# Mechanics of Lipid Membranes under the Influence of Intramembrane Viscosity

Authors: Chun Il Kim; Zhe Liu
Journal: Mathematical Problems in Engineering
 (2019)

---

## Abstract

We discuss a continuum-based model describing the deformations of lipid membranes subjected to intramembrane viscosity. Within the frame work of the theory of an elastic surface, the membrane equilibrium equations and the expressions of viscous stress are obtained. The corresponding deformation energy of the membrane is computed via the first and second fundamental form of surface. A compatible linear model is also formulated within the prescription of superposed incremental deformations through which the deformation profiles of the membrane is obtained. It is shown that the intramembrane viscous flow gives rise to straining effects on the membranes. Further, the corresponding dynamic edge conditions reduce to purely elastic boundary conditions in the limit of vanishing viscous effects. Lastly, admissible sets of velocity fields are also examined and are used to formulate membrane shape equations and the associated dynamic boundary conditions.


---

## Body

## 1. Introduction

The study of mechanical responses of lipid membranes is of great importance in the understanding of the essential cellular functions such as fission, fusion, and budding [1–3]. Lipid bilayers are composed of transversely oriented lipid molecules containing hydrophilic head groups and hydrophobic tails. These molecules arrange themselves into a two-layered sheet (a bilayer structure) with opposing orientations that effectively shield the tail groups from the surrounding aqueous solution (Gorter and Grendel [4]). Later, in 1959, David Robertson ([5]) identified that the bilayer structure is characteristic of all biological membranes (biomembranes). The microstructures (lipid molecules) of lipid membranes are unique in that they are aligned to the normal direction of the membrane prior to the deformations. In the continuum-based modeling and analysis, the distinct configuration of individual lipid molecules is accounted for by employing a director filed “d” on a material point of the membrane surface. Within this prescription together with the symmetric array of lipid molecules about the mild plane, lipid membranes can be regarded as a simple fluid membrane where its free energy is expressed as a function of Mean and Gaussian curvatures via the first (aαβ) and second (bαβ) fundamental forms. In particular, the letter requires the computation of the second gradient of deformations (strain gradient). The concept has been widely and successfully adopted in the relevant subject of problems where the microstructures of materials dominate the general mechanical responses (see, for example, [6–9]).Within this context, much theoretical work has been developed based on Helfrich potential [10] which leads to the systems of Partial Differential Equations (PDEs) via the utilization of the virtual work statement of liquid crystal [11, 12]. The numerical solutions of PDEs, often referred to as membrane shape equation, can be uniquely determined by imposing admissible sets of boundary conditions (Neumann, Dirichlet, and/or mixed) and more importantly exhibit ensuing smooth and stable behaviors with the domain of interest. This is benefited by the positive definiteness of the Helfrich potential and the uniqueness theorem of an elastic membrane dictated by Ericksen and Toupin [13]. Facilitated by the abovementioned results, a wide range of problems has been studied such as budding formations [14, 15], thickness distensions [16], and membrane-substrate interactions [17, 18]. Authors in [19] discussed the role of Lagrange-multiplier fields in the shape equations and demonstrate that there exist particular sets of Lagrange multiplier which produces a unique converged solution. Further, the mathematical perspective of shape equations and associated boundary conditions are discussed in [20].However, little has been devoted to the deformation analysis of lipid membranes subjected to intramembrane viscosity [21, 22]. Authors in [21] present nonlinear models which accommodate the effects of intramembrane viscous flow and studied the relations between the membrane deformations and viscous flows. A particular set of viscous flow fields is examined in [22] within the approximated setting of the corresponding deformations. A limited contribution has been made for the more general class of viscous flows and the assimilations of the compatible boundary conditions. Further, the corresponding analyses are heavily omitted hindering researchers to investigate relevant subjects of problems.In this study, we present a complete analysis for the deformations of lipid membranes subjected to intramembrane viscosity. By employing the theory of an elastic surface, the membrane equilibrium equations are obtained and are used to determine the deformation profiles of the membrane. The deformation energy of the membrane is accounted for by means of the Mean and Gaussian curvatures of the surface which are the function of the first and second fundamental form [23]. Further, the viscous stresses are computed via the material time derivatives of a position vector leading to the expression of the time rate of the surface matric (the first fundamental form) [21, 22]. In particular, we formulate a viable, mathematically tractable, linear prediction model within the prescription of superposed incremental deformations, while maintaining the rigor and sufficient generality in the corresponding analyses. The associated edge conditions and admissible sets of viscous flow fields are also examined. A local straining of the membrane is observed due to the presence of intramembrane viscous flow. We also demonstrate that the dynamic edge condition reduces to those in the case of purely elastic surfaces in the limit of vanishing intramembrane viscous flows (i.e., constant and/or zero velocity fields over the domain of interest). Furthermore, we examined the admissible sets of nonlinear periodic distributions of velocity fields and present associated formulations in the cases of rectangular and circular portions of the membranes. It is observed that symmetric velocity field results in vanishing tangential (shear) forces at the boundary while a nonzero shear force is induced in the case of nonsymmetric viscous flow. The obtained results can be employed in the deformation analysis of the membranes subjected to various types of viscous flows knowing the fact that any real valued functions can be approximated using periodic series expansions (e.g., Fourier expansions).Throughout the paper, we make use of a number of well-established symbols and conventions. Thus, unless otherwise stated, Greek indices take the values1 and 2. Einstein summation is applied for the repeated indices.


