Open Access Research

Dual solutions of stagnation-point flow of a nanofluid over a stretching surface

Peri K Kameswaran1, Precious Sibanda1*, Chetteti RamReddy2 and Prabhala VSN Murthy3

Author Affiliations

1 School of Mathematical Sciences, University of KwaZulu-Natal, Private Bag X01, Scottsville, Pietermaritzburg, 3209, South Africa

2 Department of Mathematics, National Institute of Technology, Rourkela, Odisha, 769008, India

3 Department of Mathematics, Indian Institute of Technology, Kharagpur, 721302, India

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Boundary Value Problems 2013, 2013:188  doi:10.1186/1687-2770-2013-188


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/188


Received:10 May 2013
Accepted:7 August 2013
Published:22 August 2013

© 2013 Kameswaran et al.; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The paper discusses the effects of homogeneous-heterogeneous reactions on stagnation-point flow of a nanofluid over a stretching or shrinking sheet. The model presented describes mass transfer in copper-water and silver-water nanofluids. The governing system of equations is solved numerically, and the study shows that dual solutions exist for certain suction/injection, stretching/shrinking and magnetic parameter values. Comparison of the numerical results is made with previously published results for special cases.

Keywords:
homogeneous-heterogeneous reactions; nanofluids; volume fraction model; stretching or shrinking sheet; magnetic field strength

1 Introduction

Problems involving fluid flow over stretching or shrinking surfaces can be found in many manufacturing processes such as in polymer extrusion, wire and fiber coating, foodstuff processing, etc. Crane [1] was the first to consider the steady two-dimensional flow of a Newtonian fluid driven by a stretching elastic flat sheet which moved in its own plane with velocity varying linearly with the distance from a fixed point. This study was subsequently extended by many authors to explore various aspects of heat transfer in a fluid surrounding a stretching sheet (Tsou et al.[2], Erickson et al.[3], Mucoglu and Chen [4], Grubka and Bobba [5], Karwe and Jaluria [6], Chen [7], Abo-Eldahab and El-Aziz [8], Salem and El-Aziz [9], Ali [10], Ishak et al.[11]).

The magnetohydrodynamic effect has important engineering applications in electrical motors. Heat transfer over a stretching or shrinking sheet subject to an external magnetic field, viscous dissipation and joule effects was studied by Jafar et al.[12]. They observed that the flow and heat transfer characteristics for a shrinking sheet were quite different from those of a stretching sheet. Lok et al.[13] analyzed MHD stagnation-point flow from a shrinking sheet. They found that dual solutions existed for small values of the magnetic parameter. The stagnation-point flow over a stretching or shrinking sheet in a nanofluid was investigated by Bachok et al.[14]. They showed that adding nanoparticles to a base fluid increased the skin friction and heat transfer coefficients. Recently, Narayana and Sibanda [15] investigated the laminar flow of a nanoliquid film over an unsteady stretching sheet. They noticed that the effect of an increase in the nanoparticle volume fraction was to reduce the axial velocity and free stream velocity in the case of a Cu-water nanoliquid. However, the opposite appeared to be true in the case of an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M1">View MathML</a>-water nanofluid. Kameswaran et al.[16] studied the effects of viscous dissipation and a chemical reaction in hydromagnetic nanofluid flow due to a stretching or shrinking sheet. They found that the velocity profiles decreased with an increase in the nanoparticle volume fraction, while the opposite was true in the case of temperature and concentration profiles. The study further showed that liquids with nanoparticle suspensions were better suited for effective cooling of the stretching sheet due to their enhanced conductivity and thermal properties.

Most chemically reacting systems involve both homogeneous and heterogeneous reactions (combustion, catalysis and biochemical systems). The simple combustion model helps us to understand the combustion phenomenon in many complex engineering applications such as in aircraft and rocket engines. A model for isothermal homogeneous-heterogeneous reactions in the boundary layer flow of a viscous fluid past a flat plate was presented by Merkin [17]. He modeled the homogeneous reaction by a cubic autocatalysis process and the heterogeneous reaction by a first-order process. Chaudhary and Merkin [18] analyzed homogeneous-heterogeneous reactions in boundary layer flow. They presented a numerical solution of the boundary layer equations near the leading edge of a flat plate. Ziabakhsh et al.[19] studied the diffusion of a chemically reactive species into a nonlinearly stretching sheet immersed in a porous medium. Chambre and Acrivos [20] studied isothermal chemical reactions on laminar boundary layer flow. The two-dimensional stagnation-point flow near an infinite permeable wall with a homogeneous-heterogeneous reaction was studied by Khan and Pop [21], while Khan and Pop [22] and Bachok et al.[23] studied the effects of homogeneous-heterogeneous reactions on fluid flow due to a stretching sheet. Recently, the effects of homogeneous-heterogeneous reactions in nanofluid flow due to a porous stretching sheet were studied by Kameswaran et al.[24]. They found that the concentration at the surface decreased with the strength of the heterogeneous reaction. In the case of a shrinking sheet, they showed that the velocity profiles decreased with increasing nanoparticle volume fraction in the case of a Cu-water nanofluid.

