1.10 The Stokes-Oseen formula [10] for drag on a sphere at low velocity V is:

2 29F 3 DV V D

16π

πµ ρ= +

where D = sphere diameter, µ = viscosity, and ρ = density. Is the formula homogeneous?

Solution: Write this formula in dimensional form, using Table 1-2:

2 29{F} {3 }{ }{D}{V} { }{V} {D} ?

16π

π µ ρ= +

22

232

ML M L M Lor: {1} {L} {1} {L } ?

LT T TLT= +

where, hoping for homogeneity, we have assumed that all constants (3,π,9,16) are pure,i.e., {unity}. Well, yes indeed, all terms have dimensions {ML/T 2}! Therefore the Stokes-Oseen formula (derived in fact from a theory) is dimensionally homogeneous.

1.22* According to the theory of Chap. 8, as a uniform stream approaches a cylinder of radius R along the line AB shown in Fig. P1.22, – ∞ < x < –R, the velocities are

2 2u U (1 R /x ); v w 0∞= − = =Fig. P1.22

Using the concepts from Ex. 1.5, �nd (a) the maximum �ow deceleration along AB; and (b) its location.

Solution: We see that u slows down monotonically from U∞ at A to zero at point B, x = −R, which is a �ow “stagnation point.” From Example 1.5, the acceleration (du/dt) is

2 2 2

2 3 3 5

du u u R 2R U 2 2 x,U1U0u

RRxttd x x∂ ∂

− ζ∂ ∂ ζ ζ

∞∞ ∞= + = + − + = =

This acceleration is negative, as expected, and reaches a minimum near point B, which is found by di�erentiating the acceleration with respect to x:

2max decel.

min

d du 5 x)b( .ro,fi0

dx dt 3 R

du)a( . sevig )td/ud( otni 192.1 gnitutitsbuS

dt

Ans

Ans

ζ

ζ

= = ≈

=−=

|

| ∞

−

−2

1.291

U0.372

RA plot of the �ow deceleration along line AB is shown as follows.

1.26 A tire has a volume of 3.0 ft3 and a ‘gage’ pressure of 32 psi at 75°F. If the ambient pressure is sea-level standard, what is the weight of air in the tire?

Solution: Convert the temperature from 75°F to 535°R. Convert the pressure to psf:

22222p (32 lbf/in )(144 in /ft ) 2116 lbf/ft 4608 2116 6724 lbf/ft≈+=+=

From this compute the density of the air in the tire:

23

airp 6724 lbf/ft

0.00732 slug/ftRT (1717 ft lbf/slug R)(535 R)

ρ ===? ?° °

Then the total weight of air in the tire is

3 2 3airW g (0.00732 slug/ft )(32.2 ft/s )(3.0 ft ) Ans.ρ υ ≈== 0.707 lbf

1.45 A block of weight W slides down an inclined plane on a thin �lm of oil, as in Fig. P1.45 at right. The �lm contact area is A and its thickness h. Assuming a linear velocity distribution in the �lm, derive an analytic expression for the terminal velocity V of the block. Fig. P1.45

Solution: Let “x” be down the incline, in the direction of V. By “terminal” velocity we mean that there is no acceleration. Assume a linear viscous velocity distribution in the �lm below the block. Then a force balance in the x direction gives:

xx

terminal

VF W sin A W sin A ma 0,

h

or: V .Ans

θ τ θ µ= − = − = =

=hW sin

Aθ

µ

1.65 The system in Fig. P1.65 is used to estimate the pressure p1 in the tank by measuring the 15-cm height of liquid in the 1-mm-diameter tube. The �uid is at 60°C. Calculate the true �uid height in the tube and the percent error due to capillarity if the �uid is (a) water; and (b) mercury. Fig. P1.65

Solution: This is a somewhat more realistic variation of Ex. 1.9. Use values from that example for contact angle θ:(a) Water at 60°C: γ ≈ 9640 N/m3, θ ≈ 0° :

