Three dimensional flow of couple stress fluid over nonlinear stretching surface

Abstract

The boundary layer flow over a continuous stretching surface has various engineering and industrial applications. Such flow commonly involves in the paper production, wire drawing, hot rolling, glass fiber production, extrusion of plastic sheets, cooling ofmetallic plate in a cooling bath and many others. Many researchers have discussed different problems through the linear stretching of the surface but there are various situations in the industrial and technological processes where the stretching of the surface is not necessary linear. Particularly the flow induced by a nonlinear stretching surface has played important role in the polymer extrusion process. With this viewpoint Vajravelu [1] provided a study to examine the flow and heat transfer characteristics of viscous fluid induced by a nonlinearly stretching surface. Cortell [2] performed a numerical study to investigate the flow of viscous fluid over a nonlinearly stretching surface. He studied the two cases of heat transfer namely the constant surface temperature and the prescribed surface temperature. Cortell [3] also explored the flow of viscous fluid over a nonlinearly stretching surface in the presence of viscous dissipation and radiation effects. Hayat et al. [4] addressed the magnetohydrodynamic (MHD) flow over a nonlinearly stretching surface by using the modified Adomian decomposition and Padé approximation techniques. Flow and heat transfer properties of nanofluid over a nonlinearly stretching surface is reported by Rana and Bhargava [5]. Mukhopadhyay [6] discussed the boundary layer flow over a permeable nonlinearly stretching surface subject to partial slip condition. Mabood et al. [7] studied the MHD flow of water-based nanofluid over a nonlinear stretching surface in the presenceof viscous dissipation. Recently Mustafa et al. [8] investigated the flow of nanofluid over a nonlinearly stretching surface subject to the convective surface boundary condition. Insertion of ultrafine nanoparticles (<100 nm) in the base liquid is termed as nanofluid. The nanoparticles utilized in nanofluids are basically made of metals (Cu, Al, Ag), oxides (Al₂O₃), carbides (SiC), nitrides (AlN, SiN) or nonmetals (graphite, carbon nanotubes) and the base liquids like water, oil or ethylene glycol. Addition of nanoparticles in the base liquids greatly enhances the thermal properties of the base liquids. Due to such interesting properties, nanofluids are useful in various industrial and technological processes such as the cooling of electronic

devices, transformer cooling, vehicle cooling, heat exchanger, nuclear reactor, biomedicine and many others. Especially the magneto nanofluids are useful in MHD power generators, removal of blockage in the arteries, hyperthermia, cancer tumor treatment, wound treatment, magnetic resonance imaging etc. The term nanofluid was first introduced by Choi and Eastman [9] and they illustrated that the thermal properties of base liquids are enhanced when we add up the nanoparticles into it. Boungiorno [10] constructed a mathematical model to explore the thermal properties of base fluids. Here the effects of Brownian motion and thermophoresisare utilized to enhance the thermal properties of base liquids. Khan and Pop [11] employed the Boungiorno model [10] to analyze the boundary layer flow of nanofluid over a stretching surface. Afterwards various attempts have been made in this direction. Few of these can be quoted through the investigations [12-22] and several refs. therein. Most of the studies in the literature explain viscous materials by the classical Navier-Stokes equations. There are several complex rheological materials such as paints, shampoos, slurries, toothpastes, polymer solutions, ketchup, paper pulp, blood, greases, drilling muds, lubricating oils and many others that cannot be characterized through the classical Navier-Stokes expressions. Such materials are known as the non-Newtonian fluids. However there is no single relation that can predict the properties of all non-Newtonian fluids. Hence various models of non-Newtonian fluids are developed in the literature. The couple stress fluid model [23-28] is one of such materials. This model has important features due to the presence of couple stresses, body couples and non-symmetric stress tensor. Some interesting examples of the couple stress fluid are blood, suspension fluids, lubricants and electro rheological fluids. The present dissertation consists of three chapters. Chapter one contains some basic concepts, definitions and equations. Chapter two addresses the two-dimensional flow of viscous nanofluid over a nonlinearly stretching surface with convective surface boundary condition. Thermophoresis and Brownian motion effects are considered. Boundary condition with the zero nanoparticles mass flux at the surface is incorporated. Homotopy analysis technique is adopted to solve the governing nonlinear differential system. Graphs are plotted to see the effects of various physical parameters on the temperature and concentration distributions. This chapter provides the detailed review of article by Mustafa et al. [8]. Chapter three is the generalization of chapter two into three directions. Firstly to consider the three-dimensional flow of couple stress nanofluid. Effects of Brownian motion and thermophoresis are taken into account. We imposed the thermal

convective [29,30] and zero nanoparticles mass flux conditions at the surface [31,32]. Secondly to analyze the influence of non-uniform magnetic field under low magnetic Reynolds number assumption. Thirdly to compute the convergent series solutions through the homotopy analysis method (HAM) [33-40]. Effects of various pertinent flow parameters on the velocities, temperature and concentration distributions are sketched and discussed. Numerical values of