## 2. Inviscid Fluid Membranes

In the present study, we adopt the frame work of differential geometry [24] in the deformation analysis of fluid membrane. Let w represent two-dimensional surface, parameterized by two embedded surface coordinate (θ1,θ2), such that the parametric position vector r∈R3 of a point on the surface is defined by the mapping r=r(θα). The local surface coordination is then given by n(θα)=1/2εαβaα×aβ where n is a unit normal to the deformed surface w and aα and aβ are associated tangent vectors defined by r,α(θα,t)=∂r/∂θα=aα; and eαβ/a refers to the two-dimensional permutation tensor with a=det(aαβ). Thus, for example, we evaluate eαβ as e11=e22=0, e12=-e21=1. Here and henceforth, Greek indices take values of 1 and 2 and Einstein summation is applied for repeated indices. The Levi-Civita contraction of the surface metric aαβ is defined by aαβ=aα·aβ, where the dot stands for the Euclidean inner product. The matrix aαβ of the surface metric is a positive-definite, with a>0, leading to the existence of dual metric aαβ which is the inverse of the surface metric (i.e., aαβ=(aαβ)-1), if a, the inverse of the surface metric aαβ, exists (i.e., aαβ=(aαβ)-1) and is referred to as the dual matrix components of aαβ. In particular, a semicolon denotes surface covariant differentiation defined by [24] (1)aα;β=aα,β-Γαβλaλ,whereΓαβλ=aα,β·aλ are the Christoffel symbols induced by the local surface coordinate on w and aλ is the dual basis, defined as (2)aα=aαβaβ.Equations (1) and (2) furnish well know Gauss-Weigard equation:(3)aα,β=aα,β·aγaγ+aα,β·nn=Γαβλaγ+bαβn;bαβ≡aα,β·nGauss,(4)n,α=n,α·aβaβ+n,α·nn=-bαβaβ;bαβ≡-n,α·aβWeingarten,where bαβ are the coefficient of the second fundamental form (curvature) and their symmetric properties can be seen as (5)bαβ=-n,α·aβ=aβ,α·n=bβα.Also, the contravariant cofactor of the curvature tensor is given by [24] (6)b~αβ=εαλεβγbλγwhich is again symmetric (i.e., b~αβ=b~βα) from the result of (5). We note here that (5) is valid for the sufficient smoothness of r up to the second-order (i.e., r∈C). Also, substituting (3) into (1) yields (7)aα;β=bαβn.Within this prescription, the equilibrium state of a purely elastic surface, subjected to normal pressure p, is given by [23](8)T;αα+pn=0,where Tα and n are thestress vectors and the local surface unit normal, respectively. Further, the associated deformation energy of the surfaces can be expressed by the two primary parameters: the coefficient of the first fundamental form aαβ (the surface metric) and the second fundamental form bαβ (the curvature). In the case of fluid films whose energy density per unit mass of the surface is F=F(aαβ,bαβ), Tα takes the following compact form [23]:(9)Tα=σβα+bμβMμαaβ-M;βαβn,and (10)σβα=ρ∂F∂aαβ+∂F∂aβαandMβα=ρ2∂F∂bαβ+∂F∂bβα.In a typical environment, where fluid films are subjected to morphological transitions, the reference velocity of the system is low and therefore the corresponding Reynolds numbers are sufficiently small [25, 26]. Further, it is widely accepted that lipid membranes are relatively stiff against areal dilation in comparison with bending or shear motions [27, 28]. This implies that deformations preserve surface area. In other words, fluid films are subjected to the two-dimensional incompressibility J=1 which can be incorporated by replacing F as [23] (11)F=F¯-γρ,where F¯,ρ, and γ are, respectively, a constitutive function, mass density, and the Lagrange-multiplier field associated with surface pressure;(12)γ=ρ2Fρ.The energy density per unit area of the surface w is then defined by [21](13)WH,K,ρ;aαβ,bαβ=ρF¯,where the expressions of Mean and Gaussian curvatures of surfaces (i.e., H and K) are given explicitly as [24] (14)H=12aαβbαβandK=12εαβελμbαλbβμ.In particular, K satisfies (15)bμαb~μβ=Kaαβ,which can be easily seen from (6). Therefore, we find F=1/ρ(W-γ) and subsequently obtain from (10) that (16)σβα=ρ∂1/ρW-γ∂aαβ+∂1/ρW-γ∂aβαandMβα=ρ2∂1/ρW-γ∂bαβ+∂1/ρW-γ∂bβα.Using chain rules, we evaluate (17)ρ∂1/ρW-γ∂aαβ=ρ-ρ-2∂ρ∂aαβW+1ρWρ∂ρ∂aαβ+WH∂H∂aαβ+WK∂K∂aαβ+ρ-2∂ρ∂aαβγ-1ρ∂γ∂ρ∂ρ∂aαβ,where WH=∂W/∂H and similarly for other terms. The derivatives of ρ, H and K with respect to aαβ can be evaluated as [23](18)∂ρ∂aαβ=-ρ2aαβ,∂H∂aαβ=-12bαβand∂K∂aαβ=-Kaαβ,Accordingly, (17) reduces to (e.g., -ρ-2∂ρ/∂aαβW=-ρ-2-ρ/2aαβW, etc.) (19)ρ∂1/ρW-γ∂aαβ=ρ12ρaαβW-12aαβWρ-12ρbαβWH-1ρWKKaαβ-12ρaαβγ+γρ12aαβ.Also from (12) and (13), Fρ can be computed as Fρ=(W/ρ-γ/ρ),ρ=Wρ/ρ-W/ρ2-γρ/ρ+γ/ρ2, and thus it yields (20)Wρ=ρFρ+Wρ+γρ-γρSubstituting (20) into (19) furnishes(21)ρ∂1/ρW-γ∂aαβ=-12bαβWH-WKKaαβ-12aαβρ2Fρ=-12γaαβ-WHH+WKKaαβ+12WHb~αβ,In the above we make use of the identity bαβ=2Haαβ-b~αβ from the Cayley-Hamilton theorem and ρ2Fρ=γ (see (12)). Thus, from (21) and the symmetry of aαβ and b~αβ, (10)1 becomes (22)σβα=λ+Waαβ-2WHH+2WKKaαβ+WHb~αβ,where(23)λ=-γ+W.Now, we evaluate ρ∂1/ρW-γ/∂bαβ as(24)ρ∂1/ρW-γ∂bαβ=ρ-ρ-2∂ρ∂bαβW+1ρWρ∂ρ∂bαβ+WH∂H∂bαβ+WK∂K∂bαβ+ρ-2∂ρ∂bαβγ-1ρ∂γ∂ρ∂ρ∂bαβ.Since, (25)∂ρ∂bαβ=0,∂H∂bαβ=12aαβand∂K∂bαβ=b~αβ,Equation (24) becomes (e.g., WH∂H/∂bαβ=WH1/2aαβ)(26)∂1/ρW-γ∂bαβ=12WHaαβ+WKb~αβ.Invoking (26) and (10)2 furnishes (27)Mβα=12WHaαβ+WKb~αβ,where bαβ=bβα from (5).Further, by combining (8) and (9), we find(28)σβα+bμβMμα;αaβ+σβα+bμβMμαaβ;α-M;βαβ;αn-M;βαβn,α+pn=0.Lastly, we project (28) into normal direction by applying dot product of unit normal n and subsequently obtain(29)σβα+bμβMμα;αaβ·n+σβα+bμβMμαaβ;α·n-M;βαβ;αn·n-M;βαβn,α·n+pn·n=0.But, since aβ·n=0, (n·n),α=0, and aα;β=bαβn (see (7)), the above reduces to (30)σβα+bμβMμαbβα-M;βαβ;α+p=0.Similarly, for tangential projection (e.g., aβ·aα=δβα, aβ;α·aα=bαβn·aα=0, etc.), we have(31)σβα+bμβMμα;α+M;μαμbαβ=0.where n,α·aβ=-bαγaγ·aβ=-bαγaγβ=-bαβ from Weingarten equation.


