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==Abstract==
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Single walled carbon nanotube, alumina and copper nanoparticles on convective mass transfer in the presence of base fluid (water) over a horizontal plate are investigated numerically. The governing partial differential equations with auxiliary conditions are reduced into the system of coupled ordinary differential equations via similarity transformation and it has been solved numerically using fourth or fifth order Runge–Kutta–Fehlberg method with shooting technique. The results display that the diffusion boundary layer thickness of the water based Cu and SWCNTs is stronger than Al<sub>2</sub> O<sub>3</sub> –water with increase of chemical reaction.  
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==Keywords==
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SWCNTs–water ; Boundary layer slip ; Chemical reaction ; Nanoparticle volume fraction
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==Nomenclature==
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<math display="inline">C</math>- Concentration of the fluid, <math display="inline">K</math>
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<math display="inline">C_w</math>- Concentration of the wall, <math display="inline">K</math>
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<math display="inline">C_{\infty }</math>- Concentration of the fluid far away from the wall, <math display="inline">K</math>
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<math display="inline">D_f</math>- Specific diffusivity of the base fluid, <math display="inline">m^2\mbox{ }s^{-1}</math>
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<math display="inline">D_{nf}</math>- Specific diffusivity of the nanofluid, <math display="inline">m^2\mbox{ }s^{-1}</math>
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<math display="inline">g</math>- Acceleration due to gravity, <math display="inline">ms^{-2}</math>
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<math display="inline">Gc</math>- Grashof number, <math display="inline">\frac{g{\beta }_c\mbox{ }\left(C_w-C_{\infty }\right)x^3}{{\nu }_f^2}</math> , <math display="inline">\frac{ms^{-2}K^{-1}Km^3}{{\left(m^2s^{-1}\right)}^2}</math>  (−)                          
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<math display="inline">k_1</math>- First order rate of chemical reaction, <math display="inline">s^{-1}</math>
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<math display="inline">Re</math>- Reynolds number, <math display="inline">\frac{u_{\infty }x}{{\nu }_f}</math> , <math display="inline">\frac{m\mbox{ }s^{-1}m}{m^2s^{-1}}</math>  (−)                          
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<math display="inline">Sc</math>- Schmidt number, <math display="inline">\frac{{\nu }_f}{D_f}</math> ,<math display="inline">\frac{m^2\mbox{ }s^{-1}}{m^2\mbox{ }s^{-1}}</math>  (−)                          
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<math display="inline">x,\mbox{ }y</math>- Streamwise coordinate and cross-stream coordinate, <math display="inline">m</math>
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<math display="inline">u,\mbox{ }v</math>- Velocity components in x and y directions, <math display="inline">m\mbox{ }s^{-1}</math>
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<math display="inline">u_{\infty }</math>- Flow velocity of the fluid away from the plate, <math display="inline">m\mbox{ }s^{-1}</math>
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===Greek symbols===
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<math display="inline">{\beta }_c</math>- Volumetric expansion coefficients of the base fluid, <math display="inline">K^{-1}</math>
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<math display="inline">{\rho }_f</math>- Density of the base fluid, <math display="inline">kg\mbox{ }m^{-3}</math>
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<math display="inline">{\rho }_s</math>- Density of the nanoparticle, <math display="inline">kg\mbox{ }m^{-3}</math>
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<math display="inline">{\rho }_{nf}</math>- Effective density of the nanofluid, <math display="inline">kg\mbox{ }m^{-3}</math>
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<math display="inline">{\left({\beta }_c\right)}_{nf}</math>- Volumetric coefficient of thermal expansion of nanofluid, <math display="inline">K^{-1}</math>
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<math display="inline">{\mu }_f</math>- Dynamic viscosity of the base fluid, <math display="inline">kg\mbox{ }m^{-1}\mbox{ }s^{-1}</math>
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<math display="inline">{\mu }_{nf}</math>- Effective dynamic viscosity of the nanofluid, <math display="inline">kg\mbox{ }m^{-1}\mbox{ }s^{-1}</math>
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<math display="inline">\gamma </math>- Buoyancy ratio, <math display="inline">\frac{Gc}{R_e^2}</math>  (−)                          
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<math display="inline">\gamma 1</math>- Chemical reaction parameter, <math display="inline">\frac{k_1x}{u_{\infty }}\left(\frac{s^{-1}m}{m\mbox{ }s^{-1}}\right)</math>  (−)                          
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<math display="inline">{\gamma }_2</math>- Velocity slip parameter, <math display="inline">\frac{v_0u_{\infty }}{{\nu }_f}</math> ,<math display="inline">\frac{m\mbox{ }(m\mbox{ }s^{-1})}{m^2s^{-1}}</math>  (−)                          
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<math display="inline">{\nu }_{nf}</math>- Dynamic viscosity of the nanofluid, <math display="inline">m^2\mbox{ }s^{-1}</math>
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<math display="inline">\Omega </math>- Resistance, <math display="inline">kg\mbox{ }m^2s^{-3}A^{-2}</math>
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<math display="inline">\zeta </math>- Nanoparticle volume fraction, (−)
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<math display="inline">\psi </math>- Dimensionless stream function, (−)
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<math display="inline">\eta </math>- Similarity variable, (−)
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<math display="inline">f</math>- Dimensionless stream function, (−)
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<math display="inline">\chi </math>- Dimensionless stream function, (−)
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==1. Introduction==
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Due to the low diffusion conductivity of mass transfer fluids used in power generation, microelectronics cooling, chemical production, refrigeration and air-conditioning, transportation, and many other applications, it is necessary to enhance effective diffusion conductivity of these fluids to improve mass transfer rate. One of the techniques, to enhance effective diffusion conductivity of these mass transfer fluids, is to add nanoparticles or nanotubes in the base fluids. Particularly with respect to mass transfer, and compared with more conventional mass transfer fluids (i.e. coolants) currently available, nanofluidic coolants exhibit enhanced diffusion conductivity.
