Rotational and Irrotational Motion

When a fluid element is subjected to a shearing force, a velocity gradient is produced perpendicular to the direction of shear, that is, a relative motion occurs between two layers. To encounter this relative motion the fluid elements have to undergo rotation. A typical example of this type of motion is the motion between two roller chains rubbing each other, but moving at different velocities. It is convenient to use an abstract quantity called circulation Г, defined as the line integral of velocity vector between any two points (to define rotation of the fluid element) in a flow field. By definition:

where dl is an elemental length, c is the path of integration.
Circulation per unit area is known as vorticity Z,

Z = Г/A.

In vector form, Z becomes:

where V is the flow velocity, given by V = i Vx + j Vy, and:

d d

V = i——- + j —.

dx dy

For a two-dimensional flow in xy-plane, vorticity Z becomes:

dVL_dV1

z dx dy

where Zz is the vorticity about the z-direction, which is normal to the flow field. Likewise, the other components of vorticity about x – and y-directions are:

_ dVz dVy

Zx ^ …

dy dz xz

y dz dx

If Z = 0, the flow is known as irrotational Bow. Inviscid flows are basically irrotational flows.