## 3. Viscous Fluid Membranes

The straining effects of the fluid attribute viscous stresses which can be incorporated via the formulation of the time derivative on the evolving surface matric [29, 30]. The expression of the corresponding stress is given by [31] (32)σαβ=-γaαβ+νaαλaβμa˙λμ,where ν is the intramembrane shear viscosity and the superscript dot (∗˙) refers to the time derivative. Since aαβ=aα·aβ, we find that (33)a˙λμ=(aλ·aμ)˙=a˙λ·aμ+aλ·a˙μ.In order to compute a˙λμ, we require to evaluate a˙λ=(∂r/∂θλ)˙; the velocity of a material point on initial surface Ω with respect to the referential frame (Lagrangian). Here we conveniently denotes this via the convected coordinate ξα such that the referential position vector x(ξα)∈Ω. While ξα is fixed coordinate, the material coordinate θα constantly moves on the current surface w over time interval t (see Figure 1). They occupy the same point in the space at the particular time instant, say t=t0, and thus x(ξα)=r(ξα,t0). However, at the instant moment later they are displaced (i.e., θα≠ξα at t0+t) and so do x and r. Thus we observe that the surface coordinate θα is a function of ξα and t which satisfies θα(ξα,t0)=ξα [29–31]. Accordingly, the corresponding velocity field can be computed via the material time derivative of a position vector as(34)u=r˙=∂r∂t+∂r∂θα∂θα∂t=rt+aαvα.In the above, vα=∂θα/∂t are the components of the tangential velocity of a material point. While vα are well defined, the physical implication of rt remains to be clarified. To see this, we make use of normal and tangential decomposition of u as(35)u=u·aαaα+u·nn=aαvα+wn,where u·aα=(∂r/∂t)·(∂θα/∂r)=∂θα/∂t=vα(θβ,t). Comparing (34) and (35), we obtain(36)rt=wn,and thereby we observe that rt is the normal velocity of a material point with magnitude rt=w.Figure 1
Material point on the constantly moving surface.Now, we evaluatea˙μ in terms of convected coordinates(37)a˙μ=(∂rθβξλ,t∂θμ)˙=(∂r∂ξλ∂ξλ∂θμ)˙=(∂r∂ξλ)˙∂ξλ∂θμ+∂r∂ξλ(∂ξλ∂θμ)˙=(∂r∂ξλ)˙∂ξλ∂θμ+∂r∂ξλ∂2ξλ∂θμ∂t.Since θαt0=ξα and ξλ are fixed coordinates (i.e., ξ˙λ=0), it is now trivial to show that (38)(∂r∂ξλ)˙=∂r˙∂ξλ=u,λand∂ξλ∂θμt0=δμλ.Therefore, (38) reduces to (39)a˙μ=u,μ,where u,λδμλ=u,μ and ∂δμλ/∂t=0. In other words u,μ=∂u/∂θμ, at the considered instant (t=t0).In addition, from (35), we compute (40)u,λ=vαaα+wn,λ=v,λαaα+vαaα,λ+w,λn+wn,λ=vα;λ-wbαλaα+vαbαλ+w,λn,where n,α=-bαβaβ from (4) and vα;β=vα,β-vβΓαβλ.Now, from (33), (39), and (40), we obtain (41)a˙λμ=vα;λ-wbαλaα+vαbαλ+w,λn·aμ+aλ·vα;μ-wbαμaα+vαbαμ+w,μn=vμ;λ+vλ;μ-2wbλμ,∵bλμ=bμλ.Substituting the above into (32) yields(42)σαβ=-γaαβ+νaαλaβμvμ;λ+vλ;μ-2wbλμ.Since aαλaβμbλμ=bαβ and bαβ=2Haαβ-b~αβ (Cayley-Hamilton identity), (42) can be rewritten as (43)σαβ=-γaαβ+νaβλaαμvμ;λ+vλ;μ-4wHaβα+2wb~βα.Therefore, we obtain the expression of the stress in the case of viscous fluid membranes by combining (22), (23), and (43) that (44)σβα=λ+Waβα-2WHH+WKKaβα+WHb~βα+νaβλaαμvμ;λ+vλ;μ-4wHaβα+2wb~βα.From (27), it is not difficult to show (45)M;βαβ;α=12WH;βaβα+WK;βb~βα;α,and (46)bμβMμα=12WHaαμbμβ+WKb~αμbμβ=WHHaβα-12WHb~βα+WKKaβα.While the second term in the above is relatively straight forward (i.e., b~αμbμβ=Kaβα), the first term is not obvious which can be seen via the Cayley-Hamilton identity as (47)12WHaαμbμβ=12WHaαμaλμbβλ=12WHaαμaλμ2Haβλ-b~βλ=12WHδλα2Haβλ-b~βλ=12WH2Haβα-b~βα.Thus, the equation of motion for the viscous fluid membrane can be obtained from (30) and (45)-(47) that (48)p=-λ+Waβαbβα+WHH+WKKaβαbβα-12WHb~βαbβα+12WH;βaβα+WK;βb~βα;α-νaβλaαμvμ;λ+vλ;μ-4wHaβα+2wb~βαbβα.Now, in view of (14), (48) can be rewritten as (49)p=-λ+W2H+WHH+WKK2H-WHK+12WH;βaβα+WK;βb~βα;α-νbλμvμ;λ+vλ;μ-4wH2H+2w2K,where we compute b~βαbβα using (15); (50)b~βαbβα=b~βαbβλaλα=Kaαλaλα=Kδαα=2K.By interchanging indices (i.e., λ↔α and μ↔β) and evaluating covariant derivatives, we reduce (49) to(51)p=WH2H2-K+2HWKK-W-2λH+Δ12WH+WK;αβb~αβ-2ν12bαβvα;β+vβ;α-2w2H2-K,where Δ is the Laplace-Beltrami operator (i.e., Δϕ=ϕ;αβaαβ) on the surface ω.In order to obtain tangential equations of motion, we compute(52)σβα+bμβMμα;α=λ,α+W,aaβα-WH;αH+WHH,α+WK;αK+WKK,αaβα+12WH;αb~βα+νaβλaαμvμ;λ+vλ;μ;α-4w,αHaβα-4wH,αaβα+2w,ab~βα,In the above, we use the fact that the covariant derivative of scalar fields is a scalar derivative (i.e.,λ;α=λ,α, W;α=W,α, etc.). Applying the Cayley-Hamilton identity, we find (53)b~αβ=2Haαβ-bαβand thereby reduce (52) to (54)σβα+bμβMμα;α=λ,α+W,aaβα-WH;αH+WHH,α+WK;αK+WKK,αaβα+12WH;αb~βα+νaβλaαμvμ;λ+vλ;μ;α-4wH,αaβα+2w,abβα.Further, combining (45) and (46) yields(55)M;μαμbαβ=WH;αHaβα-12WH;αb~βα+WK;αKaβα+WKK,αaβα.By substituting (54) and (55) into (31), we obtain (56)-γ,α+WKK,α+WHH,αaβα-4νwH,αaβα+2ν12aβλaαμvμ;λ+vλ;μ;α-w,αbβα=0.The above further reduces to (57)aβαλ,α-4vwH,α+2ν12aλμvμ;α+vα;μ;λ-w,λbαλ=0,where bβα=bλαaλβ and γ,α=-λ,α-WKK,α-WHH,α. Since aβα≠0 (57) furnishes the following tangential equilibrium equation:(58)λ,α-4vwH,α+2ν12aλμvμ;α+vα;μ;λ-w,λbαλ=0.For the implementation of the obtained models, we consider the Helfrich type of membranes where the corresponding energy density function is given by [10](59)W=kH2+k-K,where k and k- are empirical constants related to the bending stiffness of membranes. Thus, we evaluate WH=2kH and WK=k- and obtain from (51) that (60)p=kΔH+2HH2-K-2λH-2ν12bαβvα;β+vβ;α-2w2H2-K.Similarly, the viscous stress (see (44)) becomes(61)σβα=-γaβα-22kH2+k-Kaβα+2kHb~βα+νaβλaαμvμ;λ+vλ;μ-4wHaβα+2wb~βα.Finally, the condition of an incompressible fluid membrane requires J˙/J=1/2aαβa˙αβ=0 [32]. Therefore, from (41), we formulate (62)12aαβa˙αβ=12aαβvα;β+vβ;α-2wbαβ=v;αα-2wH=0,∵H=12aαβbαβ.


