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
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle C} - Concentration of the fluid, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle K}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle C_w}
- Concentration of the wall, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle K}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle C_{\infty }}
- Concentration of the fluid far away from the wall, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle K}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle D_f}
- Specific diffusivity of the base fluid, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle m^2\mbox{ }s^{-1}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle D_{nf}}
- Specific diffusivity of the nanofluid, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle m^2\mbox{ }s^{-1}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle g}
- Acceleration due to gravity, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle ms^{-2}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Gc}
- Grashof number, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \frac{g{\beta }_c\mbox{ }\left(C_w-C_{\infty }\right)x^3}{{\nu }_f^2}}
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \frac{ms^{-2}K^{-1}Km^3}{{\left(m^2s^{-1}\right)}^2}}
(−)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle k_1} - First order rate of chemical reaction, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle s^{-1}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Re}
- Reynolds number, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \frac{u_{\infty }x}{{\nu }_f}}
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \frac{m\mbox{ }s^{-1}m}{m^2s^{-1}}}
(−)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Sc} - Schmidt number, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \frac{{\nu }_f}{D_f}}
,Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \frac{m^2\mbox{ }s^{-1}}{m^2\mbox{ }s^{-1}}}
(−)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x,\mbox{ }y} - Streamwise coordinate and cross-stream coordinate, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle m}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u,\mbox{ }v}
- Velocity components in x and y directions, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle m\mbox{ }s^{-1}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u_{\infty }}
- Flow velocity of the fluid away from the plate, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle m\mbox{ }s^{-1}}
Greek symbols
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\beta }_c} - Volumetric expansion coefficients of the base fluid, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle K^{-1}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\rho }_f}
- Density of the base fluid, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle kg\mbox{ }m^{-3}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\rho }_s}
- Density of the nanoparticle, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle kg\mbox{ }m^{-3}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\rho }_{nf}}
- Effective density of the nanofluid, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle kg\mbox{ }m^{-3}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\left({\beta }_c\right)}_{nf}}
- Volumetric coefficient of thermal expansion of nanofluid, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle K^{-1}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\mu }_f}
- Dynamic viscosity of the base fluid, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle kg\mbox{ }m^{-1}\mbox{ }s^{-1}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\mu }_{nf}}
- Effective dynamic viscosity of the nanofluid, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle kg\mbox{ }m^{-1}\mbox{ }s^{-1}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \gamma }
- Buoyancy ratio, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \frac{Gc}{R_e^2}}
(−)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \gamma 1} - Chemical reaction parameter, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \frac{k_1x}{u_{\infty }}\left(\frac{s^{-1}m}{m\mbox{ }s^{-1}}\right)}
(−)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\gamma }_2} - Velocity slip parameter, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \frac{v_0u_{\infty }}{{\nu }_f}}
,Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \frac{m\mbox{ }(m\mbox{ }s^{-1})}{m^2s^{-1}}}
(−)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\nu }_{nf}} - Dynamic viscosity of the nanofluid, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle m^2\mbox{ }s^{-1}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \Omega }
- Resistance, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle kg\mbox{ }m^2s^{-3}A^{-2}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \zeta }
- Nanoparticle volume fraction, (−)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \psi } - Dimensionless stream function, (−)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \eta } - Similarity variable, (−)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f} - Dimensionless stream function, (−)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \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
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( 1) |
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( 2) |
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( 3) |
with the boundary conditions
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( 4) |
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![Physical model of the flow and coordinate system. (a) Singh and Kumar [44]. (b) ...](/wd/index.php/File:Draft_Garc%C3%ADa_697624778-1-s2.0-S2215098615301129-jestch193-fig-0001.jpg)
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Fig. 1. Physical model of the flow and coordinate system. (a) Singh and Kumar [44] . (b) Present result. |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \rho \left(kg/m^3\right)} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle c_p\mbox{ }\left(J/kgK\right)} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle k\left(W/mK\right)} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\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 |
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle v_{sf}=v_0\sqrt{\frac{u_{\infty }x}{{\nu }_{nf}}}}
is the velocity slipping factor with an initial value Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle v_0}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \lambda }
is the power index. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\rho }_{nf}}
is the effective density of the nanofluid, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\mu }_{nf}}
is the effective dynamic viscosity of the nanofluid, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle D_{nf}}
is the mass diffusivity of the nanofluid and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\beta }_{nf}}
is the volumetric expansion coefficient, which are defined by Magyari [46] and Mamut [47] as
|
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( 5) |
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle k_f}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle k_s}
are the thermal conductivity of the base fluid and nanoparticle, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \zeta }
is the nanoparticle volume fraction, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\mu }_f}
is the dynamic viscosity of the base fluid, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\beta }_f}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\beta }_s}
are the volumetric expansion coefficients of the base fluid and nanoparticle, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\rho }_f}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\rho }_s}
are the density of the base fluid and nanoparticle, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle k_{nf}}
is the effective thermal conductivity of the nanofluid. The similarity transformation and stream function are defined as
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( 6) |
The system of Eqs. (2) , (3) and (4) become
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( 7) |
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( 8) |
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with the boundary conditions
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( 9) |
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\gamma }_2=\frac{v_0u_{\infty }}{{\nu }_f}}
is the velocity slip parameter with initial value Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle v_0}
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \gamma =\frac{Gc}{{Re}^2}}
is the Buoyancy ratio, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Sc=\frac{{\nu }_f}{D_f}}
is the Schmidt number, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Gc=\frac{g{\beta }_f\mbox{ }\left(C_w-C_{\infty }\right)x^3}{{\nu }_f^2}}
is the Grashof number, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Re=\frac{u_{\infty }x}{{\nu }_f}}
is the Reynolds number and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\gamma }_1=\frac{k_1x}{u_{\infty }}}
is the Chemical reaction parameter.
