Solved Problems
Example
Problem: Water flows through a pipe that tapers from a diameter of 200 mm to 100 mm. If the velocity at the entrance is 2 m/s, calculate the discharge and the velocity at the exit.
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Example
Problem: A 300 mm pipe carrying 100 L/s branches into two pipes of 150 mm and 200 mm diameters. If the velocity in the 150 mm pipe is 2 m/s, what is the velocity in the 200 mm pipe?
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Example
Problem 3: Continuity Equation in a Converging Pipe
Water flows through a converging pipe. At section 1, the diameter is and the velocity is . At section 2, the diameter is . Calculate the velocity at section 2 and the discharge in .
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Example
Problem 4: Acceleration Field
The velocity field of a fluid is given by . Find the acceleration of a fluid particle at the point .
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Case Study 1: River Convergence and Erosion
Context: Natural channels often feature converging sections.
Application: As a river narrows (converges), the continuity equation dictates that the flow velocity must increase. This increased velocity leads to higher shear stress on the riverbed and banks, accelerating erosion in that specific reach. Engineers use kinematic principles to predict these zones of high velocity and implement countermeasures like riprap or retaining walls to protect vulnerable infrastructure.
Case Study 2: Wind Tunnel Testing of Vehicles
Context: Understanding flow patterns around objects is crucial in aerodynamics.
Application: By introducing smoke trails (streaklines) into a wind tunnel, engineers visualize the flow field around a car or aircraft model. Analyzing these kinematic patterns helps identify areas of flow separation and turbulence, which contribute to aerodynamic drag. Streamlining the vehicle's shape to maintain attached, smooth streamlines improves fuel efficiency and stability at high speeds.