# Manifold Solutions to Navier–Stokes Equations

David V. Svintradze<EMAIL_ADDRESS>School of Medicine, New Vision
University, Bokhua 11, 0159 Tbilisi, Georgia  Niels Bohr Institute, University
of Copenhagen, Blegdamsvej 17, 2100 Copenhagen, Denmark

###### Abstract

We have developed dynamic manifold solutions for the Navier-Stokes equations
using an extension of differential geometry called the calculus for moving
surfaces. Specifically, we have shown that the geometric solutions to the
Navier-Stokes equations can take the form of fluctuating spheres, constant
mean curvature surfaces, generic wave equations for compressible systems, and
arbitrarily curved shapes for incompressible systems in various scenarios.
These solutions apply to predominantly incompressible and compressible systems
for the equations in any dimension, while the remaining cases are yet to be
solved. We have demonstrated that for incompressible Navier-Stokes equations,
geometric solutions are always bound by the curvature tensor of the closed
smooth manifold for every smooth velocity field. As a result, solutions always
converge for systems with constant volumes.

††preprint: APS/123-QED

Introduction– The Navier-Stokes (NS) equations are well-known and widely
studied in fluid dynamics. However, solving these equations for generic cases
is a severe problem. To address this problem, we use an extension of
differential geometry to moving manifolds known as the calculus for moving
surfaces (CMS). This approach allows us to solve the equations for extreme
limiting cases. We focus on a moving fluid volume enclosed by a moving
surface, which acts as an interface separating the fluid from the environment.
By projecting the motion of the fluid onto the surface, the system gains a
moving shape. The volume and the surface are treated as a single system, and
finding a solution to the NS equation determines the shapes in motion or vice
versa.

More specifically, let us assume a moving surface surrounds the fluid object,
and the whole system is subject to inertial motion. Then, the system’s action
is:

$\displaystyle\mathcal{L}$
$\displaystyle=\mathcal{L}_{S}+\mathcal{L}_{\Omega}$
$\displaystyle\mathcal{L}_{S}$
$\displaystyle=\int_{S}\frac{\rho_{S}V_{S}^{2}}{2}dS-\int_{\Omega}P_{S}d\Omega$
(1) $\displaystyle\mathcal{L}_{\Omega}$
$\displaystyle=\int_{\Omega}\frac{\rho_{\Omega}V_{\Omega}^{2}}{2}d\Omega-\int_{\Omega}E_{\Omega}d\Omega$
(2)

where $\rho_{S}$ represents the surface mass density, $\rho_{\Omega}$ denotes
the volumetric mass density, $V_{S}$ stands for the surface velocities,
$V_{\Omega}$ indicates the system’s inertial velocity in a specific inertial
reference frame, $E_{\Omega}$ is the energy defining inertial motion, and
$P_{S}$ refers to the volumetric energy (surface pressure) inducing surface
motion. Here, $\Omega$ refers to the spatial system, while $S$ represents the
surface boundary throughout this paper. Following the minimum action
principle, the variation of the system action must vanish, therefore:

$\delta\mathcal{L}_{S}=-\delta\mathcal{L}_{\Omega}$ (3)

The variation of the $\mathcal{L}_{\Omega}$ inertial term leads to the NS
equations, while the variation of the $\mathcal{L}_{S}$ surface action gives
uncoupled moving manifold equations. Therefore, the solution to the
$\delta\mathcal{L}_{\Omega}=0$ NS must be the $\delta\mathcal{L}_{S}=0$ moving
manifold’s equations solutions and vice versa. In this paper, we first clarify
that variation of the $\delta\mathcal{L}_{\Omega}$ inertial term indeed leads
to the NS equations. Then we, as a reminder, re-clarify that variation of
$\delta\mathcal{L}_{S}$ dynamic manifolds leads to our surface dynamics
equations [1, 2]. Next, we demonstrate the specific solutions to the equations
of the moving manifold we derived earlier [3, 4, 5] that also must be the NS
solutions. Namely, we show that approximate solutions to the equations are
fluctuating spheres along with generic wave equations when the surface normal
velocity $C$ largely dominates over surface tangent velocities $V_{i}$. The
exact solutions become practically any shape if the system is incompressible
or tangent velocities largely dominate on normal ones. This implies that the
solutions to incompressible Navier-Stokes equations are always bounded above
by the curvature tensor of a moving, smooth, closed manifold.

Considering that the reader might need to become more familiar with
mathematical formalism, we re-introduce the basics of the extension of
differential geometry to moving manifolds, known as CMS, in the appendix. We
did not directly contribute to the development of mathematical formalism. Our
role is in application to physical problems [1, 2, 3, 4, 5], see also
conference abstracts that preceded publication of papers [6, *Svintradze2023a,
*Svintradze2022a, *Svintradze2021a, *Svintradze2020a, *Svintradze2017a,
*Svintradze2016a, *Svintradze2015a, *Svintradze2014a, *Svintradze2013a,
*Svintradze2011a, *Svintradze2009a]. Next, we proceed with providing
narratives of taking variations of (1, 2), even though for (2) is largely
known, and for (1) is published [1, 2]. Then, we provide specific solutions to
the $\delta\mathcal{L}_{S}=0$ surface dynamics equations and, therefore,
according to (3), show some specific solutions to the NS problem. The
solutions presented here partially and approximately solve NS for dominantly
compressible systems while exactly solving the NS problem for incompressible
systems.

The paper is written using tensorial notations and follows the Einstein
summation convention. The appearance of the same upper and lower indices
indicates summation. Greek indices generally indicate ambient space tensorial
components, while Latin indices indicate surface tensors. Capital letters in
lower indices stand for designations and are unrelated to tensorial
nomenclature.

Calculus for Moving Surfaces (CMS) – To briefly define the extension of
differential geometry, called the calculus for moving surfaces, we note that
the extension was initiated by Hadamard nearly a century ago and developed by
a generation of mathematicians [18, *Thomas, *MGrinfeld, 21]. Following the
CMS developments, P. Grinfeld proposed equations of motions for massive thin
fluid films [21, 22, *PhysRevLett.105.137802, *Grinfeld2012] (see Appendix A,
B). We have generalized P. Grinfeld equations for any manifolds by deriving
surface dynamics equations (referred to as moving manifolds equations as well)
[1, 2], solved shape-dynamics problems for water drops [3], and extended
Young-Laplace, Kelvin, and Gibbs-Thomson equations to arbitrarily curved
dynamic surfaces [4, 5], see also meeting abstracts that preceded the papers
[7, *Svintradze2022a, *Svintradze2021a, *Svintradze2020a, *Svintradze2017a,
*Svintradze2016a, *Svintradze2015a, *Svintradze2014a, *Svintradze2013a,
*Svintradze2011a, *Svintradze2009a]. Here, we claim that those solutions are
also some particular and generic solutions to the NS problem.