## 4. Linear Theory within the Description of Monge Parameterization

The deformed configurations of lipid membrane subjected to intramembrane viscous flow can be completely determined by the membrane shape equation together with incompressibility conditions (i.e., (60) and (62)). This involves the implementation of the corresponding equations into a desired coordinate system which results in a highly nonlinear nature of Partial Differential Equations (PDEs). The solutions of the resulting PDEs can be accommodated via numerical schemes (e.g., Finite Element Methods). However, the procedures often require massive computational resources. In this section, we present viable alternatives via the utilization of Monge parameterization of surfaces and the methods of “admissible linearization” in an effort to obtain mathematically tractable systems with minimum loss of generality.


### 4.1. Monge Surface Parameterization

Within the description of Monge parameterization, the material points on the membrane surfacew can be identified as (63)rθα,t=θ+zθ,tk,where k is the unit normal to the surface and θ is material coordinate defined on the surface. z(θ,t) is scalar-valued function describing the elevation of the bilayer membrane midplane. For example, θ=θαeα in orthonormal Cartesian basis and, unless otherwise specified, the subscripts of the surface coordinates are dropped and replaced by 1=x, 2=y. Thus, we evaluate(64)r,t=z,tk,aα=∂r∂θα=eα+z,αk,a=det⁡aαβ=1+z,12+z,22,aαβ=δαβ+z,αz,β,H=1+z,22z,11+1+z,12z,22-2z,1z,2z,122a3/2,K=z,11z,22-z,122a2andn=k-∇za,where δαβ Kronecker delta and ∇z=z,αeα is the surface gradient. In particular, the expressions of curvature tensor can be computed as (65)b=bαβaα⊗aβ=5z,αβaaα⊗aβwith aα being dual basis defined by (see (2))(66)a1=1a1+z,22e1+z,1k-z,1z,2e2+z,2k,anda2=1a1+z,12e2+z,2k-z,1z,2e1+z,1k.Lastly, we compute the normal velocity and the Christoffel symbols as(67)w=wn·n=r,t·n=z,tk·k-∇za=z,ta,andΓαβλ=z,λz,αβa,where from (34) r,t=wn.


### 4.2. Superposed Incremental Deformations of Lipid Membranes

The concept of the superposed incremental deformations has been successfully adopted in a number of relevant problems through which one could expect to obtain explicit expressions describing the deformations of the membranes. In the present context, this means that the derivatives ofz(θ,t) of all orders are considered to be “small” (e.g., z,α≪1.), and thus their products can be neglected (e.g., z,αz,β≃0). Accordingly, we compute (68)a=1+z,12+z,22≃1,aαβ=δαβ+z,αz,β≃δαβ,a1=1a1+z,22e1+z,1k-z,1z,2e2+z,2k≃e1+z,1k,anda2≃e2+z,2kand thereby find (69)aα≃aα=eα+z,αkandaαβ=aα·aβ≃aα·aβ=aαβ=δαβ.Further, by invoking (68) and (69), (64), (65), (66), and (67) can be approximated as(70)H=1+z,22z,11+1+z,12z,22-2z,1z,2z,122a3/2≃12z,11+z,22=12Δpz,K=z,11z,22-z,122a2≃0,w=z,ta≃z,t,andΓαβλ=z,λz,αβa≃0.In particular, (65) reduces to(71)b=z,αβaaα⊗aβ≃69z,αβaα⊗aβ=z,αβeα+z,αk⊗eβ+z,βk≃z,αβeα⊗eβ=∇p2z.Here, the subscript (∗)p refers to the projected counterparts of (∗) on the reference plane ωp and ∇p2z is the second gradient of z which satisfies Δpz=tr(∇p2z) (the surface Laplacian).Using the obtained results through (68)-(71), we now approximate the straining (41), viscous stress (61), and incompressibility condition (62), respectively, as(72)a˙αβ=vα;β+vβ;α-2wbαβ=vα,β-vλΓαβλ+vβ,α-vλΓβαλ-2z,taz,αβa≃vα,β+vβ,α,(73)σβα=-γaβα-22kH2+k-Kaβα+2kHb~βα+νaβλaαμvμ;λ+vλ;μ-4wHaβα+2wb~βα≃-γδβα+νvβ,α+vα,β,and(74)v;αα-2wH=vβ,αaαβ+vλΓαλα-2z,taz,αβa≃vβ,αδαβ=vα,α=0.Lastly, the normal and tangential equations of motion (see (58) and (60)) can be reduced to (e.g., ≃1/2Δpz, vα;β≃vα,β, etc.)(75)12kΔpΔpz-λΔpz-νz,αβvα,β+vβ,α≃p,andλ,α+vvλ,α+vα,λ,λ≃0.The above may be recast as(76)12kΔpΔpz-λΔpz-2νz,αβvα,β≃pandλ,α+vΔpvα≃0,where we applied the following identities (e.g., z,αβ=z,βα, vλ,αλ=vλ,λα...)(77)z,αβvα,β+vβ,α=z,αβvα,β+z,βαvβ,α=2z,αβvα,β,and (78)vvλ,α+vα,λ,λ=vvλ,λα+vα,λλ=vΔpvα∵vλ,λα=vλ,λ,α=740.


