Abstract
In this article, a comprehensive analysis is performed for fly ash and paramagnetic ( T a ) nanoparticles with a hybrid base and micro polar fluid with momentum slip conditions over a stretching sheet. With the aid of the similarity transformation, the PDEs obtained are transmuted in the nonlinear ODE system, after which the numerical simulation is performed with the finite element method. The effect of physical parameters on velocity, micro rotation, and temperature are analyzed graphically. Results reveal that the velocity boundary layer thickness for fly ash nanoparticles is higher than paramagnetic and thermal conductivity of paramagnetic is higher than fly ash nanoparticles. The paramagnetic nanoparticles are, therefore, the best coolant. The comparison of the finite element method with previous researches is tabled and found in a good agreement.
Keywords: Magnetic dipole; Micro polar nanofluid flow; Slip condition; Finite element method 2010 MSC: 00-01; 99-00
Introduction
Ferrofluids (a portmanteau of liquid and ferromagnetic particles) are liquids in the presence of an external magnetic field magnetized. These types of fluids are liquids colloidal made by nanoscale ferromagnetic or ferromagnetic particles suspended in a fluid carrier (usually a water or organic solvent). The particle suspension is caused by Brownian movement and the particles do not settle under normal circumstances. In addition, each ferromagnetic particle is covered with a surfactant to prevent agglutination. Magnetic attraction of ferromagnetic particles at the nanoscale is low when the surfactant has sufficient strength to inhibit agglomeration or magnetic aggregation. There are various applications of ferromagnetic liquids. They are often used in many industrial applications, food preservation, aerodynamics, nuclear power plants, crystal processing, laser, avionics, robotics, coolant, semiconductor processing, fiber optics, speakers, cooling, filtration, plastic drawing, aerodynamics and computer peripherals, etc. Countless applications from many scientists and researchers have accelerated the study of ferrous fluid. Neuringer [1] investigated the effects of gradients of magnetic and thermal fields on saturated viscous ferrofluid flow. Crane [2] initiated the Newtonian fluid flow with a linear stretching sheet, which was then reduced to non-Newtonian fluids. Anderson and Valnes [3] investigated the special effects of the magnetic dipole on the ferrous liquid. Titus and Abraham [4] studied ferrofluid and heat transfer beyond an expandable sheet. Some research on ferrofluid is given in the references [5-11].
The rapid growth of engineering and technology, especially in the field of electronics, energy production and energy efficiency brings the need to develop different cooling techniques. Nano fluids, in this case as a new type of heat transfer fluid are recently a popular research area for researchers and the aims to overcome the disadvantages of poor thermal conductivity conventional heat transfer fluids by improving their thermal resistance conductivity using new methods [12]. Nanofluids are essentially fluids that contain particles with a high thermal conductivity, suspended in a basic fluid [13]. Mixed convection on MHD flow of casson nanofluid over a non-linearly permeable stretching sheet has been investigated and analyzed numerically by Ghadikolaei et al. [14]. Nor Azizah Yacob et al. [15] studied the characteristics on a forced convection boundary layer flow towards a horizontal permeable stretching sheet in four types of nanofluids which are ZnO-water, ZnO-Kerosene, MgO-water and MgO-Kerosene. Hussanan et al. [16] focused on the energy transfer with the effects of carbon nanotubes (CNTs) of magneto hydrodynamic (MHD) nanofluids flow past a stretching sheet under thermal radiation and Newtonian heating. Single and multi-wall CNTs are considered in water as convectional based fluid. Numerical simulation for mixed convective threedimensional slip flow of water-based nanofluids with temperature jump boundary condition is presented by Mahanthesh et al. [17]. Nanoparticle shapes on radiative MHD water based aluminum alloy nanoparticles, AA7072 and AA7075 flow up a non-linear wall with distinct flow condition is analyzed by Kandasamy et al. [18]. The unsteady flow and heat transfer past a stretching/shrinking sheet in a hybrid nanofluid is studied by Waini et al. [19].
Microstructured fluids are called micro polar fluids. Micro flow fluids belong to a class of non-asymmetrical fluids stress tensor, called polar fluids. Micropolar fluid mechanics, de- rived from the theory developed by Eringen [20], is an interesting research area because of the wide range of applications in the industry. For example, polymeric liquids, actual liquids with suspensions, liquid crystals, animal blood, and exotic lubricants are modeled by micro polar fluids. In general, these fluids indicate fluids that cover unpredictably oriented fluids suspended in a viscous medium. In the book by Lukaszewicz [21], micro polar fluids are briefly studied. Many studies consider micro polar fluids, but those dealing with nanoparticles are limited. The suspension of nanoparticles will result in more complex nanofluids.
The thermo physical properties of the fly ash nanofluid are the main factors of influence of heat, friction and automotive cooling system energy performance. Thermo physical properties are thermal conductivity, density, dynamic viscosity and specific heat of base fluid Sozen et al. [22] reported that fly ash exhausted from thermal power plants has a mixture of metal oxides like Al2O3, Fe2O3, TiO2, SiO2, CaO and MgO in different proportions, which could be used for preparing nanofluids suitable for heat exchanger applications. They have proven that fly ash nanoparticles with water have improved thermo physical and friction properties with good improved thermal performance in a concentric parallel flow heat tube exchanger
Present work concentrates on the analysis of fly ash and paramagnetic ( T a ) nanoparticles with hybrid base fluid (60% water, and 40% ethylene glycol) due to applied magnetic dipole on a boundary layer slip flow over a stretching sheet. The effect of ferrohydrodynamic interaction parameter, thermal radiation, viscous dissipation, micro polar parameter, and Prandtl number were studied. Computations of nonlinear system are presented after non dimensionalization using the finite element method. The comparison of finite element method with previous researches is tabled.
Mathematical Formulation
Considering a steady two-dimensional electrically nonconducting and the incompressible flow of magnetic micro polar nanofluid consists of paramagnetic ( T a ) and fly ash nanoparticles suspended with a hybrid base fluid (60% water, and 40% ethylene glycol) over a stretching sheet. Partial velocity slip is considered at the surface and the suction/injection effect is also taken into account. The sheet is stretched along x axis with linear velocity uw = cx . Where c > 0 is for stretching and c < 0means shrinking of the sheet. The magnetic dipole of sufficient strength to fill the ferrofluid is located at a distance a below than the x-axis and it centers on the y-axis whose magnetic field in the positive x-direction. It is assumed that the Curie temperature c T is higher than the ambient temperature w T , while the free flow temperature is far from the surface of the stretching/shrinking sheet c T T ∞ = and therefore cannot magnetize until it begins to cool down after it has penetrated into the region of thermal boundary layer. We thus have [23, 24].
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