Commun Nonlinear Sci Numer Simulat 15 (2010) 3435–3443

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Commun Nonlinear Sci Numer Simulat

journal homepage: www.elsevier .com/locate /cnsns

Flow near the two-dimensional stagnation-point on an infinitepermeable wall with a homogeneous–heterogeneous reaction

W.A. Khan a,*, I. Pop b

a Department of Engineering Sciences, PN Engineering College, National University of Sciences and Technology, PNS Jauhar, Karachi 75350, Pakistanb Faculty of Mathematics, University of Cluj, R-3400 Cluj, CP 253, Romania

a r t i c l e i n f o a b s t r a c t

Article history:Received 11 May 2009Received in revised form 21 December 2009Accepted 23 December 2009Available online 4 January 2010

Keywords:SteadyStagnation-point flowForced convectionHomogeneous–heterogeneous reactionsNumerical solution

1007-5704/$ - see front matter � 2009 Elsevier B.Vdoi:10.1016/j.cnsns.2009.12.022

* Corresponding author. Tel.: +92 21 48503023.E-mail address: wkhan_2000@yahoo.com (W.A.

This paper studies the effects of mass transfer (suction and injection) on the problem ofsteady-state boundary layer flow near the forward stagnation-point of an infinite perme-able wall with a homogeneous (bulk) reaction given by isothermal cubic autocatalatorkinetics and the heterogeneous (surface) reaction by first-order kinetics. The case of animpermeable wall has been considered by Chaudhary and Merkin (1994) [1]. It is foundthat the mass transfer parameter considerably affects the flow characteristics.

� 2009 Elsevier B.V. All rights reserved.

1. Introduction

There is a wide range of chemical reactions, many of which have important practical applications, which proceed onlyvery slowly, or not at all, except in the presence of a catalyst. Some chemical reacting systems involve both homogeneousand heterogeneous reactions, with examples occurring in combustion, catalysis and biochemical systems [1]. The interactionbetween the homogeneous reactions in the bulk of the fluid and the heterogeneous reactions occurring on some catalyticsurface is generally very complex, involving the production and consumption of reactant species at different rates both with-in the fluid and on the catalytic surface as well as the feedback on these reaction rates through temperature variations withinthe reacting fluid, which in turn modify the fluid motion. Valuable reviews of the chemical aspects of surface, or heteroge-neous, reactions have been given by Gray and Scott [2], and Scott [3].

Chaudhary and Merkin [4] have proposed a simple model of homogeneous–heterogeneous reactions for the forced con-vection boundary layer flow near the two-dimensional forward stagnation-point on a non-permeable infinite wall withinwhich an isothermal cubic autocatalytic reaction takes place, given schematically by

Aþ 2B! 3B; rate ¼ k1�a�b2 ð1Þ

while on the catalyst surface we have the single, isothermal, first-order reaction

A! B; rate ¼ ks�a ð2Þ

where �a and �b are concentrations of chemical species A and B, and k1 and k2 are constants. It is assumed that there is noautocatalyst B in the external flow and that A has a constant concentration a0. Thus, the reaction scheme (1) guarantees

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Khan).

3436 W.A. Khan, I. Pop / Commun Nonlinear Sci Numer Simulat 15 (2010) 3435–3443

in a natural way that the reaction rate will be zero in the external flow and thus zero at the outer edge of the boundary layer.It is also assumed (for simplicity) that there is negligible heat release by the reaction. This enables the effect of thermalexpansion to be ignored with the basic boundary layer flow than being unaffected by the reactions. It is worth mentioningthat the ambient fluid cannot always be pure and there are situations where it is mixed, owing to natural or artificial pro-cesses, with some foreign species, e.g. air mixed with H2O, benzene or ammonia. Due to the concentration difference of suchspecies it can diffuse from the surface to the ambient fluid or vice versa. Consequently, for considerable concentration dif-ferences, convective flow can occur not only due to temperature difference but also due to concentration difference or thecombination of these two [5]. Hence, it can be deduced that the combined effect of heat and mass transfer (chemical reac-tions) can play an important role in some industrial operations.

The aim of this paper is to extend the case of the steady boundary layer flow near to the stagnation-point on a non-permeable infinite wall to the case of a permeable wall with suction and injection (blowing). It is worth mentioning thatone method of controlling boundary layer flows is to inject or to withdraw fluid through a permeable (porous) boundingwall.