3

4Y cos 4(0.0662 N/m)cos(0 ),m 5720.0h

D (9640 N/m )(0.001 m)θ

γ°

= = =

or: ∆htrue = 15.0 – 2.75 cm ≈ 12.25 cm (+22% error) Ans. (a)

(b) Mercury at 60°C: γ ≈ 132200 N/m3, θ ≈ 130° :

3

4Y cos 4(0.47 N/m)cos 130,m 1900.0h

D (132200 N/m )(0.001 m)θ

γ°

−===

trueor: h 15.0 0.91∆ = + ≈15.91cm ( 6%error)− Ans. (b)

1.75 Oil, with a vapor pressure of 20 kPa, is delivered through a pipeline by equally-spaced pumps, each of which increases the oil pressure by 1.3 MPa. Friction losses in the pipe are 150 Pa per meter of pipe. What is the maximum possible pump spacing to avoid cavitation of the oil?

Solution: The absolute maximum length L occurs when the pump inlet pressure is slightly greater than 20 kPa. The pump increases this by 1.3 MPa and friction drops the pressure over a distance L until it again reaches 20 kPa. In other words, quite simply,

max1.3 MPa 1,300,000 Pa (150 Pa/m)L, or L Ans.≈== 8660 m

It makes more sense to have the pump inlet at 1 atm, not 20 kPa, dropping L to about 8 km.

2.1 For the two-dimensional stress �eld in Fig. P2.1, let

xx yy3000 psf 2000 psfσ σ= =

xy 500 psfσ =

Find the shear and normal stresses on plane AA cutting through at 30° .

Solution: Make cut “AA” so that it just hits the bottom right corner of the element. This gives the freebody shown at right. Now sum forces normal and tangential to side AA. Denote side length AA as “L.”

AAn,AA LF 0(3000 sin 30 500 cos30)L sin 30(2000 cos30 500 sin 30)L cos30

σ==− +− +

Fig. P2.1

AA )a( .rof evloS Ansσ ≈ 22683 lbf/ft

t,AA AAF 0 L (3000 cos30 500 sin 30)L sin 30 (500 cos30 2000 sin 30)L cos30τ= = − − − −

AA )b( .rof evloS Ansτ ≈ 2683 lbf/ft

2.4 Given a �ow pattern with isobars po − Bz + Cx 2 = constant. Find an expression x = fcn(z) for the family of lines everywhere parallel to the local pressure gradient ? p.Solution: Find the slope (dx/dz) of the isobars and take the negative inverse and integrate: 2

tsnocpogradient

gradient

1Bxdxdd(p Bz Cx ) B 2Cx 0, or:

)zd/xd(xC2zdzdzd

dx 2Cx dx 2C dz., etargetni,suhT

dz B x BAns

=−

− + = − + = = =

−= − =

|

| 2Cz/Bx const e−=

2.8 A diamond mine is 2 miles below sea level. (a) Estimate the air pressure at this depth. (b) If a barometer, accurate to 1 mm of mercury, is carried into this mine, how accurately can it estimate the depth of the mine?

Solution: (a) Convert 2 miles = 3219 m and use a linear-pressure-variation estimate: 3

aThen p p h 101,350 Pa (12 N/m )(3219 m) 140,000 Pa . (a)Ansγ≈ + = + = ≈140 kPa

Alternately, the troposphere formula, Eq. (2.27), predicts a slightly higher pressure: 5.26 5.26

a op p (1 Bz/T ) (101.3 kPa)[1 (0.0065 K/m)( 3219 m)/288.16 K]. (a)Ans

≈ − = − −= 147 kPa

(b) The gage pressure at this depth is approximately 40,000/133,100 ≈ 0.3 m Hg or 300 mm Hg ±1 mm Hg or ±0.3% error. Thus the error in the actual depth is 0.3% of 3220 m or about ±10 m if all other parameters are accurate. Ans. (b)

2.14 The closed tank in Fig. P2.14 is at 20°C. If the pressure at A is 95 kPa

absolute, determine p at B (absolute). What percent error do you make by neglecting the speci�c weight of the air?

