~irmeund Stoffi~bertragung
Wfirme- und Stofffibertragung 25, 377-380 (1990)
© Springer-Verlag 1990
Heat transfer to non-Newtonian power-law fluid past a continuously moving porous flat plate with heat flux B.P. Jadhav and B.B. Waghmode, Kothapur, India
Abstract. The analysis of heat transfer to the non-Newtonian powerlaw fluid flow past a continuously moving flat porous plate in presence of suction/injection with heat flux has been presented. We have obtained the solution using the method of successive approximations, starting with zero approximation. It has been observed that the results obtained for n = 1 are in good agreement with the corresponding results for Newtonian fluid. For various values of flow index n and the suction/injection parameter, temperature profiles and rate of heat transfer have been presented graphically. The effect of suction is to decrease in temperature and the rate of heat transfer, while reverse nature occurs for injection. W/irmeiibertragung bei Nicht-Newton'schen Flniden, die dem Potenz-Ansatz gehorchen, enflang einer stetig bewegten, flachen, por6sen Platte mit W/irmestrom Zusammenfassung. Hier ist die Berechnung der Wfirmetibertragung bei Nicht-Newton'schen Fluiden entlang einer sich stetig bewegenden, flachen Platte mit W/irmestrom dargestellt warden, bei der die Grenzschicht abgesaugt bzw. eingeblasen wird. Die L6sung erhielten die Verfasser mit dem Verfahren der sukzessiven Approximation, beginnend mit der Nullapproximation. Es ist festgestellt worde, da6 die erhaltenen Ergebnisse fiir n = 1 mit den Ergebnissen f/~ Newton'sche Fluide iibereinstimmen. Ffir verschiedene Werte des Str6mungsindexes n und des Absaug-/Einblasparameters sind die Temperaturprofile und die Wfirmetibertragungsraten graphisch dargestellt warden. Beim Absaugen ergibt sich eine Erniedrigung der Temperatur und der W/irmetibertragungsrate, w/ihrend genau das Gegenteil beim Einblasen eintritt.
Hisio-Tsung Lin and Yen-Ping Shih [8]. Payvar [9] has obtained an analytical solution of the heat transfer with constant heat flux taking into account the viscous dissipation. Sundaram and Nath [10] have considered the heat-transfer to power-law fluid in the thermal entrance region with viscous dissipation and constant heat flux. The effect of suction/injection is important in boundary layer control. Murthy and Sharma [11] studied the effect suction/injection on the flow past a continuously moving flat plate with heat flux. Here we have extended the problem of [11] to the power-law fluid.
2 Mathematical analysis Consider the flow of a steady, laminar, incompressible nonNewtonian power-law fluid past a continuously moving porous flat plate, x-axis is taken along the direction of the flow and y-axis normal to it. The plate is assumed to be moving with uniform velocity U. The governing boundary layer equations are 0.
"~
a.
a,_,/_a.\°
+"a--~ = -?~k
ay)
au a-~ + ~av = o
(11 (2)
1 Introduction The heat transfer problems for the flow of continuously moving flat plate were studied by Tsou et al. [1], Sakiadis [2]. Soundalgekar and Murthy [3] have considered the problem for isothermal plate. The flow and heat transfer to the micropolar fluid past a porous plate has been studied by Takhar and Soundalgekar [4]. They have shown that the effect of suction is to fall in temperature and velocity and increase in the rate of heat transfer. The solution of heat transfer to non-Newtonian powerlaw fluid past a static and moving plate has been considered by Acrious et al. [5], Schowalter [6], Lee and Ames [7] and
aT u ax
aT + v
a--y =
827 , ~ ay ~
where T - temperature of the fluid 7 = K/Q kinematic viscosity K - power-law fluid parameter ~o- density k a = the thermal diffusivity QCp' k = thermal conductivity Cp - sp. heat at constant pressure n - power law flow index
(3)
378
W/irme- und Stoff/ibertragung 25 (1990) where fl is a constant to be determined such that for the first approximation f~ (0) = 1 i.e. fl is a real root of the equation
with boundary conditions ~T v = V o(x),Sy
u=U, u=0,
q~ k aty=0
v=0, T=T~asy-~oo
~=Y\
(4)
k ( T - T~) ( U 2 - " ) 1/"+1 . -
-
(5)
.
