# Closed, Two Dimensional Surface Dynamics

David V. Svintradze  Present address: Max Plank Institute for the Physics of
Complex System, 01187 Dresden, Germany.
<EMAIL_ADDRESS><EMAIL_ADDRESS>Department of Physics, Tbilisi State
University, Chavchavadze Ave. 03, 0179 Tbilisi, Georgia

###### Abstract

We present dynamic equations for two dimensional closed surfaces and
analytically solve it for some simplified cases. We derive final equations for
surface normal motions by two different ways. The solution of the equations of
motions in normal direction indicates that any closed, two dimensional,
homogeneous surface with time invariable surface energy density adopts
constant mean curvature shape when it comes in equilibrium with environment.
As an example, we apply the formalism to analyze equilibrium shapes of
micelles and explain why they adopt spherical, lamellar and cylindrical
shapes. We show that theoretical calculation for micellar optimal radius is in
good agreement with all atom simulations and experiments.

††preprint: APS/123-QED

##  I Introduction

Biological systems exhibit a variety of morphologies and experience large
shape deformations during a motion. Such ’choreography’ of shape motility is
characteristic not only for all living organisms and cells Cahalan and Parker
(2008) but also for proteins, nucleic acids and to all biomacromolecules in
general. Shape motility, which is a motion of two-dimensional surfaces, may be
a result of active (by consuming energy) or passive (without consuming energy)
processes. The time scale for shape dynamics may vary from slow (nanometer per
nanoseconds) to very fast (nanometer per femtosecond) Pal  _et al._ (2002);
Wan  _et al._ (2000). Slowly moving surfaces are considered as over-damped
systems. An example is cell motility. In that case one may use well developed
the Helfrich formalism to describe the motion. This is a coarse-grained
description of membranes with an expansion of the free energy in powers of the
curvature tensor Helfrich (1973). However, while the formalism Helfrich (1973)
are applicable to slowly moving surfaces they are not applicable to fast
moving surfaces, where biomolecules maybe fitted. Surface dynamics for
proteins or DNA Pal  _et al._ (2002); Wan  _et al._ (2000) may reach $nm/fs$
range. So that surfaces may be represented as virtual three dimensional pseudo
Riemannian manifolds. We derived fully generic equations of motions for three
manifolds Svintradze (2017a), but purposefully omitted lengthy discussion
about motion of two-dimensional surfaces, which is a topic for this paper.

Currently, significant progress on fluidic models of membrane dynamics has
already been made. The role of geometric constraints in self-assembly have
been elucidated by linking together thermodynamics, interaction free energies
and geometry Israelachvili  _et al._ (1976, 1977). The Helfrich formalism
provides the foundation for a purely differential geometric approach whereby
the membrane surface potential energy density is considered as a functional of
the static curvature Helfrich (1973), see also review papersDeserno (2015);
Lipowsky and Sackmann (1995); Seifert (1997). The model has been improved by
adding force and torque balance equations Capovilla and Guven (2002); Fournier
(2007). Specific dynamical equations accounting for bending as well as
electrodynamic effects have also been reported Scriven (1960); Arroyo and
DeSimone (2009); Gao  _et al._ (2008). Furthermore, active membrane theories
have extended our understanding of passive membranes. Active membrane theories
include external forces Salbreux and Jülicher (2017); Ramaswamy  _et al._
(2000); Chen (2004); Gov (2004) and provide a framework for the study of
active biological or chemical processes at surfaces, such as the cell cortex,
the mechanics of epithelial tissues, or reconstituted active systems on
surfaces Salbreux and Jülicher (2017).

Among the remarkable aspects of fluid lipid membranes deduced from the large
body of theoretical work Deserno (2015); Lipowsky and Sackmann (1995); Seifert
(1997), is that the physical behavior of a membrane on the length scale not
much bigger than its own thickness, can be described with high accuracy by a
purely geometric Hamiltonian Canham (1970); Helfrich (1973); Evans (1974).
Associated Euler-Lagrange equations Zhong-Can and Helfrich (1987, 1989), so
called “shape equations”, are fourth order partial nonlinear differential
equations, and finding a general analytical solution is typically difficult,
even though it has been analytically Mladenov  _et al._ (2013) and numerically
solved for some specific Svetina and Žekš (1989); Seifert and Lipowsky (1990);
Lipowsky (1991); Seifert  _et al._ (1991); Jülicher and Lipowsky (1993, 1996);
Jülicher and Seifert (1994); Miao  _et al._ (1994) and general cases Heinrich
_et al._ (1993); Kralj-Iglič  _et al._ (1993).

In fluid dynamics, material particles can be treated as a vertex of geometric
figure and virtual layers as surfaces and equations of motion for such
surfaces can be searched. We refer to the formalism as differentially
variational surfaces (DVS) (or DVS formalism) Svintradze (2017a).

In this paper, we propose different approach to the ’shape choreography’
problem. We use DVS formalism, tensor calculus of moving surfaces and the
first law of thermodynamics to derive the final equation for the closed 2D
surface dynamics (later on referred as surface) and to solve it analytically
for the equilibrium case. In other words, we derive generic equations of
motions for closed two-dimensional surfaces and without any a priori symmetric
assumptions, we show that constant mean curvature shapes are equilibrium
solutions. In contrast to the Young-Laplace law these solutions, are
universally correct descriptions of capillary surfaces as well as molecular
surfaces. In addition, our equations of motions (20-25) are generic and exact.
It advances our understanding of fluid dynamics because generalizes ideal
magneto-hydrodynamic and Naiver-Stokes equations Svintradze (2017a) and in
contrast to Navier-Stokes, as we demonstrate in this paper, are trivially
solvable for equilibrium shapes. To demonstrate the validity of these
equations and their analytical solutions we apply them to micelles. Within our
formalism it becomes simple task to show micelles lamellar, cylindrical,
spherical shapes and assert their optimal spherical radius.

For clarity, we shall give brief description of micelles and their structures.
A micelle consists of monolayer of lipid molecules containing hydrophilic head
and hydrophobic tail. These amphiphilic molecules, in aqueous environment,
aggregate spontaneously into a monomolecular layer held together due to a
hydrophobic effect Chandler (2005); Leikin  _et al._ (1993) (see also
Svintradze (2017a, 2010, 2015, 2016, b)) by weak non-covalent forces Tanford
(1973). They form flexible surfaces that show variety of shapes of different
topology, but remarkably in thermodynamic equilibrium conditions they are
spherical, lamellar (plane) or cylindrical in shape.

##  II Methods

In the section we provide basics of tensor calculus for moving surfaces and
summarize the theorems we used directly or indirectly to derive equations for
two-dimensional surface dynamics. Differential geometry preliminaries we used
here are available in tensor calculus textbook Grinfeld (2010) and in our work
Svintradze (2017a).

###  II.1 Basics of differential geometry.

Suppose that $S^{i}$ $(i=1,2)$ are the surface coordinates of the moving
manifold (or the surface) $S$ and the ambient Euclidean space is referred to
coordinates $X^{\alpha}$ (Figure 1).

