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Unformatted text preview: 210 CHAPTER 5 I IIqunucnON TD DIFFERENTIAL ANALYSIS OF FLUID MOTION $5.26 Does the velocity ﬁeld of Problem 5.22 represent a possible 5.37 COHSidef the ﬂow ﬁEId given by ‘7 =xy2i — h“; + xyl. D0
incompressible llow case? If so. evaluate and sketch the stream lemme (a) the number OfdiI‘neﬂSiOUS Oflhﬁ ﬁ0W. (b) if“ is apossr
function for the ﬂow. If not. evaluate the rate of change of density ible incompreSSibIC ﬂow. and (C) lhe acceleration Ufa ﬂUid panic
in the ﬂow ﬁeld. at point (x.y.: = (l. 2. 3).
$5.27 Consider a ﬂow with velocity components a = 0. U = 533 conﬁde" fhe ﬂow ﬁeld given by V :a‘ll’iHbl’j'i'rzz ‘
y(_v2 — 3:2), and w = 5 a: — 3f) wherea = I mi‘vsLEJ: : 3 s' I. and c = 2 m_1s’l. Determ’
a, [5 [his a 0113., mm, or mree_dimensionat ﬂow? (a) the number of dimensions of the ﬂow, (b) if it is a possible in
b. Demonstrate whether this is an incompressible or com— compressible ﬂow all“ (C) [he acceleration 0f a mild Panic“? ='
pressible ﬂow. Win! (313’. Z} = (3. l. 2). c. If possible. deriveastream function for this flow. 5.39 The velocity ﬁeld within a laminar boundary layer r “$.28 An incompressible frictionless ﬂow ﬁeld is speciﬁed by the aPPmXimale‘l by the expressmn
stream function If) =2Ax  My. where A = 1 mix. and x and y .. g AUy . AUy2 5
are coordinates in meters. Sketch the streamlines ti; = l) and If: = V — rm; f 4“ 4x3]; J 5. Indicate the direction of the velocity vector at the point (0. 0)
on the sketch. Determine the magnitude of the ﬂow rate between
the streamlines passing through the points (2, 2) and (4, 1). In this expression. A = 141 m' “2. and U = 0.240 m/s is the free
stream velocity. Show that this velocity ﬁeld represents a possib I.
incompressible ﬂow. Calculate the acceleration of a ﬂuid panic
at point (.r. y) = (0.5 m. 5 mm). Determine the slope of u
streamline through the point. $5.29 In a parallel onedimensional flow in the positive I direc—
tion. the velocity varies linearly from zero at y = O to 30 m/s at
y = LS Tn. Determine an expression for the stream function. I11.
Also determine the y coordinate above which the volume ﬂow rate
is halfthe total between _v I 0 and _v = 1.5 m. 5.40 The x component of velocity in a steady. incompressi 
ﬂow ﬁeld in the xy plane is u = A(.r5~10x3y2+5xy4), where A .
_‘ . _ ‘ I 2 m'4 s ' and x is measured in meters. Find the simplest y c n
“530 A “"63" “3"le Pmﬁlc was “sad [0 "1056] ﬂow 1" a “mum” ponent of velocity for this ﬂow ﬁeld. Evaluate the acceleration ii
incompressible boundary layer in Problem 5.”). Derive the stream a ﬂuid Daniele at point (x. v) : (1, 3)_ function for this ﬂow held. Locate streamlines at onequarter and 5.“ Consider the velocity ﬁeld V = mm; +y2)f+Ayl(r, "H2" one—half the total volume ﬂow rate in the boundary layer. . i 2 . In the Ajv plane. where A — l0 m Is, and x and y are measured _ $5.31 A parabolic velocity proﬁle was used to model ﬂow in *1 meters. Is this an incompressible flow ﬁeld? Derive an expressi
laminar incompressible boundary layer in Problem SJ 1. Derive for [ha ﬂuid acceleration Evajuate the velocity and accelerati
the stream function for this ﬂow ﬁeld. Locate streamlines at one— along the x axis. the y axis. and along a ﬁne deﬁned by y = 11.
