Rayleigh–Taylor and Richtmyer-Meshkov instability induced flow, turbulence, and mixing. II

Boundary Integral and Singularity Methods for Linearized Viscous Flow: Green's functions

The deformation of an initially spherical capsule, freely suspended in simple shear flow, can be computed analytically in the limit of small deformations [D. Barthes-Biesel, J.M. Rallison, The time-dependent deformation of a capsule freely suspended in a linear shear flow, J. Fluid Mech. 113 (1981) 251-267]. Those analytic approximations are used to study the influence of the mesh tessellation method, the spatial resolution, and the discrete delta function of the immersed boundary method on the numerical results obtained by a coupled immersed boundary lattice Boltzmann finite element method. For the description of the capsule membrane, a finite element method and the Skalak constitutive model [R. Skalak, A. Tozeren, R.P. Zarda, S. Chien, Strain energy function of red blood cell membranes, Biophys. J. 13 (1973) 245-264] have been employed. Our primary goal is the investigation of the presented model for small resolutions to provide a sound basis for efficient but accurate simulations of multiple deformable particles immersed in a fluid. We come to the conclusion that details of the membrane mesh, as tessellation method and resolution, play only a minor role. The hydrodynamic resolution, i.e., the width of the discrete delta function, can significantly influence the accuracy of the simulations. The discretization of the delta function introduces an artificial length scale, which effectively changes the radius and the deformability of the capsule. We discuss possibilities of reducing the computing time of simulations of deformable objects immersed in a fluid while maintaining high accuracy.

Efficient and accurate simulations of deformable particles immersed in a fluid using a combined immersed boundary lattice Boltzmann finite element method

The dynamic properties of interfaces often play a crucial role in the macroscopic dynamics of multiphase soft condensed matter systems. These properties affect the dynamics of emulsions, of dispersions of vesicles, of biological fluids, of coatings, of free surface flows, of immiscible polymer blends, and of many other complex systems. The study of interfacial dynamic properties, surface rheology, is therefore a relevant discipline for many branches of physics, chemistry, engineering, and life sciences. In the past three to four decades a vast amount of literature has been produced dealing with the rheological properties of interfaces stabilized by low molecular weight surfactants, proteins, (bio)polymers, lipids, colloidal particles, and various mixtures of these surface active components. In this paper recent experiments are reviewed in the field of surface rheology, with particular emphasis on the models used to analyze surface rheological data. Most of the models currently used are straightforward generalizations of models developed for the analysis of rheological data of bulk phases. In general the limits on the validity of these generalizations are not discussed. Not much use is being made of recent advances in nonequilibrium thermodynamic formalisms for multiphase systems, to construct admissible models for the stress-deformation behavior of interfaces. These formalisms are ideally suited to construct thermodynamically admissible constitutive equations for rheological behavior that include the often relevant couplings to other fluxes in the interface (heat and mass), and couplings to the transfer of mass from the bulk phase to the interface. In this review recent advances in the application of classical irreversible thermodynamics, extended irreversible thermodynamics, rational thermodynamics, extended rational thermodynamics, and the general equation for the nonequilibrium reversible-irreversible coupling formalism to multiphase systems are also discussed, and shown how these formalisms can be used to generate a wide range of thermodynamically admissible constitutive models for the surface stress tensor. Some of the generalizations currently in use are shown to have only limited validity. The aim of this review is to stimulate new developments in the fields of experimental surface rheology and constitutive modeling of multiphase systems using nonequilibrium thermodynamic formalisms and to promote a closer integration of these disciplines.

Dynamic properties of interfaces in soft matter: Experiments and theory

Three-dimensional (3D) computational modeling and simulation are presented on the motion of a large number of deformable cells in microchannels. The methodology is based on an immersed boundary method, and the cells are modeled as liquid-filled elastic capsules. The model retains two important features of the blood flow in the microcirculation, that is, the particulate nature of blood and deformation of the erythrocytes. The tank-treading and tumbling motion and the lateral migration, as observed for erythrocytes in dilute suspension, are briefly discussed. We then present results on the motion of multiple cells in semidense suspension and study how their collective dynamics leads to various physiologically relevant processes such as the development of the cell-free layer and the Fahraeus-Lindqvist effect. We analyze the 3D trajectory and velocity fluctuations of individual cell in the suspension and the plug-flow velocity profile as functions of the cell deformability, hematocrit, and vessel size. The numerical results allow us to directly obtain various microrheological data, such as the width of the cell-free layer, and the variation in the apparent blood viscosity and hematocrit over the vessel cross section. We then use these results to calculate the core and plasma-layer viscosity and show that the two-phase (or core-annular) model of blood flow in microvessels underpredicts the blood velocity obtained in the simulations by as much as 40%. Based on a posteriori analysis of the simulation data, we develop a three-layer model of blood flow by taking into consideration the smooth variation in viscosity and hematocrit across the interface of the cell-free layer and the core. We then show that the blood velocity predicted by the three-layer model agrees very well with that obtained from the simulations.

Three-dimensional computational modeling of multiple deformable cells flowing in microvessels.

