Two-Fluid Viscous Flow in a Corner

We consider steady, two-dimensional viscous flow of two fluids near a corner. The two fluids meet at the wedge vertex and are locally in contact with each other along a straight line emanating from the corner. The double wedge, treated in polar coordinates, admits separable solutions with bounded velocities at the corner. We seek local solutions which satisfy all local boundary conditions, as well as partial local solutions which satisfy all but the normal-stress boundary conditions. We find that local solutions exist for a wide range of total wedge angles and that a class of individual wedge angles and stress exponents is selected. Partial local solutions exist for all combinations of individual wedge angles and the stress exponents are determined as functions of these angles and the viscosity ratio. In both cases, Moffatt vortices can be found. Our aim in this work is to describe local two-fluid flow by determining for which wedge angles solutions exist, identifying singularities in the stress at the corner and identifying conditions under which Moffatt vortices can be present in the flow. Furthermore, for the single-wedge geometry, we identify for small capillary number non-uniformities present in solutions valid near the corner.