THE UN I VE RSITY OF MI CHI GAN
COLLEGE OF ENGINEERING
Department of Mechanical Engineering
Heat Transfer and Thermodynamics Laboratory
Technical Report No. 2
THE INFLUENCE OF LOCALIZED, NORMA.L SURFACE OSCILLA.TIONS
ON THE STEA.DY, LAMINAR FLOW OVER A. FLAT PLATE
Tsung Yen Na
Vedat S. Arpaci
John A.. Clark
ORA Project 05065
under contract with:
AERONA.UTICAL RESEARCH LABORATORY, OAR
AERONAUTICAL SYSTEMS DIVISION
A.IR FORCE SYSTEMS COMMAND
CONTRA.CT No. AF 33 (657)-8368
WRIGHT-PATTERSON AIR FORCE BASE, OHIO
administered through:
OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR
Apri 1 1964

This report was also a dissertation submitted by
the first author in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The
University of Michigan, 1964.

ACKNOWLEDGMENTS
The author is indebted to the many individuals who have contributed to
the success of this work. In particular, he would like to express his sincere
appreciation to his thesis adviser, Professor Vedat S. Arpaci, for his continued advice and encouragement and to Professors John A. Clark, Arthur G.
Hansen, Wen Jei Yang, and Young King Liu, who served as members of the thesis
committee.
This research is supported by the Aeronautical Research Laboratory, Office
of Aerospace Research,Aeronautical Systems Division, Air Force Systems Command,
Wright-Patterson Air Force Base, Ohio. The assistance of Dr. Max G. Scherberg
of this laboratory is appreciated.
ii

TA.BILE OF CONTENTS
Page
LIST OF ILLUSTRATIONS iv
NOMENCLATURE vi
ABSTRACT x
CHAPTER
I. INTRODUCTION 1
II. THEORETICAL ANALYSIS 5
Statement of the Problem 5
The Formulation of the Problem 5
Kinematic Relations 8
Formulation of the Problem in the Accelerating
Coordinate System 15
Simplification of the Formulation 20
Perturbation of the Differential Equations 28
Solutions of the Differential Equations 36
Zeroeth-order approximation 36
First-order approximation 38
Second-order approximation 43
Results of the Analysis 53
III. CONCLUSIONS 77
APPENDIX
I. DERIVATION OF EQUATIONS (43) AND (44) 79
II. SOLUTION OF EQUATION (114) 85
III. A. NUMERICAL EXAMPLE 89
Limitations of Xf 89
Order-of-Magnitude 91
The First-Order Velocity 91
Phase Angles 94
REFERENCES 96
iii

LIST OF ILLUSTRATIONS
Table Page
III-1. Tabulations of Flr, Fli, and b2 93
Figure
1. Model of the flat plate. 5
2. Analytical model of the disturbance. 6
3. The fixed and accelerating coordinate systems. 9
4. Illustration of the local Cartesian coordinate system. 12
5(a). F3(X') as a function of X'(m'=l). 47
5(b). F3(X') as a function of X'(m'=2). 48
5(c). F3(X') as a function of X'(m'=3). 49
5 (d). F3(X') as a function of X'(m'=4). 50
5(e). Comparison of F3(X') for different m's. 51
6(a). F5(X') as a function of X'(m'=l). 54
6(b). F5(X') as a function of X'(m'=2). 55
6(c). F5(X') as a function of X'(m'=3). 56
6(d). F5(X') as a function of X'(m'=4). 57
6(e). Comparison of F5(X') for different m's. 58
7(a). Velocity profile at X' = 0.5(m'=l). 60
7(b). Velocity profile at X' = l.l(m'=l). 61
7(c). Velocity profile at X' = 1.6(m'=l). 62
7(d). Velocity profile at X' = 2.4(m'=1). 63

LIST OF ILLUSTRATIONS (Concluded)
Figure Page
8(a). Local drag coefficient as a function of X'(m'=l). 64
8(b). Local drag coefficient as a function of X'(m'=2). 65
8(c). Local drag coefficient as a function of X'(m'=3). 66
8(d). Local drag coefficient as a function of X'(m'=4). 67
9(a). Temperature profile at X' = l(m'=l). 70
9(b). Temperature profile at Xv = 2(m'=2). 71
10(a). Local Nusselt number as a function of X'(m'=l). 73
10(b). Local Nusselt number as a function of X'(m'=2). 74
10(c). Local Nusselt number as a function of X'(m'=3). 75
10(d). Local Nusselt number as a function of X'(m'=4). 76

NOMENCLATURE
a thermal diffusivity of the fluid
al.. a5 coefficients of ro in the velocity and temperature profiles
blo..b5 coefficients of r1 in the velocity and temperature profiles
Cloo.C5 coefficients of i2 in the velocity and temperature profiles
Cfx local drag coefficient
dlo.od5 coefficients of r3 in the velocity and temperature profiles
el... e5 coefficients of r4 in the velocity and temperature profiles
e base of natural logarithm, 2.718281828
Flo..F5 functions of X' or Pr
hl,h2 elements of metric tensor
h coefficient of natural convection
i -
i unit vector in the x'-direction
j unit vector in the y.-direction
k unit vector in the zy-direction
k thermal conductivity
~ displacement vector in the (',y') coordinate system
L distance of the disturbance from the leading edge of the plate
m a constant
n a constant
Nux local Nusselt number
vi

NOMENCLATURE (Continued
0( ) order of magnitude
p pressure
Pr Prandtl number
r distance vector
R distance vector
Re Reynolds number
R( 3 real part of a complex quantity
t time in (x,y) coordinate system
T time in (X,Y) coordinate system
u velocity in x-direction
U velocity in X-direction
UO velocity in the mainstream
v velocity in the y-direction
V velocity in the Y-direction
x abscissa in (x,y) coordinate system
x' abscissa in (xl,yl) coordinate system
X abscissa in (X,Y) coordinate system
y ordinate in (x,y) coordinate system
y' ordinate in (xl,yT) coordinate system
Y ordinate in (X,Y) coordinate system
A inverse of the cubic root of Pr
vii

NOMENCLATURE (Continued)
Greek Letters
0a phase angle
P an angle
6 boundary layer thickness
C maximum amplitude of oscillation
r dimensionless distance, Y*/6
It dimensionless distance, Y*/St
O temperature
dimensionless parameter, e'%'4/Re
XT dimensionless parameter, e'2't2/Re
t1 viscosity
p density of fluid
a normal stress
T shearing stress
disturbance function, exp[-m(X-L)2+iutt]
Os steady disturbance function, exp [-m'(X'-1)2]
a function of X, defined by Eqo (152)
X frequency
Subscripts
0,1,2 zeroth, first and second approximation
20 steady component of the second approximation
condition at infinity
viii

NOMENCLATURE (Concluded)
c complex function
I imaginary part of a complex quantity
R real part of a complex quantity
w condition at wall
x',y' local Cartesian coordinates
X, Y accelerating coordinates
Superscripts
dimensionless quantities
* dimensionless quantities multiplied by Re /2
ix

ABSTRACT
The steady, laminar flow of an incompressible fluid overa semi-infinite
flat plate with a localized vibration on the surface of the plate is analyzed.
Formulation of the problem is made with reference to a set of coordinate
frames fixed on the surface of the plate. The complete momentum, continuity,
and energy equations are then simplified by the boundary layer assumptions.
The simplified equations are then linearized by the perturbation procedures
and the first three approximations are considered. The zeroth-order approximation is the well-known Blasius' solution. The first- and second-order approximations are solved by the integral method. Integration of the solutions with
respect to time gives a steady component of both skin friction and heat transfer. This steady change depends orn (e'2'4)/Re in the case of skin friction
and (E'2uw'2)/Re in the case of heat transfer as well as the distance from the
leading edge of the flat plate. The phase of skin friction is approximately
in phase with the input disturbance, while that of heat transfer lags approximately t/2 radians. Integration of the steady change over the distance along
the surface of the plate gives a net increase in skin friction and decrease
in heat transfer.
x

CHAPTER I
INTRODUCTION
In the last two decades the unsteady boundary layer literature on
the periodic and starting-ending type transients has grown rapidly. The
existing literature may be divided into two classes, the problems dealing with the acoustic streaming and the problems having vibration. Actually, when an object vibrates, a sound wave is emitted. Therefore,
these two areas are very closely related.
The phenomenon of acoustic streaming was first observed by Andrade2
when he made a photographic study of isothermal streaming in a standing
wave tube using tobacco smoke for visual identification. The existence
of these steady streaming (secondary) flows was mathematically established
24
by Schlichting4 using a successive approximation technique. He predicted
the existence of two regions of streaming in each quadrant around the cylinder; a thin layer next to the cylinder, called the d-c boundary layer,
with streaming directed toward the surface along the propagation axis,
and an outer streaming with direction away from the surface along this
axis. The outer region forms a vortex when the fluid is bounded, but
in unlimited space the outer core moves to infinity.
West28 in his experiments on streaming patterns around small cylinders vibrating in air and in water, observed a very interesting phenomenono Beginning with a feeble motion, streaming is increased gradually
with increasing frequency and increasing amplitude to a certain point
1

