 Solution

ID: past-exam-of-the-flows-fluctuations-and-complexity-and-symmetry-and-relativity-paper-of-part-b-of-the-physics-course-of-the-university-of-oxford/2011/2/3/solution

A parcel at fixed

z

spends time

a / u (z)

ρ c d T / d t = q_{R}

. Equivalently the steady advection equation is

ρ c u (z) T_{x} = q_{R} (z)

with

T (0, z) = T_{0}

. Therefore

T (a, z) = T_{0} + \frac{a F _{0} k}{c u ( z )} e^{- k ρ z} = T_{0} + \frac{2 μ a F _{0} k}{c f} \frac{e ^{- k ρ z}}{z ( b - z )} .

The no-slip boundary condition makes residence time diverge as either wall is approached. The resulting unbounded temperature is a failure of the approximations, rather than a physically steady wall prediction. Even small thermal conductivity transfers heat away from these layers; conduction into real perspex adds another heat sink. Thermal expansion can alter mass density, create buoyancy-driven motion and change the background profile, while temperature-dependent dynamic viscosity changes the assumed residence time. With exactly insulating walls, finite fluid thermal conductivity still regularizes the transverse profile; an entrance region and, sufficiently near a stagnant wall, longitudinal conduction also matter.

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