Velocity distribution in viscous flow through a circular pipe In internal laminar flow of a viscous fluid through a straight circular pipe (steady and fully developed), the axial velocity profile across the diameter is best described as:

Introduction / Context:
Understanding the shape of the velocity profile in pipe flow is fundamental for computing discharge, wall shear, and energy losses. For steady, fully developed laminar flow of a Newtonian fluid in a circular pipe, the profile has a distinctive mathematical form.

Given Data / Assumptions:

• Newtonian fluid with constant viscosity.
• Steady, incompressible, fully developed internal flow.
• Straight, circular pipe; no swirl; no acceleration along the axis.

Concept / Approach:

From the Navier–Stokes equations with the above assumptions, the solution is the Hagen–Poiseuille profile: velocity u decreases from a maximum at the centreline to zero at the wall due to the no-slip condition, following a quadratic (parabolic) dependence on radius.

Step-by-Step Solution:

Verification / Alternative check:

The average velocity u_avg equals one-half of u_max in laminar pipe flow, which is consistent only with a parabolic profile.

Why Other Options Are Wrong:

Elliptical, hyperbolic, or circular are not the functional form of the solution; a uniform (plug) profile contradicts the no-slip condition in laminar regime.

Common Pitfalls:

Confusing turbulent (flatter) profiles with laminar ones; forgetting that the profile becomes parabolic only under fully developed conditions.

Final Answer:

Parabolic