Internal laminar pipe flow: how does shear stress vary across the cross-section of a circular pipe carrying a Newtonian fluid?

Introduction:
Understanding radial distributions of velocity and shear stress in laminar pipe flow is essential for predicting pressure drop and wall shear stress, which affect energy consumption and scaling/erosion risks.

Given Data / Assumptions:

• Newtonian, incompressible, steady, fully developed laminar flow in a circular pipe.
• No-slip at the wall.

Concept / Approach:
For fully developed laminar flow, the velocity profile is parabolic and the shear stress varies linearly with radius. Shear stress is zero at the pipe centerline (symmetry) and reaches a maximum at the wall, equal to the wall shear τ_w.

Step-by-Step Solution:
Momentum balance gives dP/dx = constant; τ(r) = −(r/2)(dP/dx).Because dP/dx is constant, τ(r) ∝ r.At r = 0 (centre), τ = 0; at r = R (wall), τ = τ_w = −(R/2)(dP/dx).Hence linear increase from centre to wall.

Verification / Alternative check:
The parabolic velocity profile u(r) = U_max (1 − r^2/R^2) differentiates to du/dr ∝ −r; for a Newtonian fluid τ = μ du/dr ∝ r, confirming linear variation.

Why Other Options Are Wrong:

• Parabolic variation: applies to velocity, not shear stress.
• Uniform: contradicts the linear relation.
• Zero at wall: violates no-slip driven gradient; shear is maximal there.
• Maximum at centre: opposite of reality.

Common Pitfalls:
Mixing up velocity and shear stress shapes; forgetting symmetry implies zero shear at the centreline.

Final Answer:
Zero at the centre and increases linearly to a maximum at the wall