Extended Stokes' Problems for Relatively Moving Porous Half-Planes

A shear flow motivated by relatively moving half-planes is theoretically studied in this paper. Either the mass influx or the mass efflux is allowed on the boundary. This flow is called the extended Stokes' problems. Traditionally, exact solutions to the Stokes' problems can be readily obtained by directly applying the integral transforms to the momentum equation and the associated boundary and initial conditions. However, it fails to solve the extended Stokes' problems by using the integral-transform method only. The reason for this difficulty is that the inverse transform cannot be reduced to a simpler form. To this end, several crucial mathematical techniques have to be involved together with the integral transforms to acquire the exact solutions. Moreover, new dimensionless parameters are defined to describe the flow phenomena more clearly. On the basis of the exact solutions derived in this paper, it is found that the mass influx on the boundary hastens the development of the flow, and the mass efflux retards the energy transferred from the plate to the far-field fluid.