Solution: First compute ρA = pA/RT =(95000)/[287(293)] ≈ 1.13 kg/m3, hence γA ≈(1.13)(9.81) ≈ 11.1 N/m3. Then proceed around hydrostatically from point A to point B: Fig. P2.14

3 BB

p95000 Pa (11.1 N/m )(4.0 m) 9790(2.0) 9790(4.0) (9.81)(2.0) p

RT+ + − − =

B . prof evloS Accurate answer≈ 75450 Pa

If we neglect the air e�ects, we get a much simpler relation with comparable accuracy:

B95000 9790(2.0) 9790(4.0) p Approximate answer.+ − ≈ ≈75420 Pa

2.15 In Fig. P2.15 all �uids are at 20 °C.Gage A reads 15 lbf/in2 absolute and gage B reads 1.25 lbf/in2 less than gage C. Com-pute (a) the speci�c weight of the oil; and (b) the actual reading of gage C in lbf/in2

absolute.

Fig. P2.15 Solution: First evaluate γair = (pA/RT)g = [15 × 144/(1717 × 528)](32.2) ≈ 0.0767 lbf/ft3.Take γwater = 62.4 lbf/ft3. Then apply the hydrostatic formula from point B to point C:

BClioBp (1.0 ft) (62.4)(2.0 ft) p p (1.25)(144) psfγ+ + = = +

oil )a( rof evloS Ans.γ ≈ 355.2 lbf/ft

With the oil weight known, we can now apply hydrostatics from point A to point C:

C A

2C

p p gh (15)(144) (0.0767)(2.0) (55.2)(2.0) (62.4)(2.0)

or: p 2395 lbf/ft (b)Ans.

ρ= + = + + +

= = 16.6 psi

2.20 The hydraulic jack in Fig. P2.20 is �lled with oil at 56 lbf/ft3. Neglecting piston weights, what force F on the handle is required to support the 2000-lbf weight shown?

Fig. P2.20 Solution: First sum moments clockwise about the hinge A of the handle:

AM 0 F(15 1) P(1),= = + −

or: F = P/16, where P is the force in the small (1 in) piston.

Meanwhile �gure the pressure in the oil from the weight on the large piston:

oil 23-in

W 2000 lbf,fsp 44704p

A ( /4)(3/12 ft)π= = =

2

oil small1

Hence P p A (40744) 222 lbf4 12π

= = =

Therefore the handle force required is F = P/16 = 222/16 ≈ 14 lbf Ans.2.45 Determine the gage pressure at point A in Fig. P2.45, in pascals. Is it higher or lower than Patmosphere?

Solution: Take γ = 9790 N/m3 for water and 133100 N/m3 for mercury. Write the hydrostatic formula between the atmosphere and point A:

atm

A

p (0.85)(9790)(0.4 m)

(133100)(0.15 m) (12)(0.30 m)

(9790)(0.45 m) p ,

+

− −

+ = Fig. P2.45

A atm .aP 00221pp:ro Ans= − =12200 Pa (vacuum)

2.60 The pressure in the air gap is 8000 Pa gage. The tank is cylindrical. Calculate the net hydrostatic force (a) on the bottom of the tank; (b) on the cylindrical sidewall CC; and (c) on the annular plane panel BB.

Solution: (a) The bottom force is simply equal to bottom pressure times bottom area:

bottom air water

3

p p g z 8000 Pa

(9790 N/m )(0.25 0.12 m)11622 Pa-gage

ρ= + ∆ =

+ += Fig. P2.60

2bottom bottom bottomF p A (11622 Pa)( /4)(0.36 m) . (a)Ansπ === 1180 N

(b) The net force on the cylindrical sidewall CC is zero due to symmetry. Ans. (b)(c) The force on annular region CC is, like part (a), the pressure at CC times the area of CC:

3CC air water CCp p g z 8000 Pa (9790 N/m )(0.25 m) 10448 Pa-gageρ= + ∆ = + =

2 2CC CC CCF p A (10448 Pa)( /4)[(0.36 m) (0.16 m) ] . (c)Ansπ =−== 853 N

2.61 Gate AB in Fig. P2.61 is a homo-geneous mass of 180 kg, 1.2 m wide into the paper, resting on smooth bottom B. All �uids are at 20°C. For what water depth hwill the force at point B be zero?

