Understanding Velocity Gradient

The velocity gradient is a concept used in fluid dynamics. It describes how the velocity of a fluid changes as you move perpendicular to the direction of flow. Imagine layers of fluid flowing over each other; the velocity gradient tells us how much the velocity differs between adjacent layers.

More formally, the velocity gradient is defined as the rate of change of velocity with respect to the distance measured perpendicular to the direction of flow. If \(v\) is the velocity and \(y\) is the distance perpendicular to the flow, the velocity gradient is given by:

\( \text{Velocity Gradient} = \frac{dv}{dy} \)

This means the velocity gradient is calculated by dividing the change in velocity (\(dv\)) by the change in distance (\(dy\)).

Formula of Velocity Gradient

Based on the definition, the formula for velocity gradient involves velocity divided by distance. We can express this relationship dimensionally or conceptually as:

• Conceptually: Velocity Gradient \( \propto \) Velocity / Distance
• Dimensionally: Velocity has dimensions \( [\text{L}][\text{T}]^{-1} \). Distance has dimensions \( [\text{L}] \).

Therefore, the dimensions of velocity gradient are \( \frac{[\text{L}][\text{T}]^{-1}}{[\text{L}]} = [\text{T}]^{-1} \). This means velocity gradient has the dimension of inverse time.

Looking at the options provided, we need to find the one that represents velocity divided by distance. The expression "velocity \( \times \) [distance]\(^{-1}\)" is mathematically equivalent to \( \frac{\text{velocity}}{\text{distance}} \).

• Option 1: velocity \( \times \) [distance]\(^{-1}\) \( = \frac{\text{velocity}}{\text{distance}} \)
• Option 2: force \( \times \) [distance]\(^{-2}\) involves force, not typically part of velocity gradient formula.
• Option 3: velocity \( \times \) [distance]\(^{1}\) \( = \text{velocity} \times \text{distance} \) which is incorrect.
• Option 4: force \( \times \) [distance]\(^{2}\) involves force and distance squared, which is incorrect.

Thus, the formula of velocity gradient corresponds to velocity multiplied by the inverse of distance, or velocity divided by distance.

The formula aligns with the definition and the dimensions of velocity gradient.