Self-similar solutions for the stress-constrained boundary layer

We analyze the boundary-layer flow that develops in a quiescent fluid near the surface of a semi-infinite flat plate when a tangential stress is imposed at the boundary. This force induces motion confined to a layer that grows downstream of the leading edge. We find that a family of self-similar solutions exists for stress distributions that follow a power law with the streamwise coordinate. This family of solutions allows us to describe scenarios that involve increasing or decreasing stresses from an initial value. Beyond their academic relevance, the solutions to the governing equation are of interest in pumping devices that use magnetic or electric forces acting on a thin layer adjacent to the surface.