RESPONSE OF HEAT TRANSFER FROM A MOVING FLAT PLATE IN A PARABOLIC FLOW By T. R. GuPTA * [ Department of Mathematics and Statistics, U.P. Agricultural University, Pantnagar (Indid) I
Received November 22, 1971 (Communicated by Dr. B. R. Seth,
F.A.SC.)
ABSTRACT
This paper deals with the solutions of steady as well as unsteady three-dimensional incompressible thermal boundary layer equations and the study of the response of heat transfer when there is a parabolic flow over a moving flat plate. The components of velocity in boundary layer are discussed by Sarma and Gupta and those results are used to analyse thermal boundary layer equations. A general analysis is made from which we deduce (i) Solutions of two-dimensional thermal boundary layer on a moving flat plate, (ii) Solutions of thermal boundary layer on a yawed flat plate, (iii) Solutions of thermal boundary layer when there is a parabolic flow over a moving flat plate by giving different values to fi and Cx. Solutions are developed for large and small times and curves are drawn representing the variations of heat transfer from the plate with time for all the cases. The limiting time is also calculated.
1. INTRODUCTION LIGHTHILL 1 has studied the unsteady two-dimensional thermal boundary layer equation assuming that the unsteady motion is a perturbed one about a steady mean. He made use of the results of the velocity field in which the main stream is fluctuating in magnitude but not in direction. Later Sarma 3 extended the analysis to include the fluctuation in direction also. In this paper, we study the solution of three-dimensional incompressible thermal boundary layer equation when there is a parabolic flow over a moving flat plate and extend the analysis of two-dimensional thermal boundary layers to solve the three-dimensional energy equation. The components of velocity in boundary layer when there is a parabolic flow over a moving flat plate, * Present address , Department of Mathematics, University of Roorkee, Roorkee (India),
158
Heat Transfer from a Moving Flat Plate
159
are studied by Sarma and Gupta 4 and those results are used to analyse the thermal boundary layer equation considered in this paper. Here we find the solution of the energy equation when the temperature of the main stream is TM (a constant) and that of the plate is: Tw (x, t) = T M + ax'm'i + Ebxm2 Tb (t),
where x is the distance along the wall, t is the time, Tb (t) is arbitrary function of time, m,, m 2 , a and b are constants and E is a small reference parameter. Solutions are obtained for steady as well as for unsteady equations. The equations are linearized following Lighthill', and Sarma 2, 3 , and are subjected to Laplace transformation following Watson 7 . In unsteady motion, two types of solutions, one for large times, and the other for small times, are obtained assuming the temperatures as asymptotic series expansions for large times and series expansions for small times. Sets of general differential equations are obtained in a single variable for the terms from which we can deduce (i) solutions of two-dimensional thermal boundary layer on a moving flat plate, (ii) solutions of thermal boundary on a yawed flat plate. (iii) solutions of thermal boundary layer when there is a parabolic flow over a moving flat plate, by giving different values to ,8 and Cx. The heat transfer from the plate for large times comes out to be series expansion in time derivatives of the chordwise velocity component of the plate and the plate temperature. In the case of motions in which the temperatures and the velocities are increased impulsively and become steady for large times, the results are obtained by simply giving steady values to the time dependent functions. In such impulsive motions curves are drawn representing the variations of heat transfer from the plate with time. These variations are studied for twodimensional flow, for yawed flat plate and for parabolic flow over a flat plate. The time when the curve for small times joins with that for large times can be taken approximately as the limiting time taken for the unsteady quantity to become steady. This limiting time is calculated approximately for all the three cases. Finally, the author wishes to state that solutions of three-dimensional energy equation can also be found when there is a parabolic flow over a flat plate which is being at rest and the stream is in unsteady motion.
