Theoretical investigation of unsteady forced bio-convection slip flow of an exponentially stretching sheet

Abstract

The transfer of heat is the elementary aspect of the great machine made applications. The heat transfer depends upon the thermal conductivity of the working fluids as compared to the capability of thermal devices and systems. In conventional heat transfer, transfer of heat in the fluids are obtained by liquefying nanoparticles which are nanometer sized particles (1 - 100) nm. The structure of nanofluid particles is formed by nitride ceramics, metals, and semi conductor. Nanofluid is found to be a good medium for thermal conductivity. In the base fluids, the existence of the nanoparticles increases the efficiency and thermal conductivity of the fluid. An innovative technique for improving heat transfer by using ultra fine solid particles in the fluids has been used extensively during the last several years, and one of it, is nanofluid. The term nanofluid was used by Choi [1]. Das et al. [2] exposed sundary applications of nanofluids. The application of nanofluid in a convective boundary layer flow were studied by Buongiorno [3]. Brownian diffusion and the thermophoresis described the behavior of nanofluids. In Buongiorno model, we studied that the velocity of flow is the sum of the base fluid velocity to the slip velocity. Wang and Fan [4] studied that nanofluid have four scales which are the molecular, the micro, the macro, and the megascale. Zhang et al. [5] analyzed the MHD radiative flow of a nanofluid past a surface that have variable heat flux and chemical reaction. Haq et al. [6] described the stagnation point flow of a nanofluid past a stretching sheet with slip effects and thermal radiation. Makinde and Aziz [7] observed the convective boundary condition with boundary layer flow of nanofluid over a stretching sheet. Nadeem and Lee [8] analyzed the boundary layer flow past a stretching sheet in the presence of nanofluid. Nazar et al. [9] investigated the unsteady boundary layer flow of a nanofluid past a stretching sheet induced by impulsive motion. Das et al. [10] investigated entropy analysis of unsteady magneto hydrodynamics nanofluid flow past an accelerating stretching sheet with convective boundary condition. Nadeem and Haq [11] elaborated the effect of thermal radiation for magnetohydrodynamics (MHD) boundary layer flow of a nanofluid over a stretching sheet with convective boundary conditions. Nadeem and Lee [12] presented the steady boundary layer flow of nanofluid over an exponential stretching surface. Turkyilmazoglu [13] illustrated the heat and mass transfer analysis for MHD flow of viscous nanofluid with slip effect. He analyzed the closed form solutions of velocity, temperature, and concentration profiles. Ramzan et al. [14] studied unsteady second grade MHD flow of nanofluid induced by vertical sheet with thermal radiation and mixed convection. Ramzan and Ashraf [15] analyzed three-dimensional flow of an elastico-viscous nanofluid with chemical reaction and magnetic field effects. Micropolar fluids are fluids with microstructures. They belong to the nonsymmetric stress tensor. Micropolar fluid consist of rigid randomly oriented or spherical particles. They have their own spins and microrotations, suspended in viscous medium. Miccropolar fluids are polar fluids, which have some microsize effects such as rotation and shrinking etc. Physical examples of micropolar flows are, blood flow, bubby liquids, liquid crystals and so on. A latest discovery for micropolar fluid is to combine nanofluid with bioconvection development was studied by Xu and Pop [16]. Aziz et al. [17] studied theoretically the natural bio - convection boundary layer flow of a nanofluid. Agarwal et al. [18] investigated the solution of finite element flow and transfer of heat of a micropolar fluid past a stretching sheet. Hassanian and Gorla et al. [19] and few other explored the steady boundary layer flow of a micropolar flow drive by permeable and non-permeable sheets. Eringen [20] proposed the theory of a micropolar fluids. El - Aziz et al. [21] investigated that the micropolar fluids can preserve rotation with individual motion. They support stress and body moments are effects by spin inertia. Nadeem et al. [22] studied axisymmetric stagnation flow of a micropolar nanofluid in a moving cylinder. Balram and Sastry [23] studied free convection flow of a micropolar fluids in a parallel plate vertical channel. Lok