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Carbon nanotubes (CNTs) are allotropes of carbon with a cylindrical nanostructure. Nanotubes have been designed significantly larger than for any other material and these cylindrical carbon molecules have extraordinary properties, which are important for Nanoscience and Nanotechnology. In particular, owing to their extraordinary diffusion conductivity and mechanical and electrical properties, carbon nanotubes find applications as additives to enhance mass transfer in various industrial applications.
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Carbon nanotubes are classified as single-walled nanotubes (SWNCTs) and multi-walled nanotubes (MWNCTs) and the carbon nanotubes naturally align themselves into “ropes” retained together by van der Waals forces, more specifically, pi-stacking. Nanofluids act enhanced diffusion properties by diffusing nanoparticles into base fluids [[#bib0010|[1]]] , [[#bib0015|[2]]]  and [[#bib0020|[3]]] . Nanofluids with stronger diffusion conductivity and mass transfer coefficients associated to the base fluid can be significantly useful in many applications [[#bib0025|[4]]] , [[#bib0030|[5]]] , [[#bib0035|[6]]]  and [[#bib0040|[7]]] .      
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Single walled carbon nanotubes (SWCNTs) with high diffusion conductivity have attracted significantly important attention from researchers [[#bib0045|[8]]] , [[#bib0050|[9]]]  and [[#bib0055|[10]]] . In particular, research on different divisional features of SWCNTs–nanofluids are certainly necessary to advance their potential applications in science and technology. Recently, it is investigated that the nanoparticles upgraded the mass transfer inside binary nanofluids (Xuan [[#bib0060|[11]]] , Bhattacharyya [[#bib0065|[12]]] , Sridhara and Satapathy [[#bib0070|[13]]] , Uddin et al. [[#bib0075|[14]]] , Pang et al. [[#bib0080|[15]]] , Kumar et al. [[#bib0085|[16]]] , Rout et al. [[#bib0090|[17]]] , Ibrahim and Reddy [[#bib0095|[18]]]  and Gangadhar et al. [[#bib0100|[19]]] ). Recently several authors investigated about nanofluid flow and mass transfer [[#bib0105|[20]]] , [[#bib0110|[21]]] , [[#bib0115|[22]]] , [[#bib0120|[23]]] , [[#bib0125|[24]]] , [[#bib0130|[25]]] , [[#bib0135|[26]]] , [[#bib0140|[27]]] , [[#bib0145|[28]]] , [[#bib0150|[29]]] , [[#bib0155|[30]]] , [[#bib0160|[31]]] , [[#bib0165|[32]]] , [[#bib0170|[33]]] , [[#bib0175|[34]]] , [[#bib0180|[35]]] , [[#bib0185|[36]]] , [[#bib0190|[37]]] , [[#bib0195|[38]]] , [[#bib0200|[39]]] , [[#bib0205|[40]]] , [[#bib0210|[41]]] , [[#bib0215|[42]]] , [[#bib0220|[43]]] , [[#bib0225|[44]]] , [[#bib0230|[45]]] , [[#bib0235|[46]]]  and [[#bib0240|[47]]] .      
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We consider the two-dimensional boundary slip flow over a flat plate with water as base fluid encompassing single walled carbon nanotubes. Carbon nanotubes are shown to have special diffusion properties with very high diffusion conductivity. The objective of the present study is to find the approximate numerical solutions for the problem and to compare the diffusion behavior of SWCNTs–water with Cu and Al<sub>2</sub> O<sub>3</sub> –water in the presence of chemical reaction.      
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==2. Mathematical analysis==
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Consider the steady two-dimensional laminar boundary layer slip flow of water based SWCNTs, Al<sub>2</sub> O<sub>3</sub>  and Cu with coordinate system that is given in [[#f0010|Fig. 1]]  and the thermophysical properties of the fluid and nanoparticles are presented in [[#t0010|Table 1]] . Under the boundary layer approximation, the basic steady conservation of mass, momentum and diffusion equations can be written (Singh and Kumar [[#bib0225|[44]]] , Magyari [[#bib0235|[46]]]  and Mamut [[#bib0240|[47]]] ) as
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{| class="formulaSCP" style="width: 100%; text-align: center;" 
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|-
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| 
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{| style="text-align: center; margin:auto;" 
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|-
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| <math>\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=</math><math>0</math>
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| style="width: 5px;text-align: right;white-space: nowrap;" | ( 1)
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<span id='e0015'></span>
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{| class="formulaSCP" style="width: 100%; text-align: center;" 
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|-
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| 
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{| style="text-align: center; margin:auto;" 
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| <math>u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=</math><math>{\nu }_{nf}\frac{{\partial }^2u}{\partial y^2}+g{\beta }_{nf}\mbox{ }\left(C-\right. </math><math>\left. C_{\infty }\right)</math>
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| style="width: 5px;text-align: right;white-space: nowrap;" | ( 2)
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<span id='e0020'></span>
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{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <math>u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial y}=</math><math>D_{nf}\frac{{\partial }^2C}{\partial y^2}-k_1\mbox{ }\left(C-\right. </math><math>\left. C_{\infty }\right)</math>
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| style="width: 5px;text-align: right;white-space: nowrap;" | ( 3)
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with the boundary conditions
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<span id='e0025'></span>
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{| class="formulaSCP" style="width: 100%; text-align: center;" 
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| 
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{| style="text-align: center; margin:auto;" 
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| <math>u=v_{sf}\frac{\partial u}{\partial y},\mbox{ }v=</math><math>0,\mbox{ }C=C_w=C_{\infty }+C_0x^{\lambda }\mbox{ }at\mbox{ }y=</math><math>0;</math>
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|<math>\mbox{ }\overline{u}\rightarrow u_{\infty }\mbox{},\mbox{ }C\rightarrow C_{\infty }\mbox{ }as\mbox{ }\overline{y}\rightarrow \infty </math>
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| style="width: 5px;text-align: right;white-space: nowrap;" | ( 4)
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<span id='f0010'></span>
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{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
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[[Image:draft_Content_969564504-1-s2.0-S2215098615301129-jestch193-fig-0001.jpg|center|282px|Physical model of the flow and coordinate system. (a) Singh and Kumar [44]. (b) ...]]
