Numerical analysis of laminar flow over a step by a finite element method with divergence free elements

Abstract

The stationary Navier Stokes equations in the absence of temperature gradients are given by the momentum equations:

$$ - {\sigma _{ij,j}} + \rho {({u_i}{u_j})_{,j}} = \rho {f_i}$$

(1)

and the continuity equation:

$${u_{i,i}} = 0i = 1,2$$

(2)

For a Newtonian fluid the Cauchy stress tensor can be written as:

$${\sigma _{ij}} = - p{\delta _{ij}} + \eta ({u_{i,j}} + {u_{j,i}})$$

(3)

with n the dynamic viscosity.