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Mechanics of Fluids 5th Edition By Potter – Test Bank

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Mechanics of Fluids 5th Edition By Potter – Test Bank

Mechanics of Fluids, 5th Edition Chapter 4: The Integral Forms of the
Fundamental Laws
20
© 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Solutions for Section 4.6
1. If p1 = 900 kPa, the force of the water on
the nozzle is nearest:
(C) 5120 N
The energy equation along with the continuity equation will allow the velocities to
be determined. Continuity demands that 2 2
2 1 1 V V V = = (10 /4 ) 6.25 so the energy
equation gives
2
1 1
1
2
V p
z
g γ
+ +
( )2
1 2
6 25
2
V p
g γ
= +
.
2 + z
2
1
1
900 000 38 06 6 88 m/s
9810 2 9 81
V
= ∴ = V
×
.
. .
.
The momentum equation (4.6.6) applied to the control volume shown (if you don’t
sketch the c.v., you could forget that pressure force) is used to find the force Fx of
the water acting on the contraction (actually, the force of the water on the contraction
is equal and opposite to the force shown which is the force of the contraction on the
water)
1 1 2 1 1 1 2 1
2 2
( ) ( )
900 000 ( 0 05 ) 1000 ( 0 05 ) 6 88 (6 25 1) 6 88
5120 N
x
x
x
F p A m V V A V V V
F
F
ρ
π π
− + = − = −
− + × × = × × × − ×    
∴ =
ɺ
. . . . .
Make sure you check the units, especially on the right-hand side. They are
2
kg m N s /m m [ ][ ] N
s s s
m V
s

ɺ = × = × =
Remember, if you use kg, m, s, and N in the equations, the units will check out.
There’s no real need to always work two problems: one with the numbers and
another with the units. Check the units a couple times and then make sure you use
kg, m, s, and N when you input numbers into the equations.
V1
p1
4 cm dia.
10 cm dia.
V2
.
A1
Fx
Mechanics of Fluids, 5th Edition Chapter 4: The Integral Forms of the
Fundamental Laws
21
© 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
2. Water flows over a 2-m-wide flat plate with
the same velocities as in Problem 4 in the
mini-exam for Section 4. The frictional
force F acting on the horizontal surface is
nearest:
(C) 1700 N

The momentum equation (4.6.2) for this steady flow applied in the x-direction allows
us to calculate F:
2 2
c.s. top
0 1 0 1
2 2 2 2
top top
0 0
( ) ( )
160 ( 5 ) 2 8 2 ( )
. .
ˆ ˆ
x x x x
B A
x
F V dA V dA V dA V dA
y y dy dy m V
ρ ρ ρ ρ
ρ ρ
− = ⋅ = − + ⋅
= − − × +
∫ ∫ ∫ ∫
∫ ∫
V n V n
ɺ
[ ]
0 1
6 2 3 4
0
51 2 10 ( 10 25 ) 12 800 1000 (4 2) 0 0667 8
6830 12 800 4270 1700 1700

Mechanics of Fluids, 5th Edition Chapter 5: The Differential Forms of the
Fundamental Laws
22
© 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Solutions for Sections 5.1 and 5.2
1. Which of the following conditions is never a boundary condition in fluid mechanics?
(C) The tangential component of velocity is zero in an inviscid flow at a boundary
The tangential component of velocity must be tangential to the boundary at a solid
boundary, but usually, if not always, in an inviscid flow it is nonzero. A porous
boundary would provide a normal component of velocity to the fluid at the
boundary.
The pressure is zero (gage) at a boundary if the boundary is a free surface, the
atmosphere.
In a viscous flow, the fluid sticks to the boundary so it takes on the velocity of
the boundary, which is usually zero.

2. What four equations provide for the four unknowns u, v, w, p that are most often of interest in
fluid mechanics?
(A) Continuity, momentum
The momentum equation is a vector equation containing the three velocity components
u, v, w, and p. Continuity provides the fourth equation. The energy differential equation
enters only when temperature is an unknown.
3. If the x-component of velocity u depends only on y in a plane flow, the y-component of
velocity v is:
(C) f x( )
The differential continuity equation ∂ ∂ + ∂ ∂ = u x v y / / 0 is used:
0 ( ) and ( ) .
u v v f y v f x
x y
∂ ∂
= = − ∴ ≠ =
∂ ∂
It could be a constant, or 0, but not necessarily so. It is not a function of y.
Mechanics of Fluids, 5th Edition Chapter 5: The Differential Forms of the
Fundamental Laws
23
© 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
4. Three measurements are made of the x-component velocity in the diffuser of an
incompressible plane flow to be 32 m/s, 28 m/s and 20 m/s. The measurement points are
along the centerline of the symmetrical diffuser and are 4 cm apart. The y-component of the
velocity 2 cm above the centerline is approximated to be:
(B) 3 m/s
We chose to use the more accurate central difference to approximate the derivative
since the information given allows such an approximation:

20 32 m/s 150
0 08 m
u u
x x
∂ ∆ −
≅ = = −
∂ ∆ .
The continuity equation then provides
2
2
0
150 or 150 3 m/s
0 02
.
.
v u v v v
v
y x y y
∂ ∂ ∂ ∆ −
= − = ≅ = = ∴ =

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