Deformation of elastic particles in viscous shear flow

In this paper, the dynamics of two dimensional elastic particles in a Newtonian viscous shear flow is studied numerically. To describe the elastic deformation, an evolution equation for the Eulerian Almansi strain tensor is derived. A constitutive equation is thus constructed for an incompressible ''Neo-Hookean'' elastic solid where the extra stress tensor is assumed to be linearly proportional to the Almansi strain tensor. The displacement field does not appear in this formulation. A monolithic finite element solver which uses Arbitrary Lagrangian-Eulerian moving mesh technique is then implemented to solve the velocity, pressure and stress in both fluid and solid phase simultaneously. It is found that the deformation of the particle in the shear flow is governed by two non-dimensional parameters: Reynolds number (Re) and Capillary number (Ca, which is defined as the ratio of the viscous force to the elastic force). In the Stokes flow regime and when Ca is small (Ca<0.65), the particle deforms into an elliptic shape while the material points inside the particle experience a tank-treading like motion with a steady velocity field. The deformation of the elastic particle is observed to vary linearly with Ca, which agrees with theoretical results from a perturbation analysis. Interactions between two particles in a viscous shear flow are also explored. It is observed that after the initial complicated interactions, both particles reach an equilibrium elliptic shape which is consistent with that of a single particle.