Thermophysical properties of different base fluid and CNTs.
The effects of different types of base fluids on carbon nanotube (CNT) nanofluids flow over a circular stretching sheet are numerically analyzed. The nonlinear variation of radial velocity in radial direction is assumed at surface of stretching sheet. The temperature dependent fluid viscosity is taken into consideration. Two different types of flows (assisting flow and opposing flow) are discussed under the buoyant force effects. Single walled CNT and multi walled CNT are considered as nanoparticles for better thermal conductivity of the nanofluids. A set of similarity transformations to convert the partial differential equations into ordinary differential equations is hired. The non-linear ODEs are numerically solved by employing fourth order Runge-Kutta method. Discussions of numerical simulations for flow characteristics have been made appropriately. A comparative study for various type of base fluids like kerosene, engine oil and ethylene glycol is also presented. From the predicted simulation, it is observed that the variation in Nusselt number is maximum for engine oil and minimum for kerosene oil however, the variation in skin friction coefficient is largest for kerosene oil and least for engine oil. Furthermore, numerical results are also validated with achieving a good correlation with existing results.
- CNT nanofluids
- nonlinear stretching sheet
- Runge-Kutta method
- buoyancy force
- heat transfer
- similarity transformation
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 of micro/nano-sized particles in fluids have also been tried since many decades. However, the word ‘nanofluids’ was primarily introduced by Choi . 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, 3, 4] have been reported.
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, 6, 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.  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.  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 . And this study noted that the maximum thermal conductivity enhances up to 19.75% for the nanofluid containing 0.45 vol.% MWCNT at 40°C. 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 . 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, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30]. The first boundary layer flow model for nanofluids flow past stretching sheet was presented by Khan and Pop  which was extended for a convective boundary condition , nonlinear stretching sheet , unsteady stretching surface , micropolar nanofluid flow , magneto-convective non-Newtonian nanofluid slip flow over permeable stretching sheet , Non-aligned MHD stagnation point flow of variable viscosity nanofluids , Stagnation electrical MHD mixed convection , exponential temperature-dependent viscosity and buoyancy effects , thermo-diffusion and thermal radiation effects on Williamson nanofluid flow , magnetic dipole and radiation effects on viscous ferrofluid flow , transient ferromagnetic liquid flow , magnetohydrodynamic Oldroyd-B nanofluid , spherical and non-spherical nanoparticles effects , three dimensional free convective magnetohydrodynamics . Moreover, some works [31, 32, 33, 34] extended Khan and Pop’s model for CNTs nanofluids where combined effects of slip and convective boundary conditions have been discussed , convective heat transfer in MHD slip flow has been studied , nonlinear stretching sheet with variable thickness has been considered  and variable thermal conductivity and thermal radiation effects have been discussed.
After reviewed many investigations [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 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.
2. Mathematical model
We consider the laminar incompressible flow of CNTs nanofluids over a circular sheet aligned with the
The flow regime is considered as in the half space
The following boundary conditions are to be employed:
Introducing the following similarity transformations:
Reynolds model of viscosity expression can be taken as:
where is the Prandtl number.
The skin friction coefficients (
And the local Nusselt number (
The dimensionless form of skin friction coefficients can be obtained as:
And the dimensionless form of skin friction coefficients can be obtained as:
3. Numerical scheme
Numerical solutions of ordinary differential Eqs. (8) and (9) subject to boundary conditions (10) are obtained using a shooting method. First we have converted the boundary value problem (BVP) into initial value problem (IVP) and assumed a suitable finite value for the far field boundary condition, i.e.
4. Numerical simulations and discussion
This section presents the numerical simulations for dimensionless velocity, temperature, skin friction, Nusselt numbers and streamlines under the influence of the flow parameters which are illustrated through the (Figures 2–11). Table 1 is also given for thermophysical properties of base fluids and CNTs (MWCNT and SWCNT).
|Physical properties||Base fluid||Nanoparticles|
|Water||Kerosene||Engine oil||Ethylene glycol||MWCNT||SWCNT|
Figures 2 and 3 illustrate the velocity profiles for assisting and opposing flows where multi walled CNT is considered as nanoparticle and Kerosene is taken as base fluid. The Grashof number is fixed at 0.5. From all the figures, it is depicted that
The effects of viscosity parameter (
The variations in skin-friction coefficient against nanoparticle volume fraction are computed in Figures 7 and 8 for assisting and opposing flows. It is found that the skin friction coefficient is least at
Figure 8(a and b) are illustrated for the effects of different types of base fluids (kerosene, ethylene glycol and engine oil) and also effects of different CNTs on skin friction coefficient at fixed values
It is observed that the magnitude of Nusselt number is minimum at
The stream lines are plotted through the Figure 11(a–c) to see the effects of power law index at fixed values
5. Concluding remarks
A numerical investigation for boundary layer flow of CNTs nanofluids with temperature dependent viscosity over a circular stretching sheet is presented. Effects of Three types of base fluids, power law index, viscosity parameter, nanoparticle volume fraction and Grashof number on flow characteristics, and also skin friction coefficient and Nusselt number are discussed appropriately with numerical computations. The concluding remarks of present discussions are précised as:
The nature of velocity and temperature profiles for assisting and opposing flows are similar for all values of pertinent parameters except Grashof number.
The velocity boundary layer thickness elaborates with increasing the magnitude of nanoparticle volume fraction, viscosity parameter and power law index for both type of flows.
The thermal boundary layer thickness expands with reducing the magnitude of viscosity parameter, power law index and nanoparticle volume fraction for both type of flows.
The magnitude of skin friction coefficient enhances with increasing the nanoparticle volume fraction and power law index and it also enhances with decreasing the viscosity parameter for both type of flows.
The sequence for skin friction coefficient of different nanoparticles is observed as: (SWCNT) > (MWCNT).
The sequence for skin friction coefficient of different base fluids is also noted as: (kerosene) > (ethylene glycol)> (engine oil).
The Nusselt number diminishes with increasing the viscosity parameter and it also diminishes with decreasing the power law index and nanoparticle volume fraction for both type of flows.
The sequence for Nusselt number of kerosene, ethylene glycol and engine oil is noted as: (kerosene) < (ethylene glycol) < (engine oil).