content_image/338760.jpg content_image/338766.jpg content_image/338768.jpg content_image/338769.jpg content_image/338767.jpg content_image/338770.jpg
<img src='content_image/338760.jpg'>

distance between layers remains very small this implies that the integrated mean curvature also remains constant. 

## B. Determination of the vesicle shape 

Assuming an axisymmetric vesicle shape, the body surface S ( t ) is parameterized at each time t as illustrated in Fig. 2. The arc-length measured along the surface in the x - z plane is denoted by s ∈ [0 , L ], with ˆ t the unit tangent vector, ˆ n the outward pointing normal vector, and ψ the angle between the x-axis and ˆ t . The body surface is represented in cylindrical polar coordinates, 

<img src='content_image/338766.jpg'>

where φ ∈ [0 , 2 π ) is the azimuthal angle, the surface ˜ x is taken to have its center of volume at the origin, and z$_{0}$ ( t ) is a translation of that center of volume which depends upon the fluid in- teraction. Under this parameterization, the principal membrane curvatures are C$_{1}$ = ∂ψ/∂s and C$_{2}$ = sin ψ/r . Upon insertion into either of the enthalpy functionals F or G , and performing a vari- ational extremization, we obtain the following system of first-order ordinary differential equations to describe the energetically stationary vesicle shapes at time t [43] 

<img src='content_image/338768.jpg'>

<img src='content_image/338769.jpg'>

<img src='content_image/338767.jpg'>

<img src='content_image/338770.jpg'>

Here K is an auxiliary function used to make the system of equations first-order (physically it corresponds to the curvature), γ is the Lagrange multiplier that enforces the interdependence of ψ 

