Consider the thermal energy conservation equation given below for unsteady state

convective and conductive heat transfer to a vertically falling film of thickness $$ with

negligible viscous dissipation.

$(\frac{\text{d}\text{e}\text{l}\text{T}}{\text{d}\text{e}\text{l}\text{t}}+v_z\frac{\text{d}\text{e}\text{l}\text{T}}{\text{d}\text{e}\text{l}\text{z}})={}_T(\frac{de{l}^{2}T}{del{x}^2}+\frac{de{l}^{2}T}{del{z}^2})$

$v_z=U[1-(\frac{x}{}{)}^2]$

The coordinate $x$ is zero at the liquid$-$air interface $($see Figure$)$ and the temperature

of the air matches the inlet temperature of the film, $T_0.$

The solid wall is heated with a time varying temperature given by

$T_w=T_1(1+0\mathrm{.}1sin2\frac{t}{t_p})$

Here $t_p$ denotes the time period of oscillation.

Carry out a scaling analysis of the problem using $$ as the scale for $x,t_p$ as the scale

for time $t$ and setting $=\frac{T-T_0}{T_W-T_0}.$

$($a$)$ Identify other appropriate scales and dimensionless groups from the scaling

analysis. What is the other time scale arising in the scaled equation?

$($b$)$ What is the quasi$-$steady state approximation $($QSSA$)?$ Identify conditions for

applying the QSSA.

$($c$)$ Identify the conditions for eliminating the axial conduction term from the energy

balance.

$($d$)$ Write the required number of boundary conditions for the simplified differential

equation.

Note: the thermal diffusivity ${}_T=\frac{k}{C_p}$