\pi
r\sqrt{1+(r/L)^{2}})$; the integrated kinetic energy in the $z=0$ plane of
such a source is then convergent at infinity for any finite $L$. In the
presence of sources, the energy can be made finite at infinity by regulating
each source according to the prescription $m/(2\pi r)\rightarrow m/(2\pi
r\sqrt{1+(r/L)^{2}})$ for some finite regularizing parameter, $L$.

A second approach, which does not require invoking 3 dimensions, is described
by Saffman (1993, pp.125). In this approach, a small circle of radius
$\epsilon>0$ around each singularity is excluded from the domain when
computing the energy. The boundary at infinity is also replaced by a circle of
large radius $R$. In the limit of $\epsilon\rightarrow 0$ and
$R\rightarrow\infty$, the energy can be expressed as the sum of a finite part
and a divergent constant $C\propto\log{\left(R/\epsilon\right)}$. Since
constant shifts in energy are not observable, only the finite part of the
energy needs to be retained. This procedure is reminiscent of the procedure of
dimensional regularization used in quantum field theory, a connection nicely
illustrated by Olness & Scalise (2011).

The key takeaways from this section are as follows. The absolute kinetic
energy of a source flow is not finite in 2 dimensions, relative to a static
configuration, since the kinetic energy integral diverges both at the source
location and at infinity. Measuring the energy relative to that of an isolated
source eliminates the divergence at the source location. The divergence at
infinity can be regulated in two ways, and a finite regularized energy can be
defined. In terms of the regularized energy, all of the discussions of §2
apply, along with the interesting force behaviours predicted when boundaries
approximate streamlines, even when there exists a source singularity in the
flow. In what follows, we analyse the source flows of figure 3, and examine
the force behaviour when boundaries approximate the source streamlines.

###  4.2 Single Slit and Source

Consider an ellipse (which may degenerate to a slit) centered at the origin in
the complex plane, with some orientation $\alpha$, as shown in figure 3A. When
$x=0$, the source is located directly above the plate and Bernoulli suction
attracts the ellipse to the source, as shown in figure 4A. However, when
$x\neq 0$ is fixed, there is a value of $y=x\tan{\alpha}$ where the ellipse
falls on a natural source streamline. For a perfect slit, $y=x\tan{\alpha}$
corresponds to an energy minima as a function $y$, as follows from the
discussion of §2.2. We thus expect the vertical steady force to flip sign near
this point for sufficiently slender ellipses.

Figure 3: A) Geometry of a source above a boundary. A source of strength $m$
is located at $x+\mathrm{i}y$ and a stationary ellipse centered on the origin
and tilted at angle $\alpha$. When $\alpha=0$ and $x=0$, this is a 2D model of
a Bernoulli gripper above a plate extending into the page. B) Geometry of a
source $m$ above a doubly-connected body, consisting of a pair of slits. When
$\alpha=\pi/2$, this is a 2D model of a Bernoulli gripper above an object with
a central hole.

We employ a conformal mapping approach. The simpler problem of a source
exterior to a unit disk is solved and mapped to the problem of interest
(figure 3A) by a conformal map. We denote the physical domain coordinates by
$z$ and the pre-image coordinates by $\zeta$. The complex potential for a
source of strength $m=2\pi$ at location $\zeta_{s}$, where $|\zeta_{s}|>1$, is
$\log{\left(\zeta-\zeta_{s}\right)}$. The complex potential exterior to a
stationary unit disc is then given by the Milne-Thomson theorem (Milne-
Thomson, 1962, §6.21),

$W=\log{\left(\zeta-\zeta_{s}\right)}+\log{\left((1-\overline{\zeta_{s}}\zeta)/\zeta\right)}.$
(7)

The unit disk exterior maps to the tilted ellipse exterior by a Joukowsky-type
map,

$z(\zeta)=\left(\zeta+\frac{s}{\zeta}\right)e^{\mathrm{i}\alpha},$ (8)

where $s\in[0,1]$ for univalency (Llewellyn Smith et al., 2008); $s$ defines a
homotopy between the circle ($s=0$) and the slit of length $D=2$ ($s=1$).
Intermediate shapes are ellipses. To achieve the configuation in figure 3A,
$\alpha$ in (8) corresponds directly to $\alpha$ in the figure. However, the
source pre-image, $\zeta_{s}$, must be chosen to ensure that
$z(\zeta_{s})\equiv z_{s}=x+\mathrm{i}y$. After some algebra, one finds the
pre-image location,
$\zeta_{s}(z_{s})=\left(z_{s}e^{-\mathrm{i}\alpha}+\sqrt{\left(z_{s}e^{-\mathrm{i}\alpha}\right)^{2}-4s}\right)/2$.

