$(25$ points$)$ Consider a thermal boundary layer profile for flow over a flat plate to be of the form $T^{**}=\frac{T-T_s}{T_{}-T_s}=C_1+C_2(\frac{y}{{}_t}).$ The velocity profile is of the form $v^{**}=\frac{v_x}{v_{}}=\frac{y}{}$ with velocity boundary layer thickness $=(\frac{12vx}{v_{}}{)}^{\frac{1}{2}}.$

$($a$)(7$ points$)$ List the boundary conditions and obtain $T^{**}$ in terms of $y$ and the thermal boundary layer thickness ${}_t.$

$($b$)(18$ points$)$ Please express ${}_t$ as a function of Prandtl number and local Reynolds number.