T. R. GuPTA 2. GENERAL ANALYSIS AND SOLUTION OF STEADY ENERGY EQAUTION
The boundary layer equation of energy for three-dimensional incompressible flow is: ^T
?u \ 2^w 2 ^TT _ v^ 2Tv +-t() ^t +u ^x +v ^y Q^y 2 + l ^Y^ 1' (
(1)
where x and z denote the co-ordinates in the wall surface, y denoting the co-ordinate which is perpendicular to the wall. u, v and w are the velocity components in the x. y, z directions respectively, T the temperature, t the time, Cp the specific heat at constant pressure, v the coefficient of Kinematic viscosity and a the prandtl number. The dependence of temperature on the z-coordinate is neglected. In this paper we want to find the solutions of (I) when the temperature of the main stream is constant and that of the wall is unsteady, i.e., when the boundary conditions are: T = T (x, t) at y = 0 and T --± T M (a constant) as y—) oo (2)
where T,,, (x, t) is the temperature of the wall and T M is the constant temperature of the main stream. Following the work of Lighthill' and Sarma 2 ' 3 , we write: T = To (x, y) + ET 1 (x, y, t), Tw = T wo (x) + €Tw1(x, t) , Tx = THO (x) + ETH1 (x, t), u = Uo (x, Y) + €u1 (x, Y, t),
(3)
I
v = vo (x, Y) + ev 1 (x, Y, t), I w = wo (x, Y) -F Ewe. (x, Y, t),
where the '0' suffixed quantities denote the steady state and '1 ' suffixed quantities denote the time dependent part of motion and E being a small reference parameter. Substituting (3) in (1) and neglecting E 2 terms, we get: ^T °
^T °V _ ^ 2T °v
^ u0 2
+v° PY ++
( P ;o
,
(4)
Heat Transfer from a Moving Flat Plate
161
for steady state and: bT 1T1
± v
bTo
x +
uo 2
u1
T 12v
aye
o
^T1
bTo
^x + D + ul ^J
f )uo but
c^wo tbw1
y ^y 1, y 3y -^- ^ + Cp l ^
(^)
for the unsteady part of the motion. The boundary conditions are: T o = Two (x), Tl = T wi (x, t), at y = 0,
(6)
and T o --* TM , Ti --- 0 as y —> oo.
Substituting t,. = Ut, where U. > 0 and multiplying (5) by e st, and integrating with respect to t l from 0 to 0o (i.e., applying Laplace transforma-
tion), we get:
U.sT1+uo ^-+ul ^x +vo ^y +vi ^yo _
vc^ 2T1
+
2v Ziuo Sul Zwocowl (7 ) Cp t by + by ^Y1'
where i
(x, Y, s) =
f 0
T1 (x, y, t1) e -st, dtj,
a (x, y, s) = f u1 (x, y, t1) e st,dti , etc., R (s) > 0. -
0
assuming that: Tl e Sti --* 0 as tz —* oo, T l = 0 at t, = 0. -
(8)
Then the boundary conditions are: T^ = Tw ,, (x, s) at y = 0, T -* 0 as y - - oo.
(9)
In preparation for the detailed analysis of unsteady thermal boundary layer equation, we shall write down the solution of (4) along with the boun-
162
T. R. GUPTA
dary conditions given by (6). Since we are studying the thermal boundary layers when there is a parabolic flow over a moving flat plate, so x co-ordinate is along the chordwise direction of the plate. The velocity in boundary layers along chord and spanwise directions have been studied by Sarma and Gupta4 and we shall make use of those results in the analysis of thermal boundary layers considered in this paper. The velocity components u o , v o and w o were given by (10) in Sarma and Gupta4 . Instead of temperature T wo (x), let us assume that temperature difference [T wo (x) — TM ] (just for simplicity) as in Isao Imai 17 to be: Two (x) — TM = axm ,
(10)
and
To (x, y) — T,,
_ axm v Oi ()+ Cp f(1 + fl ) 2
[
8
2 (^) + (Cx) 2 03 (^)
+PCx 0 4(')}]
(11)
where a, ml are constants and q is defined by (10) in Sarma and Gupta 4 . Substituting (10) from Sarma and Gupta 4 , and (11) in (4) and equating the coefficients of various terms, we obtain the following differential equations: 2m1 01 (n).f' e7) — 0 1' ()f() = 0" e7)