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| <span style="text-align: center; font-size: 75%;">
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Fig. 1.
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Physical model of the flow and coordinate system. (a) Singh and Kumar [[#bib0225|[44]]] . (b) Present result.                  
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</span>
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<span id='t0010'></span>
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{| class="wikitable" style="min-width: 60%;margin-left: auto; margin-right: auto;"
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|+
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Table 1.
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Thermophysical properties of nanofluids, Singh and Kumar [[#bib0225|[44]]]  and Talley et al. [[#bib0230|[45]]] .                  
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|-
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! 
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! <math display="inline">\rho \left(kg/m^3\right)</math>
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! <math display="inline">c_p\mbox{ }\left(J/kgK\right)</math>
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! <math display="inline">k\left(W/mK\right)</math>
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! <math display="inline">{\beta }_c\times {10}^{-5}\mbox{ }\left(K^{-1}\right)</math>
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|-
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| Pure water
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| 997.1
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| 4179
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| 0.613
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| 63
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|-
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| Copper (Cu)
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| 8933
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| 385
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| 401
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| 4.89
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|-
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| Alumina (Al<sub>2</sub> O<sub>3</sub> )                                                    
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| 3970
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| 765
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| 40
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| 2.55
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|-
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| SWCNTs
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| 2600
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| 425
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| 6600
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| 0.99
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|}
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<math display="inline">v_{sf}=v_0\sqrt{\frac{u_{\infty }x}{{\nu }_{nf}}}</math>  is the velocity slipping factor with an initial value <math display="inline">v_0</math>  and <math display="inline">\lambda </math>  is the power index. <math display="inline">{\rho }_{nf}</math>  is the effective density of the nanofluid, <math display="inline">{\mu }_{nf}</math> is the effective dynamic viscosity of the nanofluid, <math display="inline">D_{nf}</math>  is the mass diffusivity of the nanofluid and <math display="inline">{\beta }_{nf}</math>  is the volumetric expansion coefficient, which are defined by Magyari [[#bib0235|[46]]]  and Mamut [[#bib0240|[47]]]  as
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{| style="text-align: center; margin:auto;" 
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| <math>{\rho }_{nf}=\left(1-\zeta \right){\rho }_f+\zeta {\rho }_s\mbox{},\mbox{ }{\mu }_{nf}=</math><math>\frac{{\mu }_f}{{\left(1-\zeta \right)}^{2.5}},\mbox{ }{\beta }_{nf}=</math><math>\left(1-\zeta \right){\beta }_f+\zeta {\beta }_s\mbox{},</math>
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|<math>\mbox{ }{\alpha }_{nf}=\frac{k_{nf}}{{\left(\rho c_p\right)}_{nf}},\mbox{ }D_{nf}=</math><math>\left(1-\zeta \right)D_f</math>
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{| style="text-align: center; margin:auto;" 
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| <math>\frac{k_{nf}}{k_f}=\frac{\left(k_s+2k_f\right)-2\zeta \left(k_f-k_s\right)}{\left(k_s+2k_f\right)+2\zeta \left(k_f-k_s\right)}</math>
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| style="width: 5px;text-align: right;white-space: nowrap;" | ( 5)
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<math display="inline">k_f</math>  and <math display="inline">k_s</math>  are the thermal conductivity of the base fluid and nanoparticle, <math display="inline">\zeta </math>  is the nanoparticle volume fraction, <math display="inline">{\mu }_f</math>  is the dynamic viscosity of the base fluid, <math display="inline">{\beta }_f</math>  and <math display="inline">{\beta }_s</math>  are the volumetric expansion coefficients of the base fluid and nanoparticle, <math display="inline">{\rho }_f</math>  and <math display="inline">{\rho }_s</math>  are the density of the base fluid and nanoparticle, <math display="inline">k_{nf}</math>  is the effective thermal conductivity of the nanofluid. The similarity transformation and stream function are defined as
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{| style="text-align: center; margin:auto;" 
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| <math>\eta =\frac{y}{x}\sqrt{\frac{xu_{\infty }}{{\nu }_{nf}}},\mbox{ }\psi =</math><math>\sqrt{u_{\infty }x{\nu }_{nf}}f\left(\eta \right),\mbox{ }\chi \left(\eta \right)=</math><math>\frac{C-C_{\infty }}{C_w-C_{\infty }},</math>
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|<math>\mbox{ }u=\frac{\partial \psi }{\partial y}\mbox{ }\mbox{and}\mbox{ }v=</math><math>-\frac{\partial \psi }{\partial x}</math>
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| style="width: 5px;text-align: right;white-space: nowrap;" | ( 6)
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The system of Eqs. [[#e0015|(2)]] , [[#e0020|(3)]]  and [[#e0025|(4)]]  become
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<span id='e0045'></span>
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{| style="text-align: center; margin:auto;" 
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| <math>f^{{'''}}+\frac{1}{2}ff^{{''}}+\frac{A_1}{A_3}\gamma \mbox{}\chi =</math><math>0</math>
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| style="width: 5px;text-align: right;white-space: nowrap;" | ( 7)
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<span id='e0050'></span>
