
(a)
The differential equation for the free surface height during draining with respect to time.

Answer to Problem 3.142P
The differential equation for the free surface height during draining with respect to time is
Explanation of Solution
Given information:
The diameter of the cylindrical tank is
The Figure-(1) shows a cylinder of diameter
Figure-(1)
Write the expression for the Bernoulli’s equation between the section (1) and section (2) of the given system for no losses in the system.
Here, the pressure at section (1) is
Since the pressure at the section (1) and section (2) is atmospheric, so the difference between the pressures is zero.
Substitute
Write the expression for the flow rate at section (1).
Here, the area at section (1) is
Write the expression for the area at section (1).
Here, the diameter at section (1) is
Substitute
Write the expression for the flow rate at section (2).
Here, the flow rate is
Write the expression for the area at section (2).
Here, the diameter at section (2) is
Substitute
Since the flow rate at both the sections is equal, it satisfies the continuity equation.
Substitute
Write the expression for velocity at section (2).
Here, the water level is
Substitute
Substitute
The negative sign is due to decrease in the head level in the tank.
Conclusion:
The differential equation for the free surface height during draining with respect to time is
(b)
The expression for the time

Answer to Problem 3.142P
The expression for the time
Explanation of Solution
Given information:
The diameter of the cylindrical tank is
Write the differential equation for the free surface height.
Integrate the above expression on both the sides.
Apply boundary conditions to Equation (VIII).
At
Substitute
Substitute
Since the tank is empty after time
Conclusion:
The force per unit width of the water is
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Chapter 3 Solutions
Fluid Mechanics
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