Continuity for incompressible steady flow in a pipe True or False: For an incompressible liquid flowing continuously through a pipe, the discharge (quantity per second) is different at different sections.
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ATrue
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BFalse
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CDepends only on viscosity
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DTrue only in laminar regime
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ETrue only in turbulent regime
Answer
Correct Answer: False
Explanation
Introduction:The continuity equation is a cornerstone of fluid mechanics. For incompressible steady flow, it asserts constancy of volumetric flow rate along a streamline (and through any cross-section of a pipe), regardless of local changes in area and velocity.
Given Data / Assumptions:
- Incompressible liquid (density constant).
- Steady flow (no accumulation with time).
- No leakage or branching between the two sections being compared.
Concept / Approach:Continuity for incompressible steady flow: A1 * V1 = A2 * V2 = Q (constant). If the pipe diameter changes, velocity adjusts inversely with area to keep Q the same. Therefore, the statement that discharge is “different at different sections” is false.
Step-by-Step Solution:
1) Write continuity: Q = A * V for incompressible steady flow.2) Between any two sections, Q1 = A1 * V1 and Q2 = A2 * V2.3) With no sources/sinks, Q1 = Q2, hence discharge is the same.Verification / Alternative check:Measurement devices placed at different diameters show different velocities but the same volumetric flow rate when the system is closed and steady, validating continuity.
Why Other Options Are Wrong:
- True: contradicts continuity.
- Depends only on viscosity: viscosity affects losses and profiles, not mass conservation.
- True only in laminar/turbulent: regime does not change mass conservation.
Common Pitfalls:Confusing velocity changes with discharge changes. Remember that velocity can vary with area while Q remains constant.
Final Answer:False