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This article examines a mathematical model to analyze the rotating flow of three-dimensional water based nanofluid over a convectively heated exponentially stretching sheet in the presence of transverse magnetic field with additional effects of thermal radiation, Joule heating and viscous dissipation. Silver (Ag), copper (Cu), copper oxide (CuO), aluminum oxide (Al2O3) and titanium dioxide (TiO2) have been taken under consideration as the nanoparticles and water (H2O) as the base fluid. Using suitable similarity transformations, the governing partial differential equations (PDEs) of the modeled problem are transformed to the ordinary differential equations (ODEs). These ODEs are then solved numerically by applying the shooting method. For the particular situation, the results are compared with the available literature. The effects of different nanoparticles on the temperature distribution are also discussed graphically and numerically. It is witnessed that the skin friction coefficient is maximum for silver based nanofluid. Also, the velocity profile is found to diminish for the increasing values of the magnetic parameter.
In the present fast growing and developing computer age, the transportation, communication, heavy mechanical industries, electronics industries and house hold appliances, all are running by some mechanical and electronic devices. Almost in all such devices, according to the requirements of devices, a system of cooling or heating is built-in, by which a fluid flows through or around the device to prevent these devices from overheating or cooling down from certain temperature threshold. To meet the human requirements and demand of the market, it is essential that these devices work round the clock. To keep the devices at a constant temperature, the heat dissipated must be equal to the heat generated. The conventional fluids with low thermal conductivity do not meet the temperature requirements of many mechanical and electronics devices, which results in poor performance of these devices and reduces their efficiency and working age. Therefore, it is imperative to improve the thermal conductivity of the conventional fluids.
The conventional fluids used for the transfer of heat energy were first time replaced by the nanofluids by Choi[1] followed by many researchers. A nanofluid is a mixture of nanoparticles in a conventional heat transfer fluid. The nanoparticles (1–100 nm) in size are usually metals, metallic oxides, nanofibers, etc. Choi[1] experimentally found that the nanoparticles when added to the base fluids, considerably improve the thermal conductivity of the base fluid. Magyari and Keller[2] focused on the heat and mass transfer analysis in the boundary layer flow due to an exponentially continuous stretching sheet. Eastman et al.[3] observed that the thermal conductivity of pure ethylene glycol is much increased when copper nanoparticles are added to it. Li et al.[4] investigated the MHD nanofluid flow in a thin film through unsteady stretching sheet with additional effects of thermal radiation, heat generation, Brownian motion, and thermophoresis. They used the MATLAB built-in bvp4c solver to solve their ODEs. It is found form their investigation that temperature and nanoparticle concentration have opposite behavior for the thermophoresis parameter. Nadeem et al.[5] analyzed the influence of nanoparticles on the two-dimensional flow of Maxwell nanofluid over a stretching sheet for the heat and mass transfer effects. By applying the boundary-layer approximation, they also incorporated the effects of MHD and elasticity parameter. Sheikholeslami et al.[6] presented an analysis focusing on the unsteady squeezing flow of electrically conducting nanofluid using the homotopy perturbation method. Two phase simulation model for the nanofluid is considered along with the magnetohydrodynamics effects. They concluded that the Nusselt number is a decreasing function of the squeezing parameter. Another useful contribution of Sheikholeslami and Ganji,[7] is a review work addressing both the single and the double phase models for the nanofluids. They describe briefly the various attempts of different scientists on heat transfer of convective nanofluids. It is further analyzed that while increasing the Reynolds number and Rayleigh number, the rate of heat transfer is increased. Sheikholeslami et al.[8] discussed the thermal radiation on MHD free convection of Al2O3-water nanofluid. Chopkar et al.[9] studied the effect of the size of the nanoparticles on the thermal conductivity of nanofluid and found that the thermal conductivity decreases by increasing the size of the particles. In last half decade, many articles related to the nanofluid dynamics are published in literature.[10–24]
A reasonable number of applications emphasizing the role of steady and unsteady rotating flows may be found in chemical and geophysical fluid mechanics. These all are of applied nature like in the thermal power generating systems, food processing, the skins of high speed air crafts and in rotor stator systems. The pioneering work highlighting the rotating flow was done by Wang.[25] Takhar et al.[26] discussed the effects of magnetohydrodynamic in a rotating flow. They concluded that the skin friction along the x-axis increases for the higher values of the magnetic parameter and has a reverse relation for the y-axis skin friction coefficient. Zaimi et al.[27] applied the numerical technique to examine the rotating flow of viscoelastic fluid. Turkyilmazoglu[28] applied the spectral numerical integration method for the problem related to the shrinking rotating disk with the effect of magnetohydrodynamic. Some recent attempts emphasizing the rotating flow can be found in Refs. [29–31].
