Consider a vertical heated flat plate having a constant surface temperature $\backslash (\backslash$mathrm$\{$T$\}\_\{$s$\}\backslash )$ and is placed in a quiescent medium where the ambient temperature is $\backslash (\backslash$mathrm$\{$T$\}\_\{\backslash$infty$\}\backslash ),$ and density $\backslash (\backslash$rho$\_\{\backslash$infty$\}\backslash ).$ The origin of the coordinate system is fixed at the bottom of the plate. The x$-$direction is along the plate in the vertical direction whereas the y $-$direction is normal to the plate. The acceleration due to gravity is acting in the negative x $-$direction downward.

$($i$)$ Discuss the Boussinesq approximation in this case. For what purpose is it utilized in this scenario?

$($ii$)$ Derive the expression for buoyancy force per unit volume in terms of densities. Then represent the buoyancy force in terms of the volumetric thermal expansion coefficient and the temperature difference.

$($iii$)$ By non$-$dimensionalization of the governing differential equation, derive the Grashof number.

$($iv$)$ What is the Rayleigh number and write its expression. What physical information does its value provide?

$($v$)$ For this case of natural convection, sketch of the temperature profile from the surface of the vertical plate, extending far into the ambient. Write the $3$ boundary conditions needed for this temperature profile.

$($vi$)$ For the same case, sketch the velocity profile from the surface of vertical plate and extending far into the ambient. Write at least $3$ boundary conditions needed for this velocity profile.