Acta Mechanica 147, 4 5 - 5 7 (2001)
ACTA M E C H A N I C A 9 Springer-Verlag 2001
F i r s t - o r d e r c h e m i c a l r e a c t i o n on flow past an impulsively started vertical plate with u n i f o r m heat and mass flux R. M u t h u c u m a r a s w a m y , S r i p e r u m b u d u r and P. Ganesan, C h e n n a i , I n d i a (Received February 9, 2000)
Summary. A finite-difference solution of the transient natural convection flow of an incompressible viscous fluid past an impulsively started semi-infinite plate with uniform heat and mass flux is presented here, taking into account the homogeneous chemical reaction of first order. The velocity profiles are compared with the available theoretical solution and are found to be in good agreement. The steady-state velocity, temperature and concentration profiles are shown graphically. It is observed that due to the presence of first order chemical reaction the velocity decreases with increasing values of the chemical reaction parameter. The local as well as average skin-friction, Nusselt number and Sherwood number are shown graphically.
List of symbols C!
CU C D Gc Gr 9 j!f K
Kz k Nux Nu Pr q Sc Shx Sh T!
Ts
Tw !
T t! t U0
concentration species concentration in the fluid far away from the plate species concentration near the plate dimensionless concentration mass diffusion coefficient mass Grashof number thermal Grashof number acceleration due to gravity mass flux per unit area at the plate dimensionless chemical reaction parameter chemical reaction parameter thermal conductivity dimensionless local Nusselt number dimensionless average Nusselt number Prandtl number heat flux per unit area at the plate Schmidt number dimensionless local Sherwood number dimensionless average Sherwood number temperature temperature of the fluid far away from the plate temperature of the plate dimensionless temperature time dimensionless time velocity of the plate
46 U, V X X
Y y
9 # /) 7-x
R. Muthucumaraswamy and P. Ganesan dimensionless velocity components in X, Y-directions, respectively velocity components in x, y-directions, respectively dimensionless spatial coordinate along the plate spatial coordinate along the plate dimensionless spatial coordinate normal to the plate spatial coordinate normal to the plate thermal diffusivity volumetric coefficient of thermal expansion volumetric coefficient of expansion with concentration coefficient of viscosity kinematic viscosity dimensionless local skin-friction dimensionless average skin-friction
1 Introduction The effects of a chemical reaction depend on whether the reaction is heterogeneous or homogeneous. This depends on whether they occur at an interface or as a single phase volume reaction. A few representative fields of interest in which combined heat and mass transfer plays an important role are design of chemical processing equipment, formation and dispersion of fog, distribution of temperature and moisture over agricultural fields and groves of fruit trees, damage of crops due to freezing, food processing and cooling towers. Cooling towers are the cheapest way to cool large quantities of water. They are a very common industrial sight, especially in power plants. The tower is packed with inert material, commonly with wooden slats. Hot water sprayed into the top of the tower trickles down through wood, evaporating as it goes. Air enters the bottom of the tower and rises up through the packing. In smaller towers, the air can be pumped with a fan; in larger ones, it is often allowed to rise by natural convection. Stokes [1] presented an exact solution to the Navier-Stokes equations which is the flow of a viscous incompressible fluid past an impulsively started infinite horizontal plate in its own plane. It is often called Rayleigh's problem in the literature. Following Stokes [1] analysis, Soundalgekar [2] was the first to present an exact solution to the flow of a viscous fluid past an impulsively started infinite isothermal vertical plate. Soundalgekar and Patil [3] have studied Stokes problem for an infinite vertical plate with constant heat flux. Many transport processes exist in nature and in industrial applications in which the simultaneous heat and mass transfer occur as a result of combined buoyancy effects of thermal diffusion and diffusion of chemical species. However, in nature, along with the free-convection currents caused by temperature differences, the flow is also affected by the differences in concentration. It is found useful in chemical processing industries such as food processing and polymer production. Soundalgekar [4] studied the problem of the flow past an impulsively started isothermal infinite vertical plate with mass transfer effects. Muthukumaraswamy and Ganesan [5] have analysed the above problem numerically. In this study, the plate is considered to be semi-infinite, and the dimensionless governing equations are solved using the implicit-finite difference scheme of Crank-Nicolson type. Soundalgekar et al. [6] have studied the flow past an impulsively started infinite vertical plate with constant heat flux and mass transfer. Das et al. [7] have studied effects of homogeneous first order chemical reaction on the flow past an impulsively started infinite vertical plate with constant heat
First-order chemical reaction
47
flux and mass transfer. The dimensionless governing equations are solved by the usual Laplace-transform technique. The present investigation, involving the simultaneous effects of heat and mass transfer, is concerned with a numerical study of transient natural convection flow past an impulsively started semi-infinite vertical plate which is subjected to uniform heat and mass flux. A homogeneous first order chemical reaction occurs between the diffusing species and the fluid. The fluids considered in this study are air and water. The governing equations are solved by an implicit finite-difference scheme of Crank-Nicolson type. In order to check the accuracy of our numerical results, the present study is compared with the available theoretical solution of Soundalgekar and Patil [3], and they are found to be in good agreement.
