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Group solution of a time dependent chemical convective process
Applied Mathematics and Computation 215 (2009) 1671–1684
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Group solution of a time dependent chemical convective process M.M. Kassem *, A.S. Rashed Mathematics and Physics Department, Faculty of Engineering, Zagazig University, Egypt
a r t i c l e
i n f o
Keywords: Time dependent chemical convection Group transformation method Perturbation method
a b s t r a c t The time dependent progress of a chemical reaction over a flat vertical plate is here considered. The problem is solved using the two parameter group method which reduces the number of independent by one and leads to a set of nonlinear ordinary differential equation. The behavior of the process is numerically investigated for the chemical reaction order n – 1 and different Schmidt numbers. As the problem shows a singularity at n = 1, the nonlinear system of ordinary differential equation resulting from the transformation of the problem are analytically solved through the perturbation method. The velocity and concentration of chemicals based on the analytical and numerical solutions are presented, compared and discussed. Ó 2009 Elsevier Inc. All rights reserved.
1. Introduction Recently Rashed and Kassem [12] analyzed the problem of steady stated chemical coatings past a vertical plate. Here, the study is concerned with time dependant natural convection. This problem reported by Levich [9] and Gebhart and Pera [5] concerns the immersion of a plate is in a fluid solution having a concentration c0 ðx; tÞ > 0. When the plate touches the solution a chemical reaction takes place inducing a change of concentration and implying density gradients in the presence of gravitational field [3]. Ganesan and Rani [4] analyzed the diffusion of chemically reactive species for a convective unsteady flow along a vertical cylinder using an implicit finite difference method. Makinde [10] investigated a convective flow with thermal radiation and mass transfer past a moving vertical porous plate and assumed a time-dependency for the vertical velocity. The resultant similarity equations were solved numerically using a superposition method. Ibrahim et al. [6] analytically derived the heat and mass transfer of a chemical convective process assuming an exponentially decreasing suction velocity at the surface of a porous plate and a two terms harmonic function for the rest of the variables. The mathematical technique used in the present analysis is a two parameter group transformation of the variables. This method developed by Morgan [11] reduces the number of variables by one and generates a set of ordinary differential equations. This method adopted by Kassem [7,8] and Abdel Malek et al. [1] proved to be efficient for an analysis of various flow problems. In the present work we reduce the field equations and related boundary conditions through a two parameters group. This transformation results in a system of nonlinear differential equations with appropriate boundary conditions. The obtained system of equations shows a singularity at the chemical reaction order n = 1. At this value the concentration profile of chemicals in the boundary layer is analytical derived using the perturbation method. The behavior of the flow is then numerically investigated for n – 1 and different Schmidt numbers using the shooting method and the results are plotted. For n = 1 the concentration is analytically evaluated and plotted for various chemical molecular velocities D.
* Corresponding author. E-mail address: [email protected] (M.M. Kassem). 0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.07.018
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Nomenclature Latin characters unity elements a1 ; a2 c dimensional species concentration C non-dimensional concentration concentration next to the plate c1 ambient concentration c1 D chemical molecular diffusivity. 0 vertical velocity after transformation F g gravitational acceleration G group k chemical rate constant n chemical reaction order Q, T real valued coefficients S subgroup Sc Schmidt number m/D u velocity in x direction v velocity in y direction x vertical axis y horizontal axis Greek symbols b volumetric coefficient of expansion with concentration m ¼ l=q kinematic viscosity of fluid q fluid density w stream function g similarity variable
2. Mathematical formulation The following study is concerned with time dependant convection and diffusion within a thin boundary layer adjacent to an vertical plate immersed in a fluid, having a chemical reaction of order n. For this, it is convenient to consider an idealized system illustrated in Fig. 1 and composed of a semi infinite plate set in a fluid of infinite extent. The natural convection is described by the following equations;
x
g
u
c
c
u
v
y Fig. 1. Illustration of the flow over a semi-infinite horizontal plate.
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@u @ v þ ¼ 0; @x @y
1673
ð2:1Þ 2
@u @u @u @ u þu þv gbc m 2 ¼ 0; @t @x @y @y
ð2:2Þ
@c @c @c @2c n þu þv þ kc D 2 ¼ 0; @t @x @y @y
ð2:3Þ
subjected to the following initial and boundary conditions
uðx; y; 0Þ ¼ u0 ðx; yÞ;
v ðx; y; 0Þ ¼ v 0 ðx; yÞ;
uðx; 0; tÞ ¼ v ðx; 0; tÞ ¼ 0;
cðx; y; 0Þ ¼ 0;
ð2:4Þ
cðx; 0; tÞ ¼ c0 ðx; tÞ;
ð2:5Þ
lim uðx; y; tÞ ¼ lim v ðx; y; tÞ ¼ lim cðx; y; tÞ ¼ 0;
y!1
y!1
ð2:6Þ
y!1
Eq. (2.1) is eliminated and concentration at the wall (2.5) is normalized through the transformations
@w ; @x cðx; y; tÞ Cðx; y; tÞ ¼ : c0 ðx; tÞ
u¼
@w ; @y
v ¼
ð2:7Þ ð2:8Þ
In this case the flow equations (2.1)–(2.6) reduce to
@ 2 w @w @ 2 w @w @ 2 w @3w gbc0 C m 3 ¼ 0; þ 2 @y @t @y @x @y @x @y @y
ð2:9Þ 2
@C C @c0 @w @C C @w @c0 @w @C @ C n1 þ þ þ kc0 C n D 2 ¼ 0; þ @t c0 @t @y @x c0 @y @x @x @y @y @wðx; y; 0Þ @wðx; y; 0Þ ¼ u0 ðx; yÞ; ¼ v 0 ðx; yÞ; Cðx; y; 0Þ ¼ 0; @y @x @wðx; 0; tÞ @wðx; 0; tÞ ¼ ¼ 0; Cðx; 0; tÞ ¼ 1; @y @x @w @w ¼ ¼ 0; lim Cðx; y; tÞ ¼ 0: y!1 @y @x y!1
ð2:10Þ ð2:11Þ ð2:12Þ ð2:13Þ
y!1
3. Group formulation of the problem From the group definition [2]
G : S ¼ Q s ða1 ; a2 ÞS þ T s ða1 ; a2 Þ;
ð3:1Þ
where G is a two parameter group, S and S stand for the system variables (t, x; w, C and c0 Þ before and after transformation, Q s ; T s are real valued coefficients at least differentiable in the group parameters ða1 ; a2 Þ. First and second partial derivatives are defined as;
Si ¼ Sij ¼
s Q Si Qi s Q QiQj
9 > = i; j ¼ x; y; ; Sij >
ð3:2Þ
where S stands for the dependent variables ðw; C and c0 Þ. From the above definitions (2.9) is transformed to
! @2w @2w @2w @w @w @3w @ 2 w @w @ 2 w @w @ 2 w @3w gbc0 C m 3 ¼ H1 ða1 ; a2 Þ gbc0 C m 3 ; þ þ @ x @ y @ x @ y 2 ; @t @ y @y @t @y @x @y @x @y2 @y @y @y ! ! ! @2w Qw Qw @w @ 2 w Qw Qw @w @ 2 w Qw @3w c0 C gbQ Q c C m þ 0 y x y x @y @t @y @x @y Q Q Q Q ðQ y Þ2 @x @y2 ðQ y Þ3 @y3 ! @ 2 w @w @ 2 w @w @ 2 w @3w þ R1 ¼ H1 ða1 ; a2 Þ gbc0 C m 3 ; þ @y @t @y @x @y @x @y2 @y
Qw Q yQ t
ð3:3Þ
!