2. Basic equations

Consider the steady boundary layer flow near the stagnation-point of an infinite permeable wall. It is assumed that a sim-ple homogeneous–heterogeneous reaction model given by the reaction scheme (1) exists as proposed by Chaudhary andMerkin [4]. Under this assumptions and boundary layer approximations, the basic equations are,

@�u@�xþ @

�v@�y¼ 0 ð3Þ

�u@�u@�xþ �v @

�v@�y¼ �ue

d�ue

d�xþ t

@2�u@�y2 ð4Þ

�u@�a@�xþ �v @

�a@�y¼ DA

@2�a@�y2 � k1�a�b2 ð5Þ

�u@�b@�xþ �v @

�b@�y¼ DB

@2�b@�y2 þ k1�a�b2 ð6Þ

where �x and �y are Cartesian coordinates along the wall and normal to it, respectively, �u and �v are the velocity componentsalong the �x- and �y-axes, t is the kinematic viscosity, �ueð�xÞ is the velocity outside the boundary layer and DA and DB are therespective diffusion coefficients. The boundary conditions of Eqs. (3)–(6) to be applied are

�v ¼ �vw; �u ¼ 0; DA@�a@�y¼ ks�a; DB

@�b@�y¼ �ks�a at �y ¼ 0

�u ¼ �ueð�xÞ ! U0ð�x=lÞ; �a! a0;�b! 0 as y!1

ð7Þ

where �vw is the constant mass flux velocity with �vw > 0 for injection and �vw < 0 for suction, respectively, U0 and l are veloc-ity and length scales and a0 is a constant.

We look for a similarity solution of Eqs. (3)–(6) of the following form

x ¼ �x=l; y ¼ Re1=2ð�y=lÞ; �w ¼ ðlU0tÞ1=2xf ðyÞgðyÞ ¼ �a=a0; hðyÞ ¼ �b=a0; ueðxÞ ¼ �ueð�xÞ=U0

ð8Þ

where �w is the stream function which is defined in the usual way as �u ¼ @�w=@�y and �v ¼ @�w=@�x;Re ¼ U0l=t is the Reynoldsnumber and ue(x) = x for the stagnation-point flow. Thus, we have

�u ¼ @�w@�y¼ U0xf 0ðyÞ; �v ¼ � @

�w@�x¼ �ðU0m=lÞ1=2f ðyÞ ð9Þ

where prime denotes differentiation with respect to y. Substituting (8) and (9) into Eqs. (4)–(6), we obtain the following or-dinary differential equations

f 000 þ ff 00 þ 1� f 02 ¼ 0 ð10Þ1Sc

g00 þ fg0 � Kgh2 ¼ 0 ð11ÞdSc

h00 þ fh0 þ Kgh2 ¼ 0 ð12Þ

where the dimensionless parameter K ¼ k1a20l=U0 gives a measure of the strength of the homogeneous reaction, Sc = (t/DA) is

the Schmidt number, d = (DB/DA) is the ratio of the diffusion coefficients and primes denote differentiation with respect to y.The boundary conditions (7) of Eqs. (10)–(12) become

Table 1Compar

f0

�3�2�1

01234

W.A. Khan, I. Pop / Commun Nonlinear Sci Numer Simulat 15 (2010) 3435–3443 3437

f ð0Þ ¼ f0; f 0ð0Þ ¼ 0; f 0ðyÞ ! 1 as y!1 ð13aÞg0ð0Þ ¼ Ksgð0Þ; dh0ð0Þ ¼ �Ksgð0ÞgðyÞ ! 1; hðyÞ ! 0 as y!1

ð13bÞ

where f0 ¼ ��vw=ðl=U0tÞ1=2 is the constant suction parameter (f0 > 0) or constant injection parameter (f0 < 0), respectively,and Ks = (kslRe�1/2/DA) measures the strength of the heterogeneous (surface) reaction.

In most applications we expect the diffusion coefficients of chemical species A and B to be of a comparable size. This leadsus to make the further assumption that the diffusion coefficients DA and DB are equal, i.e. to take d = 1 [4]. In this case, wehave from (13b),

gðyÞ þ hðyÞ ¼ 1 ð14Þ

Thus, Eqs. (11) and (12) reduce to

1Sc

g00 þ fg0 � Kgð1� gÞ2 ¼ 0 ð15Þ

and are subject to he boundary conditions

g0ð0Þ ¼ Ksgð0Þ; gðyÞ ! 1 as y!1 ð16Þ

Therefore, we have to solve numerically Eqs. (10) and (15) along with the boundary conditions (13a) and (16) for somevalues of the parameters f0, Sc, K and Ks.

