Influence of chemically reactive species and a

Received: 14 February 2017

Revised: 1 May 2017

Accepted: 1 May 2017

DOI: 10.1002/htj.21294

RESEARCH ARTICLE

Influence of chemically reactive species and a volumetric heat source or sink on mixed convection over an exponentially decreasing mainstream Prabhugouda M. Patil1

Nafisabanu Kumbarwadi1

Ebrahim Momoniat2 1 Department of Mathematics, Karnatak

University, Pavate Nagar, Dharwad-580 003, Karnataka, India 2 DST-NRF Centre of Excellence in Mathe-

matical and Statistical Sciences (CoE-MaSS), School of Computer Science and Applied Mathematics, University of Witwatersrand, Private Bag-3, Wits-2050, Johannesburg, South Africa

Abstract In this paper, we investigate mixed convection flow over an exponentially decreasing freestream velocity in presence of nonlinear chemically reactive species and a volumetric heat source or sink. Nonsimilar transformations are used to reduce the boundary layer equations into dimensionless equations and are further solved by the implicit finite difference scheme in combination with the quasilinearization technique. The influence of various governing parameters such as the volumetric heat source/sink parameter (Q), the ratio of buoyancy forces (N), the Richardson number (Ri), and the chemical reaction parameter (Δ) on the flow, thermal and species concentration fields are discussed and presented in terms of graphs. The numerical

2𝑣0 ); C, species concentration (kg m−3 ); 𝐶𝑓 , local skin-friction coef(𝑈∞ 𝜈)1∕2 −1 −1 ficient; 𝐶𝑝 , specific heat at constant pressure (J K kg ); 𝐶𝑤 , concentration at the wall (kg m−3 ); 𝐶𝑤0 , reference concentration;

Nomenclature: A, suction/injection parameter (𝐴 = −

𝐶∞ , ambient species concentration; 𝑓 , dimensionless stream function; 𝐹 , dimensionless velocity; 𝑔, acceleration due to gravity (m s−2 ); 𝐺, dimensionless temperature; 𝐺𝑟, 𝐺𝑟∗ , Grashof numbers due to temperature and species concentration, respectively; 𝐻, dimensionless species concentration; 𝐾0 , chemical reaction rate; L, characteristic length (m); 𝑛, exponent parameter; N, ratio of buoyancy forces; 𝑁𝑢, Nusselt number; Pr, Prandtl number (𝜈∕𝛼); 𝑄0 , heat generation coefficient; 𝑄, volumetric heat source 𝑄0 𝜈 𝑈∞ 𝐿 ); Ri, Richardson number (𝑅𝑖 = 𝐺𝑟2 ); 𝑆𝑐, Schmidt number or sink parameter (𝑄 = 2 ); 𝑅𝑒, Reynolds number (𝑅𝑒 = 𝜈 𝜌𝐶𝑝 𝑈∞

Re

(𝜈∕𝐷𝑚 ); 𝑇 , temperature (K); 𝑇𝑤 , temperature at the wall (K); 𝑇∞ , ambient temperature (K); 𝑢, velocity component in the 𝑥 direction (m s−1 ); 𝑣, velocity component in the 𝑦 direction(m s−1 ); 𝑥, 𝑦, Cartesian coordinates (m) Greek Symbols: 𝛼, thermal diffusivity (m2 s−1 ); 𝛽, 𝛽 ∗ , volumetric coefficients of the thermal and concentration expansions, respectively (K−1 ); 𝜉, 𝜂, transformed variables; 𝜇, dynamic viscosity (kg m−1 s−1 ); Δ, Chemical reaction parameter (Δ =

𝐾0 (𝐶−𝐶∞ )𝑛−1 𝜈 ); 2 𝑈∞

𝜈, kinematic viscosity (m2 s−1 ); 𝜌, density (kg m−3 ); 𝜓, stream function

Subscripts: w, condition at the wall; e, mainstream condition; 𝜉, 𝜂, partial derivatives with respect to these variables, respectively. Heat Transfer—Asian Res. 2018;47:111–125.

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© 2017 Wiley Periodicals, Inc.

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investigation reveals that the increase in volumetric heat source/sink parameter Q increases the temperature profile about 69% in presence of injection and the concentration profile decreases about 56% for Δ = −0.5 and increases around 53% for Δ = 0.5 as n increases from 1 to 2. KEYWORDS chemical reaction, mixed convection, nonsimilar solution, suction/ injection, volumetric heat source/sink

