Governing equations

Stokes equations

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

where is the deviatoric stress tensor, is pressure, is the external forces vector, is the velocity vector, is the compressibility coefficient, is the thermal expansivity coefficient and is temperature.

The ρg argument of solve! is , one array per dimension on the cell centers. For a body force that is a gravitational acceleration this is , hence the name.

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:

where is the shear viscosity and is the deviatoric strain tensor.

Heat diffusion

The pseudo-transient heat-diffusion equation is:

where is density, is specific heat capacity, is thermal conductivity, is temperature   is the energy dissipated by viscous deformation (shear heating),   is adiabatic heating, and is the sum any other source term, such as radiogenic heat production.