Figure 1: Graphical illustration of the arbitrary surface and its’ local
tangent plane. $\vec{S}_{1},\vec{S}_{2},\vec{N}$ are local tangent plane base
vectors and local surface normal respectively.
$\vec{X}_{1},\vec{X}_{2},\vec{X}_{3}$ are arbitrary base vectors of the
ambient Euclidean space and $\vec{R}=\vec{R}(X)=\vec{R}(t,S)$ is radius vector
of the point. $\vec{V}$ is arbitrary surface velocity and $C,V_{1},V_{2}$
display projection of the velocity to the $\vec{N},\vec{S}_{1},\vec{S}_{2}$
directions respectively.

Coordinates $S^{i},X^{\alpha}$ are arbitrarily chosen so that sufficient
differentiability is achieved in both, space and time. Surface equation in
ambient coordinates can be written as $X^{\alpha}=X^{\alpha}(t,S^{i})$. Let
the position vector $\vec{R}$ be expressed in coordinates as

$\vec{R}=\vec{R}(X^{\alpha})=\vec{R}(t,S^{i})$ (1)

Latin letters in indexes indicate surface related tensors. Greek letters in
indexes show tensors related to Euclidean ambient space. All equations are
fully tensorial and follow the Einstein summation convention. Covariant bases
for the ambient space are introduced as
$\vec{X}_{\alpha}=\partial_{\alpha}\vec{R}$, where
$\partial_{\alpha}=\partial/\partial X^{\alpha}$. The covariant metric tensor
is the dot product of covariant bases

$X_{\alpha\beta}=\vec{X}_{\alpha}\vec{X}_{\beta}$ (2)

The contravariant metric tensor is defined as the matrix inverse of the
covariant metric tensor, so that
$X^{\alpha\beta}X_{\beta\gamma}=\delta_{\gamma}^{\alpha}$, where
$\delta_{\gamma}^{\alpha}$ is the Kronecker delta. As far as the ambient space
is set to be Euclidean, the covariant bases are linearly independent, so that
the square root of the metric tensor determinant is unit. Furthermore, the
Christoffel symbols given by
$\Gamma^{\alpha}_{\beta\gamma}=\vec{X}^{\alpha}\cdot\partial_{\beta}\vec{X}_{\gamma}$
vanish and set the equality between partial and curvilinear derivatives
$\partial_{\alpha}=\nabla_{\alpha}$.

Now let’s discuss tensors on the embedded surface with arbitrary coordinates
$S^{i}$. Latin indexes throughout the text are used exclusively for curved
surfaces and curvilinear derivative $\nabla_{i}$ is no longer the same as the
partial derivative $\partial_{i}=\partial/\partial S^{i}$. Similar to the
bases of ambient space, covariant bases of an embedded manifold are defined as
$\vec{S}_{i}=\partial_{i}\vec{R}$ and the covariant surface metric tensor is
the dot product of the covariant surface bases:

$S_{ij}=\vec{S}_{i}\cdot\vec{S}_{j}$ (3)

The contravariant metric tensor is the matrix inverse of the covariant one.
The matrix inverse nature of covariant-contravariant metrics gives
possibilities to raise and lower indexes of tensors defined on the manifold.
The surface Christoffel symbols are given by
$\Gamma^{i}_{jk}=\vec{S}^{i}\cdot\partial_{j}\vec{S}_{k}$ and along with
Christoffel symbols of the ambient space provide all the necessary tools for
covariant derivatives to be defined at tensors with mixed space/surface
indexes:

$\displaystyle\nabla_{i}T_{\beta k}^{\alpha j}=\partial_{i}T_{\beta k}^{\alpha
j}+X_{i}^{\gamma}\Gamma_{\gamma\nu}^{\alpha}T_{\beta k}^{\nu
j}-X_{i}^{\gamma}\Gamma_{\gamma\beta}^{\mu}T_{\mu k}^{\alpha j}+$
$\displaystyle\Gamma_{im}^{j}T_{\beta k}^{\alpha m}-\Gamma_{ik}^{m}T_{\beta
m}^{\alpha j}$ (4)

where $X_{i}^{\gamma}$ is the shift tensor which reciprocally shifts space
bases to surface bases, as well as space metric to surface metric; for
instance, $\vec{S}_{i}=X^{\alpha}_{i}\vec{X}_{\alpha}$ and
$S_{ij}=\vec{S}_{i}\cdot\vec{S}_{j}=X_{i}^{\alpha}\vec{X}_{\alpha}X^{\beta}_{j}\vec{X}_{\beta}=X_{i}^{\alpha}X^{\beta}_{j}X_{\alpha\beta}$.
Note that in (II.1) Christoffel symbols with Greek indexes are zeros.

Using (2,II.1), one may directly prove metrilinic property of the surface
metric tensor $\nabla_{i}S_{mn}=0$, from where follows
$\vec{S}_{m}\cdot\nabla_{i}\vec{S}_{n}=0$, meaning that
$\vec{S}_{m}\bot\nabla_{i}\vec{S}_{n}$ are orthogonal vectors and as so
$\nabla_{i}\vec{S}_{n}$ must be parallel to $\vec{N}$ the surface normal

$\nabla_{i}\vec{S}_{j}=\vec{N}B_{ij}$ (5)

$\vec{N}$ is a surface normal vector with unit length and $B_{ij}$ is the
tensorial coefficient of the relationship and is generally referred as the
symmetric curvature tensor. The trace of the curvature tensor with upper and
lower indexes is the mean curvature and its determinant is the Gaussian
curvature. It is well known that a surface with constant Gaussian curvature is
a sphere, consequently a sphere can be expressed as:

$B_{i}^{i}=\lambda$ (6)

where $\lambda$ is some non-zero constant. According to (5,6), finding the
curvature tensor defines the way of finding covariant derivatives of surface
base vectors and as so, defines the way of finding surface base vectors which
indirectly leads to the identification of the surface.

###  II.2 Basics of tensor calculus for moving surfaces.

All Equations written above are generally true for moving surfaces. We now
turn to a brief review of definitions of coordinate velocity $V^{\alpha}$,
interface velocity $C$ (which is the same as normal velocity), tangent
velocity $V^{i}$ (Figure 1), time $\dot{\nabla}$-derivative of surface tensors
and time differentiation of the surface integrals. The original definitions of
time derivatives for moving surfaces were given in Hadamard (1968) and
recently extended in tensor calculus textbook Grinfeld (2010).

Let’s start from the definition of coordinate velocity $V^{\alpha}$ and show
that the coordinate velocity is $\alpha$ component of the surface velocity.
Indeed, by the definition

$V^{\alpha}=\frac{\partial X^{\alpha}}{\partial t}$ (7)

On the other hand the position vector $\vec{R}$ given by (1) is tracking the
coordinate particle $S^{i}$. Taking into account partial time differential of
(1) and definition of ambient base vectors, we find

$\vec{V}=\frac{\partial\vec{R}(t,S^{i})}{\partial
t}=\frac{\partial\vec{R}}{\partial X^{\alpha}}\frac{\partial
X^{\alpha}(t,S^{i})}{\partial t}=V^{\alpha}\vec{X}_{\alpha}$ (8)

Therefore, $V^{\alpha}$ is ambient component of the surface velocity
$\vec{V}$. Taking into account (8), normal component of the surface velocity
is dot product with the surface normal, so that

$C=\vec{V}\cdot\vec{N}=V_{\alpha}\vec{X}^{\alpha}N^{\beta}\vec{X}_{\beta}=V_{\alpha}N^{\beta}\delta_{\beta}^{\alpha}=V_{\alpha}N^{\alpha}$
(9)

