By Avinash Singh (Ex IES, B.
Tech IITR)
Tutorial Sheet: Fluid Mechanics-III
Sources and Sinks
1. Prove that the radius of curvature R at any point of the streamline 𝜓𝜓 = 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 is given
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�𝑢𝑢2 +𝑣𝑣 2 �2
by 𝑅𝑅 = 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 where u, v are respectively the velocity components of a fluid
�𝑢𝑢2 � �−2𝑢𝑢𝑢𝑢 −𝑣𝑣 2 �
𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕
motion along OX and OY.
2. Show that 𝑢𝑢 = −𝜔𝜔𝜔𝜔, 𝑣𝑣 = 𝜔𝜔𝜔𝜔, 𝑤𝑤 = 0 represents a possible motion of inviscid fluid. Find
the stream function and sketch streamlines. What is basic difference between this
motion and one represented by the potential 𝜙𝜙 = 𝐴𝐴 log 𝑟𝑟, where
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𝑟𝑟 = (𝑥𝑥 2 + 𝑦𝑦 2 )2 .
3. A two-dimensional flow field is given by 𝜓𝜓 = 𝑥𝑥𝑥𝑥.
a. Show that the flow is irrotational.
b. Find the velocity potential.
c. Verify that 𝜓𝜓 𝑎𝑎𝑎𝑎𝑎𝑎 𝜙𝜙 satisfy the Laplace equation
d. Find the streamlines and potential lines.
4. Find the stream function of the two-dimensional motion due to two equal sources
and an equal sink situated midway between them.
Vortex Motion
1. Verify that the stream function 𝜓𝜓 and velocity potential 𝜙𝜙 of a two-dimensional
vortex flow satisfies the Laplace equation.
2. If two vertices are of same strength and the spin is same in both, show that the
𝑟𝑟 2
relative stream lines are given by log(𝑟𝑟 4 + 𝑎𝑎4 − 2𝑎𝑎2 𝑟𝑟 2 cos 2𝜃𝜃) − � � = 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐, 𝜃𝜃 being
2𝑎𝑎
measured from the join of vertices, the origin being its middle point, 2a being the
distance between the vortices.
By Avinash Singh (Ex IES, B.Tech IITR) 1|Page
3. Three parallel rectilinear vortices of the same strength k and in the same sense meet
any plane perpendicular to them in an equilateral triangle of side a. show that the
vortices all move round the same cylinder with uniform speed in time (4𝜋𝜋 2 𝑎𝑎2 )/3𝑘𝑘.
4. Two point vortices each of strength k are situated at (±𝑎𝑎, 0) and a point vortex of
strength −𝑘𝑘/2 is situated at the origin. Show that the fluid motion is stationary and
find the equations of streamlines. Show that the streamline which passes through
the stagnation point meet the x -axis at (±𝑏𝑏, 0) where 3√3 (𝑏𝑏 2 − 𝑎𝑎2 )2 = 16𝑎𝑎3 𝑏𝑏.
5. An infinite long line vortex of strength m, parallel to the axis of z, is situated in
infinite liquid bounded by a rigid wall in the plane 𝑦𝑦 = 0. Prove that, if there be no
field of force, the surfaces of equal pressure are given by {(𝑥𝑥 − 𝑎𝑎)2 +
(𝑦𝑦 − 𝑏𝑏)2 }{(𝑥𝑥 − 𝑎𝑎)2 + (𝑦𝑦 + 𝑏𝑏)2 } = 𝑐𝑐{−(𝑥𝑥 − 𝑎𝑎)2 + (𝑦𝑦 + 𝑏𝑏)2 }, where (a, b) are the coordinates
of the vortex and c is a parametric constant.
6. A vortex pair is situated within a cylinder. Show that it will remain at rest if the
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distance of either from the centre is given by �√5 − 2�2 𝑎𝑎, where a is the radius of the
cylinder.
By Avinash Singh (Ex IES, B.Tech IITR) 2|Page
