Impact of chemical reaction on Cu, Al2 O3 and SWCNTs–nanofluid flow under slip conditions

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 Al₂ O₃ –water with increase of chemical reaction.

Keywords

SWCNTs–water ; Boundary layer slip ; Chemical reaction ; Nanoparticle volume fraction

Nomenclature

${\textstyle C}$- Concentration of the fluid, ${\textstyle K}$

${\textstyle C_{w}}$- Concentration of the wall, ${\textstyle K}$

${\textstyle C_{\infty }}$- Concentration of the fluid far away from the wall, ${\textstyle K}$

${\textstyle D_{f}}$- Specific diffusivity of the base fluid, ${\textstyle m^{2}{\mbox{ }}s^{-1}}$

${\textstyle D_{nf}}$- Specific diffusivity of the nanofluid, ${\textstyle m^{2}{\mbox{ }}s^{-1}}$

${\textstyle g}$- Acceleration due to gravity, ${\textstyle ms^{-2}}$

${\textstyle Gc}$- Grashof number, ${\textstyle {\frac {g{\beta }_{c}{\mbox{ }}\left(C_{w}-C_{\infty }\right)x^{3}}{{\nu }_{f}^{2}}}}$ , ${\textstyle {\frac {ms^{-2}K^{-1}Km^{3}}{{\left(m^{2}s^{-1}\right)}^{2}}}}$ (−)

${\textstyle k_{1}}$- First order rate of chemical reaction, ${\textstyle s^{-1}}$

${\textstyle Re}$- Reynolds number, ${\textstyle {\frac {u_{\infty }x}{{\nu }_{f}}}}$ , ${\textstyle {\frac {m{\mbox{ }}s^{-1}m}{m^{2}s^{-1}}}}$ (−)

${\textstyle Sc}$- Schmidt number, ${\textstyle {\frac {{\nu }_{f}}{D_{f}}}}$ ,${\textstyle {\frac {m^{2}{\mbox{ }}s^{-1}}{m^{2}{\mbox{ }}s^{-1}}}}$ (−)

${\textstyle x,{\mbox{ }}y}$- Streamwise coordinate and cross-stream coordinate, ${\textstyle m}$

${\textstyle u,{\mbox{ }}v}$- Velocity components in x and y directions, ${\textstyle m{\mbox{ }}s^{-1}}$

${\textstyle u_{\infty }}$- Flow velocity of the fluid away from the plate, ${\textstyle m{\mbox{ }}s^{-1}}$

Greek symbols

${\textstyle {\beta }_{c}}$- Volumetric expansion coefficients of the base fluid, ${\textstyle K^{-1}}$

${\textstyle {\rho }_{f}}$- Density of the base fluid, ${\textstyle kg{\mbox{ }}m^{-3}}$

${\textstyle {\rho }_{s}}$- Density of the nanoparticle, ${\textstyle kg{\mbox{ }}m^{-3}}$

${\textstyle {\rho }_{nf}}$- Effective density of the nanofluid, ${\textstyle kg{\mbox{ }}m^{-3}}$

${\textstyle {\left({\beta }_{c}\right)}_{nf}}$- Volumetric coefficient of thermal expansion of nanofluid, ${\textstyle K^{-1}}$

${\textstyle {\mu }_{f}}$- Dynamic viscosity of the base fluid, ${\textstyle kg{\mbox{ }}m^{-1}{\mbox{ }}s^{-1}}$

${\textstyle {\mu }_{nf}}$- Effective dynamic viscosity of the nanofluid, ${\textstyle kg{\mbox{ }}m^{-1}{\mbox{ }}s^{-1}}$

${\textstyle \gamma }$- Buoyancy ratio, ${\textstyle {\frac {Gc}{R_{e}^{2}}}}$ (−)

${\textstyle \gamma 1}$- Chemical reaction parameter, ${\textstyle {\frac {k_{1}x}{u_{\infty }}}\left({\frac {s^{-1}m}{m{\mbox{ }}s^{-1}}}\right)}$ (−)

${\textstyle {\gamma }_{2}}$- Velocity slip parameter, ${\textstyle {\frac {v_{0}u_{\infty }}{{\nu }_{f}}}}$ ,${\textstyle {\frac {m{\mbox{ }}(m{\mbox{ }}s^{-1})}{m^{2}s^{-1}}}}$ (−)

${\textstyle {\nu }_{nf}}$- Dynamic viscosity of the nanofluid, ${\textstyle m^{2}{\mbox{ }}s^{-1}}$

${\textstyle \Omega }$- Resistance, ${\textstyle kg{\mbox{ }}m^{2}s^{-3}A^{-2}}$

${\textstyle \zeta }$- Nanoparticle volume fraction, (−)

${\textstyle \psi }$- Dimensionless stream function, (−)

${\textstyle \eta }$- Similarity variable, (−)

${\textstyle f}$- Dimensionless stream function, (−)

${\textstyle \chi }$- 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 [1] , [2]  and [3] . Nanofluids with stronger diffusion conductivity and mass transfer coefficients associated to the base fluid can be significantly useful in many applications [4] , [5] , [6]  and [7] .

