From: bangerth Date: Fri, 11 Sep 2009 00:36:33 +0000 (+0000) Subject: Fix a couple things. Explain more. X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=0f6657a29c1490024247833a6a2947f51bea30c0;p=dealii-svn.git Fix a couple things. Explain more. git-svn-id: https://svn.dealii.org/trunk@19433 0785d39b-7218-0410-832d-ea1e28bc413d --- diff --git a/deal.II/examples/step-32/doc/intro.dox b/deal.II/examples/step-32/doc/intro.dox index a231f39d74..7edc6e6790 100644 --- a/deal.II/examples/step-32/doc/intro.dox +++ b/deal.II/examples/step-32/doc/intro.dox @@ -72,14 +72,14 @@ equation then looks properly written like this: \rho_{\text{ref}} [1-\beta(T-T_{\text{ref}})] \mathbf{g}. @f} Now note that the gravity force results from a gravity potential as -$\mathbf g=\nabla \varphi$, so that we can re-write this as follows: +$\mathbf g=-\nabla \varphi$, so that we can re-write this as follows: @f{eqnarray*} -\nabla \cdot (2 \eta \varepsilon ({\mathbf u})) + \nabla p &=& - -\rho_{\text{ref}} \; beta\; T\; \mathbf{g} + - \rho_{\text{ref}} [1+\beta T_{\text{ref}}] \nabla\varphi. + -\rho_{\text{ref}} \; \beta\; T\; \mathbf{g} + -\rho_{\text{ref}} [1+\beta T_{\text{ref}}] \nabla\varphi. @f} The second term on the right is time independent, and so we could -introduce a new "dynamic" pressure $p_{\text{dyn}}=p-\rho_{\text{ref}} +introduce a new "dynamic" pressure $p_{\text{dyn}}=p+\rho_{\text{ref}} [1+\beta T_{\text{ref}}] \varphi=p_{\text{total}}-p_{\text{static}}$ with which the Stokes equations would read: @f{eqnarray*} diff --git a/deal.II/examples/step-32/doc/results.dox b/deal.II/examples/step-32/doc/results.dox index 5e9d3dad23..dce0e7882e 100644 --- a/deal.II/examples/step-32/doc/results.dox +++ b/deal.II/examples/step-32/doc/results.dox @@ -321,11 +321,27 @@ in the @ref step_31 "step-31", here are a few more ideas: + {\mathbf u} \cdot \nabla T - - \nabla \cdot \kappa \nabla T &=& \gamma + \tau\frac{\partial - p}{\partial t} + \mathbf u \cdot \nabla p. + \nabla \cdot \kappa \nabla T &=& \gamma + + \tau\left\{\frac{\partial + p}{\partial t} + \mathbf u \cdot \nabla p \right\}. @f} In other words, as pressure increases in a rock volume ($\frac{Dp}{Dt}>0$) we get an additional heat source, and vice versa. - \ No newline at end of file + The time derivative of the pressure is a bit awkward to + implement. If necessary, one could approximate using the fact + outlined in the introduction that the pressure can be decomposed + into a dynamic component due to temperature differences and the + resulting flow, and a static component that results solely from the + static pressure of the overlying rock. Since the latter is much + bigger, one may approximate $p\approx p_{\text{static}}=-\rho_{\text{ref}} + [1+\beta T_{\text{ref}}] \varphi$, and consequently + $\frac{Dp}{Dt} \approx \left\{- \mathbf u \cdot \nabla \rho_{\text{ref}} + [1+\beta T_{\text{ref}}]\varphi\right\} = \rho_{\text{ref}} + [1+\beta T_{\text{ref}}] \mathbf u \cdot \mathbf g$. + In other words, if the fluid is moving in the direction of gravity + (downward) it will be compressed and because in that case $\mathbf u + \cdot \mathbf g > 0$ we get a positive heat source. Conversely, the + fluid will cool down if it moves against the direction of gravity. +