Unsteady flows of fluids with pressure dependent viscosity

In this short paper we consider the unsteady flows of fluids with pressure dependent viscosities when the effect of gravity has to be taken into account. Such a situation presents itself in many technologically relevant flows. After developing the appropriate governing equations, we establish qualitative results concerning the nature of the solutions to the governing equations as well as uniqueness results. We then obtain explicit exact solutions for two initial-boundary value problems, namely modified Stokes' first and second problems. Such explicit solutions serve a dual purpose, they present exact results for idealized problems that reflect practical situations, and they provide a means for checking the numerical schemes that are developed for flows in more complex situations.