2 Analysis Here the flow of a viscous incompressible fluid past an impulsively started semi-infinite vertical plate with uniform heat and mass flux is considered. It is assumed that the effect of viscous dissipation is negligible in the energy equation and there is a first order chemical reaction between the diffusing species and the fluid. The x-axis is taken along the semi-infinite plate in the vertically upward direction, and the y-axis is taken normal to the plate. Initially, it is assumed that the plate and the fluid are of the same temperature and concentration in a stationary condition. At time t ~ > 0, the plate starts moving impulsively in the vertical direction with constant velocity u0 against the gravitational field. The temperature and concentration level near the plate are raised at a constant rate. It is also assumed that there exists a homogeneous first order chemical reaction between the fluid and species concentration. Then, under usual Boussinesq's approximation the unsteady flow past the semi-infinite vertical plate is governed by the following equations: Ou Ov + ~ = o, 0-7 uy On
Ou
(~)
Ou
-ovt + ~ Ux + v ~
02u = gg(T 1- T;) + gg*(c' - c')
(2)
+ ~ --
Oy2 '
OT ~ OT ~ OT I 02T I 0 ~-w+'~ T ; * + V o T v = ~ OY~ '
(3)
OC' OC' OC' 02C ' --at, + ~ ~ + ~ ~-v = D ~ - KzC'.
(4)
The initial and boundary conditions are: t' <_O:
~=0,
v-0,
r'--z;o,
t/>0:
u=uo,
v=O,
u=0,
T'=T~,
u + O,
T' + 7 " ,
OT 1 Oy
c'=c:,
q k '
C'=G~ C' --+ C L
OC' Oy
j" D
at
y=O,
at
x=0,
as
y~oc.
(5)
48
R. Muthucumaraswamy and P. Ganesan
On introducing the following nondimensional quantities: y : X'll,0
Y=
P
T-
Y~O 12
U=--,
V=--,
~0
Gr-
qP
u
,
~0
~Zo 2
t-
%0
C - C' - C ~'
g~qv2
,
/]
Gc
-
g /3*v2j"
(G)
\Duo] Pr=-,
12
Sc
(2
v D '
vKl
K=
%0 2
,
Eqs. (1) to (4) are reduced to the following nondimensional form: OU
OV
(7)
ax ~ = 0 , OU OU 0--~+ U ~ +
OU __02U V ~ = GrT + GcC + Oy2 '
(8)
OT OT OT 1 02T O~ ~- U ~ + V OY - Pr OY 2 '
(9)
OC OC OC 1 02C O~ + U ~ + V OY - Sc OY ~
(lO)
KC.
The corresponding initial and boundary conditions in nondimensional form are: t_<0:
U=0,
V=0,
T=0,
t>0:
U=I,
V=0,
OT OY
U=0,
T=0,
C=0,
U-+0,
T-+0,
C=0, OC OY
1,
-1,
C--+0
at
Y=0,
at
X=0,
as
Y-+oc.
