Effect of Area Reduction on Velocity in Incompressible Fluid Flow

In incompressible fluid flow, the velocity of the fluid is inversely proportional to the cross-sectional area of the flow channel. This relationship is described by the principle of continuity, which states that the mass flow rate is constant along a streamline.

Principle of Continuity
The principle of continuity states that the mass flow rate of an incompressible fluid remains constant as it flows through a pipe or conduit, provided there are no sources or sinks of fluid along the streamline. Mathematically, it can be expressed as:

A1V1 = A2V2

Where:
- A1 and A2 are the cross-sectional areas of the flow at points 1 and 2, respectively.
- V1 and V2 are the velocities of the fluid at points 1 and 2, respectively.

Effect of Area Reduction on Velocity
When the cross-sectional area of a flow channel decreases, the velocity of the fluid increases to maintain the constant mass flow rate. This can be explained using the principle of continuity.

Explanation:
1. As the cross-sectional area of the flow channel decreases, the available space for the fluid to flow through is reduced.
2. According to the principle of continuity, the mass flow rate must remain constant. Therefore, to compensate for the reduced area, the fluid must flow at a higher velocity.
3. This higher velocity allows the same amount of fluid to pass through the smaller cross-sectional area in the same amount of time, maintaining a constant mass flow rate.
4. Conversely, if the cross-sectional area of the flow channel increases, the fluid velocity decreases to maintain the constant mass flow rate.

Conclusion
In summary, when the cross-sectional area of an incompressible fluid flow channel is reduced, the velocity of the fluid increases to maintain the constant mass flow rate. This relationship is governed by the principle of continuity, which ensures that the mass flow rate remains constant along a streamline.