Solution: Let γ = 12360 N/m3 for glycerin and 9790 N/m3 for water. The centroid of

Fig. P2.61 AB is 0.433 m vertically below A, so h CG =2.0 − 0.433 = 1.567 m, and we may compute the glycerin force and its line of action:

g

3

CP,g

F hA (12360)(1.567)(1.2) 23242 N

(1/12)(1.2)(1) sin 60m 1640.0y

(1.567)(1.2)

γ= = =

°= − = −

These are shown on the freebody at right. The water force and its line of action are shown without numbers, because they depend upon the centroidal depth on the water side:

w CG

3

CPGCGC

F (9790)h (1.2)

(1/12)(1.2)(1) sin 60 0.0722y

h (1.2) h

=

°= − = −

The weight of the gate, W = 180(9.81) = 1766 N, acts at the centroid, as shown above. Since the force at B equals zero, we may sum moments counterclockwise about A to �nd the water depth:

A

GCGC

M 0 (23242)(0.5461) (1766)(0.5cos60 )

(9790)h (1.2)(0.5 0.0722/h )

= = + °

− +

GCretaw,GCSolve for h 2.09 m, or: h 0.433 .snAh= = + =2.52 m

2.62 Gate AB in Fig. P2.62 is 15 ft long and 8 ft wide into the paper, hinged at B with a stop at A. The gate is 1-in-thick steel, SG = 7.85. Compute the 20°C water level h for which the gate will start to fall.

Solution: Only the length (h csc 60°) of the gate lies below the water. Only this part

Fig. P2.62

contributes to the hydrostatic force shown in the freebody at right:

CG

2

hF h A (62.4) (8h csc 60 )

2

288.2h (lbf)

γ= = °

=3

CP(1/12)(8)(h csc 60 ) sin 60

y(h/2)(8h csc 60 )

hcsc 60

6

° °= −

°

= − °

The weight of the gate is (7.85)(62.4 lbf/ft3)(15 ft)(1/12 ft)(8 ft) = 4898 lbf. This weight acts downward at the CG of the full gate as shown (not the CG of the submerged portion). Thus, W is 7.5 ft above point B and has moment arm (7.5 cos 60° ft) about B.

We are now in a position to �nd h by summing moments about the hinge line B:

2BM (10000)(15) (288.2h )[(h/2)csc 60 (h/6) csc 60 ] 4898(7.5cos60 ) 0,= − ° − ° − ° =

3/13or: 110.9h 150000 18369, h (131631/110.9) .= − = = Ans10.6 ft

2.66 Dam ABC in Fig. P2.66 is 30 m wide into the paper and is concrete (SG ≈ 2.40). Find the hydrostatic force on surface AB and its moment about C. Could this force tip the dam over? Would �uid seepage under the dam change your argument?

Solution: The centroid of surface AB is 40 m deep, and the total force on AB is

CGF h A (9790)(40)(100 30)1.175E9 N

γ= = ×=

The line of action of this force is two-thirds of the way down along AB, or 66.67 m from A. This is seen either by inspection (A is at the surface) or by the usual formula:

Fig. P2.66

3xx

CPCG

I sin (1/12)(30)(100) sin(53.13 )m 76.61y

h A (40)(30 100)θ °

−=−=−=×

to be added to the 50-m distance from A to the centroid, or 50 + 16.67 = 66.67 m. As shown in the �gure, the line of action of F is 2.67 m to the left of a line up from C normal to AB. The moment of F about C is thus

CM FL (1.175E9)(66.67 64.0) .Ans= = − ≈3.13E9 N m?

This moment is counterclockwise, hence it cannot tip over the dam. If there were seepageunder the dam, the main support force at the bottom of the dam would shift to the left of point C and might indeed cause the dam to tip over.