quarter and onehalf the total volume ﬂow rate in the boundary can you conclude abouuhis ﬂow ﬁeld?
layer‘ 5.42 They component of velocity in a twodimensional. inco I *5.32 Derive the stream function that represents the sinusoidal pressible ﬂow ﬁeld is given by v = —Axy. where U is in m/SJaﬂd
approximation used to model. the 1: component of velocity for the are in meterS. and A is a dimensional constant. There is no veloci a:
boundary layer Of Problem 5.12. Locate streamlines at onequarter component or variation in the z direction, Determine the dime‘
and onehalf the total volume ﬂow rate in the boundary layer. sions of the constant. A. Find the simplest .r component of veloci . 215‘ 3 3 A cubic velocity proﬁle was used to model ﬂaw in a laminar in this flow ﬁeld. Calculate the acceleration of a ﬂuid particle incompressible boundary layer in Problem 5.13. Derive the stream mint (3" 3') = (1* 2)'
function for this ﬂow ﬁeld. Locate streamlines at onequarter and 5.43 An incompressible liquid with negligible viscosity ll
onehalf the total volume ﬂow rate in the boundary layer. steadin through a horizontal pipe of constant diameter. In a n ous section of length L = 0.3 m. liquid is removed at a cons zu‘
rate per unit length. so the uniform axial velocity in the pipe I.
nix) = Utl  .n'ZL). where U = 5 m/s. Develop an expressi
for the acceleration of a ﬂuid panicle along the centerline of I:
porous section. i“:5..l4 A rigidbody motion was modeled in Example 5.6 by the
velocity ﬁeld l7 = mic”. Find the stream function for this ﬂow.
Evaluate the volume ﬂow rate per unit depth between r. = 0. I0 m
and r1 = 0.2 m, if m = 0.5 radls. Sketch the velocity proﬁle
along a line of constant 0. Check the. flow rate calculated from the
stream function by integrating the velocity proﬁle along this line. \9‘ 5.44 An incompressible liquid with negligible viscosity steadily through a horizontal pipe. The pipe diameter linearly :
ies from a diameter of It} cm to a diameter of 2.5 cm over :1 Ian
of 2 m. Develop an expression for the acceleration of a fluid =I=
ticle along the pipe centerline. Plot the centerline velocity and : '
celeration versus position along the pipe. if the inlet centerl'
velocity is 1 m/s. 35.3.5 Example 5.6 showed that the velocity ﬁeld for a free vortex
in the rt) plane is 17 = éuC/r. Find the stream function for this ﬂow.
Evaluate the volume [low rate per unit depth between r1 = 0.10 In
and r2 = 0. l 2 m. if C = 0.5 mln’s. Sketch the velocity proﬁle along
a line of constant 0. Check the ﬂow rate calculated from the stream
function by integrating the velocity proﬁle along this line. 536 Consider the velocity ﬁeld I? : ADA _ 612).: + yit); + 5.45 Solve Problem 4.] 18 to show that the radial velocity in up. I narrow gap is V, = Q/anh. Derive an expression for the accele Al4xy" —4l.t3_v)yi in the xy plane. where A = 0.25 m'is . and  .
