ADDITIONAL

PROBLEMS NOT IN

VISCOUS FLUID

FLOW TEXT

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VISCOUS FLUID FLOW – Third Edition - Frank M. White

ADDITIONAL PROBLEMS NOT IN THE TEXT

Chapter 1. Preliminary Concepts

(Text problems in Chapter 1 are solved on pages 1-16 of the Manual)

1-27 Consider the tilted free surface of a liquid,

as in Fig. P1-27. Show that, if the fluid weight is

taken into account, the tilting causes shear stresses

in the liquid and hence the liquid will flow and

cannot be in a hydrostatic condition.

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Gas, po

Liquid, density 0

Fig. P1-27

1-28 Following up on Prob. 1-4, consider laminar flow in a pipe of radius R, vz = K(R2 – r2),

for 0 r R, where K is a constant. Determine (a) the vorticity distribution; (b) the strain

rates; and (c) the average velocity V = Q/Apipe, where Q is the volume flow through the pipe.

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1-29 A realistic steady flow field, in Cartesian coordinates, has the following strain rates:

where K is a constant. Deduce formulas for the velocity components (u, v, w) and state

whether your result is unique. Sketch a few streamlines of your velocity distribution.

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1-30 Consider a wide film of liquid flowing steadily

down an inclined plane, as shown in Fig. P1-30.

The density and viscosity of both liquid and gas

are constant. The film depth h is constant.

List some simplifying assumptions that would

help us to analyze this problem and to find the film

velocity distribution without the use of a parallel

set of supercomputers. Pay especial attention to boundary conditions.

Do not solve the problem, but give a nice discussion.

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1-31 Consider a flat interface between a gas and a liquid. Let x and y denote coordinates

parallel and normal to the interface, respectively. Properties do not vary in the z direction. Gas

shear on the interface is negligible. Suppose that the surface tension coefficient varies due to

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x, u

y, v

hGAS

LIQUID

Fig. P1-30

temperature or concentration gradients, T = T (x). By summing forces in the x direction along an

elemental thin piece of interface, derive a boundary condition that relates liquid velocity u to T (x).

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1-32 Data for the viscosity of ethyl alcohol (C2H6O) are given as follows:

T, C -40 -20 0 20 40 60 80

, kg/m-s 4.81E-3 2.83E-3 1.77E-3 1.20E-3 8.34E-4 5.92E-4 4.30E-4

Fit this data to a formula similar to Eq. (1-56), plot it, and comment upon the accuracy.

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1-33 Consider the cone-plate viscometer in Fig. P1-33. The fluid to be tested fills the gap and

one measures the moment M required to steadily rotate the cone at angular velocity . Suppose

that R = 8 cm and = 4. Assuming a locally linear velocity profile and laminar flow, estimate

the fluid viscosity if a moment of 0.9 N-m is required to rotate the cone at 650 rpm.

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1-34 Analyze the interface (x) between a liquid and a gas near a plane wall, as in Fig. P1-34.

Assume constant surface tension T and small curvature, 1/Rx d2/dx2 (see Eq. 1-107 of the

text). The contact angle is at x = 0. (a) What are the appropriate boundary conditions? (b)

Find a formula for (x) and the interface height H at the wall.

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R

Heavy fluid,

Fig. P1-33

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Chapter 2. Fundamental Equations of Compressible Viscous Flow

(Text problems in Chapter 2 are solved on pages 17-32 of the Manual)

2-22 A simple demonstration of how the Grashof number arises in free convection is to

consider a small parcel of light fluid, < o, rising a distance x due to buoyancy. Equate the

decrease in potential energy of the parcel to its increase in kinetic energy to form a characteristic

“velocity” of the parcel. Use this velocity-scale to form a Reynolds number Rex and then

interpret its square, Rex2.

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2-23 In slip flow of a gas near a fixed wall, R. Raju and S. Roy, “Hydrodynamic Model for

Microscale Flows in a Channel with Two 90 Bends”, J. Fluids Engineering, vol. 126, May 2004,

pp. 489-492, propose the following model for slip velocity at the wall:

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y

x

=H Fig. P1-34

where x and y are parallel and normal to the wall, respectively, T is absolute gas temperature, and

is a tangential momentum “accommodation coefficient” whose value is approximately unity.

(a) Note the tangential temperature gradient term. What might this represent? (b) Using typical

reference properties to define dimensionless variables, rewrite this boundary condition in

dimensionless form and discuss any nondimensional parameters which arise.

