The behavior of the fluid is described by dependent variables for the pressure p and velocity v. In the fluid, the density ρ is determined by the pressure p and temperature T through the equation of state,

where κT is the isothermal compressibility and αp is the (isobaric) coefficient of thermal expansion. The pressure and velocity fields are governed by the transport equations for mass and momentum. The continuity equation governs the conservation of mass,

Here σ is the stress tensor as is given as

Where the pressure is on the diagonal and τ is the viscous stress tensor.