Page 406 - Introduction to Continuum Mechanics

390 irrotational Flow

6.20 Irrotational Flow

If the vorticity vector (or equivalently, vorticity tensor) corresponding to a velocity field, is
zero in some region and for some time interval, the flow is called irrotational in that region
and in that time interval.
Let <P(XI, x 2, *3, t) be a scalar function and let the velocity components be derived from <p
by the following equation:

i.e.,

Then the vorticity component

and similarly

That is, a scalar function <P(XI, x 2, x$, t) defines an irrotational flow field through the
Eq. (6.20.2). Obviously, not all arbitrary functions <p will give rise to velocity fields that are
physically possible. For one thing, the equation of continuity, expressing the principle of
conservation of mass, must be satisfied. For an incompressible fluid, the equation of continuity
reads

Thus, combining Eq. (6.20.2) with Eq. (6.20.3), we obtain the Laplacian equation for <f>,

i.e.,