CookBook%2FHeatTransfer

This example was originally provided by fred2 <at></at> qnet.com in http://thread.gmane.org/gmane.comp.python.fipy/911

The model problem of a (compressible) gas confined between two infinite plates is a useful test case for a Navier-Stokes solver. For stationary walls maintained at fixed (but different) temperatures, and assuming continuum no-slip boundary conditions, the the Navier-Stokes equations can be solved analytically.

The energy equation simplifies to the (1D) heat equation (i.e., a `DiffusionTerm`) to be solved for the temperature profile.

$0.5 \le \omega \le 1.0$ is the exponent of the "variable hard sphere (VHS)" viscosity/temperature relationship (i.e., $\kappa(T) \approx T^{\omega}$)

which is solved with no-slip boundary conditions (strictly applicable only for Knudsen number $K_n \rightarrow 0$) such that $T|_\text{right} = T|_\text{left} \chi$ with a wall temperature ratio $1 &lt; \chi &lt; 50$.

The normal momentum equation simplifies to $p = \text{const.}$ from which the density profile can be obtained.

The exact steady-state temperature profile is $$T(\hat{x}) = T_1 \left [\left]^{1/(\omega+1)},$$ where $\hat{x} = x/H$ and $H$ is the plate separation distance.

Starting from a linear profile, it takes only a few sweeps to reach convergence.

Image(1dht.png)

Reference `Wadsworth:1993` compares finite difference Navier-Stokes and particle simulation method predictions in the rarefied flow (i.e., slip) regime for which some experimental data are available.