In what follows, we analyse the vertical component of force as $y$ is varied
for a given $x$. The force is computed in the $\zeta$-plane since the complex
potential is known there according to (7). The steady force is given by the
expression of Tchieu et al. (2010, equation 18),

$F=\overline{\frac{\mathrm{i}}{2}\oint_{|\zeta|=1}{\left(\frac{dW}{d\zeta}\right)^{2}\left(\frac{dz}{d\zeta}\right)^{-1}d\zeta}},$
(9)

after setting the fluid density to be equal to 1. The vertical component is
$F_{s}\equiv\Imag\left\\{F\right\\}$. Using (7) and (8), the residue theorem
yields the vertical force on the ellipse,

$F_{s}=\Imag\left\\{-\frac{2\pi\zeta_{s}^{2}(\zeta_{s}-s\overline{\zeta_{s}})}{(\zeta_{s}^{2}-s)^{2}(|\zeta_{s}|^{2}-1)}e^{-\mathrm{i}\alpha}\right\\},$
(10)

which is plotted in figure 4A for the case of $x=0$ and $s=1$. The usual
Bernoulli suction effect is captured there; a low pressure on the top of the
plate leads to a net vertical (attractive) force. The suction persists
regardless of the angle $\alpha<\pi/2$ when $x=0$. Note that taking $s<1$ does
not qualitatively affect the shape of the curve.

We now examine the case where the slit may fall on a streamline of the source,
$x\neq 0$. Figure 4B plots the case of $x=2$ and $\alpha=\pi/4$ for various
slenderness parameters, $s$. When $s=1$, the slit coincides with a natural
(radial) streamline of the source at $y/D=1$. As per the discussion of §2.2,
this value of $y$ corresponds to a local energy minimum as a function of y,
and the vertical force should thus change sign when $y/D=1$. The plot in
figure 4B shows the force changes sign at precisely this point for a perfect
slit, $s=1$. There is a local region of repulsion for $y/D>1$. For large
$y/D$, the force becomes attractive again. Meanwhile, the onset of sign-
reversal of the force is shifted for an ellipse of finite slenderness ($s<1$)
until the effect vanishes for $s=0.79$ where there is no repulsive region. For
large $y/D$, all force plots are positive, indicating attraction.

Figure 4: A) Plot of the vertical steady force on the slit in the geometry
shown in figure 3A when $x=0$, $s=1$, for various $\alpha$. The force is
positive for $y>0$, indicating attraction as in the standard Bernoulli
gripper. B) Same plot when $\alpha=\pi/4$, $x=2$, for various slenderness
parameters $s$. The slit approximates a streamline when $y/D=1$, leading to an
energy minima and local sign-reversal of the force only for slender ellipses
($s\approx 1$).

###  4.3 Two Slits and Source

The vertical steady force is computed by a similar procedure in the two-body
geometry shown in figure 3B. The canonical domain for the doubly connected
problem is the annulus, where the problem can be solved simply and then mapped
to the physical domain by a conformal map (see figures 6A and 6D). The
conformal map from the annulus, $|\zeta|\in[\rho,1]$, to the two-slit geometry
shown in figure 3B is

$z(\zeta)=-\mathrm{i}Ae^{3\mathrm{i}\alpha}\frac{\omega(\zeta,\sqrt{\rho}e^{2\mathrm{i}\alpha})\omega(\zeta,\sqrt{\rho}^{-1}e^{2\mathrm{i}\alpha})}{\omega(\zeta,\sqrt{\rho})\omega(\zeta,\sqrt{\rho}^{-1})},$
(11)

where $\omega$ is the Schottky–Klein prime function defined by Crowdy (2020,
pp. 64,75); a similar map was used by Crowdy (2009) to analyse the Weis-Fogh
mechanism. Note that we consider figure 3B with zero-thickness slits. To
accommodate a source, a sink of equal strength is included in the annulus for
mass conservation, with the sink located at the pre-image of infinity,
$\zeta_{\infty}=\sqrt{\rho}$. The complex potential in the $\zeta$-plane is
(Crowdy, 2013)

$W(\zeta)=\frac{m}{2\pi}\log{\left(\frac{\omega(\zeta,\zeta_{s})\omega(\zeta,\overline{\zeta_{s}}^{-1})}{\omega(\zeta,\zeta_{\infty})\omega(\zeta,\overline{\zeta_{\infty}}^{-1})}\right)},$
(12)

up to a constant that does not enter our calculation. The source location in
the annulus, $\zeta_{s}$, must be chosen such that its image lies on the
imaginary axis as in figure 3B. Conveniently, we notice the circle
$|\zeta|=\sqrt{\rho}$ maps to the imaginary axis. We proceed by resolving the
vertical component of force on the boundary. By symmetry, we can simply double
the force on one slit; thus, the total force is twice that given in (9). Note
that the map of (11) has zeros at the locations of the sharp slit edges. To
avoid the corresponding poles while evaluating (9), the contour may be
deformed slightly into the fluid domain, as was noted by Tchieu et al. (2010).

