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 $a/u(z)$ in the light. With no thermal conduction or thermal expansion, its energy balance is $\rho cdT/dt=q_R$. Equivalently the steady advection equation is $\rho cu(z)T_x=q_R(z)$ with $T(0,z)=T_0$. ThereforeThe 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.