ð3:4Þ
where
R1 ¼ gbðQ c0 T C c0 þ T c0 Q C C þ T c0 T C Þ:
ð3:5Þ
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The invariant transformation of (3.4) implies that H1 ða1 ; a2 Þ ¼ 1 and R1 ¼ 0 giving;
T c0 ¼ T C ¼ 0
ð3:6Þ
a comparison of coefficients on both sides of (3.4) results in;
Qw ðQ w Þ2 Qw ¼ Q C Q c0 ¼ y 3 : y t ¼ x y 2 Q Q Q ðQ Þ ðQ Þ
ð3:7Þ
Following the same procedure (2.10) is transformed leading to;
Q C Q wQ C QC c0 n1 ðQ C Þn ¼ y 2 ; y x ¼ ðQ Þ t ¼ Q Q Q ðQ Þ
ð3:8Þ
similarly the transformation of initial and boundary conditions implies
Q C ¼ 1;
Ty ¼ Tt ¼ 0
ð3:9Þ
and (3.7)–(3.9) ratios reduce to
Q t ¼ ðQ y Þ2 ;
Q c0 ¼ 1;
Q w ¼ ðQ y Þ3 ;
Q x ¼ ðQ y Þ4 ;
ð3:10Þ
finally the system group structure is written as
8 8 y 4 x > > > < x ¼ ðQ Þ x þ T > > y > ¼Q y G1 y > > > > > < : t ¼ ðQ y Þ2 t 8 ; G y 3 w > > < w ¼ ðQ Þ w þ T > > > > > > > G2 > C ¼ C > : : c0 ¼ c0
ð3:11Þ
where G1 and G2 are subgroups describes the independent and dependant variables and dashes stand for their transformation. 3.1. Group transformation of the system variables The flow equations order is reduced by one if it satisfies Morgan’s theorem [11]
9 > ðiÞ u ½ai uðiÞ þ aiþ1 @@uðiÞ ¼ 0> > = i¼1 ; 6 > P > ðiÞ u ½bi uðiÞ þ biþ1 @@uðiÞ ¼0> ; 6 P
ð3:12Þ
i¼1
ðiÞ; i ¼ 1; 2; . . . 6 stand for the six system variables ðx; y; t; w; C; c0 Þ before and after transformation where uðiÞ; u
@Q si ða1 ; a2 Þ ; @a1 si @Q ða1 ; a2 Þ ; bi ¼ @a2
ai ¼
@T si ða1 ; a2 Þ ; @a1 si @T ða1 ; a2 Þ ¼ : @a2
aiþ1 ¼
ð3:13Þ
biþ1
ð3:14Þ
3.2. Transformation of the independent variables The similarity variable g(x,y,t) is obtained applying (3.12)
@g @g @g þ a3 y þ a5 t ¼ 0; @x @y @t @g @g @g ðb1 x þ b2 Þ þ b3 y þ b5 t ¼ 0: @x @y @t
ða1 x þ a2 Þ
Eliminating
@g @y
and
@g , @x
ð3:15Þ ð3:16Þ
respectively, from (3.15) and (3.16);
@g @g ðk13 x þ k23 Þ þ k53 t ¼ 0; kij ¼ ai bj aj bi ; @x @t @g @g ðk31 x þ k32 Þy þ ðk51 x þ k52 Þt ¼ 0; @y @t then invoking the group structure described in (3.11) we obtain;
ð3:17Þ ð3:18Þ
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a5 ¼ 2a3 ;
ð3:19Þ
b5 ¼ 2b3 ;
ð3:20Þ
i.e. k53 ¼ a5 b3 a3 b5 ¼ 0 and (3.17) reduces to
ðk13 x þ k23 Þ
@g ¼ 0; @x
ð3:21Þ
@g ¼ 0; @x g ¼ gðy; tÞ;
ð3:22Þ ð3:23Þ
from (3.23) in (3.18) we obtain
g ¼ ypðtÞ;
ð3:24Þ
pðtÞ ¼ atb ;
ð3:25Þ
where
without loss of generality let a = 1 while b will be determined later. 3.3. Transformation of the dependant variables Transformation of C and c0 is obtained directly from (3.11) group structure
Þ ¼ CðgÞ; Cðx; y c0 ðx; tÞ ¼ Cðx; tÞ;
ð3:26Þ ð3:27Þ
C(x,t) will be determined later. Similarly w is transformed applying (3.12)
x;y;t;wÞ ða1 x þ a2 Þ @g3 ðx@x;y;t;wÞ þ a3 y @g3 ðx@y;y;t;wÞ þ a5 t @g3 ðx@t;y;t;wÞ þ ða7 w þ a8 Þ @g3 ð@w ¼ 0;
x;y;t;wÞ ¼ 0; ðb1 x þ b2 Þ @g3 ðx@x;y;t;wÞ þ b3 y @g3 ðx@y;y;t;wÞ þ b5 t @g3 ðx@t;y;t;wÞ þ ðb7 w þ b8 Þ @g3 ð@w
ð3:28Þ
the solution of (3.28) gives
¼ / ðw=xðxÞÞ ¼ FðgÞ ; t; wÞ g 3 ðx; y 1
ð3:29Þ
and for /1 ¼ 1 the flux w transforms to
x; y ; tÞ ¼ xðx; tÞFðgÞ: wð
ð3:30Þ
Replacing for the system variables (3.26), (3.27) and (3.29) after transformation in (2.9) and (2.10) we obtain
2 3 2 2 d F d F d F dF dF A1 g 2 þ A2 F 2 ðA1 þ A3 Þ A2 þ A4 C ¼ 0; 3 dg dg dg dg dg
ð3:31Þ
2
d C m dC m dC dF g þ F A5 C A 6 C A7 C n ¼ 0; dg2 D dg D dg dg
ð3:32Þ
where
dp=dt
@ x=@x ; A2 ¼ ; mp3 mp @ x=@t gbC ; A4 ¼ ; A3 ¼ mxp2 mxp3 @ C=@t x @C A5 ¼ ; A6 ¼ ; DCp2 DpC @x A1 ¼
A7 ¼
kCn1 : Dp2
ð3:33Þ ð3:34Þ ð3:35Þ ð3:36Þ
In order to reduce (3.31) and (3.32) to a system of linear ordinary equations, the A’s must be function of g or constants. First let in (3.36) A7 ¼ Dk . This leads to; 2
2
Cðx; tÞ ¼ ðpðtÞÞn1 ¼ ðtb Þn1 ; where C(t) is the concentration of the fluid next to the plate. As C(t) is not function of ‘x’ hence
ð3:37Þ
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A6 ¼ 0;
ð3:38Þ
2b A5 ¼ t ð2bþ1Þ : ðn 1ÞD
ð3:39Þ
A5 is constant if b = 1/2.
A5 ¼
1 ; Dðn 1Þ
ð3:40Þ
thus (3.24) reduces to;
y t
g ¼ pffiffi :
ð3:41Þ
For A4 ¼ gb m we obtain 3n5
xðx; tÞ ¼ xðtÞ ¼ t2ðn1Þ ; n – 1;
ð3:42Þ
where x(t) is part of the stream function wðgÞ described in (3.30). Replacing for x; p; C in (3.33) and (3.34) in the remaining constants we obtain
A3 ¼
3n 5 ; 2ðn 1Þm
n – 1;
ð3:43Þ
A2 ¼ 0;
ð3:44Þ
1 A1 ¼ : 2m
ð3:45Þ
Substituting for the constants in (3.31) and (3.32)
3 2 d F g d F 2n 4 dF gb þ þ C ¼ 0; dg3 2ðn 1Þm dg m 2m dg2
ð3:46Þ
d C g dC 1 k þ þ C C n ¼ 0; dg2 D 2D dg Dðn 1Þ
ð3:47Þ
2
results in a system of differential equations subjected to the boundary conditions;
for g ¼ 0;
Fð0Þ ¼ F 0 ð0Þ ¼ 0;
Cð0Þ ¼ 1
0
lim g ! 1; F ð1Þ ¼ Cð1Þ ¼ 0:
ð3:48Þ ð3:49Þ
4. Numerical solution The numerical solution of (3.46) and (3.47) is investigated for different chemical orders n using Runge–Kutta method. The boundary conditions (3.48) and (3.49) are completed by guessing two additional conditions at g ¼ 0; F 00 ð0Þ; C 0 ð0Þ and the solution is iterated on through the shooting method so that the upper boundary conditions at g ¼ 1 are satisfied. 2 The parameters values adopted here are; gb ¼ 1; k ¼ 103 =s; m ¼ 10 in =s and (3.47) is rewritten in terms of the Schmidt number Sc ¼ Dm
2 d C Sc g dC 1 n ¼ þ C kC : dg2 m 2 dg ðn 1Þ
ð4:1Þ
The velocity F 0 ðgÞ and the concentration of chemicals CðgÞ are plotted versus g. The problem is solved for different reaction orders n. For n = 1.2 the results are depicted in Fig. 2 showing a very high velocity F0 = 800 next to the wall for Sc = 0. This might be explained by a very high mobility of molecules in the case Sc = 0. For larger Schmidt number Sc = .01, 1 and 0.5 the velocity fluctuations are smaller. This is due to a decrease in the molecular mobility D as the Schmidt number increases. Figs. 3–5 display the effect of Schmidt number on the momentum profile for n ranging from 2 to 3. No orders higher than the third order in any reactant are known. As the Schmidt number increases the viscosity reduces the buoyancy effect, yielding a reduction in the fluid velocity, a decrease in the maximum velocity as well as a reduction in the convection layers widths and heights. These behaviors appear clearly in Figs. 2–5. Numerical solution of Eq. (3.47) results in the concentration of chemicals. Fig. 6 displays for n = 2.5 different concentration profiles. For Sc = 0 a linear distribution of reaction species is obtained. With the increase of the Schmidt number ðSc ¼ 0:01 ! 0:5Þ the width of the concentration layer tends to decrease. This is due to an increase in the fluid viscosity. Further investigations for n = 3 (Fig. 7) show a decrease in the concentration layer widths as the Schmidt number increases.