\ 7x /
e¢
0x .
(6)
--i
n+l\
~rr2n--l\
(__],+l x" J
J~
_
1
n(n +
,
t! 2 -- n
1) j ° ( - f S )
"
(14)
Using the boundary conditions and Eq. (11) in integration we get,
We get /
03)
For the first approximation, we have it! __ _
e~
For similarity solution u - - -0 y ' v =
u = [)c,, v =
1
(t/f --f).
(7)
A (q) -
1 f"__ . . . . . f( f'l 2-n n(n+ly "-~" =0
(8)
0" + ~ -P~ ~
(9)
( f 0' - f ' 0) = 0
f~ (rl) = (A1 + A2) e (n- 2)#n - -
With the help of Eq. (7), the Eqs. (1) and (3) reduces to
+ Az) e(n-z)p. (n - 2) fl
+f,~
A3
A3 " 3 (n -- 3) f i e " - ) a,
where
--, 7
the modified Reynolds number.
( n + l ) Vo(x ) (Re) 1/"+1 U =f~
f ' (0) = i, 0'(0) = -
0(oo) =
o
(18)
where
(say)
f ' (oo) = 0
I,
z/3"+1' (17)
t 0(t/) = ~ e - "
With boundary conditions f(0)=-
1 Az=n(n+l)(n_2)
Using the values off1 and f{ in Eq. (9), we get the solution of (9), under given boundary conditions (10) as
g 2 -n X n
R~ -
(15)
(16)
1 A3 = n(n + 1 ) ( n - 3) z3 "+1"
-2
after
e (n-3)fln
j;
U | __},+~ the modified Prandtl number. c~x \ yx j
(14),
(A1 + A2) AB + - - ~ (n - 2)fl (n - 3)fl
At = n(n + l) ( n _ 2) 2 3 " '
where i/ g 2 _ . , ~
(12)
1 J)"' - n(n + 1)f~_ 1 ( - f [ ' - 1) 2-" where i = l , 2, 3 , . . .
~b(q) = (yx U z"- 1)1/,+ if(q)
qw
z
The different approximations can be obtained from
1fn+1
7-~-/
001) =
I
_
n(n+l)(n-2) a n(n+t)(n-2) 1 + n ( n ~ 1) (n - 3) 2 = 0.
where Vo(x) is suction velocity and qw is heat flux. Introducing the similarity transformation (u2-n~
f,~/~
B.+I _
(lO)
P~fl(~1) i(P~fl(tl)~a P,J; (t/) 2 - 2(n + 1) + ,~t \ 2 - ~ + 1 - ) / + (n +
I--~)"
where
f~, > 0 for suction and f~, < 0 for injection.
4 Discussions
3 Solution of the equations The Eq. (8) is solved using the method of successive approximations. Taking the zero th approximation satifying the boundary conditions (10) as 1 e_S, + 1
(11)
For various values of n andf~, the values of fl are obtained from the Eq. (12). Hence fl(t/) and fi(t/) can be known for various values of n and fw. Choosing particular values of Prandtl number P,, the temperature distribution and rate of heat transfer are obtained from (18). F r o m Table 1, we observe that for n = 1 and P~ = 0.7 the temperature is decreasing with increase in suction, while reverse nature occurs for injection. The Table 2 shows that the rate of heat transfer -0'(t7) is decreasing for increasing suction and increasing for injection.
