From: Wolfgang Bangerth Date: Mon, 12 Nov 2007 20:33:43 +0000 (+0000) Subject: Commit a few formulas X-Git-Tag: v8.0.0~9610 X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=e6f9f8f56538f279fc012bf6815a297b9562ebe6;p=dealii.git Commit a few formulas git-svn-id: https://svn.dealii.org/trunk@15489 0785d39b-7218-0410-832d-ea1e28bc413d --- diff --git a/deal.II/examples/step-22/doc/intro.dox b/deal.II/examples/step-22/doc/intro.dox index 7708deecc1..f9444b1735 100644 --- a/deal.II/examples/step-22/doc/intro.dox +++ b/deal.II/examples/step-22/doc/intro.dox @@ -1,6 +1,65 @@

Introduction

+

The Boussinesq equations

+ +This program deals with an interesting physical problem: how does a +fluid (i.e. a liquid or gas) behave if it experiences differences in +buoyance caused by temperature differences? It is clear that those +parts of the fluid that are hotter (and therefore lighter) are going +to rise up and those that are cooler (and denser) are going to sink +down against gravity. + +In cases where the fluid moves slowly enough such that inertia effects +can be neglected, the equations that describe such behavior are the +Boussinesq equations that read as follows: +@f{eqnarray*} + -\nabla \cdot \eta \varepsilon ({\mathbf u}) + \nabla p &=& + \mathrm{Ra} \; T \mathbf{g}, + \\ + \nabla \cdot {\mathbf u} &=& 0, + \\ + \frac{\partial T}{\partial t} + + + {\mathbf u} \cdot \nabla T + - + \nabla \cdot \kappa \nabla T &=& \gamma. +@f} +Here, ${\mathbf u}$ is the velocity field, $p$ the pressure, and $T$ +the temperature of the fluid. As can be seen, velocity and pressure +solve a Stokes equation describing the motion of an incompressible +fluid. The forcing term of the fluid motion is the buoyancy of the +fluid, expressed as the product of the Raleigh number $\mathrm{Ra}$, +the temperature $T$ and the gravity vector ${\mathbf g}$. (A possibly +more intuitive formulation would use $\mathrm{Ra} \; (T-\bar T) +\mathbf{g}$ as right hand side where $\bar T$ is the average +temperature, and the right hand side then describes the forces due to +local deviations from the average density; this formulation is +entirely equivalent if the gravity vector results from a gravity +potential $\phi$, i.e. $\mathbf{g}=-\nabla\phi$, and yields the exact +same solution except for the pressure which will now be $p+\mathrm{Ra} +\;\bar T \phi$.) + +While the first two equations describe how the fluid reacts to +temperature differences by moving around, the third equation states +how the fluid motion affects the temperature field: it is an advection +diffusion equation, i.e. the temperature is attached to the fluid +particles and advected along in the flow field, with an additional +diffusion (heat conduction) term. + +In these equations, $\eta$ and $\kappa$ denote the viscosity and +diffusivity coefficients. In the more general case and in many +physical applications, they may and $\eta$ often will depend +on the temperature, but we will neglect this dependence for the +purpose of this tutorial program. The term $\gamma$ on the right hand +side denotes the heat sources and may be a spatially and temporally +varying function. + + + + + + @code template