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1000
n = 1.2
800 600
F'(η )
400
Sc= 0,
0.01,
6
10
0.1, 0.5
200 0 0
2
4
8
12
14
16
18
20
22
24
-200
η
-400 -600
Fig. 2. Velocity profile F 0 ðgÞ for n = 1.2 and various Schmidt numbers.
3.5 3
n=2
2.5
F'( η )
2
Sc=0, 0.01, 0.1, 0.5
1.5 1 0.5 0
0
20
40
60
80
100
η
-0.5
Fig. 3. Velocity profiles for various Schmidt numbers and n = 2.
5. Perturbation analysis at n = 1 The differential equations (3.46) and (3.47) are singular at n = 1 and are solved setting three term series for the concentration is assumed
C ¼ C 0 þ eC 1 þ e2 C 2 þ
e ¼ n 1. Starting with (3.47) a ð5:1Þ
the boundary condition (3.48) thus reduces to
C 0 ð0Þ þ eC 1 ð0Þ þ e2 C 2 ð0Þ þ e3 C 3 ð0Þ þ ¼ 1
ð5:2Þ
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1.8 1.6
n = 2.5
1.4 1.2
F'(η )
1
Sc = 0, 0.01, 0.1, 0.5
0.8 0.6 0.4 0.2
η
0 -0.2
0
20
40
60
80
100
Fig. 4. Velocity profiles for various Schmidt numbers and n = 2.5.
1.6 1.4
n=3
1.2
F'(η)
1
Sc= 0, 0.01, 0.1, 0.5
0.8 0.6 0.4 0.2 0 -0.2
0
20
40
60
80
100
120
η Fig. 5. Velocity profiles for various Schmidt numbers n = 3.
equating the coefficient of
e0 ; e1 . . . to zero leads to
C 0 ð0Þ ¼ 1;
ð5:3Þ
C 1 ð0Þ ¼ C 2 ð0Þ ¼ C 3 ð0Þ ¼ ¼ 0;
ð5:4Þ
Eq. (3.47) is then multiplied by (n 1)
ðn 1Þ
" 2 # d C g dC k n C þ C þ ¼ 0; dg2 D D 2D dg
ð5:5Þ
where n ! 1 ¼ e
" 2 # d C 1 dC k C þ g C þ ¼ 0; dg2 2D dg D D
e
ð5:6Þ
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1.2
n=2.5
1
C(η)
0.8
Sc= 0, 0.01, 0.1,
0.5
0.6 0.4 0.2 0
0
20
40
η
60
80
100
Fig. 6. Concentration profiles for different Schmidt numbers and n = 2.5.
1.2
1
n =3
C(η)
0.8
Sc= 0, 0.01, 0.1, 0.5
0.6
0.4
0.2
0
0
20
40
60
80
100
120
η Fig. 7. Concentration profiles for different Schmidt numbers and n = 3.
substituting for (5.1) in (5.6)
! ! 2 2 2 d C0 d C1 g dC 0 dC 1 k 2 d C2 2 dC 2 2 þe þe þ þ þe þe þ ðC 0 þ eC 1 þ e C 2 þ Þ D dg2 dg2 dg2 2D dg dg dg
e þ
1 ðC 0 þ eC 1 þ e2 C 2 þ Þ ¼ 0; D
then equating the coefficients of
C 0 ðgÞ ¼ 0;
ð5:7Þ
e0 to zero leads to ð5:8Þ
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this solution is rejected as it does not satisfy the boundary conditions (3.48). Then equating the power of sides we obtain;
e1 ; e2 ; e3 on both
2
d C0 g dC 0 k 1 þ C 0 þ C 1 ¼ 0; D dg2 2D dg D
ð5:9Þ
2
d C1 g dC 1 k 1 þ C 1 þ C 2 ¼ 0; D dg2 2D dg D
ð5:10Þ
2
d C2 g dC 2 k 1 þ C 2 þ C 3 ¼ 0: D dg2 2D dg D
ð5:11Þ
These equations are subjected to the boundary conditions
C 0 ð0Þ ¼ 1; lim C 0 ðgÞ ¼ 0;
ð5:12Þ
C 1 ð0Þ ¼ 0; lim C 1 ðgÞ ¼ 0;
ð5:13Þ
C 2 ð0Þ ¼ 0; lim C 2 ðgÞ ¼ 0;
ð5:14Þ
g!1
g!1 g!1
(5.9)–(5.11) are solved backwardly. In the last equation C 3 accounts for zero according to the three term expansion while (5.10) solution results in
g2 ð1 þ 2kÞ 1 g2 g C 2 ðgÞ ¼ e4D A1 F 1 ; ; þ BH12k pffiffiffiffi ; 2 2 4D 2 D where 1 F 1
h
i
2
1þ2k 1 g ; 2 ; 4D 2
and H12k
h
g
i
pffiffiffi 2 D
ð5:15Þ
are Hypergeometric and Hermite functions and A, B are constants of integrations.
Applying the boundary conditions (5.14) these constants are found equal to zero. Similarly we obtain C 1 ðgÞ ¼ 0 and finally the solution of (5.9) yields
C 0 ðgÞ ¼
1 g2 e 4D 2
1F1
ð1 þ 2kÞ 1 g2 g þ H12k pffiffiffiffi : ; ; 2 2 4D 2 D
ð5:16Þ
A simplified form for C 0 is investigated in order to solve the differential equation (3.46). This is obtained by omitting the third term in (5.9) k/D which is usually very small as k ¼ 103 and D is large 2
d C0 g dC 0 þ ¼ 0: dg2 2D dg
ð5:17Þ
This equation solution is;
C 0 ðgÞ ¼ erfc
rffiffiffiffiffi! g g Sc pffiffiffiffi ¼ erfc : 2 m 2 D
ð5:18Þ
Replacing for (5.18) in (3.46) we obtain 3
2
d F g dF n 2 dF gb g þ ¼ erfc dg3 2m dg2 ðn 1Þm dg m 2
rffiffiffiffiffi! Sc
m
:
ð5:19Þ
This equation is singular at n = 1 and is solved reducing its differential order by setting uðgÞ ¼ ddFg. 2
d u g du n2 gb g þ u ¼ erfc dg2 2m dg ðn 1Þm m 2
rffiffiffiffiffi! Sc
m
;
ð5:20Þ
using the perturbation method and substituting in the boundary condition (3.48) results in
u0 ð0Þ ¼ u1 ð0Þ ¼ u2 ð0Þ ¼ u3 ð0Þ ¼ 0; then substituting for u ¼ u0 þ eu1 þ e2 u2 þ and
ð5:21Þ
e ¼ n 1 in (5.20)
rffiffiffiffiffi!! 2 2 2 d u0 d u1 g du0 du1 gb g Sc 2 d u2 2 du2 þe þe þ þ þe þe þ þ erfc dg2 dg2 dg2 2m dg dg dg m 2 m n2 ðu0 þ eu1 þ e2 u2 þ e3 u3 þ Þ ¼ 0: !
e
m
ð5:22Þ
M.M. Kassem, A.S. Rashed / Applied Mathematics and Computation 215 (2009) 1671–1684
Equating the coefficients of
1681
e0 ; e1 ; e2 ; . . . on both sides, we obtain a system of differential equations
2
d u0 g du0 gb g þ þ erfc dg2 2m dg m 2
rffiffiffiffiffi! Sc
m
n2
m
u1 ¼ 0;
ð5:23Þ
2
d u1 g du1 n 2 þ u ¼ 0; dg2 2m dg m 2
ð5:24Þ
2
d u2 g du2 n 2 þ u ¼ 0: dg2 2m dg m 3
ð5:25Þ
A solution of (5.23)–(5.25) backwardly results in u3 ¼ u2 ¼ u1 ¼ 0 and from (5.21) in (5.23) we obtain 2
d u0 g du0 gb g þ þ erfc dg2 2m dg m 2
rffiffiffiffiffi! Sc
m
¼ 0:
ð5:26Þ
The Mathematica solution of (5.26) considering only two terms of erfc expansion results in
rffiffiffiffiffiffi gbg2 F f1; 1g; f3=2; 2g; g2 pffiffiffiffiffiffi p Q 4m Sc g u0 ðgÞ ¼ 2gb g þ pmErf pffiffiffi ; 2m mp 2 m
ð5:27Þ
where P F Q ½fa1 ; . . . ; ap g; fb1 ; . . . ; bq g; z is the generalized Hypergeometric function. The concentration of chemicals CðgÞ at n = 1 (5.18) is investigated for different values of Schmidt number (Sc = 0.01–0.5). Fig. 8 displays the concentration profile. In this figure it appears that the layer thickness gets smaller with the decrease of the Schmidt number. This is due to an increase in the molecular mobility D of the fluid and thus of friction forces. In Fig. 9 the velocity profile decreases in thickness for decreasing Schmidt number. This reduction is due to an increase of particle mobility. 6. Parameter analysis and comparison of numerical and analytical results 6.1. Skin friction coefficient Skin friction arises from the friction of the fluid against the wall and formulate as follows:
Cf ¼ where
2sw ; qU 21
ð6:1Þ
sw is the shear stress at the wall
@u sw ¼ l @y
ð6:2Þ
y¼0
from (2.7) and (3.30) in (6.2) we obtain
Cf ¼ 2
n3 l 2ðn1Þ t F 00 ðgÞjg¼0 ; q
C 0.9
Sc= 0.5, 0.1, 0.01 0.8 0.7 0.6
10
20
30
40
50
60
Fig. 8. Concentration for n = 1 and Sc = 0.5–0.01.