tion ofa ﬂuid pamcle In the gap. the coordinates are measured in meters. Is this a possible incom pressible flow ﬁeld? Calculate the acceleration of a ﬂuid particle 5.46 Consider the lowspeed ﬂow of air between parallel di
a! Will! (l'n'l = {2. l). as shown. Assume that the ﬂow is incompressible and invisc *These problems require material from sections that may be omitted without loss of continuity in the text material. 5’6 212 CHAPTER 5 I INTRODUCTION TO DIFFERENTIAL ANALYSIS OF FLUID MOTION 5.60 Air ﬂows into the narrow gap. of height h. between closely
spaced parallel disks through a porous surface as shown. Use a
control volume. with outer surface located at position r. to show
that the uniform velocity in the r direction is V : our/2h. Find
an expression for the velocity component in the z direction
(U0 < V). Evaluate the components of acceleration for a ﬂuid
particle in the gap. P560 5.61 The velocity ﬁeld for steady inviscid flow from left to right
over a circular cylinder, of radius R, is given by l7=Ucosfl [1— (5)]é,—Usin6 [1+ (5)]60
r 1' Obtain expressions for the acceleration of a ﬂuid Particle moving
along the stagnation streamline (H = 1t) and for the acceleration
along the cylinder surface (r : R). Plot a, as a function of n'R for
9 : It, and as a function of 6 for r = R; plot an as a function of 6
for r = R. Comment on the plots. Determine the locations at
which these accelerations reach maximum and minimum values. \96562 Consider the incompressible flow of a fluid through a noz— zle as shown. The area of the nozzle is given by A = Anll 7 bx)
and the inlet velocity varies according to U = Unll — 97"”).
where A0 = 0.5 mg. L = 5 m, b = 0.! m_[, t = 0.2 s“. and
U0 : 5 mfs. Find and plot the acceleration on the centerline. with
time as a parameter. P562 63‘563 Consider the one—dimensional. incompressible flow through the circular channel shown. The velocity at section (D is given by
U = U0 + U] sin mt, where U.) : 20 m/s, Ul : 2 m/s. and w :
0.3 rad/s. The channel dimensions are L I l to. RI 2 0.2 m, and
R2 = 0.1 m. Determine the particle acceleration at the channel
exit. Plot the results as a function of time over a complete cycle.
On the same plot. show the acceleration at the channel exit if the
channel is constant area. rather than convergent. and explain the
difference between the curves. 135.63 5.64 Consider again the steady. twodimensional velocity ﬁeld
of Problem 5.53. Obtain expressions for the particle coordinates.
xi, : fit!) and vi, = ﬁtt). as functions of time and the initial par
ticle position. txn, Iv”) at t z 0. Determine the time required for a
particle to travel from initial position. (x0. yo) 2 Q. 2) to positions
(x, y) 2 (l. l) and (2, l). Compare the particle accelerations de
termined by differentiating fitt) and ﬁt!) with those obtained in
Problem 5.53. 5.65 Expand (l7  V)? in cylindrical coordinates by direct sub
stitution of the velocity vector to obtain the convective accelera
tion ot‘a ﬂuid particle. (Recall the hint in footnote 1 on page 169.) Verify the results given in Eqs. 5.12. 5.66 Which. if any. of the ﬂow fields of Problem 5.1 are
irrotational? 5.67 A flow is represented by the velocity ﬁeld 17 =
(.r7 — 2119);2 + 35x52“ — 7m“)? + (Tray  35x4yl + 2113}!5 * yllj.
Determine if the ﬁeld is (a) a possible incompressible ﬂow and (b) irrotational. 5.68 Consider again the sinusoidal velocity proﬁle used to mode]
the .r component of velocity for a boundary layer in Problem 5.12.
Neglect the vertical component of velocity. Evaluate the circula
tion around the contour bounded by x = 0.4 m. x = 0.6 m. y I 0.