B.P. Jadhav and B.B. Waghmode: Heat transfer to non-Newtonian power-law fluid 0.8
Table 1. Values of 0(t/) for n = l , Pr=0.7
tt/f~
--0.5
--0.2
-0.1
0.0
0.1
0.2
0.5
0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0
1.9587 1.3990 0.9872 0.6895 0.4777 0.3291 0.2259 0.1548 0.106t 0.0728 0.0500 0.0345 0.0238 0.0165 0.0115 0.0080 0.0056
1.7932 1.2765 0.9029 0.6360 0.4471 0.3140 0.2207 0.1552 0.1093 0.0771 0.0545 0.0386 0.0274 0.0194 0.0138 0.0098 0.0070
1.7410 1.2370 0.8749 0.6175 0.4355 0.3079 0.2171 0.1535 0.1088 0.0772 0.0548 0.0390 0.0278 0.0198 0.0141 0.0100 0.0073
1.6904 1.1983 0.8471 0.5986 0.4233 0.2997 0.2126 0.1510 0.t074 0.0765 0.0545 0.0389 0.0278 0.0199 0.0142 0.0102 0.0073
1.6410 1.1600 0.8193 0.5792 0.4102 0.2911 0.2070 0.1474 0.1051 0.0750 0.0535 0.0383 0.0274 0.0196 0.0140 0.0100 0.0072
1.5932 1.1226 0.7916 0.5595 0.3966 0.2818 0.2006 0.1430 0.1020 0.0728 0.0520 0.0372 0.0265 0.0190 0.0136 0.0097 0.0069
1.4587 1.01497 0.70998 0.4993 0.3526 0.2497 0.17697 0.1255 0.08899 0.0631 0.0447 0.03164 0.0224 0.0159 0.0112 0.0079 0.0056
379
m
0.7 0.6 ~ 0.5
t
I -0.5 ]1 -0.2 m - 0.1 5/ 0 --g 0.1 "7I 0.2 7II 0.5
~ 0.4 0.3 0.2 k
0.1 I
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Fig. 1. Temperature profiles for n=0.5, Pr= 10 Table 2. Values of -0'(r/) for n = l , P,=0.7
rl/f~
-0.5
-0.2
-0.1
0.0
0.1
0.2
0.5
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4,0 4.5 5.0 5,5 6.0 6.5 7.0 7.5 8,0
1.0000 0.7617 0.5645 0.4099 0.2931 0.2072 0.1453 0.1013 0.0704 0.0489 0.0339 0.0235 0.0163 0.0114 0.0079 0.0055 0.0039
1.0000 0.7465 0.5461 0.3945 0.2825 0.2013 0.1430 0.1015 0.0719 0.0510 0.0362 0.0257 0.0183 0.0130 0.0092 0.0066 0.0047
1.0000 0.7412 0.5397 0.3889 0.2784 0.1987 0.1415 0.1007 0.0717 0.0510 0.0363 0,0260 0.0186 0.0133 0.0095 0.0067 0.0048
t.0000 0.7357 0.5330 0.3830 0.2740 0.1956 0.1395 0.0995 0.0710 0.0507 0.0362 0.0259 0.0185 0.0t32 0.0095 0.0068 0,0048
t.0000 0.7300 0.5267 0.3768 0.2692 0.1921 0.1371 0.0979 0.0699 0.0494 0,0357 0.0255 0.0183 0.0131 0.0093 0.0067 0.0048
1.0000 0.7243 0.5197 0.3704 0.2641 0.1882 0.1343 0.0958 0.0684 0.0489 0.0349 0.0249 0.0178 0.0127 0.0091 0.0065 0.0047
1.0000 0.7062 0.4968 0.3497 0.2467 0.1744 0.1234 0.0874 0.0619 0.0438 0.0310 0.0220 0.0155 0.0170 0.0078 0.0052 0.0039
0.8 0.7 0.6
L
fw
-
0.5
~ 0.5
m Ig
-- 0.4
0
&3 0.2 0.1
0.2
The Fig. 1 shows the temperature profiles for suction and injection with Pr = 10 and n = 0.5. It is observed that increase in suction leads to decrease in temperature and increase in injection leads to increase in temperature. The Fig. 2 shows the temperature profiles for n = 1.5, Pr = 10 and various values of suction/injection. Temperature falls with increase in suction and reverse nature occurs with increase in injection. The Fig. 3 shows the comparison of temperature profiles for n = 0.5, n = 1,0, n = 1.5 with suction/injection. It is observed that the temperature increases with increasing value of flow index n for constant suction/injection. The Fig. 4 shows the rate of heat transfer for various of values of n with P, = 10, fw = --%0.5. It has been observed that the rate of heat transfer - 0 ' (~/) is increasing with n.