Single walled carbon nanotubes (SWCNTs) with high diffusion conductivity have attracted significantly important attention from researchers [8] , [9]  and [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 [11] , Bhattacharyya [12] , Sridhara and Satapathy [13] , Uddin et al. [14] , Pang et al. [15] , Kumar et al. [16] , Rout et al. [17] , Ibrahim and Reddy [18] and Gangadhar et al. [19] ). Recently several authors investigated about nanofluid flow and mass transfer [20] , [21] , [22] , [23] , [24] , [25] , [26] , [27] , [28] , [29] , [30] , [31] , [32] , [33] , [34] , [35] , [36] , [37] , [38] , [39] , [40] , [41] , [42] , [43] , [44] , [45] , [46]  and [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 Al₂ O₃ –water in the presence of chemical reaction.

2. Mathematical analysis

Consider the steady two-dimensional laminar boundary layer slip flow of water based SWCNTs, Al₂ O₃ and Cu with coordinate system that is given in Fig. 1 and the thermophysical properties of the fluid and nanoparticles are presented in Table 1 . Under the boundary layer approximation, the basic steady conservation of mass, momentum and diffusion equations can be written (Singh and Kumar [44] , Magyari [46] and Mamut [47] ) as

${\frac {\partial u}{\partial x}}+{\frac {\partial v}{\partial y}}=$$0$

( 1)

$u{\frac {\partial u}{\partial x}}+v{\frac {\partial u}{\partial y}}=$${\nu }_{nf}{\frac {{\partial }^{2}u}{\partial y^{2}}}+g{\beta }_{nf}{\mbox{ }}\left(C-\right.$$\left.C_{\infty }\right)$

( 2)

$u{\frac {\partial C}{\partial x}}+v{\frac {\partial C}{\partial y}}=$$D_{nf}{\frac {{\partial }^{2}C}{\partial y^{2}}}-k_{1}{\mbox{ }}\left(C-\right.$$\left.C_{\infty }\right)$

( 3)

with the boundary conditions

$u=v_{sf}{\frac {\partial u}{\partial y}},{\mbox{ }}v=$$0,{\mbox{ }}C=C_{w}=C_{\infty }+C_{0}x^{\lambda }{\mbox{ }}at{\mbox{ }}y=$$0;$ ${\mbox{ }}{\overline {u}}\rightarrow u_{\infty }{\mbox{​}},{\mbox{ }}C\rightarrow C_{\infty }{\mbox{ }}as{\mbox{ }}{\overline {y}}\rightarrow \infty$

( 4)

Fig. 1. Physical model of the flow and coordinate system. (a) Singh and Kumar [44] . (b) Present result.

Table 1. Thermophysical properties of nanofluids, Singh and Kumar [[#bib0225|[44]]] and Talley et al. [[#bib0230|[45]]] .

	${\textstyle \rho \left(kg/m^{3}\right)}$	${\textstyle c_{p}{\mbox{ }}\left(J/kgK\right)}$	${\textstyle k\left(W/mK\right)}$	${\textstyle {\beta }_{c}\times {10}^{-5}{\mbox{ }}\left(K^{-1}\right)}$
Pure water	997.1	4179	0.613	63
Copper (Cu)	8933	385	401	4.89
Alumina (Al2 O3 )	3970	765	40	2.55
SWCNTs	2600	425	6600	0.99

${\textstyle v_{sf}=v_{0}{\sqrt {\frac {u_{\infty }x}{{\nu }_{nf}}}}}$ is the velocity slipping factor with an initial value ${\textstyle v_{0}}$ and ${\textstyle \lambda }$ is the power index. ${\textstyle {\rho }_{nf}}$ is the effective density of the nanofluid, ${\textstyle {\mu }_{nf}}$ is the effective dynamic viscosity of the nanofluid, ${\textstyle D_{nf}}$ is the mass diffusivity of the nanofluid and ${\textstyle {\beta }_{nf}}$ is the volumetric expansion coefficient, which are defined by Magyari [46] and Mamut [47] as