A physical quantity of interest is the skin friction coefficient Cf, which can be very easily shown to be given by

Re1=2x Cf ¼ f 00ð0Þ ð17Þ

where Rex ¼ �ueð�xÞ�x=t is the local Reynolds number.

3. Results and discussion

The system of ordinary differential equations (10) and (15) subject to the boundary conditions (13a) and (16) has beensolved numerically for some values of the parameters f0, Sc, K and Ks using an implicit finite difference method. We noticethat the results presented in this study are valid only in the small region near the forward stagnation-point of an infinitewall, and are not applicable outside this region. Values of the dimensionless skin friction coefficient f

00(0) are given in Table

1 for some values of the parameter f0. The values reported by Katagiri [6] using an iterative numerical quadratures and by Loket al. [7] using the Keller-box method were also included in this table. It is seen that the present results are in excellentagreement with both results obtained by Katagiri [6] and Lok et al. [7]. We notice that for an impermeable wall (f0 = 0)the values of f

00(0) reported by Hiemenz [8] is f0 = 1.233. Further, in Tables 2 and 3 are given some data for the ratios of

numerical results to those of an asymptotic analysis given by (34) for f0� 1 (large suction) and for f0��1 (large injection),respectively. It can be concluded from these tables that asymptotic analyses might be valid for f0 P 5 (suction) and f0 6 �3(injection), respectively. We are therefore confident that the results presented in this paper are very accurate.

Fig. 1 presents the variation of the velocity f0(y) with y for f0 = 0 (impermeable wall), f0 = 2 and 4 (suction) and f0 = �2 and

�4 (injection), respectively. It is concluded from this figure that, as expected, the boundary layer thickness is greater forinjection (f0 < 0) than for suction (f0 > 0). The variation of the concentration profiles g(y) with y is shown in Fig. 2 for severalvalues of the parameter f0 when Sc = 1 and Sc = 5. It can be seen that the profiles becomes flatten as the suction parameterincreases. However, they are much flatten for Sc = 5 than for Sc = 1.

The variation of g(0) with the parameters Ks and K are shown in Figs. 3–5 for Sc = 1 and 5. Fig. 3 is for f0 = 0 (impermeablewall), while Figs. 4 and 5 are for f0 = �3 (injection) and f0 = 1 (suction), respectively. One can see from these figures that g(0)decreases with the increase of the parameters K(>0) and Ks, and its values are greater for Sc = 5 than for Sc = 1 when f0 = 0(Fig. 3) and are almost the same when f0 = �3 (injection). It is worth mentioning that the results of Chaudhary and Merkin[4] are for an impermeable wall (f0 = 0) and in that case it is possible to go further with values of g(0) in Fig. 4 for positive

ison of f00(0) for selected values of f0.

Katagiri [6] Lok et al. [7] Present results

0.329456 0.3295 0.329450.47581 0.4759 0.475810.756574 0.7567 0.756571.232588 1.2327 1.232591.889303 1.8895 1.889312.670006 2.6703 2.670063.526497 3.5268 3.526644.428637 4.4291 4.42895

Table 2Comparison of numerical and approximate values of f

00(0) and g(0) for large values of the suction parameter f0.

f0 f00(0) g(0) g(0)

Sc = 1, K = 1 Sc = 5, K = 1

Numerical Eq. (23) Numerical Eq. (24) Numerical Eq. (24)

2 2.670 3.000 0.67813 0.500 0.90958 0.9004 4.429 4.500 0.80310 0.750 0.95246 0.9506 6.308 6.333 0.85837 0.833 0.96777 0.9678 8.239 8.250 0.88949 0.875 0.97562 0.975

10 10.194 10.200 0.90943 0.900 0.9804 0.98012 12.163 12.167 0.92329 0.917 0.98361 0.98314 14.141 14.143 0.93347 0.929 0.98592 0.98616 16.123 16.125 0.94127 0.938 0.98766 0.98818 18.110 18.111 0.94744 0.944 0.98901 0.98920 20.099 20.100 0.95243 0.950 0.99010 0.990

