= 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 $\rho c\,dT/dt=q_R$. Equivalently the steady <advection equation> is $\rho c\,u(z)T_x=q_R(z)$ with $T(0,z)=T_0$. Therefore
$$
\boxed{T(a,z)=T_0+\frac{aF_0k}{c\,u(z)}e^{-k\rho z}
=T_0+\frac{2\mu aF_0k}{cf}\frac{e^{-k\rho z}}{z(b-z)}.}
$$
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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