${\rho }_{nf}=\left(1-\zeta \right){\rho }_{f}+\zeta {\rho }_{s}{\mbox{​}},{\mbox{ }}{\mu }_{nf}=$${\frac {{\mu }_{f}}{{\left(1-\zeta \right)}^{2.5}}},{\mbox{ }}{\beta }_{nf}=$$\left(1-\zeta \right){\beta }_{f}+\zeta {\beta }_{s}{\mbox{​}},$ ${\mbox{ }}{\alpha }_{nf}={\frac {k_{nf}}{{\left(\rho c_{p}\right)}_{nf}}},{\mbox{ }}D_{nf}=$$\left(1-\zeta \right)D_{f}$

${\frac {k_{nf}}{k_{f}}}={\frac {\left(k_{s}+2k_{f}\right)-2\zeta \left(k_{f}-k_{s}\right)}{\left(k_{s}+2k_{f}\right)+2\zeta \left(k_{f}-k_{s}\right)}}$

( 5)

${\textstyle k_{f}}$ and ${\textstyle k_{s}}$ are the thermal conductivity of the base fluid and nanoparticle, ${\textstyle \zeta }$ is the nanoparticle volume fraction, ${\textstyle {\mu }_{f}}$ is the dynamic viscosity of the base fluid, ${\textstyle {\beta }_{f}}$ and ${\textstyle {\beta }_{s}}$ are the volumetric expansion coefficients of the base fluid and nanoparticle, ${\textstyle {\rho }_{f}}$ and ${\textstyle {\rho }_{s}}$ are the density of the base fluid and nanoparticle, ${\textstyle k_{nf}}$ is the effective thermal conductivity of the nanofluid. The similarity transformation and stream function are defined as

$\eta ={\frac {y}{x}}{\sqrt {\frac {xu_{\infty }}{{\nu }_{nf}}}},{\mbox{ }}\psi =$${\sqrt {u_{\infty }x{\nu }_{nf}}}f\left(\eta \right),{\mbox{ }}\chi \left(\eta \right)=$${\frac {C-C_{\infty }}{C_{w}-C_{\infty }}},$ ${\mbox{ }}u={\frac {\partial \psi }{\partial y}}{\mbox{ }}{\mbox{and}}{\mbox{ }}v=$$-{\frac {\partial \psi }{\partial x}}$

( 6)

The system of Eqs. (2) , (3)  and (4) become

$f^{'''}+{\frac {1}{2}}ff^{''}+{\frac {A_{1}}{A_{3}}}\gamma {\mbox{​}}\chi =$$0$

( 7)

${\chi }^{''}+{\frac {Sc}{A_{2}}}\left({\frac {1}{2}}f{\chi }^{'}-\right.$$\left.\lambda f^{'}\chi -{\gamma }_{1}\chi \right)=$$0$

( 8)

$A_{1}=\left(1-\zeta +\zeta {\frac {{\beta }_{s}}{{\beta }_{f}}}\right),{\mbox{ }}A_{2}=$${\left(1-\zeta \right)}^{3.5}{\mbox{ }}\left(1-\zeta +\right.$$\left.\zeta {\frac {{\rho }_{s}}{{\rho }_{f}}}\right),$ ${\mbox{ }}A_{3}={\left(1-\zeta \right)}^{2.5}{\mbox{ }}\left(1-\right.$$\left.\zeta +\zeta {\frac {{\rho }_{s}}{{\rho }_{f}}}\right)$

with the boundary conditions

$f\left(0\right)=0,{\mbox{ }}f^{'}\left(0\right)=$${\gamma }_{2}{\mbox{ }}\left(A_{4}\right)f^{''}\left(0\right),{\mbox{ }}\chi \left(0\right)=$$1;{\mbox{ }}f^{'}\left(\infty \right)=1,{\mbox{ }}\chi \left(\infty \right)=$$0$

( 9)

${\textstyle {\gamma }_{2}={\frac {v_{0}u_{\infty }}{{\nu }_{f}}}}$ is the velocity slip parameter with initial value ${\textstyle v_{0}}$ , ${\textstyle \gamma ={\frac {Gc}{{Re}^{2}}}}$ is the Buoyancy ratio, ${\textstyle Sc={\frac {{\nu }_{f}}{D_{f}}}}$ is the Schmidt number, ${\textstyle Gc={\frac {g{\beta }_{f}{\mbox{ }}\left(C_{w}-C_{\infty }\right)x^{3}}{{\nu }_{f}^{2}}}}$ is the Grashof number, ${\textstyle Re={\frac {u_{\infty }x}{{\nu }_{f}}}}$ is the Reynolds number and ${\textstyle {\gamma }_{1}={\frac {k_{1}x}{u_{\infty }}}}$ is the Chemical reaction parameter.

