Water flows through a horizontal pipe that narrows from area A₁ = 0.04 m² to A₂ = 0.01 m². If the inlet velocity is V₁ = 2 m/s, what is the outlet velocity V₂?
A0.5 m/s — the velocity decreases as the pipe narrows to maintain pressure
B2 m/s — velocity is constant through a steady flow pipe
C8 m/s — volume flow rate is conserved, so A₁V₁ = A₂V₂
D4 m/s — cross-sectional area decreases by factor 4, so velocity doubles
For steady, incompressible flow, Q = A₁V₁ = A₂V₂. With A₁ = 0.04 m² and V₁ = 2 m/s, Q = 0.08 m³/s. At outlet, V₂ = Q/A₂ = 0.08/0.01 = 8 m/s. The area decreased by a factor of 4, so velocity increased by a factor of 4. Option A inverts the relationship — velocity increases, not decreases, when area decreases. Option D has an arithmetic error: it correctly identifies the factor-of-4 area decrease but incorrectly says velocity only doubles.
Question 2 Multiple Choice
A pipe branches at a T-junction into two outlet pipes. The inlet carries water at ṁ_in = 10 kg/s. One outlet carries ṁ_out,1 = 6 kg/s. Without knowing the pipe geometry or velocities, what is ṁ_out,2?
A4 kg/s — from Σṁ_in = Σṁ_out for steady flow
B6 kg/s — the second outlet matches the first by symmetry
C10 kg/s — each outlet receives the full inlet mass flow
DCannot be determined without knowing the pipe diameters
For steady flow, mass cannot accumulate inside the control volume: Σṁ_in = Σṁ_out. Therefore 10 = 6 + ṁ_out,2, giving ṁ_out,2 = 4 kg/s. No knowledge of pipe geometry, fluid velocity, or internal flow details is needed — the control volume method only requires boundary values. Option D is wrong precisely because the control volume method eliminates the need for interior knowledge; only what crosses the boundary matters.
Question 3 True / False
In steady, incompressible flow through a pipe that narrows, the fluid velocity decreases at the narrower section to conserve momentum.
TTrue
FFalse
Answer: False
In steady, incompressible flow, it is VOLUME FLOW RATE (Q = AV) that is conserved, not momentum. When area A decreases, velocity V must INCREASE to maintain constant Q. A narrowing (nozzle) accelerates the flow; a widening (diffuser) decelerates it. The misconception often arises from intuition about traffic — fewer lanes, slower speed. But fluid is incompressible and continuous: it cannot queue up at a narrowing, so each element speeds up to pass through the smaller opening at the same volumetric rate.
Question 4 True / False
The steady-flow mass balance for a control volume applies regardless of the complexity of the flow inside the boundary, including turbulence, swirling, and heat transfer.
TTrue
FFalse
Answer: True
The control volume mass balance is a macroscopic conservation law — it only requires that mass is conserved (always true) and that conditions are steady (nothing accumulates inside). The internal flow structure, however complex, is irrelevant. Turbulence, swirling, mixing, chemical reactions, heat transfer inside the CV — none of these affect the boundary accounting. This is the fundamental power of the control volume framework: it reduces complex interior problems to boundary-condition bookkeeping.
Question 5 Short Answer
Explain why the control volume method is useful for fluid mechanics problems, and what 'steady flow' allows you to simplify.
Think about your answer, then reveal below.
Model answer: The control volume method replaces the need to track every fluid particle with a boundary accounting approach: you define an imaginary surface around a region and only track what crosses it. This is useful because real fluid flows inside complex geometries are often intractable to solve in detail. The 'steady flow' simplification eliminates time-dependence: with steady conditions, nothing accumulates inside the CV, so whatever flows in must immediately flow out. This gives Σṁ_in = Σṁ_out — a simple algebraic statement that can solve for unknown velocities or areas at any port without solving the interior flow field at all.
For incompressible flow, density cancels and the equation becomes Σ(AV)_in = Σ(AV)_out. This is why engineers can design pipe networks, pumps, and nozzles using only inlet and outlet conditions — the control volume framework makes the interior a black box.