This article presents a study of homogeneous-heterogeneous reactions on MHD nanofluid stagnation point flow due to a stretching or shrinking sheet. The transformed nonlinear conservation equations are solved numerically.

2 Mathematical formulation

Consider a two-dimensional steady boundary layer flow of an incompressible nanofluid over a stretching or shrinking sheet. A Cartesian co-ordinate system is used with the x-axis along the sheet and the y-axis normal to the sheet. The flow configuration and the coordinate system are shown in Figure 1. The velocity of the outer flow is of the form <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M2">View MathML</a> and the velocity of the stretching or shrinking sheet is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M3">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M4">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M5">View MathML</a> are constants. The fluid is a water-based nanofluid containing copper (Cu) or silver (Ag) nanoparticles. The base fluid and the nanoparticles are in thermal equilibrium with no slip occurring between them. We assume the simple homogeneous-heterogeneous reaction model proposed by Chaudhary and Merkin [18] of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M6">View MathML</a>

(1)

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M7">View MathML</a>

(2)

where a and b are the concentrations of the chemical species A and B, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M8">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M9">View MathML</a> are the rate constants. We assume that both reaction processes are isothermal. Under these assumptions, the boundary layer equations governing the flow can be written in the dimensional form [18,25]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M10">View MathML</a>

(3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M11">View MathML</a>

(4)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M12">View MathML</a>

(5)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M13">View MathML</a>

(6)

The boundary conditions for equations (3)-(6) are given in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M14">View MathML</a>

(7)

where u, v are the velocity components in the x and y directions, respectively, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M15">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M16">View MathML</a> are the respective diffusion species coefficients of A and B, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M17">View MathML</a> is a positive constant. The effective dynamic viscosity of the nanofluid was given by Brinkman [26] as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M18">View MathML</a>

(8)

where ϕ is the solid volume fraction of nanoparticles. The effective density of the nanofluids is given as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M19">View MathML</a>

(9)

thumbnailFigure 1. Physical model and coordinate system.

Here, the subscripts nf, f and s represent the thermophysical properties of the nanofluid, the base fluid and nanoparticles, respectively.

The continuity equation (3) is satisfied by introducing a stream function ψ such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M20">View MathML</a>

(10)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M21">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M22">View MathML</a> is the dimensionless stream function and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M23">View MathML</a>.

The velocity components are given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M24">View MathML</a>

(11)

The concentrations of the chemical species A and B are represented as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M25">View MathML</a>

(12)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M26">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M27">View MathML</a> are dimensionless concentrations. On using equations (8), (9), (11) and (12), equations (4)-(7) transform to the following two-point boundary value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M28">View MathML</a>

(13)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M29">View MathML</a>

(14)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M30">View MathML</a>

(15)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M31">View MathML</a>

(16)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M32">View MathML</a>

(17)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M33">View MathML</a>

(18)

The non-dimensional parameters in equations (13)-(18) are the magnetic parameter M, the Schmidt number Sc, the measure of the strength of the homogeneous reaction K, the ratio of diffusion coefficients δ, the mass transfer parameter <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M34">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M35">View MathML</a> for suction and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M36">View MathML</a> for injection, the measure of the strength of the heterogeneous reaction <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M37">View MathML</a>, the Reynolds number Re and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M38">View MathML</a> is the stretching parameter, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M39">View MathML</a> for stretching and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M40">View MathML</a> for shrinking, respectively. These parameters are respectively defined as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M41">View MathML</a>

(19)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M42">View MathML</a>

(20)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M43">View MathML</a>

(21)

In most applications, we expect the diffusion coefficients of chemical species A and B to be of a comparable size. This leads us to making a further assumption that the diffusion coefficients <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M15">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M16">View MathML</a> are equal, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M46">View MathML</a> (Chaudhary and Merkin [18]). In this case we have, from equations (16), (17) and (18),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M47">View MathML</a>

(22)

Thus equations (14) and (15) reduce to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M48">View MathML</a>

(23)

subject to the boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M49">View MathML</a>

(24)

It is quite straightforward to show that the skin friction coefficient <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M50">View MathML</a>, which characterizes the surface drag, is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M51">View MathML</a>

(25)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M52">View MathML</a> is the local Reynolds number.

3 Results and discussion

The system of ordinary differential equations (13) and (23) with boundary conditions (16) and (24) was solved numerically using Matlab bvp4c routine. We considered Cu-water and Ag-water nanofluids. The thermophysical properties of the nanofluids used in this paper are given in Table 1.

