Acta Mechanica 177, 1–18 (2005) DOI 10.1007/s00707-005-0249-8

Acta Mechanica Printed in Austria

Finite-element method for the effects of chemical reaction, variable viscosity, thermophoresis and heat generation/absorption on a boundary-layer hydromagnetic flow with heat and mass transfer over a heat surface M. A. Seddeek, Buriedah, Kingom of Saudi Arabia Received October 1, 2003; revised March 9, 2005 Published online: June 20, 2005 Ó Springer-Verlag 2005

Summary. The effects of variable viscosity, thermophoresis and heat generation or absorption on hydromagnetic flow with heat and mass transfer over a heat surface are presented here, taking into account the homogeneous chemical reaction of first order. The fluid viscosity is assumed to vary as an inverse linear function of temperature. The velocity profiles are compared with previously published works and are found to be in good agreement. The governing fundamental equations are approximated by a system of nonlinear ordinary differential equations and are solved numerically by using the finite element method. 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 concentration decreases with increasing values of the chemical reaction parameter. The results also showed that the particle deposition rates were strongly influenced by thermophoresis and buoyancy force, particularly for opposing flow and hot surfaces. Numerical results for the skin-friction coefficient, wall heat transfer and particle deposition rate are obtained and reported graphically for various parametric conditions to show interesting aspects of the solution.

1 Introduction Chemical reactions can be codified as either heterogeneous or homogeneous processes. 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. In nature, the presence of pure air or water is impossible. Some foreign mass may be present either naturally or mixed with air or water. The equations of motion for the flow of gases or water, taking into account the presence of foreign mass of low level [1], free convection effects on Stokes problems [2], [3] and the effect of foreign mass on the free-convection flow past a semi-infinite vertical plate were studied [4]. Diffusion and chemical reaction in an isothermal

2

M. A. Seddeek

laminar flow along a soluble flat plate was studied, and appropriate mass-transfer analogous to the flow along a flat plate that contains a species A slightly soluble in the fluid B has been discussed [5]. Soundalgekar et al. [6] have studied the flow past an impulsively started infinite vertical plate with constant heat flux and mass transfer. Muthucumaraswamy et al. [7] have studied the impulsive motion of a vertical plate with heat flux and diffusion of chemically reactive species. Recently, Ghaly and Seddeek [8] have investigated the effects of chemical reaction, heat and mass transfer on laminar flow along a semi infinite horizontal plate with temperature dependent viscosity. Over the past two decades studies in aerosol particle deposition due to thermophoresis have gained importance for engineering applications. Technological problems include particle deposition onto a surface from a condensing vapor-gas mixture, a semi-conductor wafer in the electronic industry, blade surfaces of gas turbines, and problems for nuclear reactor safety. It has been also shown that thermophoresis is the dominant mass transfer mechanism in the modified chemical vapor deposition process used in the fabrication of optical fiber and is also important in view of its relevance to postulated accidents by radioactive particle deposition in nuclear reactors. Goren [9] was one of the first to study the role of thermophoresis in laminar flow of a viscous and incompressible fluid. Tsai [10] discussed the effect of wall suction and thermophoresis on aerosol particle deposition from a laminar flow over a flat plate. Chang et al. [11] developed the effect of thermophoresis on particle deposition from a mixed convection flow onto a vertical flat plate. Chamkha et al. [12] reported effects of heat generation or absorption and thermophoresis on hydromagnetic flow with heat and mass transfer over a flat surface. Recently, Chamkha and Pop [13] discussed the effect of thermophoresis particle deposition in a free convection boundary layer from a vertical plate embedded in a porous medium. Most of the existing analytical studies for this problem are based on the constant physical properties of the ambient fluid. However, it is known (see Herwing and Gresten [14]) that these properties may change with temperature, especially fluid viscosity. To accurately predict the flow and heat transfer rates, it is necessary to take this variation of viscosity into account. Seddeek [15] studied the effect of variable viscosity on hydromagnetic flow and heat transfer past a continuously moving porous boundary with radiation. Also, Seddeek [16] analyzed the effects of magnetic field and variable viscosity on forced non–Darcy flow about a flat plate with variable wall temperature in porous media in the presence of suction and blowing. The aim of the present study is to analyze the combined effects of chemical reaction and thermophoresis on hydromagnetic flow with heat and mass transfer over a flat surface in the presence of temperature dependent viscosity. We have reduced the two-dimensional continuity, energy and concentration equations to a system of nonlinear ordinary differential equations which are solved numerically by using the finite element method. The effects of Hartmann number, chemical reaction, variable viscosity, dimensionless wall mass transfer coefficient, Schmidt number, thermophoresis parameter and the dimensionless heat generation/absorption coefficient on the flow and heat transfer have been shown graphically.

