C. Pozrikidis

Deformation of liquid capsules enclosed by elastic membranes in simple shear flow: large deformations and the effect of fluid viscosities

The deformation of a liquid capsule enclosed by an elastic membrane in an infinite simple shear flow is studied numerically at vanishing Reynolds numbers using a boundary-element method. The surface of the capsule is discretized into quadratic triangular elements that form an evolving unstructured grid. The elastic membrane tensions are expressed in terms of the surface deformation gradient, which is evaluated from the position of the grid points. Compared to an earlier formulation that uses global curvilinear coordinates, the triangular-element formulation suppresses numerical instabilities due to uneven discretization and thus enables the study of large deformations and the investigation of the effect of fluid viscosities. Computations are performed for capsules with spherical, spheroidal, and discoidal unstressed shapes over an extended range of the dimensionless shear rate and for a broad range of the ratio of the internal to surrounding fluid viscosities

Effect of membrane bending stiffness on the deformation of capsules in simple shear flow

The effect of interfacial bending stiffness on the deformation of liquid capsules enclosed by elastic membranes is discussed and investigated by numerical simulation. Flow-induced deformation causes the development of in-plane elastic tensions and bending moments accompanied by transverse shear tensions due to the non-infinitesimal membrane thickness or to a preferred configuration of an interfacial molecular network. To facilitate the implementation of the interfacial force and torque balance equations involving the hydrodynamic traction exerted on either side of the interface and the interfacial tensions and bending moments developing in the plane of the interface, a formulation in global Cartesian coordinates is developed. The balance equations involve the Cartesian curvature tensor defined in terms of the gradient of the normal vector extended off the plane of the interface in an appropriate fashion. The elastic tensions are related to the surface deformation gradient by constitutive equations derived by previous authors, and the bending moments for membranes whose unstressed shape has uniform curvature, including the sphere and a planar sheet, arise from a constitutive equation that involves the instantaneous Cartesian curvature tensor and the curvature of the resting configuration. A numerical procedure is developed for computing the capsule deformation in Stokes flow based on standard boundary-element methods. Results for spherical and biconcave resting shapes resembling red blood cells illustrate the effect of the bending modulus on the transient and asymptotic capsule deformation and on the membrane tank-treading motion.

Motion and deformation of liquid drops, and the rheology of dilute emulsions in simple shear flow

Abstract The steady and transient deformation of a drop that is immersed in an ambient simple shear flow is studied in the limit of Stokes flow. The flow is examined as a function of the viscosity ratio λ between the drop and the suspending fluid, and the capillary number Ca. The motion of the drop is considered in an unbounded fluid and in proximity to a bounding plane wall. The problem is formulated in terms of the boundary integral method and is solved using a boundary element numerical procedure. The results provide information on the shape of stationary and transient deformed drops for a wide range of λ and Ca. The streamlines inside and outside deformed drops are analyzed and are shown to exhibit different patterns depending on λ and to a lesser degree on Ca. The kinematics of the flow on the surface of the drop is studied including the travel time of interfacial markers and the straining motion of interfacial patches. The stationary drop shapes are used to compute the effective stress tensor of a dilute emulsion of drops in shearing motion. It is found that a dilute emulsion behaves like an elastic, shear-thinning medium, for all values of λ. The effect of a plane wall is examined with reference to the time-scales of drop deforamation and migration away from the wall. The computed migration velocities are compared with those predicted by available asymptotic theories and experimental data and some differences are identified and discussed.

The effect of surfactants on drop deformation and on the rheology of dilute emulsions in Stokes flow

The effect of an insoluble surfactant on the transient deformation and asymptotic shape of a spherical drop that is subjected to a linear shear or extensional flow at vanishing Reynolds number is studied using a numerical method. The viscosity of the drop is equal to that of the ambient fluid, and the interfacial tension is assumed to depend linearly on the local surfactant concentration. The drop deformation is affected by non-uniformities in the surface tension due to the surfactant molecules convection-diffusion. The numerical procedure combines the boundary-integral method for solving the equations of Stokes flow, and a finite-difference method for solving the unsteady convection diffusion equation for the surfactant concentration over the evolving interface. The parametric investigations address the effect of the ratio of the vorticity to the rate of strain of the incident flow, the Peclet number expressing the ability of the surfactant to diffuse, the elasticity number expressing the sensitivity of the surface tension to variations in surfactant concentration, and the capillary number expressing the strength of the incident flow.

