Spin$-$Coating of a Viscous Fluid $($Fixed Volume$)$

Consider an axisymmetric drop of a Newtonian fluid of density $\backslash$rho and viscosity $\backslash$mu placed on a

rotating horizontal substrate as illustrated in the figure below. The substrate rotates at an angular velocity $\backslash$Omega $,$ which causes the drop to spread $($see Figure$).$

$($a$)$ Assuming a balance between viscous and centrifugal forces, write the equation of motion governing the z$-$dependence of the radial velocity ur$.$ What are the boundary conditions for ur$?$

$($b$)$ Show that the radial flow q per unit length is given by

q $=(\backslash$rho $(\backslash$Omega $^2)$r$($h$^3))/(3\backslash$mu $)$

What is the resulting evolution equation for the height of the film, h$($r; t$)?$

$($c$)$ Assume that the drop has already spread sufficiently so that the profile is quite flat. Derive the power law in time for the film height, h~tn$.$ How does this result compare to the time$-$dependence of a film thinning on a vertical substrate under the action of gravity and to the time$-$dependence of gravitational spreading of a drop?

$($d$)$ As the radius of the current becomes large, instabilities develop at the advancing front of the current. The liquid no longer spreads uniformly in the radial direction but forms protuberances that spread in a starfish$-$like pattern. What is causing the instability?