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GATE 2026 MT – Question 40

Transport Phenomena and Rate Processes · Fluid flow and momentum transfer · 2 marks · Multiple choice

The equation below represents steady-state laminar flow of a fluid through a horizontal tube: $$\eta\frac1r\frac{d}{dr}\left(r\frac{dv_z}{dr}\right)-\frac{dP}{dz}=0.$$

Here $P$ is fluid pressure, $\eta$ is dynamic viscosity, $(r,z)$ are cylindrical polar coordinates, and $v_z$ is axial velocity in the $z$ direction. Which one of the following statements is NOT correct?

Diagram for GATE 2026 MT question 40
  1. The fluid is Newtonian.
  2. Shear stress is maximum at the center ($r=0$).
  3. Radial velocity is zero.
  4. There is no variation of $v_z$ in the $z$ direction.

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Correct answer: (B) Shear stress is maximum at the center ($r=0$).

Explanation

Integrating the momentum equation once gives $r\,dv_z/dr=(r^2/2\eta)(dP/dz)+C$. A finite velocity gradient at the center requires $C=0$.

Thus $\tau_{rz}=\eta\,dv_z/dr=(r/2)(dP/dz)$. Its magnitude is zero at the center and greatest at the wall. The equation otherwise describes Newtonian, fully developed axial flow with zero radial velocity. Therefore **B** is false.