B.Sc (Hons) VI (Sixth) Semester Examination 2015-16
Course Code: BAS602
Paper ID: 0986409
Hydrodynamics
Time: 3 Hours
Max. Marks: 70
Note: Attempt six questions in all. Q. No. 1 is compulsory.
1.
a)
b)
c)
Answer any five of the following (limit your answer to 50
words).
(4x5=20)
Find expressions for the acceleration in Cartesian
Coordinates of an element of fluid in motion.
Each particle of a mass of liquid moves in a plane through the
axis of z; find the equation of continuity.
Determine the pressure, if the velocity field
, satisfies the equation of
motion
d)
e)
f)
g)
h)
2. a)
b)
3. a)
, where A and B are arbitrary constants.
Show that the velocity vector is everywhere tangent to lines
in the xy-plane along which
.
Show that
times the difference of the values of Stoke's
stream function at two points in the same meridian plane is
equal to the flow across the angular surface obtained by the
revolution around the axis of curve joining the points.
Find the necessary and sufficient condition that vortex lines
may be at right angles to the stream lines.
Define the following:
i)
Vortex line ii)
vortex tube and vortex filament
Obtain the stream lines of a flow
Given
; show that the surfaces
intersecting the stream lines orthogonally exist and are the
planes through z-axis, although the velocity potential does
not exist.
(5)
Find the equation of continuity by vector approach for a nonhomogeneous incompressible fluid.
(5)
Obtain Lagrange's equation of motion.
(5)
b) Derive Bernoulli's
motion.
4.
equation
for
unsteady
irrotational
(5)
Find the complex potential for the two dimensional source of
strength m placed at the origin.
(10)
5. a) For a liquid streaming past a fixed sphere, obtain the lines of
flow relative to the sphere.
(5)
b) A and B are a simple source and sink of strengths and
respectively in an infinite liquid. Show that the equation of
stream lines is
where and are the angles which AP, BP make with AB,
P being any point. Prove also that if
, the cone
defined by the equation
divides the
stream lines issuing from A into two sets, one extending to
infinity and other terminating at B.
(5)
6. a) Show that the product of the cross section and vorticity (or
angular velocity) at any point on a vortex filament is constant
along the filament and for all time when the body forces are
conservative and the pressure is a single valued function of
density only .
(5)
b) Assuming that in an infinite unbounded mass of
incompressible fluid, the circulation in any closed circuit is
independent of time, show that the angular velocity of any
element of the fluid moving rotationally varies as the length
of the element measured in the direction of the axis of
rotation.
(5)
7. a) Derive equation of motion under impulsive force.
(5)
b) The velocity components in a three-dimensional flow field
for an incompressible fluid are
. Is it a possible
field? Determine the equations of the stream lines passing
through the point
.
(5)
8. a) Show that the curves of constant potential and constant
stream functions cut orthogonally at their point of
intersection.
(5)
b) A liquid is moving in a frictionless liquid at rest at infinity.
Calculate the velocity potential and the equations of lines of
flow.
(5)