2
where suddenly a new type of circulation is started. Beyond this point,
if the frequency is held constant and the amplitude is increased, the
vortex system shrinks to a region near the cylinder surrounded by a vigorous circulation of the opposite sense. This phenomena has also been
22 26
observed by Raney, et al., and Skavlem and Tjotta;~ however, it has
not been able to be predicted theoretically. Other experimental and analytical work on streaming in its general aspects include those of West e 29,. 30 20 o 20
tervelt,30 Nyborg, and Holtsmark, et al.10
The study of the effect of small oscillations in the main stream on
the boundary layer is initiated by Lighthill.l8 He considered the problem of two-dimensional flow about a fixed cylindrical body when fluctuations in the external flow are produced by harmonic fluctuations in
magnitude, but not in the direction of the oncoming streamo. von KarmanPolhausen s technique was used to solve the resulting equations. He
also discussed the temperature fluctuations using successive approximations. Rott and Rosenzweig23 extended Lighthill's analysis in several
ways. A practical method for obtaining the response to the larinar
boundary layer to an impulsive change in velocity is presented. Hill
and Stenning9 consider the case of the boundary layer flow over a cylinder which undergoes a rotational oscillation. The amplitudes and frequencies of oscillation.in all three papers are assumed to be small and
the results are presented in terms of universal functions.
Lemlichl7 investigated the effect of transverse vibrations upon the
heat transfer rates from horizontal heated wires to air. Anatanarayanan

3
and Ramachandranl experimentally studied the effect of vibration on heat
transfer from wires to air in forced parallel flow. An increase in heat
transfer rates of 130% are observed.
Kubanskil4'l5 has studied experimentally the influence of stationary sound fields on free convection from an electrically heated horizontal cylinder in airo The direction of sound wave in his experiments
was longitudinal, ioe., parallel to the axis. For a test cylinder of
204 cm in diameter and 32.5 cm long subjected to an intensity of radiated vibrations in the center of the beam from 0.03 to 0.16 W/cm2 and a
frequency range of 8 to 30 kc, the free convective heat transfer coefficient is increased by approximately 75%. He also obtained heat transfer data for the case of a horizontal cylinder in a standing sound field
with a superimposed horizontal cross flow. 6 The sound wave is perpendicularto the direction of forced flow and also to the axis of the cylinder. Fand and Kaye6'7 performed photographic studies of the boundary
layer flow near a horizontal cylinder in the presence of sound fields
whose direction of propagation was horizontal and perpendicular to the
axis of the cylinder. A new type of streaming was identified called
"thermoacoustic streaming" which is characterized by the formation of
two vortexes above the cylinder when the sound pressure level reaches a
certain critical valueo
Schoenhals and Clark5 considered the response of velocity and
temperature of a laminar incompressible fluid to a semiinfinite flat
plate oscillating harmonically in a horizontal direction. The method of

successive approximation and perturbation technique were employed-to
obtain analytical results. Experimental data on free convection over
a finite plate were presented and the heat transfer coefficient is found
to be increased as a result of the vibration. For the semi-infinite
plate no increase could be observed experimentally. The analysis for'
this case was carried out to the second. approximation only which is harmonic in charactero Hence, it discloses no steady alternation to temperature and velocity profiles nor the heat transfer rate and shear
stresso This work was continued by Blankenship and Clark4,5 who considered an isothermal finite plate vibrating horizontally in a compressible fluid. Two cases are consideredo In one case the buoyancy forces
predominate over the inertia forces; in the other case- the situation is
reversed. They also extended the work of Schoenhals to a higher approximation for the semi-infinite plate. Eshghy3 recently considered the
same problem as Schoenhals and Clark except that the direction of vibration is verticalo
Hori published recently a series of three papers in which the
method of series expansion is used to solve various oscillation problems. Nanda and Sharma19 solved the free convection problem with an
oscillating wall temperature.
The problem of boundary layer flow with localized vibration apparently remained untreated. It is the purpose of this research to investigate the influence of localized vibration on the laminar flow of
incompressible fluid over a semi-infinite flat plate.

CHAPTER II
THEORETICAL ANALYSIS
STATEMENT OF THE PROBLEM
The analytical model is shown in Figure 1. It consists of a semiinfinite flat plate extending from 0 to X in the x-direction. A viscous
fluid flows steadily over this plate producing a laminar boundary layer.
At distance L from the leading edge a localized and periodic surface
disturbance is introduced. The effect of this localized disturbance on
the rate of heat transfer and the wall shearing stress are desired.
Y —— _ U
y
A-I
LI L
Figure 1. Model of the flat plate.
THE FORMULATION OF THE PROBLEM
The following assumptions are made in the formulation of the problem:

6
a. Distance L of the localized disturbance from the leading edge
is taken to be much greater than the boundary layer thickness 6. Thus
the effect of this disturbance on the boundary layer can be investigated within the restrictions of the boundary layer theory.
b. The fluid is incompressible so that the induced pressure waves
travel with infinite velocity.
c. The surface boundary condition is simplified by replacing the
plate and the disturbance by an analytical model consisting of a continuous curve, as shown in Figure 2. Using the probability curve, for
example, for spacewise variations, the localized oscillations of the
plate can be represented in the form of
r,( - L) + ~f
Y =-e (1)
where ce m(X-L) is the amplitude of oscillation, and m is a parameter
Figure 2. Analytical model of the disturbance.

7
to be selected according to the desired shape of the disturbance. The
amplitude of oscillation is assumed to be small compared with the boundary layer thickness. The value of m is chosen such that the curvature
of the wall remains small, allowing a curvilinear orthogonal system of
coordinates to be used whose X-axis is in the direction of the wall, the
Y-axis perpendicular to it. The desired steady periodic solutions suggest the use of the complex form of the surface oscillations. The selection of the probability curve is arbitrary, and can well be replaced
by another curve if desired. The solution of the problem is a function
of the selected curve. However, the effect of these classes of disturbances can very probably be demonstrated by any one of these curves.
do The effect of localized oscillation on the potential flow is
neglected. Hence, the potential flow is assumed to be the streaming
flow of velocity U~.
e. The velocity component of the plate in the x-direction is zeroo
With the above assumptions, the equations of momentum, continuity,
and energy with respect to the Cartesian coordinates may be written as
follows:
Momentum:
P(~t +at+v~') - 1/ __
+LU sax'+ / Xay X2 = _()
p (a A a~u ay = y
4 v II -4 V? ~ 4. 4 v

Continuity:
+ -- o (4)
ax x/
Energy:
U a.X VDy a + y) (+ )
The boundary conditions are
y= e-m(X-L)2~+ B = + —o, = m J/ - L L%,'.y-E e, 6 W =e / u= e
(6)
KINEMATIC RELATIONS
A reference frame attached to the moving surface is used. Thus, the
boundary conditions are greatly simplified at the expense of increased
complexity of the governing differential equationso This compromise is
usually more convenient for the mathematical solution. The formulation
of the problem in the moving coordinate system (X,Y) (Figure 3) which
accelerates with respect to fixed coordinate system (x,y) is as follows:
First, the relation between the coordinates of the fixed and accelerating coordinate axes will be obtained.
At time t, the boundary is assumed to be at the position OVAXo
The arc length of OVA, Figure 3,
XY I
using the particular value of ( from Eq (l), can be written in the

V
V
y Pu
\Y
Y U
\ l \y
Il
0'
I I
I I
Figure 3. The fixed and accelerating coordinate system.
form:
( = f (/ +r(X-L JJdX (8)
Expanding the integrand of Eq. (8) binominally,
X C) X +( m'x
j (X-L. d (9)
On the other hand, we have
x = x,- - s,, y, (10)
r ( J(lo)
and
y = y, + ycos~ = y, + Y 21 (11)
where subscript "1" indicates the values of x, y, and d) at location A.

10
Hence,
(y m-2X,-L (x-L)e' - (12)
Putting Eqo (12) into Eqso (10) and (11), expanding them binominally
and rearranging gives:
X XI + t (XI-L) Y J + O(6i) (13)
and,
= y + y- + 2Me(A-L)ZY02 + O(E ) (14)
Also, since (xl,y1) is on the surface of the plate, we can write
- m,(X,-L -I- +,uY = e: C (15)
Therefore, we get four Eqs~ (9), (13), (14), and (15), which may
be used to get the relation between the two coordinate systems (x,y)
and (X,Y) by simply eliminating the parameters xl and yl. The integration of the second term of Eq. (9) is simplified by neglecting terms
2
of order E o Eliminating the parameters xl and Yl from the four equations, we get
X:=I + - Lm( YL0y (16)
y =Y + E (17)

11
where
-m.(Z-L) - L'
43 ( e (18)
Equations (16) and (17) give the relations between these two coordinate
systemso It should be remarked that the assumption of neglecting terms
of order e2 and higher, or X = x, eliminates the change of length of
the X-axiso
The velocity components of the boundary in the X- and Y-directions
may now be evaluated~ Since the velocity of the boundary is assumed to
be in the y-direction, we readily have,,: jCs/~ / -(ZX-L) 2+t L(r
P/W 4-O =, LW e - (19)
The velocity components in the X- and Y-directions are
- = 6Vk cos = (EL'j)
(+ - ( 2j
= t(co/){I 62 - 2 ( ---— )
= 0toc + 0(E3) (20)
~Vwr~~~-= s ~~~ ok + ~~(22)
V2x 0 (25)
(21)

12
Next, let us find the relations between the velocity components
relative to the fixed coordinate system (u,v) and that relative to the
moving coordinate system (U,V). It is clear that the absolute velocities (u,v) equal to the sum of the relative velocities (U,V) and the
velocities of the coordinate axes due to translation and rotation. To
find these, consider Figure 4 in which a local Cartesian coordinate
(x',y') is drawn at point Q with its axes tangential and normal to the
Y' P'
Y! XIs t ocX
Figure 4. Illustration of the local
Cartesian coordinate system.
X-axis. Assume that i and j be unit vectors in the x'- and y'-directions. Then
2 -= x +.j (24)
The absolute velocity of the point P' can be obtained by differentiat

g13
ing the vector equation
r = — R + Q (25)
with respect to time t, ioe.O
d? dR d~
dt d t + t
= r + ( )
cIR C(/ d/j + +y)d
d tu't +' ('t'~ +' )27
- dt +i!~1L+VJ -x- (X -+Y 6
and
=f- Vx' L + j/y'- j
Th.en, Eq (26) becomes,
V,7 u'= L/ ~1 -', (28)
Vp'yv= V/ +Kyr X }p (29)

