\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*}
+
{\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.
-</ul>
\ 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.
+</ul>