1

I N T RO D U C T I O N

The contemporary effects of species concentration and temperature gradients in the fluid velocity are known as double diffusive flows. This is due to the increasing applications in various areas such as in heating and cooling processes, heat exchangers, crystal growth, nuclear reactor technology, chemical processes, and drying processes. A number of investigators have studied double diffusive mixed convection flows with various geometry combinations.1–4 The surface velocity is kept constant and the mainstream velocity is assumed to be decreased exponentially to which the adverse pressure gradient is established. This adverse pressure gradient results in boundary layer separation. However, there have been numerous studies that took place on the characteristics of boundary layer separation and control of flow separation. For example, Chiam5 presented the numerical solution of time-independent boundary layer flow with an exponentially decreasing velocity distribution supplementing the analytical study of Curle.6 Roy and Saikrishnan7 have reported that boundary layer separation can be controlled or delayed by nonuniform suction. Patil and colleagues8 investigated the influence of a heat source/sink and suction/injection on mixed convection flow over exponentially decreasing external flow velocity. The problem of nonsimilarity in the flow is more important in practical applications. The nonsimilarity in the flow occurs due to the surface mass transfer or main stream velocity or curvature of the surface or possibly due to all of these effects.9–11 Many investigators confined their studies to either steady, nonsimilar solutions or unsteady, self-similar solutions due to their mathematical complexities. The heat and mass transfer with chemical reactions have attracted the interest of many investigators due to versatile applications in engineering, science, and industry. Examples include spray drying of milk,12,13 burning of haystacks,12,13 and cooling towers,12,13 and so on. Ramzan and Bilar14 discussed the effects of chemical reactions and mass transfer on MHD three-dimensional flow of a nanofluid and have solved the problem by employing the homotopy analysis method. The simultaneous effects of chemical reactions and volumetric heat generation/absorption on micropolar fluid flow have been studied by Damseh and colleagues.15 The volumetric heat source/sink has immense applications in science and engineering including in electronic chips, fire and safety engineering, nuclear reactors, chemical engineering, semiconductor wafers, and geophysical sciences. Some of the interesting studies on heat sources/sinks are described by Hayat and colleagues,16 Ravindran and colleagues,17 Mohamed and Abo-Dahab,18 Singh and colleagues,19 and Hayat and colleagues.20 This study focuses on the steady, double diffusive, mixed convection flow over exponentially decreasing mainstream velocity with a nonlinear homogeneous chemical reaction. The influence of a volumetric heat source/sink on heat transfer phenomena is also analyzed. Nonsimilar transformations

PATIL ET AL.

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considered in the problem add more complexity to the numerical analysis. However, in the absence of volumetric heat source or sink, this research paper seems to yield the same results as in Patil and colleagues.21 But the effects of this physical parameter on different geometries may exhibit a new dimension in the area of heat and mass transfer research. That is, the study of exponentially decreasing mainstream velocity mainly concentrates on controlling the separation of boundary layer. Thus, by utilizing the technique of volumetric heat source or sink, one can control the boundary layer separation. In particular, in undersea applications, heating enhances the stability by balancing the thermal and momentum boundary layers near the wall.7 Overall, this research paper finds itself more interesting as it has practical importance in many boundary layer problems, for example, in controlling transition and/or delaying the boundary layer separation over control surfaces and in suppressing the recirculating bubbles. The implicit finite difference scheme along with the quasi-linearization technique22–24 is used to solve the set of highly coupled nonlinear partial differential equations. The graphical results are presented to explain the effects of various physical parameters on velocity, temperature, and species concentration fields.

2

A NA LYSI S

We consider the steady, double diffusive, mixed convection flow over an exponentially decreasing external flow velocity. We also consider the nonlinear chemically reactive species and volumetric heat source or sink in the analysis. The wall velocity is constant and free stream velocity 𝑈𝑒 is assumed to decrease exponentially, that is, 𝑈𝑒 = 𝑈∞ (1 − 𝜀𝑒𝜉 ), where 𝜉 is the stream-wise coordinate or nonsimilarity variable is defined by 𝜉 = 𝐿𝑥 , where L is the characteristic length and 𝜀 denotes the decelerating parameter such that 0 < 𝜀 < 1. We have taken the coordinate system in such a way that the sheet is moving vertically upward in the x-axis and the y-axis is normal to it, as shown in Fig. 1. All thermophysical properties of the fluid are assumed to be constant except for the variation in density. We also assumed that 𝑇𝑤 , 𝐶𝑤 and 𝑇∞ , 𝐶∞ are the temperature and concentration fields of the fluid at the wall and far away from the wall, respectively. Using Boussinesq approximation,25 the governing boundary layer equations are given by5,8 𝜕𝑢 𝜕𝑣 + = 0, 𝜕𝑥 𝜕𝑦

𝑢

(1)

[ ] 𝑑𝑈 𝜕𝑢 𝜕𝑢 𝜕2𝑢 + 𝑔 𝛽 (𝑇 − 𝑇∞ ) + 𝛽 ∗ (𝐶 − 𝐶∞ ) , +𝑣 = 𝑈𝑒 𝑒 + 𝜈 2 𝜕𝑥 𝜕𝑦 𝑑𝑥 𝜕𝑦

(2)

𝑄 𝜕𝑇 𝜕𝑇 𝜕2𝑇 + 0 (𝑇 − 𝑇∞ ), +𝑣 =𝛼 𝜕𝑥 𝜕𝑦 𝜌𝐶𝑝 𝜕𝑦2

(3)