14
Therefore, at point P of Figure 3, the absolute velocities in the
X- and Y-directions are
VP7 =I k + r-H (30)
]pZ - T BVWy (31)
Vwx and vwy are given in Eqs. (22) and (23). Also, the angular velocity
P may be found by differentiating P, Figure 3, with respect to time to
Since,
x - Y2 M (X- ) e - -2 m (j-L ) (32)
we get,
P = - C <X-L>)L'# 6 (33)
Finally, the X and Y components of the absolute velocity at point P may
be obtained by introducing Eqso (22), (23), and (33) into Eqso (30) and
(31)o The result is
Vg = - +42m (I-L)Y L'Cu E (4
Vpy = E ~oC f. (35)
It is now only a matter of geometry to write the absolute velocities in
the x- and y-directions as a function of Vpx and Vpyo Hence,
u = Vp~ co0S3 - VpyS/, 3 (56)

15
V = VPJ sl/B + I/Rp COQ y (37)
As before, we assume
Cos( =7=,_ (38)
and
stn,, =, 1 + +,f // (d'-. _
Inserting Eqso (34), (35), (38), and (39) into Eqso (36) and (37), we
get,
u = -(1 +.me(-LJ)YL P - 2m(I-L)e P7 + 0 (E2)
v/ = (~+2m('-L ) Y L J o]-2r Z-L) 6 0] + ( V+ 65t') + 0 (62)
These, neglecting terms of order 0(E2), may be reduced to
u = U t E m(Z'-L)bJ((L6 Y + 7) (40)
V = 7f+ ( CLc - 2m (I-L)Uj10 (41)
FORMULATION OF THE PROBLEM IN THE ACCELERATING COORDINATE SYSTEM
This formulation is first written with reference to a general orthogonal curvilinear coordinate system (X,Y) as follows:
Continuity:
( /hyr) -t ( Vh) =(42)
64~~~~~~~~~~~~~~~~~~~~a

16
Momentum:
f (T h.Z -h + h h +r ay hx -.) + __ _
=- 7 CX & hh x Crx ) y /xLxh )
+ hxhr k Y - hh (4y)
aT hx a hx hy Y hxhr xh )
P sh' l ahr __ C (hxOr' _
hrhy ay h hxr Y (44)
(X,Y) o
hx.,hy: elements of the metric tensor~ Now,
X I + ( nm (I-L) )(45)
(46)
and
hy= (Ty-r~~v+(dt)~ ~(48)
Putting Eqcs (45 ) and (46 ) into Eqs (47) and (48), taking

17
square roots and considering terms of cO and E', we get,
hg = / + ((2 M Y) [/-2m (Z-L)2 (49)
/hyr= 1 (50)
_ ( U h' 7
a;T -2Y ( h4 aV + hxhy ax
With Eqo (50), these equations become
oxx a =2dy) ( 1)
"r =- 2 ( y) (52)
With Eq. (50), this equation becomes
a. =+ aY hx Y (53)
In Eqso (51), (52), and (53),
rT'= Y + m(.Z-L )Y t
This is derived in Appendix Io
aXay. accelerations due to movement of coordinate axeso They are
derived as follows: At any point P, there exists four ac

18
celerations,
(angular acceleration)
p x (1e x R ) (centrifugal acceleration)
2 X 1v (coriolis acceleration)
dFj?
dt J (linear acceleration)
Noting that
= y
Ry - 6 cdo
these accelerations become
(54)
3x(/xQ= - Y/6J (55)
2? — (2 V 2 - (2p t)j (56)
dRy _j = E2~ ~'
Therefore, accelerations due to the movement of coordinate axes in the X-direction
CI, = - YIB -S26 (58)

19
and in the Y-direction
ay = -y 2 lT - 6A2P (59)
where 3 is given in Eqo (33)o
Introducing Eqso (51)-(53), (58), and (59), into Eqs. (42), (43), and
(44), we get:
Continuity:,3X) U4- 2 2mz)-/-2m J-0z21( 7 Y8Wy) = (60)
Momentum:?/( - + u y +7 y -) + C, (I —L), j (.2'. - Y 2)
_-_f +/8 (sR + _ Y2_ 2 +P2 Tr,-+ 12(n J- ) [ 1-62 n ( - X /.l ( )2' - L+t J(-)+ Pm Z 37)
ft12/LgL) + w V E{67-r TL -r4 I-L) 2W 4/nY- e coL q:
-I n (:-
# k26/n(I-2n(ZL)-2.) Ef21s7) +- YV 2r V
~aF agz Y a
5 —~T g ~Pi) -4 _ -L) \4Uf - E4OW'(
+ E2*n4)/)- 2mn(%-L)JI (-' +Y77 ~- -'-,— LC' +-2mY(Cl- L) g —L + / c' (62)
4-I-p2n7(X -Lq5( E (4, Y + */fl if)

20
Similarly, the energy equation written in terms of the accelerating coordinate system is
T X + - -T y)
- aZ2I - "2- + # (2mrn)I)/l -21(Z- L)21 (2m (-L ) Y
- Y aX2 - + Y a 2+ e- ( m&) Y (4r (-L)J (63)
SIMPLIFICATION OF THE FORMULATION
At this stage it is expedient to introduce dimensionless variables
by means of
7, =xV, L
Y' Y,, T' 0L ='=(The formulation of the problem then appears to be:
Continuity:
z'/ + a' +'(2~) 9(/-dm.c - l/)2|( Yt ) (64)
oy, ))=o (6C)

21
Momentum:
-+ CE'l2mr'crZ)~(2Lc 7)' YrCA'2),~g,,e ag' C
~Z' i Re - -- - - dJ[/2 + /c3;If& RRe
f mq,(I -2- t/ Y,~J2m - t- q ) Z2' he
f-. 2E7'C b(2,C /)](3 "./2 aL (65)
F~'eJRe
6'v'y'(X'E ~~ ~ ~(~ R'e (-.65)
=-(c3/2 y +iw Re.<.) (m Er1 IV r
+ R-+ m'4rnl) + 42m6'7I'),* f)
-"W +C + 2 n r(,, + (66)
= Z)'+' ~oo'U'U,-.) - dY )
P,-/i'e f a) I)
TE/2 -/ E(2r' tTIY(24)Y,.+, -' (.2 r -/+)'' t]2 /" K % + W -,
— 2(~. 4-2'"'(Zi)~4-K~,) (66)
Energya
PtI Re e' e''' J
P,rs- Y' + ya,. + 6'm'c',{- 2r2 CZ'(2){-'IzY a('7
We are interested in the asymptotic form of the foregoing formula

22
tion corresponding to small viscosity or large Reynolds numbers. It is
a well-known fact, however, that the terms having 1 as coefficient canRe
not be eliminated from the formulation because of the reduction of momentum equations to Eulerian equations. This difficulty may be circumvented by a new variable Y* defined such that
y* ='/ Re (68)
au, 1 82U'
Let us now consider the terms, V and - Employing Eq.
6y, Re 6y,.
(68), these terms may be rearranged as
V )Y,, Y*Re,,* (69)
__ ___1T _ / (F' _____ -(I+.2n) =)2U'
Fce y," l e (e ) y =)2 RR'e 2(70)
2'U
From Eqo (70), it is seen that the term is retained as Re + o if
ay*2
n = - o In order to retain the term V' I-, the following transfor2 6y,
mation is introduced
7i* = Re' W' (71)
which transforms Eqo (69) to V* a o Based on this argument, the
basic equations are transformed according to the relations:
Y* = Re' Y' (72)
and
Y = RiV' (73)

The result is:
Continuity:,t' /_ v* + (2' )( C -2m'(% )J(V Y oy*T) - 0 (74)
Momentum:
(87jU 4_T1;, L7~2,L)+ e (2m/()I-.2Z71 2](nA + #pfU )g/7*) e' ~ * /
+ ('C.2m'c/(-/)) (.2 L'- u Y'(/) 2 e
a._+, _, pg/I>2 + Re R t<2
-- (~Jc r (7m)' + m ( +_,__ e ZT
+. 2~7'rz.'m~ + 2/_' 4t/ (.7, /y
-.,n —'O( X /j) (.2 Y ]+ +Y
-E'(:z~''*~~~')rm'(z)]{- 2m'( -'/] (75),Csm'cxL3 ( (~' Z6 - -, U"")
t~?e 6) —V / _

24
Energy:
2 2
0ZT TT'-V
d28. Re)
-= p { le + + 2:z2'P{/-2n (%')2}(21m?- ReY/"
\,* c2O' / __ 8 __ yx/ 2_
-/
t 22 Re3' Je sY_ JRe c)Y X2
e (2 )4 (I-) aZ 1 (77)
Since we are looking for the asymptotic solutions of the above
equations for large values of Re, this suggests the expansion of. U', V*,
p P, and GQ in terms of 1R/ as follows:
U'" = U_' - K U/ / U' -i - ----— + (78)
Re2Re R
V = 17 Re _ + (79)
=%'Re2 * / Re.~-lb=+0w' e, - (80)
Re'j, Re' 0e9 -'+/ + (81)
ReO +,
Inserting Eqs, (78) through (81) into Eqs. (74) through (77), the
successive orders of approximation are obtainedo The zeroth-order
equations' are:
Continuitys t + & V* = O (82)

