Consider parallel flow over a flat plate for air at $300$ K and engine oil at $380$ K $.$ The free stream velocity is $u_{}=1\mathrm{.}6\frac{m}{s}.$ The temperature difference between the surface and the free stream is the same in both cases, with $T_s>T_{}.$

$($a$)$ Determine the location, in m $,$ where transition to turbulence occurs, $x_c,$ for both fluids.

$($b$)$ For laminar flow over a flat plate, the velocity boundary layer thickness is given by $\frac{}{x}=\frac{5}{\sqrt[\phantom{2}]{R{e}_x}}.$ Determine the velocity boundary layer thickness $,$ in m $,$ for $x=1$ for each fluid.

$($c$)$ Determine the thermal boundary layer thickness ${}_t,$ in m $,$ for $x=1$ for each fluid. At an $x-$location where both fluids experience laminar flow conditions, which fluid has the largest temperature gradient at the plate surface, $-$del$\frac{T}{d}$ely${|}_y||=0?$ Which fluid is associated with the largest local Nusselt number $?$ Which fluid is associated with the largest local heat transfer coefficient $h?$