The correct answer is option 'C' - vorticity.

Vorticity is a concept in fluid dynamics that measures the local spinning motion of fluid particles. It is a vector quantity defined as the curl of the velocity field. Vorticity is an important parameter for understanding the flow characteristics of a fluid.

To understand why vorticity is the correct answer in this context, let's break down the information given in the question:

1. The limiting value of circulation: Circulation is a measure of the flow of fluid around a closed contour. It is defined as the line integral of the velocity field along the contour. The limiting value of circulation refers to the maximum value that circulation can reach.

2. Dividing the limiting value of circulation by area of closed contour: Dividing the limiting value of circulation by the area of the closed contour gives us a quantity that represents the circulation per unit area.

Now, let's see how this relates to vorticity:

- Vorticity is directly related to circulation. The circulation around a closed contour can be expressed as the double integral of the vorticity over the area enclosed by the contour.
- Dividing the circulation by the area of the closed contour gives us the average vorticity per unit area.

Therefore, the value obtained from dividing the limiting value of circulation by the area of the closed contour is the average vorticity per unit area. This is why the correct answer is option 'C' - vorticity.

Other options provided in the question:

- Option 'A' - potential function: A potential function is a scalar field that satisfies certain conditions in fluid dynamics, but it is not directly related to circulation or vorticity.
- Option 'B' - stream function: A stream function is another scalar field used to describe fluid flow, but it is also not directly related to circulation or vorticity.
- Option 'D' - None of these: This option is incorrect because the correct answer is option 'C' - vorticity.