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{| style="text-align: center; margin:auto;" 
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| <math>{\chi }^{{''}}+\frac{Sc}{A_2}\left(\frac{1}{2}f{\chi }^{{'}}-\right. </math><math>\left. \lambda f^{{'}}\chi -{\gamma }_1\chi \right)=</math><math>0</math>
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| style="width: 5px;text-align: right;white-space: nowrap;" | ( 8)
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{| style="text-align: center; margin:auto;" 
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| <math>A_1=\left(1-\zeta +\zeta \frac{{\beta }_s}{{\beta }_f}\right),\mbox{ }A_2=</math><math>{\left(1-\zeta \right)}^{3.5}\mbox{ }\left(1-\zeta +\right. </math><math>\left. \zeta \frac{{\rho }_s}{{\rho }_f}\right),</math>
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|<math>\mbox{ }A_3={\left(1-\zeta \right)}^{2.5}\mbox{ }\left(1-\right. </math><math>\left. \zeta +\zeta \frac{{\rho }_s}{{\rho }_f}\right)</math>
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with the boundary conditions
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{| style="text-align: center; margin:auto;" 
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| <math>f\left(0\right)=0,\mbox{ }f^{{'}}\left(0\right)=</math><math>{\gamma }_2\mbox{ }\left(A_4\right)f^{{''}}\left(0\right),\mbox{ }\chi \left(0\right)=</math><math>1;\mbox{ }f^{{'}}\left(\infty \right)=1,\mbox{ }\chi \left(\infty \right)=</math><math>0</math>
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| style="width: 5px;text-align: right;white-space: nowrap;" | ( 9)
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<math display="inline">{\gamma }_2=\frac{v_0u_{\infty }}{{\nu }_f}</math>  is the velocity slip parameter with initial value <math display="inline">v_0</math> , <math display="inline">\gamma =\frac{Gc}{{Re}^2}</math>  is the Buoyancy ratio, <math display="inline">Sc=\frac{{\nu }_f}{D_f}</math>  is the Schmidt number, <math display="inline">Gc=\frac{g{\beta }_f\mbox{ }\left(C_w-C_{\infty }\right)x^3}{{\nu }_f^2}</math>  is the Grashof number, <math display="inline">Re=\frac{u_{\infty }x}{{\nu }_f}</math>  is the Reynolds number and <math display="inline">{\gamma }_1=\frac{k_1x}{u_{\infty }}</math>  is the Chemical reaction parameter.      
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Physical quantities are <math display="inline">C_f=\frac{{\tau }_w}{{\rho }_fU^2}</math>  (Skin friction coefficient) and <math display="inline">Sh_x=\frac{{\varphi }_mx}{D_f\mbox{ }\left(C_w-C_{\infty }\right)}</math>  (Local Sherwood number). <math display="inline">{\tau }_w</math> , <math display="inline">q_w</math>  and <math display="inline">{\varphi }_m</math>  are defined as      
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<math display="inline">C_f\mbox{ }{\left({Re}_x\right)}^{\frac{1}{2}}=\frac{f^{{''}}\left(0\right)}{{\left(1-\zeta \right)}^{2.5}},\mbox{ }\frac{Sh_x}{{Re}_x^{\frac{1}{4}}}=</math><math>-\frac{{\chi }^{{'}}\left(0\right)}{\left(1-\zeta \right)}</math> ; <math display="inline">{Re}_x=\frac{Ux}{{\nu }_f}</math>  is the local Reynolds number.      
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==3. Results and discussion==
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Eqs. [[#e0045|(7)]]  and [[#e0050|(8)]]  subjected to the boundary condition (9) are converted into the following system of first order differential equations, as follows:
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{| style="text-align: center; margin:auto;" 
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| <math>A_1=\left(1-\zeta +\zeta \frac{{\beta }_s}{{\beta }_f}\right),\mbox{ }A_2=</math><math>{\left(1-\zeta \right)}^{3.5}\mbox{ }\left(1-\zeta +\right. </math><math>\left. \zeta \frac{{\rho }_s}{{\rho }_f}\right),</math>
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|-
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|<math>\mbox{ }A_3={\left(1-\zeta \right)}^{2.5}\mbox{ }\left(1-\right. </math><math>\left. \zeta +\zeta \frac{{\rho }_s}{{\rho }_f}\right)</math>
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| style="width: 5px;text-align: right;white-space: nowrap;" | ( 10)
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<span id='e0070'></span>
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| <math>f^{{'}}\left(\eta \right)=u\left(\eta \right),\mbox{ }u^{{'}}\left(\eta \right)=</math><math>v\left(\eta \right),\mbox{ }v^{{'}}\left(\eta \right)=</math><math>-\frac{1}{2}f\left(\eta \right)v\left(\eta \right)-</math><math>\frac{A_1}{A_3}\gamma \mbox{}\chi \left(\eta \right)</math>
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| style="width: 5px;text-align: right;white-space: nowrap;" | ( 11)
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| <math>{\chi }^{{'}}\left(\eta \right)=p\left(\eta \right),\mbox{ }p^{{'}}\left(\eta \right)-</math><math>\frac{Sc}{A_2}\left(\frac{1}{2}f\left(\eta \right)p\left(\eta \right)-\right. </math><math>\left. \lambda u\left(\eta \right)\chi \left(\eta \right)-\right. </math><math>\left. {\gamma }_1\chi \left(\eta \right)\right)</math>
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| style="width: 5px;text-align: right;white-space: nowrap;" | ( 12)
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The boundary conditions are
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|-
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| <math>f\left(0\right)=0,\mbox{ }u\left(0\right)={\gamma }_2\mbox{ }\left(A_4\right)v\left(0\right),\mbox{ }\chi \left(0\right)=</math><math>1;\mbox{ }v\left(0\right)=\alpha ,\mbox{ }p\left(0\right)=</math><math>\beta </math>
340
|}
341
| style="width: 5px;text-align: right;white-space: nowrap;" | ( 13)
342
|}
343
344
<math display="inline">\alpha </math>  and <math display="inline">\beta </math>  are priori unknowns to be determined as a part of the solution of Eqs. [[#e0070|(11)]]  and [[#e0075|(12)]]  with conditions (13), using DSolve subroutine in MAPLE 18. The values of <math display="inline">\alpha </math>  and <math display="inline">\beta </math>  are determined upon solving the boundary conditions <math display="inline">v\left(0\right)=\alpha ,\mbox{ }and\mbox{ }p\left(0\right)=</math><math>\beta </math>  with trial and error basis, and for the benefit of the readers, the Maple 18 worksheet is listed in the Appendix. The numerical results are represented in the form of the velocity and concentration in the presence of water based SWCNT, Cu and Al<sub>2</sub> O<sub>3</sub> . Buoyancy ratio, <math display="inline">\gamma \gg 1.0</math>  is a free convection, <math display="inline">\gamma =1.0</math>  is a mixed convection and <math display="inline">\gamma \ll 1.0</math>  is a forced convection. In this work, <math display="inline">\gamma =2.0</math>  unless otherwise specified.      