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2-24 For one-dimensional flow through a porous medium of variable thickness h(x), the

differential equation for thickness relates to the pressure gradients:

where p is the fluid pressure and K(x) is the variable permeability of the medium (usually

measured in m2). Using typical reference properties to define dimensionless variables, rewrite

this equation in dimensionless form and discuss any nondimensional parameters that arise.

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Chapter 3. Solutions of the Newtonian Viscous Flow Equations

(Text problems in Chapter 3 are solved on pages 33-87 of the Manual)

3-59 In Section 3-5, for Stokes’ 1st problem of the suddenly accelerated plate, the solution for

u(y, t) in Eq. (3-107) is given subject to the initial condition u(y, 0) = 0 in Eq. (3-106). In

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Stokes’ 2nd problem, the oscillating plate, the solution of Eq. (3-111) is found without a listed

initial condition. Is this an unfortunate omission or not? Please explain.

different transient, which develops after that into the same steady oscillation, Eq. (3-111).

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3-60 The solution to Stokes’ 1st problem, Eq. (3-107), was given without any ceremony. Let

/z 0 in Eq. (3-105). Show that the similarity variable u/U0 = f(), where = y/[2(t)],

reduces Eq. (3-105) to an ordinary differential equation whose solution is an error function.

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3-61 A belt moves upward at velocity V, dragging a film

of viscous liquid of thickness h, as in Fig. P3-61. Near

the belt, the film moves upward due to no-slip. At its

outer edge, the film moves downward due to gravity.

Assuming that the only non-zero velocity is v(x), with

zero shear stress at the outer film edge, derive a formula

For (a) v(x); (b) the average velocity Vavg in the film;

and (c) the velocity Vo for which there is no net flow

either up or down. (d) Sketch v(x) for case (c).

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3-62 Consider the following proposed incompressible flow field in cylindrical coordinates:

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V

h const

,

x, u

y,v

Fig. P3-61

BELT

Air

g

where B and C are constants. (a) Determine if this distribution satisfies the continuity equation.

Does a stream function exist? (b) Does it satisfy the Navier-Stokes equation if gravity is

neglected? If so, find the pressure distribution p(r, , z).

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3-63 Consider low-speed laminar flow between two walls,

caused by natural convection. The walls are at (y = +h) and

(y = -h) respectively, as shown in the figure. The boundary

conditions are no-slip at each wall, T (x, -h) = To , and

T (x, +h) = To . Assume u = u(y), v = 0, T = T(y) and

neglect viscous dissipation. Include buoyancy effects.

Find the velocity and temperature distributions.

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3-64 The height of the equilateral triangular duct in Fig. 3-9

is h = (31/2/2)a 0.866a. In terms of height instead

of side length, investigate the following velocity

distribution for fully developed laminar flow:

where C is a constant.

(a) Determine if this distribution satisfies the no-slip condition in the duct. (b) If your answer to

part (a) is Yes, find the value of C which satisfies the fully-developed-flow x-momentum

equation. (c) Show that the volume flow rate is proportional to h4 and the pressure gradient.

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x, u

y, v

Fig. P3-63

g

h

y

z ah

Fig. 3-9

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3-65 A dashpot can be modeled as a cylinder moving

through a tube filled with oil, as in Fig. P1-33. The

clearance is small, << R, and laminar flow induced in

the annular region can be approximated as an average

velocity Vc >> V. Find an expression for the force F

required to push the piston through the tube at velocity V,

if the resistance is due to pressure drop in the clearance.

HINT: Use the one-dimensional continuity relation as part of the solution.

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3-66 If you have already solved Prob. 3-65, skip this one. You are asked to analyze the system

of Prob. 3-65 using just numerical data, not deriving a force formula. Calculate the force F for

the following data: R = 8 cm, = 4 mm, L = 40 cm, for SAE 30 oil, = 891 kg/m3, = 1.5

kg/m-s, and dashpot velocity V = 20 cm/s. Do not attempt to solve for the annular flow with a

moving wall, just assume an average velocity for fully developed laminar flow in an annulus.

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3-67 For some similarity-theory practice, try this problem, modified from a suggestion of

Clement Kleinstreuer, Engineering Fluid Dynamics, Cambridge University Press, 1997. Figure

P3-67 below shows a simplified model of laminar incompressible boundary layer convection.

The boundary layer thickness is constant and the velocity profile is simplified to be linear.

Boundary values U, Tw, and Te are constant. Dissipation is negligible. Balance x-directed

convection with conduction and find a suitable similarity variable. Solve as best you can and

plot the temperature profile and calculate the wall heat transfer. [HINT: The similarity variable

is proportional to y/x1/3.]

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FV

R

R+

LFig. P1-33

y TeU

u(y)

= constant

Fig. P3-67

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3-68 SAE 30 oil at 20C (See Table A-3) flows through a smooth pipe of diameter 2 cm.

