Consider a solid plate as shown below, two solid plates are connected and they have different

temperatures, thermal conductivities, and densities. At the top of surface plate $2,$ there is

convective heat transfer to the ambient air $($temperature $T,$ and heat transfer coefficient $h).$

Assume $T$ is a function of $y$ only The rate of frictional heating at the contact surface between

two solids $(H_S)$ is expressed by:

$H_s=fU$

The interfacial balance:

$q_y^{(i)}(L_i)+H_s=q_y^{(i+1)}(L_{i+1})$

When

$i=$ plate $i$

$U=$ the relative velocity of the solids,

$=$ the force per unit area holding the solids in contact $($i$-$e$.,$ normal to the interface$)$

$={}_i{L}_{i}g$

$f=$ the coefficient of dry friction for the materials involved. $({}_i=$ density of plate i$,L_i=$ Height

of plate i$,g=$ gravity$)$

$($a$)$ Determine temperature profiles in both plates at the steady$-$state when both plates are in the

rest $($not moving$)$

$($b$)$ Plot temperature profile in both plates

$($c$)$ If plate $2$ moves horizontally at speed $U,$ plate $1$ is in the rest. Determine the steady$-$state

temperature profile in the plates affected from the rate of interfacial energy generation