2 Governing equations The steady laminar, two-dimensional boundary-layer flow, of a viscous incompressible and homogeneous fluid, with temperature dependent viscosity, chemical reactions, thermophoresis and heat generation/absorption on hydromagnetic flow are considered. For a viscous fluid, Ling et al. [17] suggested a viscosity dependence on temperature T of the form

Finite-element method for the effects of chemical reaction

l¼

l1 ; ½1 þ dðT
T1 Þ

3 ð1Þ

so that viscosity is an inverse linear function of temperature T. Equation (1) can be rewritten as 1 ¼ AðT
Tr Þ; l where A ¼

d l1

ð2Þ 1 and Tr ¼ T1
: d

ð3Þ

In the above relations (3), both A and Tr are constants and their values depend on the reference state and d, a thermal property of the fluid. In general, A > 0 for fluids such as liquids and A<0 for gases. The most common working fluids found in engineering applications are air and water. To further demonstrate the appropriateness of Eq. (1), correlations between viscosity and temperature for air and water are given below [18]. For air, 1 ¼
123:2ðT
742:6Þ; l

ð4Þ

based on T1 ¼ 294K ð20 CÞ. For water 1 ¼ 29:83 ðT
258:6Þ; l

ð5Þ

based on T1 ¼ 288K ð15 CÞ. A heat source or sink is placed within the flow to allow for possible heat generation or absorption effects. The effect of thermophoresis is taken into account as it helps in understanding mass deposition on surface. In addition, the nonuniform magnetic field is applied in the vertical y-direction normal to the flow direction. Under the usual boundary-layer approximation, the governing equation for this problem can be written as follows: @u @v þ ¼ 0; @x @y

ð6Þ

u


 @u @u 1 @ @u rB2 þv ¼ l
ðu
u1 Þ; @x @y q @y @y q

ð7Þ

u


 @T @T kg @ 2 T l @u 2 rB2 Qo þv ¼ þ ð Þ þ ðu
u1 Þ2 þ ðT
T1 Þ; 2 @x @y qcp @y qcp @y qcp qcp

ð8Þ

u

@c @c @2c @ þv ¼D 2
ðVT cÞ
k1 c; @x @y @y @y

ð9Þ

where x and y are the horizontal and vertical directions, respectively; u, v and T are the fluid x-component (horizontal) of velocity, y-component (vertical) of velocity, and temperature, respectively. c is the mass or particle concentration in the fluid; q; l; m; kg ; cp and r are the fluid density, dynamic viscosity, kinematic viscosity, thermal conductivity, specific heat at constant pressure and electrical conductivity, respectively; BðxÞ and Q0 are the magnetic induction and heat generation/absorption coefficients, respectively; D and VT are the diffusion coefficient and the thermophoretic velocity, respectively; k1 ; u1 and T1 are the chemical reaction parameter, the free stream velocity and temperature, respectively.

4

M. A. Seddeek

The boundary conditions of Eqs. (6)–(9) are uðx; 0Þ ¼ 0; vðx; 0Þ ¼
v0 ðxÞ; Tðx; 0Þ ¼ Tw ; cðx; 0Þ ¼ Cw ; uðx; 1Þ ¼ u1 ; Tðx; 1Þ ¼ T1 ; cðx; 1Þ ¼ C1 ;

ð10Þ

where v0ðxÞ is the wall suction (>0) or blowing (<0) velocity, Tw ; Cw and C1 are the fluid wall temperature, wall mass concentration, and the free stream mass concentration, respectively. In consequence the thermophoretic velocity VT can be expressed in the form [19] VT ¼
kv

rT 1 @T ¼
kv : T T @y

ð11Þ

The value of kv represents the thermophoretic diffusivity, and k is the thermophoretic coefficient defined using k¼

