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

Write $\mu$ for dynamic viscosity and $\nu =\mu /\rho$ for kinematic viscosity, consistently with the introductory Navier-Stokes equation. The phrase “dynamic viscosity $\nu$” in this request mixes these conventions. A fully developed plane Poiseuille flow, $\mathbf{u}=(u(z),0,0)$, has zero convective acceleration. The Navier-Stokes equation gives $\mu u^{\prime \prime }=-f$ and $p_z=-\rho g$. Applying the no-slip boundary condition at both stationary plates givesThe material coefficients must be effectively constant and positive. The profile is an exact parallel-flow solution for every $\mu >0$; experimental validity additionally requires that the realized flow remain in the laminar flow regime. A sufficient conservative condition isEquivalently $\nu \gg \sqrt{f{b}^3/(12\rho )}$, or $\mu \gg \sqrt{\rho f{b}^3/12}$. This sufficient bound is stronger than necessary; there is no need to discard convective acceleration by an approximation here because it vanishes identically for the assumed profile.

Here $k$ is a mass absorption coefficient, because $k\rho$ has units of inverse length. The Beer-Lambert law is $dF/dz=-k\rho F$, giving $F(z)=F_0{e}^{-k\rho z}$. A horizontal slab of area $A$ and thickness $dz$ absorbs power $A[F(z)-F(z+dz)]$. Dividing by its volume and taking the limit gives

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.

Put $A=2\mu a{F}_{0}k/(cf)$. In the specified near-wall limit the outer temperature is $T-T_0\simeq A/(bz)$, soThus the near-wall radiative heating layer has scaleThe comparison is an estimate of the breakdown of the outer solution, not a new inner temperature solution. It requires ${\delta }_C\ll b$ and $k\rho {\delta }_C\ll 1$; neglecting longitudinal relative to transverse thermal conduction additionally requires ${\delta }_C\ll a$ in the region to which the estimate is applied.