It is easy to show that the normal component $C$ of the coordinate velocity,
generally referred as interface velocity, is invariant in contrast with
coordinate velocity $V^{\alpha}$ and its sign depends on a choice of the
normal. The projection of the surface velocity on the tangent space (Figure 1)
is tangential velocity and can be expressed as

$V^{i}=V^{\alpha}X_{\alpha}^{i}$ (10)

Taking (9,10) into account one may write surface velocity as
$\vec{V}=C\vec{N}+V^{i}\vec{S}_{i}$. Graphical illustrations of coordinate
velocity $V^{\alpha}$, interface velocity $C$ and tangential velocity $V^{i}$
are given on Figure 1. There is a clear geometric interpretation of the
interface velocity Grinfeld (2010). Let the surfaces at two nearby moments of
time $t$ and $t+\Delta t$ be $S_{t}$, $S_{t+\Delta t}$ correspondingly.
Suppose that $A\in S_{t}$ (point on $S_{t}$) and the corresponding point $B\in
S_{t+\Delta t}$, $B$ has the same surface coordinates as $A$ (Figure 2), then
$\vec{AB}\approx\vec{V}\Delta t$. Let $P$ be the point, where the unit normal
$\vec{N}\in S_{t}$ intersect the surface $S_{t+\Delta t}$, then for small
enough $\Delta t$, the angle $\angle APB\rightarrow\pi/2$ and
$AP\rightarrow\vec{V}\cdot\vec{N}\Delta t$, therefore, $C$ can be defined as

$C=\lim_{\Delta t\rightarrow 0}\frac{AP}{\Delta t}$ (11)

and can be interpreted as the instantaneous velocity of the interface in the
normal direction. It is worth of mentioning that the sign of the interface
velocity depends on the choice of the normal. Although $C$ is a scalar, it is
called interface velocity because the normal direction is implied.

###  II.3 Invariant time differentiation.

Among the key definitions in calculus for moving surfaces, perhaps one of the
most important is the invariant time derivative $\dot{\nabla}$. As we have
already stated, invariant time derivative is already well defined in the
literature Hadamard (1968); Grinfeld (2010). In this paragraph, we just give
geometrically intuitive definition.

Figure 2: Geometric interpretation of the interface velocity $C$ and of the
curvilinear time derivative $\dot{\nabla}$ applied to invariant field $F$. $A$
is arbitrary chosen point so that it lays on $F(S_{t})\in S_{t}$ curve and $B$
is its’ corresponding point on the $S_{t+\Delta t}$ surface. $P$ is the point
where $S_{t}$ surface normal, applied on the point $A$, intersects the surface
$S_{t+\Delta t}$. By the geometric construction, for small enough $\Delta
t\rightarrow 0$, $\angle APB\rightarrow\pi/2$, $\vec{AB}\approx\vec{V}\Delta
t$ and $AP\approx\vec{V}\vec{N}\Delta t$. On other hand, by the same geometric
construction the field $F$ in the point $B$ can be estimated as $F(B)\approx
F(A)+\Delta t\partial F/\partial t$, while from viewpoint of the $S_{t+\Delta
t}$ surface the $F(B)$ value can be estimated as $F(P)+\Delta
tV^{i}\nabla_{i}F$, where $\nabla_{i}F$ shows rate of change in $F$ along the
surface $S_{t+\Delta t}$ and along the directed distance $BP\approx\Delta
tV^{i}$.

Suppose that invariant field $F$ is defined on the surface at all time. The
idea behind the invariant time derivative is to capture the rate of change of
$F$ in the normal direction. Physical explanation of why the deformations
along the normal direction are so important, we give below in integration
section. This is similar to how $C$ measures the rate of deformation in the
normal direction. Let for a given point $A\in S_{t}$, find the points $B\in
S_{t+\Delta t}$ and $P$ the intersection of $S_{t+\Delta t}$ and the straight
line orthogonal to $S_{t}$ (Figure 2). Then, the geometrically intuitive
definition dictates that

$\dot{\nabla}F=\lim_{\Delta t\rightarrow 0}\frac{F(P)-F(A)}{\Delta t}$ (12)

As far as (12) is entirely geometric, it must be an invariant (free from
choice of a reference frame). From the geometric construction one can estimate
value of $F$ in point $B$, so that

$F(B)\approx F(A)+\Delta t\frac{\partial F}{\partial t}$ (13)

On the other hand, $F(B)$ is related to $F(P)$ because $B,P\in S_{t+\Delta t}$
and are nearby points on the surface $S_{t+\Delta t}$, then according to
definition of covariant derivative

$F(B)\approx F(P)+\Delta tV^{i}\nabla_{i}F$ (14)

since $\nabla_{i}F$ shows rate of change in $F$ along the surface and $\Delta
t\cdot V^{i}$ captures the directed distance $BP$. Determining $F(A),F(P)$
values from (13,14) and putting it in (12), gives

$\dot{\nabla}F=\frac{\partial F}{\partial t}-V^{i}\nabla_{i}F$ (15)

Extension of the definition (15) for any arbitrary tensors with mixed space
and surface indexes is given by the formula

$\displaystyle\dot{\nabla}T_{\beta j}^{\alpha i}=\frac{\partial T_{\beta
j}^{\alpha i}}{\partial t}-V^{k}\nabla_{k}T_{\beta j}^{\alpha
i}+V^{\gamma}\Gamma_{\gamma\mu}^{\alpha}T_{\beta j}^{\mu
i}-V^{\gamma}\Gamma_{\gamma\beta}^{\mu}T_{\mu j}^{\alpha i}$
$\displaystyle+\dot{\Gamma}_{k}^{i}T_{\beta j}^{\alpha
k}-\dot{\Gamma}_{j}^{k}T_{\beta k}^{\alpha i}$ (16)

The derivative commutes with contraction, satisfies sum, product and chain
rules, is metrinilic with respect to the ambient metrics and does not commute
with the surface derivative Grinfeld (2010). Also from (12) it is clear that
the invariant time derivative applied to time independent scalar vanishes.
Christoffel symbol $\dot{\Gamma}^{i}_{j}$ for moving surfaces is defined by
the formula $\dot{\Gamma}^{i}_{j}=\nabla_{j}V^{i}-CB^{i}_{j}$.

###  II.4 Time differentiation of integrals.

The remarkable usefulness of the calculus of moving surfaces becomes evident
from two fundamental formulas for integrations that govern the rates of change
of volume and surface integrals due to the deformation of the domain Grinfeld
(2010). For instance, in evaluation of the least action principle of the
Lagrangian there is a central role for time differentiation of the surface and
space integrals, from where the geometry dependence is rigorously clarified.

