Natural convection—M. Fig. P6.8 shows two infinite parallel vertical walls that are separated by a distance 2d. A fluid of viscosity μ and volume coefficient of expansion β fills the intervening...

Natural convection—M. Fig. P6.8 shows two infinite parallel vertical walls that are separated by a distance 2d. A fluid of viscosity μ and volume coefficient of expansion β fills the intervening space. The two walls are maintained at uniform temperatures T1 (cold) and T2 (hot), and you may assume (to be proved in a heattransfer course) that there is a linear variation of temperature in the x direction. That is: T = T + x d T2 − T1 2 , where T = T1 + T2 2 .

The density is not constant but varies according to: ρ = ρ 1 − β T − T , where ρ is the density at the mean temperature T, which occurs at x = 0. If the resulting natural-convection flow is steady, use the equations of motion to derive an expression for the velocity profile vy = vy(x) between the plates. Your expression for vy should be in terms of any or all of x, d, T1, T2, ρ, μ, β, and g. Hints: In the y momentum balance, you should find yourself facing the following combination: −∂p ∂y + ρgy, in which gy = −g. These two terms are almost in balance, but not quite, leading to a small—but important—buoyancy effect that “drives” the natural convection. The variation of pressure in the y direction may be taken as the normal hydrostatic variation: ∂p ∂y = −gρ. We then have: −∂p ∂y + ρgy = gρ + ρ 1 − β T − T (−g) = ρβg T − T , and this will be found to be a vital contribution to the y momentum balance.