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==Abstract==
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.
==Keywords==
SWCNTs–water ; Boundary layer slip ; Chemical reaction ; Nanoparticle volume fraction
==Nomenclature==
<math display="inline">C</math>- Concentration of the fluid, <math display="inline">K</math>
<math display="inline">C_w</math>- Concentration of the wall, <math display="inline">K</math>
<math display="inline">C_{\infty }</math>- Concentration of the fluid far away from the wall, <math display="inline">K</math>
<math display="inline">D_f</math>- Specific diffusivity of the base fluid, <math display="inline">m^2\mbox{ }s^{-1}</math>
<math display="inline">D_{nf}</math>- Specific diffusivity of the nanofluid, <math display="inline">m^2\mbox{ }s^{-1}</math>
<math display="inline">g</math>- Acceleration due to gravity, <math display="inline">ms^{-2}</math>
<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> (−)
<math display="inline">k_1</math>- First order rate of chemical reaction, <math display="inline">s^{-1}</math>
<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> (−)
<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> (−)
<math display="inline">x,\mbox{ }y</math>- Streamwise coordinate and cross-stream coordinate, <math display="inline">m</math>
<math display="inline">u,\mbox{ }v</math>- Velocity components in x and y directions, <math display="inline">m\mbox{ }s^{-1}</math>
<math display="inline">u_{\infty }</math>- Flow velocity of the fluid away from the plate, <math display="inline">m\mbox{ }s^{-1}</math>
===Greek symbols===
<math display="inline">{\beta }_c</math>- Volumetric expansion coefficients of the base fluid, <math display="inline">K^{-1}</math>
<math display="inline">{\rho }_f</math>- Density of the base fluid, <math display="inline">kg\mbox{ }m^{-3}</math>
<math display="inline">{\rho }_s</math>- Density of the nanoparticle, <math display="inline">kg\mbox{ }m^{-3}</math>
<math display="inline">{\rho }_{nf}</math>- Effective density of the nanofluid, <math display="inline">kg\mbox{ }m^{-3}</math>
<math display="inline">{\left({\beta }_c\right)}_{nf}</math>- Volumetric coefficient of thermal expansion of nanofluid, <math display="inline">K^{-1}</math>
<math display="inline">{\mu }_f</math>- Dynamic viscosity of the base fluid, <math display="inline">kg\mbox{ }m^{-1}\mbox{ }s^{-1}</math>
<math display="inline">{\mu }_{nf}</math>- Effective dynamic viscosity of the nanofluid, <math display="inline">kg\mbox{ }m^{-1}\mbox{ }s^{-1}</math>
<math display="inline">\gamma </math>- Buoyancy ratio, <math display="inline">\frac{Gc}{R_e^2}</math> (−)
<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> (−)
<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> (−)
<math display="inline">{\nu }_{nf}</math>- Dynamic viscosity of the nanofluid, <math display="inline">m^2\mbox{ }s^{-1}</math>
<math display="inline">\Omega </math>- Resistance, <math display="inline">kg\mbox{ }m^2s^{-3}A^{-2}</math>
<math display="inline">\zeta </math>- Nanoparticle volume fraction, (−)
<math display="inline">\psi </math>- Dimensionless stream function, (−)
<math display="inline">\eta </math>- Similarity variable, (−)
<math display="inline">f</math>- Dimensionless stream function, (−)
<math display="inline">\chi </math>- Dimensionless stream function, (−)
==1. Introduction==
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.
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.
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]]] .
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]]] .
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.
==2. Mathematical analysis==
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
{| class="formulaSCP" style="width: 100%; text-align: center;"
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{| style="text-align: center; margin:auto;"
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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)
|}
<span id='e0015'></span>
{| 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 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>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | ( 2)
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<span id='e0020'></span>
{| 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>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | ( 3)
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with the boundary conditions
<span id='e0025'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;"
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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>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | ( 4)
|}
<span id='f0010'></span>
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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%;">
Fig. 1.