(11)
3 Finite-difference technique
The unsteady, non-linear coupled Eqs. (7)- (10) with the condition (11) are solved by employing an implicit finite-difference scheme of Crank-Nicolson type. The finite-difference equations corresponding to Eqs. (7)- (10) are as follows: U~L1, j Jr- u,n+I , i,j-1
[[;n+l. __ u n + l U n. L~ i,j i 1,j -F %3
un+l i 1,j-1
Jr- UTiZj ,
1
- - V ~n- l , j - 1 ]
4AX 9 -- V: n + l + [V/.~+1 i,j-1 Jr- V//'; - V/~_I 1 = O,
(12)
2AY [~]-n+l vi, j
~
{fn+l _ un+l
n
- - U~,j]
~v5
~i,j
i-l,j
2~x
G C [(~'~-1
c ~ T~+ 1 + ri,~ ] + T -
= 5-
un
+
n
z,3 -
~-~,, + c:i,51 +
n
g~-l,j]
+ v~
Tpz+l yTn+l Un un ~i,j+l - - ~i,j-1 -F i,j+l -- i,j-1]
4AY
[lyn+l __ 2U~i;1 ~ _ / T n + l
k~i,J - 1
-- ~i,j+l
-}-
un
i,j 1
_ 2~f}a,. jr_ S n . + l ] ,3
,3
2(~y) ~
(la/
First-order chemical reaction
--
-- r i - l , j
At
~- U~,j
r~Tz+l [~i,j_l
1
49
s
Tn+l ~/d+l
zd -- T i - l , j ]
__
2AX
Tn+l Tn n i,j-1 + i,j+l - T i , j - 1 ] 4AY
2TP+l %7 -F Tn+l i , j + l - F Tni,j-1 -- 2T~,j § Ti,j+l] n 2(AY) 2
_
Pr
(14)
The thermal boundary condition at Y = 0 in the finite-difference form is Tn+l
1 L~i:I
_L
Tn
_ Tn+l
i,1
i, 1 --T,),
n
1]
2AY
2
(15)
= --1.
At Y = 0 (i.e., j = 0), Eq. (14) becomes
[T,~+I. :~,o
n
-
r,pn+z ~:~,o
m;,o]
At
T/,+lo, + T~,o _ T7~_1,o] ,
_
u},%
F
2zxx
~+1 + <7, - ~ - 2T5 + T&] 1 L~~+ i _1 l _ 2mn+~ i,0 -~ T i,1
Pr
(16)
2(AY) 2
After eliminating Tn+ti,-1+ T~, 1 using Eq. (15), Eq. (16) reduces to the form: Tn+l
~7,0
--
Ti,%]
at
r~n+l
~ u7,%
m n + l - - T i,O n+l L~i,1
1
_
[~{,o
Pr
Tn+l
i-1,0 _L '~T~/7~0__
2~x +Tir~l-Tin.o + 2 A Y ]
(17)
'
r,,--,n+l ('~n+l _ c n + l cn . n ~i,j i 1,j ~- z,3 - - C~--l,j] ~_ ~/~n [ ~ i , j + l
"n
~+1 - C},j] L[C~ %7 -
+ u<'~
1 ~7i ,nj+- 1l Sc
n
,
(Ay) 2
at
T~ 1,0]
2aX
w
- - 2 C ~ ,,; 1 @ {~Ti ,nj ++ll Jr- C i,j-1 n -
2 C ~ j Jr- C ~ , j + I ]
2(AY) 2
_ cn+l
i,j-1 -~-
Cn n i,j+i -- C~,j-1]
4AY K 2
( ( T n T 1 @ C~,j) "-w
,
(18) "
The concentration boundary condition at Y = 0 in the finite-difference form is
r~+:t + C~1,
1 L~i,1
_
cn+l <-1
-
n
0~'-1] : - 1 .
(19)
2z~Y
At Y = 0 (i.e., j = 0) Eq. (18) becomes [~n+l
~,o
~
- C;,o]
At
n
F~n+l L~i,0
cn+l C~ i - 1 , 0 Jr- i,O -- C ~ - 1 , 0 ]
~- U~,~
2AX
_ 20n+1 i rrTn+l t~i_i i,0 -~ C~+1 i,1 + C[I, - 1 - 2Ci% + C~1] , Sc
2(AY) 2
K (C~,ol + q ,n0 ) . 2
(20)
After eliminating Cn+a<-i+ Cn<-i using Eq. (19) and (20) reduces to the form: [~n+l
~i,0
n
-- C~,0]
At 1
• Sc
n
#- U~,~
rg~n+l
[~%0
_
cn+l
i 1,0 +
Cn
n
i,0 -- C ~ - 1 , 0 ]
2AX
r,v~+z _ C~+1 + C~,1 _ C~<0 + 2AY] [~/~1 i,0 , (Ay) 2
K l~n+ 1 + C~.0) 2 ~,0 , "
(21)
The region of integration is considered as a rectangle with sides Xm~x(= 1) and Ymo.~(= 14), where Yma~ corresponds to Y = oc, which lies very well outside the momentum,
50
R. Muthucumaraswamy and P. Ganesan
energy and concentration boundary layers. A 20 x 56 uniform grid for the fluid flow was selected. These grid sizes were chosen, after considering several grids, to ensure grid independence of the results while obtaining the fastest convergence. The maximum of Y was chosen as 14 after some preliminary investigations so that the last two of the boundary conditions (11) are satisfied within the tolerance limit 10 -'5. Here, the subscript i designates the grid point along the X-direction, j along the Y-direction, and the superscript n along the t-direction. During any one time step, the coefficients U~~. and E~. appearing in the difference equations are treated as constants. The values of C, T, U, V at the time level (n + 1) using the known values at the