Flow-induced deformation of an elastic membrane adhering to a wall

Abstract

a b s t r a c tThe flow-induced deformation of a two-dimensional membrane with a circular unstressed shape clamped at the two ends on a plane wall at an arbitrary contact angle is considered. Working under the auspices of generalized shell theory, the membrane is allowed to develop in-plane tensions, transverse tensions, and bending moments determined by the curvature of the resting and deformed shapes. A system of ordinary differential equations governing the membrane shape is formulated, and the associated boundary-value problem is solved by numerical methods. Numerical results are presented to illustrate the deformation of a clamped membrane due to gravity or a negative transmural pressure. The shell formulation is coupled with a boundary-integral formulation for Stokes flow, and an efficient iterative scheme is developed to describe deformed equilibrium shapes of a membrane attached to a plane wall in the presence of an overpassing shear flow. Computations for different contact angles and shear rates reveal a wide variety of profiles and illustrate the distribution of the membrane tension developing due to the flow-induced deformation.

A particle‐based moving interface method (PMIM) for modeling the large deformation of boundaries in soft matter systems

“…To obtain the dicretized weak form of the governing equations, let us turn to the weak form (17) Next, we replace the different fields v; p; N v k ; p ; C , and with their discretized form (18), (19) and (20). In order to minimize the length of the discretized governing equations, we first rewrite the interpolation equations as the product of a shape function matrix and the element nodal values vector as follows.…”

“…Another approach developed to study fluid/interface interaction within the creeping flow regime is the boundary integral method, where only the surface of the interface needs to be discretized. The method is very successful at simulating drops in viscous flows and was extended to elastic interfaces by projecting the velocity gradient of the surrounding fluid to find the deformation rate of the interface .…”

“…We show the effect of slip length on the shear flow past a two-dimensional capsule and simulate the compression of an elastic membrane lying on a viscous fluid substrate.Another approach developed to study fluid/interface interaction within the creeping flow regime is the boundary integral method, where only the surface of the interface needs to be discretized. The method is very successful at simulating drops in viscous flows [18,19] and was extended to elastic interfaces by projecting the velocity gradient of the surrounding fluid to find the deformation rate of the interface [20,21].Most of these methods, however, have in common a Lagrangian mesh for the structure interacting with the fluid and cannot easily handle the large deformations or viscoelastic behaviors observed in many biological systems. Indeed, when the membrane is described in a Lagrangian framework, extreme deformations often lead to severe distortions of the finite element mesh, an issue that can only be approached by complicated and computationally expensive mesh regularization techniques [22].A solution to this problem was presented by Cottet et al in [23] via the introduction of a fully Eulerian description of the system, in which kinematic quantities representing the interface motion (such as position or dilation) are implicitly described by a level-set function, which is defined and updated on a fixed underlying Eulerian grid.…”