𝜕𝐶 𝜕𝐶 𝜕2𝐶 − 𝐾0 (𝐶 − 𝐶∞ )𝑛 , +𝑣 =𝐷 𝜕𝑥 𝜕𝑦 𝜕𝑦2

(4)

𝑢

𝑢

with appropriate boundary conditions 𝑦 = 0 ∶ 𝑢(𝑥, 0) = 0, 𝑣(𝑥, 0) = 𝑣𝑤 (𝑥), 𝑇 = 𝑇𝑊 , 𝐶(𝑥, 0) = 𝐶𝑊 , 𝑦 → ∞ ∶ 𝑢 → 𝑈𝑒 (𝑥) = 𝑈∞ (1 − 𝜀𝑒𝜉 ), 𝑇 → 𝑇∞ , 𝐶 → 𝐶∞ .

(5)

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FIGURE 1

Schematic diagram and coordinate system

Using the nonsimilar transformations8,21 ( )1∕2 𝑈𝑒 𝑥 𝑦, , 𝜂= 𝐿 𝜈𝑥 𝜓(𝑥, 𝑦) = (𝜈𝑈𝑒 𝑥)1∕2 𝑓 (𝜉, 𝜂),

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ 𝑇 − 𝑇∞ = (𝑇𝑤 − 𝑇∞ )𝐺(𝜉, 𝜂), 𝐶 − 𝐶∞ = (𝐶𝑤 − 𝐶∞ )𝐻(𝜉, 𝜂), ⎬ ⎪ 𝜕𝜓 𝜕𝜓 𝜉 𝑢= , 𝑣=− , 𝑈𝑒 = 𝑈∞ (1 − 𝜀𝑒 ), 𝒇 𝜼 =𝐹 , ⎪ 𝜕𝑦 𝜕𝑥 ( )1∕2 { } ⎪ ⎪ 𝜂 𝑓 1 𝜈𝑈𝑒 𝑢 = 𝑈𝑒 𝐹 , 𝑣 = − (1 + 𝑚) + 𝜉 𝑓𝜉 + (𝑚 − 1)𝐹 .⎪ 2 𝑥 2 2 ⎭ 𝜉=

(6)

From Eqs. (1) to (4), we find that Eq. (1) is trivially satisfied and Eqs. (2) to (4) reduce to 𝐹𝜂𝜂 + (𝑚 + 1)

𝐺𝜂𝜂

( ) } { 𝑓 𝑅𝑖 𝜉 𝐹 + 𝑚 1 − 𝐹2 + (𝐺 + 𝑁𝐻) = 𝜉 𝐹 𝐹𝜉 − 𝑓𝜉 𝐹𝜂 , 2 2 𝜂 𝜉 (1 − 𝜀𝑒 )

{ } } { 𝑓 Pr 𝑄 𝜉 Re + Pr ( 𝑚 + 1 ) 𝐺𝜂 + 𝐺 = Pr 𝜉 𝐹 𝐺𝜉 − 𝑓𝜉 𝐺𝜂 , 𝜉 2 (1 − 𝜀𝑒 )

(7)

(8)

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{ 𝐻𝜂𝜂 + 𝑆𝑐

(𝑚 + 1 )

𝑓 2

} 𝐻𝜂 −

{ } 𝑆𝑐 Δ 𝜉 Re 𝐻 𝑛 = 𝑆𝑐 𝜉 𝐹 𝐻𝜉 − 𝑓𝜉 𝐻𝜂 . 𝜉 (1 − 𝜀𝑒 )

(9)

From Eq. (5), the dimensionless boundary conditions are 𝜂=0∶ 𝜂 = 𝜂∞ ∶

𝐹 = 0, 𝐹 = 1,

𝐺 = 1, 𝐺 = 0,

𝐻 = 1, 𝐻 = 0.

(10)

Here, 𝜂∞ is the edge of the boundary layer. Furthermore, m denotes the nondimensional pressure 𝜕𝑈 gradient parameter which is given by 𝑚 = 𝑈𝜉 𝜕𝜉𝑒 , where 𝜉 is the stream-wise coordinate (or nonsim𝑒 ilarity variable) and Ue is (free stream) external flow velocity. 𝜂 Moreover, 𝑓 (𝜉, 𝜂) = ∫0 𝐹 𝑑𝜂 + 𝑓𝑊 and 𝑓𝑊 can be obtained from transformations as ( )1∕2 { } 𝜂 𝑓 1 𝜈𝑈𝑒 (1 + 𝑚) + 𝜉 𝑓𝜉 + (𝑚 − 1)𝐹 , 𝑣=− 2 𝑥 2 2 In view of boundary condition (5) and 𝑣𝑤 = 𝑣0 𝜉 −1∕2 , we get 𝑣0 𝜉 −1∕2 = −