previous time level (n) are calculated as follows: The finite-difference equations (18) and (21) at every internal nodal point on a particular /-level constitute a tridiagonal system of equations. Such a system of equations is solved by using Thomas' algorithm as discussed in Carnahan et al. [8]. Thus, the values of C are known at every internal nodal point on a particular i at the (r~ + 1) ~h time level. Similarly, the values of T are calculated from Eqs. (14) and (17). The numerical values of U are evaluated using the finite-difference equation (13). Finally, the values of V are calculated explicitly using Eq. (12) at every nodal point at the particular/-level at the (r~ + 1) th time level. This process is repeated for various/-levels. Thus, the values of C, T, U and V are known at all grid points in the rectangular region at the (r~ -- 1) th time level. Co/nputations are carried out until the steady-state is reached. The steady-state solution is assumed to have been reached when the absolute difference between the values of U as well as the temperature T and concentration C at two consecutive time steps are less than 10 .5 at all grid points. After experimenting with few sets of mesh sizes, they have been fixed at the level zSX = 0.05, B Y = 0.25, and the time step ~A = 0.01. In this case, the spatial mesh sizes are reduced by 50% in one direction, and then in both directions, and the results are compared. It is observed that, when the mesh size is reduced by 50% in the X-direction and Y-direction the results differ in the fourth decimal places. Hence, the above mentioned sizes have been considered as appropriate mesh sizes for calculation. The local truncation error is O ( A t 2 + A Y 2 + A X ) , and it tends to zero as At, A X and A y tend to zero. Hence, the scheme is compatible. The finite-difference scheme is unconditionally stable as discussed in Muthukumaraswamy and Ganesan [5]. Stability and compatibility ensures convergence.
4 Results and discussion Representative numerical results for the uniform surface heat and mass flux will be discussed in this section. In order to ascertain the accuracy of the numerical results, the present study is compared with the previous study. The numerical results for the special case of uniform surface heat flux (Gc = 0) are compared with the available theoretical solution in the literature. The velocity profiles for Gr = 5, 10, Gc = 0, Pr = 0.71, ~ = 0.2 and K = 0 are compared with the available theoretical solutions of Soundalgekar and Patil [3] (corresponding to rl = Y / 2 v ~ ) as shown in Fig. 1. It is observed that the agreement with the theoretical solution of velocity is excellent. The finite-difference equations (18) and (21) can be adjusted to meet these circumstances if one takes (i) K > 0 for the destructive reaction, (ii) K < 0 for the generative reaction, and (iii) K = 0 for no reaction. The transient velocity profiles for different chemical reaction parameters and Schmidt numbers are shown in Fig. 2. The velocity profiles presented are those at X = 1.0. It is
First-order chemical reaction
1'0
,
\ \~
-9
0.8
51 I
Present resutt Soundatgekar 8, Patit [3]
10 5
0-6
Gc:O Pr:O.71
\\
U
0.4
0-2
0
1
0
1
u
2
Fig. 1. Comparison of velocity profiles
2.2! 2.0
1.5 U 1.0
0-5
2
3
4
Fig. 2. Velocity profiles at X = 1.0 for different Sc and K (* steady-state value)
