Effects of induced magnetic field and homogeneous–heterogeneous reactions on stagnation flow of a Casson fluid

Abstract

In this study, we analyzed the induced magnetic field effect on the stagnation-point flow of a non-Newtonian fluid over a stretching sheet with homogeneous–heterogeneous reactions and non-uniform heat source or sink. The transformed ordinary differential equations are solved numerically using Runge–Kutta and Newtons method. For physical relevance we analyzed the behavior of homogeneous and heterogeneous profiles individually in the presence of induced magnetic field. The effects of different non-dimensional governing parameters on velocity, induced magnetic field, temperature and concentration profiles, along with the skin friction coefficient and local Nusselt number, are discussed and presented through graphs. The results of the present study are validated by comparing with the existed literature. Results indicate that induced magnetic field parameter and stretching ratio parameter have the tendency to enhance the heat transfer rate.

Keywords

Induced magnetic field ; Homogeneous–heterogeneous reactions ; Non-uniform heat source/sink ; Casson fluid ; Stagnation flow

Nomenclature

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 directions

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 directions

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Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x} - Distance along the surface

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Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle c_p} - Specific heat capacity at constant pressure

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f} - Dimensionless velocity

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Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle T} - Temperature of the fluid

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,Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle B\ast }

- Non-uniform heat source/sink

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Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Nu} - Local Nusselt number

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {Re}_x} - Local Reynolds number

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Pr} - Prandtl number

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Greek symbols

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Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \eta } - Similarity variable

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Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \beta } - Casson parameter

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \lambda } - Reciprocal magnetic Prandtl number

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \Lambda } - Magnetic parameter

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Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \mu } - Dynamic viscosity

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Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \alpha } - Thermal diffusivity

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 stretching ratio parameter

Subscripts

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1. Introduction

The study of convection flows over stretching sheet has extensive applications in the field of manufacturing production of plastic, polythene, paper, polymer extrusion, cooling of elastic sheets, and fiber technology, science and engineering technology. The applications of stretching surface were broadly discussed by Chaim [1] . The flow of a non-Newtonian fluid over a stretching sheet has potential applications in the biosciences, blood flows, jelly and engineering. Casson is a shear thinning liquid. It can exhibit yield stress. If applied yield stress is greater than the shear stress, then it acts as solid, whereas if yield stress is lesser than the applied shear stress then it starts to move. The application of Casson fluid was discussed by Hayat et al. [2] . The homogeneous–heterogeneous reactions on stagnation-point flow past a stretching surface was discussed by Bachok et al. [3] . Peristaltic flow of a Carreau fluid flow with convective boundary conditions was analyzed by Hayat et al. [4] , and concluded that brinkman number and hall parameters help to enhance the thermal boundary layer thickness. Khan and Pop [5] investigated the viscoelastic fluid flow through a stretching surface with homogeneous–heterogeneous effects. Hayat et al. [6] illustrated the three-dimensional MHD bidirectional flow of a nanofluid past a permeable stretching sheet in the presence of slip condition and homogeneous–heterogeneous reactions.

Heat transfer analysis of magneto hydrodynamic stagnation-point flow toward a stretching surface in the presence of induced magnetic field was done by Ali et al. [7] . Abbas et al. [8] considered the slip conditions and homogeneous–heterogeneous effects on MHD stagnation-point flow of viscous fluid past a stretching or shrinking surface. Sandeep and Sulochana [9] discussed the mixed convection flow of an unsteady micropolar fluid flow over a stretching or shrinking sheet in the presence of magnetic and non-uniform heat source/sink. Raju et al. [10] analyzed the non-uniform heat source or sink effects on ferrofluid flow through a flat plate with thermal radiation and aligned magnetic field effects. The stagnation-point flow toward a vertical stretching surface was explained by Ishak et al. [11] . Mallikarjuna et al. [12] investigated the variable porosity regime on convection flow over a vertical cone with chemical reaction and magnetic field effect. An unsteady MHD nanofluid flow through a stretching surface in the presence of non-uniform heat source/sink and magnetic field effect was studied by Sandeep et al. [13] . Pal and Mandal [14] analyzed the induced magnetic field effect on stagnation point flow of a nanofluid past a non-isothermal stretching surface.