Physical model of the flow and coordinate system. (a) Singh and Kumar [[#bib0225|[44]]] . (b) Present result.
</span>
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<span id='t0010'></span>
{| class="wikitable" style="min-width: 60%;margin-left: auto; margin-right: auto;"
|+
Table 1.
Thermophysical properties of nanofluids, Singh and Kumar [[#bib0225|[44]]] and Talley et al. [[#bib0230|[45]]] .
|-
!
! <math display="inline">\rho \left(kg/m^3\right)</math>
! <math display="inline">c_p\mbox{ }\left(J/kgK\right)</math>
! <math display="inline">k\left(W/mK\right)</math>
! <math display="inline">{\beta }_c\times {10}^{-5}\mbox{ }\left(K^{-1}\right)</math>
|-
| Pure water
| 997.1
| 4179
| 0.613
| 63
|-
| Copper (Cu)
| 8933
| 385
| 401
| 4.89
|-
| Alumina (Al<sub>2</sub> O<sub>3</sub> )
| 3970
| 765
| 40
| 2.55
|-
| SWCNTs
| 2600
| 425
| 6600
| 0.99
|}
<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
{| class="formulaSCP" style="width: 100%; text-align: center;"
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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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{| class="formulaSCP" style="width: 100%; text-align: center;"
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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>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | ( 5)
|}
<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
{| class="formulaSCP" style="width: 100%; text-align: center;"
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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>
|}
| 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
<span id='e0045'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;"
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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>
{| class="formulaSCP" style="width: 100%; text-align: center;"
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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>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | ( 8)
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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>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>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" |
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with the boundary conditions
{| class="formulaSCP" style="width: 100%; text-align: center;"
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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>
|}
| 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.
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
<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.
==3. Results and discussion==
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:
{| class="formulaSCP" style="width: 100%; text-align: center;"
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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>
|-
|<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>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | ( 10)
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<span id='e0070'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;"
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{| style="text-align: center; margin:auto;"
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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>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | ( 11)
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<span id='e0075'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;"
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{| style="text-align: center; margin:auto;"
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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>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | ( 12)
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The boundary conditions are
{| class="formulaSCP" style="width: 100%; text-align: center;"
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{| style="text-align: center; margin:auto;"
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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>
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| style="width: 5px;text-align: right;white-space: nowrap;" | ( 13)
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<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.
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]]] .
<span id='f0015'></span>
{| 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-0002.jpg|center|395px|Comparison of concentration profiles for Sc with Fig. 8 of Singh and Kumar [44].]]
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| <span style="text-align: center; font-size: 75%;">
Fig. 2.
Comparison of concentration profiles for Sc with [[#f0045|Fig. 8]] of Singh and Kumar [[#bib0225|[44]]] .
</span>
|}
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.
<span id='f0020'></span>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;"
|-
|
[[Image:draft_Content_969564504-1-s2.0-S2215098615301129-jestch193-fig-0003.jpg|center|546px|Density of the nanofluids on concentration profiles with ...]]
|-
| <span style="text-align: center; font-size: 75%;">
Fig. 3.
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> .
</span>
|}
<span id='f0025'></span>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;"
|-
|
[[Image:draft_Content_969564504-1-s2.0-S2215098615301129-jestch193-fig-0004.jpg|center|546px|Volumetric expansion of the nanofluids on concentration profiles with ...]]
|-
| <span style="text-align: center; font-size: 75%;">
Fig. 4.
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> .
</span>
|}
<span id='f0030'></span>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;"
|-
|
[[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.]]
|-
| <span style="text-align: center; font-size: 75%;">
Fig. 5.
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> .
</span>
|}
<span id='f0035'></span>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;"
|-
|
[[Image:draft_Content_969564504-1-s2.0-S2215098615301129-jestch193-fig-0006.jpg|center|546px|Buoyancy ratio on velocity and concentration profiles with ...]]
|-
| <span style="text-align: center; font-size: 75%;">
Fig. 6.