70
h
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F'(η )
2 1.5
Sc= 0.5, 0.1, 0.01
1
5
10
15
20
25
35
η Fig. 9. Velocity F’ðgÞ for n = 1, k ¼ 103 and Sc = 0.5–0.01.
Table 1 Skin friction coefficient for different parameter values. Reaction order (n)
Schmidt number (Sc)
C f (at t = 1 s), ðU 1 ¼ 1Þ
2
0 0.01 0.1 0.5 0 0.01 0.1 0.5 0 0.01 0.1 0.5
19.02 18.26 10.58 5.84 11.32 11 9.08 6.36 9.84 9.76 8.52 6.56
2.5
3
2
where l=q ¼ m. The skin friction coefficients is evaluated for m ¼ 10 in =s, t = 1 s and U 1 ¼ 1 and different (Sc, n) values. The listed values in Table 1, shows that for large Schmidt number ðm=DÞ; C f decreases. This is due to a decrease on the particle mobility coefficient D. The Sherwood also called the mass transfer Nusselt number is a dimensionless number used in mass transfer operation. It represents the ratio of convective to diffusive mass transport, and is defined as follows:
Sh ¼ kL=D where L is a characteristic length, D is mass diffusivity and k is the mass transfer coefficient. The evaluation of this coefficient for different parameter values is listed in Table 2, where it appears that Sh inversely varies with the molecular mobility D. 6.2. Comparison of Sections 4 and 5 results A comparison of results for the Newtonian case (n = 1) and numerical results obtained for n = 1.2 (the nearest to the singular case n = 1) takes place in Figs. 10 and 11 where it appears that the velocity analytically derived (5.27) is larger than the numerical result. A further percentage analysis of the differences is listed in Tables 3.a and 3.b for the velocity F 0 ðgÞ and concentration of chemicals CðgÞ at different Schmidt numbers. The observed differences are due to the following reasons: 1. The chemical reaction order is n = 1 for the analytic solution and n = 1.2 for the numerical one. 2. The omission of the 3rd term in (5.9) Dk C 0 ¼ mk ScC 0 causes a reduction of C 0 damping effect which reflect on the final result (5.27).
Table 2 Sherwood number for different parameter values k = 0.001 mol/s,
m ¼ 10, L = 1.
Dðin =sÞ
Sc ¼ m=D
Sh = kL/D
1 1000
0 0.01
0
100
0.1
20
0.5
105 0.00005
2
106
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10 9
F'(η)
8 7
Numerical Solution at n=1.2
6
Analytical Solution at n=1
5 4 3 2 1 0
0
1
2
3
4 η
5
6
7
8
Fig. 10. Comparison between numerical and analytical results for Sc = 0.1.
0.4 0.35 0.3
Numerical Solution at n=1.2
0.25
Analytical Solution at n=1
F'( η)
0.2 0.15 0.1 0.05 0 -0.05
0
2
4
6
8
10
η Fig. 11. Comparison between analytical and numerical results for Sc = 0.5.
Table 3.a Comparison of results for both F 0 ðgÞ and CðgÞ using the numerical and analytical approach at Sc = 0.1.
g
F 0 ðgÞ Num
F 0 ðgÞ Anal
% Difference
g
CðgÞ Num
CðgÞ Anal
0 0.5 1
0 2.2 5
0 2.2 5.01
0 0 2 103
0 1 2
1 0.869 0.695
1 0.95 0.9
1.5
6.9
6.96
8:69 103
3
0.488
0.85
2
7.7
8
3:89 103
4
0.26
0.84
2.5
8
8.6
5
0.025
0.82
3 3.5 4
7.4 6.1 4.3
9 8.6 8
7:5 102 0.2 0.4 0.8
3. The difference between analytical and numerical results is small in Fig. 10 for Sc = 0.1, while in Fig. 11 the difference grows larger for Sc = 0.5. This increase in error is due to the omission of the third term in (5.9).
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Table 3.b Comparison of results for both F’ðgÞ and CðgÞ using the numerical and analytical approach at Sc = 0.5.
g
F 0 ðgÞ Num
F 0 ðgÞ Anal
% Difference
g
CðgÞ Num
CðgÞ Anal
% Difference
0 0.5
0 0.075
0 0.12
0 0.6
0 21
1 0.873
1 0.88
0
1 1.5 2 2.5 3
0.12 0.14 0.12 0.08 0.025
0.24 0.305 0.34 0.34 0.33
1 1.178 1.833 3.25 12.2
2 3 5 10 15
0.539 0.09
0.85 0.8 0.7 0.3 0.02
7:95 103 0.36 0.88
7. Conclusions This work presents the analytical and numerical solution time dependent chemical convective process within a boundary layer using the similarity group method. The problem is reduced to a system of ordinary differential equations and solved numerically for different chemical reaction orders ‘n’ n – 1 and Schmidt numbers ‘Sc’. As no mathematical analysis of the problem could be found in the literature we did solve analytically the problem at n = 1 (singular case) and compare the results with the case n = 1.2. From the parametric analysis of the problem we conclude that using the group method preserves the system physical properties. Acknowledgement The authors do thank Zagazig University for their financial support of the work and the reviewers for their remarks. References [1] M.B. Abd-el-Malek, M.M. Kassem, M.L. Mekky, Similarity solutions for unsteady free-convection from a continuous moving vertical surface, J. Comput. Appl. Math. 165 (2004) 11–24. [2] F.W. Ames, Nonlinear Partial Differential Equations in Engineering, Acadmic Press, 1972. [3] T.S. Chen, F.A. Strobel, Buoyancy effects in boundary layer adjacent to a continuous moving horizontal flat plate, J. Heat Transfer 102 (1980) 170–172. [4] P. Ganesan, H.P. Rani, On diffusion of chemically reactive species in convective flow along a vertical cylinder, Chem. Eng. Process. 39 (2000) 93–105. [5] B. Gebhart, L. Pera, The nature of vertical natural convection flow resulting from the combined buoyancy effects of thermal and mass diffusion, J. Heat Mass Transfer 14 (1972) 25–50. [6] F.S. Ibrahim, A.M. Elaiw, A.A. Bakr, Effect of the chemical reaction and radiation absorption on the unsteady MHD free convection flow past a semi infinite vertical permeable moving plate with heat source and suction, Commun. Nonlinear Sci. Numer. Simul. 13 (6) (2008) 1056–1066. [7] M. Kassem, Group solution for unsteady free-convection flow from a vertical moving plate subjected to constant heat flux, J. Comput. Appl. Math. 187 (2006) 72–86. [8] M. Kassem, Group Analysis of a non-Newtonian flow past a vertical plate subjected to a heat constant flux, Int. J. Appl. Mech. Eng. 88 (8) (2008) 661– 673. [9] V.G. Levich, Physico-Chemical Hydrodynamics, Prentice-Hall, Englewood, NJ, 1962. [10] O.D. Makinde, Free convection flow with thermal radiation and mass transfer past a moving vertical porous plate, Int. Commun. Heat Mass Transfer 32 (10) (2005) 1411–1419. [11] A.J.A. Morgan, The reduction by one of the number of independent variables in some systems of partial differential equations, Quart. J. Math. 3 (1952) 250–259. [12] A. S Rashed, M.M. Kassem, Group analysis for natural convection from a vertical plate, J. Comput. Appl. Math. 222 (2008) 392–403.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Group solution of a time dependent chemical convective process M.M. Kassem *, A.S. Rashed Mathematics and Physics Department, Faculty of Engineering, Zagazig University, Egypt
a r t i c l e
i n f o
Keywords: Time dependent chemical convection Group transformation method Perturbation method
a b s t r a c t The time dependent progress of a chemical reaction over a flat vertical plate is here considered. The problem is solved using the two parameter group method which reduces the number of independent by one and leads to a set of nonlinear ordinary differential equation. The behavior of the process is numerically investigated for the chemical reaction order n – 1 and different Schmidt numbers. As the problem shows a singularity at n = 1, the nonlinear system of ordinary differential equation resulting from the transformation of the problem are analytically solved through the perturbation method. The velocity and concentration of chemicals based on the analytical and numerical solutions are presented, compared and discussed. Ó 2009 Elsevier Inc. All rights reserved.
