MDOODZ governing equations and numerical methods — conservation of momentum (Stokes), energy, and mass, constitutive relations, non-dimensionalisation and scaling, marker-in-cell advection, free surface algorithm, pure shear ALE mode, and theoretical background.
∂τ_ij/∂x_j - ∂P/∂x_i + ρg_i = 0
Inertia is neglected (infinite Prandtl number assumption, appropriate for geological time scales). This gives the Stokes equations for slow viscous flow.
In component form:
Incompressible: ∂v_i/∂x_i = 0
Weakly compressible (when β > 0): ∂v_i/∂x_i = -1/K · DP/Dt
where K = 1/β is the bulk modulus, controlled by the bet parameter per phase.
ρCp · DT/Dt = ∂/∂x_i(k · ∂T/∂x_i) + H_shear + H_adiab + Qr
| Source term | Expression | Switch |
|---|
| Shear heating | H_shear = τ_ij · ε̇_ij | shear_heating = 1 |
| Adiabatic heating | H_adiab = α·T·v_z·ρ·g | adiab_heating = 1 |
| Radiogenic production | Qr (per phase, W/m³) | Always active if Qr > 0 |
τ_ij = 2η · ε̇_ij
where ε̇_ij is the deviatoric strain rate tensor.
τ_ij + (η/G) · Dτ_ij/Dt = 2η · ε̇_ij
Discretised using an objective stress rate:
τ^new_ij = 2η_ve · (ε̇_ij + τ⁰_ij,rotated / (2·G·Δt))