Viscous Flows and Boundary Layer Theory

$6\mathrm{.}1.$ For steady, zero heat release conditions, reduce the following equation

$\frac{\text{d}\text{e}\text{l}\text{T}}{\text{d}\text{e}\text{l}\text{t}}+\frac{\text{d}\text{e}\text{l}\text{u}\text{T}}{\text{d}\text{e}\text{l}\text{x}}+\frac{dvT}{dy}=\frac{1}{C_p}[k(\frac{de{l}^{2}T}{del{x}^2}+\frac{de{l}^{2}T}{del{y}^2})+Q_v]$

to the usual thin shear layer form given below by following an order of magnitude estimate of the individual terms. Assume that the thickness of the velocity and thermal boundary layers are similar $($both of order $)$ and that a scale for the order of magnitude of temperature is represented by $.$

$\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{x}}(uT)+\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{y}}(vT)=\frac{de{l}^{2}T}{del{y}^2}$

$)=(\frac{k}{Cp}$

Follow the steps as outlined for the momentum equations in the lectures, viz

$($i$)$ assume velocity scale for $u$ velocity is $U_{}$ and for $x-$length is $L,$ and for $y-$length is $.$

$($ii$)$ deduce a velocity scale for the $v$ velocity from continuity

$($iii$)$ write down an order of magnitude for each term in the heat transfer equation

$($iv$)$ examine magnitude of terms as $Re$ for finite value of the fluid PrandtI number $)=(\frac{C_p}{k}$