Unsteady Flow and Heat Transfer O ver an Exponentially Stretching Surface

Abstract

n view of the substantial practical applications in industrial and manufacturing fields, a lot of work has been done on boundary layer flow over a stretching surface. The interminable list of its engineering applications comprise extrusion of polymer sheet, paper production, manufacturing of metal wires and plastic sheets, annealing and tinning of copper wires, cooling of metallic plate inside a cooling reservoir and so forth. Sakiadis [1] propounded the idea of the boundary layer flow over a continuous stretched surface and devised the boundary layer equations for two-dimensional flows. Tsou et al. [2] extended the idea of the effects of heat transfer on boundary layer flow over a stretching surface. Gupta et al. [3] incorporated mass transfer analysis over stretching sheet by considering suction or blowing. Later many researchers have further probed into boundary layer flows over stretching surfaces [4-5]. In the contemporary period, boundary layer flow over an exponentially stretching sheet is gaining much importance due to its vast applications. For instance; in the process of drawing and tinning of copper wires, the rate of heat transfer past a continuously stretching surface which has exponential modifications in its stretching velocity and temperature, influences the form of the final product. In this regard, Magyari and Keller [6] forwarded the comparison of numerical and analytical solutions. Nadeem et al. [7] discussed the heat transfer analysis of nanofluid over an exponentially stretching sheet. Ibrahim et al. [8] considered the MHD effects on nanofluid over an exponentially stretching sheet. At the same time, the study of heat transfer gained momentum due to its importance in the industry for maintaining the quality of final product which significantly depends on the rate of cooling. Grubka and Bobba [9] presented the analysis of heat transfer over a linearly stretching surface with power law variations of surface temperature. Char [10] extended this work by considering the effects of suction or blowing. Elbashbeshy [11] presented the heat transfer analysis over an unsteady stretching surface. Lately Malvandi [12] analyzed the heat transfer of nanofluid over a permeable stretching sheet. Later on, researchers embarked on the effects of viscous dissipation in different fluids. Partha et al. [13] contemplated these effects on mixed convection heat transfer. Currently, investigation is being sought on the influence of viscous dissipation on nanofluids by different researchers [14-15]. Along with viscous dissipation, heat generation/absorption results on nanofluid have also been explored by Pal et al. [16]. Awais et al. [17] evaluated the heat generation/absorption effects on third grade nanofluid. A substantial research has been done on thermal radiation and heat transfer due to its ample significance in electrical power generation, solar energy, space vehicles, astrophysical flows, cooling of nuclear reactors and industrial sector. Moreover, thermal radiation is the pivot around which the polymer industry moves. Initially radiation effects were studied by Hayat et al. [18] and Biliana et al. [19]. Nowadays the impact of thermal radiation on nanofluid is vastly being investigated. Prominent work has been done by Hayat et al. [20] and Sheikholeslami et al. [21]. Fluids which allow electrical conduction are frequently used in power generators, in the pattern of heat exchanges, MHD accelerators and electrostatic filters. This factor is of vital importance especially in metallurgical procedures for example, the process of cooling of filaments and strips being pulled out from an inert fluid and exclusion of non-metallic materials from molten metals. The filaments are being drawn through an electrically conducting fluid exposed to a magnetic field; this helps in controlling the cooling rate. Therefore, MHD fluid flows are of great importance, which were analyzed by Andersson [22] in 1992, after that by Damseh et al. [23] in 2006 and then by Ishak [24] in 2011. Currently research is being sought on MHD flows over exponentially stretching sheets [25-26]. It is worth mentioning that unsteady flow conditions are also encountered in case when flow becomes time dependent due to unusual change in temperature or stretching of the sheet or heat flux of the sheet. Elbashbeshy [27] discussed the thermal radiation and magnetic field effects on unsteady flow. Mansur et al. [28] discussed the unsteady boundary layer flow over a stretching sheet. Fluid particle suspension or dusty fluid deals with the motion of liquid or gas comprising inert, immiscible solid particles such as dust in gas cooling systems, blood flow in arteries etc. Their immense application is traced in cement process and steel manufacturing industry, fluidized bed, magneto hydrodynamic generators (MHG), gas purification, sedimentation pipe flows and bio fluids. The foremost task of formulating the governing equations for the flow of dusty fluids was undertaken by Saffman [29] who also brought into consideration the stability of the laminar flow of dusty fluid with uniformly distributed dust particles. Vajravelu et al. [30] carried out investigation on hydro magnetic flow of dusty fluid and highlighted the outcome of fluid particle interaction and suction on fluid properties. Extensive research has been done on fluid with dust particles by Gireesha et al. [31-33], on various aspects such as for nanofluids, convective boundary conditions, unsteady flow over an exponentially stretching sheet, and MHD flow with heat transfer analysis. It was the need of the hour to enhance the thermal conductivity of some important conventional heat transfer fluids such as water, mineral oil and ethylene glycol which possess poor heat transfer characteristics. A novel solution was found by introducing small metallic solid particles in the fluid; this revolutionized the realm of technology and industry. These nanoparticles elevate the thermal conductivity of fluids improving the heat transfer properties. The fluids so obtained were termed as nanofluids. Therefore, nanofluids are characterized as a solid-fluid mixture with base fluid of low conductivity and nano-meter sized particles with high thermal conductivity. This fluid was first introduced by Choi [34] in 1995. A comprehensive study of convective transport within nanofluid was done by Buongiorno [35]. Boundary layer flow over an exponentially stretching sheet of nanofluid was initially investigated by Nadeem et al. [36]. Bachok et al. [37] examined the unsteady boundary layer flow of nanofluid. Nanoparticles with their low volume fraction, stability and remarkable useful applications in optical, biomedical and electronic fields have opened up new horizons of research recently. Naramgari et al. [38] conducted the research on dual solutions of MHD nanofluid flow on an exponentially stretching sheet. Nadeem et al. [39-40] investigated nanofluid model in different geometries with pertinent physical properties of fluid. Gireesha et al. [41] put forward the solution for MHD flow and heat transfer of a dusty nanofluid over a stretching sheet. HAM is a frequently used approach among scientists to find explicit analytical solutions of non-linear problems. The optimal HAM (OHAM) is a modified and advanced version of HAM. OHAM uses the technique of minimizing the squared residual error through which the optimal values of the convergence control parameters are obtained. Yubashita et al. [42] were the first to propose the optimization method for finding convergence control parameters. Then, it was Liao [43] who suggested that by minimized squared residual error one can acquire the optimized value of convergence control parameters. In the vast field of literature, there is hardly any research conducted on the effects of nanoparticles on an unsteady flow of dusty fluid over an exponentially stretching sheet. In the present thesis, effect of dust particles on an unsteady boundary layer flow and heat transfer characteristics of a viscous nanofluid over an exponentially stretching sheet have been critically examined. Moreover, thermal radiation, viscous dissipation and internal heat generation absorption effects are also brought into consideration. For heat transfer analysis, two heating processes (i) Variable Exponential order Surface Temperature (VEST) case and (ii) Variable Exponential order Heat Flux (VEHF) case have been profoundly studied. Similarity transformations are defined to reduce governing equations to coupled non-linear Ordinary Differential Equations. Solution of the resulting ODE’s is computed by OHAM. The Optimal Convergence Control parameters are acquired. Finally, non-dimensional velocities, temperature and concentration profiles and the effects of various physical parameters on them are exhibited by plotting graphs and are also presented in tabular form.