The derivatives $dW/d\zeta$ and $dz/d\zeta$, and hence the integrand of (9),
can be written analytically in terms of the functions
$K(\zeta,\alpha)=\zeta\partial\log\omega(\zeta,\alpha)/\partial\zeta$. The
integrand may then be explicitly evaluated using the series representations
given by Crowdy (2020, pp. 278,280), and easily integrated numerically. The
result is plotted in figure 5 for the case of two slits near-parallel to the
real axis.

For large positive $y$, the force on the plates is positive, indicating
attraction between the source and slits. For large negative $y$, the force on
the plates is negative, indicating attraction again. However, a small region
of repulsion is encountered near the point $y=0$, when the source is aligned
with the slits. At small distances above the point $y=0$, when $y\gtrsim 0$,
the source is evidently repelled by the slits as the force on the slits is
negative. Just below the point $y=0$, when $y\lesssim 0$, the force on the
slits becomes positive, indicating attraction again since the source is still
above the slits. When $y$ becomes sufficiently negative, so that it lies below
the slits, the force on the slits again becomes negative, indicating
attraction as we already noted.

All force plots vanish when $y=0$, and the force switches from attractive to
repulsive. The location of this zero is precisely predicted by the discussion
of §2.2: when the natural source streamlines align with the slits, at $y=0$ as
seen in figure 3B, there exists a local energy minima as a function of $y$.

When the slits are parallel to the real axis, $\alpha=\pi/2$, the force is an
odd function of $y$. Interestingly, the odd symmetry is quickly disrupted for
deviations from $\alpha=\pi/2$. For $\alpha<\pi/2$, the repulsive region is
retained, but the repulsive force magnitude is diminished. Since decreasing
$\alpha$ pushes the reference point $y=0$ further away from the plates (see
figure 3B), it is to be expected that the magnitude of the repulsive force
should decrease since the source flow decays with distance. Despite its
diminishing magnitude, the repulsive region persists for $\alpha<\pi/2$, as is
seen in figure 5.

As was found in the single body calculation of §4.2, we expect the repulsion
found in the two-body problem to be valid up to some finite slit widths.
However, we do not present computations for finite slit widths in the two-body
problem.

Additionally, we note that no Kutta condition was imposed in the calculation
of §4.3. The forces in figure 5 should thus be interpreted as those on
slightly inflated slits with zero circulation and smooth corners. If the slits
were truly perfect, a Kutta condition would be necessary, which would affect
the vertical force. By slightly inflating the ellipse, the singularity in the
conformal map is pushed off of the object boundary, into the exterior of the
fluid domain; hence, this interpretation is consistent with our method of
calculating (9) by pushing the contour slightly into the fluid domain to
exclude the map singularity.

Figure 5: Vertical force on the slits shown in figure 3B, near $\alpha=\pi/2$
(slits on real axis). For large $|y/D|$ there is attraction as in the usual
Bernoulli gripper. When $y=0$, slits lie on streamlines and there is local
repulsion. For tilted slits, there is only repulsion for $y/D\gtrsim 0$ as
indicated by the shaded region.

##  5 Conclusion

It follows from the minimum energy theorem of Kelvin (1849) that boundaries
cannot reduce the kinetic energy and hence the added mass of a translating
body. However, boundaries that resemble streamlines can create local energy
minima as a function of the boundary-separation distance, resulting in non-
intuitive effects on fluid forces.

By a similar analysis in the theory of electrostatics, Levin & Johnson (2011)
predicted a maximum in the electromagnetic energy when conducting boundaries
approximate potential contours, which manifested as a repulsive electrostatic
force of induction. Below, we outline the analogous effects found in an ideal
fluid, due to the energy minimum occurring when boundaries approximate
streamlines.

In a system where Lamb (1975, §137) found a repulsive unsteady force, we show
a transition to attraction when boundaries approximate streamlines. In two
models of a Bernoulli gripper, where the steady force is typically attractive,
we show that the steady force reverses sign when the boundary approximates a
streamline. Effects are most prominent with slit-type boundaries, but are
shown in §4.2 to persist for bodies with finite thicknesses. The geometry-
dependent sign-reversal of the steady force in the Bernoulli gripper might be
useful in future engineering design. Furthermore, similar effects may be
relevant in electromagnetic power flows in 2D near-zero-index media, which
were recently shown to be mathematically equivalent to ideal fluid flows
(Liberal et al., 2020).

####  Acknowledgements. The author is grateful for valuable discussions with
Peter Baddoo, Bavand Keshavarz, Valeri Frumkin, Ousmane Kodio, and John Bush.

####  Funding. The author was funded by an MIT Presidential Fellowship during
this work.

####  Declaration of Interests.

The authors report no conflict of interest.

##  Appendix A Proof of Added Mass Increase by Conformal Maps

We prove that the added mass force may not decrease in the presence of a
stationary boundary. The analysis follows that of Llewellyn Smith et al.
(2008) and Tchieu et al. (2010).