Physical quantities are ${\textstyle C_{f}={\frac {{\tau }_{w}}{{\rho }_{f}U^{2}}}}$ (Skin friction coefficient) and ${\textstyle Sh_{x}={\frac {{\varphi }_{m}x}{D_{f}{\mbox{ }}\left(C_{w}-C_{\infty }\right)}}}$ (Local Sherwood number). ${\textstyle {\tau }_{w}}$ , ${\textstyle q_{w}}$ and ${\textstyle {\varphi }_{m}}$ are defined as

${\textstyle C_{f}{\mbox{ }}{\left({Re}_{x}\right)}^{\frac {1}{2}}=}$Failed to parse (syntax error): \frac{f^{{´´}}\left(0\right)}{{\left(1-\zeta \right)}^{2.5}},\mbox{ }\frac{Sh_x}{{Re}_x^{\frac{1}{4}}}= $-{\frac {{\chi }^{'}\left(0\right)}{\left(1-\zeta \right)}}$ ; ${\textstyle {Re}_{x}={\frac {Ux}{{\nu }_{f}}}}$ is the local Reynolds number.

3. Results and discussion

Eqs. (7) and (8) subjected to the boundary condition (9) are converted into the following system of first order differential equations, as follows:

$A_{1}=\left(1-\zeta +\zeta {\frac {{\beta }_{s}}{{\beta }_{f}}}\right),{\mbox{ }}A_{2}=$${\left(1-\zeta \right)}^{3.5}{\mbox{ }}\left(1-\zeta +\right.$$\left.\zeta {\frac {{\rho }_{s}}{{\rho }_{f}}}\right),$ ${\mbox{ }}A_{3}={\left(1-\zeta \right)}^{2.5}{\mbox{ }}\left(1-\right.$$\left.\zeta +\zeta {\frac {{\rho }_{s}}{{\rho }_{f}}}\right)$

( 10)

$f^{'}\left(\eta \right)=u\left(\eta \right),{\mbox{ }}u^{'}\left(\eta \right)=$$v\left(\eta \right),{\mbox{ }}v^{'}\left(\eta \right)=$$-{\frac {1}{2}}f\left(\eta \right)v\left(\eta \right)-$${\frac {A_{1}}{A_{3}}}\gamma {\mbox{​}}\chi \left(\eta \right)$

( 11)

${\chi }^{'}\left(\eta \right)=p\left(\eta \right),{\mbox{ }}p^{'}\left(\eta \right)-$${\frac {Sc}{A_{2}}}\left({\frac {1}{2}}f\left(\eta \right)p\left(\eta \right)-\right.$$\left.\lambda u\left(\eta \right)\chi \left(\eta \right)-\right.$$\left.{\gamma }_{1}\chi \left(\eta \right)\right)$

( 12)

The boundary conditions are

$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)=$$1;{\mbox{ }}v\left(0\right)=\alpha ,{\mbox{ }}p\left(0\right)=$$\beta$

( 13)

${\textstyle \alpha }$ and ${\textstyle \beta }$ are priori unknowns to be determined as a part of the solution of Eqs. (11) and (12) with conditions (13), using DSolve subroutine in MAPLE 18. The values of ${\textstyle \alpha }$ and ${\textstyle \beta }$ are determined upon solving the boundary conditions ${\textstyle v\left(0\right)=\alpha ,{\mbox{ }}and{\mbox{ }}p\left(0\right)=}$$\beta$ 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 Al₂ O₃ . Buoyancy ratio, ${\textstyle \gamma \gg 1.0}$ is a free convection, ${\textstyle \gamma =1.0}$ is a mixed convection and ${\textstyle \gamma \ll 1.0}$ is a forced convection. In this work, ${\textstyle \gamma =2.0}$ unless otherwise specified.

It is observed from the Fig. 2 that the agreement with the theoretical solution of the concentration profile for different values of Schmidt number is significantly correlated with Fig. 8 of Singh and Kumar [44] .

Fig. 2. Comparison of concentration profiles for Sc with Fig. 8 of Singh and Kumar [44] .