Shape Dynamics Equations– One can derive surface dynamics equations based on
the definitions of velocity field and spatiotemporal covariant derivatives,
along with integrations theorems given in CMS textbook [21]. Derivation
requires all basic knowledge of differential geometry and its CMS extension.
To avoid repetition of algebraic manipulations but maintain the consistency of
the paper, we are giving only short narratives here. For detailed derivation,
see [1, 2] and for extensions, see [3, 4, 5].

Let us start with the most generic form of the (1) action

$\mathcal{L}_{S}=\int_{S}\frac{\rho
V_{S}^{2}}{2}dS-\int_{S}\Lambda_{S}dS-\int_{\Omega}P_{S}d\Omega$ (4)

where $\Lambda_{S}$ is the surface energy density and is explicit here
$\Lambda_{S}\notin P_{S}$, while is not explicit in (1) $\Lambda_{S}\in
P_{S}$.

Next, we take variations of the kinetic and energy terms from surface action
(4).

$\delta\mathcal{L}_{S}=\delta\int_{S}\frac{\rho_{S}V_{S}^{2}}{2}dS-\delta(\int_{\mathbb{S}}\Lambda_{S}dS+\int_{\Omega}P_{S}d\Omega)$
(5)

Detailed derivations are given in [1, 2] and in its current form published in
[4, 5]. According to integration theorems (34, 33) and calculations of the
covariant time derivative (31) [1, 2, 4, 5], we have

$\displaystyle\delta\int_{S}\frac{\rho_{S}V_{S}^{2}}{2}dS=\int_{S}\rho_{S}C(\dot{\nabla}C+2V^{i}\nabla_{i}C+V^{a}V^{b}B_{ab})$
$\displaystyle+\rho_{S}V_{a}(\dot{\nabla}V^{a}+V^{k}\nabla_{k}V^{a}-C\nabla^{a}C-CV^{k}B_{k}^{a})dS$
(6)
$\displaystyle\delta\int_{S}\Lambda_{S}dS=\int_{S}(\partial_{t}\Lambda_{S}-V^{k}\nabla_{k}\Lambda_{S}-\Lambda_{S}CB_{i}^{i})dS$
(7)
$\displaystyle\delta\int_{\Omega}P_{S}d\Omega=\int_{\Omega}\partial_{t}P_{S}d\Omega+\int_{S}P_{S}CdS$
(8)

Here, $\rho_{S}$ is the surface mass density, $C$ is the surface normal
velocity, $V^{i}$ is the surface tangent velocities, $V^{\alpha}$ is the
ambient $\alpha=1,2,3...$ component of the $V_{S}$ surface velocity,
$\nabla_{i},\dot{\nabla}$ stands for tangent space and time covariant
derivatives, and small Latin indexes $i=1,2...$ refer to the surface-
associated tensors. Note that the first integral in (6) represents the normal
component of the variation, while the second integral represents the tangent
component. $\partial_{t}\Lambda_{S}$ is mainly governed by normal deformations
while $V^{k}\nabla_{k}\Lambda_{S}$ is tangential one so that
$N^{\alpha}\partial_{t}\Lambda_{S}=\partial_{t}F^{\alpha}$, where
$F_{S}^{\alpha}$ is some force acting on the surface in $\alpha=1,2,3$ spacial
direction. A variation of the (8) is normal because the surface pressure acts
in the normal direction.

Because (7) has mixed normal and tangent components and (8) has only normal
variation, by applying the Gauss theorem to the normal components and taking
into account the conservation of surface mass the surface dynamics equations
$\delta\mathcal{L}_{S}=0$ follow:

$\displaystyle\dot{\nabla}\rho_{S}+\nabla_{i}(\rho_{S}V^{i})=\rho_{S}CB_{i}^{i}$
$\displaystyle\partial_{\alpha}[V^{\alpha}(\rho_{S}(\dot{\nabla}C+2V^{i}\nabla_{i}C+V^{a}V^{b}B_{ab})-P_{S}+\Lambda_{S}B_{i}^{i})]$
$\displaystyle=\partial_{t}P_{S}+\partial_{\alpha}\partial_{t}F_{S}^{\alpha}$
(9)
$\displaystyle\rho_{S}(\dot{\nabla}V^{i}+V^{k}\nabla_{k}V^{i}-C\nabla^{i}C-CV^{k}B_{k}^{i})=-\nabla^{i}\Lambda_{S}$

The first equation expresses the surface mass conservation, the second depicts
manifold evolution in the normal direction, and the last illustrates manifold
dynamics in the tangent space. Since $i$ is the manifold’s dimension, the
number of the last equation also indicates the dimensions of the tangent
space. In this framework, the manifold is one less-dimensional surface than
the space, and the number of dimensions is not limited, even though here we
discuss the simplest case, a two-dimensional surface embedded in three-
dimensional space. The analyses and results can be readily extended to any
dimensional manifold. Details of derivations are given in Appendix C.

Navier-Stokes (NS) Equations– In the previous section, we presented the
derivation narratives of the surface dynamics equations. This section derives
the NS equations from $\delta\mathcal{L}_{\Omega}=0$, even though the
derivation of the NS equations has been well established and does not require
any special attention [25]. Considering equation (2) and the continuity
equation for mass density, we have

$\displaystyle\partial_{t}\rho_{\Omega}$
$\displaystyle=-\partial_{\alpha}(\rho_{\Omega}V_{\Omega}^{\alpha})$
$\displaystyle\delta\int_{\Omega}\frac{\rho_{\Omega}V^{2}_{\Omega}}{2}d\Omega$
$\displaystyle=\int_{\Omega}\rho_{\Omega}\bm{V_{\Omega}}(V_{\Omega}^{\alpha}\partial_{\alpha}\bm{V_{\Omega}}+\partial_{t}\bm{V_{\Omega}})d\Omega$
(10)

See derivation in Appendix D. This is exactly the kinetic term in the NS
equation. The surface ambient velocities for inertial motion
$V_{\Omega}^{\alpha}$ is the $\alpha$ component of $\bm{V_{\Omega}}$ in flat
ambient space, and the surface normal velocity for inertial motion is
$\bm{V_{N}}=\bm{V_{\Omega}}\cdot\bm{N}$. Note here that there is no immediate
connection between $\bm{V_{\Omega}}$ and $\bm{V_{S}}$ because in the
laboratory reference frame $\bm{V_{\Omega}}=0$, while if the surface is in
motion, then there is no reference frame in which $\bm{V_{S}}$ is zero.

