Nearly touching spheres in a viscoelastic fluid

Abstract

We theoretically investigate the forces and moments acting on two nearly touching spheres immersed in a second-order fluid. We divide the problem into four sub-classes, where each class represents the translational or rotational motion of the spheres either along the line joining the centers or the axis, which is oriented perpendicular to the line joining the centers. Using a regular perturbation solution methodology with the Deborah number as the small parameter, we obtain analytical expressions for the hydrodynamic forces and the moments experienced by the spheres for each sub-class considered. We find that, while the introduction of viscoelasticity does not generate any torques on the spheres, the viscoelastic contribution to force is non-zero and acts along the line joining the sphere centers for each sub-class. For asymmetric sub-classes, the presence of viscoelasticity produces a lift force on the spheres. We validate our method with the reciprocal theorem approach and find our force estimates to be accurate for small sphere separations. The analytical expressions obtained in this study can be utilized in computational schemes to study the behavior of a suspension of particles immersed in a viscoelastic fluid.