1. Introduction

The first key point of present investigation is carbon nanotubes (CNTs) nanofluids that means the suspension of CNTs in conventional fluids such as air, helium, water, minerals oils, Freon, and ethylene glycol. The applications of nanofluids spread over a wide range of disciplines, including heat transfer, material science, physics, and chemical engineering. Many techniques have been used to enhance the thermal conductivity of the base fluids in which the suspension

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

of micro/nano-sized particles in fluids have also been tried since many decades. However, the word 'nanofluids' was primarily introduced by Choi [1]. He has studied the thermal conductive effects on convectional fluids subject to suspension of metallic nanoparticles. He has observed that with suspension of nanoparticles, the thermal conductivity of convectional fluids enhances. In these directions, numerous investigations have therefore been carried out in the past few decades, seeking to wide applications and developments, some interesting reviews [2–4] have been reported.

spherical and non-spherical nanoparticles effects [29], three dimensional free convective magnetohydrodynamics [30]. Moreover, some works [31–34] extended Khan and Pop's model for CNTs nanofluids where combined effects of slip and convective boundary conditions have been discussed [31], convective heat transfer in MHD slip flow has been studied [32], nonlinear stretching sheet with variable thickness has been considered [33] and variable

Numerical Simulation of Nanoparticles with Variable Viscosity over a Stretching Sheet

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After reviewed many investigations [16–34] on boundary layer flow of nanofluids past through linear/nonlinear stretching sheet, none of the them has been reported for boundary layer flow of CNTs nanofluids over a circular stretching sheet. Considering this gap of research, we formulate a model and also numerically simulate it to study the flow characteristic of CNTs nanofluids flow over a circular stretching sheet. Kerosene, Engine oil and Ethylene glycol are considered for base fluids. And MCNT and SWCNT are also taken as nanoparticles. The negative and positive buoyant force effects as opposing and assisting flows are also examined. The effects of fluid and flow parameters on velocity profile, temperature profile, skin-friction coefficient, Nusselt number

We consider the laminar incompressible flow of CNTs nanofluids over a circular sheet aligned with the rθ-plane in the cylindrical coordinate system (r, θ, z). The schematic representation

thermal conductivity and thermal radiation effects have been discussed.

and stream lines are computed numerically.

and coordinate system are depicted in Figure 1.

Figure 1. Schematic representation of CNTs nanofluids flow over circular stretching sheet.

2. Mathematical model

To study the heat transfer rate of water based nanofluids, the various types of nanoparticles like titanium dioxide, alumina, silica diamond, zinc-oxide and copper [5–7] have been considered. It has been depicted that the thermal conductivity enhances with suspension of the nanoparticles. Moreover, CNTs have also received great interest due to significant enhancement of thermal conductivity, unique structure and physical (mechanical and electrical) properties [8, 9]. Ding et al. [8] have prepared a nanofluids with suspension of CNTs in distilled water and measured the thermal conductivity and viscosity of CNTs nanofluids. They have reported the enhancement of thermal conductivity depends on CNTs concentration, and pH level. They also concluded that nanofluids with 0.5 wt.% CNTs, the maximum enhancement is over 350% at Re = 800. Ko et al. [9] experimentally reported the flow characteristics of CNTs nanofluids in a tube and summarized that the friction factor for CNT nanofluid is low in compare to water. Another experimental study for thermophysical properties of CNTs nanofluids with base fluid as mixture of water and ethylene glycol has been presented by Kumaresan and Velraj [10]. And this study noted that the maximum thermal conductivity enhances up to 19.75% for the nanofluid containing 0.45 vol.% MWCNT at 40C. In addition, few more experimental investigations [11, 12] with applications in a tubular heat exchanger of various lengths for energy efficient cooling/heating system and turbulent flow heat exchanger are performed. With CNTs nanofluids, some more numerical simulations [13, 14] for flow characteristics are presented in literature and discussed the physiological flows application.

The second key point is boundary layer flow over a circular stretching sheet. The boundary layer flow past a stretching sheet was initially investigated by Crane [15]. He has discussed its applications such as annealing and tinning of copper wires, polymer extrusion in melt spinning process, manufacturing of metallic sheets, paper production, and glass, fiber and plastic production etc. The heat transfer rate play important role in all manufacturing, productions and fabrication processes. This model has been explored by many investigators for various physical aspects. However, some interesting extensions of Crane's model for boundary layer flow of nanofluids have recently investigated, see examples [16–30]. The first boundary layer flow model for nanofluids flow past stretching sheet was presented by Khan and Pop [16] which was extended for a convective boundary condition [17], nonlinear stretching sheet [18], unsteady stretching surface [19], micropolar nanofluid flow [20], magneto-convective non-Newtonian nanofluid slip flow over permeable stretching sheet [21], Non-aligned MHD stagnation point flow of variable viscosity nanofluids [22], Stagnation electrical MHD mixed convection [23], exponential temperature-dependent viscosity and buoyancy effects [24], thermo-diffusion and thermal radiation effects on Williamson nanofluid flow [25], magnetic dipole and radiation effects on viscous ferrofluid flow [26], transient ferromagnetic liquid flow [27], magnetohydrodynamic Oldroyd-B nanofluid [28],

spherical and non-spherical nanoparticles effects [29], three dimensional free convective magnetohydrodynamics [30]. Moreover, some works [31–34] extended Khan and Pop's model for CNTs nanofluids where combined effects of slip and convective boundary conditions have been discussed [31], convective heat transfer in MHD slip flow has been studied [32], nonlinear stretching sheet with variable thickness has been considered [33] and variable thermal conductivity and thermal radiation effects have been discussed.

After reviewed many investigations [16–34] on boundary layer flow of nanofluids past through linear/nonlinear stretching sheet, none of the them has been reported for boundary layer flow of CNTs nanofluids over a circular stretching sheet. Considering this gap of research, we formulate a model and also numerically simulate it to study the flow characteristic of CNTs nanofluids flow over a circular stretching sheet. Kerosene, Engine oil and Ethylene glycol are considered for base fluids. And MCNT and SWCNT are also taken as nanoparticles. The negative and positive buoyant force effects as opposing and assisting flows are also examined. The effects of fluid and flow parameters on velocity profile, temperature profile, skin-friction coefficient, Nusselt number and stream lines are computed numerically.