Next, we take a variation of the second part of the (2) integral. Here, we
follow Landau&Lifshitz designation because this part does not need any special
physical explanation and can be readily obtained by modeling inertial force
[25]. Let’s assume there is no energy in/out flux of the system. That is, the
system is ”passive” fluid

$\int_{S}E_{\Omega}V_{N}dS=\int_{S}E_{\Omega}\bm{V_{\Omega}}\cdot\bm{N}dS=\int_{\Omega}\nabla_{\alpha}(EV_{\Omega}^{\alpha})d\Omega=0$
(11)

where $\bm{J}=E\bm{V_{\Omega}}=0$ is the energy flux. We applied the Gauss
theorem to convert the surface integral to the space integral. Taking the (11)
and the integration theorem (33) into account, we have

$\delta_{t}\int_{\Omega}E_{\Omega}d\Omega=\int_{\Omega}\partial_{t}E_{\Omega}d\Omega=\int_{\Omega}V_{\Omega}^{\alpha}\partial_{\alpha}E_{\Omega}d\Omega$
(12)

Because the negative gradient of the energy is the force, we can model it by
the momentum flux density tensor
$\partial_{\alpha}E_{\Omega}=-\partial_{\beta}M_{\alpha}^{\beta}$ where

$M_{\alpha}^{\beta}=p\delta_{\alpha}^{\beta}+\rho_{\Omega}V_{\alpha}V^{\beta}-\sigma_{\alpha}^{\beta\prime}$
(13)

$\sigma_{\alpha\beta}=-p\delta_{\alpha\beta}+\sigma_{\alpha}^{\beta\prime}$ is
called stress tensor and $\sigma_{\alpha}^{\beta\prime}$ is the viscous stress
tensor, $\delta_{\alpha}^{\beta}$ is the Kronecker delta. The equation (13)
demonstrates that the potential field’s pressure difference directly affects
the momentum flux density. This pressure difference triggers inertial motion,
which relies on the motion’s speed in the $\alpha$ and $\beta$ directions and
the internal frictional force. By considering angular velocities and
appropriately modeling the viscous stress tensor [25], one can derive (the
derivation is given in the appendix) the following:

$\displaystyle V^{\alpha}\partial_{\alpha}E_{\Omega}$
$\displaystyle=\bm{V_{\Omega}}\bm{\nabla}E_{\Omega}$ (14)
$\displaystyle=\bm{V_{\Omega}}\cdot(-\bm{\nabla}p+\mu\Delta\bm{V_{\Omega}}+(\xi+\frac{1}{3}\mu)\bm{\nabla}\partial_{\alpha}V_{\Omega}^{\alpha})$

(14) in combination with (10) gives a variation of the (2) action, and forcing
it to vanish trivially gives the NS equation with consideration that integral
equations stand for all inertial velocity $\bm{V_{\Omega}}$

$\displaystyle\rho_{\Omega}$
$\displaystyle(\partial_{t}\bm{V_{\Omega}}+V_{\Omega}^{\alpha}\partial_{\alpha}\bm{V_{\Omega}})=$
$\displaystyle-\bm{\nabla}p+\mu\Delta\bm{V_{\Omega}}+(\xi+\frac{1}{3}\mu)\bm{\nabla}\partial_{\alpha}V_{\Omega}^{\alpha}$
(15)

Solutions– Based on the identity established above (3), the solution to the NS
equation is also the solution to the surface dynamics equations and vice
versa. This means we can provide simple solutions to the surface dynamics
equation for two extreme cases, which will also be NS solutions. In the first
case, when the normal velocity $C$ is much larger than the tangent velocity
$V^{i}$, we refer to it as a predominantly compressible system. In the second
case, when the tangent velocity $V^{i}$ is much larger than the normal
velocity $C$, it is a predominantly incompressible system.

Constant Field Approximations and Dominantly Compressible Systems $C>>V^{i}$–
Here, we show the solution to (9) therefore to NS in the special case when
$P_{S},\Lambda_{S},\bm{F}_{S}=const$. Then, equations (9) are simplified to
the

$\displaystyle\dot{\nabla}\rho_{S}+\nabla_{i}(\rho_{S}V^{i})=\rho_{S}CB_{i}^{i}$
$\displaystyle\rho_{S}(\dot{\nabla}C+2V^{i}\nabla_{i}C+V^{a}V^{b}B_{ab})=-\Lambda_{S}B_{i}^{i}$
(16)
$\displaystyle\rho_{S}(\dot{\nabla}V^{i}+V^{k}\nabla_{k}V^{i}-C\nabla^{i}C-CV^{k}B_{k}^{i})=0$

(16) is generally known as Grinfeld equations for thin fluid films [21]. Now
let’s assume that Grinfeld equations are approximate solutions of surface
dynamics equations (9) for quasi-constant $P_{S},\Lambda_{S},\bm{F}_{S}\neq
const$ fields. Then for homogenous surfaces $\nabla^{i}\Lambda_{S}=0$ and
dominantly compressible cases $C>>V^{i}$, when tangential velocities are
infinitesimally small $V^{i}\approx 0$, the dynamic manifolds equations (9)
simplify as

$\displaystyle\partial_{t}\rho_{S}$ $\displaystyle=\rho_{S}CB_{i}^{i}$
$\displaystyle-\partial_{\alpha}(P_{S}V^{\alpha})$
$\displaystyle=\partial_{t}P_{S}+\partial_{\alpha}\partial_{t}F_{S}^{\alpha}$
(17) $\displaystyle\rho_{S}\partial_{t}C$
$\displaystyle=-\Lambda_{S}B_{i}^{i}$

Note here that for homogeneous surfaces
${\partial_{\alpha}P_{S},\partial_{\alpha}F_{S}^{\alpha}=0}$ and the immediate
solution to the second equation, therefore to the system (17) is

$\displaystyle\partial_{\alpha}V^{\alpha}$ $\displaystyle=0$ (18)
$\displaystyle\partial^{2}_{t}C$ $\displaystyle=-\gamma(\Delta C-2KC)$ (19)

(19) provides the wave equation for near-planar surfaces where Gaussian
curvature $K\approx 0$ is infinitesimally small. It is noteworthy that, in
this particular case, even though the system is predominantly compressible, it
has an approximate solution (18) characteristic of dominantly incompressible
systems. The first equation gives the spherical solution of

$\bm{V}=k\bm{R}/R^{3}$ (20)

$k$ is some coupling constant, and $\bm{R}$ is the position vector (see
derivations in Appendix E). Taking (20) into account, one can extract an
analytical solution to (18) having the form of

$\displaystyle\bm{R}=A(\bm{R}_{0}+\omega_{\alpha}R_{\alpha}\bm{S}^{\alpha}t)+BR_{0\alpha}e^{\omega_{\alpha}\bm{S}^{\alpha}t}+\psi_{\alpha}\bm{S}^{\alpha}$
(21)

where $R_{0\alpha}$ is arbitrary coordinates of the initial position vector
$\bm{R}_{0}$, $\omega_{\alpha}$ is the coupling constant indicating the
frequency of oscillation in $\alpha$ directions, and $\bm{S}^{\alpha}$ is the
vector component of $\bm{S}$ unit vector, $A,B$ are some constants that can be
defined by initial conditions, and ${{\bm{\psi}}}_{\xi}(\xi,t)$ is the wave
function defined in the Appendix E [3]. All derivations are in the appendix.

