A novel pseudo-rigid body approach to the non-linear dynamics of soft micro-particles in dilute viscous flow

Abstract

We propose a novel, demonstrably effective, utmost versatile and computationally highly efficient pseudo-rigid body approach for tracking the barycenter and shape dynamics of soft, i.e. non-linearly deformable micro-particles dilutely suspended in viscous flow. Pseudo-rigid bodies are characterized by affine deformation and thus represent a first-order extension to the kinematics of rigid bodies. Soft particles in viscous flow are ubiquitous in nature and sciences, prominent examples, among others, are cells, vesicles or bacteria. Typically, soft particles deform severely due to the mechanical loads exerted by the fluid flow. Since the shape dynamics of a soft particle - a terminology that shall here also include its orientation dynamics - also affects its barycenter dynamics, the resulting particle trajectory as a consequence is markedly altered as compared to a rigid particle. Here, we consider soft micro-particles of initially spherical shape that affinely deform into an ellipsoidal shape. These kinematic conditions are commensurate with i) the affine deformation assumption inherent to a pseudo-rigid body and ii) the celebrated Jeffery-Roscoe model for the traction exerted on an ellipsoidal particle due to creeping flow conditions around the particle. Without loss of generality, we here focus on non-linear hyperelastic particles for the sake of demonstration. Our novel numerical approach proves to accurately capture the particular deformation pattern of soft particles in viscous flow, such as for example tank-treading, thereby being completely general regarding the flow conditions at the macro-scale and, as an option, the constitutive behavior of the particle. Moreover, our computational method is highly efficient and allows straightforward integration into established Lagrangian tracking algorithms as employed for the point-particle approach to track rigid particles in dilute viscous flow.