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> .
</span>
|}
<span id='f0040'></span>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;"
|-
|
[[Image:draft_Content_969564504-1-s2.0-S2215098615301129-jestch193-fig-0007.jpg|center|546px|Nanoparticle volume fraction on velocity and concentration profiles with ...]]
|-
| <span style="text-align: center; font-size: 75%;">
Fig. 7.
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> .
</span>
|}
<span id='f0045'></span>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;"
|-
|
[[Image:draft_Content_969564504-1-s2.0-S2215098615301129-jestch193-fig-0008.jpg|center|546px|Slip parameter on velocity and concentration profiles with ...]]
|-
| <span style="text-align: center; font-size: 75%;">
Fig. 8.
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> .
</span>
|}
<span id='f0050'></span>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;"
|-
|
[[Image:draft_Content_969564504-1-s2.0-S2215098615301129-jestch193-fig-0009.jpg|center|px|Chemical reaction effects on skin friction and Sherwood number with ...]]
|-
| <span style="text-align: center; font-size: 75%;">
Fig. 9.
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> .
</span>
|}
<span id='t0015'></span>
{| class="wikitable" style="min-width: 60%;margin-left: auto; margin-right: auto;"
|+
Table 2.
<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> .
|-
! Nanofluid
! <math display="inline">{\rho }_s\mbox{ }\left(kg\mbox{ }m^{-3}\right)</math>
! <math display="inline">f^{{''}}\left(0\right)</math>
! <math display="inline">-{\chi }^{{'}}\left(0\right)</math>
|-
| Cu–water
| 100
| 0.40556786
| 1.66762743
|-
|
| 5000
| 0.39729350
| 1.49649125
|-
|
| 9000
| 0.39205250
| 1.39171507
|-
| Al<sub>2</sub> O<sub>3</sub> –water
| 100
| 0.41099025
| 2.10543257
|-
|
| 5000
| 0.39026200
| 1.57012058
|-
|
| 9000
| 0.38101065
| 1.35210936
|-
| SWCNTs–water
| 100
| 0.40318732
| 1.53028233
|-
|
| 5000
| 0.40138208
| 1.49572195
|-
|
| 9000
| 0.39999141
| 1.46928143
|}
<span id='t0020'></span>
{| class="wikitable" style="min-width: 60%;margin-left: auto; margin-right: auto;"
|+
Table 3.
<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> .
|-
! Nanofluid
! <math display="inline">{\beta }_s{10}^{-5}K^{-1}</math>
! <math display="inline">f^{{''}}\left(0\right)</math>
! <math display="inline">-{\chi }^{{'}}\left(0\right)</math>
|-
| Cu–water
| 1.0
| 0.391988185
| 1.39326051
|-
|
| 5.0
| 0.392131817
| 1.39328133
|-
|
| 10
| 0.392311344
| 1.39330737
|-
| Al<sub>2</sub> O<sub>3</sub> –water
| 1.0
| 0.393190042
| 1.64736502
|-
|
| 5.0
| 0.393673401
| 1.64743613
|-
|
| 10
| 0.394277475
| 1.64752500
|-
| SWCNTs–water
| 1.0
| 0.402251627
| 1.51233513
|-
|
| 5.0
| 0.402285124
| 1.51233986
|-
|
| 10
| 0.402326996
| 1.51234577
|}
<span id='t0025'></span>
{| class="wikitable" style="min-width: 60%;margin-left: auto; margin-right: auto;"
|+
Table 4.
