# Feel++

## Incompressible Navier-Stokes model

Navier-Stokes model is used to modelise incompressible Newtonian fluid. It can be describe by these conservative laws :

• momentum conservation equation

\rho_{f} \left. \frac{\partial\mathbf{u}_f}{\partial t} \right|_\mathrm{x} + \rho_{f} \left( \boldsymbol{u}_{f} \cdot \nabla_{\mathrm{x}} \right) \boldsymbol{u}_{f} - \nabla_{\mathrm{x}} \cdot \boldsymbol{\sigma}_{f} = \boldsymbol{f}^t_f , \quad \text{ in } \Omega^t_f \times \left[t_i,t_f \right]

• mass conservation equation

\nabla_{\mathrm{x}} \cdot \boldsymbol{u}_{f} = 0 , \quad \text{ in } \Omega^t_f \times \left[t_i,t_f \right]

We add to them the material constitutive equation :

\boldsymbol{\sigma}_{f} = -p_f \boldsymbol{I} + \mu_f (\nabla_\mathrm{x} \boldsymbol{u}_f + (\nabla_\mathrm{x} \boldsymbol{u}_f)^T)

 Notation Quantity Unit \rho_f fluid density kg/m^3 \boldsymbol{u}_f fluid velocity m/s \boldsymbol{\sigma}_f fluid stress tensor N/m^2 \boldsymbol{f}^t_f source term kg/(m ^3 \times s) p_f pressure fields kg/(m \times s^2) \mu_f dynamic viscosity kg/(m \times s) \bar{U} characteristic inflow velocity m/s \nu kinematic viscosity m^2/s L characteristic length m