###  A.1 Complex Variable Formulation

Suppose a two-dimensional body $B$ translates in an ideal fluid with velocity
$(V_{x},V_{y})$, which we write in complex notation as $V\equiv
V_{x}+\mathrm{i}V_{y}$. Suppose also that there is a stationary boundary
defined by $\partial D$, and fluid occupies the region between $\partial D$
and $\partial B$ (which may be finite or unbounded). The fluid flow between
the boundaries is characterized by a complex potential satisfying the Laplace
equation, $\nabla^{2}W(z)=0$, where $z=x+\mathrm{i}y$. The velocity field is
given by $U(x,y)=\overline{dW(z)}/dz=u(z)+\mathrm{i}v(z)$, where the bar
denotes complex conjugation. The no-flux boundary condition takes the form
$\Real\\{\overline{U}n\\}=\Real\\{\overline{V}n\\}$ for $z\in\partial B$, and
$\Real\\{\overline{U}n\\}=0$ for $z\in\partial D$, where $n$ is the complex
normal vector to the boundary.

This problem is conveniently phrased using a conformal map. By the doubly-
connected version of Riemann mapping theorem (Goluzin, 1969, pp. 208), there
exists a conformal map between the annulus and a generic two-body geometry.
Thus, the Laplace equation may be solved in the annulus, subject to
appropriate boundary conditions, and taken to the physical plane using a
conformal map (Tchieu et al., 2010). We take the annulus defined by
$|\zeta|\in[\rho,1]$ without loss of generality. We also specify that the
conformal map $z(\zeta)$ takes $|\zeta|=1$ to $\partial B$ and $|\zeta|=\rho$
to $\partial D$. The Neumann boundary conditions become

$\Imag\\{W(\zeta)\\}=\Imag\\{\overline{V}z(\zeta)\\},$ (13)

on each boundary; note that the second boundary is stationary so $V=0$ there.
We can also take the velocity of $B$ to be purely imaginary, $V=Ui$ with
$U\in\mathbb{R}$.

Figure 6: Conformal Map Geometries. (A) The annulus geometry in the
$\zeta$-plane. The conformal map $z(\zeta)$ maps the outer circle to $\partial
B$ and the inner circle to $\partial D$. (B) Example of inverted confined
geometry ($b=0$). (C) Example of a confined geometry without inversion. (D)
Example of an unbounded fluid geometry ($b\neq 0$), the conventional two-body
problem.

The conformal map may be written as a general Laurent series plus a simple
pole in the fluid domain,

$z(\zeta)=\sum_{k\in\mathbb{Z}}a_{k}\zeta^{k}+\frac{b}{\zeta-c},$ (14)

where $c$ is located in the fluid domain, $|c|\in(\rho,1)$. For the confined
problem, there is no fluid at infinity and thus no singularity, i.e., $b=0$.
The area enclosed by $\partial B$, $A_{1}$, may be computed by Green’s theorem
though it never enters direct calculations.

###  A.2 Force Calculation and Added Mass Increase

The unsteady force on $B$ is given by the Blasius formula in the
$\zeta$-plane,

$F_{US}=\mathrm{i}\frac{d}{dt}\left(\oint_{|\zeta|=1}{z(\zeta)\frac{dW(\zeta)}{d\zeta}d\zeta}\right)+A_{1}\ddot{z},$
(15)

where the fluid density has been set to 1 (Tchieu et al., 2010). The second
term contributes negatively to the added mass, while the first has been seen
in examples (Tchieu et al., 2010) to contribute positively so the added mass
is overall a positive quantity. Herein, we will analyse (15) and show that the
boundary will never decreases the added mass relative the boundary-less
problem. The transient problem with $U=0$ and $\dot{U}=1$ defines the added
mass coefficient; equivalently, we solve for the force (15) by setting $U=1$
and eliminating the time derivative. The added mass coefficient opposing the
acceleration is then defined (noting that acceleration is along the imaginary
axis) by the relation,

$C_{a}(\rho)=-\Imag\left\\{\frac{F_{US}}{A_{1}}\right\\}=-\frac{2\pi}{A_{1}}\Imag\left\\{{\mbox{Res}_{|\zeta|<1}}\left(z(\zeta)\frac{dW(\zeta)}{d\zeta}\right)\right\\}-1,$
(16)

by the residue theorem. The residue term in (16) may be expressed in terms of
the coefficients of the mapping after imposing the boundary conditions. The
boundary data is only non-zero on the unit circle $|\zeta|=1$, and is expanded
there in a Laurent series as

$\Imag\\{\overline{V}z(\zeta)\\}=-\frac{1}{2}\left(\sum_{n\in\mathbb{Z}}\left(a_{n}+\bar{a}_{-n}\right)\zeta^{n}+\frac{b}{\zeta-c}+\frac{\bar{b}\zeta}{1-\bar{c}\zeta}\right)=\sum_{k\in\mathbb{Z}}b_{k}\zeta^{k},$
(17)

where we have set $V=\mathrm{i}$ as noted earlier. The residue theorem gives
the data coefficients,