## 4.1. Monge Surface Parameterization

Within the description of Monge parameterization, the material points on the membrane surfacew can be identified as (63)rθα,t=θ+zθ,tk,where k is the unit normal to the surface and θ is material coordinate defined on the surface. z(θ,t) is scalar-valued function describing the elevation of the bilayer membrane midplane. For example, θ=θαeα in orthonormal Cartesian basis and, unless otherwise specified, the subscripts of the surface coordinates are dropped and replaced by 1=x, 2=y. Thus, we evaluate(64)r,t=z,tk,aα=∂r∂θα=eα+z,αk,a=det⁡aαβ=1+z,12+z,22,aαβ=δαβ+z,αz,β,H=1+z,22z,11+1+z,12z,22-2z,1z,2z,122a3/2,K=z,11z,22-z,122a2andn=k-∇za,where δαβ Kronecker delta and ∇z=z,αeα is the surface gradient. In particular, the expressions of curvature tensor can be computed as (65)b=bαβaα⊗aβ=5z,αβaaα⊗aβwith aα being dual basis defined by (see (2))(66)a1=1a1+z,22e1+z,1k-z,1z,2e2+z,2k,anda2=1a1+z,12e2+z,2k-z,1z,2e1+z,1k.Lastly, we compute the normal velocity and the Christoffel symbols as(67)w=wn·n=r,t·n=z,tk·k-∇za=z,ta,andΓαβλ=z,λz,αβa,where from (34) r,t=wn.


## 4.2. Superposed Incremental Deformations of Lipid Membranes

The concept of the superposed incremental deformations has been successfully adopted in a number of relevant problems through which one could expect to obtain explicit expressions describing the deformations of the membranes. In the present context, this means that the derivatives ofz(θ,t) of all orders are considered to be “small” (e.g., z,α≪1.), and thus their products can be neglected (e.g., z,αz,β≃0). Accordingly, we compute (68)a=1+z,12+z,22≃1,aαβ=δαβ+z,αz,β≃δαβ,a1=1a1+z,22e1+z,1k-z,1z,2e2+z,2k≃e1+z,1k,anda2≃e2+z,2kand thereby find (69)aα≃aα=eα+z,αkandaαβ=aα·aβ≃aα·aβ=aαβ=δαβ.Further, by invoking (68) and (69), (64), (65), (66), and (67) can be approximated as(70)H=1+z,22z,11+1+z,12z,22-2z,1z,2z,122a3/2≃12z,11+z,22=12Δpz,K=z,11z,22-z,122a2≃0,w=z,ta≃z,t,andΓαβλ=z,λz,αβa≃0.In particular, (65) reduces to(71)b=z,αβaaα⊗aβ≃69z,αβaα⊗aβ=z,αβeα+z,αk⊗eβ+z,βk≃z,αβeα⊗eβ=∇p2z.Here, the subscript (∗)p refers to the projected counterparts of (∗) on the reference plane ωp and ∇p2z is the second gradient of z which satisfies Δpz=tr(∇p2z) (the surface Laplacian).Using the obtained results through (68)-(71), we now approximate the straining (41), viscous stress (61), and incompressibility condition (62), respectively, as(72)a˙αβ=vα;β+vβ;α-2wbαβ=vα,β-vλΓαβλ+vβ,α-vλΓβαλ-2z,taz,αβa≃vα,β+vβ,α,(73)σβα=-γaβα-22kH2+k-Kaβα+2kHb~βα+νaβλaαμvμ;λ+vλ;μ-4wHaβα+2wb~βα≃-γδβα+νvβ,α+vα,β,and(74)v;αα-2wH=vβ,αaαβ+vλΓαλα-2z,taz,αβa≃vβ,αδαβ=vα,α=0.Lastly, the normal and tangential equations of motion (see (58) and (60)) can be reduced to (e.g., ≃1/2Δpz, vα;β≃vα,β, etc.)(75)12kΔpΔpz-λΔpz-νz,αβvα,β+vβ,α≃p,andλ,α+vvλ,α+vα,λ,λ≃0.The above may be recast as(76)12kΔpΔpz-λΔpz-2νz,αβvα,β≃pandλ,α+vΔpvα≃0,where we applied the following identities (e.g., z,αβ=z,βα, vλ,αλ=vλ,λα...)(77)z,αβvα,β+vβ,α=z,αβvα,β+z,βαvβ,α=2z,αβvα,β,and (78)vvλ,α+vα,λ,λ=vvλ,λα+vα,λλ=vΔpvα∵vλ,λα=vλ,λ,α=740.


