Question

For a flowing fluid, a dimensionless combination of velocity (V), length scale (l), and acceleration due to gravity (g) would be

• A. \( \frac{V^2}{gl} \)
• B. \( \frac{Vg}{l} \)
• C. \( \frac{g l^2}{V} \)
• D. \( \frac{l}{V^2g} \)

Hint:

When dealing with dimensionless quantities, always check the dimensional consistency of the equation to ensure that it is dimensionless.

Answer 1

The Correct Option is A

In fluid mechanics, dimensionless groups are often used to analyze the behavior of flowing fluids. 
The combination \( \frac{V^2}{gl} \) is a dimensionless number that describes the relationship between the velocity (\( V \)), length scale (\( l \)), and the acceleration due to gravity (\( g \)).

To check the dimensional correctness:

\[ \left[ \frac{V^2}{gl} \right] = \frac{\left[ {velocity} \right]^2}{\left[ {acceleration} \right] \times {length}} = \frac{\left( \frac{m}{s} \right)^2}{\left( \frac{m}{s^2} \right) \times m} = \frac{m^2/s^2}{m^2/s^2} = 1, \]

which confirms that the expression is dimensionless.

Therefore, the correct answer is option (A).

Other options do not form a dimensionless group, which is required for this problem.