= Solution
Take the layer depth as the length unit, its <thermal diffusion time> as the time unit, and its imposed <temperature> difference as the <temperature> unit. Then $\mathbf u=(u,0,w)$ is the <velocity field>, $P$ the <pressure> after the conductive <hydrostatic pressure> has been subtracted, $\theta$ the departure from the conductive <temperature> $T_0=1-z$, $\mathbf e_z$ the upward <unit vector>, and $R=g\alpha\Delta T d^3/(\nu\kappa_T)$ the <Rayleigh number>. Here $d$ is the dimensional depth, $\nu$ the <kinematic viscosity>, $\kappa_T$ the <thermal diffusivity> and $\alpha$ the <thermal expansion coefficient>. The <gradient>, <divergence> and <Laplacian> act on the dimensionless coordinates. The term $w$ in the <heat equation> is <advection> of the background <gradient>, since $-\mathbf u\cdot\nabla T_0=w$. <Mass density> variations enter only the <buoyancy> term under the <Boussinesq approximation>; the <incompressible flow> constraint is $\nabla\cdot\mathbf u=0$.
For fixed, impermeable, stress-free plates, the <no-penetration boundary condition>, <stress-free boundary condition> and <perfectly conducting thermal boundary condition> are
$$
\boxed{w=0,\qquad \partial_z u=0,\qquad\theta=0\quad\text{at }z=0,1.}
$$
For a nonzero horizontal <Fourier mode>, <incompressibility> gives $u=i w_z/k$, so the <velocity> conditions are equivalently $w=w_{zz}=0$ on both plates. A uniform horizontal <velocity> is not fixed by these <boundary conditions>; take its mean to be zero when discussing the stationary forced pattern.
\b[The printed small-Prandtl-number label does not generally justify these equations.] In thermal-time units the full momentum equation contains $\operatorname{Pr}^{-1}(\partial_t\mathbf u+\mathbf u\cdot\nabla\mathbf u)$, where the <Prandtl number> is $\operatorname{Pr}=\nu/\kappa_T$. Dropping that term gives <infinite-Prandtl-number convection> or another explicitly justified <Stokes flow> approximation. It is not the general $\operatorname{Pr}\to0$ limit. For example the horizontal shear $u=U_0e^{-\operatorname{Pr}\pi^2t}\cos\pi z$, $w=\theta=0$, satisfies the full linear momentum equation and the plate conditions, and persists at fixed $t$ as $\operatorname{Pr}\to0$. The printed inertia-free equation would instead require this shear to vanish. The following parts solve the equations actually displayed; applying them at small <Prandtl number> needs an additional slaving or creeping-flow ordering.
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