2Cs ðkg =kp þ Ct KnÞC : ð1 þ 3Cm KnÞð1 þ 2kf =kp þ 2Ct KnÞ

ð12Þ

Here, kg and kp are the thermal conductivity of the air and aerosol particles, respectively; constants Cs ; Ct and Cm are determined by experimental data, Cs ¼ 1:147; Ct ¼ 2:20; Cm ¼ 1:146 [20]; C ¼ 1 þ KnðC1 þ C2 expð
C3 =KnÞ and Kn is the Knudsen number, C1 ¼ 1:2; C2 ¼ 0:41 and C3 ¼ 0:88 [21]. Though the value of k can be taken as ranging from 0.2 to 1.2, a representative value for particles smaller than 1lm is 0.5. We introduced a thermophoretic parameter s ¼
kðTw
T1 Þ=T, with s ¼ 0:01; 0:05 and 0.1, in which the corresponding values for
kðTw
T1 Þ were approximately 3, 15 and 30K for a reference temperature T=293 K. By using the following similarity transformations [12]:  u 12 1 1 ; w ¼ ð2mu1 xÞ2 f ðgÞ; s¼y 2mx T
T1 c h¼ ; u¼ ; c1 Tw
T1

ð13Þ

the dimensionless temperature h can also be written as h¼

T
Tr þ hr ; Tw
T1

where hr ¼

Tr
T1 1 ¼ constant; ¼
dðTw
T1 Þ Tw
T1

and its value is determined by the viscosity/temperature characteristics of the fluid and the operating temperature difference DT ¼ Tw
T1 . W is the stream function defined in the usual @W way such that u ¼ @W @y ; v ¼
@x . Substituting Eqs. (2), (3), (11), (13) and (14) into Eqs. (6)–(10) produces the following similarity equations and boundary conditions: 

 1 h
hr 00 0 2 h
hr 0 f
f h þ Ha ðf
1Þ
ff 00 ¼ 0; h
hr hr hr 000




h i h00 þ pr f h0 þ Ecðf 00 Þ2 þ EcHa2 ðf 0
1Þ2 þ Dh ¼ 0;

ð14Þ ð15Þ

5

Finite-element method for the effects of chemical reaction

u00 þ Sc½ðf
sh0 Þu0
sh00 u
cu ¼ 0; f 0 ð0Þ ¼ 0; f 0 ð1Þ ¼ 1;

f ð0Þ ¼ f0 ; hð0Þ ¼ 1; hð1Þ ¼ 0;

ð16Þ

uð0Þ ¼ 0;

ð17Þ

uð1Þ ¼ 1; 1

where the primes denote differentiation with respect to g, f0 ¼ 2v0 ½2mux 1 2 is the dimensionless wall mass transfer coefficient such that f0 > 0 indicates wall suction and f0 <0 indicates wall injection, and Ha2 ¼

2rb2 qu1

Mcp k tk1 c¼ 2 u1 Pr ¼

Ec ¼ D¼

u1 cp ðTw
T1 Þ

2Q0 qcp u1

Sc ¼

m D

(Hartmann number); (Prandtl number); (dimensionless chemical reaction parameter); (Eckert number); (dimensionless heat generation or absorption coefficient); (Schmidt number);

D is the diffusion constant, Q0 the heat generation or absorption coefficient. The value of hr is determined by the viscosity of the fluid under consideration and the operating temperature deference. If hr is large, in other words, if d or ðT1
Tw Þ is small, the effects of variable viscosity can then be neglected. On the other hand, for a smaller value of hr either the fluid viscosity changes markedly with temperature or the operating temperature difference is high. In either case, the variable viscosity effect is expected to become very important. Also bearing in mind that the liquid viscosity varies differently with temperature than that of gas, therefore, it is important to note that hr is negative for liquids and positive for gases.

3 Method of solution 3.1 Finite element method This method basically involves the following steps: (i) (ii) (iii) (iv) (v)

Division of the domain into linear elements, called the finite element mesh, Generation of the element equations using variational formulations, Assembly of element equations as obtained in step (ii), Imposition of the boundary conditions to the equations obtained in (iii), Solution of the assembled algebraic equations.

The assembled equations can be solved by any of the numerical techniques. The details of the steps given above can be found in [23]–[25].