( )1∕2 { } 𝜂 𝑓 1 𝜈𝑈𝑒 (1 + 𝑚) + 𝜉 𝑓𝜉 + (𝑚 − 1)𝐹 2 𝑥 2 2 𝑖. 𝑒., 𝑓𝑤 =

𝐴 (1 − 𝜀𝑒𝜉 )1∕2

,

(11)

where A is the surface mass transfer (suction/injection) parameter with 𝐴 = 0 for an impermeable surface, 𝐴 > 0 for the suction and 𝐴 < 0 for the injection or blowing. Furthermore, the skin friction coefficient (Re1∕2 𝐶𝑓 ), the heat transfer (Re−1∕2 𝑁𝑢), and mass transfer (Re−1∕2 𝑆ℎ) rates are defined as 𝐶𝑓 = 𝜇

2(𝜕𝑢∕𝜕𝑦 )𝑦 = 0 𝜌 𝑈𝑒

2

= 2 Re− 1∕2 𝜉 −1∕2 (1 − 𝜀𝑒𝜉 )

−1∕2

𝐹𝜂 (𝜉, 0) ,

𝑖. 𝑒 ., (Re )1∕2 𝐶𝑓 = 2 𝜉 −1∕2 (1 − 𝜀𝑒𝜉 )−1∕2 𝐹𝜂 (𝜉, 0) .

(12)

(𝜕𝑇 ∕𝜕𝑦)𝑦=0 ( )1∕2 𝐺𝜂 (𝜉, 0) , 𝑁𝑢 = − 𝑥 ( ) = − Re𝜉 (1 − 𝜀𝑒𝜉 ) 𝑇𝑤 − 𝑇∞ 𝑖.𝑒., (Re )−1∕2 𝑁𝑢 = − 𝜉 1∕2 (1 − 𝜀𝑒𝜉 )1∕2 𝐺𝜂 (𝜉, 0) .

(13)

(𝜕𝐶∕𝜕𝑦)𝑦=0 ( )1∕2 𝐻𝜂 (𝜉, 0) , 𝑆ℎ = − 𝑥 ( ) = − Re𝜉(1 − 𝜀𝑒𝜉 ) 𝐶𝑤 − 𝐶∞ 𝑖.𝑒., (Re )−1∕2 𝑆ℎ = − 𝜉 1∕2 (1 − 𝜀𝑒𝜉 )1∕2 𝐻𝜂 (𝜉, 0) .

(14)

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Equations (13) and (14) at 𝜉 = 0 signify the similarity solutions when all solutions along x-direction are made identical using similarity transformations.

3

METHOD OF SOLUTION

The implicit finite difference scheme along with the quasi-linearization technique22–24 is employed to solve the set of nonlinear partial differential Eqs. (7) to (9) with the boundary condition in Eq. (10) to obtain the sequence of linear partial differential equations as follows: 𝑖+1 + 𝐴𝑖1 𝐹𝜂𝑖+1 + 𝐴𝑖2 𝐹 𝑖+1 + 𝐴𝑖3 𝐹𝜉𝑖+1 + 𝐴𝑖4 𝐺𝑖+1 = 𝐴𝑖5 , 𝐹𝜂𝜂

(15)

𝑖+1 + 𝐵1𝑖 𝐺𝜂𝑖+1 + 𝐵2𝑖 𝐺𝑖+1 + 𝐵3𝑖 𝐺𝜉𝑖+1 + 𝐵4𝑖 𝐹 𝑖+1 = 𝐵5𝑖 , 𝐺𝜂𝜂

(16)

𝑖+1 + 𝐶1𝑖 𝐻𝜂𝑖+1 + 𝐶2𝑖 𝐻 𝑖+1 + 𝐶3𝑖 𝐻𝜉𝑖+1 + 𝐶4𝑖 𝐹 𝑖+1 = 𝐶5𝑖 . 𝐻𝜂𝜂

(17)

The coefficient functions with iterative index i are known and the functions with iterative index (i + 1) are to be determined. The corresponding boundary conditions are given by 𝐹 𝑖+1 (𝜉, 0) = 0, 𝐺𝑖+1 (𝜉, 0) = 1, 𝐻 𝑖+1 (𝜉, 0) = 1 at 𝜂 = 0, 𝐹 𝑖+1 (𝜉, 𝜂) = 1, 𝐺𝑖+1 (𝜉, 𝜂) = 0, 𝐻 𝑖+1 (𝜉, 𝜂) = 0 at 𝜂=𝜼∞ . The coefficients in Eqs. (15) to (17) are given by 𝑓 + 𝜉 𝑓𝜉 }; 2 ( ) 𝐴𝑖2 = − 2 𝑚 𝐹 + 𝜉 𝐹𝜉 ; 𝐴𝑖1 = {(1 + 𝑚)

𝐴𝑖3 = − 𝜉 𝐹 ; 𝐴𝑖4 = 𝐴𝑖5 =

𝜉 R𝑖 (1 − 𝜀𝑒𝜉 )2 𝜉 𝑅𝑖 𝑁 (1 − 𝜀𝑒𝜉 )2

; ;