The pressure drop is 123 kPa/m. If the flow is fully developed, according to the writer, the

resulting flow rate will be 6 m3/h. If the pressure drop and fluid remain the same, determine the

size of a smooth equilateral triangle that would pass the same flow rate. Is the flow laminar?

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Chapter 4. Laminar Boundary Layers(Text problems in Chapter 4 are solved on pp. 88-137 of the Manual.)

4-56 For laminar boundary layer separation, a purely algebraic alternative to Thwaites’

method was given by Stratford (1954). Define the local freestream pressure coefficient by

Then Stratford’s inner-outer velocity-matching scheme predicts separation when

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xTw

T(x,y)

0

where xB is the position of the freestream velocity maximum. The formula is applied in the

decelerating region downstream of xB. Test this criterion for the Howarth (1938) flow,

Eq. (4-114), and compare with the Thwaites separation estimate.

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4-57 Test the Stratford separation formula, from Prob. 4-56, for one of the Tani (1949)

freestream velocity distributions in Table 4-5 and compare the separation prediction with

Thwaites’ method.

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4-58 Boundary layer displacement effects are

important when a flow enters a duct near a wall,

as in Fig. P4-58. The flow is two-dimensional,

at U = 6 m/s of air at 20C and 1 atm. The duct

entrance is 50 cm downstream of a flat plate tip.

If the duct height is h = 3 mm, estimate the

volume flow of air into the duct per unit

width into the paper.

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4-59 As a variation of Prob. 4-58 above, if the geometry and air velocity are the same, find the

spacing h for which the volume flow into the duct will be 0.017 m3/s per meter of plate width.

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x = 0

U

x = 50 cm

DUCT

(x)

h

4-60 Consider two-dimensional steady laminar boundary layer flow with zero pressure gradient

and constant boundary thickness. The velocity profile does not change shape, that is, u/U =

fcn(y/). Show that such a flow would be related to a uniformly permeable wall at y = 0.

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4-61 In Table 4-5, the laminar-flow test cases of Tani (1949) are missing a sixth power case.

Use the method of Thwaites to predict the separation point for U = Uo(1 – x6).

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4-62 Consider laminar flat plate flow with the following approximate velocity profile:

which satisfies the conditions u = 0 at y = 0 and u = 0.993U at y = . (a) Use this profile in

the two-dimensional momentum integral relation to evaluate the approximate boundary layer

thickness variation (x). Assume zero pressure gradient. (b) Now explain why your result in

part (a) is deplorably inaccurate compared to the exact Blasius solution.

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4-63 A proposed approximate velocity profile for a boundary layer is a 4th order polynomial:

(a) Does this profile satisfy the same three conditions as the 2nd order polynomial of Eq. (4-11)

of the text? (b) What additional condition is satisfied by the 4th order polynomial at y = which

is not satisfied by the 2nd order form? (c) What pressure gradient dp/dx is implied by this profile?

(d) Evaluate the displacement and momentum thicknesses as compared to . (e) Use the 4th order

profile to evaluate the momentum integral relation for flat-plate flow, Cf = 2(d/dx), and show

that the boundary layer thickness is approximated by

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4-64 In Chapter 3 of the text we used the simple quadratic laminar flat plate profile, Eq. (4-11),

to make an integral analysis that had about a 10% error for thickness and friction:

Now use this approximate profile to estimate the velocity normal to the wall, v(x, y), assuming

two-dimensional steady laminar flow. Plot this (approximate) normal velocity profile and

compare its value at y = with the Blasius result, v(x, ) = 0.86/(Rex)1/2. HINT: You will

need to use the thickness estimate (x) from Eq. (4-14) to obtain v.

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4-65 Consider laminar boundary layer flow of air with the Howarth (1938) linear freestream

velocity distribution, U/Uo = 1 – x/L. As a text example of Thwaites’ method, we found

separation at x/L 0.123. For this same adverse pressure gradient, use a similar integral method

to calculate the local heat transfer for a constant wall temperature, Tw Te. Note what happens

at the separation point and comment.

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Chapter 5. The Stability of Laminar Flows (Text problems in Chapter 5 are solved on pp. 138-163 of the Manual.)

5-32 For transition in accelerating pipe flow, fit the data of Fig. 5-37b to a power law and then

discuss how transition time varies with pipe diameter, acceleration, and kinematic viscosity.

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5-33 Consider water at 20ºC which fills a horizontal pipe of diameter 7 cm. If the water

accelerates from rest uniformly, with a constant acceleration of 2 g’s, estimate (a) the time of

transition to turbulence, and (b) the Reynolds number at this time. Comment if appropriate.

