From a3673829630ea1a07bb6de2dbac9f049e736fa0e Mon Sep 17 00:00:00 2001 From: Wolfgang Bangerth Date: Fri, 11 Sep 2009 00:15:03 +0000 Subject: [PATCH] Correct a few formulas. git-svn-id: https://svn.dealii.org/trunk@19432 0785d39b-7218-0410-832d-ea1e28bc413d --- deal.II/examples/step-32/doc/intro.dox | 10 +++++----- 1 file changed, 5 insertions(+), 5 deletions(-) diff --git a/deal.II/examples/step-32/doc/intro.dox b/deal.II/examples/step-32/doc/intro.dox index d4ffb3cdc6..a231f39d74 100644 --- a/deal.II/examples/step-32/doc/intro.dox +++ b/deal.II/examples/step-32/doc/intro.dox @@ -75,7 +75,7 @@ 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: @f{eqnarray*} -\nabla \cdot (2 \eta \varepsilon ({\mathbf u})) + \nabla p &=& - -\beta\; T\; \mathbf{g} + + -\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 @@ -84,7 +84,7 @@ introduce a new "dynamic" pressure $p_{\text{dyn}}=p-\rho_{\text{ref}} with which the Stokes equations would read: @f{eqnarray*} -\nabla \cdot (2 \eta \varepsilon ({\mathbf u})) + \nabla p_{\text{dyn}} &=& - -\rho \; \beta \; T \mathbf{g}, + -\rho_{\text{ref}} \; \beta \; T \; \mathbf{g}, \\ \nabla \cdot {\mathbf u} &=& 0. @f} @@ -97,7 +97,7 @@ On the other hand, we will here use the form of the Stokes equations that considers the total pressure instead: @f{eqnarray*} -\nabla \cdot (2 \eta \varepsilon ({\mathbf u})) + \nabla p &=& - \rho_{\text{ref}} [1-\beta(T-T_{\text{ref}})] \mathbf{g}, + \rho(T)\; \mathbf{g}, \\ \nabla \cdot {\mathbf u} &=& 0. @f} @@ -222,7 +222,7 @@ just so happens to be the order of magnitude that would make the two equations numerically about the same. So, we now get this for the Stokes system: @f{eqnarray*} -\nabla \cdot (2 \eta \varepsilon ({\mathbf u})) + \nabla p &=& - -\rho \; \beta \; T \mathbf{g}, + \rho(T) \; \mathbf{g}, \\ \frac{\eta}{L} \nabla \cdot {\mathbf u} &=& 0. @f} @@ -233,7 +233,7 @@ scaled pressure $\hat p = \frac{L}{\eta}p$, and we get the scaled equations @f{eqnarray*} -\nabla \cdot (2 \eta \varepsilon ({\mathbf u})) + \nabla \left(\frac{\eta}{L} \hat p\right) &=& - -\rho \; \beta \; T \mathbf{g}, + \rho(T) \; \mathbf{g}, \\ \frac{\eta}{L} \nabla \cdot {\mathbf u} &=& 0. @f} -- 2.39.5