1. Introduction Recently Rashed and Kassem [12] analyzed the problem of steady stated chemical coatings past a vertical plate. Here, the study is concerned with time dependant natural convection. This problem reported by Levich [9] and Gebhart and Pera [5] concerns the immersion of a plate is in a fluid solution having a concentration c0 ðx; tÞ > 0. When the plate touches the solution a chemical reaction takes place inducing a change of concentration and implying density gradients in the presence of gravitational field [3]. Ganesan and Rani [4] analyzed the diffusion of chemically reactive species for a convective unsteady flow along a vertical cylinder using an implicit finite difference method. Makinde [10] investigated a convective flow with thermal radiation and mass transfer past a moving vertical porous plate and assumed a time-dependency for the vertical velocity. The resultant similarity equations were solved numerically using a superposition method. Ibrahim et al. [6] analytically derived the heat and mass transfer of a chemical convective process assuming an exponentially decreasing suction velocity at the surface of a porous plate and a two terms harmonic function for the rest of the variables. The mathematical technique used in the present analysis is a two parameter group transformation of the variables. This method developed by Morgan [11] reduces the number of variables by one and generates a set of ordinary differential equations. This method adopted by Kassem [7,8] and Abdel Malek et al. [1] proved to be efficient for an analysis of various flow problems. In the present work we reduce the field equations and related boundary conditions through a two parameters group. This transformation results in a system of nonlinear differential equations with appropriate boundary conditions. The obtained system of equations shows a singularity at the chemical reaction order n = 1. At this value the concentration profile of chemicals in the boundary layer is analytical derived using the perturbation method. The behavior of the flow is then numerically investigated for n – 1 and different Schmidt numbers using the shooting method and the results are plotted. For n = 1 the concentration is analytically evaluated and plotted for various chemical molecular velocities D.
* Corresponding author. E-mail address: [email protected] (M.M. Kassem). 0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.07.018
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Nomenclature Latin characters unity elements a1 ; a2 c dimensional species concentration C non-dimensional concentration concentration next to the plate c1 ambient concentration c1 D chemical molecular diffusivity. 0 vertical velocity after transformation F g gravitational acceleration G group k chemical rate constant n chemical reaction order Q, T real valued coefficients S subgroup Sc Schmidt number m/D u velocity in x direction v velocity in y direction x vertical axis y horizontal axis Greek symbols b volumetric coefficient of expansion with concentration m ¼ l=q kinematic viscosity of fluid q fluid density w stream function g similarity variable
2. Mathematical formulation The following study is concerned with time dependant convection and diffusion within a thin boundary layer adjacent to an vertical plate immersed in a fluid, having a chemical reaction of order n. For this, it is convenient to consider an idealized system illustrated in Fig. 1 and composed of a semi infinite plate set in a fluid of infinite extent. The natural convection is described by the following equations;
x
g
u
c
c
u
v
y Fig. 1. Illustration of the flow over a semi-infinite horizontal plate.
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@u @ v þ ¼ 0; @x @y
1673
ð2:1Þ 2
@u @u @u @ u þu þv gbc m 2 ¼ 0; @t @x @y @y
ð2:2Þ
@c @c @c @2c n þu þv þ kc D 2 ¼ 0; @t @x @y @y
ð2:3Þ
subjected to the following initial and boundary conditions
uðx; y; 0Þ ¼ u0 ðx; yÞ;
v ðx; y; 0Þ ¼ v 0 ðx; yÞ;
uðx; 0; tÞ ¼ v ðx; 0; tÞ ¼ 0;
cðx; y; 0Þ ¼ 0;
ð2:4Þ
cðx; 0; tÞ ¼ c0 ðx; tÞ;
ð2:5Þ
lim uðx; y; tÞ ¼ lim v ðx; y; tÞ ¼ lim cðx; y; tÞ ¼ 0;
y!1
y!1
ð2:6Þ
y!1
Eq. (2.1) is eliminated and concentration at the wall (2.5) is normalized through the transformations
@w ; @x cðx; y; tÞ Cðx; y; tÞ ¼ : c0 ðx; tÞ
u¼
@w ; @y
v ¼
ð2:7Þ ð2:8Þ
In this case the flow equations (2.1)–(2.6) reduce to
@ 2 w @w @ 2 w @w @ 2 w @3w gbc0 C m 3 ¼ 0; þ 2 @y @t @y @x @y @x @y @y
ð2:9Þ 2
@C C @c0 @w @C C @w @c0 @w @C @ C n1 þ þ þ kc0 C n D 2 ¼ 0; þ @t c0 @t @y @x c0 @y @x @x @y @y @wðx; y; 0Þ @wðx; y; 0Þ ¼ u0 ðx; yÞ; ¼ v 0 ðx; yÞ; Cðx; y; 0Þ ¼ 0; @y @x @wðx; 0; tÞ @wðx; 0; tÞ ¼ ¼ 0; Cðx; 0; tÞ ¼ 1; @y @x @w @w ¼ ¼ 0; lim Cðx; y; tÞ ¼ 0: y!1 @y @x y!1
ð2:10Þ ð2:11Þ ð2:12Þ ð2:13Þ
y!1
3. Group formulation of the problem From the group definition [2]
G : S ¼ Q s ða1 ; a2 ÞS þ T s ða1 ; a2 Þ;
ð3:1Þ
where G is a two parameter group, S and S stand for the system variables (t, x; w, C and c0 Þ before and after transformation, Q s ; T s are real valued coefficients at least differentiable in the group parameters ða1 ; a2 Þ. First and second partial derivatives are defined as;
Si ¼ Sij ¼
s Q Si Qi s Q QiQj
9 > = i; j ¼ x; y; ; Sij >
ð3:2Þ
where S stands for the dependent variables ðw; C and c0 Þ. From the above definitions (2.9) is transformed to
! @2w @2w @2w @w @w @3w @ 2 w @w @ 2 w @w @ 2 w @3w gbc0 C m 3 ¼ H1 ða1 ; a2 Þ gbc0 C m 3 ; þ þ @ x @ y @ x @ y 2 ; @t @ y @y @t @y @x @y @x @y2 @y @y @y ! ! ! @2w Qw Qw @w @ 2 w Qw Qw @w @ 2 w Qw @3w c0 C gbQ Q c C m þ 0 y x y x @y @t @y @x @y Q Q Q Q ðQ y Þ2 @x @y2 ðQ y Þ3 @y3 ! @ 2 w @w @ 2 w @w @ 2 w @3w þ R1 ¼ H1 ða1 ; a2 Þ gbc0 C m 3 ; þ @y @t @y @x @y @x @y2 @y
Qw Q yQ t
ð3:3Þ
!