Heat transfer analysis of a stagnation point viscoelastic fluid flow toward a stretching sheet was investigated by Mahapatra and Gupta [15] . Gorla [16] depicted the stagnation point flow of non-Newtonian fluid flow in the presence of transverse magnetic field. Heat transfer characteristics of MHD Casson fluid flow past a permeable an exponentially stretching sheet was considered by Raju et al. [17] . Hayat et al. [18] analyzed the two-dimensional flow of Eyring–Powell fluid flow toward a stretching sheet with homogeneous–heterogeneous reactions and concluded that the Eyring–Powell fluid material parameter enhances the boundary layer thickness. Heat transfer characteristics of Casson fluid flow past a permeable exponentially stretching sheet in the presence of thermal radiation was discussed by Pramanik [19] and concluded that an increasing value of Casson parameter suppresses the velocity profiles. Animasaun [20] discussed the MHD Casson fluid flow in the presence of viscous dissipation. Stagnation point flow of optimized exact solution for oblique flow of Casson-nanofluid with convective boundary conditions was discussed by Nadeem et al. [21] . Makinde and Aziz [22] and [23] examined the boundary layer analysis of a nanofluid flow through a stretching surface in the presence of convective boundary conditions and buoyancy effects. MHD heat transfer characteristics of an electrically conducting nanofluid flow through nonlinearly stretching surface in the presence of viscous dissipation was examined numerically by Mabood et al. [24] . Nadeem and Saleem [25] proposed an analytical solution for an unsteady mixed convection MHD flow on a rotating cone. Dissipation and magnetic field effects on the boundary layer flow of a power-law nanofluid induced by a permeable stretching/shrinking sheet were investigated numerically by Dhanai et al. [26] . Chaudhary and Merkin [27] proposed a simple isothermal model for homogeneous–heterogeneous reactions for boundary layer flow. Very recently, Hayat et al. [28] and Mahanta and Shaw [29] investigated the heat transfer characteristics of Casson fluid through different channels. Hayat et al. [30] studied the homogeneous and heterogeneous reactions on the flow over a nanotube with homogeneous heating. Further, Farooq et al. [31] discussed the homogeneous–heterogeneous characteristics on the flow of Jeffrey fluid.

In this study, we investigated the induced magnetic field and non-uniform heat source or sink effects on the stagnation-point flow of a non-Newtonian fluid toward a stretching sheet in the presence of homogeneous–heterogeneous reactions. The emerging set of governing nonlinear partial differential equations is transformed into a set of ordinary differential equations using similarity transformation, which are then solved numerically using Runge–Kutta and Newtons method. The effects of different non-dimensional governing parameters on velocity, induced magnetic field, temperature and concentration profiles, along with the friction factor and local Nusselt number, are discussed and presented through graphs and tables.

2. Flow analysis

Consider a steady, incompressible, electrically conducting stagnation point flow of a Casson fluid toward a stretching sheet in the presence of induced magnetic field, non-uniform heat source/sink and homogeneous–heterogeneous reactions. The stretching sheet is considered along the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x-}

axis and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle y-}
axis is normal to it as displayed in Fig. 1 .

Fig. 1.

Physical model and coordinate system.

The governing boundary layer equations for the flow and induced magnetic field profiles are given by [7] .

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( 1)

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( 2)

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( 3)

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( 4)

Subjected to the boundary conditions

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( 5)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\mu }_e}

 is the magnetic diffusivity of the fluid, which is given by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\mu }_e=1/4\pi \sigma }
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u}
 and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle v}
 are the velocity components along the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x}
 and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle y}
directions, respectively, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \rho }
 is the density, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \nu }
 is the viscosity.

To convert the governing equations of the flow into a set of nonlinear ordinary differential equations, we now introduce the following similarity transformation:

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( 6)

Equation (6) identically satisfies the continuity equations (1) and (2) , and equations (3) and (4) become

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( 7)

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( 8)

Subjected to the transformed boundary conditions

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( 9)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \beta }

 is the Casson parameter, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \Lambda }
 is the magnetic parameter, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \lambda }
 is the reciprocal magnetic Prandtl number, which are given by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \Lambda =\frac{\mu H_0^2}{4\pi \rho c^2},\mbox{ }\lambda =}

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3. Heat transfer analysis

The governing equation for energy in the presence of no-uniform heat source/sink is given by

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( 10)

With the boundary conditions

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( 11)