observed that for P r = 0.71, G r = 2, Gc = 5, Sc = 0.6 and K = - 2 . 0 , the velocity increases with time, reaches a temporal m a x i m u m a r o u n d time t -- 0.95 and becomes steady at time - 6.7. It is observed that the velocity increases during generative reaction a n d decreases in destructive reaction. It is clear that the velocity increases with decreasing values of the Schmidt number or the chemical reaction parameter. The time taken to reach the steady-state increases with increasing Schmidt number or chemical reaction parameter. However, the time required for the velocity to reach steady-state depends u p o n b o t h the Schmidt number and the chemical reaction parameter. This shows that the contribution o f mass diffusion to the b u o y a n c y force increases the m a x i m u m velocity significantly. The steady-state velocity pro-
52
R. Muthucumaraswamy and P. Ganesan
2.25 Gr Gc Pr t 5 100.714.3 5 5 0'71 7"8 2 50.7111.6 5 7.0
2-0
1-5 k-4V
Sc = 0.6
u 1-0
0.5
0 1.25
I
I
i
1
2
3
Y
i
/,
Fig. 3. Steady-state velocity profiles at X = 1.0 for different Gr, Gc and Pr
i
K Pr t 2.0 0.7113-4" 0.2 0.71 11-6" - 2 ' 0 0'71 6"7* 0.20"710'41 0"2 0"71 0"18 0.2 7"0 8.6"
1.0
~/~F
Gr=2
0.5 \
0
Gc =5
2 Y
3
3"5
Fig. 4. Temperature profiles at X = 1.0 for different K and Pr (* steady-state value)
files for different thermal Grashof number, mass Grashof number and Prandtl number are shown in Fig. 3. It is observed that the velocity increases with increasing thermal Grashof number or mass Grashof number and decreases with increasing Prandtl number. The transient and steady-state temperature for different values of the Prandtl number and chemical reaction parameter are shown in Fig. 4. The effect of the Prandtl number is very important in temperature profiles. The thermal-boundary-layer thickness decreases with increasing Prandtl number. It is observed that the temperature increases with increasing values of the chemical reaction parameter.
First-order chemical reaction
53
1-5
1.0
C
0-5
0
1
2
3 Y
3-5
Fig. 5. Concentration profiles at X 1.0 for different K and Sc (* steady-state value)
The effects of the chemical reaction parameter and the Schmidt number are very important in the concentration field. The transient and steady-state concentration profile for different chemical reaction parameters and Schmidt numbers are shown in Fig. 5. The reaction reduces the local concentration, thus increasing its concentration gradient and its flux. It is observed that there is a fall in concentration due to increasing values of the chemical reaction parameter or Schmidt number. Knowing the velocity, the temperature and the concentration field, it is customary to study the rate of shear stress, the rate of heat transfer and the rate of concentration in their transient and steady-state conditions. The dimensionless local as well as average values of the skin-friction, the Nusselt number and the Sherwood number are given by the following expressions:
Tx
--
~
y=0 ~
1
"Y:--
~
0
Y=0
Nux = - X
=0 ,
(24)
N~ = _ 111
:o dX,
(25)
o
R. Muthucumaraswamy and P. Ganesan
54 2-0
Pr = 0.71 Gr = 0.4
1"5
Gr = 0.2
"IX Kc 2.0 -2-0 2-0 0,2
1"0
Sc 0-6 0-6 0.16 0.16
0.5
I
I
I
o2
0.4
0'-6
t~,
0.8
~,
%
I
1-o
X 1-0
,
Fig. 6. Local skin-friction
, Gr Gc Sc 5 10 0.6 5 5 0.6 2 5 0.6 2 5 2.0
0"75
NU x 0-5
K-- 0.2 Pr =0'71 0.25
0.2
Shx=-X/
=
Cy=o
0.4
'
j[(V/ol Cy:o- J
0
dX.
X
0-6
0.g
1-0 Fig. 7. Local Nusselt number
(26)
(27)
First-order chemical reaction
55
1.25 K
Sc
2
0.6 - - x
0.2 o.s-x\
1.0
/
0-6 0"6
- 1
-2
/
/t
.4
Shx
0.5 / /
///"
/
/
Gr --2
G~=s
, 0
0-2
,
0.4
Pr =011
0.6
0.8
1.0
X
Fig. 8. Local Sherwood number
1.5
~
.
.
.