ð3:4Þ
where
R1 ¼ gbðQ c0 T C c0 þ T c0 Q C C þ T c0 T C Þ:
ð3:5Þ
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The invariant transformation of (3.4) implies that H1 ða1 ; a2 Þ ¼ 1 and R1 ¼ 0 giving;
T c0 ¼ T C ¼ 0
ð3:6Þ
a comparison of coefficients on both sides of (3.4) results in;
Qw ðQ w Þ2 Qw ¼ Q C Q c0 ¼ y 3 : y t ¼ x y 2 Q Q Q ðQ Þ ðQ Þ
ð3:7Þ
Following the same procedure (2.10) is transformed leading to;
Q C Q wQ C QC c0 n1 ðQ C Þn ¼ y 2 ; y x ¼ ðQ Þ t ¼ Q Q Q ðQ Þ
ð3:8Þ
similarly the transformation of initial and boundary conditions implies
Q C ¼ 1;
Ty ¼ Tt ¼ 0
ð3:9Þ
and (3.7)–(3.9) ratios reduce to
Q t ¼ ðQ y Þ2 ;
Q c0 ¼ 1;
Q w ¼ ðQ y Þ3 ;
Q x ¼ ðQ y Þ4 ;
ð3:10Þ
finally the system group structure is written as
8 8 y 4 x > > > < x ¼ ðQ Þ x þ T > > y > ¼Q y G1 y > > > > > < : t ¼ ðQ y Þ2 t 8 ; G y 3 w > > < w ¼ ðQ Þ w þ T > > > > > > > G2 > C ¼ C > : : c0 ¼ c0
ð3:11Þ
where G1 and G2 are subgroups describes the independent and dependant variables and dashes stand for their transformation. 3.1. Group transformation of the system variables The flow equations order is reduced by one if it satisfies Morgan’s theorem [11]
9 > ðiÞ u ½ai uðiÞ þ aiþ1 @@uðiÞ ¼ 0> > = i¼1 ; 6 > P > ðiÞ u ½bi uðiÞ þ biþ1 @@uðiÞ ¼0> ; 6 P
ð3:12Þ
i¼1
ðiÞ; i ¼ 1; 2; . . . 6 stand for the six system variables ðx; y; t; w; C; c0 Þ before and after transformation where uðiÞ; u
@Q si ða1 ; a2 Þ ; @a1 si @Q ða1 ; a2 Þ ; bi ¼ @a2
ai ¼
@T si ða1 ; a2 Þ ; @a1 si @T ða1 ; a2 Þ ¼ : @a2
aiþ1 ¼
ð3:13Þ
biþ1
ð3:14Þ
3.2. Transformation of the independent variables The similarity variable g(x,y,t) is obtained applying (3.12)
@g @g @g þ a3 y þ a5 t ¼ 0; @x @y @t @g @g @g ðb1 x þ b2 Þ þ b3 y þ b5 t ¼ 0: @x @y @t
ða1 x þ a2 Þ
Eliminating
@g @y
and
@g , @x
ð3:15Þ ð3:16Þ
respectively, from (3.15) and (3.16);
@g @g ðk13 x þ k23 Þ þ k53 t ¼ 0; kij ¼ ai bj aj bi ; @x @t @g @g ðk31 x þ k32 Þy þ ðk51 x þ k52 Þt ¼ 0; @y @t then invoking the group structure described in (3.11) we obtain;
ð3:17Þ ð3:18Þ
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a5 ¼ 2a3 ;
ð3:19Þ
b5 ¼ 2b3 ;
ð3:20Þ
i.e. k53 ¼ a5 b3 a3 b5 ¼ 0 and (3.17) reduces to
ðk13 x þ k23 Þ
@g ¼ 0; @x
ð3:21Þ
@g ¼ 0; @x g ¼ gðy; tÞ;
ð3:22Þ ð3:23Þ
from (3.23) in (3.18) we obtain
g ¼ ypðtÞ;
ð3:24Þ
pðtÞ ¼ atb ;
ð3:25Þ
where
without loss of generality let a = 1 while b will be determined later. 3.3. Transformation of the dependant variables Transformation of C and c0 is obtained directly from (3.11) group structure
Þ ¼ CðgÞ; Cðx; y c0 ðx; tÞ ¼ Cðx; tÞ;
ð3:26Þ ð3:27Þ
C(x,t) will be determined later. Similarly w is transformed applying (3.12)
x;y;t;wÞ ða1 x þ a2 Þ @g3 ðx@x;y;t;wÞ þ a3 y @g3 ðx@y;y;t;wÞ þ a5 t @g3 ðx@t;y;t;wÞ þ ða7 w þ a8 Þ @g3 ð@w ¼ 0;
x;y;t;wÞ ¼ 0; ðb1 x þ b2 Þ @g3 ðx@x;y;t;wÞ þ b3 y @g3 ðx@y;y;t;wÞ þ b5 t @g3 ðx@t;y;t;wÞ þ ðb7 w þ b8 Þ @g3 ð@w
ð3:28Þ
the solution of (3.28) gives
¼ / ðw=xðxÞÞ ¼ FðgÞ ; t; wÞ g 3 ðx; y 1
ð3:29Þ
and for /1 ¼ 1 the flux w transforms to
x; y ; tÞ ¼ xðx; tÞFðgÞ: wð
ð3:30Þ
Replacing for the system variables (3.26), (3.27) and (3.29) after transformation in (2.9) and (2.10) we obtain
2 3 2 2 d F d F d F dF dF A1 g 2 þ A2 F 2 ðA1 þ A3 Þ A2 þ A4 C ¼ 0; 3 dg dg dg dg dg
ð3:31Þ
2
d C m dC m dC dF g þ F A5 C A 6 C A7 C n ¼ 0; dg2 D dg D dg dg
ð3:32Þ
where
dp=dt
@ x=@x ; A2 ¼ ; mp3 mp @ x=@t gbC ; A4 ¼ ; A3 ¼ mxp2 mxp3 @ C=@t x @C A5 ¼ ; A6 ¼ ; DCp2 DpC @x A1 ¼
A7 ¼
kCn1 : Dp2
ð3:33Þ ð3:34Þ ð3:35Þ ð3:36Þ
In order to reduce (3.31) and (3.32) to a system of linear ordinary equations, the A’s must be function of g or constants. First let in (3.36) A7 ¼ Dk . This leads to; 2
2
Cðx; tÞ ¼ ðpðtÞÞn1 ¼ ðtb Þn1 ; where C(t) is the concentration of the fluid next to the plate. As C(t) is not function of ‘x’ hence
ð3:37Þ
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A6 ¼ 0;
ð3:38Þ
2b A5 ¼ t ð2bþ1Þ : ðn 1ÞD
ð3:39Þ
A5 is constant if b = 1/2.
A5 ¼
1 ; Dðn 1Þ
ð3:40Þ
thus (3.24) reduces to;
y t
g ¼ pffiffi :
ð3:41Þ
For A4 ¼ gb m we obtain 3n5
xðx; tÞ ¼ xðtÞ ¼ t2ðn1Þ ; n – 1;
ð3:42Þ
where x(t) is part of the stream function wðgÞ described in (3.30). Replacing for x; p; C in (3.33) and (3.34) in the remaining constants we obtain
A3 ¼
3n 5 ; 2ðn 1Þm
n – 1;
ð3:43Þ
A2 ¼ 0;
ð3:44Þ
1 A1 ¼ : 2m
ð3:45Þ
Substituting for the constants in (3.31) and (3.32)
3 2 d F g d F 2n 4 dF gb þ þ C ¼ 0; dg3 2ðn 1Þm dg m 2m dg2
ð3:46Þ
d C g dC 1 k þ þ C C n ¼ 0; dg2 D 2D dg Dðn 1Þ
ð3:47Þ
2
results in a system of differential equations subjected to the boundary conditions;
for g ¼ 0;
Fð0Þ ¼ F 0 ð0Þ ¼ 0;
Cð0Þ ¼ 1
0
lim g ! 1; F ð1Þ ¼ Cð1Þ ¼ 0:
ð3:48Þ ð3:49Þ
4. Numerical solution The numerical solution of (3.46) and (3.47) is investigated for different chemical orders n using Runge–Kutta method. The boundary conditions (3.48) and (3.49) are completed by guessing two additional conditions at g ¼ 0; F 00 ð0Þ; C 0 ð0Þ and the solution is iterated on through the shooting method so that the upper boundary conditions at g ¼ 1 are satisfied. 2 The parameters values adopted here are; gb ¼ 1; k ¼ 103 =s; m ¼ 10 in =s and (3.47) is rewritten in terms of the Schmidt number Sc ¼ Dm
2 d C Sc g dC 1 n ¼ þ C kC : dg2 m 2 dg ðn 1Þ
ð4:1Þ
The velocity F 0 ðgÞ and the concentration of chemicals CðgÞ are plotted versus g. The problem is solved for different reaction orders n. For n = 1.2 the results are depicted in Fig. 2 showing a very high velocity F0 = 800 next to the wall for Sc = 0. This might be explained by a very high mobility of molecules in the case Sc = 0. For larger Schmidt number Sc = .01, 1 and 0.5 the velocity fluctuations are smaller. This is due to a decrease in the molecular mobility D as the Schmidt number increases. Figs. 3–5 display the effect of Schmidt number on the momentum profile for n ranging from 2 to 3. No orders higher than the third order in any reactant are known. As the Schmidt number increases the viscosity reduces the buoyancy effect, yielding a reduction in the fluid velocity, a decrease in the maximum velocity as well as a reduction in the convection layers widths and heights. These behaviors appear clearly in Figs. 2–5. Numerical solution of Eq. (3.47) results in the concentration of chemicals. Fig. 6 displays for n = 2.5 different concentration profiles. For Sc = 0 a linear distribution of reaction species is obtained. With the increase of the Schmidt number ðSc ¼ 0:01 ! 0:5Þ the width of the concentration layer tends to decrease. This is due to an increase in the fluid viscosity. Further investigations for n = 3 (Fig. 7) show a decrease in the concentration layer widths as the Schmidt number increases.
