Fluid flows in the $x$-direction past the infinite plane $y=0$ with uniform but timedependent velocity $U(t)=U_{0} G\left(t / t_{0}\right)$, where $G$ is a positive function with timescale $t_{0}$. A long region of the plane, $0<x<L$, is heated and has temperature $T_{0}\left(1+\gamma(x / L)^{n}\right)$, where $T_{0}, \gamma, n$ are constants $[\gamma=O(1)]$; the remainder of the plane is insulating $\left(T_{y}=0\right)$. The fluid temperature far from the heated region is $T_{0}$. A thermal boundary layer is formed over the heated region. The full advection-diffusion equation for temperature $T(x, y, t)$ is

$$T_{t}+U(t) T_{x}=D\left(T_{y y}+T_{x x}\right),$$

where $D$ is the thermal diffusivity. By considering the steady case $(G \equiv 1)$, derive a scale for the thickness of the boundary layer, and explain why the term $T_{x x}$ in (1) can be neglected if $U_{0} L / D \gg 1$.

Neglecting $T_{x x}$, use the change of variables

$$\tau=\frac{t}{t_{0}}, \quad \xi=\frac{x}{L}, \quad \eta=y\left[\frac{U(t)}{D x}\right]^{1 / 2}, \quad \frac{T-T_{0}}{T_{0}}=\gamma\left(\frac{x}{L}\right)^{n} f(\xi, \eta, \tau)$$

to transform the governing equation to

$$f_{\eta \eta}+\frac{1}{2} \eta f_{\eta}-n f=\xi f_{\xi}+\frac{L \xi}{t_{0} U_{0}}\left(\frac{G_{\tau}}{2 G^{2}} \eta f_{\eta}+\frac{1}{G} f_{\tau}\right)$$

Write down the boundary conditions to be satisfied by $f$ in the region $0<\xi<1$.

In the case in which $U$ is slowly-varying, so $\epsilon=\frac{L}{t_{0} U_{0}} \ll 1$, consider a solution for $f$ in the form

$$f=f_{0}(\eta)+\epsilon f_{1}(\xi, \eta, \tau)+O\left(\epsilon^{2}\right) .$$

Explain why $f_{0}$ is independent of $\xi$ and $\tau$.

Henceforth take $n=\frac{1}{2}$. Calculate $f_{0}(\eta)$ and show that

$$f_{1}=\frac{G_{\tau} \xi}{G^{2}} g(\eta)$$

where $g$ satisfies the ordinary differential equation

$$g^{\prime \prime}+\frac{1}{2} \eta g^{\prime}-\frac{3}{2} g=\frac{-\eta}{4} \int_{\eta}^{\infty} e^{-u^{2} / 4} d u .$$

State the boundary conditions on $g(\eta)$.

The heat transfer per unit length of the heated region is $-\left.D T_{y}\right|_{y=0}$. Use the above results to show that the total rate of heat transfer is

$$T_{0}[D L U(t)]^{1 / 2} \frac{\gamma}{2}\left\{\sqrt{\pi}-\frac{\epsilon G_{\tau}}{G^{2}} g^{\prime}(0)+O\left(\epsilon^{2}\right)\right\}$$