For any scalar field $F=F(t,S)$ defined on a Euclidean domain $\Omega$ with
boundary $S$ evolving with the interface velocity $C$, the evolution of the
space integral and surface integral for closed surfaces are given by the
formulas

$\displaystyle\frac{d}{dt}\int_{\Omega}Fd\Omega=$
$\displaystyle\int_{\Omega}\frac{\partial F}{\partial t}d\Omega+\int_{S}CFdS$
(17) $\displaystyle\frac{d}{dt}\int_{S}FdS=$
$\displaystyle\int_{S}\dot{\nabla}FdS-\int_{S}CFB_{i}^{i}dS$ (18)

The first term in the integral represents the rate of change of the tensor
field, while the second term shows changes in the geometry. Of course there
are rigorous mathematical proofs of these formulas in the tensor calculus
textbooks. We are not going to reproduce proof of these theorems here, but
instead we give less rigorous but completely intuitive explanation of why only
interface velocity has to be count. Rigorous mathematical proof follows from
fundamental theorem of calculus

$\frac{d}{dt}\int_{a}^{b(t)}F(t,x)dx=\int_{a}^{b(t)}\frac{\partial
F(t,x)}{\partial t}dx+b^{\prime}(t)F(t,b(t))$ (19)

In the case of volume integral or surface integral it can be shown that
$b^{\prime}(t)$ is replaced by interface velocity $C$.

Intuitive explanation is pretty simple. Propose there is no interface velocity
then closed surface velocity only has tangent component. For each given time
tangent velocity (if there is no interface velocity) translates each point to
its neighboring point and therefore, does not add new area to the closed
surface (or new volume to the closed space, or new length to the closed
curve). As so, tangential velocity just induces rotational movement (or
uniform translational motion) of the object and can be excluded from additive
terms in the integration. Perhaps, it is easier to understand this statement
for one dimensional motion. Let’s assume that material point is moving along
some trajectory (some closed curve or loop), then, in each point, the velocity
of the material point is tangential to the curve. Now one can translate this
motion into the motion of the closed curve where the loop has only tangential
velocity. In this aspect, the embedded loop only rotates (uniformly translates
in the plane) without changing the length locally, therefore tangential
velocity of the curve does not add new length to the curve (Same is true for
open curve with fixed ends).

##  III General equations of surface motions

Fully non-restrained and exact equations for moving three-dimensional surfaces
in electromagnetic field, when the interaction with an ambient environment is
ignored, reads

$\displaystyle\dot{\nabla}\rho+\nabla_{i}(\rho V^{i})=$ $\displaystyle\rho
CB^{i}_{i}$ (20) $\displaystyle\partial_{\alpha}(\rho
V^{\alpha}(\dot{\nabla}C+2V^{i}\nabla_{i}C+V^{i}V^{j}B_{ij})-V^{\alpha}(\frac{1}{4\mu_{0}}F_{\mu\nu}F^{\mu\nu}+A_{\mu}J^{\mu}))=$
$\displaystyle fa$ (21) $\displaystyle\int_{S}\rho
V_{i}(\dot{\nabla}V^{i}+V^{j}\nabla_{j}V^{i}-C\nabla^{i}C-CV^{j}B_{j}^{i})dS=$
$\displaystyle\int_{\Omega}f^{i}a_{i}d\Omega$ (22)

where $\rho$ is the surface mass density, $V^{\alpha}$, $V^{i}$ are coordinate
and tangential components of the surface velocity, $C$ is interface velocity,
$\alpha=0,1,2,3$ for Minkowski four-dimensional space-time ambient space,
$i=0,1,2$ for pseudo-Riemannian manifold (surface), $B_{ij}$ is the surface
curvature tensor, $F^{\mu\nu}$ is electromagnetic tensor,
$F^{\alpha}=J^{\alpha}-\partial_{\beta}F^{\beta\alpha}$, $J^{\alpha}$ is
$\alpha$ component of $\vec{J}=(J^{\alpha})$ four current, $f,f^{i}$ are
normal and tangential components of $\vec{F}=(F^{\alpha})$, $a$, $a_{i}$ are
the normal and tangential components of the partial time derivative of the
four vector potential $\vec{A}=(A_{\alpha})$, $S$, $\Omega$ stand for surface
and space integrals respectively. Exact derivation of (20-22) is given in our
work Svintradze (2017a), we don’t reproduce derivation of this set in this
paper, rather just mention that first one is the consequence of mass
conservation, second and third equations come from minimum action principle of
a Lagrangian and imply motion in normal direction (21) and in tangent
direction (22).

For two dimensional surface dynamics, Minkowskian space becomes Euclidean, so
that $\alpha=1,2,3$ and the surface is two-dimensional Riemannian manifold
$i=1,2$. So that, after modeling the potential energy as a negative volume
integral of the internal pressure and inclusion interaction with an
environment, (20-22) further simplifies as

$\displaystyle\dot{\nabla}\rho+\nabla_{i}(\rho V^{i})=$ $\displaystyle\rho
CB^{i}_{i}$ (23) $\displaystyle\partial_{\alpha}(\rho
V^{\alpha}(\dot{\nabla}C+2V^{i}\nabla_{i}C+V^{i}V^{j}B_{ij})+V^{\alpha}(P^{+}+\Pi))=$
$\displaystyle-V^{\alpha}\partial_{\alpha}(P^{+}+\Pi)$ (24) $\displaystyle\rho
V_{i}(\dot{\nabla}V^{i}+V^{j}\nabla_{j}V^{i}-C\nabla^{i}C-CV^{j}B_{j}^{i})=$
$\displaystyle 0$ (25)

where $P^{+}$, $\Pi$ are internal hydrodynamic and osmotic pressures,
respectively. Derivation of (20-22) can be found in Svintradze (2017a). We
derive (23-25) in appendix section. It is noteworthy that from the last
equations set only the second equation (24) differs from the dynamic fluid
film equations Grinfeld (2009, 2010)

$\rho(\dot{\nabla}C+2V^{i}\nabla_{i}C+V^{i}V^{j}B_{ij})=\sigma B_{i}^{i}$ (26)

where $\sigma$ is surface tension. (26) is only valid when the surface can be
described with time invariable surface tension Grinfeld (2009, 2010), meaning
that the surface is homogeneous and the surface tension is constant, while
(24) does not have that restriction. Using (26) in (24) and taking into
account that in equilibrium processes internal pressure is the same as
external pressure, one gets exactly the same equation of motion in normal
direction (39) as we get from using the first law of thermodynamics (see
below).

$\partial_{\alpha}(\sigma
V^{\alpha}B_{i}^{i}+(P^{+}+\Pi)V^{\alpha})=-(\partial_{\alpha}P^{+}+\partial_{\alpha}\Pi)V^{\alpha}$
(27)

It is worth of mentioning that (23-25) also follows from (20-22) if one
applies same formalism as it is given in (30-32). Indeed, for relatively
slowly moving surfaces space is three dimensional Euclidean so that
$\alpha=1,2,3$, the surface is two-dimensional Riemannian ($i=1,2$) and the
potential energy becomes

$\displaystyle U=$
$\displaystyle\int_{\Omega}(\frac{1}{4\mu_{0}}F_{\mu\nu}F^{\mu\nu}+A_{\mu}J^{\mu})$
$\displaystyle=$
$\displaystyle\int_{\Omega}(-\frac{\epsilon_{0}}{2}E^{2}+\frac{1}{\mu_{0}}B^{2}-q\varphi+\vec{A}\vec{J})d\Omega$
(28)

where $\vec{E},\vec{B}$ are electric and magnetic fields and
$q,\varphi,\vec{A},\vec{J}$ are charge density, electric potential, magnetic
vector potential and current density vector respectively. Using (30-32)
formalism into account, we find

$dU=-(P^{+}+\Pi)d\Omega=(-\frac{\epsilon_{0}}{2}E^{2}+\frac{1}{\mu_{0}}B^{2}-q\varphi+\vec{A}\vec{J})d\Omega$
(29)

Taking into account (29) and that the pressure comes from the normal force
applied to the surface, we find ${fa=-V^{\alpha}\partial_{\alpha}(P^{+}+\Pi)}$
and in tangent direction $f^{i}a_{i}=0$, then (20-22) becomes (23-25).
Electromagnetic potential energy can be generalized if one takes into account
environment, which enters in energy terms as bound and free charges and
electric/magnetic fields are replaced by polarization and magnetization
vectors Svintradze (2017a).