Table 3Comparison of numerical and approximate values of f

00(0) and g(0) for large values of the injection parameter f0.

f0 f00(0) g(0) g(0)

Sc = 1, K = 1 Sc = 5, K = 1

Numerical Eq. (35) Numerical Eq. (36) Numerical Eq. (36)

�2 0.47581 0.50000 0.00113 0 0 0�4 0.24904 0.25000 0 0 0 0�6 0.16654 0.16667 0 0 0 0�8 0.12497 0.12500 0 0 0 0�10 0.09999 0.10000 0 0 0 0�12 0.08333 0.08330 0 0 0 0�14 0.07143 0.07142 0 0 0 0�16 0.06250 0.06250 0 0 0 0�18 0.05556 0.05556 0 0 0 0�20 0.05000 0.05000 0 0 0 0

0 1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1f0 = -4, -2, 0, 2, 4

f' (y)

y

Fig. 1. Velocity profiles for several values of the parameter f0.

3438 W.A. Khan, I. Pop / Commun Nonlinear Sci Numer Simulat 15 (2010) 3435–3443

values of K, but when f0 < 0 (injection), the values of g(0) become zero close to K = 0. However, it is possible to go further forK > 0 when f0 > 0 (suction).

Finally, Figs. 6 and 7 illustrate the variation of g(0) with f0 for several values of Ks and K when Sc = 1 and 5. It is seen thatthe solution of Eq. (14) subjected to the boundary conditions (15) is unique when f0 P 0 (suction), while dual solutions arefound to exist when f0c < f0 < 0 (injection) and no solutions exist when f0 < f0c < 0, where f0c(<0) is the critical value of f0.Therefore, for a particular value of f0, the solution of Eq. (14) could be obtained up to the critical value f0c(<0), where the

y0 2 4

0

0.2

0.4

0.6

0.8

1

g (y

)

K = 1

Ks = 1

Sc = 5

f 0=2f 0=1f 0=0

f 0=-

1

(b)

y0 2 4 6

0

0.2

0.4

0.6

0.8

1g

(y)

K = 1

Ks = 1

Sc = 1f 0

=2f 0=1

f 0=0

f 0=-

1

(a)

Fig. 2. Concentration profiles g(y) for several values of the parameter f0: (a) Sc = 1; (b) Sc = 5.

0 1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1

K=0, 1, 2

Sc = 5

f0 = 0

g (0

)

Ks

(b)

0 1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1

K=0, 1, 2

Sc = 1

f0 = 0

g (0

)

Ks

(a)

Fig. 3. Variation of g(0) with Ks for several values of K when f0 = 0 (impermeable wall): (a) Sc = 1; (b) Sc = 5.

W.A. Khan, I. Pop / Commun Nonlinear Sci Numer Simulat 15 (2010) 3435–3443 3439

upper branch solution meets the lower branch solution. Beyond this critical value, no solution is obtained (using the bound-ary layer approximations) since the boundary layer separates from the surface. The lower branch solution terminates at f0 = 0with values of g(0) given in Table 4. It is seen that the values of g(0) decreases for all values of K, Ks and Sc considered. Fromour computations results in that the critical points or turning points terminate at f0c � �4, where g(0) = 0 as can be seen fromFigs. 6 and 7. This is in full agreement with the asymptotic solution given by (34) for large values of injection parameter f0. Tothis end, we wish to mention that as in similar physical situations, we postulate that the upper branch solutions in the pres-ent problem are physically stable and occur in practice, whilst the lower branch solutions are not physically obtained. Thispostulate can be verified by performing a stability analysis but this is beyond the scope of the present paper.

4. Asymptotic solutions

4.1. Large suction (f0� 1)

Following Watson [9] in solving the general two-dimensional boundary layer flow with a uniform velocity of suction forf0� 1, we introduce the new variables

-4 -3 -2 -1 00

0.2

0.4

0.6

0.8

1

Ks =0.025, 0.05, 0.075

K

Sc = 5

f0 = -3

(b)

-4 -3 -2 -1 00

0.2

0.4

0.6

0.8

1

Ks =0.025, 0.05, 0.075

K

Sc = 1

f0 = -3

(a)

g (0

)

g (0

)

Fig. 4. Variation of g(0) with K for selected values of Ks when f0 = �3 (injection): (a) Sc = 1; (b) Sc = 5.