𝐵1𝑖 = Pr {(1 + 𝑚 ) 𝐵2𝑖 =

𝑓 + 𝜉 𝐹𝜉 }; 2

Pr 𝜉𝑄Re ; (1 − 𝜀𝑒𝜉 )

𝐵3𝑖 = − Pr 𝜉 𝐹 ; 𝐵4𝑖 = − Pr 𝜉 𝐺𝜉 ; 𝐵5𝑖 = − Pr 𝜉 𝐹 𝐺𝜉 ; 𝐶1𝑖 = 𝑆𝑐 {(1 + 𝑚 )

𝑓 + 𝜉 𝑓𝜉 }; 2

(18)

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TABLE 1

Values of (Re)1∕2 𝐶𝑓 , (Re)−1∕2 𝑁𝑢, and (Re)−1∕2 𝑆ℎ, for different values of N, Ri, Re, 𝜀 and Δ with Q = 0.1, A = 1, Sc = 0.94, Pr = 0.7,𝜉 = 1 in support of nonsimilar solutions N

Ri

𝜺

Re

−0.5

10

0.01

10

1.0

10

0.01

10

n

(𝐑𝐞)𝟏∕𝟐 𝑪𝒇

(𝐑𝐞)−𝟏∕𝟐 𝑵𝒖

0.5

1

10.18419

0.53009

2.4812

0.5

1

18.39392

0.63568

2.4999 2.5277

𝚫

(𝐑𝐞)−𝟏∕𝟐 𝑺𝒉

3.0

10

0.01

10

0.5

1

28.61246

0.75094

1.0

10

0.01

50

0.5

1

20.74521

−1.53546

5.1436

1.0

10

0.01

100

0.5

1

30.36012

−7.84759

7.1653

1.0

5

0.01

10

0.5

1

1.0

5

0.001

10

0.5

1

0.07536

1.9722

10.52598

9.925800

0.48532

2.4628

1.0

10

0.01

10

0.5

1

56.43818

−1.00760

−1.2030

1.0

10

0.01

10

−0.5

2

24.88136

−1.17631

−0.3087

1.0

10

0.01

10

−0.5

3

24.27273

−1.22854

0.1585

𝑛 𝑆𝑐 𝜉 Δ Re 𝑛−1 ; 𝐻 (1 − 𝜀𝑒𝜉 ) 𝐶3𝑖 = − 𝑆𝑐 𝜉 𝐹 ;

𝐶2𝑖 = −

𝐶4𝑖 = − 𝑆𝑐 𝜉 𝐻𝜉 ;

𝐶5𝑖 = − 𝑆𝑐 𝜉 𝐹 𝐻𝜉 − (𝑛 − 1)

𝜉 𝑆𝑐 Δ Re 𝑛 𝐻 . (1 − 𝜀𝑒𝜉 )

Furthermore, the set of linear partial differential equations are reduced to a system of algebraic equations with a block tridiagonal matrix, which is then solved by Varga’s algorithm.26 Moreover, a grid-independent study was carried out to examine the effect of the step size Δ𝜂 and the edge of the boundary layer 𝜂∞ on the solution in the quest for their optimization. The 𝜂max , that is, 𝜂∞ was so chosen that the solution shows negligible changes for 𝜂 larger than 𝜂max . A step size of Δ𝜂 = 0.001 was found to be satisfactory for a convergence criteria with an absolute error less than 10−4 in all cases and the value of 𝜂∞ = 10 was found to be sufficiently large for the velocity to reach the relevant free stream velocity. Thus, the numerical results are compared with previously published available literature for particular cases by Chiam5 and Patil and colleagues8 with parameter values Re = 0, 𝑅𝑖 = 0, Pr = 0, Δ = 0, 𝑆𝑐 = 0, 𝑄 = 0 , 𝑛 = 0 and 𝐴 = 0 with decelerating parameter 𝜀 = 0.1, 𝜀 = 0.001 and 𝜀 = 0.000001, the data can be read by Table 1 (corresponding with trail runs with step sizes of 𝜉 = 0.1, 0.05, 0.01, 0.001, 0.0001 are considered to minimize computing time) in Chiam5 and results are found to be in excellent agreement, as displayed in Fig. 2. In support of nonsimilar solutions, the numerical solutions for skin friction coefficient, Nusselt number, and Sherwood number are presented for various values of all physical parameters as shown in Table 1.

4

RESULTS AND DISCUSSION

The numerical computations have been carried out for various values of 𝜀 (0.0001 ≤ 𝜀 ≤ 0.1), 𝜉(0 ≤ 𝜉 ≤ 1), 𝑁(0 ≤ 𝑁 ≤ 10), 𝑅𝑖 (−1 ≤ 𝑅𝑖 ≤ 10) , Re (10 ≤ Re ≤ 250), 𝑄(−1 ≤ 𝑄 ≤ 1), Pr(0.7 ≤ Pr ≤ 7.0), 𝑆𝑐(0.22 ≤ 𝑆𝑐 ≤ 2.57), 𝐴(−1 ≤ 𝐴 ≤ 1) , Δ (−1 ≤ Δ ≤ 1), 𝑛 (1 ≤ 𝑛 ≤ 3), and 𝛼 (−0.5 ≤ 𝛼 ≤ 0.5). The physical parameters are specified in particular range to illustrate the

118

PATIL ET AL.