##  IV Results and Discussion

###  IV.1 General assumptions.

In this section we apply basics of thermodynamics and fundamental theorems of
calculus of moving surfaces to demonstrate shortest derivation of the
equation, describing motion of homogeneous, closed two dimensional surface
with time invariable surface tension at normal direction (27). We consider the
system consisted of aqueous media with the formed closed surface in it (Figure
3). The system is isolated with constant temperature and there is no absorbed
or dissipated heat on the surface; in other words, a process is adiabatic.

Figure 3: Graphical illustration of the isolated system containing aqueous
solution. Water molecules are represented as red and white sticks. The system
boundary is shown as white faces with black edges. The subsystem-micelle is
closed surface, blue blob in the center of the system.

According to the first law of thermodynamic, as far as there is no dissipated
or absorbed heat, the change of the internal energy of the surface must be

$dE=\delta W$ (30)

where $\delta W$ is infinitesimal work done on the subsystem and $dE$ is
infinitesimal change of the internal energy. Because the temperature of the
system is constant, the differential of the subsystems’ internal energy can be
remodeled as

$dE=dU$ (31)

where $U$ is the total potential energy of the surface. By the definition the
elementary work done on the subsystem is

$\delta W=(P^{-}+\Pi)d\Omega$ (32)

where , $P^{-},\Pi$ are external hydrodynamic and osmotic pressures applied on
the surface by the surroundings correspondingly and $\Omega$ is the volume
that surface encloses with boundary of $S$ surface area. Let’s propose that
the surface is homogeneous (i.e material particles are homogeneously
distributed on the surface) so that the total potential energy is integration
of the potential energy per unit area over the surface, then

$dU=\sigma dS$ (33)

where $\sigma$ is the potential energy per unit area and is called surface
tension in the paper. As far as we discuss simplest case of the system
consisted of aqueous medium and single closed surface, we can suggest that the
surface tension is not time variable. Using (30-33) after few lines of
algebra, we fined

$\int_{S}\sigma dS=\int_{\Omega}(P^{-}+\Pi)d\Omega$ (34)

Assuming the surface is moving so that (34) stays valid for any time
variations, then time differentiation of the left side must be equal to time
differentiation of right integral. As far as on the right hand side we have
space integral, time differentiation can be taken into the integral, using
general theorems for differentiation of space and surface integrals (17-18),
so that integration theorem for space integral holds and the convective and
advective terms due to volume motion are considered

$\displaystyle\frac{d}{dt}\int_{\Omega}(P^{-}+\Pi)d\Omega=\int_{\Omega}(\partial_{\alpha}P^{-}+\partial_{\alpha}\Pi)\frac{\partial
X^{\alpha}}{\partial t}d\Omega$ $\displaystyle+\int_{S}C(P^{-}+\Pi)dS$ (35)

To calculate time derivative of the surface integral we have to take into
account the theorem about time differentiation of the surface integral (18),
from which follows that for time invariable surface tension

$\frac{d}{dt}\int_{S}\sigma dS=\int_{S}-\sigma CB_{i}^{i}dS$ (36)

Where $C=V^{\alpha}N_{\alpha}$ is interface velocity, $N_{\alpha}$ is $\alpha$
component of the surface normal and ${V=\partial X^{\alpha}/\partial t}$ is
coordinate velocity, $X^{\alpha}$ is general coordinate and $B_{i}^{i}$ is the
trace of the mixed curvature tensor generally known as mean curvature. After
few lines of algebra putting (34-36) together, we find

$\int_{S}(\sigma
CB_{i}^{i}+C(P^{-}+\Pi))dS=-\int_{\Omega}(\partial_{\alpha}P^{-}+\partial_{\alpha}\Pi)V^{\alpha}d\Omega$
(37)

Generalized Gauss theorem converts the surface integral of the left hand side
of (37) into space integral, so that

$\displaystyle\int_{S}N_{\alpha}V^{\alpha}(\sigma
B_{i}^{i}+P^{-}+\Pi)dS=\int_{\Omega}\partial_{\alpha}(\sigma
V^{\alpha}B_{i}^{i}$ $\displaystyle+(P^{-}+\Pi)V^{\alpha})d\Omega$ (38)

Combination of (37) and (IV.1) immediately gives equation of motion for
surface in normal direction

$\partial_{\alpha}(\sigma
V^{\alpha}B_{i}^{i}+(P^{-}+\Pi)V^{\alpha})=-(\partial_{\alpha}P^{-}+\partial_{\alpha}\Pi)V^{\alpha}$
(39)

For equilibrium processes internal and external pressures are identical
${P^{-}=P^{+}}$, so that (39) becomes identical to the equation of motion in
normal direction observed from master equations (23-27). Also, we should note
that (39) is only valid for motion of the homogeneous surfaces with time
invariable surface tension at normal direction, therefore, it does not display
any deformation in tangent directions. (39) further simplifies when the
surface comes in equilibrium with the solvent where divergence of the surface
velocity $\partial_{\alpha}V^{\alpha}$ (stationary interface) along with
$\partial P/\partial t$ (where $P=P^{-}+\Pi$) vanishes, then the solution to
(39), taking into account the condition (IV.1), becomes

$B^{i}_{i}=-\frac{P}{\sigma}$ (40)

The result (40) shows that the solution is constant mean curvatures (CMC)
surfaces. Such CMC are rare and can be many if one relaxes the condition we
restricted to the system. We consider isolated system where the surface is
closed subsystem, these two preconditions mathematically mean that the surface
we discuss is compact embedded surface in $\mathbb{R}^{3}$. According to A. D.
Alexandrov uniqueness theorem for surfaces, a compact embedded surface in
$\mathbb{R}^{3}$ with constant non-zero mean curvature is a sphere Alexandrov
(1958). Correspondingly the solution (40) is a sphere (as far as we have
compact two-manifold in the Euclidean space). When

$\frac{P}{\sigma}\neq 0$ (41)

the surface is spheroid (or a cylinder if one relaxes compactness restriction
making the cylinder infinitely long) and becomes plane (again when compactness
argument is relaxed) or other zero mean curvature shape when compactness
argument is not relaxed but contour of the surface remains fixed. This
surprisingly simple and elegant derivation explains all the shapes surfaces
can adopt in aqueous solution at equilibrium conditions.111Even though we set
environment as aqueous, it enters into equations as osmotic pressure term,
which due to a generality of arguments can be anything. Therefore, as a medium
one may pick any liquid or gas. If the compactness condition is relaxed then
(40) predicts that in addition to cylinder and plane all other CMC surfaces
are also equilibrium shapes for moving surfaces. Taking into account that the
surface tension in general can be a function of many variables, such as
Gaussian curvature, bending rigidity, spontaneous curvature, molecules
concentration, geometry of surfactant molecules and etc., then (39) may
predict possible deformations of differently shaped surfaces and their wide
range of static shapes. In fact, if considered that surface tension, which is
defined as potential energy per unit area, can be a function of mean curvature
$\sigma=\sigma(B^{i}_{i})$, then Taylor expansion of $\sigma(B^{i}_{i})$
naturally rises all additional terms. These generalizations and temperature
fluctuations can be included in the equations, but it is not scope of this
paper and should be addressed separately. One may even propose $\sigma$ as
time independent the Helfrich Hamiltonian and then (39, 40) will become
equation of static shapes for homogeneous surfaces with time invariable
surface tension.