Dominantly Incompressible Systems, Tangent Velocity Domination $C<<V^{i}$ –
Let’s assume another extreme scenario where tangent velocities $V^{i}$ largely
dominate over the normal velocity $C\approx 0$. In these predominantly
incompressible systems, the surface dynamics equations (9) significantly
simplify

$\displaystyle\dot{\nabla}\rho_{S}+\nabla_{i}(\rho_{S}V^{i})=0$
$\displaystyle\partial_{\alpha}[V^{\alpha}(\rho_{S}V^{a}V^{b}B_{ab}-P_{S}+\Lambda_{S}B_{i}^{i})]=\partial_{t}P_{S}+\partial_{\alpha}\partial_{t}F_{S}^{\alpha}$
$\displaystyle\rho_{S}(\dot{\nabla}V^{i}+V^{k}\nabla_{k}V^{i})=-\nabla^{i}\Lambda_{S}$
(22)

The first and the last equations restrict tangent velocities (22), while the
second equation lands the analytic solution of

$B_{ab}=(2P_{S}+\partial_{\alpha}F_{S}^{\alpha}-\Lambda_{S}B_{i}^{i})(\rho_{S}V_{a}V_{b})^{-1}$
(23)

We refer to the solution as the generalized Young-Laplace law for nearly
incompressible cases [5]. If one assumes an infinitesimally thin surface with
infinitesimally small non-zero surface mass density, then the first and last
equations from (22) become identities for homogeneous surfaces
$-\nabla^{i}\Lambda_{S}=0$, therefore relaxing constraints to tangent
velocities and any moving surface with (23) curvature tensor for $\forall
V^{i}$ becomes the estimated solution.

The solution (23) extends to the generic Kelvin equation when the system shape
is defined by vapor $p_{v}$ and saturation $p_{s}$ pressures, then

$B_{ab}=(\frac{k_{B}T}{v_{m}}\ln\frac{p_{v}}{p_{s}}+\partial_{\alpha}F_{S}^{\alpha}-\Lambda_{S}B_{i}^{i})(\rho_{S}V_{a}V_{b})^{-1}$
(24)

$p_{v}/p_{s}$ is relative humidity, and $v_{m}$ is the molar volume of the
surface (see derivation in [4]). Note that homogenous and static shapes with
spherical geometry give the Kelvin equation [4, 5].

Analytically, solutions to the Navier-Stokes equations can be derived if the
temperature distribution is provided instead of surface pressure
distributions. In this case, the solution takes the form of generic Gibbs-
Thomson equations [5].

$B_{ab}=(\gamma
H+v_{m}(\partial_{\alpha}F_{S}^{\alpha}-\Lambda_{S}B_{i}^{i}))(\mu
V_{a}V_{b})^{-1}$ (25)

where $\gamma=1-T/T_{0}$, $T$ is the temperature distribution on the surface,
$T_{0}$ is the bulk temperature, $\mu=\rho_{S}v_{m}$ is the surface molar mass
($v_{m}$ is the molar volume of the surface) and $H$ is fusion enthalpy. If
$\gamma,H$ are given, one can find the surface curvature tensor and identify
the moving manifold solution to NS.

(23, 24, 25) solutions indicate that NS equations for incompressible fluids
are always bound from above by the curvature tensor of the smooth closed
manifolds. Therefore, the NS solution converges for $\forall\bm{V_{\Omega}}$
inertial velocities.

Conclusion– We have solved the Navier-Stokes equations for two extreme cases.
Namely, we have demonstrated that for predominantly compressible systems, the
estimated solutions are generic wave equations and fluctuating spheres. On the
other hand, the formation of any moving shapes with any surface patterns is
possible for predominantly incompressible fluids. We have presented solutions
for the cases where $C>>V^{i}$ (dominantly compressible) and $C<<V^{i}$
(dominantly incompressible). Despite imposing restrictions on the tangent
velocities (22), the equations (23, 24, 25) clarify that closed and smooth
manifolds always bound the solutions of the NS equations. Therefore, we refer
to them as manifold solutions. However, the case for all $\forall C,V^{i}$
remains unsolved mainly, even though we have fully solved the NS problem for
incompressible fluids and produced partial, approximated solutions for
compressible ones.

###### Acknowledgements.

This work is supported by Shota Rustaveli Science Foundation of Georgia by
grant no. STEM-22-365 and Euro Commission’s ERC Erasmus+ collaborative linkage
grant between the University of Copenhagen and New Vision University. We also
benefit from New Vision University’s internal funding. We want to acknowledge
the warm hospitality of the Biocomplexity Department of Niels Bohr Institute,
University of Copenhagen, where the paper was initiated and completed.

##  Appendix A Gaussian Differential Geometry

Here, we revive the concept of extrinsic differential geometry, called
Gaussian geometry. Gauss’s approach, in contrast to Riemannian geometry, is
fundamentally extrinsic. Embedding in ambient space is necessary to understand
better moving manifolds. Therefore, instead of describing Riemannian geometry,
we start from the Gaussian one, considering that the formalism is sufficient
for higher than two-dimensional surfaces. In mathematics, higher dimensional
surfaces are called hyper-surfaces. To minimize terminology, we refer to any
dimensional manifold as a surface and explicitly indicate whether the referred
surface is two-dimensional.

Let
$S^{i}\in\mathbb{S}^{n},i=0,1,2,...,n,X^{\alpha}\in\mathbb{R}^{n+1},\alpha=0,1,2...,n+1$
be the generic coordinates of the point on the hyper-surface $\mathbb{S}$
(referred as surface) embedded on one higher dimensional flat space
$\mathbb{R}$ so that $\bm{R}$ is a position vector
$\bm{R}\in\mathbb{R}^{(n+1)},\bm{R}=\bm{R}(X^{\alpha},t)=\bm{R}(S^{i},t)$.
Greek indexes stand for the space, and Latin for the surface. Bold letters
indicate vectors. Partial derivatives of the position vector define covariant
base vectors of ambient space and embedded surface, while the scalar product
of base vectors are metric tensors

$\begin{matrix}[l]\bm{X}_{\alpha}=\partial_{\alpha}\bm{R},&\bm{S}_{i}=\partial_{i}\bm{R}\\\
X_{\alpha\beta}=\bm{X}_{\alpha}\cdot\bm{X}_{\beta}=\delta_{\alpha\beta},&S_{ij}=\bm{S}_{i}\cdot\bm{S}_{j}\end{matrix}$
(26)

where $\partial_{\alpha}=\partial/\partial
X^{\alpha},\partial_{i}=\partial/\partial S^{i}$. Note that since ambient
space is flat $X_{\alpha\beta}=\delta_{\alpha\beta}$. Inverse to surface
$S_{ij}$ tensors are contravariant metric tensors
$S_{ij}S^{jk}=\delta_{i}^{k}$. The surface normal $\bm{N}$ is defined as the
normal vector to surface bases so that $\bm{N}\cdot\bm{S}_{i}=0,\bm{N}^{2}=1$
and to ensure convex geometry being defined as positive curvature, the
direction of the normal is set toward the surface.