FIGURE 2

Comparison of results of F𝜂 (𝜉, 0) with Chiam5 and Patil and colleagues8 for Re = 0, Ri = 0, Pr = 0, Sc = 0, Q = 0, A = 0, Δ = 0, and n = 0

special features of the solutions. To be more realistic, the values of stream-wise coordinate (or nonsimilarity parameter)𝜉 are chosen for similarity solution 𝜉 = 0 and for nonsimilar solution 𝜉 = 1. The buoyancy ratio parameter (N) is chosen to be N = −3, −1, 1, 3, 5, 10, where the positive values of N (N > 0) imply that both buoyancy forces act in the same direction. On the contrary, the negative values of N (N < 0) appear when thermal and concentration buoyancy forces act in opposite direction. Higher values of Re (Re = 10, 50, 100, 150, 200, 250) leads the inertial forces to dominate over viscous forces and fluid moves faster. The values of Prandtl number (Pr) is chosen to be Pr = 0.7 (for air) and Pr = 7.0 (for water). The values of Schmidt number (Sc) are chosen to be more realistic as Sc = 0.66, 0.94, 2.57 denoting diffusing chemical species of most common interest such as Propyl Benzene hydrogen, water vapor and Propyl Benzene at 25◦ C at one atmospheric pressure. The edge of the boundary layer 𝜂∞ has been taken between 4.0 and 10.0 based on the values of the physical parameters of the profiles. The effects of the stream-wise coordinate (or nonsimilarity parameter)𝜉 on the velocity 𝐹 (𝜉, 𝜂) , temperature 𝐺 (𝜉, 𝜂) profiles for Re = 10, 𝑅𝑖 = 10, Pr = 0.7, Δ = 0.1, 𝑆𝑐 = 0.94, 𝑄 = 0.05 , 𝑁 = 1, 𝜀 = 0.01, 𝑛 = 1.0, and 𝐴 = 1.0 are shown in Fig. 3. The velocity profile increases in Fig. 3, while the temperature profile decreases as 𝜉 increases in [0, 1]. This is because of the reason that the increase in the value of 𝜉 decreases the pressure in the flow direction by enhancing the flow rate. For instance, velocity profile rises remarkably around 261%, while temperature profile reduces about 77% with the increase of 𝜉 from 0 to 1.This reveals that nonsimilar solutions are prominent. The variations of buoyancy ratio parameter (N) on ( 𝐹 (𝜉, 𝜂) ) velocity profile are displayed in Fig. 4 for Re = 10, 𝑅𝑖 = 10, Pr = 0.7, Δ = 0.1, 𝑆𝑐 = 0.94, 𝑄 = 0.05 , 𝜉 = 1, 𝜀 = 0.01, 𝑛 = 1.0 and 𝐴 = 1.0. It is predicted that with increase in the ratio of buoyancy forces, the velocity profile and momentum boundary layer thicknesses increase. The N corresponds to the ratio between the solute and thermal buoyancy forces. In particular, for Re = 10, 𝑅𝑖 = 10, Pr = 0.7, Δ = 0.1, 𝑆𝑐 = 0.94, 𝑄 = 0.05 , 𝜉 = 1, 𝜀 = 0.01, 𝑛 = 1.0 and 𝐴 = 1.0 increase in N from -1 to 1, the velocity profile increases around 57%. The impact of the mixed convection parameter Ri (or Richardson number) on velocity profile ( 𝐹 (𝜉, 𝜂) ) for Re = 10, 𝑁 = 1, Pr = 0.7, Δ = 0.1, 𝑆𝑐 = 0.94, 𝑄 = 0.05 , 𝜉 = 0.5, 𝜀 = 0.01, 𝑛 = 1.0 and 𝐴 = 1.0 is shown in Fig. 5. The velocity overshoot is observed with the increasing values of Ri. Physically, the Ri corresponds to the ratio of the Grashof number and Reynolds number such that increasing values of Ri increase the exponentially stretching wall velocity. Thus, this results in increase of the momentum boundary layer thickness. The negative values of Ri show the backflow due

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FIGURE 3

Effects of 𝜉 on velocity and temperature profiles for Re = 10, Ri = 10, A = 1, 𝜀 = 0.01, N = 1, Sc = 0.94, Pr = 0.7, Q = 0.05, Δ = 0.1, and n = 1