3.3 For steady laminar �ow through a long tube (see Prob. 1.12), the axial velocity distribution is given by u = C(R 2 − r2), where R is the tube outer radius and C is a constant. Integrate u(r) to �nd the total volume �ow Q through the tube.

Solution: The area element for this axisymmetric �ow is dA = 2πr dr. From Eq. (3.7),

2 2

0

( )2 .R

Q u dA C R r r dr Ansπ= = − =π 4CR2

3.8 Three pipes steadily deliver water at 20°C to a large exit pipe in Fig. P3.8. The velocity V2 = 5 m/s, and the exit �ow rate Q4 = 120 m3/h. Find (a) V1; (b) V3; and (c) V4 if it is known that increasing Q3 by 20% would increase Q4 by 10%.

Solution: (a) For steady �ow we have Q1 + Q2 + Q3 = Q4, or Fig. P3.8

1 1 2 2 3 3 4 4V A V A V A V A+ + = )1(

Since 0.2Q3 = 0.1Q4, and Q4 = (120 m3/h)(h/3600 s) = 0.0333 m3/s,

34

323

(0.0333 m /s) (b)

2 (0.06 )2

Q.snAV

A π= = =5.89 m/s

Substituting into (1),

2221 (0.04 ) (5) (0.05 ) (5.89) (0.06 ) 0.0333 (a)

4 4 4.snAV

π π π+ + = 1V 5.45 m/s=

From mass conservation, Q4 = V 4A 4

3 24 )c( 4/)60.0()(V)s/m 3330.0( Ans.π= 4V 5.24 m/s=

3.16 An incompressible �uid �ows past an impermeable �at plate, as in Fig. P3.16, with a uniform inlet pro�le u = Uo and a cubic polynomial exit pro�le

3

o3

where 2

yu U

η ηη

δ−

≈ =Fig. P3.16

Compute the volume �ow Q across the top surface of the control volume.

Solution: For the given control volume and incompressible �ow, we obtain 3

ootfelthgirpot 300

30 Q Q Q Q U b dy U b dy

2 2y y δδ

δ δ= + − = + − −

o o5

Q U b U b , solve for .8

Ansδ δ= + − o3

Q U b8

= δ

3.20 Oil (SG-0.91) enters the thrust bearing at 250 N/hr and exits radially through the narrow clearance between thrust plates. Compute (a) the outlet volume �ow in mL/s, and (b) the average outlet velocity in cm/s.

Solution: The speci�c weight of the oil is (0.91)(9790) = 8909 N/m3. Then

Fig. P3.20

36

2 1 3

250/3600 N/s mQ Q 7.8 10 . (a)

s8909 N/mAns−= = = × =

mL7.8

s

6222But also Q V (0.1 m)(0.002 m) 7.8 10 , solve for V . (b)Ansπ −= = × =

cm1.24

s

3.41 In Fig. P3.41 the vane turns the water jet completely around. Find the maximum jet velocity V o for a force F o.

Solution: For a CV enclosing the vane and the inlet and outlet jets, Fig. P3.41

= − = − = − − +x o out out in in jet o jet oF F m u m u m ( V ) m ( V )

2o o o o oor: F 2 A V , solve for V Ans.ρ == o

2o o

F2 ( /4)Dρ π

3.44 Consider uniform �ow past a cylinder with a V-shaped wake, as shown. Pressures at (1) and (2) are equal. Let b be the width into the paper. Find a formula for the force Fon the cylinder due to the �ow. Also compute C D = F/(ρU2Lb).

Fig. P3.44

Solution: The proper CV is the entrance (1) and exit (2) plus streamlines above and below which hit the top and bottom of the wake, as shown. Then steady-�ow continuity yields,

L

2 1 0

U y0 u dA u dA 2 1 b dy 2 UbH,

2 Lρ ρ ρ ρ= − = + −

where 2H is the inlet height. Solve for H = 3L/4. Now the linear momentum relation is used. Note that the drag force F is to the

right (force of the �uid on the body) thus the force F of the body on �uid is to the left.We obtain,

L2

x drag2 1 0

U y U yF 0 u u dA u u dA 2 1 1 b dy 2H U b F

2 L 2 Lρ ρ ρ ρ= = − = + + − = −

2 2drag

3L 3 7Use H , then F U Lb U Lb .