The space and temperature-dependent heat generation/absorption (non-uniform heat source/sink) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle q^{{'´´}}}

is defined by Sandeep and Sulochana [9] :

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Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left. \left. T_{\infty }\right)f^{{'}}+B\ast \left(T-\right. \right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left. \left. T_{\infty }\right)\right),

( 12)

where T is the fluid temperature, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \alpha }

 is the thermal diffusivity coefficient, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \upsilon }
 is the kinematic viscosity of the base fluid, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle A\ast }
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle B\ast }
are parameters of the space and temperature-dependent internal heat generation/absorption. The positive and negative values of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle A\ast }
 and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle B\ast }
represent heat generation and absorption parameters respectively. Now introducing the similarity transformation

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \eta ={\upsilon }_f{}^{-1/2}c^{1/2}y,\mbox{ }\theta \left(\eta \right)=

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left(T-T_{\infty }\right)/\left(T_w-T_{\infty }\right),

( 13)

By using equation (13) , equations (10) , (11) and (12) transformed to

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\theta }^{{''}}+\left(A\ast f^{{'}}+B\ast \theta \right)+

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): Prf{\theta }^{{'}}=0,

( 14)

With the transformed boundary conditions

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \theta =1,\mbox{ }\mbox{at}\mbox{ }\eta =0,\mbox{ }\theta =

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 0,\mbox{ }\mbox{as}\mbox{ }\eta \rightarrow \infty ,

( 15)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Pr=\frac{v}{\alpha }}

is the Prandtl number.

4. Mass transfer analysis

We considered a simple homogeneous–heterogeneous reaction model that exists as proposed by Chaudhary and Merkin [27] in the following form:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left(u\frac{\partial a}{\partial x}+v\frac{\partial a}{\partial y}\right)=

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): D_A\frac{{\partial }^2a}{\partial y^2}-k_cab^2\mbox{},

( 16)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left(u\frac{\partial b}{\partial x}+v\frac{\partial b}{\partial y}\right)=

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): D_B\frac{{\partial }^2b}{\partial y^2}+k_cab^2\mbox{},

( 17)

Subject to the boundary conditions

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{c} D_A\frac{\partial a}{\partial y}=k_sa\left(y\right),\mbox{ }D_B\frac{\partial b}{\partial y}=-k_sa\left(y\right)\mbox{ }\mbox{at}\mbox{ }y=0,\\ a\left(y\right)=a_0\mbox{},\mbox{ }b\left(y\right)=0\mbox{ }\mbox{as}\mbox{ }y\rightarrow \infty , \end{array}\rbrace

( 18)

whereFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle D_A\mbox{ }\mbox{and}\mbox{ }D_B}

 are the diffusion coefficients. We now introduced the similarity transformations as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): G\left(\eta \right)=a/a_0\mbox{},\mbox{ }H\left(\eta \right)=

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): b/a_0\mbox{},

( 19)

Using equation (19) , equations (16) , (17) and (18) transformed to

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): G^{{''}}+Sc\left(fG^{{'}}-KGH^2\right)=0,

( 20)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \delta H^{{''}}+Sc\left(fH^{{'}}+KGH^2\right)=0,

( 21)

With the transformed boundary conditions

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{c} G^{{'}}=K_sG,\mbox{ }\delta H^{{'}}=-K_sG\mbox{ }\mbox{at}\mbox{ }\eta =0,\\ G=1,\mbox{ }H=0\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{as}\mbox{ }\eta \rightarrow \infty , \end{array}\rbrace

( 22)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Sc}

 is the Schmidt number, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle K}
 is the measure of strength of homogeneous reaction, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle K_s}
 is the strength of heterogeneous reaction, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Re=c/\nu }
 is the Reynolds number and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \delta }
 is the ratio of diffusion coefficient, which are represented below.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \delta =\frac{D_B}{D_A},\mbox{ }Sc=\frac{\nu }{D_A},\mbox{ }K=

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{k_ca_0{}^2}{c},\mbox{ }K_s=\frac{k_s}{D_A\sqrt{Re}}\rbrace

( 23)

In most of the applications, we expect the diffusion coefficient of chemical species A and B to be of comparable size. This leads us to make an assumption that the diffusion coefficients D_A and D_B are equal. That is the case to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \delta =1}

For engineering interest, the shear stress coefficient Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle C_f}