(12)
with 0(0)=1, 0 (0-)=0,
0 2' ()f() + 4 f" (^) = — 1 0 2" (n),
(13)
with 02 (0) = 02(00) = 0,
463 (71)f ' () — 63 ' ( ,J)J ( ryI) = 1 6 3 " (9) + /2' (n),
(14)
Heat Transfer from a Moving Flat Plate
163
with 03 (0) = 0(o) = 0, 20 4 (^)f' (^) — 04'()f() = 1 04 (^) — f" (^) h' (n),
(15)
with 04 (0) = 0 4 (00) =0,
where f (i) and It (n) are given in Sarma and Gupta 4 . When (3— = Cx = 0, the three-dimensional incompressible thermal boundary layer equation reduces to the two-dimensional incompressible thermal boundary layer equation and is solved by Sarma 3 , for a general flow and in particular for a flow along a flat plate. When , 0, and Cx = 0, we have a case of thermal boundary layer equation for a yawed flat plate with uniform main stream velocity components along chord and spanwise directions and jl 0, Cx 0 0, then it is a case of thermal boundary layer equation for a parabolic flow on the plate with a non-zero pressure gradient in the spanwise direction. Solution of equation (12) when ml = 0 is given in Schlichting5 and solution of (13) is given in Sarma 3 . The equations (14) and (15) are numerically integrated by taking prandtl number Q = 0.7 and values are tabulated in Appendix 1. The amount of heat transfer from the plate to the fluid is given by (32). 3. SOLUTION FOR LARGE TIMES
In this paragraph we shall solve the equation (7) along with the boundary conditions given by (9) when the steady main stream velocities along chord and spanwise directions are given by Sarma and Gupta 4 [i.e., when U o = U. (a constant) and W = U. (J3 — Cx) and the plate is moving with velocity EUb (t) in the chordwise direction]. Let us assume: Tw1(x, t) = Tw2 (x)
Tb (t),
(16)
a separable form for the unsteady part of the temperature of the plate, where Tb (t) is an arbitrary function of time. For further discussion, we assume (10) for [T wo (x) — T M ] and write: T w 2 (x) = bxrn 2
,
(17)
164
T. R. GUPTA
where b and rn 2 are constants. The velocity components u l , v 1 and w1. are given by (10) and (11) in Sarma and Gupta 4 . To solve equation (5), we expand T 1 (x, y, t) in temrs of 1tb (t) and Tb (t) with respect to t. Thus: Ti (x, y, t) 1
=
u
x i
L" U
„
mdub
i=o 2
2
(
diub
)
i=0 00
+ (Cx)2
(d2 ub
2
f LLi i mo^^^^
+j (Cx) + (U . b)
(--x-
(T YT5 , i ('f)d dcb 00
ab
r xm 2 ( x ) T 612 (j) d
(18)
i=0
Substituting for u 0 , v o , w o , ul , v 1 , w 1 , from (10) and (11) (Sarma and Gupta 4) and for T o , Tl from equations (11) and (18) in (5) and equating the coefficients of i x m _^, d2 'U b
x
^
(x
i -1
d z ub
U-. dti ' U °° .U„) x dpi ' U
2 P2
i
(x t
dit^b
at{ ,
Z d b U C x (u ^ i ib ' PcxU 2co (VX_ ^a 2
2
Z
and x
;
m _i i: c00) i d t Tb (t) (
we have the following differential equations: Ti, i-i
+ 2 (i + m l ) T1, if ' 4 (2i + 1) B11 B
— 4 .f
__ 4a T ,, 1,
+ Z O1Bi l
i for
i > 1,
(19)
Heat Transfer from a Moving Flat Plate in,TI, of' , - 4 f'T'1,
o
+
m101B0,
- el, B
165
4^
(20)
with T 1 , i (0) =T 1 ,i(oo) =0 for i -0, T2, i-1 + 4 if' T2, i -- UT' 2, i - (2i + 1) 0 2 Bi 4I
(21)
T 2,i+f'Bz" for i>1,
1 2 1 f T 2, 0 + 02'B0 + Q T 2, 0 + 2-J BO = 0, '
( 22 )
"
with
(0) = T 2, i (oo) = 0 for i > 0, i - 4 (2i + 1) Bi02 ' T3,i-1 + 2 if' T3, i - 4.fT,3, T2, i
= I T 3 ", i + 4 f" Mi', for i > 1, .f T'3,o+Bo 0 2'
+
1
(23) 24)
T 3,0 + =0, i f "Mo( "
with T 3 , i (0) = T 3 , i (oo) = 0 for i > 0, T4,'L-1 H -
2 (i + 2)f' T4, i - afT'4, i + 0 3 B i ' - I (2i + 1) BiO 3
'
= -T" 41 i + h' Ni' for i > 1, f'T4, 0 - 4.fT'4, 0 + B 0 '03 - 4 B 0 0 3 ' = 4^ T" 4 , 0 + 2 h'N o ',
