Governing equations

Stokes equations

Deformation of compressible viscous flow is described by the equations of conservation of momentum and mass:

$$\begin{array}{r}\mathrm{\nabla }\cdot \mathit{\tau }-\mathrm{\nabla }p=\mathit{f}\end{array}$$$$\begin{array}{r}\mathrm{\nabla }\cdot \mathit{v}=\beta \frac{\partial p}{\partial t}+\alpha \frac{\partial T}{\partial t}\end{array}$$

where $\mathit{\tau }$ is the deviatoric stress tensor, $p$ is pressure, $f$ is the external forces vector, $\mathit{v}$ is the velocity vector, $\beta$ is the compressibility coefficient, $\alpha$ is the thermal expansivity coefficient and $T$ is temperature.

Constitutive equation

To close the system of equations (1)-(2), we further need the constitutive relationship between stress and deformation. In its simplest linear form this is:

$$\begin{array}{r}\mathit{\tau }=2\eta \dot{\mathit{\epsilon }}\end{array}$$

where $\eta$ is the shear viscosity and $\dot{\mathit{\epsilon }}$ is the deviatoric strain tensor.

Heat diffusion

The pseudo-transient heat-diffusion equation is:

$$\begin{array}{r}\rho C_p\frac{\partial T}{\partial t}=\mathrm{\nabla }\cdot (\kappa \mathrm{\nabla }T)+\mathit{\tau }:\dot{\mathit{\epsilon }}+\alpha T(\mathit{v}\cdot \mathrm{\nabla }P)+H\end{array}$$

where $\rho$ is density, $C_p$ is specific heat capacity, $\kappa$ is thermal conductivity, $T$ is temperature $\mathit{\tau }:\dot{\mathit{\epsilon }}$ is the energy dissipated by viscous deformation (shear heating), $\alpha T(\mathit{v}\cdot \mathrm{\nabla }P)$ is adiabatic heating, and $H$ is the sum any other source term, such as radiogenic heat production.