Physical quantities are Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle C_f=\frac{{\tau }_w}{{\rho }_fU^2}}
(Skin friction coefficient) and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Sh_x=\frac{{\varphi }_mx}{D_f\mbox{ }\left(C_w-C_{\infty }\right)}}
(Local Sherwood number). Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\tau }_w}
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle q_w}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\varphi }_m}
are defined as
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle C_f\mbox{ }{\left({Re}_x\right)}^{\frac{1}{2}}=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{f^{{´´}}\left(0\right)}{{\left(1-\zeta \right)}^{2.5}},\mbox{ }\frac{Sh_x}{{Re}_x^{\frac{1}{4}}}= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): -\frac{{\chi }^{{'}}\left(0\right)}{\left(1-\zeta \right)}
; Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\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:
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( 10) |
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( 11) |
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( 12) |
The boundary conditions are
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( 13) |
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \alpha }
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \alpha }
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \beta }
are determined upon solving the boundary conditions Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle v\left(0\right)=\alpha ,\mbox{ }and\mbox{ }p\left(0\right)=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \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, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \gamma \gg 1.0}
is a free convection, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \gamma =1.0}
is a mixed convection and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \gamma \ll 1.0}
is a forced convection. In this work, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \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] .
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![Comparison of concentration profiles for Sc with Fig. 8 of Singh and Kumar [44].](/wd/index.php/File:Draft_Garc%C3%ADa_697624778-1-s2.0-S2215098615301129-jestch193-fig-0002.jpg)
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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.
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Fig. 3. Density of the nanofluids on concentration profiles with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \lambda =0.5,\mbox{ }\gamma =} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 0.1,\mbox{ }Sc=6.2,\mbox{ }{\gamma }_2=0.1,\mbox{ }{\gamma }_1= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 0.5 . |
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Fig. 4. Volumetric expansion of the nanofluids on concentration profiles with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \lambda =0.5,\mbox{ }\gamma =} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 0.1,\mbox{ }Sc=6.2,\mbox{ }{\gamma }_2=0.1,\mbox{ }{\gamma }_1= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 0.5 . |
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Fig. 5. Chemical reaction on concentration profiles with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \lambda =0.5,\mbox{ }\gamma =} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 0.1,\mbox{ }Sc=6.2,\mbox{ }{\gamma }_2=0.1 . |
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Fig. 6. Buoyancy ratio on velocity and concentration profiles with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \lambda =0.5,\mbox{ }Sc=6.2,\mbox{ }{\gamma }_2=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 0.1,\mbox{ }{\gamma }_1=0.5 . |
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Fig. 7. Nanoparticle volume fraction on velocity and concentration profiles with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\gamma }_1=0.5,\mbox{ }\lambda =} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 0.5,\mbox{ }Sc=6.2,\mbox{ }\gamma =0.1,\mbox{ }{\gamma }_2= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 0.1 . |
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Fig. 8. Slip parameter on velocity and concentration profiles with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \lambda =0.5,\mbox{ }Sc=6.2,\mbox{ }\gamma =} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 0.1,\mbox{ }{\gamma }_1=0.5 . |
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Fig. 9. Chemical reaction effects on skin friction and Sherwood number with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \lambda =0.5,\mbox{ }Sc=6.2,\mbox{ }\gamma =} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 0.1 . |
| Nanofluid | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\rho }_s\mbox{ }\left(kg\mbox{ }m^{-3}\right)} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f^{{´´}}\left(0\right)} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\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 |
| Nanofluid | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\beta }_s{10}^{-5}K^{-1}} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f^{{´´}}\left(0\right)} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\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 |
| Nanofluid | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \gamma 1} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f^{{´´}}\left(0\right)} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\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 |
| Nanofluid | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \gamma } | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f^{{´´}}\left(0\right)} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\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 |
| Nanofluid | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\gamma }_2} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f^{{´´}}\left(0\right)} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\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 |
| Nanofluid | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \zeta } | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f^{{´´}}\left(0\right)} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\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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