K-4 0 4 8 12 16

0

0.2

0.4

0.6

0.8g

(0)

(b)

Sc = 3

f0 = 1

Ks = 1, 2, 3

K-4 0 4 8 12 16

0

0.2

0.4

0.6

0.8

g (0

)

(a)

Sc = 1

f0 = 1

Ks = 1, 2, 3

Fig. 5. Variation of g(0) with K for selected values of Ks when f0 = 1 (suction): (a) Sc = 1; (b) Sc = 3.

3440 W.A. Khan, I. Pop / Commun Nonlinear Sci Numer Simulat 15 (2010) 3435–3443

f ðgÞ ¼ f0 þ1f0

/ðtÞ þ � � � ; gðgÞ ¼ uðtÞ; t ¼ f0g ð18Þ

Substituting (18) into Eqs. (10) and (15), we obtain

/000 þ /00 þ 1f 20

ð//00 þ 1� /02Þ ¼ 0 ð19Þ

1Sc

u00 þu0 þ 1f 20

½/u0 � Kuð1�uÞ2� ¼ 0 ð20Þ

with the boundary conditions

/ð0Þ ¼ 0; /0ð0Þ ¼ 0; /ð1Þ ¼ 1

u0ð0Þ ¼ Ks

f0uð0Þ; uð1Þ ¼ 1

ð21Þ

where primes denote now the differentiation with respect to t.For large suction, / and u can be expressed in power series of 1/f0 as

-4 -2 0 2 4

0

0.2

0.4

0.6

0.8

1

Sc = 5

K = 2

f0

(b)

Ks = 0.5, 1, 1.5

-4 -2 0 2 4

0

0.2

0.4

0.6

0.8

1

Ks = 0.5, 1, 1.5

Sc = 1

K = 2

f0

g (0

)

g (0

)

(a)

Fig. 6. Variation of g(0) with f0 for several values of Ks with Ks = 1: (a) Sc = 1; (b) Sc = 5.

g (0

)

g (0

)

-4 -2 0 2 4

0

0.2

0.4

0.6

0.8

1

K = 1, 2, 3

Sc = 5

Ks = 1

f0

(b)

-4 -2 0 2 4

0

0.2

0.4

0.6

0.8

1

K = 1, 2, 3

Sc = 1

Ks = 1

f0

(a)

Fig. 7. Variation of g(0) with f0 for several values of K: (a) Sc = 1; (b) Sc = 5.

Table 4Values of g(0) for an impermeable wall (f0 = 0)

Ks g(0) at K = 2 (Fig. 3) K g(0) atKs = 1 (Fig. 5)

Sc = 1 Sc = 5 Sc = 1 Sc = 5

0.5 0.29594 0.19274 1 0.28943 0.3661 0.19441 0.15382 2 0.19441 0.153821.5 0.14862 0.1296 3 0.1105 0.05891

W.A. Khan, I. Pop / Commun Nonlinear Sci Numer Simulat 15 (2010) 3435–3443 3441

/ ¼ /0ðtÞ þ1f 20

/1ðtÞ þ � � � ; u ¼ u0ðtÞ þ1f0

u1ðtÞ þ � � � ð22Þ

Substituting (22) into (19)–(21) and equating the coefficients of successive powers of 1/f0, we obtain two systems of lineardifferential equations with the corresponding boundary conditions. Solving analytically these equations, we obtain

/00 ¼ 1� e�t ; /01 ¼ 2t þ 12

t2� �

e�t

u0 ¼ 1; u1 ¼ �KSc

e�Sct

ð23Þ

3442 W.A. Khan, I. Pop / Commun Nonlinear Sci Numer Simulat 15 (2010) 3435–3443

From (23), the values of f00(0) and g(0) can be expressed as

f 00ð0Þ ¼ f0 þ2f0þ Oðf�3

0 Þ ð24Þ

gð0Þ ¼ 1� KSc

1f0þ O f�2

0

� �ð25Þ

Eqs. (19) and (20) with the boundary conditions (21) are also solved numerically by an implicit finite difference methodfor large values of the suction parameter f0 and the numerical values are compared in Table 2 with the asymptotic approx-imations (24) and (25).