###  IV.2 Physical application, micelle.

We can put equation (39) and its solution (40) under the test for homogeneous
micellar surface equilibrated with the aqueous solution. Based on (40) we can
calculate minimal value of a micelle radius. The value of the trace of the
mixed curvature tensor for a sphere is

$B^{i}_{i}=-\frac{2}{R}$ (42)

where $R$ is radius.

Figure 4:  Simulated three dimensional coordinates of the micelle in aqueous
solution display sphere with diameter $38.5{\AA}$. (Left)
dihexanoylphosphatidylcholine (DHPC phospholipids) are modeled as orange
balls. (Right) Gaussian mapping at contour resolution $8{\AA}$ of the micelle
shows spherical structure.

Let’s calculate value of the surface pressure when the micelle still can
exist. Lipids in a micelle are confined in the surface by hydrophobic
interactions with average energy in the range of hydrogen bonding. As far as
values of hydrogen bonding energy are somewhat uncertain in the literature, by
the first approximation we take average energy for the hydrogen bonding energy
interval and assign it to the lipid molecule. Low boundary of the interval
(minimum energy) for $XH\cdot\cdot\cdot Y$ hydrogen bond is about 1 kJ per mol
($CH\cdot\cdot\cdot C$ unit) and high boundary is about 161 kJ per mol
($FH\cdot\cdot\cdot F$ unit), the low and high values are taken according to
references Larson and McMahon (1984); Emsley (1980). Therefore, average energy
is about $(1+161)/2=81kJ/mol\approx 13\cdot 10^{-20}J$. To estimate hydrogen
bonding energy per molecule with the undefined shape (lipid molecule) in the
first approximation is to assign average energy to it and consider the
spherical shape with the gyration radius. Of course it is low level
approximation, but even such rough calculations produce reasonable results.
After all these rough estimations the pressure to move one lipid from the
surface, in order to induce critical deformations of the surface, is about
average energy per the average volume of the lipid molecule

$P\approx\frac{3\cdot 13\cdot 10^{-20}}{4\pi r_{G}^{3}}\approx 3.1\cdot
10^{7}N/m^{2}$ (43)

where $4\pi r_{G}^{3}/3$ is the estimated volume of a lipid molecule
considered as sphere with the gyration radius $r_{G}\approx 1nm$. On the other
hand, surface tension of a fluid monolayer at optimal packing of the lipids is
about $\sigma\approx 3\cdot 10^{-2}N/m$ Jähnig (1996); Israelachvili  _et al._
(1977); Israelachvili (2011), using these and (42,43) in (40) the estimated
micelle radius is

$R\approx\frac{2\cdot 3\cdot 10^{-2}}{3.1\cdot 10^{7}}=19.3\pm 0.1{\AA}$ (44)

Figure 5: All atom simulation of DHPC micelle. (A) The figure shows a geometry
of the DHPC surfactant molecule used in simulation and gives parametric
description of volume, surface area, sphericity and effective radius. (B)
Indicates atomistic simulation result contoured by Gaussian map and the
diameter of the micelle, measured by PyMol. The diameter of the simulated
micelle appears to be $54.0{\AA}$ with the uncertainty of the measurement
$0.1{\AA}$.

These calculations put the minimum radius in nanometer scale and is in very
good agreement with experimental as well as computational frameworks Feller
_et al._ (1995); Egberts  _et al._ (1994). To further validate the (44)
result, we ran a CHARMM based Micelle Builder simulation Jo  _et al._ (2008);
Cheng  _et al._ (2013) for 100 phospholipid molecules (DHPC lipids). The
simulation result (Figure 4) generated a spherical micelle with diameter
$38.5\pm 0.1{\AA}$. These calculations indeed indicate that even such rough
estimations produce reasonable accuracy.

To get more convincing estimations it is necessary to take into account that
neither lipids are spherical nor hydrophobic interactions per lipid are
average energy of single hydrogen bond. In the second approximation lipids are
no longer undefined spheres, but have well defined surfactant geometry. The
Hydrophobic energy is no longer average energy of single hydrogen bond, but is
1 kJ per mol per $-CH_{2}-$ unit. In all atom simulations we used
dihexanoylphosphatidylcholine (DHPC) lipid molecule having $12-CH_{2}-$ units
(Figure 5) per hydrophobic tail, so hydrophobic energy is about
$12kJ/mol\approx 1.99\cdot 10^{-20}J$. Accurate calculation of the lipid
molecule volume using cavity, channel and cleft volume calculator Voss and
Gerstein (2010), gives the volume estimation of about $894{\AA}^{3}$. Using
this value, one gets

$P\approx\frac{1.99\cdot 10^{-20}}{0.894\cdot 10^{-27}}\approx 2.22\cdot
10^{7}N/m^{2}$ (45)

On other hand, using the same surface tension of a fluid monolayer at the
optimal packing of the lipids, one gets $R=27\pm 0.1{\AA}$. All atom
simulation also generates spherical structure with diameter $54\pm 0.1{\AA}$
(Figure 5). There is still some uncertainty in this estimation because we
assigned 1 kJ/mol energy per $-CH_{2}-$ unit and we based on references data
Larson and McMahon (1984); Emsley (1980), while in other literature it is
mentioned that the hydrophobic interactions are about 4 kJ/mol per $-CH_{2}-$
unit Liu (2011). In our opinion, this discrepancy can be resolved if one
calculates hydrophobic energy based on the potential energy

$U=-\int_{\Omega}\frac{\epsilon_{0}}{2}E_{CH_{2}}^{2}d\Omega$ (46)

where $\vec{E}_{CH_{2}}$ is electric field per $-CH_{2}-$, $\epsilon_{0}$ is
dielectric constant in the vacuum and $\Omega$ stands for the volume of the
lipid molecules. (46) directly emerges from $F_{\mu\nu}F^{\mu\nu}$ term
written in the equations of motion (20-22 and III-29). For electrostatics

$U=\int_{\Omega}\frac{1}{4\mu_{0}}F_{\mu\nu}F^{\mu\nu}d\Omega=-\int_{\Omega}\frac{\epsilon_{0}}{2}E_{CH_{2}}^{2}d\Omega$
(47)

so one should go to the scrutiny of calculating electric field for each
$-CH_{2}-$ units, then take a sum of the electric field and square it (we are
not going to do it in this paper). Also, one may ask why the hydrophilic
interaction energy is not taken into account in these calculations.
Hydrophilic head of the lipid molecule is in contact with water molecules so
there is no work needed to drag it in aqueous solution from the lipids layer.
Therefore, hydrophilic interaction energy can be neglected. The most work goes
on overcoming hydrophobic interactions between lipid tails.

