Vorticity Equation in Polar Coordinates

In the polar coordinates, the vorticity equation can be expressed as:

Z qt dqt 1 dqn

r dr r дв ’

where r and в are the polar coordinates and qt and qn are the tangential and normal components of velocity, respectively. The derivation of Equation (5.2) is given in Section 5.3.

If (r, в, n) are the radial, azimuthal and normal coordinates of a polar coordinates system, the vorticity expression is given by:

Z = irZr + ІвС + ^^,

where ir, ie and in are the unit vectors in the directions of r, в and n, respectively. The vorticity components can be expressed as:

1 du„

Z _ 1 d(rue)

Zn л

r dr

where ur, ug and un are the velocity components along r, в and n directions, respectively.

Example 5.1

Find the vorticity of the following flows.

(a)

V = c(x + y)i — c(x + y)j,