Fluid dynamics equations, number of variables and number of equations

Question

Continuity and Navier-Stokes equation for fluid are, \begin{eqnarray} \frac{\partial \rho}{\partial t} + \nabla\cdot (\rho \mathbf{u}) &=& 0 \\ \rho\left(\frac{\partial \mathbf{u}}{\partial t} + \mathbf{u}\cdot \nabla u \right) &=& -\nabla p + \nabla \cdot\sigma + \mathbf{{F}_{ext}}, \end{eqnarray}

where letters have the usual meaning. In total we have 4 equations (one continuity equation and 3 momentum balance components of Navier-Stokes) available and number of unknowns are 5 (pressure+ density, 3 velocity components). How can we then determine all the 5 variables in general?

Answer 1

The missing equation is energy conservation $$ \frac{\partial}{\partial t} {\cal E} + \vec\nabla \cdot\vec\jmath^{\,\cal E}=0 $$ where ${\cal E}$ is the energy density and $\vec\jmath^{\,\cal E}$ is the energy current $$ \vec\jmath^{\,\cal E} = \vec{u}\left[ {\cal E}+P \right] -\eta u\cdot\sigma-\kappa\vec\nabla T\, . $$ Now the equations close if you have an equation of state, $P=P({\cal E}^0,\rho)$, where ${\cal E}^0={\cal E}-\frac{1}{2}\rho u^2$ is the internal energy density. Note that the equation of state also fixes $T({\cal E}^0,\rho)$ using thermodynamic identities (although this is tedious in practice; for a non-interacting gas things are simple, $T=mP/\rho$). The energy equation can be rewritten in various ways, for example as an equation for entropy production.

Answer 2

Assuming we want to calculate "standard" fluid dynamics - compressible single phase flow in 3D space - the variables to solve are:

• Pressure
• Density
• Temperature
• Velocity (3 components in 3 dimensions)

a total of 6 components.

The equations for conservative states are:

• continuity / mass conservation - one component)
• momentum conservaiton (3 components - in 3D, more general 1 component per dimension)
• energy conservation

a total of 5 equations, and (as pointed out before) the 6th equation will be the equation of state (could be ideal gas law..)

Depending on the task at hand, the total number of equation and states will change:

• For lower dimensions (1D or 2D), the velocity vector reduces components by the same amount as the momentum equation does.
• Incompressible flow removes one state and one equation
• For more sophisticated calculations, often extra equations and states are added, e.g. for turbulence modelling (see k-epsilon), multiphase flow, or when adding chemistry