4.2. Large injection

To solve this problem for large injection, Kubota and Fernandez [10] have shown that the boundary layer has to be dividedinto two regions: an inner region adjacent to the wall where the viscosity plays a minor role and an outer region where thetransition occurs from the inner layer to the inviscid flow outside the boundary layer, respectively. A uniformly valid solutioncan be obtained by matching the solutions of these two regions. In order to compare our solution with exact numerical solu-tions for large injection, especially, with the values of f

00(0) and g(0), we follow the method of Katagiri [11] and consider only

solutions of an inner region.It is convenient to look for solutions of (10) and (15) subjected to the boundary conditions (13a) and (16) by introducing

new variables defined as

Fð�f Þ ¼ ½f 0ðgÞ�2; Gð�f Þ ¼ gðgÞ; �f ¼ f ðgÞf �0¼

ffiffiffiep

f ðgÞ ð26Þ

where f �0 ¼ �f0ð> 0Þ. Then, (10) and (15) become

effiffiffiFp

F 00 þ �f F 0 þ 2ð1� FÞ ¼ 0 ð27Þ1Sc

eFG00 þffiffiffiFp

�f G0 � KGð1� GÞ2 ¼ 0 ð28Þ

subject to the boundary conditions

F ¼ 0 at �f ¼ �1

F ! 1 as �f !1ð29aÞ

ffiffiffiep ffiffiffi

Fp

G0 ¼ KsG at �f ¼ �1

G! 1 as �f !1ð29bÞ

To consider an inner (wall) solution, which is valid for fixed �f and small e(�1), we expand Fð�f Þ and Gð�f Þ in power series ofe as

F ¼ F0ð�f Þ þ eF1ð�f Þ þ � � �G ¼ G0ð�f Þ þ

ffiffiffiep

Gð�f Þ þ � � �ð30Þ

Substitution of (30) into Eqs. (27) and (28), and the boundary conditions (29), we obtain the following two sets of equationswith the corresponding boundary conditions

�f F 00 þ 2ð1� F0Þ ¼ 0

�fffiffiffiffiffiF0

pG00 � KG0 1� G2

0

� �¼ 0

F0 ¼ 0; G0 ¼ 0 at �f ¼ �1

ð31Þ

ffiffiffiffiffiF0

pF 000 þ �f F 01 � 2F1 ¼ 0ffiffiffiffiffi

F0

p�f G01 � K 1� 3G2

0

� �G1 ¼ 0

F1 ¼ 0;ffiffiffiffiffiF0

pG00 ¼ KsG1 at �f ¼ �1

ð32Þ

The analytical solution of these equations can be expressed as

F0 ¼ 1� �f 2; F1 ¼ . . .

G0 0; G1 0ð33Þ

Thus, the values of f00(0) and g(0) are given by

f 00ð0Þ ¼ 1jf0jþ Oðjf0j�5Þ; gð0Þ ¼ 0 ð34Þ

W.A. Khan, I. Pop / Commun Nonlinear Sci Numer Simulat 15 (2010) 3435–3443 3443

for large values of the injection parameter jf0j � 1. Therefore, for large suction or large injection, asymptotic analyses yield

f 00ð0Þ ¼f0 þ 2

f0þ O f�3

0

� �for suction

1jf0 jþ Oðjf0j�5Þ for injection

(ð35Þ

gð0Þ ¼1� K

Sc1f0þ O f�2

0

� �for suction

0 for injection

(ð36Þ

Eqs. (31) and (32) are also solved with the corresponding boundary conditions numerically by an implicit finite differencemethod for large values of injection parameter f0 and the numerical values are compared in Table 3 with the asymptoticapproximations (35) and (36).

5. Conclusions

In this paper we have studied the effects of the mass transfer parameter on the forced convection boundary layer flownear a forward two-dimensional stagnation-point within which an isothermal cubic autocatalytic reaction given schemati-cally by Eqs. (1) and (2). The governing nonlinear ordinary differential equations were solved numerically using an implicitfinite difference method. We discussed the effects of the governing parameters on the fluid flow and concentration charac-teristics. For convenience, we considered Schmidt number one and five. A new feature that emerges from our results is theexistence of dual solutions for the injection parameter f0(<0) in the range f0c < f0 < 0. It is found that these solutions terminateat f0 = 0 with values given in Table 2.

Acknowledgements

The authors wish to express their very sincerely thanks to the reviewers for their valuable comments and suggestions.

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