Navier-Stokes equations

The Navier-Stokes equations are the basic governing equations for a viscous, heat conducting fluid. It is a vector equation obtained by applying Newtons Law of Motion to a fluid element and is also called the momentum equation. It is supplemented by the mass conservation equation, also called continuity equation and the energy equation. Usually, the term Navier-Stokes equations is used to refer to all of these equations.

The instantaneous continuity equation (1), momentum equation (2) and energy equation (3) for a compressible fluid can be written as:

 $\frac{\partial \rho}{\partial t} + \frac{\partial}{\partial x_j}\left[ \rho u_j \right] = 0$ (1) $\frac{\partial}{\partial t}\left( \rho u_i \right) + \frac{\partial}{\partial x_j} \left[ \rho u_i u_j + p \delta_{ij} - \tau_{ji} \right] = 0$ (2) $\frac{\partial}{\partial t}\left( \rho e_0 \right) + \frac{\partial}{\partial x_j} \left[ \rho u_j e_0 + u_j p + q_j - u_i \tau_{ij} \right] = 0$ (3)

For a Newtonian fluid, assuming Stokes Law for mono-atomic gases, the viscous stress is given by:

 $\tau_{ij} = 2 \mu S_{ij}^*$ (4)

Where the trace-less viscous strain-rate is defined by:

 $S_{ij}^* \equiv \frac{1}{2} \left(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right) - \frac{1}{3} \frac{\partial u_k}{\partial x_k} \delta_{ij}$ (5)

The heat-flux, $q_j$, is given by Fourier's law:

 $q_j = -\lambda \frac{\partial T}{\partial x_j} \equiv -C_p \frac{\mu}{Pr} \frac{\partial T}{\partial x_j}$ (6)

Where the laminar Prandtl number $Pr$ is defined by:

 $Pr \equiv \frac{C_p \mu}{\lambda}$ (7)

To close these equations it is also necessary to specify an equation of state. Assuming a calorically perfect gas the following relations are valid:

 $\gamma \equiv \frac{C_p}{C_v} ~~,~~ p = \rho R T ~~,~~ e = C_v T ~~,~~ C_p - C_v = R$ (8)

Where $\gamma$, $C_p$, $C_v$ and $R$ are constant.

The total energy $e_0$ is defined by:

 $e_0 \equiv e + \frac{u_k u_k}{2}$ (9)

Note that the corresponding expression (15) for Favre averaged turbulent flows contains an extra term related to the turbulent energy.

Equations (1)-(9), supplemented with gas data for $\gamma$, $Pr$, $\mu$ and perhaps $R$, form a closed set of partial differential equations, and need only be complemented with boundary conditions.

Existence and uniqueness

The existence and uniqueness of classical solutions of the 3-D Navier-Stokes equations is still an open mathematical problem. In 2-D, existence and uniqueness of regular solutions for all time have been shown by Jean Leray in 1933. He also gave the theory for the existence of weak solutions in the 3-D case while uniqueness is still an open question.