and _v : 8 mm. What would be the results of this evaluation if it
were performed 0.2 in further downstream? Assume U : 0.5 m/s. 5.69 Consider the velocity ﬁeld for ﬂow in a rectangular “cor
ner.“ V = Axi— ij, with A = 0.3 57L. as in Example 5.8. Evalu
ate the circulation about the unit square of Example 5.8. 5.70 Consider the twodimensional ﬂow ﬁeld in which it : Axy
and U : Brig. whereA=  m‘L  s". B=—%m 'l  s". and
the coordinates are measured in meters. Show that the velocity
ﬁeld represents a possible incompressible flow. Determine the
rotation at point (.r, y) = (I. I). Evaluate the circulation about the “curve” bounded by y = 0. .r = 1. y = l. and .r = 0. *5.71 Consider the ﬂow ﬁeld represented by the stream function
if; 2 x6 * 1556i}:z + 15.3}:4 i )2“. Is this a possible twodimen
sional, incompressible How? Is the flow irrotational? $5.72 Consider a ﬂow field represented by the stream function
If; = 3.t5_v e ltbr'ly3 + 3x35“. Is this a possible twodimensional in
compressible ﬂow? Is the ﬂow irrotational? $5.73 Consider a ﬂow ﬁeld represented by the stream function
ti} =7 A1203 + Uri). whereA : constant. Is this a possible two—di
mensional incompressible flow? ls the flow irrotational? *55'4 Consider a velocity field for motion parallel to the x axis
with constant shear. The shear rate is did/(iv = A. where
A = 0.1 s' '. Obtain an expression for the velocity ﬁeld. l7. Calcu
late the rate of rotation. Evaluate the stream function for this flow
field. \9‘4‘535 A ﬂow ﬁeld is represented by the stream function If; = X2 — yl. Find the corresponding velocity ﬁeld. Show that this
ﬂow ﬁeld is irrotational. Plot several streamlines and illustrate the
velocity ﬁeld. \C’ﬁﬁﬂ'ﬁ Consider the velocity ﬁeld given by V=Axyi+ 833}. where A 2 4 m‘ ' s_ l. B = —2 m 's". and the coordinates are measured in meters. Determine the ﬂuid rotation. Evaluate the Cir
cttlation about the "curve" bounded by y I 0. x : 1.); : l. and .r *These problems require material from sections that may be omitted without loss ot’continuity in the text material. = U. Obtain an expression for the stream function. Plot several
streamlines in the first quadrant. *577 Consider the flow ﬁeld represented by the stream function
tit = At)’ 4" Avg. where A I l s7]. Show that this represents a
possible incompressible. ﬂow ﬁeld. Evaluate the rotation of the
ﬂow. Plot a few streamlines in the upper halfplane. i“5.78 Consider the [low represented by the velocity ﬁeld
i V=lAy+Blf+ij, where A 2 6 '. B = 3 ms' '. and the
coordinates are measured in meters. Obtain an expression for the
stream function. Plot several streamlines (including the stagnation
streamline] in the ﬁrst quadrant. Evaluate the circulation] about the
"curve" botmded by y : ft. x = l. _v I l. and ,r I t'). 5.79 Consider again the viscomctric flow of Example 5.7.
Evaluate the average rate of rotation of a pair of perpendicular
line segments oriented at :45 from the x axis. Show that this is
the saute as in the example. *530 The velocity ﬁeld near the core of a tornado can be
approximated as q al+ K V= ,
an Ear do is this an irrotational ﬂow ﬁeld“? Obtain the stream function for
this flow. 5.8] Consider the pressure~driven ﬂow between stationary paral
lel plates separated by distance t'J. Coordinate _v is measured from
the bottom plate. The velocity ﬁeld is given by u = Utyl'blll —
ty/btl. Obtain an expression for the circttlation about a closed con,
tour of height h and length L. Evaluate when ft = [7/2 and when
it: b. Show that the same result is obtained from the area integral
ofthe Stokes Theorem {Eq. 5.18). 5.82 The velocity proﬁle for fully developed ﬂow in a circular
tube is l/E Z Vlmx[l * (HEEL Evaluate the rates of linear and
angular deformation for this flow. Obtain an expression for the
vorticity vector. 5.33 Consider the pressuredriven flow between stationary paral
lel plates separated by distance 21). Coordinate y is measured
from the channel centerline. The velocity ﬁeld is given by
it = nmaxll * {Ir/8):]. Evaluate the rates of linear and angular
deformation. Obtain an expression for the vorticity vector, C.
Find the location where the vorticity is a maximum. 5.34 A linear velocity proﬁle was used to model ﬂow in a lami
nar incompressible boundary layer in Problem 5.10. Express the
rotation of a ﬂuid particle. Locate the maximum rate of rotation.