4 2 6Ansρ ρ= = − ≈ 21

U Lb3

ρ

The dimensionless force, or drag coe�cient F/(ρU2Lb), equals C D = 1/3. Ans.

3.54 For the pipe-�ow reducing section of Fig. P3.54, D 1 = 8 cm, D 2 = 5 cm, and p2 =1 atm. All �uids are at 20°C. If V 1 = 5 m/s and the manometer reading is h = 58 cm, estimate the total horizontal force resisted by the �ange bolts.

Fig. P3.54

Solution: Let the CV cut through the bolts and through section 2. For the given manometer reading, we may compute the upstream pressure:

1 2 merc waterp p ( )h (132800 9790)(0.58 m) 71300 Pa (gage)γ γ− = − = − ≈

Now apply conservation of mass to determine the exit velocity: 2 2

2221Q Q , or (5 m/s)( /4)(0.08 m) V ( /4)(0.05) , solve for V 12.8 m/sπ π ≈==

Finally, write the balance of horizontal forces:

x bolts 1,gage 1 2 1F F p A m(V V ),= − + = −

2 2boltsor: F (71300) (0.08) (998) (0.08) (5.0)[12.8 5.0] .

4 4Ans

π π≈−−= 163 N

3.61 A 20°C water jet strikes a vane on a tank with frictionless wheels, as shown. The jet turns and falls into the tank without spilling. If θ = 30°, estimate the horizontal force F needed to hold the tank stationary.

Solution: The CV surrounds the tank and wheels and cuts through the jet, as shown. We have to assume that the splashing into the tank does not increase the x-momentum of the water in the tank. Then we can write the CV horizontal force relation:

Fig. P3.61

( )ρ θυ= − = − = − tejninix tank

dF F u d m u 0 mV independent of

dt2 2

2j j 3

slug 2 ftThus F A V 1.94 ft 50 .

4 12 sftAns

πρ ≈== 106 lbf

3.74 Water at 20°C �ows down a vertical 6-cm -diameter tube at 300 gal/min, as in the �gure. The �ow then turns horizontally and exits through a 90° radial duct segment 1 cm thick, as shown. If the radial out�ow is uniform and steady, estimate the forces (Fx, F y, F z)required to support this system against �uid momentum changes.

Solution: First convert 300 gal/min = 0.01893 m3/s, hence the mass �ow is ρQ = 18.9 kg/s. The vertical-tube velocity (down) is Vtube = 0.01893/[(π/4)(0.06)2] = −6.69 k m/s. The exit tube area is (π/2)R∆h = (π/2)(0.15)(0.01) = 0.002356 m2, hence Vexit = Q/A exit =0.01893/0.002356= 8.03 m/s. Now estimate the force components:

θ ρ θ+ °

− °

= = = − ∆ ?45

45

sin . (a)x out out exitF u dm V hR d Ansx 0F

θ ρ θ ρ+ °

− °

= = − = − ∆ − = − ∆45

45

cos 0 2tixetixenituotuoyF v dm mv V hR d V hRyF

≈−= (8.03)(998)(0.01)(0.15) 2 . (b)snA:ro y N71F −

= = − = − − ≈( ) (18.9 / )[0 ( 6.69 / )] . (c)z out inF m w w kg s m s Ansz N621F +

3.133 The long pipe in Fig. 3.133 is �lled with water at 20°C. When valve A is closed, p1 − p2 = 75 kPa. When the valve is open and water �ows at 500 m3/h, p1 − p2 =160 kPa. What is the friction head loss between 1 and 2, in m, for the �owing condition?

