On the axial flow of an incompressible viscous fluid in a pipe with a porous boundary

Fully developed laminar flow of an incompressible viscous fluid through a porous pipe with suction and injection is considered. An exact solution of the Navier-Stokes equations is given. The solution is in a series form in terms of the modified Bessel functions. The flow properties depend on the cross-Reynolds number, V. a/ν, where V is the suction velocity, a is the radius of the pipe and ν is the kinematic viscosity of the fluid. It is found that for large values of the cross-Reynolds number the flow near the region of the suction shows a boundary-layer character. In this region, the velocity varies sharply and the vorticity is concentrated near this region, and in the other parts of the pipe the vorticity does not show an appreciable change. A complete description of the flow is presented by using the graphs of the velocity, the volume flux across a plane normal to the flow and the vorticity.