Table 1. Thermophysical properties of water,Cu-water andAg-water nanofluids, Oztop and Abu-Nada[27]

In order to determine the accuracy of our numerical results, the present results for the skin-friction coefficient <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M54">View MathML</a> were compared with the available published results of Jafar et al.[12], Bachok et al.[23] and Wang [28] in Tables 2, 3 and 4.

Table 2. Comparison of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M54">View MathML</a>from current results with Wang[28]and Jafar et al.[12]for various values ofλfor the stretching sheet when<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M56">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M57">View MathML</a>

Table 3. Comparison of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M54">View MathML</a>from current results with Wang[28]and Jafar et al.[12]for various values ofλfor the shrinking sheet when<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M59">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M57">View MathML</a>

Table 4. Comparison of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M54">View MathML</a>obtained by Bachok et al.[23]with the present results for particular values of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M62">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M63">View MathML</a>

Tables 2 and 3 give the coefficient <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M54">View MathML</a> for different parameter values. Table 2 gives a comparison of the present results with those obtained by Jafar et al.[12] and Wang [28] when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M69">View MathML</a>, for different values of the stretching parameter. We observe that for increasing λ, the present results are in good agreement with results in the literature.

Table 4 gives the values of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M54">View MathML</a> for selected λ when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M62">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M63">View MathML</a>. We note here that with decreasing λ the first solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M73">View MathML</a> decreases while the second solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M74">View MathML</a> increases. These results are in good agreement with the results obtained by Bachok et al.[23] in the absence of the particular physical parameter.

The effects of the Schmidt number, stretching, magnetic and chemical reaction parameters are shown in Figures 2-12.

thumbnailFigure 2. Effects ofλon velocity, when<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M75">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M76">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M77">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M78">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M79">View MathML</a>.

Figure 2 shows the effects of both stretching and shrinking on the velocity profiles in the case of a Cu-water nanofluid. We observe that in both cases, the velocity profiles increase with the parameter λ. Further, we note that for a shrinking sheet, the velocity in the case of a Cu-water nanofluid is larger than that of a clear fluid. The opposite is, however, true for the case of a stretching sheet. The momentum boundary layer thickness decreases as λ increases and the flow has an inverted boundary layer structure when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M80">View MathML</a>. The findings in the case of a clear fluid are similar to the results obtained by Jat and Chaudhary [29].

Figures 3(a) and 3(b) illustrate the effect of the magnetic parameter, nanoparticle volume fraction and stretching or shrinking parameters on the velocity profiles. We note that for both stretching and shrinking sheets, the fluid velocity increases with λ and M. Furthermore, increasing the value of M also causes thinning of the boundary layer. This implies an increase in the velocity gradient <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M54">View MathML</a>. Thus the magnetic field enhances the fluid motion in the boundary layer in the case of a clear fluid. The same trend is observed in the case of a Cu-water nanofluid. We also observe in the case of injection that, with increasing magnetic parameter, the increment in the momentum boundary layer is more significant than in the case of suction.

thumbnailFigure 3. Effects ofMon velocity when (a)<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M75">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M76">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M84">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M79">View MathML</a>and (b)<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M75">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M76">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M84">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M89">View MathML</a>.

Figures 4(a) and 4(b) illustrate the effects of the stretching or shrinking parameter and volume fraction on the solute concentration when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M90">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M91">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M92">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M93">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M94">View MathML</a>. We observe that the concentration profiles increase with the stretching parameter. We note also that the concentration increases as λ varies from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M95">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M96">View MathML</a> for both the clear fluid and the Cu-water nanofluid. However, beyond <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M96">View MathML</a>, the opposite is true for the clear fluid and the Cu-water nanofluid.

thumbnailFigure 4. Variation of concentration for different values ofλ, when (a)<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M75">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M76">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M84">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M101">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M79">View MathML</a>and (b)<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M75">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M76">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M84">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M101">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M89">View MathML</a>.

The variation of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M108">View MathML</a> for different values of K and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M37">View MathML</a> is shown in Figures 5 and 6, respectively. From Figure 5 we observe that the concentration at the surface decreases as the strength of the heterogeneous reaction increases. This is simply explained by the fact that the strength of the chemical reaction depends on the concentration. On the other hand, from Figure 6, we found that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M108">View MathML</a> decreases with increasing K and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M37">View MathML</a>. These findings are similar to the results reported by Kameswaran et al.[24].

thumbnailFigure 5. Effects of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M112">View MathML</a>andKon concentration, when<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M75">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M114">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M79">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M101">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M117">View MathML</a>.

thumbnailFigure 6. Effects ofKand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M112">View MathML</a>on concentration, when<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M119">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M79">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M101">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M117">View MathML</a>.