Fig. P3.133

Solution: With the valve closed, there is no velocity or friction loss:

21211 2 2 1

p p p p 75000m 66.7zz:ro,zz

)18.9(899ggg ρρρ−

+ = + − = = ≈

When the valve is open, the velocity is the same at (1) and (2), thus “d” is not needed: 2 2

1 2 1 221f

p p V V 160000With �ow: h (z z ) 0 7.66 .

g 2g 998(9.81)Ans

ρ− −

= + + − = + − ≈8.7 m

3.141 Water at 20°C is pumped at 1500 gal/ min from the lower to the upper reservoir, as in Fig. P3.141. Pipe friction losses are approximated by hf ≈ 27V 2/(2g), where V is the average velocity in the pipe. If the pump is 75 percent e�cient, what horse-power is needed to drive it?

Fig. P3.141

3.145 The large turbine in Fig. P3.145 diverts the river �ow under a dam as shown. System friction losses are hf =3.5V 2/(2g), where V is the average velocity in the supply pipe. For what river �ow rate in m3/s will the power extracted be 25 MW? Which of the two possible solutions has a better “conversion e�ciency”?

Fig. P3.145

Solution: The �ow rate is the unknown, with the turbine power known: 2 2

1 1 2 2brutfbrutf21

p V p Vz z h h , or: 0 0 50 0 0 10 h h

2g 2gγ γ+ + = + + + + + + = + + + +

2f pipe p p pipe 2

pipe

Qwhere h 3.5V /(2g) and h P /( Q) and V

( /4)Dγ

π= = =

Introduce the given numerical data (e.g. Dpipe = 4 m, P pump = 25E6 W) and solve: 3 3Q 35410Q 2.261E6 0, with roots Q 76.5, 137.9, and 214.4 m /s− + = = + + −

The negative Q is nonsense. The large Q (=137.9) gives large friction loss, hf ≈ 21.5 m. The smaller Q ( = 76.5) gives hf ≈ 6.6 m, about right. Select Qriver ≈ 76.5 m3/s. Ans.

3.169 Once it has been started by su�cient suction, the siphon in Fig. P3.169 will run continuously as long as reservoir �uid is available. Using Bernoulli’s equation with no losses, show (a) that the exit velocity V2 depends only upon gravity and the distance H and (b) that the lowest (vacuum) pressure occurs at point 3 and depends on the distance L + H . Fig. P3.169

Solution: Write Bernoulli from 1 to 2:

222221 a a1 2

2121

VVV p pp pz z , or: 0 z z

g2g2g2γ γ γ γ+ + ≈ + + + + ≈ + +

2 exit 1 2Solve for V V 2g(z z ) .Ans= = − = 2gHSince the velocity is constant throughout the tube, at any point C inside the tube,

23a C2aCCp z p z , or, at point 3: p p (z z ) .Ansγγγ =−−=+≈+ , g(L H)anim p − +ρ

3.172 The 35°C water �ow of Fig. P3.172 discharges to sea-level standard atmosphere. Neglecting losses, for what nozzle diameter D will cavitation begin to occur? To avoid cavitation, should you increase or decrease D from this critical value?

Solution: At 35°C the vapor pressure of water is approximately 5600 Pa (Table A.5). Bernoulli from the surface to point 3 gives the

Fig. P3.172

Torricelli result V 3 = √(2gh) = √2(32.2)(6) ≈ 19.66 ft/s. We can ignore section 2 and write Bernoulli from (1) to (3), with p 1 = pvap and ∆z = 0:

222231211 2

2

1 3

V V V Vp p 117 2116,:ro,

2 2 1.93 2 1.93 2

Dbut also V V

1/12

ρ ρ+ ≈ + + ≈ +

=

431

ftEliminate V and introduce V 19.66 to obtain D 3.07E 4, .

sAns= = − D 0.132 ft≈

To avoid cavitation, we would keep D < 0.132 ft , which will keep p1 > pvapor.

4.9 An idealized incompressible �ow has the proposed three-dimensional velocity distribution

V = 4xy2i + f(y)j – zy2k

Find the appropriate form of the function f(y) which satis�es the continuity relation.