(25) (26)
with T4,i(0)=T4,i_(oo)=0 for i>0, 1) B204 T5,i-I +2(r +1)J'T5,i- 4JT'5,i_{ -04B1' - (2i +
= 4v T „ 5,i - (
+ h'Mi') for i > 1,
'
(27)
T. R. GUPTA-
166
If T5, o — If T 5,0 + 2 04130' — 4 Boe4 '
'
_4- T „
5,o
-
'
2(Zf „ N°'+h'M o '),
(28)
with T 5,f(0) =T5,i(oo)=0 for i>0, T 61 _1 + 2 (i + m2).f' T6, i — 4 .f T's, i = LT"*s for i > 1, (29) 1 , 2 m2.f' Ts, o — 4 .f'I' s , ° = 4Q T"0, °i
(30)
with T 6, (0) = 1, To,o(oo) =0
and
T6, i (0) = T 6 , i ( 00) = 0 for i l l. The equations (20), (22), (24) (26) and (28) are numerically integrated for values of m l = 0, 1 and prandtl number a = 0.7 and numerical results are given in Appendix 1. The equation (30) is exactly same as equation (12) when T 6, o () is replaced by 0 1 (,7) and m 2 is replaced by m l and its solution for m 2 = I is also given in Appendix I. The amount of heat transfer from the plate to the fluid is given by: Q * (x ,
t) = QO * (x) + EQ1 * (x,1 t)
— K l ('TO)— EK E ( 2) , ^y yo
^Y l=o
(31)
where Q 0 * (x) and Q 1 * (x, t) are steady and unsteady heat transfers and K1 is the coefficient of thermal conductivity. Thus Q 0 * (x) _ — 2 Kl 1
U
J
[axmO1' ( 0) +
U Cp2 1( 1 + N 2) B2 , (0)
+ (Cx) 2 0 3 , ( 0) `F' N (Cx) 0 4 ' (0)}] +
(32)
Heat Transfer from a Moving Flat Plate
167
and Q* (x, t)
_ 2U„ K' \Ux)} a LI
( X)
„ 2 xm,T' '', ' (0) ddab
t=o
+ C 2(
du
/^
7 )T ' 2, 2 (0) atb + ('2
0
r
:-)
CO
dt b +(Cx)2
X T' (0)
(
Ux.
'' T' 4, i. (
0)
-)
ddt b
00
+(Cx) z (XYT'S,. (0)ddtba s=o
xm,
+ Ub i=o
(33)
(LT)zT'o, i (0) ddtb ,
Thus the unsteady part of temperature and heat transfer distributions are given by (18) and (33). In the case of a plate which is set impulsively in uniform motion in the chordwise direction when there is a parabolic flow with velocity E and in the case of the temperature of the plate which is increased impulsively and becomes steady for large times with the value (T M + ax'`' + Ebx n ), the equations (18) and (33) reduce to: ,
Tl (x, Y, t) 2
— U b Iax m l T1,0 ("!) -I” ^~ (T 2, 0 (?1) + 2 T3,0 ('i) + (cx) 2 T4, 0 (-1) + fl (cx) Ts, o (^l)} + Ubxm 2 T 6 , 0 (34) Q * i (x, t)
— — ^x U0 ^Uz) 1 axm T'i, o ( 0) + C *2 T'2' - o ( 0) + PT'3, o ( 0) + (Cx) 2 T'4, o ( 0) + 8 (Cx) T'5, o ( 0)}
+u b
A3
bx ,mn 'T`e,o (0)] , (35)
T. R. GUPTA
168
where ub (t) =
1.
Introducing the dimensionless Nusselt and Eckert numbers, the steady and unsteady amount of heat transfer from (32) and (35) are: ( Ux ) No (x) .
--[0'(0) + E {(l + f 2) 0 2 ' ( 0) + (cx) 2 03' (0)
+ 9 (Cx) 0 4 0)}],
(36)
' (
(U„ x) b N l (x, t) _ — iT'1, [
o (0) + E {T'2, 0 ( 0) + fl T'3, 0 ( 0) 2
+ (Cx) 2 T' 4 , 0
+
fl ( Cx) T'5, 0 ( 0)I -r- A T'6, 0 (0)],
(37)
I
where — xQ0 * , steady Nusselt number, N (x) _ Ki (Two — TnI) ° Nl (x ' t) E—
x Q 1, unsteady Nusselt number,
K1 (Two — Tyr)
U2 (T TI1) C wo —
y
.
(38)
Eckert number,
and _
A
U Tw2 (a non-dimensional variable). — T,c) Ub (Two 00
Thus we see that steady amount of heat transfer depends on the dimensionless Eckert number E and the unsteady amount of heat transfer depends on E as well as on A (a dimensionless variable arising due to the unsteady temperature of the plate) a art from and Cx. For all the cases Q is assumed to be constant and equal to 0.7. For m 1 = 0, the steady amount of heat transfer given by (36) decreases with the increase in E for all P and Cx defined earlier. For a fixed value of E, the amount of heat transfer when Cx = 2, ,B = 1, is greater than that for 8= 1, Cx = 0, which is even greater than the two-dimensional case.
