Question

Many figures in the text show streamlines. Explain why fluid velocity is greatest where streamlines are closest together. (Hint: Consider the relationship between fluid velocity and the cross-sectional area through which it flows.).

Short answer

The fluid velocity is greatest for the closest streamlines due to the lowest cross-sectional area for the closest position.

Step-by-step solution

Relationship between fluid velocity and cross-sectional area

The flow rate of a fluid flow is given as:

Q = AV

Here, A is the cross-sectional area and V is the fluid velocity.

For constant flow rate, the cross-sectional area is inversely related to fluid velocity. It shows that fluid velocity of fluid increases with a decrease in the cross-sectional area.

Reason for the greatest fluid velocity at the closest streamlines

The distance between streamlines is very less for the closest position so the cross-sectional area for the closest streamlines is very less. The fluid velocity becomes greatest at the closest position due to the lowest cross-sectional area.