In fluid mechanics, when will shear stress develop on a fluid element? Choose the condition that actually produces shear (i.e., a velocity gradient) within the fluid.
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AThe fluid and its container are subjected to uniform linear acceleration (no relative motion inside the fluid)
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BThe fluid is inviscid (zero viscosity, cannot sustain shear)
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CThe fluid is viscous and the flow is non-uniform, creating a velocity gradient
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DThe fluid is at rest under static conditions
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EAll of the above situations produce shear
Answer
Correct Answer: The fluid is viscous and the flow is non-uniform, creating a velocity gradient
Explanation
Introduction / Context:Understanding when shear stress develops inside a fluid is fundamental for civil, mechanical, and environmental engineering applications. Shear arises from relative motion between adjacent fluid layers, which requires both viscosity and a velocity gradient. This concept underpins pipe-flow headloss, boundary layers on structures, and open-channel resistance.
Given Data / Assumptions:
- Shear stress in a Newtonian fluid is proportional to viscosity and the rate of deformation (velocity gradient).
- Static fluids or fluids under uniform linear acceleration experience only pressure fields, not internal shear.
- Inviscid models assume zero viscosity and therefore no shear resistance.
Concept / Approach:For Newtonian fluids, the constitutive relation is: shear_stress = mu * (du/dy). If mu = 0 (inviscid) or du/dy = 0 (no velocity gradient), shear_stress = 0. Only when mu > 0 and the velocity varies spatially does shear arise. Hence, viscous, non-uniform flow is the necessary condition for shear inside the fluid continuum.
Step-by-Step Solution:
Identify the requirement for shear: both viscosity (mu > 0) and a velocity gradient (du/dy ≠ 0).Check uniform linear acceleration: fluid moves as a rigid body; du/dy = 0 → no shear.Check inviscid assumption: mu = 0 → no shear regardless of velocity field.Conclude that viscous, non-uniform flow produces shear due to finite mu and nonzero du/dy.Verification / Alternative check:
Classic examples—Couette flow and Poiseuille flow—show shear proportional to velocity gradient; remove the gradient or viscosity and shear vanishes.Why Other Options Are Wrong:
Uniform linear acceleration: induces hydrostatic-like pressure variation only.Inviscid fluid: by definition cannot support shear stress.Fluid at rest: no motion, no gradient, hence no shear.Common Pitfalls:
Confusing body accelerations with internal deformation; only deformation rates create shear in viscous fluids.Final Answer:
The fluid is viscous and the flow is non-uniform, creating a velocity gradient