25
Momentum:
=-'t_ + - _ - (.ny.*j p'4a2m'rz(-,1)63-2m f2' —i2 (83)
mgiu/ of 1' i
-'Re1/ (84)
/' a -y
I2,2 /12 (83)
Re2U0,~~~~ Re2~~~(85)
It is seen that terms of order -— T~ and smaller are neglected,
Re, cRe
Since these terms are small compared with the terms retained, which are
magitder IO However, terms involving yl are retained since the order of
magnitude of them depends not only on Re but also on be and red The
reduce the basic equade of these terms is seen to be dependent on the fscillationllwingll have no effect on the flowrs
~ 1 >/ -~ (87)
Re 2Re Re M

26
and
J,2 J x, 8)
co > E (88)
Re'? Re 2
This means that only terms with exist. The basic equations can
be further simplified to:
Continuity:
)/ *
+ = o (89)
Momentum:
)& ~c T1 &X2 0 Y W
VI
-= -, #2 (90),E /- - R (91)
Energy:
__/ [/- / 0 0 /'2c O
c -f-T~~~ -1~~~~~~T~~~ t(92)
T' -} c;'+,,- Pr a Y)(9.2
In these equations, the subscript "o" has been omitted for reason
of simplicity. In Eq. (90), the inertia term represents the inertia
force due to rotation of the normal coordinate axis. The inertia term
in Eq. (91) is the inertia force due to the change of the velocity of
vibration in the Y*-directiono Since this inertia force is a function
of XI, it will influence the pressure gradient in the X'-direction.
Complexity of the formulation of the problem suggests the KarmanPolhausen integral procedure be used most conveniently. Thus Eq. (91)

27
is integrated with respect to Y* and, based on assumption d,
=~' - -';~ (93)
where
~' = L 1;E (94)
Differentiating Eqo (93) with respect to X', we get
-ADZ*>( 1 ~ (95)
Equation (95) is then substituted into Eq. (90) and the formulation becomes:
Continuity:
__) +? = _ 0 (96)
Momentum:
4- ~~ 4-.2 Y*
T''TT,*U' E -+ 2n
- tU2J co 2do (97)
Energy:
de' ___ /+O' _ / _2__
UTl+7 6s +2 (98)
with the boundary conditions
Y*:o' g' ]t o, 1
Y* a: Z7= 1, 8'= o

28
The form of the formulation shows that this problem is equivalent
to the unsteady flow of an incompressible fluid over a flat plate with
a periodic body force, since all terms relating to the curvature effects
of the coordinate system (X',Y*) are shown to be negligible.
PERTURBATION OF THE DIFFERENTIAL EQUATIONS
The basic equations are:
7bZ-r + ey* - O (99).22 0~3 (.u 2Y,-T'f- - -z'-/ y -*- R-:62m?.z/)"3,' (S )
c U 6'__ 7'____ (100)
T' a' - -' P, a3 yC* (101)
with the boundary conditions:
Yx- 0o 7 = / 9 1
= 77* --- ) O
YX= --, 8= 0
Let us assume that
(<7, T) = (. ) +, Y T)R
u (zF, (,
f3 ~(' Y., 7') -- t ( ~;' Y) + e f9,(' Y*, TJ) + G. - ~J Z,'( t T') - - - - -. - -
(XBy,T-) =E ( ) i + ), T ) + t g (, --— 1 ) (102)
By substituting Eqo (102) into Eqs. (99) through (101) and collecting

29
various powers of C, we get:
(X)e
0SR~~/
+ o (103)
u/,=/ Z1 (104)
SE' + _;_ c)- ( 105)
Y ~ 0 / ) 0
) +Z} = o/ 0 (106)
SuT' + tv + 4 + I + g aot + 22-;
-.; +___ da (107)
+T" +'* Pr Y' " (108)
P 2'*
T9-T' *+' +,r. + -- or*o
* Z,'= 0,* (,/f
2
U2 2, a P~" O (109)
+ UO Ux
c~\ Tj; (IZ)

30
OY*-' = 0* o -./ - -0
yx= 31: E~ - O / -. 02
Since we are looking for the steady periodic solutions of the problem, we let:
Z (.I'" Y T') = l? (Bj7 Yj /0T j, r t } (112)
9, fCZ <* = O / 9/ Y)C'?
where ( denotes the real part of the complex quantity in the brackets.
U1, V, and Q1 given in Eq. (112) are then substituted into Eqs. (106)
through (108). Since, in the resulting equations, the symbol6Z appears
in front of each term, we can therefore drop it and get:
t C= 0 (113)
&o'~ T, -- g-, ~' $~' o -' za zC;,,-,.- I go2
~ z'+ +,2:': )'.''. y r'")
-a'U; ITC 2 */ " (114)
Cw'g+,c, ~, cZ (115)

31
y-' T z*= O, ~- o
YX - S'' - ~, * =
if I, c — o
Equation (112) is then substituted into the right-hand side of Eqs.
(109), (110), and (111), and we get:
z oY 0 (116)
T2X' O U a 2 - y2
R l; e1 7' e I'N T'
a~ T' +' 9~',.; V + + a* Y I y, y
az e " e T,' Rfe' J~i~~c~ t,'"'r' (11 )
~1' 2: 7Y w= p J2 + O
y*= 0 z2 0 o
y*= =/:' / Ta= O, /a —- o
The right-sides of Eqso (117) and (118) need further simplificationo Now, let us write
= [1TicoSdYT'- U/I Stn CduT
Y f gce'J = iCOS j/ C /T S7 l,Si T
The ~ ~ rihtsie of os'17.an 18 edfrte ipiia
tio No,;'et u writ

32
where U' and Ui are the real and imaginary parts of UT. Similarly,
1R 1i ic
V* and V*I are the real and imaginary parts of V*. Then, we have
1R 11 lc
=ZIR, c/ os C c'S T + UTbI 5 1' T
-2 ~.I /'' )S 2(119)
Using the trigonometric relation that
Srn2 coTr = / (/- COS2C6'T/)
2
Eqo (119) becomes:
R f' e'
-( UR;' 1
2 +X
Next, the following identities are used:
x C/ 6X Uc' -V 6''
and

33
/ i 2/ /)CoS
/UI~/~u:~~C' j /
( v / ) S/1n 20 T
Equation (120) then reduces to:
/T /.I c{ ~/, ( z -/ —-' + $ZT;, ) + 1 2tc''I e z(121)
where the superscript bar (-) denotes the complex conjugate. In a similar manner, the second term on the left of Eq. (117) can be reduced to:
/f/
(~ sC 4- *U'e Cb.I2jl 2%.caaor}j 122)
The right-hand terms in the energy eqjuation, Eq. (118). can be redeuced in a similar manner. Therefore, we can write:
+ ax.a, +~c, R I Cz' 1

34
and
a ( u1 c e f 6 c It T j
= 4 ( Zc J y* +:) a Y C e )
__ elW lI (124)
Substituting Eqso (121) through (124) into Eqs. (117) and (118) resuits in~
0. O"
d272/ Tt/dlJ2/ (ZT~ ZT,,,~iZT',i o., TT' ___ Z. cU' a' Y 7 t C y_2/
=Z7 + -' "-'e'r (125)
and
e+2/ #g / y*61 U-2
T/ Pi wr'' / __
= 00 + (126)
twhere
U7/= - ('' 4-ZZ/-4-T (127)
no 4 Cc Yo 7/c Y yY
7J - -'(UI'C + *Tc oy" (128)

35
2 7c <C ) 7/ +cP r (130)
Inspection of Eqs. (125) and (126) reveals the steady periodic
solutions of U2, V2, and 92, may be written in the form
L= 20 (Z: Y -+- R {, (X )e'" ] (1315)
- 20 (132)
6o (T, Y ) e (133)
in order to comply with the right-hand sides of Eqs. (125) and (126).
In this way, the second approximations are separated into two parts:,
namely, the nonoscillatory and the oscillatory components~ Substituting Eqso (131) through (133) into Eqso (116), (125), and (126), the
following sets of equations are obtained.
Nonoscillatory components:
&9g + s = 0 (134)
_ ctj20 20"' ** WI"'*t
=)V T2z _ i (1 P5)

36
with. the boundary conditions
y* o: &woj40 - O. yro'
Y* - J; 720 ~ y 620 - 0
Os cillatory components:
-+O (137)
2 2X' 3g - +7+77cd U- (138)
~? / /02/
C P r'.21 (l T' ~ o (159)
with boundary conditions
-y -'/: 7, = J 02/ O0
SOLIUT:IONS OF THE DIFFERENTIAL EQUATIONS
Zeroethh-Order Approximation
The momentum equation (104) is integrated, over the velocity boundary
layer thickness 6Vo
J a 1I cz j -r-J4 0 = - (I y*) (140)

37
From the continuity equation (103), we have:
y*
CIfUo _ & /(141)
Equation (141) is substituted into Eq. (140) and it reduces to:
a___ ~ 1(ZU2 - ZG/) =, jy, (142)
Similarly, the energy equation (105) is integrated over the thermal
boundary layer thickness 6t:
/ +d Y, + ( da)y (aY (143).o
Again, using Eq. (141) and rearranging the terms, we get:
ifo r = 5- (/o3r.) - (1-o (144)
0
The solution of the momentum integral equation (142) is given first by
Pohlhausen21 and later by Holstein and Bohleno8 The velocity profile
is found to be:
= - (14)
where T: and 5v is the velocity boundary layeir thickness
The solution of the thermal boundary is given by Squire.27 The
temperature profile is:
90= / -2 + 2a - X (146)