6

M. A. Seddeek

To solve the differential equations (14)–(16), with the boundary conditions (17), we assume f 0 ¼ h ) Eqs. (14)–(17) become


 1 h
hr 00 0 0 2 h
hr ð18Þ h
h h þ Ha ðh
1Þ
fh0 ¼ 0; h
hr hr hr h00 þ pr½f h0 þ Ecðh0 Þ2 þ EcHa2 ðh
1Þ2 þ Dh ¼ 0;

ð19Þ

u00 þ Sc½ðf
sh0 Þu0
sh00 u
cu ¼ 0;

ð20Þ

subject to hð0Þ ¼ 0; hð1Þ ¼ 1;

hð0Þ ¼ 1;

f ð0Þ ¼ f0 ; hð1Þ ¼ 0;

uð0Þ ¼ 0;

ð21Þ

uð1Þ ¼ 1:

For computational purposes and without any loss of generality, 1 has been fixed as 6. The whole domain is divided in to a set of 60 line elements of width 0.1, each element being two nodded.

3.2 Variational formulation Over two nodded linear elements ðge ; geþ1 Þ is given by Zgeþ1

W1 ðf 0
hÞ dg ¼ 0;

ð22Þ

ge

Zgeþ1 ge

Zgeþ1



 1 h
hr 2 h
hr 0 0 h h þ Ha ðh
1Þ
fh0 dg ¼ 0; W2 ½h
h
hr hr hr

ð23Þ

W3 ½h00 þ PRðf h0 þ Ecðh0 Þ2 þ Ec Ha2 ðh
1Þ2 þ DhÞdg ¼ 0;

ð24Þ

W4 ½u00 þ Scððf
sh0 Þu0
sh00 u
cuÞdg ¼ 0;

ð25Þ

00




ge

Zgeþ1 ge

where W1 ; W2 ; W3 and W4 are arbitrary test functions or variations in f ; h; h and u, respectively.

3.3 Finite element formulation The finite element model may be obtained from Eqs. (22)–(25) by substituting finite element approximations of the form f ¼

2 X j¼1

fj wj ;

h¼

2 X j¼1

hj w j ;

h¼

2 X j¼1

hj wj ;

/¼

2 X j¼1

/j wj ;

ð26Þ

7

Finite-element method for the effects of chemical reaction

with W1 ¼ W2 ¼ W3 ¼ W4 ¼ wi ði ¼ 1; 2Þ; where wi are the shape functions for element ðge ; geþ1 Þ and are defined as g
g g
ge ðeÞ ðeÞ ; w2 ¼ ; ge  g  geþ1 : ð27Þ w1 ¼ eþ1 geþ1
ge geþ1
ge The finite element model of the equations is 2

½k11  ½k12 

½k13  ½k14 

32

ff g

3

2

fr1 g

3

7 6 76 7 ½k23  ½k24  76 fhg 7 6 fr2 g 7 7¼6 76 7; 6 7 6 7 ½k33  ½k34  7 54 fhg 5 4 fr3 g 5 f/g fr4 g ½k43  ½k44 

6 21 6 ½k  ½k22  6 6 ½k31  ½k32  4 ½k41  ½k42 

ð28Þ

where ½kmn and ½rm  ðm; n ¼ 1; 2; 3; 4Þ are defined as k11 ij

¼

Zgeþ1

dwj dg; wi dg

k12 ij

¼


ge

k21 ij

Zgeþ1

14 k13 ij ¼ kij ¼ 0;

wi wj dg; ge

¼ 0;

k22 ij

¼

Zgeþ1 

 dwj dwi dwj
Ha2 ðwi ; wj Þ þ
f wi dg ¼ 0; dg dg dg

ge

k23 ij

Zgeþ1 " ¼
ge

k24 ij ¼ 0; k32 ij

¼

! #
f h
0
dwj 1 2 ðh
1Þ 0 þ Ha h wi wi wj
w w dg ¼ 0; wi wj
hr hr dg hr i j

k31 ij ¼ 0; 