The concentration of the water based SWCNTs, Cu and Al₂ O₃ increases with increases of the density of the nanoparticles (Fig. 3 ) and the rate of mass transfer of SWCNTs and Cu–water decreases (Table 2 ) with increase of density. It is interesting to note that the concentration of Al₂ O₃ –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 Fig. 4 and Table 3 ). The concentration of the nanofluids (water based SWCNTs, Cu and Al₂ O₃ ) decreases (Fig. 5  and Fig. 6 ) but the rate of mass transfer increases (Table 4  and 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 Al₂ O₃ –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 Al₂ O₃ –water decreases and the concentration of the Cu–water is uniform with the increase of the nanoparticle volume fraction (Fig. 7 ) whereas the diffusion boundary layer thickness of SWCNTs–water is stronger than the water based Cu and Al₂ O₃ with the increase of the nanoparticle volume fraction. The velocity/concentration of the nanofluids (water based SWCNTs, Cu and Al₂ O₃ ) 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 Al₂ O₃ –water is stronger than SWCNTs and Cu–water (Fig. 8 and Table 6 ) with the increase of the velocity slip parameter. It is revealed that the diffusion conductivity of the water based Al₂ O₃ plays a significant role on the enhancement of the mass transfer rate of nanofluids as compared with Cu and SWCNTs–water (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 Al₂ O₃ –water with the increase of nanoparticle volume fraction (Table 7 ). From the Fig. 9 , it is observed that the skin friction and Sherwood number of SWCNTs–water and Al₂ O₃ –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 Al₂ O₃ –water.

Fig. 3. Density of the nanofluids on concentration profiles with ${\textstyle \lambda =0.5,{\mbox{ }}\gamma =}$$0.1,{\mbox{ }}Sc=6.2,{\mbox{ }}{\gamma }_{2}=0.1,{\mbox{ }}{\gamma }_{1}=$$0.5$ .

Fig. 4. Volumetric expansion of the nanofluids on concentration profiles with ${\textstyle \lambda =0.5,{\mbox{ }}\gamma =}$$0.1,{\mbox{ }}Sc=6.2,{\mbox{ }}{\gamma }_{2}=0.1,{\mbox{ }}{\gamma }_{1}=$$0.5$ .

Fig. 5. Chemical reaction on concentration profiles with ${\textstyle \lambda =0.5,{\mbox{ }}\gamma =}$$0.1,{\mbox{ }}Sc=6.2,{\mbox{ }}{\gamma }_{2}=0.1$ .

Fig. 6. Buoyancy ratio on velocity and concentration profiles with ${\textstyle \lambda =0.5,{\mbox{ }}Sc=6.2,{\mbox{ }}{\gamma }_{2}=}$$0.1,{\mbox{ }}{\gamma }_{1}=0.5$ .

Fig. 7. Nanoparticle volume fraction on velocity and concentration profiles with ${\textstyle {\gamma }_{1}=0.5,{\mbox{ }}\lambda =}$$0.5,{\mbox{ }}Sc=6.2,{\mbox{ }}\gamma =0.1,{\mbox{ }}{\gamma }_{2}=$$0.1$ .

Fig. 8. Slip parameter on velocity and concentration profiles with ${\textstyle \lambda =0.5,{\mbox{ }}Sc=6.2,{\mbox{ }}\gamma =}$$0.1,{\mbox{ }}{\gamma }_{1}=0.5$ .

Fig. 9. Chemical reaction effects on skin friction and Sherwood number with ${\textstyle \lambda =0.5,{\mbox{ }}Sc=6.2,{\mbox{ }}\gamma =}$$0.1$ .

Table 2. Failed to parse (syntax error): {\textstyle f^{{´´}}\left(0\right)} and ${\textstyle -{\theta }^{'}\left(0\right)}$ for different values of ${\textstyle {\rho }_{s}}$ with ${\textstyle \lambda =0.5,{\mbox{ }}\gamma =}$$0.1,{\mbox{ }}Sc=6.2,{\mbox{ }}{\gamma }_{2}=0.1,{\mbox{ }}{\gamma }_{1}=$$0.5$ .

