(Solved): (a) The Boussinesq equations of thermal convection can be written in the dimensionless form u ...

(a) The Boussinesq equations of thermal convection can be written in the dimensionless form $\begin{array}{rl}??\mathbf{u}& =0,\\ \frac{1}{\mathrm{Pr}}\left[{\mathbf{u}}_t+(\mathbf{u}??)\mathbf{u}\right]& =??p+{?}^2\mathbf{u}+\mathrm{Ra}T\hat{\mathbf{k}},\\ T_t+\mathbf{u}??T& ={?}^{2}T.\end{array}$ Explain the meaning of these equations and define $\mathbf{u},p,T,\mathrm{Pr}$, and Ra. Assuming heating from the bottom, write down appropriate dimensionless boundary conditions assuming stress-free, impermeable boundaries, with prescribed temperatures on top and bottom, and no heat flux through the sides. (b) Use the stream function to define $u=?{?}_z$ and $w={?}_x$ and explain why this is a reasonable change of variables. Show that the Boussinesq equations can be written in the form $\begin{array}{rl}\frac{1}{\mathrm{Pr}}\left[{?}^2{?}_t+{?}_x{?}^2{?}_z?{?}_z{?}^2{?}_x\right]& =\mathrm{Ra}{T}_x+{?}^4?,\\ T_t+{?}_x{T}_z?{?}_z{T}_x& ={?}^{2}T,\end{array}$ with the associated boundary conditions $\begin{array}{rl}?={?}^2?=0& \text{ at }z=0,1,\\ T=0& \text{ at }z=1,\\ T=1& \text{ at }z=0,\\ ?={?}^2?=0& \text{ at }x=0,?,\\ T_x=0& \text{ at }x=0,?,\end{array}$ and write down the steady state solution $\left({?}^{?},T^{?}\right)$ corresponding to no-flow. (c) By perturbing the steady state, $?={?}^{?}+?,T=T^{?}+?,$ and linearising show that $\begin{array}{rl}{?}_t?{?}_x& ={?}^2?,\\ \frac{1}{\mathrm{Pr}}{?}^2{?}_t& ={?}^4?+\mathrm{Ra}{?}_x.\end{array}$ Explain why normal mode solutions of the form $\begin{array}{rl}?=& A{\mathrm{e}}^{?t}\sin kx\cos m?z,\\ ?=& B{\mathrm{e}}^{?t}\cos kx\cos m?z,\\ & 5\end{array}$ with $k=\frac{n?}{?},$ where $m$ and $n$ are integers and $A,B$ are constants, are not appropriate, and suggest normal mode solutions that would be appropriate. Deduce that the eigenvalue, $?$, satisfies $\left(?+K^2\right)\left(\frac{?}{K^2\mathrm{Pr}}+1\right)?\frac{{\mathrm{Ra}}^2}{K^4}=0,$ where $K^2=k^2+m^2{?}^2.$ Show that the steady state will be unstable if $\mathrm{Ra}>27{?}^4/4$, regardless of the value of $\mathrm{Pr}$.