A disk of radius $\backslash ($ R $\backslash )$ rotates at an angular velocity $\backslash (\backslash$Omega $\backslash )$ inside a disk$-$shaped container filled with oil of viscosity $\backslash (\backslash$mu $\backslash ),$ as shown in Figure QA$1.$ The shear stress of a viscous fluid is $\backslash (\backslash$tau$=\backslash$mu d u $/$ d y $\backslash ).$ The shear stress on the outer disk edges can be neglected. Assuming a linear velocity profile of the oil in the $\backslash ($ y $\backslash )$ direction, and using a three$-$dimensional cylindrical coordinate system $\backslash (($r$-\backslash$theta$-$y$)\backslash ),$

$($a$)$ Derive the equation that formulates the velocity profile of oil between the top surface of the disk and the top wall of the container, using $\backslash ($ r $\backslash )$ and $\backslash ($ y $\backslash )$ as independent variables.

$[10]$

$($b$)$ Derive a formula for the viscous shear stress at the top surface of the disk, using $\backslash ($ r $\backslash )$ and $\backslash ($ y $\backslash )$ as independent variables.

$[10]$

$($c$)$ Derive a formula for the overall viscous torque on the disk.

$[10]$