The definition of base vectors allows the introduction of Christoffel symbols
for embedded surfaces. For ambient flat space, the symbols
$\Gamma_{\alpha\beta}^{\gamma}=\bm{X}^{\gamma}\cdot\partial_{\alpha}\bm{X}_{\beta}=0,\partial_{\alpha}=\nabla_{\alpha}$
are vanishing, while for embedded surface
$\Gamma_{ij}^{k}=\bm{S}^{k}\cdot\partial_{i}\bm{S}_{j}\neq 0$ and
$\partial_{i}\neq\nabla_{i}$.

While all symbols are unchanged in differential geometry for moving manifolds,
they are not the only ones and are enhanced by extrinsic parameters.
Christoffel symbols allow the definition of invariant to reference frame
covariant derivatives

$\nabla_{k}T_{i}^{j}=\partial_{k}T_{i}^{j}-\Gamma^{m}_{ki}T_{m}^{j}+\Gamma_{mk}^{j}T_{i}^{m}$
(27)

Definition (27) provides metrilinic property of metric tensors
$\nabla_{\gamma}X_{\alpha\beta},\nabla_{k}S_{ij}=0$, consequently the surface
base vectors are orthogonal to their covariant derivatives
${\bm{S}_{i}\cdot\nabla_{k}\bm{S}_{j}=0}$, therefore

$\nabla_{i}\bm{S}_{j}=\bm{N}B_{ij}$ (28)

$\bm{N}$ is the surface normal and $B_{ij}$ is the curvature tensor and
according to definition (28) the curvature tensor is symmetric and extrinsic
parameter. Trace of the mixed curvature tensor $B_{i}^{i}$ is the mean
curvature, while the determinant is the Gaussian curvature $K=\lvert
B^{\cdot}_{\cdot}\rvert$.

##  Appendix B Calculus for Moving Surfaces

All definitions from Gaussian geometry (26 – 28) are also applicable to moving
manifolds. However, extensions are required to parameterize the surface
motion. The introduction of parametric time immediately adds to the necessity
of defining time derivatives to preserve covariance. For that, we first
introduce surface velocities: $C$ the surface normal velocity, and $V^{i}$
tangent ones as components of ambient velocity

$\bm{V_{S}}=\partial_{t}\bm{R}=C\bm{N}+V^{i}\bm{S}_{i}$ (29)

Note here that $\bm{V_{S}}=(V^{\alpha}),C,V^{i}$ are only defined in ambient
flat space and, therefore, are tensors only in the space. Note that the
interface velocity

$C=\bm{N}\cdot\bm{V_{S}}=N^{\alpha}\bm{X}_{\alpha}V_{\beta}\bm{X}^{\beta}=N^{\alpha}V_{\alpha}$
(30)

normal component, the surface velocity is invariant. Therefore, if the surface
is assumed to be moving initially, it will retain motion in all reference
frames. The invariance of interface velocity straightforwardly indicates that,
in general, for moving manifolds, a reference frame cannot be found where
$\bm{V_{S}}$ vanishes.

Introduction of the velocity field (29), without further elaboration (for
details, see [1, 21]), opens the path toward defining the covariant time
derivative

$\dot{\nabla}T_{a}^{b}=\partial_{t}T_{a}^{b}-V^{k}\nabla_{k}T_{a}^{b}+\dot{\Gamma}_{k}^{b}T^{k}_{a}-\dot{\Gamma}^{k}_{a}T_{k}^{b}$
(31)

where $\dot{\Gamma}_{a}^{b}=\nabla_{a}V^{b}-CB_{a}^{b}$ is the Christoffel
symbols analog for invariant time derivatives first introduced by P. Grinfeld
[21]. $\dot{\nabla}$ is invariant and has all properties as covariant
derivatives. According to metrilinic property of covariant space/time
derivatives $\nabla_{k}S_{ij},\dot{\nabla}S_{ij}=0$, therefore taking into
account (31) we have

$\displaystyle\dot{\nabla}S_{ij}=\partial_{t}S_{ij}-V^{k}\nabla_{k}S_{ij}-\dot{\Gamma}^{k}_{i}S_{kj}-\dot{\Gamma}_{j}^{k}S_{ik}$
$\displaystyle=\partial_{t}S_{ij}-(\nabla_{i}V^{k}-CB^{k}_{i})S_{kj}-(\nabla_{j}V^{k}-CB_{j}^{k})S_{ik}$
$\displaystyle=\partial_{t}S_{ij}-(\nabla_{i}V_{j}-CB_{ij})-(\nabla_{j}V_{i}-CB_{ji})$
$\displaystyle=\partial_{t}S_{ij}-\nabla_{i}V_{j}-\nabla_{j}V_{i}+2CB_{ij}=0,\Rightarrow$
$\displaystyle\partial_{t}S_{ij}=\nabla_{i}V_{j}+\nabla_{j}V_{i}-2CB_{ij}$
(32)

Note that even though the invariant time derivative (31) and covariant space
derivatives (27) have the same properties, they are fundamentally different
because one is intrinsic. The last one is extrinsic and includes ambient
surface velocities and curvature tensors.

Definitions (27–28, 29, 31) lead to integration theorems

$\displaystyle\partial_{t}\int_{\Omega}Fd\Omega$
$\displaystyle=\int_{\Omega}\partial_{t}Fd\Omega+\int_{S}FCdS$ (33)
$\displaystyle\partial_{t}\int_{S}FdS$
$\displaystyle=\int_{S}\dot{\nabla}FdS-\int_{S}FCB_{i}^{i}dS$ (34)

Equations (26, 31) along with integration theorems (33, 34) form the
foundation of CMS. Note here that for incompressible systems, due to the
conservation of volume, according to (33) and Gauss theorem, one must have

$\partial_{t}\int_{\Omega}d\Omega=\int_{S}CdS=\int_{\Omega}\nabla_{\alpha}V^{\alpha}d\Omega=0$
(35)

From where the condition $C,\nabla_{\alpha}V^{\alpha}=0$ follows. We
frequently use (35) when discussing incompressible systems.

The proofs of (33) are known in the literature. Here, we present a short and
elegant proof of the (34) theorem that, to our knowledge, was not known
before, though it was presented in [21, 20].