F I G U R E 4 Effects of N on velocity profile for Re = 10, Ri = 10, A = 1, 𝜀 = 0.01, Sc = 0.94, Pr = 0.7, Q = 0.05, Δ = 0.1, 𝜉 = 1, and n = 1

to the buoyancy opposing force which reduces the magnitude of the velocity profile within the exponentially decreasing main stream boundary layer flow. Furthermore, the positive values of Ri (Ri > 0) show the velocity overshoot due to favourable pressure gradient. The velocity profile increases about 73% and temperature profile decreases approximately 10% with the increase of Ri from −3 to 1. The temperature distribution for various values of volumetric heat source/sink parameter Q and suction/injection parameter A are plotted in Fig. 6 for Re = 10, 𝑁 = 1, Pr = 0.7, Δ = 0.1, 𝑆𝑐 = 0.94, 𝑅𝑖 = 10, 𝜉 = 1, 𝜀 = 0.01, 𝑛 = 1.0, where the temperature profile increases with decreasing values of A. The thickness of the thermal boundary layer increases with the injection parameter (A0) parameter enhances the temperature distribution in which heat is generated within the boundary layer, whereas the heat absorption parameter (Q 1 (i.e., n = 2, 3) the chemical reaction is nonlinear. On a theoretical point of view, even though n = 1 implies the linear chemical reaction; the same cannot be seen in Fig. 8. It is because of the fact that the considered governing equations in this research paper are generally coupled in nature. In view of this, other parameters also show the significant effects over fluid flow. Thus, in Fig. 8, the chemical reaction parameter exhibits its impact along with the order of the chemical reaction on concentration profile. Moreover, in the presence of a destructive chemical reaction parameter (Δ < 0), the concentration profile increases with decreasing values of n and in the presence of a constructive chemical reaction parameter (Δ > 0) the concentration profile increases with increasing values of n. Physically,

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FIGURE 9

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Effects of Ri and 𝜀 on skin friction coefficient for Re = 10, N = 1, Sc = 0.94, Δ = 0.5, Q = 0.01, Pr =

0.7, and n = 1

the constructive chemical reaction (Δ > 0) parameter generates the species concentration in the fluid, leading to enhanced concentration distribution. The reaction order of constructive chemical reaction parameter (Δ > 0) enhances the concentration of the fluid, whereas the order of the destructive chemical reaction parameter (Δ < 0) reduces the concentration of the fluid. For example, the concentration profile decreases about 56% for Δ = −0.5 and increases around 53% for Δ = 0.5 with increases of n from 1 to 2. The influence of the mixed convection parameter Ri and decelerating parameter 𝜀 on (Re1∕2 𝐶𝑓 ) skin friction coefficient for Re = 10, 𝑁 = 1, Pr = 0.7, 𝑄 = 0.1, 𝑛 = 1, Δ = 0.5, 𝑆𝑐 = 0.94 is shown in Fig. 9. It is observed that the increase in decelerating parameter increases the skin friction coefficient, because it increases the exponentially decreasing mainstream velocity and hence the skin friction parameter. Also, the value of (Re1∕2 𝐶𝑓 ) increases with increasing values of Ri. For instance, the skin friction coefficient increases about 30% and the rate of heat transfer decreases approximately71% at 𝜀 = 0.001 with the increase of Ri from 3 to 5. The variations of the volumetric heat source/sink parameter Q and suction parameter A for Re = 10, 𝑁 = 1, Pr = 0.7, 𝑅𝑖 = 10, 𝑛 = 1, Δ = 0.1, 𝜀 = 0.001, 𝑆𝑐 = 0.22 on heat transfer rate (Re−1∕2 𝑁𝑢) are shown in Fig. 10. The heat transfer rate (Re−1∕2 𝑁𝑢) increases with the decrease of Q because the heat absorption or heat sink (Q< 0) absorbs the thermal energy, enhancing the rate of heat transfer, whereas a heat source or heat generation (Q>0) reduces the heat transfer rate. Moreover, suction improves the heat transfer rate and injection reduces the heat transfer rate. For instance, (Re−1∕2 𝑁𝑢) increases 20% and 29% in the presence of suction with increasing values of Q from -0.1 to 0 and 0 to 0.1, respectively. Additionally, in the presence of injection, the (Re−1∕2 𝑁𝑢) decreases by 31% and 51% with the increase of Q from −0.1 to 0 and 0 to 0.1 at 𝜉 = 1 . For the sake of brevity, the graph of Q and A on skin friction coefficient is not included in this research paper. However, the results show that the skin friction coefficient enhances for suction and heat generation. The variations of the chemical reaction parameter (Δ) and the exponent parameter n on (Re−1∕2 𝑆ℎ) mass transfer rate are shown in Fig. 11. It is observed that the mass transfer rate increases with increasing values of Δ. Physically, higher values of Δ increases the rate of mass diffusion in the flow. It is predicted that in the case of a constructive chemical reaction parameter (Δ > 0), the mass transfer coefficient increases with decreasing values of exponent parameter n. For a destructive chemical