et al. [24] carried out the steady two-dimensional asymmetric stagnation point flow of a micropolar fluid Bio-convection arrangement are mostly occur due to the upswimming of microorganism (very small living organism such as bacteria that are visible under a microscope) that are a little dense. By the upswimming the upper portion of the fluid layers of suspensions become to dense. Due to the gathering of microorganisms it become unstable. The microorganisms fall down to cause bio - convection. In geophysical phenomena thermo - bio - convection plays an important role e.g. the phenomena thermophiles in which hot springs immigrate by motile microorganism i.e. heat loving microorganisms. Another application is in the field of microbial increased oil recovery. In oil-bearing, layers are added microorganisms and nutrients to arrange permeability deviation. Nield and Kuznetsov [25] reported that microorganisms may contribute toward the development in bio-micro-systems, they show a significant aspect in mixing and increase in mass movement. Uddin et al. [26] recently studied bio-convection flow of a nanofluid past a moving plate in the presence of stefan blowing and influence of multiple slip. Uddin et al. [27] analyzed the MHD free convective boundary layer flow of Newtonian heating boundary condition with nanofluid over a flat vertical plate. Khan et al. [28] analyzed the boundary layer flow of a nanofluid consist of gyrotactic microorganisms over a vertical plate with magnetic field. Kuznetsov [29] is reported the onset of thermo - bio - convection in a shallow fluid, saturated porous layer heated from below in a suspension of oxytactic microorgansims. Zaimi et al. [30] studied the stagnation flow of a nanofluid over a stretching sheet in the existence of gyrotactic microorganisms. Beg et al. [31] investigated boundary layer bio - convective non - Newtonian nanofluid flow from a horizontal flat plate in a porous medium. Kuznetsov [32] studied the interaction of oxytactic microorganisms in a shallow horizontal layer of finite depth. Mandal and Mukhopadhyay [33] carried out the heat transfer effects on boundary layer flow embedded in a porous medium with variable heat flux. Here the flow generation is caused due to an exponential stretching of sheet. It appears that Wang [34] indicated an exact similarity solution for the steady three-dimensional flow of a viscous and incompressible fluid due to a stretching flat surface. Pop and Merkin[35] analyzed an unsteady three-dimensional free convection flow near a general stagnation point placed in a fluid-saturated porous medium. However, only one few studies are available in the literature which discussed a three-dimensional boundary-layer flow over an exponentially stretching surface. Suction/injection (blowing) of a fluid through the bounding surface can significantly change the flow field. In general, suction tends to increase the skin friction coefficient, whereas injection acts in the opposite manner. The mechanism of suction has also importance in several engineering activities such as in the design of thrust bearing, thermal oil recovery, and radial diffusers [36]. The bioconvection parameters have significant effects on flow, mass and heat transfer, and motile microorganism density number Contents of this dissertation consist of three chapters. Chapter first is related to some basic and conceptual definitions of the fluid, basic governing equations and laws. In chapter two we have reviewed the paper of “Nur Amalina Abd. Latiff” i.e. “Unsteady forced bioconvection slip flow of a micropolar nanofluid from a stretching/shrinking sheet”. Convergent series solutions are developed via homotopy analysis method. The results for velocity, micro rotation, temperature, concentration, and microorganism are established and analyzed. Chapter three is my extension work in which we have investigated the unsteady forced bioconvection slip flow of an exponentially stretching sheet. The analysis is carried out for the three dimensional unsteady, laminar, and an incompressible flow of a micropolar nanofluid. The governing partial differential equations of the flow are converted into non-linear ordinary differential equations using similarity transformation. These equations are solved numerically by using shooting/bvp4c method. The effects of different parameters are studied graphically. Further, the graphical behavior of skin friction, Nusselt number, microorganism flux is presented.