$b_{n}=\left\\{\begin{array}[]{ll}-\frac{1}{2}\left(a_{n}+\bar{a}_{-n}+\bar{b}\,\bar{c}^{\;n-1}\right)&n>0\\\
-\frac{1}{2}\left(a_{n}+\bar{a}_{-n}+bc^{-n-1}\right)&n\leq
0,\end{array}\right.$ (18)

noting that $|c|<1$. The complex potential is obtained by matching the data of
(18) onto $W(\zeta)$ at the boundaries. We expand in a Laurent series,
$W(\zeta)=\sum_{k\in\mathbb{Z}}w_{k}\zeta^{k}$. As noted by Tchieu et al.
(2010), the imaginary part of $W(\zeta)$ can be written as an analytic
function using the Schwartz conjugate on each boundary giving two equations,

$\Imag\\{W(\zeta)\\}=\frac{1}{2i}\left(W(\zeta)-\bar{W}(\rho_{i}^{2}\zeta^{-1})\right)=\frac{1}{2i}\sum_{k\in\mathbb{Z}}\left(w_{k}-\overline{w}_{-k}\rho_{i}^{-2k}\right)\zeta^{k}.$
(19)

where $\rho_{i}\in\\{\rho,1\\}$. The null boundary data on $|\zeta|=\rho$
relates the coefficients by $\rho^{2k}w_{k}=\bar{w}_{-j}$. Meanwhile, matching
the $\rho_{i}=1$ equation of (19) onto the data in (18) yields
$w_{k}=2ib_{k}/(1-\rho^{2k})$, which is explicitly

$w_{n}=\left\\{\begin{array}[]{ll}-\mathrm{i}\left(a_{n}+\bar{a}_{-n}+\bar{b}\,\bar{c}^{\;n-1}\right)/(1-\rho^{2n}),&n>0\\\
-\mathrm{i}\left(a_{n}+\bar{a}_{-n}+bc^{-n-1}\right)/(1-\rho^{2n})&n\leq
0.\end{array}\right.$ (20)

The derivative of the potential is then obtained by term-by-term
differentiation, $dW/d\zeta=\sum_{k\in\mathbb{Z}}kw_{k}\zeta^{k-1}$. Now
$C_{a}(\rho)$ can be computed explicitly using the residue theorem, in terms
of the coefficients of the map of (14); the explicit expression for $C_{a}$ is
suppressed for brevity. We now show that the effect of a boundary is never to
decrease the added mass (16), for a given shape $B$, relative to the boundary-
less problem. Note that $\rho\rightarrow 0$, corresponds to the boundary-less
problem (Crowdy, 2020). The problem can be phrased as: _can one choose
coefficients $\\{a_{i},b\\}$ in_ (14) _such that $C_{a}(\rho)/C_{a}(0)<1$ for
some $\rho\in(0,1)$?_ We begin by assuming $C_{a}(\rho)/C_{a}(0)<1$ holds and
proceed toward a contradiction.

After letting $B\equiv bc^{k-1}$, some algebra reveals that
$C_{a}(\rho)/C_{a}(0)<1$ is equivalent to

$\displaystyle\Real\left\\{\sum_{k>0}\frac{k\rho^{2k}}{1-\rho^{2k}}\left(B\bar{B}+2\bar{B}(\bar{a}_{k}+a_{-k})+(a_{k}+\bar{a}_{-k})(a_{-k}+\bar{a}_{k})\right)\right\\}<0.$
(21)

The left hand side may be bounded below by applying the Cauchy–Schwarz
inequality. It follows that (21) implies the weakened inequality

$\sum_{k>0}\frac{k\rho^{2k}}{1-\rho^{2k}}\left(|B|^{2}-2|B||\bar{a}_{k}+a_{-k}|+|a_{k}+\bar{a}_{-k}|^{2}\right)<0,$
(22)

which is impossible since the left side is positive semi-definite. Therefore,
it is true that $C_{a}(\rho)\geq C_{a}(0)$.

##  Appendix B Kelvin’s Theorem With Circulation

The main assumption in the proof of Theorem 1 was that the velocity potential
$\phi$ was single-valued. This assumption implies that the corresponding
irrotational flow $\boldsymbol{v}=\bnabla\phi$ has zero circulation around any
closed loop. Here, we will demonstrate how Kelvin’s energy theorem, as
presented in Theorem 1, may be generalized in the presence of non-zero
circulation. Note that a statement resembling the theorem to be proved here
was given in words by Lamb (1975, §56), but not proved there.

