Elastic Displacements and Viscous Flows in Wedge-Shaped Geometries with a Straight Edge: Green’s Functions for Perpendicular Forces

Edges are abundant when elastic solids glide in guiding rails or fluids are contained in vessels. We here address induced displacements in elastic solids or small-scale flows in viscous fluids in the vicinity of one such edge. For this purpose, we solve the governing elasticity equations for linearly elastic, potentially compressible solids, as well as the low-Reynolds-number flow equations for incompressible fluids. Technically speaking, we derive the associated Green’s functions under confinement by two planar boundaries that meet at a straight edge. The two boundaries both feature no-slip or free-slip conditions, or one of these two conditions per boundary. Previously, we solved the simpler case of the force being oriented parallel to the straight edge. Here, we complement this solution by the more challenging case of the force pointing into a direction perpendicular to the edge. Together, these two cases provide the general solution. Specific situations in which our analysis may find application in terms of quantitative theoretical descriptions are particle motion in confined colloidal suspensions, dynamics of active microswimmers near edges, or actuated distortions of elastic materials due to activated contained functionalized particles.