## 5. Admissible Boundary Conditions and Linear Approximations

The admissible sets of boundary conditions for the membranes subjected to intramembrane viscous flows are discussed in the works of [17, 21]. For the uniform Helfrich type of membranes (i.e., W=kH2+k-K), these are given by [17](79)M=12kκν+κτ+k-τ,fn=k-τ′-kH,νfν=14kκν2-κτ2-k-τ2+λ+νaβλaαμvμ;λ+vλ;μ-4wHaβα+2wb~βανβνα,fτ=νaβλaαμvμ;λ+vλ;μ-4wHaβα+2wb~βανβνα-12kτκν-12k+k-τκτ,where v and τ are the magnitudes of unit tangent and normal to the parametric curves on w (see Figure 2). The expression of τ can be obtained by taking arc length derivative of r(s) with respect to s as (80)τ=drsds,s:archlengthparameteronw.Together, they form a local orthonormal basis (i.e., v×τ=n). Hence, the normal curvatures in the directions of ν and τ yield (81)κν=v·bv,κτ=τ·bτandτ=τ·bvsee,also,Figure2.We now introduce an orthonormal basis on the projected plane such that (82)vp×τp=k,where τp is the unit tangent to the projected curve defined by (83)τp=dθdS,S:archlengthparameteronwp,and k is the unit normal to the plane ωp. We note that, here and henceforth, the subscript (∗)p refers to the projected counterparts of (∗) on the projected plane ωp. Up to the leading order approximation, the two coordinates are related by (see, for example, [19, 20]) (84)v≃vp+∇pz×τp,τ≃τp+τp·∇pzk,(85)να=aα·v≃ναp+eαβz,βandτα=aα·τ≃ταp,where (να)p=eα·vp and (να)p=eα·τp and eαβ is the unit alternator defined by e12=-e21=1 and e11=e22=0. Thus we find from (71), (81), and (84) that(86)κν=v·bv≃vp+∇pz×τp·∇p2zvp+∇pz×τp≃vp·∇p2zvp,(87)κτ=τ·bτ≃τp+τp·∇pzk·∇p2zτp+τp·∇pzk≃τp·∇p2zτp,and(88)τ=τ·bv≃τp+τp·∇pzk·∇p2zvp+∇pz×τp≃τp·∇p2zvp.Consequently, the edges conditions (see (79)) reduce to (89)M≃12kΔpz+k-τp·∇p2zτp,fν≃λ+νvα,β+vβ,ανβνα,(90)fτ≃νvα,β+vβ,ατβνα,andfn≃k¯∇pτ·τp-kνp·∇pH,where(91)νανβ≃νανβp+eαλz,λνβp+eβλz,λναpandτβνα≃τβναp+eαλz,λτβ.Figure 2
Local coordinatev-τ-n and associated normal curvatures κν and κτ.


### 5.1. Rectangular Portion of Membranes under the Effects of Viscous Flow

We now consider the membrane flows over a rectangular portion of the plane (see Figure3). During such motion, lipid molecules constantly flow in through the boundary, while the boundary remains clamped. The compatible boundary conditions are (92)z=0andn=konw.In view of (64), the latter condition requires n=(k-∇z)/a=k, and thus the second of (92) leads to(93)∇z=z,αaα=0→z,α=0,∵aα≠0,on the boundary. Further, (93) is equivalent to (94)∇z=z,vv+z,ττ=0→z,v=z,τ=0,which can be seen by using chain rule (e.g., z,v=∂z/∂v=(∂z/∂θα)(∂θα/∂v)=z,αva=0). Therefore, (89)1 reduces to (95)M≃12kΔpz.The imposition of kinetic conditions is completely arbitrary depending on the desired applications and yet must satisfy the condition of incompressibility (vα,α=0). For example, one could propose the following velocity potential: (96)Vx,y=Ax+y,where A is constant real number. We then evaluate (97)V,x=vx=A,andV,y=vy=A.Equation (97) satisfies vα,α=0, since both vx,x and vy,y are individually nulled out and therefore, no viscous effects are created (e.g., vx,x=A,x=0). One of the physically meaningful (and admissible) velocity potentials is (98)Vx,y=12Ax2-y2,which yields(99)V,x=vx=Ax,V,y=vy=-Ay,andvx,x+vy,y=A-A=0see,also,Figure3.The corresponding edge forces are now computed as (i.e., vx,y=A,y=0 and vy,x=-A=0)(100)fν≃λ+νvα,β+vβ,ανβνα=-q,andfτ≃νvα,β+vβ,ατβνα=0,where q is interpreted as a prescribed surface pressure (i.e., fν=-q). In the case of nonviscous fluid membrane (i.e., purely elastic boundary: vx=vy=0), (100)1 reduces to (see, for example, [16, 21]) (101)fν≃λ+q=λ-.Finally, the shape equation (see (76)) becomes (102)12kΔpΔpz-λΔpz-2νzx,xA-zy,yA≃p.We also find that the tangential equation of motion is automatically satisfied (103)λ,α+vΔpvα=0,since λ,α=0 for constant λ and Δpvα=vx,x+vy,y=0. Thus the solution of (102) can be completely determined by imposing (92), (93), and (95). Here we intentionally omit the details for the sake of conciseness which can be found in [22]. The deformation of the lipid membrane subjected to the normal flow on the surface of the membrane and the internal pressure field is presented in Figure 4. The resulting deformation contour is elongated in y direction due to the straining effects of the applied viscous flow. In the limit of vanishing viscous flow, the solution recovers to the results in [18].Figure 3
Viscous flow over a rectangular portion of membranes.Figure 4
Membrane straining in the presence of intramembrane viscosity.It is also noted here that other sets of velocity field can be considered. For example, in the assimilation of nonuniform velocity fields, one could use periodic distributions as(104)vx=cos⁡xcos⁡yandvy=sin⁡xsin⁡y,which satisfies (105)vx,x+vy,y=-sin⁡xcos⁡y+sin⁡xcos⁡y=0.Lastly, knowing the fact that eitherτxνy or τyνx vanishes on the boundary (e.g., νp=-vxe1=-e1 and τp=-τye2=-e2 so that vy=τx=0 on ∂Ω1 (see, Figure 3)), we find(106)νvα,β+vβ,ατβνα=ν2vx,xτxνx+2vy,yτyνy+vx,y+vy,xτxνy+vy,x+vx,yτyνx=νvx,y+vy,x,∵τxνx=τyνy=τxνy=0andτyνx=1.However, since(107)vx,y=-cos⁡xsin⁡yandvy,x=cos⁡xsin⁡y,Equation (106) identically vanishes (i.e., ν(vα,β+vβ,α)τβνα=0). Thus, (100)2 yields fτ=0 which implies that no tangential (shear) force is acting on the boundary (the velocity field is symmetric).