345
346
It is observed from the [[#f0015|Fig. 2]]  that the agreement with the theoretical solution of the concentration profile for different values of Schmidt number is significantly correlated with [[#f0045|Fig. 8]]  of Singh and Kumar [[#bib0225|[44]]] .
347
348
<span id='f0015'></span>
349
350
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
351
|-
352
|
353
354
355
[[Image:draft_Content_969564504-1-s2.0-S2215098615301129-jestch193-fig-0002.jpg|center|395px|Comparison of concentration profiles for Sc with Fig. 8 of Singh and Kumar [44].]]
356
357
358
|-
359
| <span style="text-align: center; font-size: 75%;">
360
361
Fig. 2.
362
363
Comparison of concentration profiles for Sc with [[#f0045|Fig. 8]]  of Singh and Kumar [[#bib0225|[44]]] .                  
364
365
</span>
366
|}
367
368
The concentration of the water based SWCNTs, Cu and Al<sub>2</sub> O<sub>3</sub>  increases with increases of the density of the nanoparticles ([[#f0020|Fig. 3]] ) and the rate of mass transfer of SWCNTs and Cu–water decreases ([[#t0015|Table 2]] ) with increase of density. It is interesting to note that the concentration of Al<sub>2</sub> O<sub>3</sub> –water is stronger than SWCNTs–water and Cu–water with increase of density of the nanoparticles. The concentration and the rate of mass transfer of the nanofluids are uniform with increase of volumetric expansion of the nanoparticles (see [[#f0025|Fig. 4]]  and [[#t0020|Table 3]] ). The concentration of the nanofluids (water based SWCNTs, Cu and Al<sub>2</sub> O<sub>3</sub> ) decreases ([[#f0030|Fig. 5]]  and [[#f0035|Fig. 6]] ) but the rate of mass transfer increases ([[#t0025|Table 4]]  and [[#t0030|Table 5]] ) with increase of chemical reaction and buoyancy ratio because of the combined effect of diffusion conductivity and kinematic viscosity of the nanofluids. It is also observed that the diffusion boundary layer thickness of water based Cu and SWCNTs are stronger than Al<sub>2</sub> O<sub>3</sub> –water with increase of chemical reaction since the rate of chemical reaction and the diffusion conductivity play a dominant role on the Cu and SWCNTs–water. The concentration of the SWCNTs–water and Al<sub>2</sub> O<sub>3</sub> –water decreases and the concentration of the Cu–water is uniform with the increase of the nanoparticle volume fraction ([[#f0040|Fig. 7]] ) whereas the diffusion boundary layer thickness of SWCNTs–water is stronger than the water based Cu and Al<sub>2</sub> O<sub>3</sub>  with the increase of the nanoparticle volume fraction. The velocity/concentration of the nanofluids (water based SWCNTs, Cu and Al<sub>2</sub> O<sub>3</sub> ) increases/decreases with the increase of the velocity slip parameter. It is interesting to note that the strength of the diffusion boundary layer thickness of Al<sub>2</sub> O<sub>3</sub> –water is stronger than SWCNTs and Cu–water ([[#f0045|Fig. 8]]  and [[#t0035|Table 6]] ) with the increase of the velocity slip parameter. It is revealed that the diffusion conductivity of the water based Al<sub>2</sub> O<sub>3</sub>  plays a significant role on the enhancement of the mass transfer rate of nanofluids as compared with Cu and SWCNTs–water ([[#t0025|Table 4]] ) with the increase of chemical reaction. Finally, it is noticed that the SWCNTs–water leads to an average convective mass transfer enhancement higher than Cu and Al<sub>2</sub> O<sub>3</sub> –water with the increase of nanoparticle volume fraction ([[#t0040|Table 7]] ). From the [[#f0050|Fig. 9]] , it is observed that the skin friction and Sherwood number of SWCNTs–water and Al<sub>2</sub> O<sub>3</sub> –water is stronger than the other two nanofluids with increase of chemical reaction respectively. This is due to the combined effect of nanoparticle volume friction and diffusion expansion of SWCNTs–water and Al<sub>2</sub> O<sub>3</sub> –water.
369
370
<span id='f0020'></span>
371
372
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
373
|-
374
|
375
376
377
[[Image:draft_Content_969564504-1-s2.0-S2215098615301129-jestch193-fig-0003.jpg|center|546px|Density of the nanofluids on concentration profiles with ...]]
378
379
380
|-
381
| <span style="text-align: center; font-size: 75%;">
382
383
Fig. 3.
