Semi-analytical solutions for two-dimensional elastic capsules in Stokes flow

Abstract

Elastic capsules occur in nature in the form of cells and vesicles and are manufactured for biomedical applications. They are widely modelled, but there are few analytical results. In this paper, complex variable techniques are used to derive semi-analytical solutions for the steady-state response and time-dependent evolution of two-dimensional elastic capsules with an inviscid interior in Stokes flow. This provides a complete picture of the steady response of initially circular capsules in linear strain and shear flows as a function of the capillary number Ca. The analysis is complemented by spectrally accurate numerical computations of the time-dependent evolution. An imposed nonlinear strain that models the far-field velocity in Taylor's four-roller mill is found to lead to cusped steady shapes at a critical capillary number Ca c for Hookean capsules. Numerical simulation of the time-dependent evolution for Ca > Ca c shows the development of finite-time cusp singularities. The dynamics immediately prior to cusp formation are asymptotically self-similar, and the similarity exponents are predicted analytically and confirmed numerically. This is compelling evidence of finite-time singularity formation in fluid flow with elastic interfaces.

Effects of membrane hardness and scaling analysis for capsules in planar extensional flows

“…For all capsule hardness, this edge curvature shows a very fast (exponential-like) increase with the capsule elongation L c which is higher for the less-strain-hardening capsules (such as the Skalak C = 0.1 and neo-Hookean capsules) owing to their lamellar shape. It is of interest to note that Higley, Siegel & Booty (2012) also found that the steady-state edge curvature of Hookean two-dimensional capsules increases exponentially with the flow rate in extensional flows. This suggests that the existence of a very small (in particular, exponentially small) edge length scale is a general feature of capsules in strong extensional flows, facilitated by the development of very small membrane tensions at the capsule tips (Dodson & Dimitrakopoulos 2008, 2009Higley et al 2012).…”

“…It is of interest to note that Higley, Siegel & Booty (2012) also found that the steady-state edge curvature of Hookean two-dimensional capsules increases exponentially with the flow rate in extensional flows. This suggests that the existence of a very small (in particular, exponentially small) edge length scale is a general feature of capsules in strong extensional flows, facilitated by the development of very small membrane tensions at the capsule tips (Dodson & Dimitrakopoulos 2008, 2009Higley et al 2012). Figure 7(b) shows the elongation dependence of the edge curvature C xz that characterizes the local edge length scale along the lateral z-axis.…”

Convergence of the boundary integral method for interfacial Stokes flow

“…They have been widely applied to compute the evolution of interfaces in potential flow, including the Kelvin-Helmholtz and Rayleigh-Taylor instabilities [5], [29], Hele-Shaw flow [26], [33], and water waves [7], [10]. They have also been extensively applied in Stokes flow to simulate the evolution of drops, bubbles, elastic capsules and vesicles [23], [42], [47], [50]. Boundary integral methods have been particularly important in micro-and bio-fluidic applications in which viscous forces are dominant over inertial ones.…”

“…Spectrally accurate discretizations of boundary integral formulations are now routinely implemented for 2D interfacial Stokes flow of drops and bubbles, see, e.g., [15], [30], [31], [40], [41], [53] and references therein. Spectral or high-order boundary integral methods for inextensible vesicles or elastic capsules are provided by [23], [37], [46], and [50]. High order accurate discretizations of axisymmetric and 3D flow problems, although still a subject of current research, are increasingly common [16], [48], [49], [51], [54].…”

Microfluidic probing of the complex interfacial rheology of multilayer capsules

“…New experimental techniques have been developed to characterize elastic polymer membranes at liquid interfaces [14][15][16][17][18] . The rheology of such membranes has been studied in model geometries 15,17,[19][20][21][22][23] and also on real capsules in viscous flows to mimic real conditions of fabrication and use, through various experiments 16,[23][24][25][26] , theories [27][28][29] and simulations [30][31][32] . From this perspective, microfluidics is an ideal tool which allows to integrate measurement units directly in situ in the devices and to perform automated measurements on a large number of capsules.…”