Let $F\in S$ be a smooth scalar function on the closed smooth manifold $S$.
Then, across any closed stationary contour $\gamma$, the contour integral
vanishes because the contour normal $n_{i}$ lying in the tangent plane always
stays perpendicular to the surface tangent velocity $\bm{V}_{T}$, so that

$v=\bm{n}\cdot\bm{V}_{T}=n_{i}V^{i}=0$ (36)

Using this and that, the total time derivative of the integral has two parts.
The first part reflects that $F$ changes as time evolves, and the second part
reflects that the surface changes as time evolves. Therefore

$\displaystyle\frac{d}{dt}$
$\displaystyle\int_{S}FdS=\int_{S}\partial_{t}FdS+\int_{S}F\partial_{t}dS$
$\displaystyle=\int_{S}(\partial_{t}F+F\nabla_{i}V^{i}-F\nabla_{i}V^{i})dS+\int_{x}\int_{y}F\partial_{t}\sqrt{S}dxdy$
$\displaystyle=\int_{S}(\partial_{t}F+F\nabla_{i}V^{i}-F\nabla_{i}V^{i})dS$
$\displaystyle+\int_{x}\int_{y}F(\nabla_{i}V^{i}-CB_{i}^{i})\sqrt{S}dxdy$
$\displaystyle=\int_{S}(\partial_{t}F+F\nabla_{i}V^{i}-F\nabla_{i}V^{i})dS$
$\displaystyle+\int_{S}F(\nabla_{i}V^{i}-CB_{i}^{i})dS$
$\displaystyle=\int_{S}(\partial_{t}F+\nabla_{i}(FV^{i})-V^{i}\nabla_{i}F-CFB_{i}^{i})dS$
(37)

By Gauss theorem, integral from $\nabla_{i}(FV^{i})$ is converted to the
contour integral and, according to the (36) boundary condition, vanishes. If
one relaxes the condition that the surface is closed and is bounded by static
contour, then according to (37), the integration theorem (34) updates as

$\partial_{t}\int_{S}FdS=\int_{S}\dot{\nabla}FdS-\int_{S}FCB_{i}^{i}dS+\int_{\gamma}vFd\gamma$
(38)

Since we discuss only closed surfaces here, the last term from (38) becomes
irrelevant.

Note that, for calculation of $\partial_{t}\sqrt{S}$ in (37), we used Jacobi’s
theorem about taking derivative for determinants ${\partial S/\partial
S^{ij}=SS_{ij}}$ and (32) we have

$\displaystyle\partial_{t}S=\frac{\partial S}{\partial
S^{ij}}\partial_{t}S^{ij}$
$\displaystyle=SS_{ij}(\nabla^{i}V^{j}+\nabla^{j}V^{i}-2CB^{ij})$
$\displaystyle=2S(\nabla_{i}V^{i}-CB_{i}^{i})$ (39)

From (39) according to the complex function derivative rule, the identity

$\partial_{t}\sqrt{S}=\frac{1}{2\sqrt{S}}\partial_{t}S=\sqrt{S}(\nabla_{i}V^{i}-CB_{i}^{i})$
(40)

trivially follows. Next, using the (31) definition written for scalar
functional

$\dot{\nabla}F=\partial_{t}F-V^{i}\nabla_{i}F$ (41)

Applying (41) to the (37) with taking into account the Gauss theorem and the
boundary condition (36), the integration theorems (34, 38) trivially follows.

##  Appendix C Derivation of Shape Dynamics Equations

We start by taking the variation of the kinetic term first. In the initial
steps, the derivation looks like a derivation of the NS kinetic part. We drop
superscript $S$ in the index to avoid interference with tensorial indexes and
recover at the end of derivations.

To deduce the equations of motion, we first derive the simplest one from the
set (9). According to the conservation of mass law, the following boundary
conditions must be satisfied. At the end of the variation, the total mass of
the surface is unchanged $dm/dt=0$. The pass integral along any closed contour
$\gamma$ from the surface must vanish because contour normal velocity
$v=n_{i}V^{i}=0$ ($n_{i}$ is the contour unit normal situated on the tangent
plain) when the surface reaches static shape at the end of the variation.
Considering these boundary conditions, the Gauss theorem, conservation of
mass, and integration formulas (33, 34), we have

$\displaystyle 0$ $\displaystyle=\int_{\gamma}\rho vd\gamma=\int_{\gamma}\rho
n_{i}V^{i}d\gamma=\int_{S}\nabla_{i}(\rho V^{i})dS$
$\displaystyle=\int_{S}(\nabla_{i}(\rho V^{i})+\rho CB_{i}^{i}-\rho
CB_{i}^{i})dS$ $\displaystyle=\int_{S}(\nabla_{i}(\rho V^{i})-\rho
CB_{i}^{i})dS+\int_{S}\dot{\nabla}\rho dS-\frac{d}{dt}\int_{S}\rho dS$
$\displaystyle=\int_{S}(\dot{\nabla}\rho+\nabla_{i}(\rho V^{i})-\rho
CB_{i}^{i})dS$ (42)

Since (42) holds for every integrand, the first equation follows. To deduce
the second and third equations, we note

$\displaystyle\delta(\int_{S}$
$\displaystyle\Lambda_{S}dS+\int_{\Omega}P_{S}d\Omega)=\int_{S}(\dot{\nabla}\Lambda_{S}-\Lambda_{S}CB_{i}^{i})dS$
$\displaystyle+\int_{\Omega}\partial_{t}P_{S}d\Omega+\int_{S}P_{S}CdS$
$\displaystyle=\int_{S}(\partial_{t}\Lambda_{S}-V_{i}\nabla^{i}\Lambda_{S}-\Lambda_{S}CB_{i}^{i})dS$
$\displaystyle+\int_{\Omega}\partial_{t}P_{S}d\Omega+\int_{S}P_{S}CdS$ (43)

Next, we note that $V^{i}\nabla_{i}\Lambda_{S}$ governs the surface tangent
deformation while all other terms in (43) are responsible for normal
deformations. Therefore, by modeling and Gauss theorem

$\displaystyle\partial_{t}\Lambda_{S}$
$\displaystyle=N_{\alpha}\partial_{t}F^{\alpha}$ (44)
$\displaystyle\int_{S}\partial_{t}\Lambda_{S}dS$
$\displaystyle=\int_{\Omega}\partial_{\alpha}\partial_{t}F^{\alpha}d\Omega$
(45)

Taking the (44, 45) into account, the tangent and normal parts of the
variation straightforwardly follows

$\displaystyle\delta_{T}(\int_{S}\Lambda_{S}dS+\int_{\Omega}P_{S}d\Omega)=-\int_{S}V_{i}\nabla^{i}\Lambda_{S}dS$
(46)
$\displaystyle\delta_{N}(\int_{S}\Lambda_{S}dS+\int_{\Omega}P_{S}d\Omega)$
(47)
$\displaystyle=\int_{\Omega}[\partial_{t}P_{S}+\partial_{\alpha}\partial_{t}F^{\alpha}+\partial_{\alpha}(P_{S}V^{\alpha})+\partial_{\alpha}(\Lambda_{S}B_{i}^{i}V^{\alpha})]d\Omega$

Next, let’s proceed with taking the variation of the kinetic part

$\displaystyle\delta\int_{S}$
$\displaystyle\frac{\rho_{S}V^{2}_{S}}{2}dS=\int_{S}(\dot{\nabla}\rho\cdot\frac{V^{2}}{2}+\dot{\nabla}\frac{V^{2}}{2}\cdot\rho-\frac{\rho
V^{2}}{2}CB_{i}^{i})dS$ $\displaystyle=\int_{S}((\rho
CB_{i}^{i}-\dot{\nabla}_{i}(\rho
V^{i}))\cdot\frac{V^{2}}{2}+\dot{\nabla}\frac{V^{2}}{2}\cdot\rho)dS$
$\displaystyle-\int_{S}\frac{\rho V^{2}}{2}CB_{i}^{i}dS$
$\displaystyle=\int_{S}(-\nabla_{i}(\rho V^{i}\frac{V^{2}}{2})+\rho
V^{i}\nabla_{i}\frac{V^{2}}{2}+\dot{\nabla}\frac{V^{2}}{2}\cdot\rho)dS$
$\displaystyle=\int_{S}(\rho
V^{i}\nabla_{i}\frac{V^{2}}{2}+\dot{\nabla}\frac{V^{2}}{2}\cdot\rho)dS$
$\displaystyle=\int_{S}\rho\bm{V}(V^{i}\nabla_{i}\bm{V}+\dot{\nabla}\bm{V})dS$
(48)