384
385
Density of the nanofluids on concentration profiles with <math display="inline">\lambda =0.5,\mbox{ }\gamma =0.1,\mbox{ }Sc=6.2,\mbox{ }{\gamma }_2=</math><math>0.1,\mbox{ }{\gamma }_1=0.5</math> .                  
386
387
</span>
388
|}
389
390
<span id='f0025'></span>
391
392
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
393
|-
394
|
395
396
397
[[Image:draft_Content_969564504-1-s2.0-S2215098615301129-jestch193-fig-0004.jpg|center|546px|Volumetric expansion of the nanofluids on concentration profiles with ...]]
398
399
400
|-
401
| <span style="text-align: center; font-size: 75%;">
402
403
Fig. 4.
404
405
Volumetric expansion of the nanofluids on concentration profiles with <math display="inline">\lambda =0.5,\mbox{ }\gamma =0.1,\mbox{ }Sc=6.2,\mbox{ }{\gamma }_2=</math><math>0.1,\mbox{ }{\gamma }_1=0.5</math> .                  
406
407
</span>
408
|}
409
410
<span id='f0030'></span>
411
412
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
413
|-
414
|
415
416
417
[[Image:draft_Content_969564504-1-s2.0-S2215098615301129-jestch193-fig-0005.jpg|center|546px|Chemical reaction on concentration profiles with λ=0.5, γ=0.1, Sc=6.2, γ2=0.1.]]
418
419
420
|-
421
| <span style="text-align: center; font-size: 75%;">
422
423
Fig. 5.
424
425
Chemical reaction on concentration profiles with <math display="inline">\lambda =0.5,\mbox{ }\gamma =0.1,\mbox{ }Sc=6.2,\mbox{ }{\gamma }_2=</math><math>0.1</math> .                  
426
427
</span>
428
|}
429
430
<span id='f0035'></span>
431
432
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
433
|-
434
|
435
436
437
[[Image:draft_Content_969564504-1-s2.0-S2215098615301129-jestch193-fig-0006.jpg|center|546px|Buoyancy ratio on velocity and concentration profiles with ...]]
438
439
440
|-
441
| <span style="text-align: center; font-size: 75%;">
442
443
Fig. 6.
444
445
Buoyancy ratio on velocity and concentration profiles with <math display="inline">\lambda =0.5,\mbox{ }Sc=6.2,\mbox{ }{\gamma }_2=</math><math>0.1,\mbox{ }{\gamma }_1=0.5</math> .                  
446
447
</span>
448
|}
449
450
<span id='f0040'></span>
451
452
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
453
|-
454
|
455
456
457
[[Image:draft_Content_969564504-1-s2.0-S2215098615301129-jestch193-fig-0007.jpg|center|546px|Nanoparticle volume fraction on velocity and concentration profiles with ...]]
458
459
460
|-
461
| <span style="text-align: center; font-size: 75%;">
462
463
Fig. 7.
464
465
Nanoparticle volume fraction on velocity and concentration profiles with <math display="inline">{\gamma }_1=0.5,\mbox{ }\lambda =0.5,\mbox{ }Sc=</math><math>6.2,\mbox{ }\gamma =0.1,\mbox{ }{\gamma }_2=0.1</math> .                  
466
467
</span>
468
|}
469
470
<span id='f0045'></span>
471
472
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
473
|-
474
|
475
476
477
[[Image:draft_Content_969564504-1-s2.0-S2215098615301129-jestch193-fig-0008.jpg|center|546px|Slip parameter on velocity and concentration profiles with ...]]
478
479
480
|-
481
| <span style="text-align: center; font-size: 75%;">
482
483
Fig. 8.
484
485
Slip parameter on velocity and concentration profiles with <math display="inline">\lambda =0.5,\mbox{ }Sc=6.2,\mbox{ }\gamma =0.1,\mbox{ }{\gamma }_1=</math><math>0.5</math> .                  
486
487
</span>
488
|}
489
490
<span id='f0050'></span>
491
492
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
493
|-
494
|
495
496
497
[[Image:draft_Content_969564504-1-s2.0-S2215098615301129-jestch193-fig-0009.jpg|center|px|Chemical reaction effects on skin friction and Sherwood number with ...]]
498
499
500
|-
501
| <span style="text-align: center; font-size: 75%;">
502
503
Fig. 9.
504
505
Chemical reaction effects on skin friction and Sherwood number with <math display="inline">\lambda =0.5,\mbox{ }Sc=6.2,\mbox{ }\gamma =0.1</math> .                  
506
507
</span>
508
|}
509
510
<span id='t0015'></span>
511
512
{| class="wikitable" style="min-width: 60%;margin-left: auto; margin-right: auto;"
513
|+
514
515
Table 2.
516
517
<math display="inline">f^{{''}}\left(0\right)</math> and <math display="inline">-{\theta }^{{'}}\left(0\right)</math>  for different values of <math display="inline">{\rho }_s</math>  with <math display="inline">\lambda =0.5,\mbox{ }\gamma =0.1,\mbox{ }Sc=6.2,\mbox{ }{\gamma }_2=</math><math>0.1,\mbox{ }{\gamma }_1=0.5</math> .                  
518
519
|-
520
521
! Nanofluid
522
! <math display="inline">{\rho }_s\mbox{ }\left(kg\mbox{ }m^{-3}\right)</math>
523
! <math display="inline">f^{{''}}\left(0\right)</math>
524
! <math display="inline">-{\chi }^{{'}}\left(0\right)</math>
525
|-
526
527
| Cu–water
528
| 100
529
| 0.40556786
530
| 1.66762743
531
|-
532
533
| 
534
| 5000
535
| 0.39729350
536
| 1.49649125
537
|-
538
539
| 
540
| 9000
541
| 0.39205250
542
| 1.39171507
543
|-
544
545
| Al<sub>2</sub> O<sub>3</sub> –water                                                    
546
| 100
547
| 0.41099025
548
| 2.10543257
549
|-
550
551
| 
552
| 5000
553
| 0.39026200
554
| 1.57012058
555
|-
556
557
| 
558
| 9000
559
| 0.38101065
560
| 1.35210936
561
|-
562
563
| SWCNTs–water
564
| 100
565
| 0.40318732
566
| 1.53028233
567
|-
568
569
| 
570
| 5000
571
| 0.40138208
572
| 1.49572195
573
|-
574
575
| 
576
| 9000
577
| 0.39999141
578
| 1.46928143
579
|}
580
581
<span id='t0020'></span>
582
583
{| class="wikitable" style="min-width: 60%;margin-left: auto; margin-right: auto;"
584
|+
585
586
Table 3.