 (friction factor) and the local Nusselt number Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Nu_x}
 are given by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): C_f=\frac{{\tau }_w}{{\rho }_fU_w^2},\mbox{ }\mbox{ }Nu_x=

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{xq_w}{k_f\mbox{ }\left(T_w-T_{\infty }\right)},\rbrace

( 24)

Here, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\tau }_w}

 is the wall shear stress and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle q_w}
is the wall heat flux, which are given by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\tau }_w=\mu {\left(\frac{\partial u}{\partial y}\right)}_{y=0}\mbox{},\mbox{ }q_w=

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): -k{\left(\frac{\partial T}{\partial y}\right)}_{y=0}\mbox{},\rbrace

( 25)

Using (6) and (13) we get

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {Re}_x^{1/2}C_f=\left(1+\frac{1}{\beta }\right)f^{{''}}\left(0\right),

( 26)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {Re}_x^{-1/2}Nu_x=-{\theta }^{{'}}\left(0\right),

( 27)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {Re}_x=u_wx/v_f}

 is the local Reynolds number.

5. Results and discussion

The set of nonlinear ordinary differential equations (7) , (8) , (10) , (20) and (21) subjected to the boundary conditions (9), (11) and (22) are solved numerically using Runge–Kutta and Newtons method (Mallikarjuna et al. [12] ). The influence of non-dimensional governing parameters on velocity, induced magnetic field, temperature and concentration profiles, along with the friction factor coefficient and local Nusselt number, are discussed and presented through graphs and tables. For numerical calculations, we considered Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle K=K_S=1,\mbox{ }a/c=0.5,\mbox{ }\delta =} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 3,\mbox{ }\lambda =0.2,\mbox{ }Pr=1

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \Lambda =A\ast =B\ast =0.1,\mbox{ }Sc=}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 0.6

. In the entire study we kept these values as common except the variations in the corresponding graphs and tables. In this paper green color indicates the Newtonian fluid case and red color indicates the non-Newtonian fluid case.

Fig. 2 , Fig. 3 , Fig. 4 and Fig. 5 depict the influence of stretching ratio parameter on velocity, induced magnetic field and concentration profiles for both Newtonian and non-Newtonian fluid cases. It is evident that the velocity and homogeneous concentration profiles are encouraged with increasing values of stretching ratio parameter, and reverse profiles are observed in induced magnetic field and heterogeneous concentration profiles. Basically, an increase in the stretching ratio parameter initially develops the more pressure on the flow; due to this reason, we have seen an enhancement in the velocity profiles.

Fig. 2.

Velocity profiles for different values of stretching ratio parameter.

Fig. 3.

Induced magnetic field profiles for different values of stretching ratio parameter.

Fig. 4.

Concentration profiles for different values of stretching ratio parameter.

Fig. 5.

Concentration profiles for different values of stretching ratio parameter.

The effects of induced magnetic field parameter on velocity, induced magnetic field and concentration profiles for both Newtonian and non-Newtonian cases are plotted in Fig. 6 , Fig. 7 , Fig. 8 and Fig. 9 . It is noticed that the momentum, thermal and concentration boundary layer thicknesses are enhanced with an increase in the induced magnetic field parameter. Usually an increase in the induced magnetic field develops the electric current. This electric force can help to enhance the momentum and thermal boundary layer thickness. This leads to an increase in the momentum and thermal boundary layer thickness.

Fig. 6.

Velocity profiles for different values of induced magnetic field parameter.

Fig. 7.

Induced velocity profiles for different values of induced magnetic field parameter.

Fig. 8.

Concentration profiles for different values of induced magnetic field parameter.

Fig. 9.

Concentration profiles for different values of induced magnetic field parameter.

The influence of homogeneous reaction parameter on concentration profiles for both Newtonian and non-Newtonian cases is depicted in Fig. 10 and Fig. 11 . It displays that an increasing value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle K}

enhances the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle H}
concentration profiles and depreciates theFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle G}
 concentration profiles. It may be due to the fact that the reaction rates dominate diffusion coefficients. Quite similar results are displayed in Fig. 12  and Fig. 13 , with an increase in the heterogeneous parameter. This agrees with the general physical behavior of homogeneous and heterogeneous reactions.

Fig. 10.

Concentration profiles for different values of strength of homogeneous reaction parameter.

Fig. 11.

Concentration profiles for different values of strength of homogeneous reaction parameter.

