A fluid of constant viscosity, with density given by $)=(ar()\frac{-}{b}ar\frac{}{b}ar()(T\frac{-}{b}ar(T)),$ is confined in a rectangular slot. The slot has vertical walls at $x=+-B,y=+-W,$ and a top and bottom at $z=+-H,$ with $HWB.$ Gravity acts in the $-z$ direction. The walls are nonisothermal, with temperature distribution $T_w\frac{?}{b}=ar(T)+Ay,$ so that the fluid circulates by free convection. The velocity profiles are to be predicted, for steady laminar$-$flow conditions and small deviations from the mean density $\frac{?}{\text{b}\text{a}\text{r}}().$

a$)$ Simplify the equations of continuity, motion, and$/$energy according to the postulates: $v={}_z{v}_z(x,y),de{l}^2\frac{v_z}{d}el{y}^{2}de{l}^2\frac{v_z}{d}el{x}^2,$ and $T=T(y).$ These postulates are reasonable for slow flow, except near the edges $y=+-W$ and $z=+-H.$

b$)$ List the boundary conditions to be used with the problem as simplified in $($a$).$

c$)$ Solve for the temperature, and velocity profiles.

d$)$ Calculate the of constant A using velocity profile equation at $)=(0,$ for an experiment with water at $20C$ in a chamber with $B=0\mathrm{.}1mm,W=2\mathrm{.}0mm\frac{,}{b}ar()=0\mathrm{.}998\frac{g}{c}{m}^3\frac{,}{b}ar()=0\mathrm{.}00021K^{-1},=1\mathrm{.}0019{10}^{-2}\frac{g}{c}m.s,$ and $H=2cm,$ if the maximum permissible convective movement is $0\mathrm{.}1\%$ of $H$ in a one$-$hour experiment.