Nanofluid	${\textstyle {\rho }_{s}{\mbox{ }}\left(kg{\mbox{ }}m^{-3}\right)}$	Failed to parse (syntax error): {\textstyle f^{{´´}}\left(0\right)}	${\textstyle -{\chi }^{'}\left(0\right)}$
Cu–water	100	0.40556786	1.66762743
	5000	0.39729350	1.49649125
	9000	0.39205250	1.39171507
Al2 O3 –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

Table 3. Failed to parse (syntax error): {\textstyle f^{{´´}}\left(0\right)} and ${\textstyle -{\theta }^{'}\left(0\right)}$ for different values of ${\textstyle {\beta }_{s}}$ with ${\textstyle \lambda =0.5,{\mbox{ }}\gamma =}$$0.1,{\mbox{ }}Sc=6.2,{\mbox{ }}{\gamma }_{2}=0.1,{\mbox{ }}{\gamma }_{1}=$$0.5$ .

Nanofluid	${\textstyle {\beta }_{s}{10}^{-5}K^{-1}}$	Failed to parse (syntax error): {\textstyle f^{{´´}}\left(0\right)}	${\textstyle -{\chi }^{'}\left(0\right)}$
Cu–water	1.0	0.391988185	1.39326051
	5.0	0.392131817	1.39328133
	10	0.392311344	1.39330737
Al2 O3 –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

Table 4. Failed to parse (syntax error): {\textstyle f^{{´´}}\left(0\right)} and ${\textstyle -{\theta }^{'}\left(0\right)}$ for different values of ${\textstyle \gamma 1}$ with ${\textstyle \lambda =0.5,{\mbox{ }}\gamma =}$$0.1,{\mbox{ }}Sc=6.2,{\mbox{ }}{\gamma }_{2}=0.1$ .

Nanofluid	${\textstyle \gamma 1}$	Failed to parse (syntax error): {\textstyle f^{{´´}}\left(0\right)}	${\textstyle -{\chi }^{'}\left(0\right)}$
Cu–water	0.1	0.41054960	0.72599331
	3.0	0.37832265	2.30159323
	10.0	0.36615522	4.15126063
Al2 O3 –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

Table 5. Failed to parse (syntax error): {\textstyle f^{{´´}}\left(0\right)} and ${\textstyle -{\theta }^{'}\left(0\right)}$ for different values of ${\textstyle \gamma }$ with ${\textstyle \lambda =0.5,{\mbox{ }}Sc=6.2,{\mbox{ }}{\gamma }_{2}=}$$0.1,{\mbox{ }}{\gamma }_{1}=0.5$ .

Nanofluid	${\textstyle \gamma }$	Failed to parse (syntax error): {\textstyle f^{{´´}}\left(0\right)}	${\textstyle -{\chi }^{'}\left(0\right)}$
Cu–water	0.01	0.33580455	1.38400682
	10	3.34902280	1.71832502
	100	17.6203525	2.57510459
Al2 O3 –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

Table 6. Failed to parse (syntax error): {\textstyle f^{{´´}}\left(0\right)} and ${\textstyle -{\theta }^{'}\left(0\right)}$ for different values of ${\textstyle {\gamma }_{2}}$ with ${\textstyle \lambda =0.5,{\mbox{ }}Sc=6.2,{\mbox{ }}\gamma =}$$0.1,{\mbox{ }}{\gamma }_{1}=0.5$ .

Nanofluid	${\textstyle {\gamma }_{2}}$	Failed to parse (syntax error): {\textstyle f^{{´´}}\left(0\right)}	${\textstyle -{\chi }^{'}\left(0\right)}$
Cu–water	0.1	0.39212786	1.39328076
	1.0	0.31282789	1.52795081
	3.0	0.18904557	1.64059657
Al2 O3 –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

Table 7. Failed to parse (syntax error): {\textstyle f^{{´´}}\left(0\right)} and ${\textstyle -{\theta }^{'}\left(0\right)}$ for different values of ${\textstyle \zeta }$ with ${\textstyle {\gamma }_{1}=0.5,{\mbox{ }}\lambda =}$$0.5,{\mbox{ }}Sc=6.2,{\mbox{ }}\gamma =0.1,{\mbox{ }}{\gamma }_{2}=$$0.1$ .

Nanofluid	${\textstyle \zeta }$	Failed to parse (syntax error): {\textstyle f^{{´´}}\left(0\right)}	${\textstyle -{\chi }^{'}\left(0\right)}$
Cu–water	0.05	0.39212786	1.39328076
	0.1	0.38564620	1.35442510
	0.2	0.37843858	1.38488721
Al2 O3 –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 Al₂ O₃ –water with increase of chemical reaction. The diffusion boundary layer thickness of SWCNTs–water is stronger than Al₂ O₃ 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 Al₂ O₃ –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 Al₂ O₃ –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 FRGS1208/2013 .

Appendix

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