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Fluid Mechanics - Engineering

Q1:

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

A shear stress over the cross-section is proportional to the distance from the surface of the pipe.

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

C velocity profile varies hyperbolically and the shear stress remains constant over the cross-section.

D average flow occurs at a radial distance of 0.5 r from the centre of the pipe (r = pipe radius).

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.