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1000
n = 1.2
800 600
F'(η )
400
Sc= 0,
0.01,
6
10
0.1, 0.5
200 0 0
2
4
8
12
14
16
18
20
22
24
-200
η
-400 -600
Fig. 2. Velocity profile F 0 ðgÞ for n = 1.2 and various Schmidt numbers.
3.5 3
n=2
2.5
F'( η )
2
Sc=0, 0.01, 0.1, 0.5
1.5 1 0.5 0
0
20
40
60
80
100
η
-0.5
Fig. 3. Velocity profiles for various Schmidt numbers and n = 2.
5. Perturbation analysis at n = 1 The differential equations (3.46) and (3.47) are singular at n = 1 and are solved setting three term series for the concentration is assumed
C ¼ C 0 þ eC 1 þ e2 C 2 þ
e ¼ n 1. Starting with (3.47) a ð5:1Þ
the boundary condition (3.48) thus reduces to
C 0 ð0Þ þ eC 1 ð0Þ þ e2 C 2 ð0Þ þ e3 C 3 ð0Þ þ ¼ 1
ð5:2Þ
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1.8 1.6
n = 2.5
1.4 1.2
F'(η )
1
Sc = 0, 0.01, 0.1, 0.5
0.8 0.6 0.4 0.2
η
0 -0.2
0
20
40
60
80
100
Fig. 4. Velocity profiles for various Schmidt numbers and n = 2.5.
1.6 1.4
n=3
1.2
F'(η)
1
Sc= 0, 0.01, 0.1, 0.5
0.8 0.6 0.4 0.2 0 -0.2
0
20
40
60
80
100
120
η Fig. 5. Velocity profiles for various Schmidt numbers n = 3.
equating the coefficient of
e0 ; e1 . . . to zero leads to
C 0 ð0Þ ¼ 1;
ð5:3Þ
C 1 ð0Þ ¼ C 2 ð0Þ ¼ C 3 ð0Þ ¼ ¼ 0;
ð5:4Þ
Eq. (3.47) is then multiplied by (n 1)
ðn 1Þ
" 2 # d C g dC k n C þ C þ ¼ 0; dg2 D D 2D dg
ð5:5Þ
where n ! 1 ¼ e
" 2 # d C 1 dC k C þ g C þ ¼ 0; dg2 2D dg D D
e
ð5:6Þ
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M.M. Kassem, A.S. Rashed / Applied Mathematics and Computation 215 (2009) 1671–1684
1.2
n=2.5
1
C(η)
0.8
Sc= 0, 0.01, 0.1,
0.5
0.6 0.4 0.2 0
0
20
40
η
60
80
100
Fig. 6. Concentration profiles for different Schmidt numbers and n = 2.5.
1.2
1
n =3
C(η)
0.8
Sc= 0, 0.01, 0.1, 0.5
0.6
0.4
0.2
0
0
20
40
60
80
100
120
η Fig. 7. Concentration profiles for different Schmidt numbers and n = 3.
substituting for (5.1) in (5.6)
! ! 2 2 2 d C0 d C1 g dC 0 dC 1 k 2 d C2 2 dC 2 2 þe þe þ þ þe þe þ ðC 0 þ eC 1 þ e C 2 þ Þ D dg2 dg2 dg2 2D dg dg dg
e þ
1 ðC 0 þ eC 1 þ e2 C 2 þ Þ ¼ 0; D
then equating the coefficients of
C 0 ðgÞ ¼ 0;
ð5:7Þ
e0 to zero leads to ð5:8Þ
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M.M. Kassem, A.S. Rashed / Applied Mathematics and Computation 215 (2009) 1671–1684
this solution is rejected as it does not satisfy the boundary conditions (3.48). Then equating the power of sides we obtain;
e1 ; e2 ; e3 on both
2
d C0 g dC 0 k 1 þ C 0 þ C 1 ¼ 0; D dg2 2D dg D
ð5:9Þ
2
d C1 g dC 1 k 1 þ C 1 þ C 2 ¼ 0; D dg2 2D dg D
ð5:10Þ
2
d C2 g dC 2 k 1 þ C 2 þ C 3 ¼ 0: D dg2 2D dg D
ð5:11Þ
These equations are subjected to the boundary conditions
C 0 ð0Þ ¼ 1; lim C 0 ðgÞ ¼ 0;
ð5:12Þ
C 1 ð0Þ ¼ 0; lim C 1 ðgÞ ¼ 0;
ð5:13Þ
C 2 ð0Þ ¼ 0; lim C 2 ðgÞ ¼ 0;
ð5:14Þ
g!1
g!1 g!1
(5.9)–(5.11) are solved backwardly. In the last equation C 3 accounts for zero according to the three term expansion while (5.10) solution results in
g2 ð1 þ 2kÞ 1 g2 g C 2 ðgÞ ¼ e4D A1 F 1 ; ; þ BH12k pffiffiffiffi ; 2 2 4D 2 D where 1 F 1
h
i
2
1þ2k 1 g ; 2 ; 4D 2
and H12k
h
g
i
pffiffiffi 2 D
ð5:15Þ
are Hypergeometric and Hermite functions and A, B are constants of integrations.
Applying the boundary conditions (5.14) these constants are found equal to zero. Similarly we obtain C 1 ðgÞ ¼ 0 and finally the solution of (5.9) yields
C 0 ðgÞ ¼
1 g2 e 4D 2
1F1
ð1 þ 2kÞ 1 g2 g þ H12k pffiffiffiffi : ; ; 2 2 4D 2 D
ð5:16Þ
A simplified form for C 0 is investigated in order to solve the differential equation (3.46). This is obtained by omitting the third term in (5.9) k/D which is usually very small as k ¼ 103 and D is large 2
d C0 g dC 0 þ ¼ 0: dg2 2D dg
ð5:17Þ
This equation solution is;
C 0 ðgÞ ¼ erfc
rffiffiffiffiffi! g g Sc pffiffiffiffi ¼ erfc : 2 m 2 D
ð5:18Þ
Replacing for (5.18) in (3.46) we obtain 3
2
d F g dF n 2 dF gb g þ ¼ erfc dg3 2m dg2 ðn 1Þm dg m 2
rffiffiffiffiffi! Sc
m
:
ð5:19Þ
This equation is singular at n = 1 and is solved reducing its differential order by setting uðgÞ ¼ ddFg. 2
d u g du n2 gb g þ u ¼ erfc dg2 2m dg ðn 1Þm m 2
rffiffiffiffiffi! Sc
m
;
ð5:20Þ
using the perturbation method and substituting in the boundary condition (3.48) results in
u0 ð0Þ ¼ u1 ð0Þ ¼ u2 ð0Þ ¼ u3 ð0Þ ¼ 0; then substituting for u ¼ u0 þ eu1 þ e2 u2 þ and
ð5:21Þ
e ¼ n 1 in (5.20)
rffiffiffiffiffi!! 2 2 2 d u0 d u1 g du0 du1 gb g Sc 2 d u2 2 du2 þe þe þ þ þe þe þ þ erfc dg2 dg2 dg2 2m dg dg dg m 2 m n2 ðu0 þ eu1 þ e2 u2 þ e3 u3 þ Þ ¼ 0: !
e
m
ð5:22Þ
M.M. Kassem, A.S. Rashed / Applied Mathematics and Computation 215 (2009) 1671–1684
Equating the coefficients of
1681
e0 ; e1 ; e2 ; . . . on both sides, we obtain a system of differential equations
2
d u0 g du0 gb g þ þ erfc dg2 2m dg m 2
rffiffiffiffiffi! Sc
m
n2
m
u1 ¼ 0;
ð5:23Þ
2
d u1 g du1 n 2 þ u ¼ 0; dg2 2m dg m 2
ð5:24Þ
2
d u2 g du2 n 2 þ u ¼ 0: dg2 2m dg m 3
ð5:25Þ
A solution of (5.23)–(5.25) backwardly results in u3 ¼ u2 ¼ u1 ¼ 0 and from (5.21) in (5.23) we obtain 2
d u0 g du0 gb g þ þ erfc dg2 2m dg m 2
rffiffiffiffiffi! Sc
m
¼ 0:
ð5:26Þ
The Mathematica solution of (5.26) considering only two terms of erfc expansion results in
rffiffiffiffiffiffi gbg2 F f1; 1g; f3=2; 2g; g2 pffiffiffiffiffiffi p Q 4m Sc g u0 ðgÞ ¼ 2gb g þ pmErf pffiffiffi ; 2m mp 2 m
ð5:27Þ
where P F Q ½fa1 ; . . . ; ap g; fb1 ; . . . ; bq g; z is the generalized Hypergeometric function. The concentration of chemicals CðgÞ at n = 1 (5.18) is investigated for different values of Schmidt number (Sc = 0.01–0.5). Fig. 8 displays the concentration profile. In this figure it appears that the layer thickness gets smaller with the decrease of the Schmidt number. This is due to an increase in the molecular mobility D of the fluid and thus of friction forces. In Fig. 9 the velocity profile decreases in thickness for decreasing Schmidt number. This reduction is due to an increase of particle mobility. 6. Parameter analysis and comparison of numerical and analytical results 6.1. Skin friction coefficient Skin friction arises from the friction of the fluid against the wall and formulate as follows:
Cf ¼ where
2sw ; qU 21
ð6:1Þ
sw is the shear stress at the wall
@u sw ¼ l @y
ð6:2Þ
y¼0
from (2.7) and (3.30) in (6.2) we obtain
Cf ¼ 2
n3 l 2ðn1Þ t F 00 ðgÞjg¼0 ; q
C 0.9
Sc= 0.5, 0.1, 0.01 0.8 0.7 0.6
10
20
30
40
50
60
Fig. 8. Concentration for n = 1 and Sc = 0.5–0.01.
