5.5 Water enters a cylindrical tank through two pipes at
rates of 250 and 100 gal/min (see Fig. P5.5). If the level
of the water in the tank remains constant, calculate the
average velocity of the flow leaving the tank through an
8-in. inside-diameter pipe.
5.18 Two rivers merge to form a larger river as shown in Fig. P5.18.
At a location downstream from the junction (before the two streams
completely merge), the nonuniform velocity profile is as shown and
the depth is 6 ft. Determine the value of V.
5.25 Air at standard conditions enters the
compressor shown in Fig. P5.25 at a rate of 10 ft3/s.
It leaves the tank through a 1.2-in.-diameter pipe
with a density of 0.0035 slugs/ft3 and a uniform
speed of 700 ft/s. (a) Determine the rate (slugs/s) at
which the mass of air in the tank is increasing or
decreasing. (b) Determine the average time rate of
change of air density within the tank.
5.29 A hypodermic syringe (see Fig. P5.29) is used to apply
a vaccine. If the plunger is moved forward at the steady rate
of 20 mm/s and if vaccine leaks past the plunger at 0.1 of the
volume flowrate out the needle opening, calculate the
average velocity of the needle exit flow. The inside
diameters of the syringe and the needle are 20 mm and 0.7 mm
5.38 A 10-mm diameter jet of water is deflected by a
homogeneous rectangular block (15 mm by 200 mm by 100 mm)
that weighs 6 N as shown in Video V5.6 and Fig. P5.38.
Determine the minimum volume flowrate needed to tip the block.
5.45 Determine the magnitude and direction of the anchoring force
needed to hold the horizontal elbow and nozzle combination shown
in Fig. P5.45 in place. Atmospheric pressure is 100 kPa(abs). The
gage pressure at section (1) is 100 kPa. At section (2), the water exits
to the atmosphere.
5.47 A converging elbow (see Fig. P5.47) turns water through
an angle of 135° in a vertical plane. The flow cross section
diameter is 400 mm at the elbow inlet, section (1), and 200
mm at the elbow outlet, section (2). The elbow flow passage
volume is 0.2 m3 between sections (1) and (2). The water
volume flowrate is 0.4 m3/s and the elbow inlet and outlet
pressures are 150 kPa and 90 kPa. The elbow mass is 12 kg.
Calculate the horizontal (x direction) and vertical (z direction)
anchoring forces required to hold the elbow in place.
5.64 A Pelton wheel vane directs a horizontal, circular
cross-sectional jet of water symmetrically as indicated in
Fig. P5.64 and Video V5.6. The jet leaves the nozzle
with a velocity of 100 ft/s. Determine the x direction
component of anchoring force required to (a) hold the
vane stationary, (b) confine the speed of the vane to a
value of 10 ft/s to the right. The fluid speed magnitude
remains constant along the vane surface.
A vane slider assembly moves under the
influence of a liquid jet as shown below. The
coefficient of kinetic friction for motion of the
slider along the surface is
= 0.49. Calculate:
a) the acceleration of the slider (in m/sec2) at
the instant when U = 7.90 m/sec.
b) the terminal speed of the slider (in m/sec).
5.92 A 100-ft-wide river with a flowrate of
2400 ft3/s flows over a rock pile as shown in
Fig. P5.92. Determine the direction of flow
and the head loss associated with the flow
across the rock pile.
5.100 A water siphon having a constant inside diameter of 3 in. is
arranged as shown in Fig. P5.100. If the friction loss between A and B is
0.8V2/2g, where V is the velocity of flow in the siphon, determine the
flowrate involved.
5.104 For the 180° elbow and nozzle flow shown in Fig.
P5.104, determine the loss in available energy from
section (1) to section (2). How much additional available
energy is lost from section (2) to where the water comes
to rest?
5.116 Water is pumped from the large tank shown in Fig. P5.116.
The head loss is known to be equal to 4V2/2g and the pump head
is hp= 20 − 4Q2, where hp is in ft when Q is in ft3/s. Determine
the flowrate.
5.130 Two water jets collide and form one homogeneous
jet as shown in Fig. P5.130. (a) Determine the speed, V,
and direction, θ, of the combined jet. (b) Determine the
loss for a fluid particle flowing from (1) to (3), from (2) to
(3). Gravity is negligible.
6.8 For a certain incompressible, two-dimensional flow field the velocity component in the y direction is given
by the equation
v = 3xy + x2y
Determine the velocity component in the x direction so that the volumetric dilatation rate is zero
6.14 The velocity components of an incompressible, two-dimensional velocity field are given by the equations
Show that the flow is irrotational and satisfies conservation of mass.
6.35 Determine the stream function corresponding to the velocity potential
Φ = x3 – 3 x y2
6. 37 It is known that the velocity distribution for two-dimensional flow of a viscous fluid between wide parallel
plates (Fig. P6.37) is parabolic; that is,
with v = 0. Determine, if possible, the corresponding stream function and velocity potential.
6.54 Water flows over a flat surface at 4 ft/s, as shown in Fig. P6.54. A pump draws off water through a narrow
slit at a volume rate of 0.1 ft3/s per foot length of the slit. Assume that the fluid is incompressible and inviscid
and can be represented by the combination of a uniform flow and a sink. Locate the stagnation point on the wall
(point A) and determine the equation for the stagnation streamline. How far above the surface, H, must the fluid
be so that it does not get sucked into the slit?
6.63 One end of a pond has a shoreline that resembles a half-body as shown in Fig. P6.63. A vertical porous
pipe is located near the end of the pond so that water can be pumped out. When water is pumped at the rate of
0.08 m3/s through a 3-m-long pipe, what will be the velocity at point A? Hint: Consider the flow inside a halfbody.
6.65 A Rankine oval is formed by combining a source–sink pair, each having a strength of 36 ft2/s and separated by
a distance of 12 ft along the x axis, with a uniform velocity of 10 ft/s (in the positive x direction). Determine the
length and thickness of the oval.
6.67 An ideal fluid flows past an infinitely long, semicircular “hump” located along a plane boundary, as shown in
Fig. P6.67. Far from the hump the velocity field is uniform, and the pressure is p0. (a) Determine expressions for the
maximum and minimum values of the pressure along the hump, and indicate where these points are located. Express
your answer in terms of ρ, U, and p0. (b) If the solid surface is the ψ = 0 streamline, determine the equation of the
streamline passing through the point θ = π/2,r = 2a.
6.79 Determine the shearing stress for an incompressible Newtonian fluid with a velocity distribution of
V = (3xy2 − 4x3)î + (12x2y − y3)ĵ.
6.90 A layer of viscous liquid of constant thickness (no velocity perpendicular to plate) flows steadily down an
infinite, inclined plane. Determine, by means of the Navier–Stokes equations, the relationship between the
thickness of the layer and the discharge per unit width. The flow is laminar, and assume air resistance is
negligible so that the shearing stress at the free surface is zero.
6.102 An infinitely long, solid, vertical cylinder of radius R is located in an infinite mass of an incompressible
fluid. Start with the Navier–Stokes equation in the θ direction and derive an expression for the velocity
distribution for the steady flow case in which the cylinder is rotating about a fixed axis with a constant angular
velocity ω. You need not consider body forces. Assume that the flow is axisymmetric and the fluid is at rest at
infinity.