By Avinash Singh (Ex IES, B.
Tech IITR)
Tutorial Sheet: Fluid Mechanics-I
1. For a two-dimensional flow the velocities at a point in a fluid may be expressed in the
Eulerian coordinates by 𝑢𝑢 = 𝑥𝑥 + 𝑦𝑦 + 2𝑡𝑡 𝑎𝑎𝑎𝑎𝑎𝑎 𝑣𝑣 = 2𝑦𝑦 + 𝑡𝑡. Determine the Lagrange
coordinates as the functions of the initial positions 𝑥𝑥0 𝑎𝑎𝑎𝑎𝑎𝑎 𝑦𝑦0 and the time t.
2. � , where A, B, C are constants,
If the velocity distribution is 𝒒𝒒 = 𝐴𝐴𝑥𝑥 2 𝑦𝑦𝒊𝒊̂ + 𝐵𝐵𝑦𝑦 2 𝑧𝑧𝑧𝑧𝒋𝒋̂ + 𝐶𝐶𝐶𝐶𝑡𝑡 2 𝒌𝒌
then find the acceleration and velocity components.
3. The particles of a fluid move symmetrically in space with regard to a fixed centre; prove
𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜌𝜌 𝜕𝜕(𝑟𝑟 2 𝑢𝑢)
that the equation of continuity is + 𝑢𝑢 + = 0, where u is the velocity at
𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝑟𝑟 2 𝜕𝜕𝜕𝜕
distance r.
4. A mass of fluid moves in such a way that each particle describes a circle on one plane
𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕𝜕𝜕
about a fixed axis; show that the equation of continuity is + = 0, where 𝜔𝜔 is the
𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕
angular velocity of a particle whose azimuthal angle is 𝜃𝜃 at time t.
5. A mass of fluid is in motion so that the lines of motion lie on the surface of co-axial
𝜕𝜕𝜕𝜕 1 𝜕𝜕𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕𝜕𝜕
cylinders. Show that the equation of continuity is + + = 0, where u,v are the
𝜕𝜕𝜕𝜕 𝑟𝑟 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕
velocity perpendicular and parallel to z.
6. Consider a two-dimensional incompressible steady flow field with velocity components
3 𝑟𝑟0 1 𝑟𝑟03
in spherical coordinates (𝑟𝑟, 𝜃𝜃, 𝜙𝜙) given by 𝑣𝑣𝑟𝑟 = 𝑐𝑐1 �1 − 2 𝑟𝑟
+
2 𝑟𝑟 3
� cos 𝜃𝜃 , 𝑣𝑣𝜙𝜙 = 0, 𝑣𝑣𝜃𝜃 =
3 𝑟𝑟0 1 𝑟𝑟03
−𝑐𝑐1 �1 − − � sin 𝜃𝜃 , 𝑟𝑟 ≥ 𝑟𝑟0 > 0 where 𝑐𝑐1 𝑎𝑎𝑎𝑎𝑎𝑎 𝑟𝑟0 are arbitrary constants. Is the equation
4 𝑟𝑟 4 𝑟𝑟 3
of continuity satisfied?
7. Liquid flows through the pipe whose surface is the surface of revolution of the curve 𝑦𝑦 =
𝑘𝑘𝑥𝑥 2
𝑎𝑎 +
𝑎𝑎
about the x axis (−𝑎𝑎 ≤ 𝑥𝑥 ≤ 𝑎𝑎). If the liquid enters at the end 𝑥𝑥 = −𝑎𝑎 of the pipe
with velocity V, show that the time taken by a liquid particle to traverse the entire length
2𝑎𝑎 2𝑘𝑘 𝑘𝑘 2
of the pipe from 𝑥𝑥 = −𝑎𝑎 𝑡𝑡𝑡𝑡 𝑥𝑥 = 𝑎𝑎 is � � �1 + 3 + �. Assume that k is so small that the
𝑉𝑉(1+𝑘𝑘 2 ) 5
fluid remains one dimensional throughout.
𝑥𝑥 2 𝑦𝑦 2 𝑧𝑧 2
8. Show that the surface 𝑎𝑎 2 𝑘𝑘 2 𝑡𝑡 4
+ 𝑘𝑘𝑡𝑡 2 �𝑏𝑏2 + 𝑐𝑐 2 � = 1 is a possible form of boundary surface of
a liquid at time t.
By Avinash Singh (Ex IES, B.Tech IITR) 1|Page
𝑥𝑥 2 𝑦𝑦 2 𝑧𝑧 2
9. Determine the restrictions on 𝑓𝑓1 , 𝑓𝑓2 , 𝑓𝑓3 if �𝑎𝑎2 � 𝑓𝑓1 (𝑡𝑡) + �𝑏𝑏2 � 𝑓𝑓2 (𝑡𝑡) + �𝑐𝑐 2 � 𝑓𝑓3 (𝑡𝑡) = 1 is a possible
boundary surface of a liquid.
𝑥𝑥 2 𝑦𝑦 2
10. Show that �𝑎𝑎2 � tan2 𝑡𝑡 + �𝑏𝑏2 � cot 2 𝑡𝑡 = 1 is a possible form for the bounding surface of a
liquid, and find an expression for the normal velocity.
11. Find the equation of the streamlines for the flow 𝒒𝒒 = −(3𝑦𝑦 2 )𝚤𝚤̂ − 6𝑥𝑥𝚥𝚥̂ at the point (1,1).
12. Determine the streamlines and path lines of the particle when the components of
𝑥𝑥 𝑦𝑦 𝑧𝑧
velocity field are given by 𝑢𝑢 = , 𝑣𝑣 = 𝑎𝑎𝑎𝑎𝑎𝑎 𝑤𝑤 = . Also state the condition for which
1+𝑡𝑡 2+𝑡𝑡 3+𝑡𝑡
the streamlines are identical with path lines.
13. If the velocity of an incompressible fluid at the point (𝑥𝑥, 𝑦𝑦, 𝑧𝑧) is given by
3𝑥𝑥𝑥𝑥 3𝑦𝑦𝑦𝑦 3𝑧𝑧 2 −𝑟𝑟 2
� , , �, prove that the liquid motion is possible and that the velocity potential
𝑟𝑟 5 𝑟𝑟 5 𝑟𝑟 5
cos 𝜃𝜃
is 𝑟𝑟 2
. Also determine the streamlines.
14. Show that if the velocity potential of an irrotational fluid motion is equal to
3
𝑦𝑦
𝐴𝐴(𝑥𝑥 2 + 𝑦𝑦 2 + 𝑧𝑧 2 )−2 𝑧𝑧 tan−1 �𝑥𝑥 �, the lines of flow will be on the series of the surfaces 𝑥𝑥 2 + 𝑦𝑦 2 +
2 2
𝑧𝑧 2 = 𝑐𝑐 3 (𝑥𝑥 2 + 𝑦𝑦 2 )3 .
If the fluid be in motion with a velocity potential 𝜙𝜙 = 𝑧𝑧 log 𝑟𝑟, and if the density at a point fixed
in space be independent of the time, show that the surfaces of equal density are of the
1
forms 𝑟𝑟 2 �log 𝑟𝑟 − � − 𝑧𝑧 2 = 𝑓𝑓(𝜃𝜃, 𝜌𝜌), 𝑤𝑤ℎ𝑒𝑒𝑒𝑒𝑒𝑒 𝜌𝜌 is the density at (𝑧𝑧, 𝑟𝑟, 𝜃𝜃).
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By Avinash Singh (Ex IES, B.Tech IITR) 2|Page
