Wall shear distribution in fully developed laminar pipe flow: For a viscous fluid in a circular pipe, where is the shear stress magnitude maximum across the cross-section?

Introduction / Context:

Understanding shear stress distribution is crucial for estimating friction losses and wall stresses in internal flows. In steady, fully developed laminar flow through a circular pipe, velocity is parabolic and shear stress varies linearly from the centre to the wall.

Given Data / Assumptions:

• Newtonian fluid; steady, incompressible, fully developed laminar flow.
• No slip at the wall; axisymmetric conditions.

Concept / Approach:

From the momentum balance, shear stress τ(r) varies with radius r as τ(r) = −(dp/dx) * r / 2, where dp/dx is the (negative) pressure gradient. Thus, τ increases linearly with r, reaching zero at the centre (r = 0) and maximum at the pipe wall (r = R).

Step-by-Step Solution:

Verification / Alternative check:

The parabolic velocity profile v(r) = v_max * (1 − (r/R)^2) has derivative dv/dr proportional to −r, which is largest in magnitude at the wall, consistent with τ = μ dv/dr being maximum there.

Why Other Options Are Wrong:

• Uniform shear contradicts the linear τ(r) dependence on r.
• Maximum at centreline is incorrect; τ(0) = 0.
• Zero at wall (no-slip means zero velocity, not zero shear).

Common Pitfalls:

• Confusing velocity (zero at wall) with shear stress (maximum at wall).

Final Answer:

Maximum at the inner wall (pipe surface)