During the study of nanofluid, the thermal radiative properties of Newtonian and non-Newtonian fluids for the heat transfer phenomenon have got much attention. Because of the insertion of the nanoparticles in the base fluid, the thermal properties are enhanced which resultantly rises the temperature of the nanofluids and for the higher temperature differences, the effects of the thermal radiation cannot be neglected. The operating systems performing the energy conversion at high temperature show a comparable effect of the thermal radiation. In other engineering and chemical processes such as solar water technology, fossil fuel combustion, astrophysical flows, hypersonic flights, gas turbines, space vehicles, nuclear reactors etc., the effects of thermal radiations are quite phenomenal. Many researchers have considered the influence of thermal radiation on the boundary layer flow of Newtonian and non-Newtonian fluids. Mushtaq et al.[32] considered the nonlinear thermal radiation in the two-dimensional stagnation point flow with additional effects of Joule heating and viscous dissipation over a convectively heated surface. They deduced that both the temperature and its gradient are increasing functions of thermal radiation parameter. Pourmehran et al.[33] numerically investigated the MHD boundary layer flow of nanofluid through convectively heated vertical stretching sheet. During the study, the influence of thermal radiation and buoyancy effects got special attention. Pourmehran et al. considered three different types of base fluid i.e., pure water, ethylene glycol 30% and ethylene glycol 50% while the four types of nanoparticles i.e., copper, silver, alumina, and titanium oxide.
Motivated by the above mentioned literature, the primary objective of the present study is to examine the effects of thermal radiation and viscous dissipation on heat transfer flow over a bi-directional convectively heated exponentially stretching sheet in the presence of transverse magnetic field and volumetric rate of heat generation. Five different nanoparticles (silver, copper, copper oxide, titanium oxide, alumina) are assumed to be suspended in the pure water. A detailed comparative study of these nanofluids for the flow and heat transfer is presented and discussed graphically and numerically. Boundary layer approximations are used to govern the partial differential equations, which are then transformed to the ordinary differential equations with the help of transformations. The modeled problem is solved numerically by the shooting method using Runga-Kutta integration scheme of order 4. Effects of emerging parameters on velocity and temperature profiles are discussed in detail. Nusselt number and skin friction coefficient are also calculated. In the limiting case, the results are verified by reproducing the results of previously published article[34–35]
A laminar, incompressible and steady water-based electrically conducted nanofluid flow over an exponentially bidirectional stretching sheet is considered. Sheet temperature Tf is controlled via convection by considering hot fluid below it. The temperature faraway from the surface where its difference is negligible is known as the ambient temperature and is denoted by T∞. The fluid is assumed to rotate with angular velocity Ω = Ω0 ex/L along z-axis having the coriolis effect. A transverse variable magnetic field B(x) = B0 ex/2L is applied along z-axis with the assumption of small Reynolds number, ignoring the induced magnetic field. The fluid has internal volumetric rate of heat generation Q0. In the x direction, the velocity of the sheet is taken as Uw = U ex/L as shown in Fig.
Further the effects of the thermal radiation, Joule heating and viscous dissipation are considered in the formulation of energy equation. For the nanofluid model, the Tiwari and Das model.[36] has been utilized. Applying the boundary layer by incorporating the Boussinesq approximations, the conservation equations of mass, momentum and energy in the mathematical form can be expressed as
An efficient numerical technique, namely the shooting method has been employed to solve the transformed ordinary differential equations along with the boundary conditions for different values of the emerging parameters. While applying the shooting method,[42] first the higher order boundary value problem is converted to a system of first order initial value problem (IVP). During the conversion, f is denoted by y1, g by y4 and θ by y6. The missing initial conditions are supposed to be ς1, ς2 and ς3. The converted first order IVP takes the following form
In this section, we discuss the influence of different parameters such as nanoparticles volume fraction ϕ, rotational parameter λ, magnetic parameter M, thermal radiation parameter R, Eckert number Ec, heat generation/absorption parameter Qh on the velocity, temperature, skin-friction and Nusselt number, both graphically and numerically in the tabular form.
In Table
Table
The effect of thermal radiation parameter R, Eckert number Ec, heat generation/absorption parameter Qh and Biot number on Nusselt number is shown in Table
To visualize the effect of different physical parameters on the velocity f′(η) and the temperature profile θ(η), Figs.
To observe the effect of the variation in the Biot number Bi, Eckert number Ec, and heat generation parameter Qh on the temperature distribution Figs.
This article encompasses the three-dimensional MHD rotating flow of electrically conducting nanofluid over an exponentially stretching sheet. The effect of heat generation, viscous dissipation and thermal radiation for five different nanoparticles is analyzed graphically and numerically. The main findings of the investigation are as follows. Al2O3-H2O nanofluid has more capacity to transfer heat as compared to the other discussed nanofluids when the thermal radiation is enhanced. The skin friction coefficient is maximum for Ag-H2O nanofluid. An increase in the Eckert number Ec and the heat generation parameter Qh reduces the Nusselt number. This reduction in the heat transfer rate is much lower for Ag-H2O nanofluid. The velocity profile diminishes for increasing values of the magnetic parameter M. Ag-H2O and Cu-H2O nanofluids have greater values of the Nusselt number as compared to Al2O3-H2O and TiO2-H2O
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