## 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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