70
h
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M.M. Kassem, A.S. Rashed / Applied Mathematics and Computation 215 (2009) 1671–1684
F'(η )
2 1.5
Sc= 0.5, 0.1, 0.01
1
5
10
15
20
25
35
η Fig. 9. Velocity F’ðgÞ for n = 1, k ¼ 103 and Sc = 0.5–0.01.
Table 1 Skin friction coefficient for different parameter values. Reaction order (n)
Schmidt number (Sc)
C f (at t = 1 s), ðU 1 ¼ 1Þ
2
0 0.01 0.1 0.5 0 0.01 0.1 0.5 0 0.01 0.1 0.5
19.02 18.26 10.58 5.84 11.32 11 9.08 6.36 9.84 9.76 8.52 6.56
2.5
3
2
where l=q ¼ m. The skin friction coefficients is evaluated for m ¼ 10 in =s, t = 1 s and U 1 ¼ 1 and different (Sc, n) values. The listed values in Table 1, shows that for large Schmidt number ðm=DÞ; C f decreases. This is due to a decrease on the particle mobility coefficient D. The Sherwood also called the mass transfer Nusselt number is a dimensionless number used in mass transfer operation. It represents the ratio of convective to diffusive mass transport, and is defined as follows:
Sh ¼ kL=D where L is a characteristic length, D is mass diffusivity and k is the mass transfer coefficient. The evaluation of this coefficient for different parameter values is listed in Table 2, where it appears that Sh inversely varies with the molecular mobility D. 6.2. Comparison of Sections 4 and 5 results A comparison of results for the Newtonian case (n = 1) and numerical results obtained for n = 1.2 (the nearest to the singular case n = 1) takes place in Figs. 10 and 11 where it appears that the velocity analytically derived (5.27) is larger than the numerical result. A further percentage analysis of the differences is listed in Tables 3.a and 3.b for the velocity F 0 ðgÞ and concentration of chemicals CðgÞ at different Schmidt numbers. The observed differences are due to the following reasons: 1. The chemical reaction order is n = 1 for the analytic solution and n = 1.2 for the numerical one. 2. The omission of the 3rd term in (5.9) Dk C 0 ¼ mk ScC 0 causes a reduction of C 0 damping effect which reflect on the final result (5.27).
Table 2 Sherwood number for different parameter values k = 0.001 mol/s,
m ¼ 10, L = 1.
Dðin =sÞ
Sc ¼ m=D
Sh = kL/D
1 1000
0 0.01
0
100
0.1
20
0.5
105 0.00005
2
106
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M.M. Kassem, A.S. Rashed / Applied Mathematics and Computation 215 (2009) 1671–1684
10 9
F'(η)
8 7
Numerical Solution at n=1.2
6
Analytical Solution at n=1
5 4 3 2 1 0
0
1
2
3
4 η
5
6
7
8
Fig. 10. Comparison between numerical and analytical results for Sc = 0.1.
0.4 0.35 0.3
Numerical Solution at n=1.2
0.25
Analytical Solution at n=1
F'( η)
0.2 0.15 0.1 0.05 0 -0.05
0
2
4
6
8
10
η Fig. 11. Comparison between analytical and numerical results for Sc = 0.5.
Table 3.a Comparison of results for both F 0 ðgÞ and CðgÞ using the numerical and analytical approach at Sc = 0.1.
g
F 0 ðgÞ Num
F 0 ðgÞ Anal
% Difference
g
CðgÞ Num
CðgÞ Anal
0 0.5 1
0 2.2 5
0 2.2 5.01
0 0 2 103
0 1 2
1 0.869 0.695
1 0.95 0.9
1.5
6.9
6.96
8:69 103
3
0.488
0.85
2
7.7
8
3:89 103
4
0.26
0.84
2.5
8
8.6
5
0.025
0.82
3 3.5 4
7.4 6.1 4.3
9 8.6 8
7:5 102 0.2 0.4 0.8
3. The difference between analytical and numerical results is small in Fig. 10 for Sc = 0.1, while in Fig. 11 the difference grows larger for Sc = 0.5. This increase in error is due to the omission of the third term in (5.9).
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Table 3.b Comparison of results for both F’ðgÞ and CðgÞ using the numerical and analytical approach at Sc = 0.5.
g
F 0 ðgÞ Num
F 0 ðgÞ Anal
% Difference
g
CðgÞ Num
CðgÞ Anal
% Difference
0 0.5
0 0.075
0 0.12
0 0.6
0 21
1 0.873
1 0.88
0
1 1.5 2 2.5 3
0.12 0.14 0.12 0.08 0.025
0.24 0.305 0.34 0.34 0.33
1 1.178 1.833 3.25 12.2
2 3 5 10 15
0.539 0.09
0.85 0.8 0.7 0.3 0.02
7:95 103 0.36 0.88
7. Conclusions This work presents the analytical and numerical solution time dependent chemical convective process within a boundary layer using the similarity group method. The problem is reduced to a system of ordinary differential equations and solved numerically for different chemical reaction orders ‘n’ n – 1 and Schmidt numbers ‘Sc’. As no mathematical analysis of the problem could be found in the literature we did solve analytically the problem at n = 1 (singular case) and compare the results with the case n = 1.2. From the parametric analysis of the problem we conclude that using the group method preserves the system physical properties. Acknowledgement The authors do thank Zagazig University for their financial support of the work and the reviewers for their remarks. References [1] M.B. Abd-el-Malek, M.M. Kassem, M.L. Mekky, Similarity solutions for unsteady free-convection from a continuous moving vertical surface, J. Comput. Appl. Math. 165 (2004) 11–24. [2] F.W. Ames, Nonlinear Partial Differential Equations in Engineering, Acadmic Press, 1972. [3] T.S. Chen, F.A. Strobel, Buoyancy effects in boundary layer adjacent to a continuous moving horizontal flat plate, J. Heat Transfer 102 (1980) 170–172. [4] P. Ganesan, H.P. Rani, On diffusion of chemically reactive species in convective flow along a vertical cylinder, Chem. Eng. Process. 39 (2000) 93–105. [5] B. Gebhart, L. Pera, The nature of vertical natural convection flow resulting from the combined buoyancy effects of thermal and mass diffusion, J. Heat Mass Transfer 14 (1972) 25–50. [6] F.S. Ibrahim, A.M. Elaiw, A.A. Bakr, Effect of the chemical reaction and radiation absorption on the unsteady MHD free convection flow past a semi infinite vertical permeable moving plate with heat source and suction, Commun. Nonlinear Sci. Numer. Simul. 13 (6) (2008) 1056–1066. [7] M. Kassem, Group solution for unsteady free-convection flow from a vertical moving plate subjected to constant heat flux, J. Comput. Appl. Math. 187 (2006) 72–86. [8] M. Kassem, Group Analysis of a non-Newtonian flow past a vertical plate subjected to a heat constant flux, Int. J. Appl. Mech. Eng. 88 (8) (2008) 661– 673. [9] V.G. Levich, Physico-Chemical Hydrodynamics, Prentice-Hall, Englewood, NJ, 1962. [10] O.D. Makinde, Free convection flow with thermal radiation and mass transfer past a moving vertical porous plate, Int. Commun. Heat Mass Transfer 32 (10) (2005) 1411–1419. [11] A.J.A. Morgan, The reduction by one of the number of independent variables in some systems of partial differential equations, Quart. J. Math. 3 (1952) 250–259. [12] A. S Rashed, M.M. Kassem, Group analysis for natural convection from a vertical plate, J. Comput. Appl. Math. 222 (2008) 392–403.