Figures 7 and 8 show the effects of K and λ on the concentration when the other parameters are fixed. We note, as expected, that the wall concentration decreases as the strength of the homogeneous reaction increases. The level of decrease is, however, more significant in the case of Ag-water than for Cu-water. Figure 8 shows the influence of stretching on the concentration profiles. It can be seen that the concentration decreases when the sheet is shrunk and increases with stretching.

thumbnailFigure 7. Effects ofKon concentration, when<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M84">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M101">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M125">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M126">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M127">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M117">View MathML</a>.

thumbnailFigure 8. Effects ofλon concentration, when<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M101">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M76">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M127">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M132">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M133">View MathML</a>.

The effects of suction/injection parameter <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M34">View MathML</a> on the velocity and concentration profiles are presented in Figure 9 for a Cu-water nanofluid. These profiles are qualitatively similar to the profiles obtained in the case of a clear fluid. Here we note dual solutions, i.e., the first and second solutions. The velocity and concentration profiles decrease with increasing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M35">View MathML</a> in the case of the first solution. The opposite is, however, true in the case of the second solution. The far field boundary conditions are asymptotically satisfied, thus supporting the validity of the numerical solutions and the existence of dual solutions. In Figure 9(b), the concentration increases more rapidly with increasing suction/injection in the case of the second solution.

thumbnailFigure 9. Effects of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M136">View MathML</a>on (a) velocity and (b) concentration profiles, when<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M137">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M76">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M84">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M75">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M141">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M142">View MathML</a>.

Figure 10 shows the effect of the magnetic parameter on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M143">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M26">View MathML</a> for both Ag-water and Cu-water nanofluids. The velocity and concentration profiles are higher for Cu particles compared to Ag nanoparticles in the absence of the magnetic parameter. Increasing the value of M causes thinning of the boundary layer. The magnetic field enhances the fluid motion at the boundary, and for clear fluids these results are similar to well-known results in the literature.

thumbnailFigure 10. Effects of magnetic parameterMon (a) velocity and (b) concentration profiles, when<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M84">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M75">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M76">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M127">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M141">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M142">View MathML</a>.

The variation of the velocity and concentration profiles with the stretching/shrinking parameter is shown in Figure 11. The dual velocity <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M143">View MathML</a> and concentration <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M26">View MathML</a> profiles decrease with an increase in the magnitude of λ in the case of the first solution and increases in the case of the second solution. It is to be noted that momentum boundary layer thickness for the second solution is thicker than for the first solution. For the case of a clear fluid, the results are similar to those obtained by Bhattacharyya [30]. We also note that the velocity gradient at the surface increases with λ, which is consistent with the results predicted from the computation of the skin friction coefficient.

thumbnailFigure 11. Effects ofλon (a) velocity and (b) concentration profiles, when<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M153">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M137">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M141">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M156">View MathML</a>.

The variation of the reduced skin friction coefficient <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M54">View MathML</a> with λ is shown in Figure 12(a). The values of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M54">View MathML</a> are positive when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M80">View MathML</a> and negative when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M160">View MathML</a>. Physically, a positive <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M54">View MathML</a> implies that the fluid exerts a drag force on the plate and a negative sign implies the opposite. It is evident that dual solutions of equations (13) and (23) subject to the boundary conditions (16) and (24) exist when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M40">View MathML</a>. There is a critical value of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M163">View MathML</a> for which the first and second solutions meet. This critical value depends on the values of the other embedded parameters, and we found, for instance, that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M164">View MathML</a> when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M165">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M166">View MathML</a> when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M167">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M168">View MathML</a> when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M169">View MathML</a>. These results show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M170">View MathML</a> increases with M. However, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M171">View MathML</a>, the solution is not unique, there being two solutions for each λ. Figure 12(b) shows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M172">View MathML</a> increases with the magnetic parameter M.

thumbnailFigure 12. Effects ofλon (a)<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M173">View MathML</a>and (b)<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M174">View MathML</a>, when<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M175">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M176">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/188/mathml/M177">View MathML</a>.

4 Conclusions

The effects of homogeneous-heterogeneous reactions in MHD nanofluid flow due to a stretching or shrinking sheet have been studied. The transformed governing nonlinear differential equations have been solved numerically. Dual solutions for the velocity and concentration distributions have been obtained for some values of the stretching/shrinking, suction/injection and magnetic parameters. The effects of physical and fluid parameters on the velocity, concentration and skin friction have been analyzed. It was observed that for both the cases of shrinking and stretching sheets, the fluid velocity increased with the magnetic parameter. The concentration at the surface decreased as the strength of heterogeneous reactions increased for both Cu-water and Ag-water nanofluids. The boundary layer thickness for the first solution is always thinner than that for the second solution.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The work including proof reading was done by all the authors.

Acknowledgements

The authors are grateful to the University of KwaZulu-Natal for financial support.

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