Fluid Mechanics PU — University of the Punjab

UNIVERSITY OF THE PUNJAB

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Section II

Q.6(a)

Usc the method of images to prove that if there be a source of strength m at

the point zo in a fluid bounded by the lines = 0 and 0 = 7/3, the solution

is o + if = -I In (z3 - zỏ) (z3 - zở), where zo = Xo + iyo and zo =

xo - iYo.

(10)

(b)

Discuss briefly the superposition of Poiseuille and Couette flows. Is this

superposition is dynamically admissible? If yes, why? Which components

of velocity are nonzero for it? Evaluate the expression for velocity field and

pressure for the same flow.

(10)

Q.7(a)

How a vortex, doublet and uniform stream are combined? Describe the

resulting flow in detail.

(10)

(b)

State and Prove Kutta and Joukowski theorem.

(10)

Q.8

(a)

Under which configuration of vortices the time required to move a distance

a is " ? Prove it. Also define the interval for which the above configuration

remains unchanged.

(10)

(b)

Derive the Navier-Stokes equations for an incompressible viscous fluid.

Show that, in the case of Couette flow, the velocity of the fluid is given by

(10)

Q.9(a)

The velocity distribution for the steady, laminar and fully developed flow

between two fixed parallel plates is given by u = - 2 ax (hy - y?), | (10)

where

p is the pressure gradient, h is the gap width between the plates

and yis the distance measured upward from the lower plate. Determine the

volumetric flow rate, maximum velocity and the shearing stress at both the

plates.

What is the Stokes first problem? Write its mathematical formulation and

solve it by Laplace transform for the velocity field

(10)