##  V Conclusion

We have presented a framework for the analysis of two dimensional surface
dynamics (identified as micelle in the text) using first law of thermodynamics
and calculus of moving surfaces. In final equations of normal motion (39,27)
we assume that a surface is homogeneous and has time invariable surface
tension. However, the general equations (23-25) don not have these constrains
and indicate arbitrary motion along normal deformation, as well as into
tangent directions, but are analytically more complex. The solution to the
normal equations of motion in equilibrium conditions are surprisingly simple
and display all possible equilibrium shapes. We applied the formalism to
estimate micelle optimal radius and compared estimations to all atom
simulations. Even for low-level approximations, we found remarkable agreement
between theoretically calculated radius and one obtained from atomistic
simulations and from experiments. One can readily apply the theory to any
closed surfaces; such are vesicles, membranes, water droplets or soap films.

As a final remark, even though the analytic solution (40) looks like
generalized Young-Laplace law, the difference is obvious. $B_{i}^{i}$ is a
trace of mixed curvature tensor, known as mean curvature, and when the mean
curvature is constant, it defines whole class of constant mean curvature (CMC)
surfaces. Generalized the Young-Laplace law is a priori formulated for
spherical morphologies and therefore in some particular cases can be obtained
from (40) constant mean curvature shapes. The condition for holding the
particular case is a compactness. However, the compactness argument can be
relaxed in our derivation if the considered system is set to be much larger
than the subsystem. Therefore, the solution (40) effectively predicts
formation of all CMC surfaces while Young-Laplace law is correct for spherical
structures alone. Also in derivation of Young-Laplace relation one of
cornerstone idea is suggestion of spherical symmetries, while our derivation
is free of symmetries and explains why CMC surfaces are such abundant shapes
in nature, observable even on molecular levels. In fact, according to the
results, any homogeneous closed surface with time invariable surface tension
adopts CMC shape when it comes in equilibrium with environment.

## Acknowledgments

We thank Dr. Frank Julicher (MPIPKS) and Dr. Erwin Frey (LMU), for stimulating
discussions. The paper in it’s current form was initiated at Aspen Center for
Physics, which is supported by National Science Foundation grant PHY-1607611
and was partially supported by a grant from the Simons Foundation.

## Appendix: Derivation of Equations of Motions for Closed, Two dimensional
Surfaces

Now we turn to the derivation of (23-25) without using any information from
(20-22) (though derivation of (23-25) from (20-22) is strightforwad and
trivial if one sets $V^{0}=0$ in (20-22 equations Svintradze (2017a)). To
deduce the equations of motion we derive the simplest one from the set (23)
first. It is direct consequence of generalization of conservation of mass law.
Variation of the surface mass density must be so that $dm/dt=0$, where
$m=\int_{S}\rho dS$ is surface mass with $\rho$ surface density. Since the
surface is closed, at the boundary condition $v=n_{i}V^{i}=0$, a pass integral
along any curve $\gamma$ across the surface must vanish ($n_{i}$ is a normal
of the curve and lays in the tangent space). Using Gauss theorem, conservation
of mass and integration formula (18), we find

$\displaystyle 0=$ $\displaystyle\int_{\gamma}v\rho
d\gamma=\int_{\gamma}n_{i}V^{i}\rho d\gamma=\int_{S}\nabla_{i}(\rho V^{i})dS$
$\displaystyle=$ $\displaystyle\int_{S}(\nabla_{i}(\rho V^{i})-\rho
CB_{i}^{i}+\rho CB_{i}^{i})dS$ $\displaystyle=$
$\displaystyle\int_{S}(\nabla_{i}(\rho V^{i})-\rho
CB^{i}_{i})dS+\int_{S}\dot{\nabla}\rho dS-\frac{d}{dt}\int_{S}\rho dS$
$\displaystyle=$ $\displaystyle\int_{S}(\dot{\nabla}\rho+\nabla_{i}(\rho
V^{i})-\rho CB_{i}^{i})dS$ (48)

Since last integral must be identical to zero for any integrand, one
immediately finds first equation from the set (23). To deduce second and third
equations, we take a Lagrangian

$L=\int_{S}\frac{\rho V^{2}}{2}dS+\int_{\Omega}(P^{+}+\Pi)d\Omega$ (49)

and set minimum action principle requesting that $\delta L/\delta t=0$.
Evaluation of space integral is simple and straightforward, using integration
theorem for space integral where the convective and advective terms due to
volume motion is properly taken into account (17), we find

$\frac{\delta}{\delta
t}\int_{\Omega}(P^{+}+\Pi)d\Omega=\int_{\Omega}\partial_{\alpha}(P^{+}+\Pi)V^{\alpha}d\Omega+\int_{S}C(P^{+}+\Pi)dS$
(50)

Derivation for kinetic part is a bit tricky and challenging that is why we do
it last. Straightforward, brute mathematical manipulations, using first
equation from (23), lead

$\displaystyle\frac{\delta}{\delta t}\int_{S}\frac{\rho
V^{2}}{2}dS=\int_{S}(\dot{\nabla}\frac{\rho V^{2}}{2}-CB_{i}^{i}\frac{\rho
V^{2}}{2})dS$
$\displaystyle=\int_{S}(\dot{\nabla}\rho\frac{V^{2}}{2}+\rho\dot{\nabla}\frac{V^{2}}{2}-CB_{i}^{i}\frac{\rho
V^{2}}{2})dS$ $\displaystyle=\int_{S}((\rho CB_{i}^{i}-\nabla_{i}(\rho
V^{i}))\frac{V^{2}}{2}+\rho\dot{\nabla}\frac{V^{2}}{2}-CB_{i}^{i}\frac{\rho
V^{2}}{2})dS$ $\displaystyle=\int_{S}(-\nabla_{i}(\rho
V^{i})\frac{V^{2}}{2}+\rho\dot{\nabla}\frac{V^{2}}{2})dS$
$\displaystyle=\int_{S}(-\nabla_{i}(\rho V^{i}\frac{V^{2}}{2})+\rho
V^{i}\nabla_{i}\frac{V^{2}}{2}+\rho\dot{\nabla}\frac{V^{2}}{2})dS$
$\displaystyle=\int_{S}(-\nabla_{i}(\rho
V^{i}\frac{V^{2}}{2})+\rho\vec{V}(V^{i}\nabla_{i}\vec{V}+\dot{\nabla}\vec{V}))dS$
(51)

At the end point of variations the surface reaches stationary point and
therefore by Gauss theorem (as we used it already in (Appendix: Derivation of
Equations of Motions for Closed, Two dimensional Surfaces)), we find

$\int_{S}-\nabla_{i}(\rho V^{i}\frac{V^{2}}{2})dS=-\int_{\gamma}\rho
V^{i}n_{i}\frac{V^{2}}{2}d\gamma=0$ (52)

$\gamma$ is stationary contour of the surface and $n_{i}$ is the normal to the
contour, therefore interface velocity for contour $v=n_{i}V^{i}=0$ and the
integral (52) vanishes, correspondingly

$\frac{\delta}{\delta t}\int_{S}\frac{\rho
V^{2}}{2}dS=\int_{S}\rho\vec{V}(V^{i}\nabla_{i}\vec{V}+\dot{\nabla}\vec{V})dS$
(53)

To decompose dot product in the integral by normal and tangential components
and, therefore, deduce final equations, we do following algebraic
manipulations

