Large deformations of a cylindrical liquid-filled membrane by a viscous shear flow

Abstract

This paper treats the steady flow fields generated inside and outside an initially circular, inextensible, cylindrical membrane filled with an incompressible viscous fluid when the membrane is placed in a two-dimensional shear flow of another viscous fluid. The Reynolds numbers of both the interior and exterior flows were assumed to be zero (‘creeping flow’), but no further approximations were made in the formulation. A series solution of the resulting free boundary-value problem in powers of a dimensionless shear rate parameter was constructed through fifth order. When combined with a conformal coordinate transformation this series gave accurate results for large deformations of the membrane (up to an aspect ratio of 2.5). The rather tedious algebraic manipulations required to obtain the series solution were done by computer with a symbol-manipulation program (reduce), which both formulated the boundary-value problems for each successive order and solved them. Results are presented which show how the shear rate and fluid viscosities influence the internal and external velocity and pressure fields, the membrane deformation and its ‘tank-treading’ frequency, and the membrane tension.This work demonstrates that classical perturbation techniques combined with computer algebra offer a useful alternative to purely numerical methods for problems of this type.

Three-Dimensional Lattice Boltzmann Simulation of Two-Phase Flow Containing a Deformable Body with a Viscoelastic Membrane

“…From this figure, a rotating flow inside the body and a circulating flow around the body are observed. These flow characteristics are similar to those found by Zahalak et al [28].…”

Large deformations of elastic cylindrical capsules in shear flows

“…Finally, 16is the tangential external velocity component on the membrane. The viscous shear stress components appearing in (1 1) can be expressed in terms off, '$ and "$; these expressions are given as equation 19of Zahalak et al (1987).…”

“…The problem we have formulated is a difficult one in which complicated boundary conditions must be imposed on an a priori unknown boundary. As in our previous study of a pressurized, inextensible membrane (Zahalak et al 1987), we will construct the solution by expanding all field variables in asymptotic series of a gage function 6(e). In contrast to the previous problem however, the forms of the expansions (including the form of the gage function) in the current problem are not obvious.…”

“…Space limitations preclude any detailed discussion of the solution algorithm, which was essentially similar to that described in Zahalak et al (1987). In this section we will merely point out a few salient features of the analysis; a more detailed exposition of the solution can be found in Rao (1990Rao ( , 1991.…”

“…In order to handle the massive algebraic manipulations inherent in this approach we employed a symbolic manipulation program (REDUCE 3.1, Rand Corporation, Santa Monica, CA, 1983) which both formulated the problems for the higher-order terms and solved them. We have previously published a report (Zahalak et al 1987) of a similar investigation which applied this technique to a simpler problem than the one considered here, namely deformations of an inextensible, initially pressurized cylindrical membrane by a viscous shear flow. In this paper, we deal with an elastic membrane which is initially unpressurized.…”