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FIGURE 10

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Effects of Q and A on heat transfer rate for Re = 10, Ri = 10, N = 1, Sc = 0.22, 𝜀 = 0.01, Δ = 0.1,

and n = 1

F I G U R E 1 1 Effects of Δ and n on heat transfer coefficient for Re = 10, Ri = 10, N = 1, A = 1, 𝜀 = 0.01, Pr = 0.7, Q = 0.5, and Sc = 0.94

reaction parameter (Δ < 0) the mass transfer rate increases with increasing values of n. Thus, the order of chemical reaction parameter n shows the significant effects on heat transfer rate. Particularly, when Re = 10, 𝑅𝑖 = 10, Pr = 0.7, 𝑆𝑐 = 0.94, 𝑄 = 0.5 , 𝜉 = 1, 𝜀 = 0.01 and 𝐴 = 1.0 , the mass transfer rate increases 17% at Δ = 0.5 and 74% at Δ = −0.5, with the increase of the exponent parameter n from 1 to 2. In the above entire section, we have discussed the effects of various parameters along with their percentage variations. Physically, these percentage descriptions indicate how fast the flow rise /fall within the exponentially decreasing free stream velocity. So, by utilizing the percentages, one can predict the behaviour of the fluid flow numerically.

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5

CONC LU SI ON S

In this study, we investigated nonsimilar solutions of double diffusive, mixed convection flow with exponentially decreasing freestream velocity. The rate of heat and mass transfer characteristics in the presence of a volumetric heat source/sink and nonlinear chemical reaction are explained and discussed graphically. From this analysis, we note some interesting results as follows: • The velocity profile increases with the increase in N and Ri. For example, 𝐹 (𝜉, 𝜂) increases approximately about 20% as N increases from 5 to 10. • The temperature distribution increases with increasing values of volumetric heat source/sink Q. For instance, 𝐺(𝜉, 𝜂) increases about 69% as Q increases from 0 to 0.5 in presence of injection. Furthermore, the increasing values of Q increases (Re−1∕2 𝐶𝑓 ), which delays the boundary layer separation. • The species concentration profile increases with a decrease in chemical reaction parameter Δ. Particularly, for Δ > 0, 𝐻(𝜉, 𝜂) increases for increasing values of n. In addition, the rise in the chemical reaction parameter increases the rate of mass transfer. • An increase in the decelerating parameter 𝜀 increases (Re1∕2 𝐶𝑓 ). For instance, (Re1∕2 𝐶𝑓 ) increases about 30% with the increase of Ri from 3 to 5 at 𝜀 = 0.001. ACKNOW LEDGMENTS Dr. P. M. Patil acknowledges the financial assistance by the University Grants Commission (UGC), New Delhi 110 002, India through the Major Research Project F. No. 43–413/2014(SR) dated 30 October 2015 and also UGC-SAP-DRS-III with Ref. No. F.510/3/DRS-III/2016 (SAP-I) dated 29 Feb. 2016. Miss Nafisabanu Kumbarwadi acknowledges the financial support by UGC, New Delhi, through National Fellowship Award letter No. 2015–16/MANF/2017-18/KAR-54842. E. Momoniat thanks the National Research Foundation of South Africa under grant number 103483 and the DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS) for their support. The opinions expressed and conclusions arrived at are those of the author and are not necessarily to be attributed to the CoE-MaSS. REFERENCES 1. Hayat T, Bilal Ashraf M, Alsaedi A, Alhuthali MS. Soret and Dufour effects in three-dimensional flow of Maxwell fluid with chemical reaction and convective condition. Int J Numer Method H. 2015;25:1–25. 2. Nithyadevi N, Yang R. Double diffusive natural convection in a partially heated enclosure with Soret and Dufour effects. Int J Heat Fluid Fl. 2009;30:902–910. 3. Kiran P. Throughflow and non-uniform heating effects on double diffusive oscillatory convection in a porous medium. Ain Shams Eng J. 2016;7:453–462. 4. Ahmed S, Mahdy A. Unsteady MHD double diffusive convection in the stagnation region of an impulsively rotating sphere in the presence of thermal radiation effect. Journal of the Taiwan Institute of Chemical Engineers. 2016;58:173–180. https://doi.org/10.1016/j.jtice.2015.06.033 5. Chiam TC. A numerical solution for the laminar boundary layer flow with an exponentially decreasing velocity distribution. Acta Mech. 1998;129:255–261. 6. Curle N. Development and separation of a laminar boundary layer with an exponentially increasing pressure gradient. Q J Mech Appl Math. 1981;34:383–395. 7. Roy S, Saikrishnan P. Multiple slot suction/injection into an exponentially decreasing free stream flow. Int Commun Heat Mass. 2008;35:163–168.

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How to cite this article: Patil PM, Kumbarwadi N, Momoniat E. Influence of chemically reactive species and a volumetric heat source or sink on mixed convection over an exponentially decreasing mainstream. Heat Transfer—Asian Res. 2018;47:111–125. https://doi.org/ 10.1002/htj.21294