Zgeþ1 pr

Ech0 wi

 dwj 2
þ Ec Ha ðwi wj h
2wi wj Þ dg ¼ 0; dg

ð29Þ

ge

k33 ij ¼

  Zgeþ1
dwj dwi dwj þ pr
f wi þ Dwi wj dg ¼ 0; dg dg dg ge

k34 ij ¼ 0; k44 ij

42 43 k41 ij ¼ kij ¼ kij ¼ 0;

  Zgeþ1
dwi dwj dw þ Sc ð
f
sh0 Þwi i
sh00 wi wj
cwi wj dg ¼ 0; ¼ dg dg dg ge

ri1 ¼ 0;


 dh geþ1 ri2 ¼
wi ; dg ge


 dh geþ1 ri3 ¼
wi ; dg ge


 du geþ1 ri4 ¼
wi ; dg ge

where
f ¼

2 X

f
i wi ;

h¼

2 X

i¼1


h ¼

2 X i¼1

hi wi ;

h0 ¼

2 X

i¼1

hi wi ;

0

h ¼

2 X i¼1

h0i wi ;

i¼1

h0i wi ;

00

h ¼

2 X i¼1

ð30Þ h00i wi :

8

M. A. Seddeek

After assembly of the elements, a system of nonlinear equations is obtained, which is solved iteratively, and the required accuracy of 0.0005 is obtained. Of special importance for this flow and heat transfer situation are the skin-friction coefficient f 00 ð0Þ, wall heat transfer
h0 ð0Þ and wall deposition flux u0 ð0Þ. These are defined as follows:  1 sf @u  Cf ¼ 2 ¼ ðRex Þ
2 f 00 ð0Þ; sf ¼ l ; ð31Þ @y y¼0 qu1 Nux ¼

1 qw x 1 ¼
ðRex Þ
2 h0 ð0Þ; ðTw
T1 Þk 2

qw ¼
k

 @T  ; @y y¼0

Fig. 1. Comparison of steady-state velocity profiles with those of White and Chamkha

Fig. 2. Effects of hr on velocity profiles

ð32Þ

Finite-element method for the effects of chemical reaction

Stx ¼

1
Js 1 ¼ ðRex Þ
2 u0 ð0Þ; u1 c1 Sc

Js ¼
D

 @c  ; @y y¼0

9

ð33Þ

where Rex ¼ 2um1 x is the local Reynolds number.

4 Results and discussion In order to assess the accuracy of our computed results, our results for steady-state values of the velocity profiles were compared to those of the values computed by White [22] and

Fig. 3. Effects of hr on temperature profiles

Fig. 4. Effects of hr on concentration profiles

10

M. A. Seddeek

Fig. 5. Effects of c on concentration profiles

Fig. 6. Effects of Ha on velocity profiles

Chamkha et al. [12] for different values of the parameters. These are shown in Fig. 1. It was observed that our results agree very well with those of White and Chamkha. To study the behavior of the velocity, temperature and concentration profiles, curves are drawn for various values of the parameters that describe the flow. Figures 2, 3 and 4 illustrate the effects of variable viscosity parameter hr on the dimensionless velocity f 0 g, temperature hðgÞ and concentration uðgÞ, respectively. It is clearly seen that as hr ! 0 the boundary-layer thickness decreases and the velocity distribution and the concentration increases while the temperature distribution decreases. This is because for a given fluid d is fixed, smaller hr implies higher temperature difference between the wall and the ambient fluid. The results presented in this paper demonstrate quite clearly that hr , which is an indicator of the

Finite-element method for the effects of chemical reaction

11

Fig. 7. Effects of Ha on temperature profiles

Fig. 8. Effects of Ha on concentration profiles

variation of viscosity with temperature, has a substantial effect on the velocity and temperature distributions within the boundary layer over a heat surface as well as the drag and heat transfer characteristics. The effects of the chemical reaction parameter with variable viscosity are very important in the concentration field. The nondimensional concentration distribution uðgÞ for different chemical reaction parameters are shown in Fig. 5. There is a fall in the concentration due to increasing values of the chemical reaction parameter. Figures 6–8 present typical profiles for the velocity, temperature and concentration for various values of the Hartmann number Ha, respectively for a physical situation with heat