Solution: Simply substitute the given velocity components into the incompressible continuity equation:

2 2 2 2(4 ) ( ) 4 0u v w f df

xy zy y yx y z x y z dy

∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂

+ + = + + − = + − =

22: 3 . Integrate: ( ) ( 3 )df

or y f y y dy Ans.dy

= − = − = +3y constant−

4.19 An incompressible �ow �eld has the cylinder components υθ = Cr , υz = K (R2 – r2),υr = 0, where C and K are constants and r ≤ R , z ≤ L . Does this �ow satisfy continuity? What might it represent physically? (a) Assuming a no-slip condition at the wall, �nd an expression for the velocity component v(x, y) for y ≤ δ. (b) Then �nd the maximum value of v at the station x = 1 m, for the particular case of air�ow, when U = 3 m/s and δ = 1.1 cm.

Solution: The two-dimensional incompressible continuity equation yields 22

32320

2 2, 2

y

x constv u y d y d d y y

U or: v U dyy x dx dx dx

∂ ∂ δ δ δ∂ ∂ δδδδ =

−= − = − + = − |

. (a)snAerehw:ro2 3

2 32 ,22 3 2

d y y d Cv U

xxdxd x= − = =

δ δ δδ δ

(b) We see that v increases monotonically with y, thus vmax occurs at y = δ :

(3 / )(0.011 ). (b)

6(1 )max ym s m

snAvvmδ== = = =| mU

0.0055s6x

δ

This estimate is within 4% of the exact vmax computed from boundary layer theory.

4.21 Air �ows under steady, approximately one-dimensional conditions through the conical nozzle in Fig. P4.21. If the speed of sound is approximately 340 m/s, what is the minimum nozzle-diameter ratio De/Do for which we can safely neglect compressibility e�ects if Vo = (a) 10 m/s and (b) 30 m/s?

Solution: If we apply one-dimensional continuity to this duct,

Fig. P4.21

222o o o e e e o e e o o eV D V D , or V V (D /D ) if

4 4π π

ρρρρ = ≈ ≈

To avoid compressibility corrections, we require (Eq. 4.18) that Ma ≤ 0.3 or, in this case, the highest velocity (at the exit) should be Ve ≤ 0.3(340) = 102 m/s. Then we compute

1/2 1/2ooeonimoe(D /D ) (V /V ) (V /102) if V 10 m/s . (a)Ans= = = =0.31

oif V 30 m/s . (b)Ans= =0.54

4.31 According to potential theory (Chap. 8) for the �ow approaching a rounded two-dimensional body, as in Fig. P4.31, the velocity approaching the stagnation point is given by u = U(1 – a2/x2), where a is the nose radius and U is the velocity far upstream. Compute the value and position of the maximum viscous normal stress along this streamline. Is this also the position Fig. P4.31

of maximum �uid deceleration? Evaluate the maximum viscous normal stress if the �uid is SAE 30 oil at 20°C, with U = 2 m/s and a = 6 cm.

Solution: (a) Along this line of symmetry the convective deceleration is one-dimensional:

2 2 2 42

x 2 3 3 5

u a 2a a aa u U 1 U 2U

x x x x x∂∂

= = − = −

xdaThis has a maximum deceleration at 0, or at (5/3) a . (a)

dxAns= = − √ =x 1.29a−

The value of maximum deceleration at this point is 2x,maxa 0.372U /a.= −

(b) The viscous normal stress along this line is given by

2

xx 3

u 2a U)b( .mumixam a htiw 22

x xAns

∂τ µ µ

∂= = max

4 Uat x a

a= = −

µτ

Thus maximum stress does not occur at the same position as maximum deceleration. For SAE 30 oil at 20°C, we obtain the numerical result

max3

kg kg 4(0.29)(2.0))b( ., 92.0, 719,lio 03 EAS

m s (0.06 m)mAnsρ µ τ= = = ≈

?39 Pa

4.50 Investigate the polar-coordinate stream function ψ = Kr 1/2sin 1

2 ,θ K = constant. Plot the streamlines in the full xy plane, �nd any stagnation points, and interpret.

Solution: Simply set ψ/K = constant and plot r versus θ. This represents inviscid �ow around a 180° turn. [See Fig. 8.14(e) of the text.]

y

x

Fig. P4.50