Which one of the following statements is correct ? When a fluid passes through the narrow part of non-uniform pipe :

Options

A. Its velocity and pressure both increase
B. Its velocity and pressure both decrease
C. Its velocity decreases but its pressure increase
D. Its velocity increases but its pressure decreases

Answer

To solve the question regarding the behavior of fluid in a non-uniform pipe, we can follow these steps:

### Step 1: Understand the Continuity Equation
In fluid mechanics, the continuity equation states that for an incompressible fluid, the mass flow rate must remain constant from one cross-section of a pipe to another. This can be expressed as:
\[ A_1 v_1 = A_2 v_2 \]
where:
- \( A_1 \) and \( A_2 \) are the cross-sectional areas of the pipe at two different points,
- \( v_1 \) and \( v_2 \) are the fluid velocities at those points.

### Step 2: Analyze the Pipe Sections
In a non-uniform pipe, if one section is narrower than the other:
- If \( A_1 > A_2 \) (the wide part), then \( v_1 < v_2 \) (the narrow part). This means that the fluid velocity increases as it moves from the wider section to the narrower section.

### Step 3: Apply Bernoulli's Principle
According to Bernoulli's principle, for a fluid flowing in a horizontal pipe, the total mechanical energy per unit volume remains constant. This can be expressed as:
\[ P + \frac{1}{2} \rho v^2 = \text{constant} \]
where:
- \( P \) is the pressure,
- \( \rho \) is the fluid density,
- \( v \) is the fluid velocity.

### Step 4: Relate Velocity and Pressure
From Bernoulli's equation, if the velocity of the fluid increases (as we established in Step 2), the pressure must decrease. Thus, if \( v_2 > v_1 \), then \( P_2 < P_1 \).

### Step 5: Conclusion
Based on the analysis:
- The fluid velocity increases in the narrow part of the pipe.
- The pressure in the narrow part decreases.

Therefore, the correct statement is: **When a fluid passes through the narrow part of a non-uniform pipe, the velocity increases and the pressure decreases.**

### Final Answer
The correct option is **D: Velocity increases, pressure decreases.**

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