I
0./-,
0.6
0.8
1.0
1.2
1.l,
Fig, 2. Temperature profiles for n = 1.5, Pr= 10 5 Conclusions (1) Although the method employed gives approximate values of 0(~/) and O'(q), the nature of profiles can be well judged. (2) The effect of suction is to decrease the temperature and the rate heat transfer. While reverse nature occurs with injection. (3) The effect of increase in flow index n is to increase in temperature and the rate of heat transfer. (4) The effect of increase in P r a n d t l n u m b e r P, is to decrease in temperature and also decrease in the rate of heat transfer.
380
W~irme-und Stoff~bertragung 25 (1990) 1.0
0.8 0.7 0.8 0.6 n
o.s \w
0,3
,, -, ,,
fw
I
-o.s
fl
-o5
I~
1,5
0.6
0.5
71.00.5
' 0.4
0.1 0.2 O0
0.2
0.4
0.6
0.8
1.0
1.2
1./,
Fig. 3. Comparison of temperature profiles for P, = 10 0
0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
L
Fig. 4. Heat transfer profiles for P~= 10 References
1. Tsou, E K.; Sparrow, E. M.; Goldstein, R. J.: Flow and heat transfer in the boundary layer on a continuous moving surface. Int. J. Heat Mass Transfer 13 (1970) 129 2. Sakiadis, B. C.: Boundary layer behaviour on continuous solid surfaces; The boundary layer on a continuous flat surface. A.I.Ch.E.J. 7 (1961) 467 3. Soundalgekar, V. M.; Murthy T. V. R.: Heat transfer in flow past a continuously moving plate with variable temperature. W/irme-Stofffibertrag. 14 (1980) 91 4. Takhar H. S.; Soundalgekar, V. M.: Flow and heat transfer of a micropolar fluid past a porous plate. Indian J. Pure appl. Maths. 16 (5) (1985) 552 558 5. Acrious, A; Shah, M. J.; Peterson, E. E.: Momentum and heat transfer in laminar boundary layer flow of non-Newtonian fluids past external surfaces. A.I.Ch.EJ, 6 (1960) 312 6. Sehowalter, W. R.: The application of boundary layer theory to power law pseudoplastic fluids. Similar solution. A.I.Ch.EJ. 6 (I970) 24 7. Lee, S. Y.; Ames, W. E: Similarity solutions for non-Newtonian fluids. A.I.Ch.EJ. 12 (1966) 700
8. Lin Hisiao-Tsung; Shih Ven-Ping: Laminar boundary layer heat transfer to power law fluids. Chem. Eng. Commun. 4 (1980) 557 - 562 9. Payvar, P.: Asymptotic Nusselt number for dissipative nonNewtonian flow through ducts. Appl. Sci. Res. 27 (1973) 297 10. Sundaram, K. M.; G. Nath: Heat transfer to electrically conducting power law fluid in the thermal entrance region with constant heat flux. Proc. Indian Acad. Sci. 83A, 2 (1976) 50-64 11. Murthy, ~I:V. R.; Sharma, Y.V.B.: Heat transfer in flow past a continuously moving porous flat plate with heat flux. W/irmeStofffibertrag. 20 (1985) 39 42 B. P. Jadhav B. B. Waghmode Department of Mathematics Shivaji University Kolhapur 416 004 India Received June 5, 1989