Here, we must decompose (48) into normal and tangent components, which
requires tricky algebraic manipulations. Instead of restating the algebra
here, we can refer to the direct derivation given in [1, 2] and give the final
form only

$\displaystyle\dot{\nabla}\bm{V}+V^{i}\nabla_{i}\bm{V}$
$\displaystyle=(\dot{\nabla}C+2V^{i}\nabla_{i}C+V^{a}V^{b}B_{ab})\bm{N}$ (49)
$\displaystyle+(\dot{\nabla}V^{j}+V^{i}\nabla_{i}V^{j}-C\nabla^{j}C-CV^{i}B_{j}^{i})\bm{S}_{j}$

Dotting (49) on the surface velocity $\bm{V}$ and taking into account the
definition (29) we get

$\displaystyle\delta\int_{S}$
$\displaystyle\frac{\rho_{S}V^{2}_{S}}{2}dS=\int_{S}C(\dot{\nabla}C+2V^{i}\nabla_{i}C+V^{a}V^{b}B_{ab})$
$\displaystyle+V_{i}(\dot{\nabla}V^{i}+V^{j}\nabla_{j}V^{i}-C\nabla^{i}C-CV^{j}B_{i}^{j})dS$
(50)

Equating (50) with (43), and considering the tangent and normal components of
the energy variation (46, 47), and applying Gauss’s theorem to the normal
components, we obtain

$\displaystyle\int_{\Omega}\partial_{\alpha}[V^{\alpha}($
$\displaystyle\rho_{S}(\dot{\nabla}C+2V^{i}\nabla_{i}C+V^{a}V^{b}B_{ab})$ (51)
$\displaystyle-
P_{S}+\Lambda_{S}B_{i}^{i})]d\Omega=\int_{\Omega}(\partial_{t}P_{S}+\partial_{\alpha}\partial_{t}F_{S}^{\alpha})d\Omega$

and

$\displaystyle\int_{S}\rho_{S}V_{i}(\dot{\nabla}V^{i}+V^{k}\nabla_{k}V^{i}-$
$\displaystyle C\nabla^{i}C-CV^{k}B_{k}^{i})dS$
$\displaystyle=-\int_{S}V_{i}\nabla^{i}\Lambda_{S}dS$ (52)

From (42, 51, 52) the moving manifold equations (22) trivially follow.

##  Appendix D Derivation of NS Equation

We start by taking the variation of the kinetic term first. Because
superscript $\Omega$ appears everywhere in this subsection, to minimize the
number of letters in the equations and avoid interference with tensorial
indexes, we drop it during calculations and recover it at the end.

$\displaystyle\delta\int_{\Omega}$
$\displaystyle\frac{\rho_{\Omega}V^{2}_{\Omega}}{2}d\Omega=\int_{\Omega}(\partial_{t}\rho\cdot\frac{V^{2}}{2}+\partial_{t}\frac{V^{2}}{2}\cdot\rho)d\Omega$
$\displaystyle+\int_{\Omega}\partial_{\alpha}(\frac{\rho
V^{2}}{2}V^{\alpha})d\Omega=$
$\displaystyle=\int_{\Omega}(-\partial_{\alpha}(\rho
V^{\alpha})\cdot\frac{V^{2}}{2}+\partial_{t}\frac{V^{2}}{2}\cdot\rho)d\Omega$
$\displaystyle+\int_{\Omega}\partial_{\alpha}(\frac{\rho
V^{2}}{2}V^{\alpha})d\Omega$
$\displaystyle=\int_{\Omega}(-\partial_{\alpha}(\rho
V^{\alpha}\frac{V^{2}}{2})+\rho
V^{\alpha}\partial_{\alpha}\frac{V^{2}}{2}+\partial_{t}\frac{V^{2}}{2}\cdot\rho)d\Omega$
$\displaystyle+\int_{\Omega}\partial_{\alpha}(\frac{\rho
V^{2}}{2}V^{\alpha})d\Omega$ $\displaystyle=\int_{\Omega}(\rho
V^{\alpha}\partial_{\alpha}\frac{V^{2}}{2}+\partial_{t}\frac{V^{2}}{2}\cdot\rho)d\Omega$
$\displaystyle=\int_{\Omega}\rho_{\Omega}\bm{V_{\Omega}}(V^{\alpha}\partial_{\alpha}\bm{V_{\Omega}}+\partial_{t}\bm{V_{\Omega}})d\Omega$
(53)

here we used the continuity equation $\partial_{t}\rho+\partial_{\alpha}(\rho
V^{\alpha})=0$ and $V^{2}=\bm{V}\cdot\bm{V}$.

Next, we model the gradient of energy, the inertial force acting on the moving
fluid, therefore clarifying (13) and its derivative. The viscous stress tensor
$\sigma_{\alpha\beta}\prime$ is the only less defined term in the (13).

$\partial_{\alpha}E_{\Omega}=-\partial_{\beta}M_{\alpha}^{\beta}$ (54)

In a fluid, internal friction processes arise when different particles of the
fluid move at different velocities. This means that various parts of the fluid
are in relative motion. Therefore, the viscous stress tensor depends on the
velocity’s space derivatives. If the velocity gradients are small, we can
assume that the momentum transfer due to viscosity depends only on the first
derivatives of the velocity. To the same approximation,
$\sigma_{\alpha\beta}\prime$ may be a linear function of the derivatives
$\partial_{\alpha}V_{\beta}$. There can be no terms
$\sigma_{\alpha\beta}\prime$ independent of $\partial_{\alpha}V_{\beta}$,
since the viscous stress tensor must vanish for constant velocities. Next, we
notice that $\sigma_{\alpha\beta}\prime$ must also vanish when the whole fluid
is in uniform rotation since it is clear that in such a motion, no internal
friction occurs in the fluid. In uniform rotation with angular velocity
$\omega$, the velocity equals the vector product $V=r\times\omega$. The sums
are the linear combination of
$\partial_{\alpha}V_{\beta}+\partial_{\beta}V_{\alpha}$ and vanish when cross-
product does [25]. Hence, the viscous stress tensor has to be proportional to
linear combinations and divergence, therefore

$\sigma_{\alpha\beta}\prime=\mu(\partial_{\alpha}V_{\beta}+\partial_{\beta}V_{\alpha}-\frac{2}{3}\delta_{\alpha\beta}\partial_{\gamma}V^{\gamma})+\xi\delta_{\alpha\beta}\partial_{\gamma}V^{\gamma}$
(55)