The dynamic response of an initially spherical capsule subject to different externally imposed flows is examined. The neo-Hookean and Skalak et al. (Biophys. J., vol. 13 (1973), pp. 245–264) constitutive laws are used for the description of the membrane mechanics, assuming negligible bending resistance. The viscosity ratio between the interior and exterior fluids of the capsule is taken to be unity and creeping-flow conditions are assumed to prevail. The capillary number , beyond the interval of stability, the membrane has two tips along the direction of elongation where the deformation is most severe, and no equilibrium shapes could be identified. For both regions outside the interval of stability, the membrane model is not appropriate and bending resistance is essential to obtain realistic capsule shapes. This pattern persists for the two constitutive laws that were used, with the Skalak et al. law producing a wider stability interval than the neo-Hookean law owing to its strain hardening nature.

Spherical capsules in three-dimensional unbounded Stokes flows: effect of the membrane constitutive law and onset of buckling

We consider a neutrally buoyant and initially uncharged drop in a second liquid subjected to a uniform electric field. Both liquids are taken to be leaky dielectrics. The jump in electrical properties creates an electric stress balanced by hydrodynamic and capillary stresses. Assuming creeping flow conditions and axisymmetry of the problem, the electric and flow fields are solved numerically with boundary integral techniques. The system is characterized by the physical property ratios R (resistivities), Q (permitivities) and λ (dynamic viscosities). Depending on these parameters, the drop deforms into a prolate or an oblate spheroid. The relative importance of the electric stress and of the drop/medium interfacial tension is measured by the dimensionless electric capillary number, CaE .F orλ = 1, we present a survey of the various behaviours obtained for a wide range of R and Q. We delineate regions in the (R, Q)-plane in which the drop either attains a steady shape under any field strength or reaches a fold-point instability past a critical CaE. We identify the latter with linear instability of the steady shape to axisymmetric disturbances. Various break-up modes are identified, as well as more complex behaviours such as bifurcations and transition from unstable to stable solution branches. We also show how the viscosity contrast can stabilize the drop or advance break-up in the different situations encountered for λ =1 .

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112007007999

Axisymmetric deformation and stability of a viscous drop in a steady electric field

We present a numerical model of the hydrodynamic interactions between two capsules freely suspended in a simple shear flow. The capsules are identical and each consists of a liquid droplet enclosed by a thin hyperelastic membrane, devoid of bending resistance and obeying a neo-Hookean constitutive law. The two capsules are slightly prestressed with a given inflation ratio in order to avoid the small deformation instability due to compression observed for a single capsule in simple shear flow. The viscosity ratio between the interior and exterior fluids of the capsule is taken to be unity and creeping flow conditions are assumed to prevail. The boundary-element method is used with bi-cubic B-splines as basis functions on a structured mesh in order to discretize the capsule surface. A new method using two grids with initially orthogonal pole axes is developed to eliminate polar singularities in the load calculation and to allow for long computation times. Two capsules suspended in simple shear flow usually have different velocities and thus eventually pass each other. We study this crossing process as a function of flow strength and initial particle separation. We find that hydrodynamic interactions during crossing lead to large shape alterations, elevated elastic tensions in the membrane and result in an irreversible trajectory shift of the capsules. Furthermore, a tendency towards buckling is observed, particularly during the separation phase where large pressure differences occur. Our results are in qualitative agreement with those obtained for a pair of interacting liquid droplets but show the specific role played by the membrane of capsules.

Hydrodynamic interaction between two identical capsules in simple shear flow

The deformation of a drop as it flows along the axis of a circular capillary in low Reynolds number pressure-driven flow is investigated numerically by means of boundary integral computations. If gravity effects are negligible, the drop motion is determined by three independent parameters: the size a of the undeformed drop relative to the radius R of the capillary, the viscosity ratio λ between the drop phase and the wetting phase and the capillary number Ca, which measures the relative importance of viscous and capillary forces. We investigate the drop behaviour in the parameter space (a/R, λ, Ca), at capillary numbers higher than those considered previously. If the fluid flow rate is maintained, the presence of the drop causes a change in the pressure difference between the ends of the capillary, and this too is investigated. Estimates for the drop deformation at high capillary number are based on a simple model for annular flow and, in most cases, agree well with full numerical results if λ ≥ 1/2, in which case the drop elongation increases without limit as Ca increases. If λ 1). A companion paper (Lac & Sherwood, J. Fluid Mech., doi:10.1017/S002211200999156X) uses these results in order to predict the change in electrical streaming potential caused by the presence of the drop when the capillary wall is charged.

Motion of a drop along the centreline of a capillary in a pressure-driven flow

We investigate the effect of an initial isotropic prestress on the mechanics of a spherical capsule freely suspended in a simple shear flow. The capsule consists of a drop of liquid enclosed by a very thin hyperelastic membrane. A sufficient prestress may prevent the buckling instability that arises at low shear rates when the membrane bending resistance is neglected. A small perturbation analysis of the capsule deformation is first obtained for small prestress and weak viscous stresses. When the viscous shear stress and the preinflation are large, a numerical model of the capsule deformation shows how prestress decreases the capsule deformation for a given shear rate and how it significantly increases the elastic tension in the membrane at a given deformation level. These results may help the experimentalists to detect the presence of an osmotic pressure difference in a semipermeable capsule.

Etienne Lac

Papers

Spherical capsules in three-dimensional unbounded Stokes flows: effect of the membrane constitutive law and onset of buckling

Axisymmetric deformation and stability of a viscous drop in a steady electric field

Hydrodynamic interaction between two identical capsules in simple shear flow

Motion of a drop along the centreline of a capillary in a pressure-driven flow

Deformation of a capsule in simple shear flow: Effect of membrane prestress