Fluid Mechanics (ME21201)
Section - 3
Tutorial – 8
Topics – Reynolds Transport theorem Control volume approach, Conservation
of linear momentum and mass, Conservation of angular momentum
1. Water flowing through an 8-cm-diameter pipe enters a porous section, as in Fig, which allows
a uniform radial velocity vw through the wall surfaces for a distance of 1.2 m. If the entrance
average velocity V1 is 12 m/s, find the exit velocity V2 if (a) vw = 15 cm/s out of the pipe walls
or (b) vw = 10 cm/s into the pipe. (c) What value of vw will make V2 = 9 m/s?
Ans : (a) 3 m/s, (b) 18 m/s (c) 0.05 m/s
2. The cylindrical container in Fig. is 20 cm in diameter and has a conical contraction at the
bottom with an exit hole 3 cm in diameter. The tank contains fresh water at standard sea-level
conditions. If the water surface is falling at the nearly steady rate dh/dt  0.072 m/s, estimate
the average velocity V out of the bottom exit
Ans : 3.2 m/s
3. An incompressible fluid flows past an impermeable flat plate, as in Fig, with a uniform inlet
profile u = U0 and a cubic polynomial exit profile
 3 −  3 
y
u  U0 
 where  =

 2 
Compute the volume flow Q across the top surface of the control volume?
Ans : 3U 0b / 8
4. Water enters the bottom of the cone in Fig. at a uniformly increasing average velocity V = Kt.
If d is very small, derive an analytic formula for the water surface rise h (t) for the condition h
= 0 at t = 0. Assume incompressible flow.
1/3
3

Ans :  Kt 2 d 2 cot 2  
8

5. In contrast to the liquid rocket in Fig, the solid-propellant rocket in Fig. is self-contained and
has no entrance ducts. Using a control volume analysis for the conditions shown in Fig,
compute the rate of mass loss of the propellant, assuming that the exit gas has a molecular
weight of 28.
Ans : -11.8 kg/s
6. An incompressible fluid in Fig. is being squeezed outward between two large circular disks by
the uniform downward motion V0 of the upper disk. Assuming one-dimensional radial outflow,
use the control volume shown to derive an expression for V (r).
Ans : V = V0 r / (2h)
7. A wedge splits a sheet of 20 C water, as shown in Fig. Both wedge and sheet are very long
into the paper. If the force required to hold the wedge stationary is F = 124 N per meter of
depth into the paper, what is the angle θ of the wedge?
Ans :  = 48o
8. Water at 20 C flows through a 5-cm-diameter pipe that has a 180 vertical bend, as in Fig. The
total length of pipe between flanges 1 and 2 is 75 cm. When the weight flow rate is 230 N/s, p1
= 165 kPa and p2 = 134 kPa. Neglecting pipe weight, determine the total force that the flanges
must withstand for this flow.
Ans : Fx = −750 N , Fy = 14 N
9. When a uniform stream flows past an immersed thick cylinder, a broad low-velocity wake is
created downstream, idealized as a V shape in Fig. Pressures p1 and p2 are approximately equal.
If the flow is two-dimensional and incompressible, with width b into the paper, derive a formula
for the drag force F on the cylinder. Rewrite your result in the form of a dimensionless drag
coefficient based on body length CD = F / ( U 2bL)
Ans : CD = 1/ 3
10. When a jet strikes an inclined fixed plate, as in Fig, it breaks into two jets at 2 and 3 of equal
velocity V = Vjet but unequal flows αQ at 2 and (1- ) Q at section 3, α being a fraction. The
reason is that for frictionless flow the fluid can exert no tangential force Ft on the plate. The
condition Ft = 0 enables us to solve for α. Perform this analysis, and find α as a function of
the plate angle θ . Why doesn’t the answer depend on the properties of the jet?
Ans :  = (1 + cos ) / 2
11. The water tank in Fig. stands on a frictionless cart and feeds a jet of diameter 4 cm and velocity
8 m/s, which is deflected 60 by a vane. Compute the tension in the supporting cable.
Ans : Fx = 40 N
12. In Fig. the jet strikes a vane that moves to the right at constant velocity Vc on a frictionless cart.
Compute (a) the force Fx required to restrain the cart and (b) the power P delivered to the cart.
Also find the cart velocity for which (c) the force Fx is a maximum and (d) the power P is a
maximum.
Ans : (a) Fx =  Aj (V j − Vc ) 2 (1 − cos  ) , (b)  AjVc (V j − Vc ) 2 (1 − cos  ) (c) 0 (d ) V j / 3
13. A vertical jet of water impinges on a horizontal disk as shown. The disk assembly mass is 30
kg. When the disk is 3 m above the nozzle exit, it is moving upward at U = 5 m/s. Compute the
vertical acceleration of the disk at this instant.
Ans : 2.28 m/s2
14. The elbow shaped tube in Fig, with constant cross-sectional area A and diameter D h , L.
Assume incompressible flow, neglect friction, and derive a differential equation for dV / dt
dV
h
=g
when the stopper is opened.
Ans :
dt
L+h
15. A steady jet of water is used to propel a small cart along a horizontal track as shown. Total
resistance to motion of the cart assembly is given by FD = kU2, where k = 0.92 Ns2/m2. Evaluate
the acceleration of the cart at the instant when its speed is U = 10 m/s.
Ans : 13.5 m/s2
16. A rocket is attached to a rigid horizontal rod hinged at the origin as in Fig. P3.104. Its initial
mass is M0, and its exit properties are m and Ve relative to the rocket. Set up the differential
equation for rocket motion, and solve for the angular velocity ω (t) of the rod. Neglect gravity,
V 
mt 
air drag, and the rod mass.
Ans :  = − e ln 1 −

R  M0 
17. The simplified lawn sprinkler shown rotates in the horizontal plane. At the center pivot, Q =
15 L/min of water enters vertically. Water discharges in the horizontal plane from each jet. If
the pivot is frictionless, calculate the torque needed to keep the sprinkler from rotating.
Neglecting the inertia of the sprinkler itself, calculate the angular acceleration that results when
the torque is removed.
Ans : 2.402*103 1/s2
18. Water flows in a uniform flow out of the 2.5-mm slots of the rotating spray system, as shown.
The flow rate is 3 L/s. Find (a) the torque required to hold the system stationary and (b) the
steady-state speed of rotation after it is released.
Ans : (a) 1.26 N.m (b) 120 rpm
19. A small lawn sprinkler is shown. The sprinkler operates at a gage pressure of 140 kPa. The
total flow rate of water through the sprinkler is 4 L/min. Each jet discharges at 17 m/s (relative
to the sprinkler arm) in a direction inclined 30 above the horizontal. The sprinkler rotates
about a vertical axis. Friction in the bearing causes a torque of 0.18 N.m opposing rotation.
Evaluate the torque required to hold the sprinkler stationary.
Ans : - 0.0161 N.m
20. A large irrigation sprinkler unit, mounted on a cart, discharges water with a speed of 40 m/s at
an angle of 30 to the horizontal. The 50-mm-diameter nozzle is 3m above the ground. The
mass of the sprinkler and cart is M = 350 kg. Calculate the magnitude of the moment that tends
to overturn the cart. What value of V will cause impending motion?
Ans : V2 = 40 m/s