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add equations symbols description
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albert-de-montserrat committed Aug 8, 2024
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Expand Up @@ -4,11 +4,11 @@ The pseudo-transient method consists in augmenting the right-hand-side of the ta
## Heat diffusion
The pseudo-transient heat-diffusion equation is:

$\widetilde{\rho}\frac{\partial T}{\partial \psi} + \rho C_p \frac{\partial T}{\partial t} = \nabla \cdot (K\nabla T) = -\nabla q$
$\widetilde{\rho}\frac{\partial T}{\partial \psi} + \rho C_p \frac{\partial T}{\partial t} = \nabla \cdot (\kappa\nabla T) = -\nabla q$

We use a second order pseudo-transient scheme were continuation is also done on the flux, so that:

$\widetilde{\theta}\frac{\partial q}{\partial \psi} + q = -K\nabla T$
$\widetilde{\theta}\frac{\partial q}{\partial \psi} + q = -\kappa\nabla T$

## Stokes equations

Expand All @@ -33,3 +33,34 @@ and
$\widetilde{V} = \sqrt{ \frac{\widetilde{K} +2\widetilde{G}}{\widetilde{\rho}}}, \qquad r = \frac{\widetilde{K}}{\widetilde{G}}, \qquad Re = \frac{\widetilde{\rho}\widetilde{V}L}{\eta}$

where the P-wave $\widetilde{V}=V_p$ is the characteristic velocity scale for Stokes, and $Re$ is the Reynolds number.

### Physical parameters

| Symbol | Parameter |
| :------------------------------- | :--------------------: |
| $T$ | Temperature |
| $q$ | Flux |
| $\boldsymbol{\tau}$ | Deviatoric stress |
| $\dot{\boldsymbol{\varepsilon}}$ | Deviatoric strain rate |
| $\boldsymbol{u}$ | Velocity |
| $\boldsymbol{f}$ | External forces |
| $P$ | Pressure |
| $\eta$ | Viscosity |
| $\rho$ | Density |
| $\beta$ | Compressibility |
| $G$ | Shear modulus |
| $\alpha$ | Thermal expansivity |
| $C_p$ | Heat capacity |
| $\kappa$ | Heat conductivity |

### Pseudo-transient parameters

| Symbol | Parameter |
| :------------------- | :---------------------------: |
| $psi$ | Pseudo time step |
| $\widetilde{K}$ | Pseudo bulk modulus |
| $\widetilde{G}$ | Pseudo shear modulus |
| $\widetilde{V}$ | Characteristic velocity scale |
| $\widetilde{\rho}$ | Pseudo density |
| $\widetilde{\theta}$ | Relaxation time |
| $Re$ | Reynolds number |

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