Page 412 - Introduction to Continuum Mechanics

396 Vorticity Transport Equation for Incompressible Viscous Fluid with a Constant Density

It should be noted that even though the viscous terms drop out from the Navier-Stokes
equations in the case of irrotational flows, it does not mean that there is no viscous dissipation
in an irrotational flow of a viscous fluid. In fact, so long as there is one nonzero rate of
deformation component, there is viscous dissipation [given by Eq. (6.17.4)] and the rate of
work done to maintain the irrotational flow exactly compensates the viscous dissipations.

6.23 Vorticity Transport Equation for Incompressible Viscous Fluid with a
Constant Density

In this section, we derive the equation governing the vorticity vector for an incompressible
homogeneous viscous fluid. First, we assume that the body force is derivable from a potential
BQ dQ
Q , i.e., B; = ——. Now, with/? = constant and B; = ——, the Navier-Stokes equation can
n
dx; dXj
be written

where v ~ p/p is called the kinematic viscosity. If we operate on Eq. (6.23,1) by the differential
operator e mm—— [i.e, taking the curl of both sides of Eq. (6.23.1)]. We have, since
ox n

and

The Navier-Stokers equation therefore, takes the form