Two rigid discs are co$-$axially pressed together resulting in a pressure $\backslash (\backslash$mathrm$\{$P$\}(\backslash$mathrm$\{$r$\})\backslash )$ at the common interface. Initially the system is stationary and the ambient temperature is $\backslash (\backslash$mathrm$\{$T$\}\_\{\backslash$infty$\}\backslash ).$ Then the upper disk suddenly is rotated at a constant angular velocity $\backslash (\backslash$omega $\backslash ).$ The coefficient of dry friction between the disks is $\backslash (\backslash$mu$\_\{\backslash$mathrm$\{$s$\}\}\backslash ).$ Formulate the problem to find the temperature at each disk considering:

a$.$ Spatial distributions for both discs using differential formulation.

b$.$ Axially lumped upper disc

c$.$ Both discs axially lumped