Finite deformation of liquid capsules enclosed by elastic membranes in simple shear flow

The transient deformation of liquid capsules enclosed by elastic membranes subject to simple shear flow is studied numerically using a new implementation of the boundary element method. The numerical results for capsules with spherical unstressed shapes and varying degrees of surface elasticity are compared with the predictions of an asymptotic theory for small deformations due to Barthes-Biesel and coworkers, and the significance of nonlinear effects due to finite deformation is assessed. It is found that the capsules exhibit continuous elongation when the dimensionless shear rate becomes larger than a critical threshold, in agreement with recent experimental observations of capsules with polymerized interfaces. Membrane failure at large deformations is discussed with respect to membrane thinning and development of excessive elastic tensions, and it is argued that the location where the membrane is likely to rupture due to continued deformation is insensitive to the precise mechanism of rupture. The numerical results suggest that a dilute suspension of capsules behaves like shear-thinning medium with some elastic properties. Results of oblate spheroidal capsules suggest that the points of maximum membrane thinning and tension coincide but their location depends upon the unstressed capsule shape.

The flow of a liquid film along a periodic wall

The creeping flow of a liquid film along an inclined periodic wall of arbitrary geometry is considered. The problem is formulated using the boundary-integral method for Stokes flow. This method is extended to two-dimensional flows involving free surfaces, and is implemented in an iterative numerical procedure. Detailed calculations for flow along a sinusoidal wall are perfomed. The free-surface profile is studied as a function of flow rate, inclination angle, wave amplitude, and surface tension, and is compared with previous asymptotic solutions. The results include streamline patterns, velocity profiles and wall-shear-stress distributions, and establish criteria for flow reversal. For specified wall geometry, the asymptotic behaviour for very small flow rates is shown to be a strong function of surface tension. It is demonstrated that these results are valid in a qualitative sense for general wall geometries. The analogy between gravity-driven flow and the flow of a liquid layer on a rotating disk (spin coating) is also discussed.

The instability of a moving viscous drop

The deformation of a moving spherical viscous drop subject to axisymmetric perturbations is considered. The problem is formulated using two different variations of the boundary integral method for Stokes flow, one due to Rallison & Acrivos, and the other based on an interfacial distribution of Stokeslets

A Finite-volume/Boundary-element Method for Flow Past Interfaces in the Presence of Surfactants, with Application to Shear Flow Past a Viscous Drop

Abstract A finite-volume method is developed for solving the convection–diffusion equation governing the transport of an insoluble surfactant over a generally evolving fluid interface, using an unstructured triangular grid. The unstructured grid has significant advantages compared with a structured grid based on global curvilinear coordinates, concerning adaptability and ability to conserve the total amount of the surfactant. The finite-volume method is combined with a boundary-element method for Stokes flow to yield an integrated procedure that is capable of describing the evolution of an interface from a specified initial state. Several series of simulations of the deformation of a neutrally buoyant viscous drop suspended in an infinite simple shear flow, or a semi-infinite shear flow bounded by a plane wall are performed. The results for the infinite flow extend those presented previously for the particular case where the ratio of the drop viscosity to the ambient fluid viscosity, λ , is equal to unity. It is shown that the effect of surfactant transport on the drop deformation and on the effective rheological properties of a dilute suspension becomes increasingly more important as λ becomes smaller and the drop reduces to an inviscid bubble. For semi-infinite flow past a drop above a plane wall, it is found that interfacial stresses due to variations in surface tension facilitate the drop migration away from the wall.

Deformation of liquid capsules with incompressible interfaces in simple shear flow

The transient deformation of liquid capsules enclosed by incompressible membranes whose mechanical properties are dominated by isotropic tension is studied as a model of red blood cell deformation in simple shear flow. The problem is formulated in terms of an integral equation for the distribution of the tension over the cell membrane which is solved using a point-wise collocation and a spectral-projection method. The computations illustrate the dependence of the deformed steady cell shape, membrane tank-treading frequency, membrane tension, and rheological properties of a dilute suspension, on the undeformed cell shape. The general features of the evolution of two-dimensional cells are found to be similar to those of three-dimensional cells, and the corresponding membrane tank-treading frequency and maximum tension are seen to attain comparable values. The numerical results are compared with previous asymptotic analyses for small deformations and available experimental observations, with satisfactory agreement. An estimate of the maximum shear stress for membrane breakup and red blood cell hemolysis is deduced on the basis of the computed maximum membrane tension at steady state.

The axisymmetric deformation of a red blood cell in uniaxial straining Stokes flow

The axisymmetric deformation of a red blood cell placed in a uniaxial straining Stokes flow is considered. The cell is modelled as a fluid capsule that contains a Newtonian fluid, and is bounded by an area-preserving membrane with negligible resistance to bending. First, it is theoretically demonstrated that spheroidal cells with isotropic membrane tension constitute stationary configurations. To compute transient cell deformations, a numerical procedure is developed based on the boundary-integral method for Stokes flow. Calculations show that initially prolate or oblate cells with isotropic membrane tension deform into stationary spheroids. Cells with a highly oblate initial shape may develop a persistent small pocket along their axis during the deformation. The shear elasticity of the membrane prevents folding, but may cause the formation of sharp corners and concave regions along the cell contour. A decrease in the membrane shear elasticity results in substantial increase in the magnitude of the transient and asymptotic membrane tensions. The maximum strain rate below which a red blood cell remains intact is estimated to be e x = 10 5 s −1 .