In a linear, viscous, incompressible fluid, the Cauchy stress is of the form
where p is the pressure, δij equals 0 if i≠j and 1 if i = j, and the comma followed
by an index indicates the partial derivative with respect to that variable and v is
the velocity. Thus vi,j = . Also,
p denotes the pressure. Show, using the balance
of mass equation that incompressible implies div v = 0. Next show that the balance
of momentum equation requires
This is the famous Navier Stokes equation for incompressible viscous linear fluids.
There are still open questions related to this equation, one of which is worth
$1,000,000 at this time.