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Chapter 6. Incompressible Turbulent Mean Flow(Text problems in Chapter 6 are solved on pp. 164-215 of the Manual.)

6-48 Let us revisit Prob. 6-45, which modifies Prob. 6-43 to be a cone.

The problem was: Consider a conical flat-walled

diffuser as in Fig. P6-8. Assume incompressible

flow with a one-dimensional freestream velocity

U(x) and entrance velocity Uo at x=0. The entrance

diameter is W . Use two-dimensional theory.

Assume turbulent flow at x=0, with momentum thickness

o/W = 0.02, H(0) = 1.3, and UoW/ = 105. Using the method of Head, Sect. 6-8.1.2, numerically

estimate the angle f for which separation occurs at x = 1.5 W. [The answer was f = 4.6.]

Now: Make this a parameter study. Experiments with conical diffusers show that, for a given

length (L/W = 1.5 here), the separation angle increases if inlet boundary layer thickness decreases

[see, e.g., White (2003), pp. 399-404]. Can Head’s theory predict this effect? Calculate and plot

the separation angle fsep for inlet momentum thicknesses in the range 0.0002 o/W 0.05.

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6-49 For turbulent boundary layer methods that cannot calculate Cf ® 0, some other criterion

is needed for predicting separation. The most common strategy is to assume H 3 at separation.

In a paper published after this 5th edition was put to bed, L. Castillo, X. Wang, and W. K.

George, “Separation Criteria for Turbulent Boundary Layers Via Similarity Analysis”, J. Fluids

Engineering, vol. 126, May 2004, pp. 297-304, show, with both theory and experiment, that

turbulent separation occurs at a dimensionless pressure gradient parameter whose value is

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Uox

f

Fig. P6-48

fW

where is the momentum thickness and U¥ is the freestream velocity. Using the turbulent

two-dimensional steady momentum integral relation, show that the above criterion is equivalent

to a shape factor Hsep 2.76 ± 0.23. Which criterion has less uncertainty?

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6-50 Show that, if the turbulent logarithmic law-of-the-wall is valid, then the two-dimensional

stream function x,y, when nondimensionalized by kinematic viscosity, can be written entirely

in terms of wall-law variables u+ and y+. Find an analytic function for this stream function.

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6-51 Let us revisit the boundary layer displacement

problem when a flow enters a duct near a wall,

similar to Prob. 4-58. The flow is two-dimensional,

with U = 40 m/s of air at 20C and 1 atm. The duct

entrance is 1.5 m downstream of a flat plate tip.

If the duct height is h = 13 mm, estimate the

volume flow of air into the duct per unit

width into the paper. Assume the flow is

turbulent from the leading edge onward.

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6-52 As a variation of Prob. 6-51 above, if the geometry and air velocity are the same, find the

spacing h for which the volume flow into the duct will be 0.5 m3/s per meter of plate width.

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6.53 Water, at 1 atm and 20C, flows through a 30-cm

smooth square duct at 0.4 m3/s as in Fig. P6-53.

Fifty thin flat plates of chord-length 3 cm are stretched

across the duct at random positions. Their wakes do not

interfere with each other. (a) How much additional

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x = 0

U = 40 m/s

x = 1.5 m

DUCT

(x)

h

Fig. P6-51

Fig. P6-53

50 plates

pressure drop do these plates contribute to the duct flow loss?

(b) By what per cent do the plates increase the pressure drop over the 1-meter length?

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6-54 A model two-dimensional hydrofoil has the following theoretical potential-flow surface

velocities on its upper surface at a small angle of attack:

x/C 0.0 0.025 0.05 0.1 0.2 0.3 0.4 0.6 0.8 1.0

V/U¥ 0.0 0.97 1.23 1.28 1.29 1.29 1.24 1.14 0.99 0.82

The stream velocity U¥ is varied to set the Reynolds number. The chord length C is 40 cm.

The fluid is water at 20C and 1 atm. Assume that x is a good approximation to the arc length

along the upper surface. Using any turbulent-boundary-layer method of your choice, find the

predicted separation point, if any, for ReC = 6E6. For simplicity, assume turbulent flow from

the leading edge onward.

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6-55 Air at 20C issues from a 2-cm-diameter nozzle at a near-uniform velocity of 20 m/s. At

section x, a considerable distance downstream, measurements of the turbulent jet velocity

distribution have been fit to the following formula:

(a) Evaluate the jet momentum at section x and compare with the nozzle momentum. (b)

Determine the volume flow of air entrained into the jet, if any, between the nozzle and section x.