38
where TT - L and ST is the thermal boundary layer thickness. Now, let
T
us define:
a 6'~~ = < ~~~~~~(147)
Squire found that for laminar flow over a flat plate at zero incidence,
= Pr 3 (148)
Therefore, the temperature profile can be written as:
/= / 2Pr2?+2Pr?-Pr? (149)
These two approximate solutions have been checked with exact solutions
and the errors are both less than 5%. Therefore, we take Eqs. (145)
and (149) as the solutions of the zeroth-order approximation.
First-Order Approximation
The momentum equation for the first-order approximation is given
in Eq. (114) as
YI'*Z' x; UicZ $- Z U1C + t70< - ~ c sI eY*
+ C2 (m' (Z )q'Jo2 (J- d2Y
It can be shown (see Appendix II) that the convective terms in
Ego (114) is small compared with the other terms. Therefore, Eq. (114)

39
becomes, I,/ 2I/-...
L A), c + 2 m'(z- ) ( y
y, ), +if (150)
The boundary conditions are
YX - - 0' Z' —;- 0
Y*=: = o
To solve Eq. (150) by means of the integral method, we assume a
velocity profile in the form:
UTc = + + C, c, +' e, 7 (151)
The constants are determined by the boundary conditions:
U/ O G C-V* O /?=.~ ~: Ti = o, ~ O
The second boundary condition is the so-called compatibility condition which is obtained by satisfying the momentum equation on the sure
face of the plateo With the boundary conditions, the constants are determined to be
a/ = 0
C,- + bt (X)
e, - 2b, d- 9(,

40
where
// J
Y(X) = c6Qo2''s a(,' ('_ I- d,'s (152)
The velocity profile is therefore
7;, = b:, +?- (3b,+)? + (153)
The profile for U' Eq. (153), is put into the integrated form
lc'
of Eq.o (150),
1'c jy U- ( ) d+ (154)
which gives
L C' 60. /5 6, + o. o33 3 zt)
t,2w' a d'(155)
Solving for bl, we get
-bX (156)
~ -I- 0./5 LcOW'J
The boundary layer thickness 5' may be found be substituting Eq. (145)
into Eq. (142), which gives
3' = 5. e493 9 (157)
For the present case, U e is assumed to be constant. Since, in nondimensionalizing the basic equations, the reference velocity is taken as UO,

thus
U,= D = 1 (158)
Using Eqso (157) and (158), Eqo (156) becomes
bi = (44- 453) )' J'fqs (C/M'(T-L /)'- 0 250)
LG) - (/9.+99)+ (8 63)M'(/i)' - 0250j (159)
It has been discussed in simplifying the basic equations that, if
the oscillation is to have any effect on the flow, w I must be. very largeo
Also, it is shown in Appendix III that the order-of-magnitude of m' is
approximately 4000 Therefore, the imaginary part in EqO (159) is small
compared with its real parto Therefore, Eqo (159) becomes
Ui = (44.453) b,'(- - t 4.5022- 6.004 73f' 2.502 7 4) (160)
where
6,/ = 6 /2 m' 0.25 (161)
The phase angle, al, is therefore zero, since Eqo (160) contains real
part onlyo
To solve the energy equation, Eqo (115), the same procedure in obtaining solution for Eqo (114) is usedo The temperature profile 9O is
lc
solv ed by'
iwU'/c1, V7 c >yO = / r yL 2 (162)

42
Integrating over thermal boundary layer thickness 6t
~{~ 8~ wY* + z7;/ 7, d,+-,-~ J -- ( 4- (:163)
eAY;cc/<+ /Pr y*/y*0 (163)
o o
or,
L JA+ dE'|c c Zr J ( gyJoUy C_ (164)
~ ~o~0
The profiles G0 and Ulc are given by Eqs. (146) and (160). Again, a
0 ~ ~ ~lc
fourth-order polynominal is assumed for
/ = C a -C h 7t + cS' te'j 4 4 e + 7td (165)
subjected to the boundary conditions that
7,= /: 8Z,- o, yc o
The constants in Eq. (165) are determined by the above boundary conditions as
Cs = -
e3 = 2 b3
The profile is therefore,c L = b3 ( 3 - 3 +.2 4) (166)
Equations (146), (160), and (166) are then substituted into Eqo

43
"(164) and we get
(O /5) b3 + (2 60.02 oJ' ) (Pr,) F (-P= (167)
where
F, (r) = 0. 834 Pr o./08Pr - 0.064 P O. o/4 Pr (168)
F, (I'X) =- i f-2m' (7-/)n —r'). - 0 o.25)+Z.'(2I / 4' (169)
Solving for:b3, we get
b (26 o02) A, F2 coI2
/,
-A.. - (0. /5)
-Prs c~ Pr 3
or, using Eqo (157),
b (296.36) Prw' F2 0(2(170)'-~3 -- $ (170)
where
C)2= bian ( 5 /3i2 X)p r I3) (171)
Since XC' must be large in order to have effect on the flow (in the order of 400) it is seen that U2 is very close to 2- This means that the
heat transfer lags approximately 2- radians behind the input disturbances
(cf., Appendix III).
Second-Order.: Approximation
It has been shown.in Eqs. (131l) through (133) ~that the second-order
velocities and temperature consist of two terms, one steady component

44
which is a function of X' and Y* only and a second term which oscillates
harmonically with a frequency of 2w'. The second term will be zero
when integrated with respect to time, and is therefore of less importance. The steady component will contribute to the skin friction and
heat transfer and are therefore solved in this paragraph.
The continuity, momentum, and energy equations of the steady components of the second approximation are given in Eqs. (134) through
(136) as
+o 2 = O (134)
4(
_K0 9X}_ 2_y' u, a, _ ___,
oo 8 a.24 +'o.,,o +-~ -t,o $ /' - p, a2 y*
too Ae ir oA? o, Pr <9 ~
/c31T( 2OX YF# C0 Y
with the boundary conditions
* * /
-= 0 7: -0 o, 0o= 0
c& it can be shown by following the procedures in obtaining solution
for Eqo. (114) that the solutions of U.o, V*o, and Q'f canbe obtained

45
by the following simplified forms of Eqs. (134) through (136):
2+ 2 y = 0 (172)
20 L' - 4.,, A a /* )
c~g<0= 4').V _F: (173)
y.,2 IC,=' + P/' (174)
IC iC IC
with the boundary conditions
y* = o -: oTJ = 0 = oL=
y$ _ /:> 20 -1 ~0 o
As before, a fourth degree polynominal is assumed for U' Thus,
20
we have
C20= + 647 fc# 2 + c3+ 4 > (175)
and the boundary conditions are
7 = ~'TJ20= ~, ~= 0
The constants in Eq. (175) are then solved to be
a4 = 0
C4 -O
e, =-.2 b4
Therefore, U20 can be written as
Zj, = b/'?- 3 -+2?) (176)

Next, Eq. (173) is integrated over the boundary layer thickness 3'
and, after using the integrated form of the equation of continuity
Y*
20 dy (177)
0'
it becomes
T /
(~= j~ -i- U/U'd (178)
Equation (176) is then substituted into Eq. (178), we then get
b,. = — (53 749 C'4') /F (179)
where
F3 (') = I Cm /(Z —)Z- 0.25).50 - 4mI'(X- /))/l'(-l Z'- 0.25)
+ 21 nl'(2ZX 1)] (180o)
F3(X') is shown in Figures 5(a) through 5(e) for different m's.
To solve the steady component of the second-order temperature profile, Eq. (174) is integrated over the thermal boundary layer thickness,
no=O ~,- ~E'o {1( O,/C: ) 0, Y) (181)
Again, a fourth-degree temperature is assumed to be in the form of
t.~o = as + bs + C5?J + s??t + et4' (182)

5
t14)~~~~~~~~~~~~~~~~c) ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 4
-5
0 1 2 3 4
X' = XIL
Figure 5(a). F3(X') as a function of X'(m'=l).

5 l
r>
10 0
oo
-5
0 1 2 3 4
X' = XIL
Figure 5(b). F3(X') as a function of X'(m'=2).

F3 (X')
CD
co
C,,
X
0
II
6t

10
8
6
4
2
UP 2
4
6
8
1 2 3=X 4
X') as a funct=XI on of X'
Figure 5(d). F3(Xt) as a function of X1(ml-J4).

10
M' =4
M' =3
M' =2
0
-10
0 1 2 3
X'-' =XIL
Figure 5(e). Comparison of F3(X') for different m's.

subjected to the following boundary conditions
/ %o?1 = 0
* ao= =
2o-0 t 20
Then the constants are solved to be
Cs= 0
d5 = - 3 b5
e ~=2 b5
Equation (182) then becomes
0. = b 5 (t -37' A 2P) (183)
Equations (160), (166), and (183) are then substituted into Eqo
(181) and the equation thus obtained is used to solve for b5, we then
get
b5 = +439/7) Go', Pr)F+ (Pr)Fs (X) (184)
where
F, (Pr) = o 834 Pr 3- /0./o08Pr - 0.064 Pr + 0. 0/4 Pr (185)
F,(Pr)=-0. 067Pr 3 0. 62 Pr - 0. /2 P- + 0. 03S Pr 3 (186)
Fs (I') = -&n 32''- (4m+ /3 n'3)'
+ (32mr'4+/7,r',)Z'5+ (8r'*-,/sm'a3 /.5n"' 2)1
+- (3m,3- /9.5mn).'J + (5/ n2. 2,).
+ ow8/2S/ 7'Z - 0. 03/3}J (187)

53
F5(X?) is plotted in Figures 6(a) through 6(e) for different m's.
RESULTS OF THE ANALYSIS
With the solutions of the basic equations obtained in the preceding sections, we can proceed to determine the time-averaged velocity
and temperature profiles, skin friction, and heat transfer.
The time-averaged velocity profile in the boundary layer is
= YTJ + 0(188)
Re Y o
in which the oscillating terms in the first- and second-order approximations become zero upon integration with respect to time. The terms
UO and UAo are given in Eqso (145) and (176), respectively. Using
them, Eqo (188) becomes
U' = ( 7-2'+ - f 53. 79) 6(g (1 - 3?7 2 ) (189)
where
f Re (190)
In Figure 5, it is seen that F3(X') is equal to zero at three
points. For example, at m' = 1 these points are at X' = 0.75, 1.18,
and 2O02~ This means the velocity profile is the same as Blasiust velocity profile. The location of these points does not depend on kf,
being dependent on m' only. The surface of the flat plate is divided
into four regions by these three points. The effect of oscillation on

50
40
30
20
10
0
it)
-10
-20
-30
-40
-50
1 2 3 4
X' =X/L
Figure 6(a). F5(X') as a function of X'(m'=l).