We consider a flow in domain $D$ as described in Theorem 1, and we define the
boundary domain $B\equiv\mathbb{R}^{2}\backslash D$. Suppose that $B$ is
comprised of $N$ disconnected parts labelled $B_{i}$, each containing some
circulation so that we can write the complex potential function as

$W_{\Gamma}(z)=\tilde{W}(z)+\mathrm{i}\sum_{k=1}^{N}\frac{\Gamma_{k}}{2\pi}\log{\left(z-z_{k}\right)},$
(23)

where $z_{k}\in B_{k}$ and $\tilde{W}(z)$ is a single valued function over
$D$. $\Gamma_{k}$ gives the circulation around the body $B_{k}$. The
multivaluedness of $\phi=\mathrm{Re}\\{W\\}$ comes from the branch cuts of the
logarithm terms. We now state and subsequently prove a generalization of
Theorem 1, applicable to a general potential of the form given in (23).

######  Theorem B.5.

Consider an ideal fluid domain $D\subseteq\mathbb{R}^{2}$, with velocity field
$\boldsymbol{v}(\boldsymbol{x})=\bnabla\phi(\boldsymbol{x})$ satisfying
$\nabla^{2}\phi=0$ with no-flux boundary conditions on $\partial D$. Further
assume the boundary $B\equiv\mathbb{R}^{2}\backslash D$ may be subdivided into
$N$ disjoint regions $B_{1},B_{2},...,B_{N}$. Also assume
$\phi=\mathrm{Re}\\{W\\}$ to single-valued up to some amount of circulation
around each boundary such that the complex potential is written as,

$W(z)=\tilde{W}(z)+\mathrm{i}\sum_{k=1}^{N}\Gamma_{k}\log{\left(z-z_{k}\right)},$
(24)

for some circulations $\Gamma_{k}\in\mathbb{R}$ and where $z_{k}\in B_{k}$. If
the fluid domain $D$ extends to infinity, it should be further assumed that
$\sum_{k=1}^{N}\Gamma_{k}=0$, so that the energy intergal converges at
infinity. Further assume that the volume of each boundary does not change in
time and the velocity vanishes at infinity. Then any other incompressible flow
$\boldsymbol{q}(\boldsymbol{x})$, satisfying the boundary conditions on
$\partial D$, possesses a kinetic energy greater than or equal to that of
$\boldsymbol{v}$,

$\frac{1}{2}\rho\int_{D}||\boldsymbol{v}||^{2}dV\leq\frac{1}{2}\rho\int_{D}||\boldsymbol{q}||^{2}dV,$
(25)

provided that

$\sum_{k=1}^{N-1}\left(\sum_{j=1}^{k}\Gamma_{j}\right)\int_{L_{k}}\left(\boldsymbol{q}-\boldsymbol{v}\right)\cdot\boldsymbol{n}dA\geq
0,$ (26)

where $L_{k}$ is a line connecting $z_{k}$ to $z_{{k+1}}$; $L_{k}$ are chosen
not to intersect one another. As in Theorem 1, we assume the bulk flow
$||\boldsymbol{v}||^{2}$ contains no non-integrable singularities.

It is clear that in the case that $\Gamma_{i}=0$ for all $i$, Theorem 1 is
reproduced. An alternative condition for the applicability of (25) is that the
flux through each line $L_{i}$ is equal for the flows being compared,
$\boldsymbol{q}$ and $\boldsymbol{v}$. In that case the integrals in (26) all
vanish. This condition is reminiscent of that proposed by Saad & Majdalani
(2017), who generalized Kelvin’s minimum theorem to simply connected domains
with porous boundaries, while assuming a single-valued potential. We now prove
Theorem B.5.

######  Proof B.6.

The flow is decomposed according to
$\boldsymbol{q}=\boldsymbol{v}+\boldsymbol{w}$, where $\boldsymbol{w}$ is
finite in $D$. We then find that
$(||\boldsymbol{q}||^{2}-||\boldsymbol{v}||^{2})=2\bnabla\phi\cdot\boldsymbol{w}+||\boldsymbol{w}||^{2}$.
The integrand is integrable and the product rule and divergence theorem apply,
yielding

$\frac{1}{2}\rho\int_{D}\left(||\boldsymbol{q}||^{2}-||\boldsymbol{v}||^{2}\right)dV=\rho\left(\int_{C}\phi\boldsymbol{w\cdot
n}dA-\int_{D}\phi\bnabla\cdot\boldsymbol{w}dV\right)+\frac{1}{2}\rho\int_{D}||\boldsymbol{w}||^{2}dV.$
(27)

The second term on the right hand side vanishes by the incompressibility
condition on $\boldsymbol{q}$ and the last term is non-negative. It is left to
show that the first term on the right hand side is greater or equal to zero.
The contour $C$, over which that integral is taken, is simply the boundary
$\partial D$ if $\phi$ is a single valued function. Otherwise, it must be
chosen carefully so as to not cross branch cuts in order for the divergence
theorem to be applicable (see figure 7B). If the net sum of circulations
vanishes, as we assumed, we can always choose the branch points to connect to
one another in the following manner so that there are $N-1$ distinct branch
cuts for $N$ disconnected boundaries.