.K
Sc
.2.0
~\\\\
1-2s
//i-2.o
\\
0.6
/ / / - 2.0 o.16
1"0
Pr : 0.71 Gr =0.4 Gc = 0.2
0-7
0-5
0
, 02
, 0.4
, 0.6
0.8 1:
110
1.2 Fig. 9. Averageskin-friction
The derivatives involved in Eqs. (22) to (27) are evaluated using a five-point approximation formula, and then the integrals are evaluated using the Newton-Cotes closed integration formula. The local skin-friction values are evaluated from Eq. (22) and plotted in Fig. 6 as a function of the axial coordinate X. The local skin-friction decreases as X increases. It is observed that the local shear stress increases with increasing the value of the chemical reaction parameter or Schmidt number. The local Nusselt number for different values of the thermal Grashof number and mass Grashof number are studied graphically in Fig. 7. It is observed that the rate of heat transfer increases with increasing values of the thermal Grashof number or mass Grashof number. The local Sherwood number for different values of the Schmidt number and the chemical reaction parameter are shown in Fig. 8. It is observed that the local Sherwood number increases with increasing Schmidt number. This is true because the concert-
R. Muthucumaraswamy and P. Ganesan
56 1.8
I ~
I
I
\\ \~,
L
Gr 5 5 2
///-
1.6
Gc 10 5 5
Sc 0.6 0.6 0.6
NU
1.4 PF : 0.71 k =0-2
1.2 0
I 0,2
~ 0,4
~ 0'6
[ 0~
Fig. 10. Average Nusselt number 2"0
~~
'
/-/.,~
Sc
K'
O.6 2-0 0.6 0.2 0.6 -1.0
Gr:2 Gc: S Pr= 0.71
1'5
1.0
0.5
~ o'.2
0.16
0'.4
0-2
;6
&
,.o Fig. 11. Average Sherwood number
tration profiles decrease with increasing values of Sc near the plate. The rate of concentration increases during destructive reaction and decreases in generative reaction. The effects of Gr, Ge, Se and the chemical reaction parameter on the average values of the skin-friction, the Nusselt number and the Sherwood number are shown in Figs. 9, 10 and 11, respectively. The average skin-friction increases with increasing K or Sc throughout the transient period. Theaverage Nusselt number increases with decreasing Sc, and it increases with increasing Gr or Gc. The average Sherwood number increases with increasing the chemical reaction parameter or the Schmidt number.
First-order chemical reaction
57
5 Conclusions A detailed numerical study has been carried out for the flow past an impulsively started semiinfinite vertical plate with uniform heat and mass flux. A homogeneous first order chemical reaction has been considered in this analysis. The dimensionless governing equations are solved by an implicit finite-difference scheme of C r a n k - N i c o l s o n type. The fluids considered in this study are air and water. A comparison between the present numerical results and the available theoretical solution for the case of uniform surface heat flux is also made. The agreement between the two is found to be very good. It is observed that the velocity and concentration increases during generative reaction and decreases in destructive reaction. It is found that the n u m b e r o f time steps to reach the steady-state depends strongly on the chemical reaction p a r a m e t e r or the Schmidt number. The reaction reduces the local concentration, thus increasing its concentration gradient and its flux. Moreover, because chemical reaction rates can be very fast, the increase in mass transfer can be large.
References [1] Stokes, G. G.: On the effect of internal friction of fluids on the motion of pendulums. Cambridge Phil.Trans. IX, 8 106 (1851). [2] Soundalgekar, V. M.: Free convection effects on the Stokes problem for an infinite vertical plate. ASME J. Heat Transfer 99, 499- 501 (1977). [3] Soundalgekar, V. M.: Effects of mass transfer and free convection on the flow past an impulsively started vertical plate. ASME J. Appl. Mech. 46, 757-760 (1979). [4] Soundalgekar, V. M., Patti, M. R.: Stokes problem for a vertical plate with constant heat flux. Astrophys. Space Sci. 70, 179 182 (1980). [5] Muthukumaraswamy, R., Ganesan, P.: Unsteady flow past an impulsively started vertical plate with heat and mass transfer. Heat Mass Transfer 34, 187-193 (1998). [6] Soundalgekar, V. M., Birajdar, N. S., Darwekar, V. K.: Mass transfer effects on the flow past an impulsively started infinite vertical plate with variable temperature or constant heat flux. Astrophysics Space Sci. 100, 159-164 (1984). [7] Das, U. N., Deka, R. K., Soundalgekar, V. M.: Effects of mass transfer on flow past an impulsively started infinite vertical plate with constant heat flux and chemicai reaction. Forschung im Ingenieurwesen Engineering Research 60, 284-287 (1994). [8] Carnahan, B., Luther, H. A., Wilkes, J. O.: Applied numerical methods. New York: Wiley 1969. [9] Byron Bird, R., Warren, E. S., Lightfoot, E. N.: Transport phenomena. New York: Wiley 1960. Authors' addresses: R. Muthucumaraswamy, Department of Mathematics and Computer Applications, Sri Venkateswara College of Engineering, Sriperumbudur 602 105; P. Ganesan, Department of Mathematics, Anna University, Chennai 600 025, India