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.

After Stokes presented the significant paper [

Due to above descriptions, the viscous flow generated by relatively moving porous half-planes, which has not been studied yet, is analyzed in this paper. Mathematical formulation is firstly given in Section

The extended Stokes’ problems for relatively moving porous planes are depicted in Figure

Diagram of the extended Stokes’ problems.

The detailed derivation for solutions to the flow systems governed by (

To solve the second subsystem governed by (

Before analyzing the solutions derived in the previous section, dimensionless parameters are required to enhance the understanding of the flow. Hence, for the first problem, applying the following dimensionless parameters:

Velocity profiles at various values of

Velocity profiles of various values of

For the second problem, the total dimensionless solution is obtained by combining (

Velocity profiles at various

Velocity profiles of various values of

A still fluid suddenly driven by relatively moving porous half-planes is theoretically analyzed in this paper. In addition to the integral transforms, it is impossible to acquire the exact solution without using an important technique which divides the original problem into two subsystems. The solution to the first subsystem is equivalent to half of the solution of the traditional Stokes’ problem. As for the second subsystem, the velocity profiles in the whole domain can be obtained by solving the flow in the positive-

The author appreciates the financial support from National Science Council of Taiwan with Grant no. NSC-97-2221-E-270-013-MY3.