Fig. 12.

Concentration profiles for different values of heterogeneous reaction parameter.

Fig. 13.

Concentration profiles for different values of strength of heterogeneous reaction parameter.

The effect of the ratio of the diffusion coefficient on the concentration profiles is depicted in Fig. 14 . Mixed profiles in the concentration are observed; initially, less diffused particles are developed in the concentration. Due to this reason, initially we have noticed depreciation in the concentration profiles. Fig. 15 and Fig. 16 illustrate the effect of non-uniform heat source or sink parameter on temperature profiles. Generally, higher values of non-uniform heat source or sink parameter enhance temperature profiles. Because it generates an internal energy in the flow, we have seen an improvement in temperature profiles.

Fig. 14.

Concentration profiles for different values of ratio of diffusion parameter.

Fig. 15.

Temperature profiles for different values of non-uniform heat source/sink parameter.

Fig. 16.

Temperature profiles for different values of non-uniform heat source/sink parameter.

Table 1 and Table 2 depict the comparison of the present results with previously published work under some special limited cases. We found an excellent agreement of the present results with existing results. This proves the validity of the present results along with the accuracy of the numerical technique we used in this study. Table 3 and Table 4 presented the effect of various emerging thermo physical parameters for skin friction coefficients and local Nusselt number for both Newtonian and non-Newtonian fluid cases. It is evident that the friction factor coefficients and heat transfer rate are enhanced with an increase in the induced magnetic field parameter. The quite similar types of results are displayed in stretching ratio parameter. It is also observed that an increase in the non-uniform heat source or sink parameter declines the rate of heat transfer.

Table 1. Comparison of the skin friction coefficient when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle K_s=K=\Lambda =K=A\ast =B\ast =} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): Sc=0\mbox{ }and\mbox{ }\beta \rightarrow \infty .

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): a/c

Ishak et al. [[#bib0060|[11]]]

Mahapatra and Gupta [[#bib0080|[15]]]

Ali et al. [[#bib0040|[7]]]

Present

0.1

−0.9694

−0.96941

0.2

−0.9181

−0.91810

0.5

−0.6673

−0.66731

2

2.0175

2.01750

3

4.7294

4.7293

4.72931

Table 2. Comparison of local Nusselt number when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle K=K_s=Sc=\Lambda =K=A\ast =} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): B\ast =0,\mbox{ }\beta \rightarrow \infty .

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): a/c

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): Pr

Mahapatra and Gupta [[#bib0080|[15]]]

Present results

0.05

0.5

1

1.5

0.05

0.5

1

1.5

0.1

−0.081

−0.383

−0.603

−0.777

−0.0811

−0.3833

−0.6031

−0.7772

0.2

−0.099

−0.408

−0.625

−0.797

−0.0991

−0.4082

−0.6252

−0.7971

0.5

−0.136

−0.473

−0.692

−0.863

−0.1360

−0.4731

−0.6921

−0.8632

1

−0.178

−0.563

−0.796

−0.974

−0.1782

−0.5632

−0.7962

−0.9741

2

−0.241

−0.709

−0.974

−1.171

−0.2413

−0.7094

−0.9743

−1.1713

3

−0.289

−0.829

−1.124

−1.341

−0.2891

−0.8290

−1.1241

−1.3411

Table 3. The physical parameter values for Newtonian fluid case.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): a/c

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \Lambda

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): K

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): K_s

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): A\ast

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): B\ast

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \delta

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): f^{{''}}\left(0\right)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\theta }^{{'}}\left(0\right)

0.5

−0.510405

1.731575

0.6

−0.458589

1.739244

0.7

−0.371571

1.753433

0.1

−0.516500

1.723825

0.15

−0.435305

1.739196

0.2

−0.321719

1.760058

1

−0.516500

1.723825

1.4

−0.516500

1.723825

1.8

−0.516500

1.723825

0.4

−0.516500

1.723825

0.6

−0.516500

1.723825

0.8

−0.516500

1.723825

0.5

−0.520685

1.268812

1

−0.520685

0.707954

1.5

−0.520685

0.147097

1

−0.475514

1.521289

2

−0.475514

1.216227

3

−0.475514

0.860203

1.5

−0.516500

1.723825

2

−0.516500

1.723825

2.5

−0.516500

1.723825

Table 4. The physical parameter values in non-Newtonian fluid case.