12

M. A. Seddeek

Fig. 9. Effects of f0 on velocity profiles

Fig. 10. Effects of f0 on temperature profiles

generation and thermophoretic effect. It is clearly shown that the velocity and concentration increases with increasing the Hartmann number Ha as shown in Figs. 6–8. The effects of the Hartmann number on the heat transfer are shown in Fig. 7. It is observed that the temperature decreases when Ha increases. Application of a magnetic field moving with the free stream has the tendency to induce a motive force which increases the motion of the fluid and decreases its boundary layer. This is accompanied by a decrease in the fluid temperature and a slight increase in the concentration. Figures 9–11 illustrate the effect of varying the wall mass transfer coefficient f0 on the velocity, temperature and concentration profiles, respectively. The imposition of wall fluid

Finite-element method for the effects of chemical reaction

13

Fig. 11. Effects of f0 on concentration profiles

Fig. 12. Effects of D on velocity profiles

suction ðf0 > 0Þ for this problem has the effect of reducing all of the hydrodynamic, thermal and concentration boundary layers causing the fluid velocity and its concentration to increase while decreasing its temperature. But imposition of wall fluid injection or blowing produces the opposite effect, namely decreases the fluid velocity and concentration and increases its temperature. The decreasing thickness of the concentration layer is caused by two effects: (i) the direct action of suction, and (ii) the indirect action of suction causing a thinner thermal boundary layer, which corresponds to higher temperature gradients, a consequent increase in the thermophoretic force and higher concentration gradients.

14

M. A. Seddeek

Fig. 13. Effects of D on temperature profiles

Fig. 14. Effects of Sc on concentration profiles

Figures 12 and 13 show the effect of dimensionless heat generation or absorption coefficient D on the velocity and the temperature distribution. As shown, increasing the values of D produces increases in the velocity g > 0:3 and temperature distributions of the fluid. This is expected since heat generation ðD > 0Þ causes the thermal boundary layer to become thicker and the fluid to become warmer. In the case that the strength of the heat source is relatively large, the maximum fluid temperature does not occur at the wall but rather in the fluid region close to it. Conversely, the presence of a heat sink or a heat absorption effect ðD < 0Þ causes a reduction in the thermal state of the fluid.

Finite-element method for the effects of chemical reaction

15

Fig. 15. Effects of s on concentration profiles

Fig. 16. Effects of hr and Ha on skin-friction coefficients

In Figs. 14 and 15, we have shown typical concentration profiles for various values of the Schmidt number Sc and the thermophoretic parameter s, respectively. It is clear from Fig. 14 that the concentration layer thickness decreases while the concentration increases as the Schmidt number Sc increases; this is analogous to the effect of increasing the Prandtl number on the thickness of a thermal boundary layer. For the parametric condition used in Fig. 15, the effect of increasing the thermophoretic parameter s is limited to increasing slightly the wall slope of the concentration profile for g < 1:1 but decreasing the concentration for values of g > 1:1. We notice that positive s indicates a cold surface, but is negative for a hot surface. The skin-friction, the wall heat transfer and the wall deposition flux coefficients values are evaluated from Eqs. (31)–(33) and plotted in Figs. 16, 17 and 18, respectively, as a function of

16

M. A. Seddeek

Fig. 17. Effects of hr and Ha on wall heat transfer

Fig. 18. Effects of hr and Ha on wall deposition flux

Hartmann number Ha. It is observed that the skin-friction, the wall heat transfer and the wall deposition flux increases with increasing the values of Ha and hr . Figure 19 illustrates the variation of wall deposition flux with the chemical reaction parameter for several values of the Hartmann number. We see that increasing values of c decrease the wall deposition flux for all values of Ha. It is hoped that the present work will serve as a vehicle for understanding more complex problems involving the various physical effects investigated in the present problem.