$\displaystyle\dot{\nabla}\vec{V}+V^{i}\nabla_{i}\vec{V}$
$\displaystyle=\dot{\nabla}\vec{V}+V^{i}\nabla_{i}\vec{V}+CV^{i}B_{i}^{j}\vec{S}_{j}-CV^{i}B_{i}^{j}\vec{S}_{j}=\dot{\nabla}\vec{V}+V^{i}\nabla_{i}\vec{V}+CV^{i}B_{i}^{j}X_{j}^{\alpha}\vec{X}_{\alpha}-CV^{i}B_{i}^{j}\vec{S}_{j}$
(54)

Now using Weingarten’s formula
$X_{j}^{\alpha}B_{i}^{j}=-\nabla_{i}N^{\alpha}$, metrinilic property of
Euclidian space base vectors $\nabla_{i}\vec{X}_{\alpha}=0$, definition of
surface normal $\vec{N}=N^{\alpha}\vec{X}_{\alpha}$ and taking into account
definition of surface velocity $\vec{V}=C\vec{N}+V^{i}\vec{S}_{i}$ and its
derivatives, we find

$\displaystyle\dot{\nabla}\vec{V}+V^{i}\nabla_{i}\vec{V}+CV^{i}B_{i}^{j}X_{j}^{\alpha}\vec{X}_{\alpha}-CV^{i}B_{i}^{j}\vec{S}_{j}$
$\displaystyle=\dot{\nabla}\vec{V}+V^{i}\nabla_{i}\vec{V}-CV^{i}\vec{X}_{\alpha}\nabla_{i}N^{\alpha}-CV^{i}B_{i}^{j}\vec{S}_{j}$
$\displaystyle=\dot{\nabla}\vec{V}+V^{i}\nabla_{i}\vec{V}-CV^{i}\nabla_{i}(N^{\alpha}\vec{X}_{\alpha})-CV^{i}B_{i}^{j}\vec{S}_{j}$
$\displaystyle=\dot{\nabla}\vec{V}+V^{i}\nabla_{i}\vec{V}-CV^{i}\nabla_{i}\vec{N}-CV^{i}B_{i}^{j}\vec{S}_{j}$
$\displaystyle=\dot{\nabla}\vec{V}+V^{i}\nabla_{i}(C\vec{N})+V^{i}\nabla_{i}(V^{j}\vec{S}_{j})-CV^{i}\nabla_{i}\vec{N}-CV^{i}B_{i}^{j}\vec{S}_{j}$
$\displaystyle=\dot{\nabla}\vec{V}+V^{i}\vec{N}\nabla_{i}C+V^{i}\nabla_{i}(V^{j}\vec{S}_{j})-CV^{i}B_{i}^{j}\vec{S}_{j}$
$\displaystyle=\dot{\nabla}(C\vec{N})+\dot{\nabla}(V^{j}\vec{S}_{j})+V^{i}\vec{N}\nabla_{i}C+V^{i}\nabla_{i}(V^{j}\vec{S}_{j})-CV^{i}B_{i}^{j}\vec{S}_{j}$
(55)

Continuing algebraic manipulations using Thomas formula
$\dot{\nabla}\vec{N}=-\nabla^{i}C\vec{S}_{i}$, the formula for surface
derivative of interface velocity $\vec{N}\nabla_{i}C=\dot{\nabla}\vec{S}_{i}$
and the definition of curvature tensor (5) yield

$\displaystyle\dot{\nabla}(C\vec{N})+\dot{\nabla}(V^{j}\vec{S}_{j})+V^{i}\vec{N}\nabla_{i}C+V^{i}\nabla_{i}(V^{j}\vec{S}_{j})-CV^{i}B_{i}^{j}\vec{S}_{j}$
$\displaystyle=\dot{\nabla}(C\vec{N})+C\nabla^{j}C\vec{S}_{j}+2V^{i}\vec{N}\nabla_{i}C+V^{i}V^{j}B_{ij}\vec{N}+\dot{\nabla}(V^{j}\vec{S}_{j})$
$\displaystyle-V^{i}\vec{N}\nabla_{i}C+V^{i}\nabla_{i}(V^{j}\vec{S}_{j})-V^{i}V^{j}B_{ij}\vec{N}-C\nabla^{j}C\vec{S}_{j}-CV^{i}B_{i}^{j}\vec{S}_{j}$
$\displaystyle=\dot{\nabla}(C\vec{N})-C\dot{\nabla}\vec{N}+2V^{i}\vec{N}\nabla_{i}C+V^{i}V^{j}B_{ij}\vec{N}+\dot{\nabla}(V^{j}\vec{S}_{j})-V^{j}\dot{\nabla}\vec{S}_{j}$
$\displaystyle+V^{i}\nabla_{i}(V^{j}\vec{S}_{j})-V^{i}V^{j}\nabla_{i}\vec{S}_{j}-C\nabla^{j}C\vec{S}_{j}-CV^{i}B^{j}_{i}\vec{S}_{j}$
$\displaystyle=(\dot{\nabla}C+2V^{i}\nabla_{i}C+V^{i}V^{j}B_{ij})\vec{N}+(\dot{\nabla}V^{j}+V^{i}\nabla_{i}V^{j}-C\nabla^{j}C-CV^{i}B_{i}^{j})\vec{S}_{j}$
(56)

Taking dot product of (Appendix: Derivation of Equations of Motions for
Closed, Two dimensional Surfaces) on $\vec{V}$ and combining it with (53) last
derivation finally reveals variation of kinetic energy, so that we finally
find

$\displaystyle\frac{\delta}{\delta t}\int_{S}\frac{\rho
V^{2}}{2}dS=\int_{S}(\rho
C(\dot{\nabla}C+2V^{i}\nabla_{i}C+V^{i}V^{j}B_{ij})+\rho
V_{i}(\dot{\nabla}V^{i}+V^{j}\nabla_{j}V^{i}-C\nabla^{i}C-CV^{j}B_{j}^{i}))dS$
(57)

Combining (Appendix: Derivation of Equations of Motions for Closed, Two
dimensional Surfaces-50) and (57) together and taking into account that the
pressure acts on the surface along the surface normal, we immediately find
first (23) and the last equation (25) of the set. To clarify second equation
(24), we have

$\displaystyle\int_{S}\rho
C(\dot{\nabla}C+2V^{i}\nabla_{i}C+V^{i}V^{j}B_{ij})dS$
$\displaystyle=\int_{\Omega}-\partial_{\alpha}(P^{+}+\Pi)V^{\alpha}d\Omega-\int_{S}C(P^{+}+\Pi)dS$
$\displaystyle\int_{S}C(\rho(\dot{\nabla}C+2V^{i}\nabla_{i}C+V^{i}V^{j}B_{ij})+P^{+}+\Pi)dS$
$\displaystyle=\int_{\Omega}-\partial_{\alpha}(P^{+}+\Pi)V^{\alpha}d\Omega$
(58)

After applying Gauss theorem to the second equation (58), the surface integral
is converted to space integral so that we finally find

$\displaystyle\partial_{\alpha}(\rho
V^{\alpha}(\dot{\nabla}C+2V^{i}\nabla_{i}C+V^{i}V^{j}B_{ij})+(P^{+}+\Pi)V^{\alpha})=-\partial_{\alpha}(P^{+}+\Pi)V^{\alpha}$
(59)

and, therefore, all three equations (23-25) are rigorously clarified.

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