### 5.2. Circular Portion of Membranes under the Effects of Viscous Flow

Lipid membranes are often subjected to interactions with other cellular organisms such as proteins and nucleus. Considering their morphological boundaries (e.g., circular, elliptic, etc.), a Cartesian coordinate may not be suitable for studying such interactions. For this purpose, we present an analysis under polar coordinate system such that the local orientation is defined by(108)k=vp×τp=-er×-eθ.The membrane-substrate interactions are driven by the binormal force fn acting on the interacting boundary (a=ri, see Figure 5) of an annular domain while outer boundary is clamped similar to the rectangular portion of membrane (i.e., z=0 and z,α=0). From (90) and (108), the binormal force can be computed as (109)fn≃k¯∇pτ·τp-kνp·∇pH=k¯∂τ∂rer+1r∂τ∂θeθ·-eθ-k-er·∂H∂rer+1r∂H∂θeθ.By invoking (70) and (88), the above further reduces to (i.e., τ=τp·(∇p2z)vp=(-eθ)·(∇p2z)(-er)=0))(110)fn=ker·∂1/2Δpz∂rer+1r∂1/2Δpz∂θeθ=k12Δpz,r.If the driven force is measured per unit area (denoted as σ), (88) requires fn=σ=k1/2(Δpz),r and thus yields (111)Δpz,r=2σk.Also, the condition of incompressibility in a polar coordinate is given by (112)div⁡v=vr,r+vθ,θr+vrr=0.Now, we consider circumferentially dominant viscous flow as (113)vθ=Aandvr=Ar,so that the incompressibility condition is met;(114)vr,r+vθ,θr+vrr=-Ar2+0+Ar2=0.Since either τθνr or τrνθ vanishes on the boundary (e.g., νp=-vrer=-er, τp=-τθeθ=-eθ and vθ=τr=0 on both ∂Ωi and ∂Ωo (see Figure 5)), we find(115)νvα,β+vβ,ατβνα=ν2vr,rτrνr+2vθ,θτθνθ+vr,θ+vθ,rτrνθ+vθ,r+vr,θτθνr=νvr,θ+vθ,r,∵τrνr=τθνθ=τrνθ=0andτθνr=1.Further, in view of (88) and (100), the above identically vanishes (i.e., ν(vr,θ+vθ,r)=0, ∵vθ,r=0 and vr,θ=0) and therefore, (100)2 is satisfied.Figure 5
Viscous flow over a circular portion of membranes.The case of radially dominant viscous flows can also be assimilated by considering the following velocity fields:(116)vθ=-Aθandvr=A.Accordingly, we evaluate(117)vθ,θ=-Aandvr,r=vr,θ=vθ,r=0.Thus, the incompressibility condition is automatically satisfied; (118)vr,r+vθ,θr+vrr=0+-Ar+Ar=0.The associated tangential force can be computed as ν(vα,β+vβ,α)τβνα=ν(vr,θ+vθ,r)=0 so that fτ=0 on the boundary.Lastly, we consider a nonlinear periodic viscous flow by introducing the following velocity fields:(119)vθ=Arcos⁡2θandvr=Arsin⁡2θ,such that (112) is satisfied (120)vr,r+vθ,θr+vrr=Asin⁡2θ+-2Arsin⁡2θr+Asin⁡2θ=0.In particular, we compute fτ as(121)fτ=νAcos⁡2θ+2Arcos⁡2θ,which further implies that the velocity field is nonsymmetric (nonzero shear forces acting on the boundary). The deformations of membranes subjected to substrate interactions and intramembrane viscosity can be uniquely determined from the shape equations (75) and the boundary conditions (93), (94), and (111). The results in Figure 6 illustrate that corrugation is observed over the domain of interest due to the intramembrane viscosity and is faded away when the viscous effects are removed. The procedures are relatively straight forward and therefore are reserved for the sake of conciseness.Figure 6
Deformations of lipid membranes subjected to radially dominant viscous flows.


## 5.1. Rectangular Portion of Membranes under the Effects of Viscous Flow

We now consider the membrane flows over a rectangular portion of the plane (see Figure3). During such motion, lipid molecules constantly flow in through the boundary, while the boundary remains clamped. The compatible boundary conditions are (92)z=0andn=konw.In view of (64), the latter condition requires n=(k-∇z)/a=k, and thus the second of (92) leads to(93)∇z=z,αaα=0→z,α=0,∵aα≠0,on the boundary. Further, (93) is equivalent to (94)∇z=z,vv+z,ττ=0→z,v=z,τ=0,which can be seen by using chain rule (e.g., z,v=∂z/∂v=(∂z/∂θα)(∂θα/∂v)=z,αva=0). Therefore, (89)1 reduces to (95)M≃12kΔpz.The imposition of kinetic conditions is completely arbitrary depending on the desired applications and yet must satisfy the condition of incompressibility (vα,α=0). For example, one could propose the following velocity potential: (96)Vx,y=Ax+y,where A is constant real number. We then evaluate (97)V,x=vx=A,andV,y=vy=A.Equation (97) satisfies vα,α=0, since both vx,x and vy,y are individually nulled out and therefore, no viscous effects are created (e.g., vx,x=A,x=0). One of the physically meaningful (and admissible) velocity potentials is (98)Vx,y=12Ax2-y2,which yields(99)V,x=vx=Ax,V,y=vy=-Ay,andvx,x+vy,y=A-A=0see,also,Figure3.The corresponding edge forces are now computed as (i.e., vx,y=A,y=0 and vy,x=-A=0)(100)fν≃λ+νvα,β+vβ,ανβνα=-q,andfτ≃νvα,β+vβ,ατβνα=0,where q is interpreted as a prescribed surface pressure (i.e., fν=-q). In the case of nonviscous fluid membrane (i.e., purely elastic boundary: vx=vy=0), (100)1 reduces to (see, for example, [16, 21]) (101)fν≃λ+q=λ-.Finally, the shape equation (see (76)) becomes (102)12kΔpΔpz-λΔpz-2νzx,xA-zy,yA≃p.We also find that the tangential equation of motion is automatically satisfied (103)λ,α+vΔpvα=0,since λ,α=0 for constant λ and Δpvα=vx,x+vy,y=0. Thus the solution of (102) can be completely determined by imposing (92), (93), and (95). Here we intentionally omit the details for the sake of conciseness which can be found in [22]. The deformation of the lipid membrane subjected to the normal flow on the surface of the membrane and the internal pressure field is presented in Figure 4. The resulting deformation contour is elongated in y direction due to the straining effects of the applied viscous flow. In the limit of vanishing viscous flow, the solution recovers to the results in [18].Figure 3
Viscous flow over a rectangular portion of membranes.Figure 4
Membrane straining in the presence of intramembrane viscosity.It is also noted here that other sets of velocity field can be considered. For example, in the assimilation of nonuniform velocity fields, one could use periodic distributions as(104)vx=cos⁡xcos⁡yandvy=sin⁡xsin⁡y,which satisfies (105)vx,x+vy,y=-sin⁡xcos⁡y+sin⁡xcos⁡y=0.Lastly, knowing the fact that eitherτxνy or τyνx vanishes on the boundary (e.g., νp=-vxe1=-e1 and τp=-τye2=-e2 so that vy=τx=0 on ∂Ω1 (see, Figure 3)), we find(106)νvα,β+vβ,ατβνα=ν2vx,xτxνx+2vy,yτyνy+vx,y+vy,xτxνy+vy,x+vx,yτyνx=νvx,y+vy,x,∵τxνx=τyνy=τxνy=0andτyνx=1.However, since(107)vx,y=-cos⁡xsin⁡yandvy,x=cos⁡xsin⁡y,Equation (106) identically vanishes (i.e., ν(vα,β+vβ,α)τβνα=0). Thus, (100)2 yields fτ=0 which implies that no tangential (shear) force is acting on the boundary (the velocity field is symmetric).


