Question

An incompressible fluid is flowing through a vertical pipe (height h and crosssectional area 𝐴_𝑜). A thin mesh, having n circular holes of area 𝐴_ℎ is fixed at the bottom end of the pipe. The speed of the fluid entering the top-end of the pipe is 𝑣_𝑜. The volume flow rate from an individual hole of the mesh is given by:
(g is the acceleration due to gravity)

• A. \(\frac{A_0}{n}\sqrt{v^2_o+2gh}\)
• B. \(\frac{A_0}{n}\sqrt{v^2_o+gh}\)
• C. \(n(A_O-A_h)\sqrt{v^2_o+2gh}\)
• D. \(n(A_O-A_h)\sqrt{v^2_o+gh_o}\)

Answer 1

The Correct Option is A

We apply the principle of conservation of energy and the equation for fluid flow to determine the flow rate from a hole in the mesh.

Step 1: Using the Bernoulli Equation

The Bernoulli equation for the flow through the pipe gives the speed of the fluid at the hole as:

\[ v = \sqrt{v_0^2 + 2gh} \]

Here:

• \( v_0 \): Speed of the fluid at the top end of the pipe
• \( h \): Height difference from the hole to the fluid’s source

Step 2: Flow Rate for an Individual Hole

The flow rate \( Q \) for an individual hole is given by the area \( A_h \) multiplied by the velocity \( v \):

\[ Q = A_h v = A_h \sqrt{v_0^2 + 2gh} \]

Step 3: Total Flow Rate for \( n \) Holes

The total flow rate through all \( n \) holes is:

\[ Q_{\text{total}} = n \times A_h \times \sqrt{v_0^2 + 2gh} \]

Step 4: Using the Total Cross-Sectional Area

Since the total cross-sectional area of the pipe is \( A_0 \), we use \( A_0 \) in place of \( A_h \) for the total flow rate per hole. The final expression for the volume flow rate from an individual hole is:

\[ \frac{A_0}{n} \]

Conclusion:

Thus, the correct answer is option (A).

The velocity of fluid flow is given by:

\[ v = \sqrt{v_0^2 + 2gh} \]