In case of laminar flow of fluid through a circular pipe, the

ANS:B - surface of velocity distribution is a paraboloid of revolution, whose volume equals half the volume of circumscribing cylinder.

In case of laminar flow of fluid through a circular pipe, the correct statement regarding the velocity profile is: surface of velocity distribution is a paraboloid of revolution, whose volume equals half the volume of circumscribing cylinder. Here's why this statement is correct:

• Velocity Profile in Laminar Flow: For laminar flow through a circular pipe, the velocity profile across the pipe's cross-section is parabolic. This means that the velocity varies with the radial distance rrr from the center of the pipe as follows: u(r)=umax(1−r2R2)u(r) = u_{max} \left( 1 - \frac{r^2}{R^2} \right)u(r)=umax​(1−R2r2​) where:
  • u(r)u(r)u(r) is the axial velocity at a radial distance rrr,
  • umaxu_{max}umax​ is the maximum axial velocity at the centerline (at r=0r = 0r=0),
  • RRR is the radius of the pipe.
• Paraboloid of Revolution: The shape of the velocity distribution across the pipe's cross-section forms a paraboloid of revolution. This means if you were to rotate the parabolic velocity profile around the axis of the pipe, it would form a shape resembling a paraboloid.
• Volume Comparison: The statement mentions that the volume of this paraboloid of revolution equals half the volume of the circumscribing cylinder. This is a geometric property related to the distribution of velocity across the cross-section of the pipe.

Therefore, in laminar flow through a circular pipe, the velocity profile is indeed a paraboloid of revolution, satisfying the condition that its volume equals half the volume of the circumscribing cylinder. This characteristic is a result of the laminar flow's smooth and orderly movement of fluid layers, which results in a predictable parabolic velocity distribution.