Flows induced by flat surfaces sheared in their own plane

Exact steady solutions of the Navier–Stokes equations for fluid flow induced by flat plates executing uniform or nonuniform shearing motion in their own plane are reported. In several cases uniform transpiration through the plate is required for existence of solutions. Solutions combining plate shear with plate stretching are also given. It is shown that hyperbolic shearing motion can give rise to damped oscillatory fluid motion above the plate while simple and spatially periodic shearing motions do not. This is interpreted as waves traveling on vertically attenuated vortex lines set up in the fluid when the hyperbolic shear is sufficiently strong.