18.6. Examples in GATE Examinations

Problem Description: In the GATE 2006 civil engineering exam, the velocity field is given with a scalar component of the velocity field where lambda is unknown. Density varies with respect to time, indicating it is an unsteady problem.

Flow Classification:

• One-dimensional
• Unsteady
• Laminar
• Fixed control volume
• Compressible

Given Data:

• Velocity field:
• Density varies with respect to time.

Objective: Find the value of lambda such that mass is conserved.

In this problem, we have a defined velocity field with a certain variable, lambda, that we need to determine. The problem states that the density of the fluid is changing over time, indicating that it is an unsteady flow situation. We classify the type of flow to set the applicable equations and assumptions. Since we are dealing with a one-dimensional, fixed control volume that experiences compressibility, we can relate the changes in density with the flow characteristics using the mass conservation equations. By applying these equations, we can calculate the value of lambda by ensuring that the mass remains conserved across the specified control volume.

Think of this problem like a bank account where money is constantly being deposited (inflows) and withdrawn (outflows). If you need to find out how much money you have at the end of a certain period, you have to account for all deposits and withdrawals made during that time. Similarly, in fluid mechanics, to keep track of the mass or fluid in a volume, you must account for the inflows and outflows, leading to a condition where you can solve for unknown variables like lambda.

Problem Description: In GATE 2012, a simple pipe problem involving a T-joint is presented. Water inflow and outflows are provided with the need to find the velocity in the branch pipe 'R'.

Flow Classification:

• One-dimensional
• Steady
• Laminar
• Fixed control volume
• Incompressible flow

Given Data:

• Pipe Branch P: Diameter (D) = 4m, Velocity (V) = 6 m/s
• Pipe Branch Q: Diameter (D) = 4m, Velocity (V) = 5 m/s
• Pipe Branch R: Diameter (D) = 2m, Velocity (V) = ? m/s

In this example, we have a branching pipe system where water flows into the main pipe and divides into two branches (Q and R). To find the velocity in branch R, we employ the principle of conservation of mass. For steady flow, the mass coming in must equal the mass going out. Here, we can state that the inflow into the control volume equals the sum of the outflows from branches Q and R. Using the equation of flow (Q = A * V), where A is the cross-sectional area, we can express the flow rates and solve for the unknown velocity in branch R. This approach simplifies the analysis due to the assumptions of incompressibility and the defined steady-state conditions.

Imagine a water park slide where you have one big slide that splits into two smaller slides. The water flowing down the big slide must split equally between the two smaller slides. If you know how fast the water is flowing down the bigger slide, you can find out how fast the water is flowing down each of the smaller slides based on how wide each slide is. Similarly, in this problem, by considering the flow rates at the junction, we can deduce the unknown velocity in one of the branches.

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