Consider the parallel disk viscometer shown below, containing two immiscible

fluids. Fluid $1$ has viscosity ${}_1$ and density ${}_1,$ fluid $2$ has viscosity ${}_2$ and

density ${}_2.$ The fluids are held in place by surface tension between the two

disks of radius R separated by a distance of $2$B$.$ The upper disk, at $z=+B$

rotates with angular velocity $$ while the lower disk is stationary.

a$)$ Show that the equation for the velocity in the direction of rotation is

separable and of the form $v_{}=f(r)g(z).$

b$)$ From the boundary condition at $z=+B$ guess what $f(r)$ should be$.$ Show

that the separation constant is then zero and obtain the velocity profile $v_{}.$

Note that the fact that you can satisfy all of the boundary conditions

proves that your guess is OK$.$

c$)$ Determine the shear stress profile, ${}_z,$ and determine the torque, $T,$

requirad in ti irn the innar dial