Finite-element method for the effects of chemical reaction

17

Fig. 19. Effects of c and Ha on wall deposition flux

References [1] Gebhart, B.: Heat transfer. New York: McGraw Hill, 1971. [2] Soundalgeker, V. M.: Free convection effects on Stokes problem for a vertical plate. J. Heat Transf. ASME 99C, 499–501 (1977). [3] Soundalgeker, V. M., Patil, M. R.: Stokes problem for vertical plate with constant heat flux. J. Astrophys. Space Sci. 64,165–172 (1980). [4] Gebhart, B., Pera L.: The nature of vertical natural convection flow resulting from the combined buoyancy effects of thermal and mass diffusion. J. Heat Mass Transf. 14, 2025–2050 (1971). [5] Fairbanks, D. F., Wike, C. R.: Diffusion and chemical reaction in an isothermal laminar flow along a soluble flat plate. Ind. Eng. Chem. Res. 42, 471–475 (1950). [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. Astrophys. Space Sci. 100, 159–164 (1984). [7] Muthucumaraswamy, R, Ganesan, P.: On impulsive motion of a vertical plate with heat flux and diffusion of chemically reactive species. Forsch. Ing.-Wes. 66, 17–23 (2000). [8] Ghaly, A. Y., Seddeek, M. A.: Chebyshev finite difference method for the effects of chemical reaction, heat and mass transfer on laminar flow along a semi infinite horizontal plate with temperature dependent viscosity. Chaos Solitons Fractals 19, 61–70 (2004). [9] Goren, S. L.: The role of thermophoresis in laminar flow of a viscous and incompressible fluid. J. Colloid Interface Sci. 61, 77 (1977). [10] Tsai, R.: A simple approach for evaluating the effect of wall suction and thermophoresis on aerosol particle deposition from a laminar flow over a flate plate. Int. Comm. Heat Mass Transf. 26, 249– 257 (1999). [11] Chang, Y. P.,Tsai, R., Sui, F. M.: The effect of thermophoresis on particle deposition from a mixed convection flow onto a vertical flat plate. J. Aerosol Sci. 30,1363–1378 (1999). [12] Chamkha, A. J., Camille, I.: Effects of heat generation/absorption and thermophoresis on hydromagnetic flow with heat and mass transfer over a flat surface. Int. J. Num. Meth. Heat Fluid Flow 10, 432–448 (2000). [13] Chamkha, A. J., Pop, I.: Effect of thermophoresis particle deposition in free convection boundary layer from a vertical flat plate embedded in a porous medium. Int. Comm. Heat Mass Transf. 31, 421–430 (2004). [14] Herwing, H., Gersten, K.: The effect of variable properties on laminar boundary-layer flow. Wa¨rme Stoffu¨bertr. 20, 47–52 (1986).

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M. A. Seddeek: Finite-element method for the effects of chemical reaction

[15] Seddeek, M. A.: The effect of variable viscosity on hydromagnetic flow and heat transfer past a continuously porous boundary with radiation. Int. Comm. Heat Mass Transf. 27, 1037–1048 (2000). [16] Seddeek, M. A.: Effects of magnetic field and variable viscosity on forced non–Darcy flow about a flat plate with variable wall temperature in porous media in the presence of suction and bellowing. J. Appl. Mech. Tech. Phys. 43, 13–17 (2002). [17] Ling, J. X., Dybbs, A.: Forced convection over a flat plate submersed in a porous medium, variable viscosity case. Paper 87-WA/HT-23. New York: ASME 1987. [18] Weast, R. C.: Hand book of chemistry and physics, 67th ed. Boca Raton: C.R.C. Press 1986–1987. [19] Talbot, L., Cheng, R. K., Scheffer, R. W., Wills, D. P: Thermophoresis of particles in a heated boundary layer. J. Fluid Mech. 101, 737–758 (1980). [20] Shen, C.: Thermophoretic deposition of particles onto cold surface of bodies in two–dimensional and axi-symmetric flows. J. Colloid Interface Sci. 127, 104–115 (1989). [21] Batchelor, G. K., Shen, C.: Thermophoretic deposition in gas flow over cold surfaces. J. Colloid Interface Sci. 107, 21–37 (1985). [22] White, F.: Viscous fluid flow, 1st ed. New York: McGraw-Hill 1974. [23] Reddy, J. N.: An introduction to the finite element method. McGraw-Hill 1984. [24] Bejan, A.: Convection heat transfer, 3rd ed. New York: Wiley 2004. [25] Bhargava, R., Kumar, L., Takhar, H. S.: Finite element solution of mixed convection micropolar flow driven by a porous stretching sheet. Int. J. Engng. Sci. 41, 2161–2178 (2003). Author’s address: M. A. Seddeek, Al-Qasseem University, College of Science, Mathematics Department, P. O. Box 237, Buriedah, 81999, Kingdom of Saudi Arabia