587
588
<math display="inline">f^{{''}}\left(0\right)</math>  and <math display="inline">-{\theta }^{{'}}\left(0\right)</math>  for different values of <math display="inline">{\beta }_s</math>  with <math display="inline">\lambda =0.5,\mbox{ }\gamma =0.1,\mbox{ }Sc=6.2,\mbox{ }{\gamma }_2=</math><math>0.1,\mbox{ }{\gamma }_1=0.5</math> .                  
589
590
|-
591
592
! Nanofluid
593
! <math display="inline">{\beta }_s{10}^{-5}K^{-1}</math>
594
! <math display="inline">f^{{''}}\left(0\right)</math>
595
! <math display="inline">-{\chi }^{{'}}\left(0\right)</math>
596
|-
597
598
| Cu–water
599
| 1.0
600
| 0.391988185
601
| 1.39326051
602
|-
603
604
| 
605
| 5.0
606
| 0.392131817
607
| 1.39328133
608
|-
609
610
| 
611
| 10
612
| 0.392311344
613
| 1.39330737
614
|-
615
616
| Al<sub>2</sub> O<sub>3</sub> –water                                                    
617
| 1.0
618
| 0.393190042
619
| 1.64736502
620
|-
621
622
| 
623
| 5.0
624
| 0.393673401
625
| 1.64743613
626
|-
627
628
| 
629
| 10
630
| 0.394277475
631
| 1.64752500
632
|-
633
634
| SWCNTs–water
635
| 1.0
636
| 0.402251627
637
| 1.51233513
638
|-
639
640
| 
641
| 5.0
642
| 0.402285124
643
| 1.51233986
644
|-
645
646
| 
647
| 10
648
| 0.402326996
649
| 1.51234577
650
|}
651
652
<span id='t0025'></span>
653
654
{| class="wikitable" style="min-width: 60%;margin-left: auto; margin-right: auto;"
655
|+
656
657
Table 4.
658
659
<math display="inline">f^{{''}}\left(0\right)</math>  and <math display="inline">-{\theta }^{{'}}\left(0\right)</math>  for different values of <math display="inline">\gamma 1</math>  with <math display="inline">\lambda =0.5,\mbox{ }\gamma =0.1,\mbox{ }Sc=6.2,\mbox{ }{\gamma }_2=</math><math>0.1</math> .                  
660
661
|-
662
663
! Nanofluid
664
! <math display="inline">\gamma 1</math>
665
! <math display="inline">f^{{''}}\left(0\right)</math>
666
! <math display="inline">-{\chi }^{{'}}\left(0\right)</math>
667
|-
668
669
! Cu–water
670
! 0.1
671
! 0.41054960
672
! 0.72599331
673
|-
674
675
| 
676
| 3.0
677
| 0.37832265
678
| 2.30159323
679
|-
680
681
| 
682
| 10.0
683
| 0.36615522
684
| 4.15126063
685
|-
686
687
| Al<sub>2</sub> O<sub>3</sub> –water                                                    
688
| 0.1
689
| 0.41416858
690
| 0.82851250
691
|-
692
693
| 
694
| 3.0
695
| 0.37888581
696
| 2.74065092
697
|-
698
699
| 
700
| 10.0
701
| 0.36666962
702
| 4.95357715
703
|-
704
705
| SWCNTs–water
706
| 0.1
707
| 0.42574385
708
| 0.77551634
709
|-
710
711
| 
712
| 3.0
713
| 0.38516728
714
| 2.50750348
715
|-
716
717
| 
718
| 10.0
719
| 0.37042421
720
| 4.52831231
721
|}
722
723
<span id='t0030'></span>
724
725
{| class="wikitable" style="min-width: 60%;margin-left: auto; margin-right: auto;"
726
|+
727
728
Table 5.
729
730
<math display="inline">f^{{''}}\left(0\right)</math> and <math display="inline">-{\theta }^{{'}}\left(0\right)</math>  for different values of <math display="inline">\gamma </math>  with <math display="inline">\lambda =0.5,\mbox{ }Sc=6.2,\mbox{ }{\gamma }_2=</math><math>0.1,\mbox{ }{\gamma }_1=0.5</math> .                  
731
732
|-
733
734
! Nanofluid
735
! <math display="inline">\gamma </math>
736
! <math display="inline">f^{{''}}\left(0\right)</math>
737
! <math display="inline">-{\chi }^{{'}}\left(0\right)</math>
738
|-
739
740
! Cu–water
741
! 0.01
742
! 0.33580455
743
! 1.38400682
744
|-
745
746
| 
747
| 10
748
| 3.34902280
749
| 1.71832502
750
|-
751
752
| 
753
| 100
754
| 17.6203525
755
| 2.57510459
756
|-
757
758
| Al<sub>2</sub> O<sub>3</sub> –water                                                    
759
| 0.01
760
| 0.33613146
761
| 1.63782625
762
|-
763
764
| 
765
| 10
766
| 3.45010806
767
| 1.99415775
768
|-
769
770
| 
771
| 100
772
| 18.5863879
773
| 2.94381782
774
|-
775
776
| SWCNTs–water
777
| 0.01
778
| 0.33707129
779
| 1.50197468
780
|-
781
782
| 
783
| 10
784
| 3.93205639
785
| 1.88208023
786
|-
787
788
| 
789
| 100
790
| 21.0538645
791
| 2.83579739
792
|}
793
794
<span id='t0035'></span>
795
796
{| class="wikitable" style="min-width: 60%;margin-left: auto; margin-right: auto;"
797
|+
798
799
Table 6.
