Applications of the image method in low-Reynolds-number hydrodynamics and linear elasticity to right-angled edges and corners

The image method is useful for solving boundary problems, known from e.g. electrostatics. Here, we present a general overview of when and how this method can be applied to the linear equations in classical continuum mechanics, the Stokes and Navier-Cauchy equations. In the two cases, our aim is to calculate the fluid velocity or displacement field in the whole domain under consideration, respectively. A Green's function method that characterizes the response to a point force is utilized.

We start with the well-known solutions in the case of planar interfaces. Afterwards, we extend this work to orthogonal edges where two boundaries meet and finally to corners with three mutually orthogonal boundaries [1]. We find that the applicability of the image method crucially depends on the type of boundary condition: We consider cases where the boundaries are (stress-)free, free-slip boundaries (only sliding along the surface is allowed) as well as no-slip boundary conditions (vanishing fluid velocities/displacements at the boundary) and all possible combinations thereof. Interestingly, we find that the image method can only be applied if all but one boundary are of free-slip type. In those cases, we also explicitly list the resulting solutions and illustrate them.