First, label each disconnected part of the boundary, $B_{1},B_{2},...B_{N}$.
Beginning with $B_{1}$, choose its branch cut, which begins at $z_{1}$, to
pass through the branch point inside $B_{2}$; label this line $L_{1}$. The
discontinuity across $L_{1}$ is $\Gamma_{1}$. Next, take both the branch cuts
associated with $B_{1}$ and $B_{2}$ to extend to $z_{3}\in B_{3}$. We choose
the the branches of $B_{1}$ and $B_{2}$ to coincide on a line labelled
$L_{2}$, which then has a discontinuity $\Gamma_{1}+\Gamma_{2}$. Repeating
this procedure, we arrive at the line $L_{N-1}$ connecting $B_{N-1}$ to
$B_{N}$, whose discontinuity is give by $\sum_{j=1}^{N-1}\Gamma_{j}$. Taking
now all the branches to extend to complex infinity along the same line
$L_{N}$, the associated discontinuity is $\sum_{j=1}^{N}\Gamma_{j}=0$, and the
branch cut cancels. This procedure for choosing branch cuts is is illustrated
for $N=3$ in figure 7. The integral around $C$ in (27) vanishes by the no flux
condition, except along the lines $L_{1},L_{2},...,L_{N-1}$.

The contributions of those integrals come from the fact that the logarithm is
$2\pi$-discontinuous across each branch cut. On the $L_{1}$ segment, the
contribution to the integral is $\rho\Gamma_{1}\int_{L_{1}}\boldsymbol{w\cdot
n}dA$. Physically, this is the flux of
$\boldsymbol{w}=\boldsymbol{q}-\boldsymbol{v}$ through $L_{1}$ times the
discontinuity across the branch cut. Similarly, the contribution from segment
$L_{k}$ is given by the integral
$\rho\left(\sum_{j=1}^{k-1}\Gamma_{j}\right)\int_{L_{k}}\boldsymbol{w\cdot
n}dA$; physically, this represents the flux $\boldsymbol{w}$ through $L_{k}$
times the discontinuity across $L_{k}$. Summing over all segments, we find

$\rho\int_{C}\phi\boldsymbol{w\cdot
n}dA=\rho\sum_{i=1}^{N-1}\left(\sum_{j=1}^{i-1}\Gamma_{j}\right)\int_{L_{i}}\boldsymbol{w\cdot
n}dA.$ (28)

We require the right hand side to be greater or equal to zero, for (25) to
hold.

Figure 7: Branch cut choices in the case $N=3$. In panel (A), we show a
possible choice of branch cuts when each body $B_{i}$ possesses a circulation
$\Gamma_{i}$. Branch cuts are drawn as zigzag lines, and labelled with its
associated discontinuity. Each colour represents a single branch cut with a
particular discontinuity. In panel (B), we take the branch cuts to overlap so
that there are 2 distinct branches. The branch connecting to complex infinity
is cancelled since $\Gamma_{1}+\Gamma_{2}+\Gamma_{3}=0$. The integration
contour $C$ around the branch cuts is drawn.

