The velocity potential function for a source is a mathematical representation of the velocity field around a source. In fluid dynamics, a source refers to a point in space where fluid is being emitted or flowing outwards. The velocity potential function describes the velocity of the fluid at any point in space.

The velocity potential function for a source varies with the distance r. This means that as we move away from the source, the velocity potential changes. The given options are:

a) 1/r
b) 1/r^2
c) e^r
d) In r

The correct answer is option 'd' - In r.

Explanation:
To understand why the correct answer is 'In r', let's consider the nature of the velocity potential function for a source.

The velocity potential function for a source can be derived using potential flow theory. In potential flow, the fluid is assumed to be inviscid (no viscosity) and irrotational (no vorticity). This means that the fluid has no internal friction and there are no swirling motions.

For a source, the velocity potential function can be expressed as:

Φ = Q * ln(r)

Where:
Φ is the velocity potential function
Q is the strength of the source
ln(r) is the natural logarithm of the distance r from the source

From this equation, we can see that the velocity potential function is proportional to the natural logarithm of the distance. This is why the correct answer is 'In r'.

The natural logarithm function, ln(r), is commonly used to describe a wide range of phenomena in physics and mathematics. It has the property that it grows slowly as the argument (r in this case) increases. This makes it suitable for describing the decrease in velocity potential as we move away from the source.

In summary, the velocity potential function for a source varies with the distance r as In r. This means that the velocity potential decreases logarithmically as we move away from the source.