Express the rate of angular deformation for a fluid particle. Locate
the maximum rate of angular deformation. Express the rates of
linear defonrtation for a lluid particle. Locate the maximum rates
oflinear deformation. Express the shear force per unit volume in
the .r direction. Locate the maximum shear force per unit volume:
interpret this result. 5.85 The x component of velocity in a laminar boundary layer in
water is approximated as it : U sin(rr'vt'2r5). where U : 3 oils and
t5: 2 mm. The _v component ofvelocity is much smaller than it. Oh 55 SUMMARY AND USEFUL EQUATIONS 213 5.86 Problem 4.3l gave. the velocity profile for fully developed
laminar ﬂow in a circular tube as it I ttnmll e (r/Rf]. Obtain an
expression for the shear force per unit volume in the x direction for this flow. Evaluate its maximum value for the conditions of
Problem 4.3 l . \06587 Use Extcl to generate the solution of Eq. 5.28 for m = 1 shown in Fig. 5.16. To do so, you need to learn how to perform linear
algebra in Excel. For example. for N : 4 you will end up with the
matrix equation of Eq. 5.34. To solve this equation for the n values.
you will have to compute the inverse of the 4 X 4 matrix. and then
multiply this inverse into the 4 X l matrix on the right of the
equation. In Excel. to do array operations. you must use the
following rules: Prerselect the cells that will contain the result; use
the appropriate Et‘t't’f army firmrim: (look at Excel's Help for def
tails); press CtrltShit‘t +Entct‘. not just Enter. For example, to invert
the 4 X 4 matrix you would: Prerseleet a blank 4 X 4 array that will
contain the inverse matrix: type : mim'r'rset [array containing matrix
to be invertedl): press CtrllShiftl—Enter. To multiply a 4 X 4 matrix
into a 4 x  matrix you would: PreAselect a blank 4 X l array that
will contain the result; type = mmut‘t([array containing 4 X 4
matrix]. [auray containing 4 X  matrixl); press CLrl+Shift+ Enter. 065.88 Following the steps to convert the differential equation Eq. 5.28 (for m I l) into a difference equation (for example. Eq.
5.34 for N : 4). solve
(In 7+. :2s'n.‘ ()S.‘
dx u tlt) t It"\ I tth) =0 for N : 4. 8. and lo and compare to the exact solution Hem” = sin(x) — cos(x) + e" ‘
Hints: Follow the rules for Excel array operations as described in
Problem 5.87. Only the right side of the difference equations will
change. compared to the solution method of Eq. 5.28 (for example. only the right side of Eq. 5.34 needs modifying). SE51?) Following the steps to convert the differential equation Eq. 5.28 (for m : 1) into a difference equation (for example. Eq.
5.34 for N I 4). solve do 1
—+tt=,r“ 05x51
(it For N : 4. 8. and 16 and Compare to the extract solution 14(0) : 2 ‘1
Lima : .t‘ i 21 + 2 Hint: Follow the hints provided in Problem 5.88. \S‘aSSU A lUcm cube of mass M = 5 kg is sliding across an oiled surface. The oil viscosity is ,u = 0.4 N  57mg. and the thickness of
the oil between the cube and surface is (5 2 0.25 mm. 1fthe initial
speed of the block is a” 2 i this. use the numerical method that
was applied to the linear form of Eq. 5.28 to predict the cube
motion for the ﬁrst second of motion. Use N i 4, 8. and 16 and compare to the exact solution 5
“mun : uni, [Alli/[Ml )t‘ where A is the area of contact. Hint: Follow the hints provided in
Problem 5.87. tairt an expression for the net shear force per unit volume in the ,r di~ SE59] Use Excel to generate the solutions of Eq. 5.28 for m : 2 rection on a ﬂuid element. Calculate its maximum value for this ﬂow. showu in Fig. 5. l9. *These problems require material from sections that may be omitted without loss ofcontinuity in lbe text material. ...
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 Fall '08
 ZOHAR
 Fluid Dynamics, Velocity, stream function, velocity field, incompressible llow

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