Helium flows through a pipe of diameter D. The flow of helium is different from that of other fluids in that the usual no-slip condition at a solid boundary does not apply. There is some sliding at...

Helium flows through a pipe of diameter D. The flow of helium is different from that of other fluids in that the usual no-slip condition at a solid boundary does not apply. There is some sliding at the pipe wall, and the helium has some velocity at that location. The boundary condition at the pipe wall is

where r is the radial coordinate, R is the pipe radius, and K is a constant.

(a) Determine the velocity profile of the helium pipe flow.

(b) Determine the shear stress distribution.

(c) Determine the relationship between the discharge and the gradient of the piezometric pressure.