80
60
40
i
20
20
20 0 1 2 3 4
X' = XIL
Figure 6(b). F5(X') as a function of X' (m':2).

1200
800
x 400
-40 I 0 I I
-400
0 1 2 3 4
X' = XIL
Figure 6(c). F5(X') as a function of X'(m'=3).

5000
4000
3000
L 2000
1000
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
X' =XIL
Figure 6(d). Fs(Xt) as a function of X'(m'=4).

58
4000
3500
3000
2500
2000.m'= 4
rn'= 3
in 1500 =
1000
500
0
0 1 2 3
X' = XIL
Figure 6(e). Comparison of F5(X') for different m's.

59
the flow being different in each region. Figures 7(a) through 7(d) show
the velocity profiles in each of the four regions.
By definition, the local skin friction is', =(Y)y=o
so the drag coefficient is
In terms of dimensionless quantities, it becomes
= ( (191)
Equation (189) is put into Eq. (191) and we then get
Cfx = 2 3-/97 -, f (9.189) F7 (192)
Cfx is plotted in Figures 8(a) to 8(d).
The net effect of the oscillation on the surface of the plate is
obtained by integrating the second term of Eq. (192) with respect to X'
between the upstream and downstream penetration depths and dividing it
by the distance between them. The result is
Cf = - Lf, J Af(9. /89) (193)
Lf"
in which the two penetration depths are determined numerically as follows: In Figure 5 it is seen that at m' = 1 the expression

1 O0
0. 8
0.6
f = 2x 10-3
0. 4
0. 2
0 0.2 0.4 0.6 0.8 1.0
U'
Figure 7(a). Velocity profile at X' = 0.5(m'=l).

1. O
0. 8
0. 6
0.4
If = O
Af 2x 103
0. 2
0 0. 2 0.4 0.6 0.8 1.0
UFiure 7(). Velocity profile at X =
Figure 7(b). Velocity profile at XI = 1-1(m =)

1.0
0. 8
0.6
Xf = 10-3:. 0.4
Af =-0,f - 2x 10-3
0. 2
0 0.2 0.4 0.6 0.8 1.0
U'
Figure 7(c). Velocity profile at X' = 1.6(m'=l).

1. 0
0.8
0. 6
0.4
yf = 0
0. 2
0.2 0.4 0.6 0.8 1.0
U'
Figure 7(d). Velocity profile at'X' 2.4(m'=1).

2.0
X' =1
1.0
Af — 10 -3 l
Xf 2x163
0 1 2 3 4
X' = XIL
Figure 8(a). Local drag coefficient as a function of X'(mt=l).

2. 0
Y Ul
\' J J,, O, -,.....
L
1. 0
0 2 4-3
0 1 2 3 4
X'= XIL
Figure 8(b). Local drag coefficient as a function of X' (m'=2).

2.0
Y
Uca
1. 0
C-)3
u ~~~~~~~~~~~~~~~Af = x10
0f = 10_ _ _
0 1 2 3 4
X' =XIL
Figure 8(c). Local drag coefficient as a function of X'(m'=3).

2.0
Y
U,
1.0
x o3 5 x X' 4 -
0 1 2 3 4
X' =XIL
Figure 8(d). Local drag coefficient as a function of X'(m' —4).

68
F (Z)
SY'
equals to its maximum value of 0.917 at X' = 1.7. It is also seen that
this expression equals to 0.01 (approximately 1% of its maximum value)
at X' = 0.07 and -0.01 at XI = 3.47. Therefore, we take Lfu =.07 and
Lfd = 3~47~
Numerical integration of Eq. (193) then yields the following results:
z Cf 02260'o
f X
-- 0./21/'12/E; ilk for n'/=2,Re_
00~55 d for'- 3:~00229 - f:
= 0022 or k 4 (194)
The positive sign for ACf means that the oscillation results in an
steady increase in skin frictiono However, if the first term on the
right of Eqo (192) is integrated over the same penetration depths, the
result will be of the order of 2.5,/oRe. In viewing of the inherent
limitation of the perturbation method, the maximum value of kf is of
the order of 10-3 (see Appendix III). Therefore, the net change is
always smaller than 1% of its value without this localized disturbance.
The time-averaged temperature profile in the thermal boundary
layer is
0' =O -8, (195)

Again, the oscillating terms in the first- and second-order approximations become zero upon integration with respect to time. The terms!'
and Q2o are given in Eqs. (149) and (183), respectively. Using them,
Eqo (195) becomes
0e (/ —.2 =T- + 2 7 T
+ (+39/7 r) F, F F/ (T F - P3 r+ 2 7r ) (196)
where
-? -Re (197)
From Figure 6, it is seen that F5(XI) changes from positive to
negative at a certain XIo For example at m' = 1 this point is at
XI = 2.02. At this point, the temperature profile is not influenced
by the oscillation. Figures 9(a) and 9(b) show the temperature profiles in the boundary layero
To find an expression for the Nusselt number, it is defined that
N ax= L _ _ _ ( _- )
In terms of dimensionless quantities, it becomes
ihxy* = - (198)

1.0
0. 8
0.6 At =4.0
At =2.0
0.4
0. 2
0.2 0.4 0.6 0.8 1.0
Figure 9(a). Temperature profile at X' = l(m'=l).

1.0
0. 8
0.6.
At = 1.0
0.4
At 0.0. 2
At 0~
0.2 0.4 0.6 0.8 1.0
e'
Figure 9(b). Temperature profile at X' 2(mt'=2).

72
Substituting Eqo (196) into Eqo (198), we then get
NuX_,f 0.34/9Pr _ (7508yPr3 (199)
Nux is plotted in Figures 10(a) through 10(d)o
As before, the average effect of the oscillation of the Nusselt
number is obtained by integrating the second term in Eq. (199) between
the two penetration depths. Then we have
/
Unl = - = L~- - L J X ALT (7568Pr3) Ft F' (200)
L - LfL.L,6e
After numerically integrating it by the same' method as in computing
Eqg (194), we get
ANu = -(3+S 2o0)/ AvT Pr SrF /=- f or'=/
=-(388,0oooR)JeAT Pr3 F, F4 o r n2
-(2,205, 000)JRe A TPr F3 Ff -4ir m/' 3
= (6,O650,oO)J, X A F, FT - Pr,,- 4 (201)
The minus sign indicates a decrease in heat transfero
Integration of the first term on the right of Eq. (199) between the
same penetration depth shows that it is of the order of Pr/3 Reo The
maximum value of Xf above which the perturbation method being no longer
-3
valid was given as 10 Therefore, the maximum value of XT is of the
order of 2_, which is 10-8 (see Appendix III). Again, the net change
in heat transfer is seento be less than 1% of its undisturbed value.

2.0O
21 0
10
0 1 2 3 4~~~At 2 t
xi = XIL
Figure 10(a). Local Nusselt number as a function of X?(mr=l).

2.0
Y
Y___U00
xix~x
Z~~~~~~~~~~~~~~~
1. 0
4-4
At -0
At.
At 2.1
0
0 1 2 3 4
X' =XIL
Figulre 10(b). Local Nusselt number as a function of X' (m'=2).

1. 0
xi=Ii, 0.5
it XA t l
At =.05
0 1 2 3 4
X' = XIL
Figure 10(c). Local Nusselt number as a function of X'(m'=3).

1.0
" —- L I!
X'= 1
0.5
zI I,~At =0
At =.05
At -.!
0 1 2 3 4
X' = XIL
Figure 10(d). Local Nusselt number as a function of X'(mI=4).