<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> .
|-
! Nanofluid
! <math display="inline">\gamma 1</math>
! <math display="inline">f^{{''}}\left(0\right)</math>
! <math display="inline">-{\chi }^{{'}}\left(0\right)</math>
|-
! Cu–water
! 0.1
! 0.41054960
! 0.72599331
|-
|
| 3.0
| 0.37832265
| 2.30159323
|-
|
| 10.0
| 0.36615522
| 4.15126063
|-
| Al<sub>2</sub> O<sub>3</sub> –water
| 0.1
| 0.41416858
| 0.82851250
|-
|
| 3.0
| 0.37888581
| 2.74065092
|-
|
| 10.0
| 0.36666962
| 4.95357715
|-
| SWCNTs–water
| 0.1
| 0.42574385
| 0.77551634
|-
|
| 3.0
| 0.38516728
| 2.50750348
|-
|
| 10.0
| 0.37042421
| 4.52831231
|}
<span id='t0030'></span>
{| class="wikitable" style="min-width: 60%;margin-left: auto; margin-right: auto;"
|+
Table 5.
<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> .
|-
! Nanofluid
! <math display="inline">\gamma </math>
! <math display="inline">f^{{''}}\left(0\right)</math>
! <math display="inline">-{\chi }^{{'}}\left(0\right)</math>
|-
! Cu–water
! 0.01
! 0.33580455
! 1.38400682
|-
|
| 10
| 3.34902280
| 1.71832502
|-
|
| 100
| 17.6203525
| 2.57510459
|-
| Al<sub>2</sub> O<sub>3</sub> –water
| 0.01
| 0.33613146
| 1.63782625
|-
|
| 10
| 3.45010806
| 1.99415775
|-
|
| 100
| 18.5863879
| 2.94381782
|-
| SWCNTs–water
| 0.01
| 0.33707129
| 1.50197468
|-
|
| 10
| 3.93205639
| 1.88208023
|-
|
| 100
| 21.0538645
| 2.83579739
|}
<span id='t0035'></span>
{| class="wikitable" style="min-width: 60%;margin-left: auto; margin-right: auto;"
|+
Table 6.
<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> .
|-
! Nanofluid
! <math display="inline">{\gamma }_2</math>
! <math display="inline">f^{{''}}\left(0\right)</math>
! <math display="inline">-{\chi }^{{'}}\left(0\right)</math>
|-
| Cu–water
| 0.1
| 0.39212786
| 1.39328076
|-
|
| 1.0
| 0.31282789
| 1.52795081
|-
|
| 3.0
| 0.18904557
| 1.64059657
|-
| Al<sub>2</sub> O<sub>3</sub> –water
| 0.1
| 0.42304686
| 1.53129203
|-
|
| 1.0
| 0.33241653
| 1.78719462
|-
|
| 3.0
| 0.21833993
| 1.92701399
|-
| SWCNTs–water
| 0.1
| 0.40225154
| 1.51233512
|-
|
| 1.0
| 0.33662154
| 1.64320141
|-
|
| 3.0
| 0.23558511
| 1.79291693
|}
<span id='t0040'></span>
{| class="wikitable" style="min-width: 60%;margin-left: auto; margin-right: auto;"
|+
Table 7.
<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> .
|-
! Nanofluid
! <math display="inline">\zeta </math>
! <math display="inline">f^{{''}}\left(0\right)</math>
! <math display="inline">-{\chi }^{{'}}\left(0\right)</math>
|-
| Cu–water
| 0.05
| 0.39212786
| 1.39328076
|-
|
| 0.1
| 0.38564620
| 1.35442510
|-
|
| 0.2
| 0.37843858
| 1.38488721
|-
| Al<sub>2</sub> O<sub>3</sub> –water
| 0.05
| 0.39873152
| 1.52782462
|-
|
| 0.1
| 0.39572694
| 1.57741139
|-
|
| 0.2
| 0.39151724
| 1.73935723
|-
| SWCNTs–water
| 0.05
| 0.40085700
| 1.57295700
|-
|
| 0.1
| 0.39935796
| 1.66314059
|-
|
| 0.2
| 0.60331953
| 1.81114233
|}
==4. Conclusion==
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.
==Acknowledgements==
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]] .
==Appendix==
[[Image:draft_Content_969564504-1-s2.0-S2215098615301129-jestch193-fig-5001.jpg|center|546px|Image for unlabelled figure]]
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Published on 10/04/17
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