Heat Transfer from a Moving Flat Plate
169
From (37), we observe that if we neglect the unsteady temperature (i.e., A = 0), then the unsteady amount of heat transfer for m, = m 2 = 0, increases with the increase in E and if the unsteady tem_ erature is also taken into consideration and we take A = 1 then the values of heat transfer is still larger than that without unsteady temperature. For m 1 = m 2 = 1, the amount of heat transfer is even greater than that for m l = m 2 = 0 for all P and Cx. 4. SOLUTION FOR SMALL TIMES
Following Sarma 3 and Sarma and Gupta4 for small times, we obtain the solution of (7) for an impulsive motion. Thus, when ub (t) and Tb (t) become the unit functions then the amount of heat transfer from the plate for small times is: Q 1 *(U C, vx)'
K1
_
-
+ ^^
2(1
)
U 2 f„ (0) —(12a^")ax'n1(N)
a
41 -f- /Q) 2 (
U
2
((
^1^
a (2m,. — 1) xm, B' (0)
^q
l
/
——, Ca0p ll I N 2) 02'(0) —3(Cx)203' ( 0) ) — N (Cx) 0 4 (0)) r N '
— bxm U^ N> ( 4m
64 1 ) f„ (0) N
I + 0 (N3/2
), (39)
where N = x . (a dimensionless variable).
(40)
Introducing dimensionless coefficients, in the form of Nusselt and Eckert number we can write (39) in dimensionless form as: \Uvx/ N. (x,
t) ub =
N l+ 2om 1 a 4 (1 { ^/v) a(2m^,-1) 8 1'( 0) + 1Va) Tr (
(1
[
)
-
— E ((l + N Z) e2 ' (0)
—
3 (Cx) 2 0 3 ' (0)
170
T. R. GUPTA — B (Cx) 0 4 ' (0)) 1 N + 2 - C1
+
A
^( N) ±
(4
64 :
+ ^^) Ef (0) ,
1 ) f „ (0) N]
+ 0 (N312),
(41)
where N l (x, t), E and A are defined in (38). From (41), we observe the following behaviours with rn l = m 2 = 0 and ff = 0.7. (i) When f = Cx = 0, the magnitude of heat transfer when the unsteady temperature is neglected (i.e., A = 0), increases with time and join with the steady values for large times given by (37) for different values of E. When E = 0, the case reduces to two-dimensional energy equation without dissipation and is discussed by Sarma 3 . Our results agree with his results. When the unsteady temperature is also taken into consideration (i.e., A= 1), then the magnitude of heat transfer is very large at N = 0, i.e., at the start and decreases and join smoothly with the steady values for large times (see Figs. I and 4) . The approximate joining values of N are 8.3, 9.2, 2.2,
1 z Ix
.o =o
FIG. 1. Variation of heat transfer with N when S=Cx=O for u=0.7 and mi=m,—O.
Heat Transfer from a Moving Flat Plate
171
O7 and 0.35 for (fl = Cx = 0, A=O, E = 0), (P = Cx = 0, A=O, E= -2), (8=Cx=O, A=1, E=2), (i4=Cx=0, A= 1, E=0) and (^6 = Cx = 0, A = 1, E = - 2) respectively.
f z
4
ix :
Fio. 2. Variation of heat transfer with N when fl=1, Cx=O for o= 0.7 and m 1 =m 2 =0.
(ii) When i4 = 1, Cx = 0, the magnitude of heat transfer when unsteady temperature is neglected, i.e., A = 0, decreases with time for E = 2 whereas for E = - 2, it increases and for both the values the magnitude joins with the steady values for large times given by (37) (see Fig. 2), When A = 1, the magnitude is very large at the start and with the increase in time, the magnitude decreases for E = - 2, 2 and joins with the steady values for large times (see Fig. 5). The approximate joining values of N are 11 0, 9.25, 4.25 and 0.25 for (fi = 1, Cx = 0, A = 0, E = 2), (6 = 1, Cx = 0, A = 0, E=-2), (fl=1, Cx=O. A=l, E=2) and (fi=1, Cx=O, A=1, E = - 2) respectively. (iii) When l4 = 1, Cx = 2, the magnitude of heat transfer when the unsteady temperature is neglected, i.e., A = 0, increases with time for E = 2 and decreases with time for E = - 2 and joins with the steady values for large time. This can be seen from Fig. 3. When A = 1 the magnitude is very large at the start and decreases with time for E = 2 and then increases
T. R. GUPTA
172
and joins with the steady values for large times. For E _ — 2, the magnitude decreases with time and joins with the steady values for large times. This
x
z afs
FIG. 3.