## References

  * Ackermann & Arbhabhirama (1964) Ackermann, Norbert L & Arbhabhirama, A. 1964 Viscous and boundary effects on virtual mass. J. Eng. Mech. 90, 123–130.
  * Basset (1888) Basset, A. B. 1888 A treatise on hydrodynamics: with numerous examples, , vol. 2. Deighton, Bell and Company.
  * Batchelor (2000) Batchelor, G.K. 2000 An introduction to fluid dynamics. Cambridge university press.
  * Brennen (1982) Brennen, C.E. 1982 A review of added mass and fluid inertial forces .
  * Buckingham (1914) Buckingham, E. 1914 On physically similar systems; illustrations of the use of dimensional equations. Physical review 4 (4), 345.
  * Bujard et al. (2021) Bujard, T., Giorgio-Serchi, F. & Weymouth, G.D. 2021 A resonant squid-inspired robot unlocks biological propulsive efficiency. Science Robotics 6 (50), eabd2971.
  * Childress (1981) Childress, S. 1981 Mechanics of swimming and flying. Cambridge University Press.
  * Chung & Chen (1984) Chung, H. & Chen, S.S. 1984 Hydrodynamic mass. Tech. Rep.. Argonne National Lab.
  * Crowdy (2009) Crowdy, D. 2009 The spreading phase in lighthill’s model of the weis-fogh lift mechanism. J. Fluid Mech. 641, 195–204.
  * Crowdy (2013) Crowdy, D. 2013 Analytical formulae for source and sink flows in multiply connected domains. Theor. Comput. Fluid Dyn. 27, 1–19.
  * Crowdy (2020) Crowdy, D. 2020 Solving problems in multiply connected domains. SIAM.
  * Davis et al. (2008) Davis, S., Gray, J.O. & Caldwell, D. G. 2008 An end effector based on the bernoulli principle for handling sliced fruit and vegetables. Robotics and Computer-Integrated Manufacturing 24, 249–257.
  * Giesen et al. (2013) Giesen, T., Bürk, E., Fischmann, C., Gauchel, W., Zindl, M. & Verl, A. 2013 Advanced gripper development and tests for automated photovoltaic wafer handling. Assembly Automation 33, 334–344.
  * Goluzin (1969) Goluzin, G.M. 1969 Geometric theory of functions of a complex variable, , vol. 26. American Mathematical Soc.
  * Gonzalez & Taha (2022) Gonzalez, C. & Taha, H.E. 2022 A variational theory of lift. J. Fluid Mech. 941.
  * Griffiths (2013) Griffiths, D.J. 2013 Introduction to electrodynamics; 4th ed.. Boston, MA: Pearson, re-published by Cambridge University Press in 2017.
  * Jackson (1999) Jackson, J.D. 1999 Classical electrodynamics, 3rd edn. New York, NY: Wiley.
  * Kelvin (1849) Kelvin, Lord 1849 On vis-viva of a liquid in motion. Camb. Dublin Math. J. 4, 107–112.
  * Lamb (1975) Lamb, H. 1975 Hydrodynamics. University Press.
  * Levin & Johnson (2011) Levin, M. & Johnson, S.G. 2011 Is the electrostatic force between a point charge and a neutral metallic object always attractive? Am. J. Phys. 79 (8), 843–849.
  * Liberal et al. (2020) Liberal, I., Lobet, M., Li, Y. & Engheta, N. 2020 Near-zero-index media as electromagnetic ideal fluids. Proceedings of the National Academy of Sciences 117 (39), 24050–24054.
  * Llewellyn Smith et al. (2008) Llewellyn Smith, S.G., Michelin, S. & Crowdy, D. 2008 The dipolar field of rotating bodies in two dimensions. J. Fluid Mech. 607, 109–118.
  * McKee & Czarnecki (2019) McKee, K. & Czarnecki, A. 2019 Acceleration due to buoyancy and mass renormalization. Am. J. Phys. 87, 165–170.
  * Milne-Thomson (1962) Milne-Thomson, L. 1962 Theoretical Hydrodynamics. London: MacMillan & Co Ltd.
  * Newman (2018) Newman, J. N. 2018 Marine hydrodynamics. The MIT press.
  * Olness & Scalise (2011) Olness, F. & Scalise, R. 2011 Regularization, renormalization, and dimensional analysis: Dimensional regularization meets freshman e&m. American Journal of Physics 79 (3), 306–312.
  * Ralph (2014) Ralph, W Earl Company Inc. 2014 _Festo Bernoulli Gripper OGGB_. https://www.youtube.com/watch?v=phSlc8B4kFo&t=6s.
  * Saad & Majdalani (2017) Saad, T. & Majdalani, J. 2017 Extension of kelvin’s minimum energy theorem to incompressible fluid domains with open regions. J. Fluid Mech. 825, 208–212.
  * Saffman (1967) Saffman, P.G. 1967 The self-propulsion of a deformable body in a perfect fluid. J. Fluid Mech. 28, 385–389.
  * Saffman (1993) Saffman, P.G. 1993 Vortex dynamics. Cambridge university press.
  * Sedov (1965) Sedov, L.I. 1965 Two-Dimensional Problems in Hydrodynamics and Aerodynamics. London: John Wiley & Sons Inc.
  * Steele et al. (2017) Steele, S.C., Weymouth, G.D. & Triantafyllou, M.S. 2017 Added mass energy recovery of octopus-inspired shape change. J. Fluid Mech. 810, 155–174.
  * Tchieu et al. (2010) Tchieu, A. A., Crowdy, D. & Leonard, A. 2010 Fluid-structure interaction of two bodies in an inviscid fluid. Physics of Fluids 22.
  * Thomson (1848) Thomson, W. 1848 Theorems with reference to the solution of certain partial differential equations. Camb. Dublin Math. J 4, 84.
  * Valentín et al. (2014) Valentín, D., Presas, A., Egusquiza, E. & Valero, C. 2014 Influence of the added mass effect and boundary conditions on the dynamic response of submerged and confined structures. In IOP Conf. Ser.: Earth Environ. Sci., , vol. 22, p. 032042. IOP Publishing.
  * Wambsganss et al. (1974) Wambsganss, M.W., Chen, S.S. & Jendrzejczyk, JA 1974 Added mass and damping of a vibrating rod in confined viscous fluids. Tech. Rep.. Argonne National Lab., Argonne, IL.
  * Wang (2004) Wang, Q. 2004 Interaction of two circular cylinders in inviscid fluid. Physics of Fluids 16, 4412–4425.
  * Weymouth & Triantafyllou (2013) Weymouth, G.D. & Triantafyllou, M.S. 2013 Ultra-fast escape of a deformable jet-propelled body. J. Fluid Mech. 721, 367–385.
  * Whitham (1974) Whitham, G.B. 1974 Linear and nonlinear waves(book). New York, Wiley-Interscience, 1974. 651 p .