Alternative derivation of the Navier-Stokes equation (skip on a first reading) •It can be shown (Faber page 196-198) that the isotropy of the fluid, the symmetry of the shear stress and the linearity between stress and strain rate imply, with similar equations for 2 and 3. •For incompressible fluids, the equations may be re-written, 27 NAVIER-STOKES EQUATION CHARLES L. FEFFERMAN The Euler and Navier-Stokes equations describe the motion of a fluid in Rn (n = 2 or 3). These equations are to be solved for an unknown velocity vector u(x,t) = (u i(x,t)) 1≤i≤n ∈ Rn and pressure p(x,t) ∈ R, defined for position x ∈ Rn and time t ≥ 0. Basic assumptions. The Navier-Stokes equations are based on the assumption that the fluid, at the scale of interest, is a continuum - a continuous substance rather than discrete particles. Another necessary assumption is that all the fields of interest including pressure, flow velocity, density, and temperature are at least weakly differentiable. Stokes equations. Later, we will turn to the text of a computer program that embodys these ideas. Going through the text of that program, we will explain how Newton's method, the Galerkin procedure, the nite element approximation, and the Naver Stokes equations all come together. 2 The Time-Dependant Compressible Navier Stokes Equations A uid