800
801
<math display="inline">f^{{''}}\left(0\right)</math> and <math display="inline">-{\theta }^{{'}}\left(0\right)</math>  for different values of <math display="inline">{\gamma }_2</math>  with <math display="inline">\lambda =0.5,\mbox{ }Sc=6.2,\mbox{ }\gamma =0.1,\mbox{ }{\gamma }_1=</math><math>0.5</math> .                  
802
803
|-
804
805
! Nanofluid
806
! <math display="inline">{\gamma }_2</math>
807
! <math display="inline">f^{{''}}\left(0\right)</math>
808
! <math display="inline">-{\chi }^{{'}}\left(0\right)</math>
809
|-
810
811
| Cu–water
812
| 0.1
813
| 0.39212786
814
| 1.39328076
815
|-
816
817
| 
818
| 1.0
819
| 0.31282789
820
| 1.52795081
821
|-
822
823
| 
824
| 3.0
825
| 0.18904557
826
| 1.64059657
827
|-
828
829
| Al<sub>2</sub> O<sub>3</sub> –water                                                    
830
| 0.1
831
| 0.42304686
832
| 1.53129203
833
|-
834
835
| 
836
| 1.0
837
| 0.33241653
838
| 1.78719462
839
|-
840
841
| 
842
| 3.0
843
| 0.21833993
844
| 1.92701399
845
|-
846
847
| SWCNTs–water
848
| 0.1
849
| 0.40225154
850
| 1.51233512
851
|-
852
853
| 
854
| 1.0
855
| 0.33662154
856
| 1.64320141
857
|-
858
859
| 
860
| 3.0
861
| 0.23558511
862
| 1.79291693
863
|}
864
865
<span id='t0040'></span>
866
867
{| class="wikitable" style="min-width: 60%;margin-left: auto; margin-right: auto;"
868
|+
869
870
Table 7.
871
872
<math display="inline">f^{{''}}\left(0\right)</math>  and <math display="inline">-{\theta }^{{'}}\left(0\right)</math>  for different values of <math display="inline">\zeta </math>  with <math display="inline">{\gamma }_1=0.5,\mbox{ }\lambda =0.5,\mbox{ }Sc=</math><math>6.2,\mbox{ }\gamma =0.1,\mbox{ }{\gamma }_2=0.1</math> .                  
873
874
|-
875
876
! Nanofluid
877
! <math display="inline">\zeta </math>
878
! <math display="inline">f^{{''}}\left(0\right)</math>
879
! <math display="inline">-{\chi }^{{'}}\left(0\right)</math>
880
|-
881
882
| Cu–water
883
| 0.05
884
| 0.39212786
885
| 1.39328076
886
|-
887
888
| 
889
| 0.1
890
| 0.38564620
891
| 1.35442510
892
|-
893
894
| 
895
| 0.2
896
| 0.37843858
897
| 1.38488721
898
|-
899
900
| Al<sub>2</sub> O<sub>3</sub> –water                                                    
901
| 0.05
902
| 0.39873152
903
| 1.52782462
904
|-
905
906
| 
907
| 0.1
908
| 0.39572694
909
| 1.57741139
910
|-
911
912
| 
913
| 0.2
914
| 0.39151724
915
| 1.73935723
916
|-
917
918
| SWCNTs–water
919
| 0.05
920
| 0.40085700
921
| 1.57295700
922
|-
923
924
| 
925
| 0.1
926
| 0.39935796
927
| 1.66314059
928
|-
929
930
| 
931
| 0.2
932
| 0.60331953
933
| 1.81114233
934
|}
935
936
==4. Conclusion==
937
938
Mass transfer of the water based SWCNTs, Alumina and Cu over a flat plate in the presence of chemical reaction under slip condition are investigated. The system of coupled nonlinear ODEs are solved numerically using the fourth or fifth order Runge–Kutta–Fehlberg method with the shooting technique. The strength of diffusion boundary layer thickness of SWCNTs and Cu–water is stronger than Al<sub>2</sub> O<sub>3</sub> –water with increase of chemical reaction. The diffusion boundary layer thickness of SWCNTs–water is stronger than Al<sub>2</sub> O<sub>3</sub>  and Cu–water with increase of nanoparticle volume fraction since the single walled carbon nanotubes (SWCNTs) have extraordinary mechanical, electrical, thermal, optical and chemical properties. The rate of chemical reaction in the presence of Cu–water and SWCNTs–water plays a dominant role on the flow field due to the combined effects of diffusion conductivity and density of the water based Cu and SWCNTs. Furthermore, the strength of diffusion boundary layer thickness of Al<sub>2</sub> O<sub>3</sub> –water is stronger than SWCNTs and Cu–water with increase of the velocity slip parameter. This is due to the combined effect of the kinematic viscosity and diffusion conductivity of the Al<sub>2</sub> O<sub>3</sub> –water.
939
940
==Acknowledgements==
941
942
The authors wish to express their cordial thanks to our beloved The Vice Chancellor and The Dean, FSTPi, UTHM, Malaysia, for their encouragements and acknowledge the financial support received from FRGS[[#gsp0010|1208/2013]] .
943
944
==Appendix==
945
946
947
[[Image:draft_Content_969564504-1-s2.0-S2215098615301129-jestch193-fig-5001.jpg|center|546px|Image for unlabelled figure]]
948
949
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