Variation of heat transfer with N when $=1, C=2 for a =0 7 and m l —m 2 -0. .
-___--GRAPH OF( 37 )WHEN A•1 GRAPHOFI 44 )WHEN A•1
1QE=2 E=0
im E=-2
al> N
•0
.4
.6
TTTTTT 1.2
1.6
-AYIR
24
28
32
36
4.0
4
bJ
4, Variation of heat transfer with N when P =1, Cx =O and m 1 =m g =0 for a=0.7,
173
Heat Transfer from a Moving Flat Plate
"
E=2
E=-2
"' N
FIG. 5.
•4
•8
12
I.6
20 2.4 N --f AXIS
8
3,2
3.-
40
44
Variation of heat transfer with N when ,S=1, Cx=O and rn 1 =m 2 =0 for a=0.7.
E_?
A
6
7
6
WHEN A =I WMFN A. I
FIG. 6.
Variation of heat transfer with N when p=1, Cx=2 and in—m 2 --=O for v=0.7,
T. R. GUPTA
174
can be seen from Fig. 6. This decrease and then increase for E = 2 is due to the curvature of the outer flow stream lines. The approximate joining values of N are 5.4, 4.5, 5.5, and 4.4 for (^ = 1, Cx =2, A = 0, E = 2), (P=I, Cx = 2, A = 0, E=-2), (E = 1, Cx = 2, A=l, E=2) and (fi = 1, Cx = 2, A = 1, E = — 2) respectively. REMARKS
The author has also done some work on the thermal boundary layer equation for a parabolic flow over a moving flat plate when the temperature of the main stream is constant and heat transfer from the plate is unsteady and this work will be published in due course. The author is highly indebted to Dr. G. N. Sarma, Reader in Mathematics, University of Roorkee and Dr. S. K. Sharma, Professor of Mathematics, U.P. Agricultural University, Pant Nagar, for their encouragement and guidance for the completion of this paper. REFERENCES 1. Lighthill, M. J.
.. "The response of laminar skin friction and heat transfer to fluctuations in the stream velocity," Proc. Roy. Soc., London, 1954, Ser. A, 224, 1-23.
2. Sarma, G. N.
.. "Solutions of unstready boundary layer equations," Cnmb.. Phill. Soc., 1964, 60, 137-58.
3. 4.
Proc.
.. "Unified theory for the solutions of the unsteady thermal boundary layer equation," Ibid., 1965. 61, 809-25. "Parabolic flow over a moving flat plate,"
and Gupta, T. R.
Roorkee Research Journal, 1970, 12 (1-2).
5. Schlichting, H. (Editor) .. 6. ISAO, IMAI
Boundary Layer Theory
University of
(Pergamon, 1955).
.. "On the heat transfer to constant property laminar boundary layer with power function free stream velocity and wall temperature distribution," Quart. App!. Math., 1958, 16, 33-45.
7. Watson, J.
.. "The two-dimensional laminar flow near the stagnation point cf a cylinder which has an arbitrary transverse motion," Quart. J. Mech. App!. Math., 1959, 12, 176-90.
175
Heat Transfer from a Moving Flat Plate APPENDIX I
In this appendix we shall give the numerical values of the functions used in this paper. The differential equations for f(17), h (n), B o (j), M o (- q) and N o (9) and their solutions are given in Sarma and Gupta 4 . The differential equations for 01(n), 03 0 4 (n), T1, , ( n), T 2 , 0 (-1)' T3, 0 (1)), T4, 0 ( 77), (71), etc., are numerically integrated for a = 0.7 and numerical values T 5, o are: (")),
Big (0)
T'1,
m1 =00
—0.58536
—0.51594
m1 = 1
—0.96088
—1.38964
0
( 0)
0 2'(0)=0 - 2446 ; 03 '(0) = 112283; 0 4 '(0) = —078712.
T' 2 , a (0) = — 024929; T' 3, o (0) = 0.23991; T' 4, o (0) = — 1.68523 ; T' 5, a (0) = — 0.06753.