Creeping Flow Through a Cone

One model for fluid flow through a porous medium is flow through a cone from its apex

outward. Thus, consider steady$-$state creeping flow in such a cone with half$-$angle $.$ The

flow is purely radial.

$($a$)$ Show that the stream function $$ depends only on the spherical coordinate $.$

$($b$)$ Write down the governing equation for the stream function.

$($c$)$ Show that the solution for $()$ has the general form, $()=a+$bcos$+csi{n}^{2}cos,$

where $a,b,$ and $c$ are constants.

$($c$)$ We need three boundary conditions to determine the constants $a,b,$ and $c.$ A smart

student suggested one boundary condition: $()=0$ on the axis of symmetry. Give a

physical explanation of this boundary condition. A second boundary condition is supplied

by knowing $Q,$ the volume flow rate in the cone outward. Specify the third boundary

condition and determine $a,b,$ and $c.$