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