$\mu$ is referred as viscosity and $\xi$ is called second viscosity. Taking
(56) into account in (13) and taking the derivative of it one gets

$\displaystyle\partial_{\alpha}E_{\Omega}=-\partial_{\beta}M_{\alpha}^{\beta}$
$\displaystyle=-\partial_{\alpha}p+\partial_{\beta}(\mu(\partial_{\alpha}V^{\beta}+\partial^{\beta}V_{\alpha}$
$\displaystyle-\frac{2}{3}\delta_{\alpha}^{\beta}\partial_{\gamma}V^{\gamma})+\xi\delta_{\alpha}^{\beta}\partial_{\gamma}V^{\gamma})$
(56)

Taking into account that the first $\mu$ and second $\xi$ viscosities are
assumed to be velocity-independent, considering that in isotropic fluids,
coefficients are scalar quantities only. Note here that our calculations are
model-free, meaning $\partial_{\alpha}E_{\Omega}$ can be modeled as necessary,
but it does not change the structure of the kinetic terms. Plugging (56) in
(54) and adding (53) one gets

$\displaystyle\mathcal{L}_{\Omega}$
$\displaystyle=\int_{\Omega}\rho_{\Omega}\bm{V_{\Omega}}\cdot[(V^{\alpha}\partial_{\alpha}\bm{V_{\Omega}}+\partial_{t}\bm{V_{\Omega}})$
$\displaystyle-(-\bm{\nabla}p+\mu\Delta\bm{V_{\Omega}}+(\xi+\frac{1}{3}\mu)\bm{\nabla}\partial_{\alpha}V_{\Omega}^{\alpha})]d\Omega$
(57)

Note that $p$ is unrelated to the surface pressure $P_{S}$. $p$ is the
pressure induced by inertial motion, while $P_{S}$ is the surface pressure the
system applies to the boundary, regardless of whether it is inertially moving.

##  Appendix E Dominantly Compressible Systems

Here, for self-consistency, we repeat the solution to (17), though it is
published [3]. Derivation of the condition (18)
$\partial_{\alpha}V^{\alpha}=0$ is trivial, and it leads to easily obtainable
by algebra the solution

$\bm{V}=k\frac{\bm{R}}{R^{3}}\Rightarrow\frac{\partial{\bm{R}}}{\partial
t}={\omega}_{\xi}{R}_{\xi}{{\bm{S}}}^{{\bm{\xi}}}$ (58)

here $k$ is some coupling constant and $\bm{R}$ is the position vector,
${\omega}_{\xi},{R}_{\xi}$ are $\xi$ component of the frequency and position
vector, ${{\bm{S}}}^{{\bm{\xi}}}$ is the $\xi$ vector component of the unit
spherical vector $\bm{S}=(\sin\phi\sin\theta,\sin\phi\cos\theta,\cos\theta)$.
The equation (58) has two trivial solutions and one, according to vector
calculus, the rotor of some angular velocity like functional $\bm{L}$ solution

$\displaystyle{\bm{R}}$
$\displaystyle={{\bm{R}}}_{{\bf{0}}}+{\omega}_{\xi}{R}_{\xi}t{{\bm{S}}}^{{\bm{\xi}}}$
(59) $\displaystyle{\bm{R}}$
$\displaystyle={R}_{0\xi}{e}^{{\omega}_{\xi}t{{\bm{S}}}^{{\bm{\xi}}}}$ (60)
$\displaystyle\bm{V}$ $\displaystyle=-\bm{\nabla}\times\bm{L}$ (61)

where
$e^{\omega_{\alpha}\bm{S}^{\alpha}t}=(e^{\omega_{x}\bm{S}^{x}t},e^{\omega_{y}\bm{S}^{y}t},e^{\omega_{z}\bm{S}^{z}t})$
by definition. Next, assuming that $\bm{L}$ behaves same as the position
vector (61) solution can be expanded as

$\displaystyle\partial_{t}\bm{R}=-\bm{\nabla}\times\bm{L}$ (62)
$\displaystyle\frac{1}{v_{0}^{2}}\partial_{t}\bm{L}=\bm{\nabla}\times\bm{R}$
(63)

$v_{0}$ is some constant wave propagation velocity. After some algebra on (62,
63), one ends up with the wave equation for the position vector

$\frac{1}{{v}_{0}^{2}}\frac{{\partial}^{2}{\bm{R}}}{\partial{t}^{2}}-{\nabla}^{2}{\bm{R}}=0$
(64)

The solution to the (64) is straightforward, considering that all spacial
directions are linearly independent, and therefore, in each direction, one has
a linear wave equation. Therefore

${\bm{R}}={{\bm{R}}}_{0}+{\psi}_{\xi}{{\bm{S}}}^{{\bm{\xi}}}$ (65)

where, the wave function ${{\bm{\psi}}}_{\xi}(\xi,t)$ is given by the formula

$\displaystyle{{\bm{\psi}}}_{\xi}(\xi,t)=\sum_{m=1}^{\infty}\,\frac{2}{l}({\int}_{0}^{l}\,{\bm{\psi}}(\xi,0)\,\sin\,\frac{m\pi\xi}{l}d\xi)$
$\displaystyle\cdot\cos(\frac{{v}_{0}m\pi t}{l})\sin(\frac{{v}_{0}m\pi t}{l})$
(66)

Some linear combination of the (59, 60, 65) solutions gives (21) final
solution.

To derive (19), we start by considering the equilibrium surface density
$\rho_{0}$. When both $C$ and $B_{i}^{i}$ are infinitesimally small, the
linearized conservation of mass can be expressed as $\partial_{t}\rho_{S}=0$,
with the solution $\rho_{S}=\rho_{0}$. This suggests that for sufficiently
small $C$, the density of the diffusive layer remains constant. Therefore, by
taking the time derivative of the third equation of (17) we have

$\displaystyle\rho_{0}\partial_{t}^{2}C=-\Lambda_{0}\partial_{t}B_{i}^{i}$
$\displaystyle=-\Lambda_{0}(\nabla_{i}\nabla^{i}C+CB_{ij}B^{ij})$ (67)
$\displaystyle\dot{\nabla}B_{i}^{i}$
$\displaystyle=\nabla_{i}\nabla^{i}C+CB_{ij}B^{ij}$ (68)

here (68) is well known theorem of the CMS [21] and in linear approximations
$\dot{\nabla}\approx\partial_{t}$. Taking into account that
$B_{ij}B^{ij}=-2K$, $K$ is the Gaussian curvature [21], then from (67)
directly follows

$\frac{{\partial}^{2}C}{\partial{t}^{2}}=-\gamma({\rm{\Delta}}C-2KC)$ (69)

where $\gamma=\Lambda_{0}/\rho_{0}$. In planar approximations, $K\approx 0$
and the wave equation follows

$\frac{{\partial}^{2}C}{\partial{t}^{2}}=-\gamma{\rm{\Delta}}C$ (70)

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