CHAPTER III
CONCLUSIONS
In the preceding chapters, the problem is solved by the perturbation method. The various orders of approximations are then solved by
the integral method which is known to be accurate in the case of uniform flow over a flat plate. In spite of the inherent limitations of
the perturbation method, the following important conclusions are drawn
as a result of this analysiso
a. The phase of skin friction is approximately in phase with the
input disturbance, while that of heat transfer lags approximately 2
radianso
bo The existence of a steady change of skin friction and heat
transfer is shown in Eqs. (192) and (199) and also plotted in Figures
8 and 10. The four branches of the skin friction is expected since
the equation of the boundary shows that it consists of two points of
inflection and a point of zero slope at X' = 1. These points divide
the plate into four parts, each producing a different effect on the
skin friction. The governing parameters for the steady change in skin
friction and heat transfer are Xf and XT, respectively. Increasing
the curvature of oscillation, indicated by an increase of ml, tends to
shrink the effect closer to the point of disturbance.
co Integration of the steady local skin friction and heat transfer over XP shows that the net effect of the prescribed oscillation is
77

78
an increase in total skin friction and a decrease in heat transfer from
the plate to the fluid. The magnitude of these net effects are always
less than 1% of the undisturbed flow within the limitations of the perturbation analysis
A numerical example is given in Appendix III which helps to check
the validity of the various assumptions made in the analysiso

APPENDIX I
DERIVATION OF EQUATIONS (43) AND (44)
At a certain instant, the boundary will be at a poistion shown in
Figure 35 The normal velocity of the boundary is equal to ecia and the
tangential component of it is of the order of O(E2)o It is therefore
neglected~ To derive the momentum equations, we have to consider the
inertia, pressure, body, and viscous forces separately.
The rate of change of any vector quantity Q may be written as
DT c T
where, in two-dimensional form,
_ x Q+ + it QY (I-2)
If we substitute Q = pV, i.e.,
= Qa - ry ( y
= L?S + cr P? (I-3)
then Eqo (I-1) becomes:
-D, V) =- ( I-4)
where
~y: = Y + (1-5)
79

8o
and for incompressible fluid, we take
0 = cons ant (I-6)
Equations (I-5) and (I-6) are substituted into Eqo (I-4) and we
get, for incompressible flow,
DT c T
= FtT - V
+ ply a7+( V )7 +7T(7 R)
+I/OU (1/ v x L* + 7B( V) ty (I-7)
The operators in this equation
y hz + r Y (-8)
and
<~( h )y ])(h
/ VVPiv txP = b-I (I +-9)
Putting into Eqo (I-7) we get
=T + h < hI y,1
+ pyT +.h-' a) + V l Z
TT
+ p J (v' V) Il + P0 7(t/ v (I-10)
Finally, we have to find out the last two terms. From vector calculus.

81
we have
aL*_k~ = il` /w ~f k ) (I-ll)
and
__ L -k L-k -. _k (I-12)
Therefore,
__x L x ~ _h_ t_ y,x Y _h
-IC.)Zh / q Lz J 2hr Yy (I-'3)
Lx hw d: h: s hA Y hy Y,
LX ly wAx (I-14)
1Y hr aY
Similarly,
___ hr L" __(I-15)
e) hy aY
and
lY Lxz a(h
Y h- -P ~(I-16)
Therefore,
(?. w) taru l —ah tP Q/7h
/ JL - h Z 4Y - th a (I-17)
(IV'V)Ly= h h aY hr 1 h,(1
and.
(7[T Tr c/i_ j L- (I-18)
7) tit 1 hx r ~<y h r h1

82
Equations (1-17) and (1-18) are put into Eqo (I-10) and the result becomes,'- U / U __U
D:P?L''T haZ + hy aY hz/7rh ay -V af~
- is t s + /7 shP (I 19)
Y;T hy cY flhtda
Therefore, the inertial force per unit volume in the X-direction is
__ - T Ir (U 7 -<U ) (I-20)? a —- hx'1 by /c:Y hx/ r gY ~.
and in the Y-direction is
PJc0 7 U 7 77 /r (AY 7h U 2h)j (I-21)
+ + h C)Y hP/, /,8
The pressure forces per unit volume is independent of the coordinate axis chosen and is therefore equal to
and / A (I-22)
respectivelyo
Finally. we have to derive the viscous force termso In two-dimensional form, we can write the viscous force per unit volume as,
1t~~S A Y _Y hXh ) (I-23)
brhr h,

83
where
6S, = L~ z + Lr,'X (I-24)
S, = tx r= r- iy (ory (I-25)
Substituting Eqso (I-24) and (I-25) into Eq. (I-23) we get
L//aS coLIS hh hy OX
+ cr//y +cx;yr
+ Lz y f+ iy y
T rx hx CY + rr r (I -26)
Using Eqs (I-ll) through ( 1-14), we get
Ther o (6X ihxe) XyX an ir'
+ 14 L y C) (*ryy hr) Y w)+ yxot - (I-27)
Therefore, in the X- and Y-directions:
~rg w (grhv) + ~(hXhx) (I-28)

.84
and
//S~CO.S, y
ffi~d(~yh la (Yhr) (YY1X + dr' _ (I-29)
hlbhy aX 1:) Y
where, from the general Stokes' law,
c37xi 2 4g.L hy j (I-30)
<y =2/( d6 h Yy j
2_:y = /2f 7.- y + I7 h p 7) (I-31)'7> ='T T'( hr hraY h x
In these three equations, the absolute velocities U9 and VI are
used which are given in Eqso (34) and (355) as:
U/ = + 2m (Z- L ) Y' t
and
7: +: ~'E
The reason for this is that any physical law must be given referring to
fixed. or absolute, coordinate system,
The acceleration terms that must be added on the left side of the
moment equation is discussed in the text,

APPENDIX II
SOLUTION OF EQUATION (114)
In solving Eq. (114), a method combining the methods of successive
approximation and momentum integral is used. The major simplification
obtained is that the convective terms are small compared with the
18
other terms. The idea was given by Lighthill, but without a detailed
mathematical proof. It is the purpose of this appendix to prove it
mathematically.
It has been discussed before that w' is a large parameter. According to the theory of differential equations containing a large
parameter, we can retain only terms having this parameter as a factor
and the highest order term to get a first approximation. This reduces
Eq. (114) to
LCA)' +.2 2,(' — /. ( 2 Y*)
2 1 /CI) s +e/)'"'~' d'" (II-B)
where the superscript (1) means first approximation to Eq. (114). Comparing Eqso (114) and (II-1) shows that the terms dropped are the convective terms. The question now is the accuracy of this approximation.
In order to investigate it, let us write the solution to Eq. (114) as
85

86
/(/) _ 2 (II 2)
The first approximation is solved from Eq. (II-1)o To solve the second
approximation, we substitute Eqo (II-2) into Eqo (114) and subtract
Eqo (II-l) from it. Then we get
s og/(2) YC y2
-/ -f$ Y/c ~ + (
=fs; _. 7c ye 0 (II-3)
The right-hand side of Eqo (II-3) is calculated approximately by
replacing U and V*c by their first approximation ul(1) and V*(1)
replac ing lUc c
lclr.c lc
Therefore, we have,62)6~ ~ 2a; - (2). t /U0T's Cg a /.- X'" I #" (II -4)
The boundary conditions are
y)=* o Uj'=7Z= 0
In order to compare the two solutions, U (1) and U,(2) let us
first integrate Elc (II4) over the boundary layer thickness
first integrate Eq. (II-4) over the boundary layer thickness 6K. Thus

87
we get
/
L6,J L, c d) + _
0
o
= jIX(2-1,' (II-5)
Again, the velocity profile is assumed to be
/r:=.2 + -. 7 + C.? + e7 (II-6)
with the boundary conditions;,t) a2/C _
0 U / tJ, 0=O / 0
The constant are therefore
02 - 0
C_ = 0
-2 = - 3b2
e2 = 2b2
The velocity profile then becomes
/ (2)
9b + 9- (II-7)
The velocity profiles of UO, U'(1) and UL 2) given by Eqso (145),
(160), and (II-7)g respectively are substituted into Eqo (II-5)o We
then get
__ _(0~~.. +( 0 =o/075). (II-8)
dB/ dX~c/'

88
Solving for b2, we get
b =(/56.486)c//27'1_+Z''+(fn, t. -.2'Sd2. m2~o.25} &',s e~4
/ I + ( 26 337)6,'2Z'
(II-9)
where
2l = 7K-l1(5 /32 ) 601 (II-10)
It can be shown numerically that UI(2) is always less than 1% of
U(l)o Therefore, we can take u{,l) as the solution to Eqo (114) and
the error introduced is thus less than 1lo This is shown numerically
in Appendix IIIo

APPENDIX III
A NUMERICAL EXAMPLE
In order to show the validity of the various assumptions made in
the analysis and also get a more clear idea of the effect of a localized vibration on the flow, a numerical example-is worked out in this
appendix~
Let us consider a semi-infinite flat plate with air flows at 48
fpso The localized vibration is put at x = 2 ft from the leading edgeo
The viscosity of air is taken to be 0.144x10-3 fps.
From the given conditions, it is seen that,e _ y L.+8 x2 - 6.6 7 x /o 5
Re o 14/ x /o10-3
EJ =?X.8 x. 0/22 ~f = 0. /06 /'n.
Following assumption c which limits c to be small compared with b,
i.e., c <<, c can be selected as 0.015 in. in this example. Then,
G 0. 00/5
G _ ~.:_ _ -- 6.25 X/0
L6 2x/2
w' =)o =()(F4)= 24
LIMITATIONS OF Xf
From Eqg (192) and Figure 8, the steady change of skin friction is
seen to be directly proportional to %f. When Xf is so small as to introduce only a maximum change of 1% or less, the oscillation may be considered to be of no effect to the unperturbed flowo This gives the lower
89

90
limit of Xf which is approximately at
-f -Re /
So,
t' = [ f6.67-) 360
(6,25 x 10-5) 360
or,
0C =2 —4.' = 8600 r/~ec d 820o0 Cp t,
Due to the limitation of the perturbation analysis, the maximum
deviation must be kept small compared with the unperturbed value in
order to make sure of the convergence of the series expansion (102).
Taking 20% maximum deviation, Xf is seen to be approximately
- ___X = 20 x /O
So,
G) -= 760
c0 = /8,200 r, /sec, -/74,oo00 C1pm
This gives the upper limit of'f in the analysis, above which the perturbation analysis being invalid~