(c) Estimate the distance that section x is downstream of the nozzle.

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6-56 Theodore von Kármán, in NACA Technical Memorandum No. 611, 1931, used a

similarity approach to derive the following model for turbulent eddy viscosity:

(a) If w near the wall, show that this model leads to the logarithmic overlap law (6-38a).

(b) In fully developed pipe flow, the shear stress is linear across the duct:

where R is the pipe radius and y is measured from the wall (y = 0) to the centerline (y = R).

Kármán (1931) applied his model above to pipe flow and integrated to the following result:

Plot this formula versus y/R for the special case Umax/v* = 25.0 and compare it with the

logarithmic law for the same value of Umax/v*. Comment if appropriate. (c) Discuss some

possible reasons why Kármán’s model and his pipe-flow result have not become popular.

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6-57 Consider fully developed turbulent flow in a pipe of radius R. Recall that the shear stress

varies linearly from zero at the centerline to a maximum, w, at the wall. Assume that the log-

law holds all the way across the pipe and (a) find an expression for the variation of eddy

viscosity t across the pipe, in terms of the coordinate y measured from the wall inward. (b) Also

find the position of maximum eddy viscosity.

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6-58 Water at 20C flows through a smooth pipe of diameter 2 cm. The pressure drop is 123

kPa/m. If the flow is fully developed, according to the writer, the resulting flow rate will be 21.3

m3/h. If the pressure drop and fluid remain the same, determine the size of a smooth equilateral

triangle that would pass the same flow rate. Is the flow turbulent?

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Chapter 7. Compressible Boundary Layer Flow(Text problems in Chapter 7 are solved on pp. 216-245 of the Manual.)

7-28 Analyze steady compressible Couette flow between an upper moving plate and a lower

fixed plate, as shown in Fig. P7-28. There is no pressure gradient, and the variables (, u, T)

vary with y only. Assume constant cp, Pr, and g. Viscosity and thermal conductivity may vary

with temperature, but their ratio /k is assumed constant. Neglect . (a) Set up continuity,

momentum, and energy and carry out the solution as far as you can – a complete solution is not

possible because the variation (T) is not known. (b) Find an expression for the heat transfer at

the lower wall. (c) Examine the special case of an adiabatic lower wall.

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H

x

yT = Te , u = U

T = Tw , u = 0

Fig. P7-28

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7-29 Air, Pr = 0.71, flows at supersonic speed on a flat plate and encounters a double-wedge

configuration as in Fig. P7-29. The boundary layers are laminar. Compare the adiabatic wall

temperature in region 3 with the corresponding value in region 1 and comment cogently.

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7-30 Most of the text emphasizes external shear layers. What about internal shear layers,

namely, duct flow? Consider a short length of steady adiabatic duct flow, as shown.

The area A is constant. Assume one-dimensional flow with local wall shear stress w. The flow is

compressible, that is, density is variable, (x). (a) Derive an x-momentum (axial)

equation which relates pressure and velocity changes to wall shear stress. (b) Add in continuity

and an ideal (perfect) gas with constant (isothermal) temperature to derive the following relation

for mass flow over a duct of length L. Assume a constant friction factor, = 8w/(V2).

- -20

15

18

Fig. P7-29

Ma1 = 3.5T1 = 0C

(2)(3)

Control volume

cross-section area A, diameter D

dx

, V, T, p

Fig. P7-30

where 1 and 2 are the upstream and downstream sections of the duct, and L = x1 – x2. The

quantity R is the ideal gas constant.

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7-31 Supersonic turbulent flow of air, Pr = 0.71,

passes through a Prandtl-Meyer expansion fan, as (1)

in Fig. P7-31. Approach conditions are Ma1 = 3.0,

p1 = 150 kPa, and T1 = 120C. Assume steady isentropic

supersonic flow through the fan [see, e.g., White (2003), Chap. 9].

(a) Estimate the adiabatic wall temperature in region (2).

(b) If the actual wall temperature in region 2 is Tw2 = 100C, estimate the wall heat transfer at a

point where the local Reynolds number is 4E7. [NOTE: If the algebra in part (b) is tiring, make

your estimate from the graphs in the text in Chap. 7.]

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7-32 Extend Prob. 7-14 as follows. What is the effect of the wedge half-angle? Assume that

the wedge side length remains at L = 19.32 cm for any angle. Assume laminar flow. The

writer’s result in Prob. 7-14, for a half angle of 15, was qw,mean 163,000 W/m2. Will increasing

the angle decrease the heat transfer? Explain. Compute the heat transfer for a zero angle.

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20

(2)

Fan

Fig. P7-31

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