Fluid Mechanics Applications/B30:Study and Analysis of vorticity

In our day to day life,Most of the fluid flows with which we are familiar like from bathtubs to swimming pools, are not rotating, or they are rotating so slowly that rotation is not important except maybe at the drain of a bathtub as water is let out. As a result, we do not have a good intuitive understanding of rotating flows. In the ocean, rotation and the conservation of vorticity strongly influence flow over distances exceeding a few tens of kilometers. The consequences of the rotation leads to results we have not seen before in our day-to-day dealings with fluids. Vorticity is something that explore some of the consequences of rotation.

In simple words, vorticity is the rotation of fluid.

Vorticity is a-

(i) fundamental property of fluid flows

(ii) is a notable feature of many flows

(iii)it leads to novel insight that can be generalized.

Mathematical Definition
The vorticity is defined as the curl of the velocity vector.It is denoted by the symbol "$$\vec{v}\,$$".

Therefore, it can be written as- $$\omega = \nabla \times \vec{v}\,$$

Thus, each point in the fluid has an associated vector vorticity, and the whole fluid space may be thought of as being threaded by vortex lines which are everywhere tangent to the local vorticity vector. These vortex lines represent the local axis of spin of the fluid particle at each point. In two dimensions, the vorticity is the sum of angular velocities of any pair of mutually perpendicular, infinitesimal fluid lines passing through the point in question. For rigid body rotation, every line perpendicular to the axis of rotation has the same velocity: therefore the vorticity is the same at every point, and is twice the angular velocity.

$$\omega = 2 \times \Omega\,$$