The impact of confinement on the deformation of an elastic particle under axisymmetric tube flow

Abstract

We study an elastic particle translating axially along the centre-line of a rigid cylindrical tube filled with a Newtonian viscous fluid. The flow is pressure-driven and an axial body force is applied to the particle. We consider the regime in which the ratio of typical viscous fluid stress to elastic stiffness is small, leading to small elastic strains in the particle. In this case, there is a one-way decoupling of the fluid-structure interaction problem. The leading-order fluid problem is shown to be pressure-driven Stokes flow past a rigid sphere, and is solved using the semi-analytical method of reflections. The traction exerted by the fluid on the particle can be computed and used to formulate a pure solid-mechanics problem for the deformation of the particle, which can be solved analytically. This framework is used to investigate the role of the background flow, an axial body force, and the tube wall on the particle’s leading-order translational velocity, resulting deformation, and induced solid stress. By considering the first-order fluid problem the next-order correction to the translational velocity of the particle is shown to be zero. Depending on the magnitude of the ratio of applied body force to viscous forces, the particle can either have a bullet-like shape, an anti-bullet shape, or retain its original spherical shape. A non-linear arbitrary Lagrangian-Eulerian finite element implementation is used, in conjunction with various existing results from the literature, to validate the method of reflections solutions and interrogate their range of validity.