## 5.2. Circular Portion of Membranes under the Effects of Viscous Flow

Lipid membranes are often subjected to interactions with other cellular organisms such as proteins and nucleus. Considering their morphological boundaries (e.g., circular, elliptic, etc.), a Cartesian coordinate may not be suitable for studying such interactions. For this purpose, we present an analysis under polar coordinate system such that the local orientation is defined by(108)k=vp×τp=-er×-eθ.The membrane-substrate interactions are driven by the binormal force fn acting on the interacting boundary (a=ri, see Figure 5) of an annular domain while outer boundary is clamped similar to the rectangular portion of membrane (i.e., z=0 and z,α=0). From (90) and (108), the binormal force can be computed as (109)fn≃k¯∇pτ·τp-kνp·∇pH=k¯∂τ∂rer+1r∂τ∂θeθ·-eθ-k-er·∂H∂rer+1r∂H∂θeθ.By invoking (70) and (88), the above further reduces to (i.e., τ=τp·(∇p2z)vp=(-eθ)·(∇p2z)(-er)=0))(110)fn=ker·∂1/2Δpz∂rer+1r∂1/2Δpz∂θeθ=k12Δpz,r.If the driven force is measured per unit area (denoted as σ), (88) requires fn=σ=k1/2(Δpz),r and thus yields (111)Δpz,r=2σk.Also, the condition of incompressibility in a polar coordinate is given by (112)div⁡v=vr,r+vθ,θr+vrr=0.Now, we consider circumferentially dominant viscous flow as (113)vθ=Aandvr=Ar,so that the incompressibility condition is met;(114)vr,r+vθ,θr+vrr=-Ar2+0+Ar2=0.Since either τθνr or τrνθ vanishes on the boundary (e.g., νp=-vrer=-er, τp=-τθeθ=-eθ and vθ=τr=0 on both ∂Ωi and ∂Ωo (see Figure 5)), we find(115)νvα,β+vβ,ατβνα=ν2vr,rτrνr+2vθ,θτθνθ+vr,θ+vθ,rτrνθ+vθ,r+vr,θτθνr=νvr,θ+vθ,r,∵τrνr=τθνθ=τrνθ=0andτθνr=1.Further, in view of (88) and (100), the above identically vanishes (i.e., ν(vr,θ+vθ,r)=0, ∵vθ,r=0 and vr,θ=0) and therefore, (100)2 is satisfied.Figure 5
Viscous flow over a circular portion of membranes.The case of radially dominant viscous flows can also be assimilated by considering the following velocity fields:(116)vθ=-Aθandvr=A.Accordingly, we evaluate(117)vθ,θ=-Aandvr,r=vr,θ=vθ,r=0.Thus, the incompressibility condition is automatically satisfied; (118)vr,r+vθ,θr+vrr=0+-Ar+Ar=0.The associated tangential force can be computed as ν(vα,β+vβ,α)τβνα=ν(vr,θ+vθ,r)=0 so that fτ=0 on the boundary.Lastly, we consider a nonlinear periodic viscous flow by introducing the following velocity fields:(119)vθ=Arcos⁡2θandvr=Arsin⁡2θ,such that (112) is satisfied (120)vr,r+vθ,θr+vrr=Asin⁡2θ+-2Arsin⁡2θr+Asin⁡2θ=0.In particular, we compute fτ as(121)fτ=νAcos⁡2θ+2Arcos⁡2θ,which further implies that the velocity field is nonsymmetric (nonzero shear forces acting on the boundary). The deformations of membranes subjected to substrate interactions and intramembrane viscosity can be uniquely determined from the shape equations (75) and the boundary conditions (93), (94), and (111). The results in Figure 6 illustrate that corrugation is observed over the domain of interest due to the intramembrane viscosity and is faded away when the viscous effects are removed. The procedures are relatively straight forward and therefore are reserved for the sake of conciseness.Figure 6
Deformations of lipid membranes subjected to radially dominant viscous flows.


## 6. Conclusions

We present a rigorous analysis for the mechanics of fluid membranes subjected intramembrane viscous flow. The induced membrane free energy is computed using Mean and Gaussian curvatures, essentially, through the first and second fundamental form (the second gradient of deformations). The corresponding normal and tangential equilibrium equation is then formulated based on the theory of elastic surface and is used to obtain the membrane shape equations. More importantly, we derive a compatible linear model within the prescription of incremental deformations superposed on large. The admissible sets of viscous flow fields are examined through which the deformation profiles of fluid membranes are obtained. It turns out that intramembrane viscous flow gives rise to straining effects of the membrane. We also show that the dynamic edge conditions reduce to purely elastic boundary conditions in the limit of vanishing viscous effects (i.e., constant flows).Lastly, we have considered nonlinear periodic distributions of viscous flow fields and demonstrated that there exist particular sets of viscous flows which satisfy the conditions of incompressibility. It is found that the corresponding tangential (shear) forces are identically vanished in the cases of symmetric velocity field while nonzero tangential forces are created for nonsymmetric flows. The results can be further extended to the morphological analysis of membranes subjected to various types of viscous flows in that velocity potential functions can be approximated via periodic series expansions (e.g., Fourier series expansion).



---

*Source: 3412129-2019-04-07.xml*