91
ORDER -OF -MAGNITUDE
The order-of-magnitude of the various terms in Eqs. (64) through
(67) can be grouped in accordance with their coefficients as follows:
[12l = 1
to
I - 6
IjEi= /0
E (/ )aY2 0-2
In the analysis, only terms of order-of-magnitude of 1 and 10-2
are retained. The other terms are seen to be small compared with these
terms. This confirms the simplification made in Eqs. (74) through (92).
THE FIRST-ORDER VELOCITY
From Eq. (II-2), Ujc is given as
U/C "7
where U' is given by Eq. (153) as,i =? + w b -( 3b, + 2) ( + (26,+ W)?f
and U'(2) given in Eq. (11-7) as:
lc
/c= b2? - 373?4)

92
To compare the order-of-magnitude of U () and U it is only
necessary to -compare bl and b2o Now, bl is given in Eq. (159) as
where
fir= (44 453)[% 6s -(-L)Y- 0.25O
=, i {19.499i ( 8' 663)(IX-1) 0o.25j
and also b2 is given by Eqo (II-9).as,_ (156.48 6_)__?V4 (-.2 r`Y-z%4 M'2r' 4o. s.2 - -
= I + (.26. 337)W2) 6 2
where
c2 =-3 -'(5. /32 ) C'
Before comparing bl and b2, bl and b2 themselves can be simplifiedo
To simplify bl, we compare flr and fli in Table III-lo The value of wct
is taken to be 360, its minimum value. It is seen that the error in
neglecting fli is only 1%, i eo
< > J
as shown in Table III-lo Therefore, we can write
b, = 00"e s(44.53),/~ci [ (X-I )]'Z 0.25]

93
TABLE III-1
TABULATION OF Flr~ Fli, AND b2
X Flr Fli b2
o200000E 00 -0429785E 01.522297E-01 -o 660317E-01
o400000E 00 -.961128E 01.467422E-01 - 752335E-01
o60o00E 00.143776E 02 o466146E-01 -o563923E-01
o800000oo E 00 -.156624E 02.475844E-01 -o556825E-02
lo00000QE 00 -111133E 02.481479E-01 -.635250E-01.120000E 01 -.467866E 00.472948E-01.124418E 00
14o0000E 01 138944E 02 ~443809E-01 l151452E o00
o16ooo0000E 01 o278531E 02 o392983E-01 o 133929E 00
o180000E 01 o374228E 02 K325421E-01.813180E-01
o200000E 01 o404725E 02 o250442E-01.165248E-01.220000E 01 ~373359E 02 o 178388E-01 - 371958E-01.240000E 01 -301682E 02 o117294E-01 -667636E-01
o260000E 01 o 216656E 02 o710763E-02 -.718796E-01
o280000E 01.139541E 02.396549Ev02 -.606875E-01
o300000E 01 o 810871E 01 o203593E-02 -o432158E-01
-320000E 01.426934E 01 961644E-03 -o267813E-01
o340000E 01 o204306E 01 o417850oE-o03 -146862E-01
o360000E 01 o 890719E 00 o167030E-03 -o719876E-02
o380000E 01 o354438E 00 o614302E-04 - 317519E-02
o400000E 01 d128920E 00 o207893E-04 -o 26610E-02
To simplify b2, it is seen that
(26. 337 7) OI)' ~
and that
-/ C.5. / 3 2 Z85/
since the minimum values of ci- and X' in this analysis is 360 and 0o1,
respectively~ Therefore, U2 is approximate L and
b2 = (3o.d/92)'A (, -— m'/K'4 4/n<X'3. (3'5i"4- m'I
- 2. 5 -. 25 3

94
The minimum value of w' is 360. For this value of V', the value of b2
is seen to be about 1% or less than flr, as shown in Table III-1. Therefore, we known that
and we can write
=ic,c = — (44'+"53)(-?4s2fa ga4+2 5o2004 (III-1)
where
-: =' 17/m (E-)Z- o 25 (III5-2)
PHASE ANGLES
From Eq. (III.1), the phase angle in skin friction is seen to be
zero~ In the case of heat transfer, the phase angle is given by Eqo
(171) as
2 L= an-/( 5./32 / Pr3 )
For Pr = 0072 and A' = 360, this equation becomes
= fcn2l/ (/655 5Z) (III-3)
The smallest value of X' in this analysis may be taken as 0o2, then
od) = /' n -/ (93 /)

Considering that tan 890 = 57.3, we see that U2 is approximately equal
to 900 for all other w' and X'.

REFERENCES
lo Anatanarayanan, R. and Ramachandran, A. "Effect of Vibration on
Heat Transfer from a Wire to Air in Parallel Flowo" Trans. ASME,
80 (1958), 1426.
2. Andrade, E. N. "On the Circulations Caused by Vibration of Air
in a Tube)" Proco Roy. Soco, A-134 (1931), 445.
3. Eshghy, S. "The Effect of Longitudinal Oscillations on Fluid Flow
and Heat Transfer from Vertical Surfaces in Free Convectiono" Ph.D.
Thesis, The University of Michigan, 19635
4. Blankenship, Vo D. and Clark, Jo A. "Effects of Oscillation on
Free Convection from a Vertical Finite Plate." ASMS paper, 63-HT-31o
5.: Blankenship, V. D. and Clark, J. A. "Experimental Effects of Transverse Oscillations on Free Convection of a Vertical, Finite Plate."
ASME paper, 63-WA-123.
6. Fand, R. M. and Kaye, J. "Acoustic Streaming Near a Heated Cylinder."
J. Acous. Soco, 32 (1960), 579.
7o Fand, R. Mo and Kaye, Jo "The Influence of Sound on Free Convection
from a Horizontal Cylinder." J. Heat Transfer, 83, C-2 (1961), 133.
8. Holstein, H. and Bohlen, T. "Ein Einfaches Verfahren zur Berechnung Laminarer Reibungsschichten, die dem Nahungsansatz von K.
Pohlhausen genugeno" Lilienthal-Bericht, S10 (1940), 5.
9. Hill, P. G. and Stenning, A. Mo "Laminar Boundary Layers in Oscillatory Flowo" J. Basic Eng., 82, D-3 (1960), 5935
10. Holtsmark, J., Johnsen, I., Sikkeland, To and Skavlem, SO "Boundary Layer Flow Near a Cylindrical Obstacle in an Oscillating, Incompressible Fluid," JO Acouso Soco, 26 (1954), 26,
11. Hori, E. "Unsteady Boundary Layers (First Report)." Bulletin
JSME, 5 (1962).
12. Hori, E. "Unsteady Boundary Layers (Second Report)." Bulletin
JSME, 5 (1962).
96

97
REFERENCES (Continued)
13. Hori, E. "Unsteady Boundary Layers (Third Report)." Bulletin
JSME, 5 (1962).
14. Kubanskii, P. N. "Currents Around a Heated Solid in a Standing
Acoustic Wave." Zhurnal Tekhnika Fizika, 22 (1952), 585.
15. Kubanskii, P. N. "Currents Around a Heated Solid in a Standing
Acoustic Wave)" Trans. USSR Academy of Science, 82 (1952), 585o
16. Kubanskii, P. No "Effects of Acoustic Vibrations of Finite
Amplitude on the Boundary Layer." Zhurnal Tekhnika Fizika, 22
(1952), 5935
17. Lemlich, R. "Effect of Vibration on Natural Convection Heat Transfero" Symposium on Pulsatory and Vibrational Phenomena." Indo and
Eng. Chemo, 47 (1955).
18. Lighthill, Mo JO "The Response of Laminar Skin Friction and Heat
Transfer to Fluctuations in the Stream Velocity." Proc. Royo Soc.,
A-224 (1954), 1.
19. Nanda, R. So and Sharma, V. PO "Free Convection Laminar Boundary
Layers in Oscillating Flow." Jo Fluid Mech., 15 (1963), 419.
20. Nyborg, W. L. "Acoustic Streaming Equations: Laws of Rotational
Motion for Fluid Elements." J. Acous. Soc., 25 (1953), 938.
21. Pohlhausen, K. "Zur Naherungsweisen Integration der Differentialgleichung der Laminaren Reibungsschicht." ZAMM, 1 (1921), 252.
22. Raney, W. P., Corelli, J. C. and Westervelt, P. JO "Acoustical
Streaming in the Vicinity of a Cylinder." J. Acous. Soc., 26
(1954), 1006.
235 Rott, N. and Rosenzweig, M. L. "On the Response of the Laminar
Boundary Layer to Small Fluctuations of the Free-Streaming Velocity."
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24. Schlichting, H. "Boundary Layer Theory." 4th Edition, Chapter XI,
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25. Schoenhals, R o J. and Clark, Jo A. "The Response of Laminar Incompressible Fluid Flow and Heat Transfer to Transverse Wall Vibrationso'
JO He~at Transfer, 84, C53 (1962), 225.

98
REFERENCES (Concluded)
26. Skavlem, S. and Tjotta, S. "Steady Rotational Flor of an Incompressible, Viscous Fluid Enclosed Between Two Coaxial Cylinders."
J. Acous. Soc., 27 (1955), 26.
27. Squire, Ho B. "Heat Transfer Calculation for Aerofoils." ARC
R and M, 1986 (1942).
28. West, G. D. "Circulations Occurring in Acoustic Phenomena."
Proc. Phy. Soc. London, B-64 (1951), 483.
29. Westervelt, PO Jo "The Theory of Steady Rotational Flow Generated
by a Sound Field." J. Acous. Soc., 25 (1953), 60.
30. Westervelt, P. Jo "Acoustic Streaming Near a Small Obstacleo" Jo
Acouso Soco, 25 (1953), 1123o

UNIVERSITY OF MICHIGAN
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