Frontiers in Heat and Mass Transfer (FHMT), 3, 033004 (2012) DOI: 10.5098/hmt.v3.3.3004 Global Digital Central ISSN: 2151-8629 1 VISCOUS DISSIPATION EFFECTS ON THE LIMITING VALUE OF NUSSELT NUMBERS FOR A SHEAR DRIVEN FLOW BETWEEN TWO ASYMMETRICALLY HEATED PARALLEL PLATES Pranab Kumar Mondal, * Sanchayan Mukherjee Department of Mechanical Engineering, Kalyani Government Engineering College, Kalyani - 741235, India ABSTRACT The present paper deals with the analytical investigation for the limiting value of Nusselt number, including the effect of viscous dissipation on heat transfer for a laminar shear driven flow between two infinite parallel plates, where the bottom plate is fixed and the top plate is moving in an axial direction at a constant speed. The study concentrates on hydro-dynamically fully developed flow of a Newtonian fluid of constant properties without considering the axial conduction in the fluid. To investigate the effect of viscous dissipation on heat transfer by defining the limiting Nusselt number, plates are kept at constant equal temperatures. Close form expressions for the limiting Nusselt numbers as a function of the Brinkman number and asymmetry parameter are evaluated. Focus is given to the viscous dissipative effect due to the shear produced by the movable top plate over and above the viscous dissipation due to internal fluid friction. The interactive effects of the Brinkman number and the degree of asymmetry on the limiting Nusselt number are investigated analytically. Specific to the cases considered for this study, the appearance of point of singularities due to the variation of Nusselt number with the Brinkman number is observed, and discussion has been made considering the energy balance, and second law analysis of thermodynamics. Keywords: analytical investigations, heat transfer, shear driven flow, degree of asymmetry, Brinkman number, axial conduction, second law of thermodynamics. * Corresponding author. Email: pranab2k3@yahoo.com 1. INTRODUCTION In the recent past, fluid flow in small devices and the corresponding heat transfer has received serious attention in view of the remarkable development in the field of microelectronics and MEMS. The dissimilarity in thermal behavior in small devices is mainly due to the rise in temperature, which is primarily attributable to the viscous dissipation effect. In small devices the effect of viscous dissipation that could play a vital role, results in inefficient heat dissipation leading to local overheating problems. The understanding of the fluid flow phenomena has been critically reviewed (Hak, 1999) through micro scale devices and explored the physics of the plow emphasizing the use of MEMS in different areas for flow control. In the domain of the macroflows, there are so many practical applications where heat transfer normally occurs in the fluid flow system involving moving boundaries. Particularly, in many material processing applications such as extrusion, hot rolling, drawing, and continuous casting, materials continuously move in a channel. In such industrial applications, it is of great importance to encounter the heat transfer from the moving boundary to the surrounding fluid and vice-versa. However, the moving boundary deforms the fluid velocity profile, and shears the fluid layer near the boundary, resulting in local changes in velocity gradient. Thus the viscous dissipation effects may not be neglected in heat transfer analysis associated with moving boundaries. The thermal energy generated due to the viscous dissipation is significant near the wall, which alters the heat transfer rates following the changes in the temperature profile. In order to obtain the actual heat transfer rate in the application of moving boundaries, it is important to take into account the effects of viscous dissipations using accurate velocity distribution. The first theoretical work (Brinkman, 1951) concerning the heat generation due to viscous dissipation has analyzed the effects of viscous heating for the flow of a single phase Newtonian fluid through a circular tube. The temperature distribution in the thermal entrance region has been examined considering the zero temperature of the wall and an insulated wall. The temperatures were found to be the highest, not surprisingly, within a small area near the wall region. The available literature in the area of convective heat transfer has, however, considered the effects of viscous dissipation to be important in two cases: flow of very viscous fluids and flow in capillary tubes. A numerical study (Cheng and Wu, 1976) reported the influence of viscous dissipation for the flow of a Newtonian fluid through a parallel plate channel. The numerical analysis explored the onset of the instability for longitudinal vortices in the entrance region of the channel with the thermal boundary condition of an isothermal heated lower plate, and cooled upper-plate. The effects of viscous dissipation on laminar forced convection through a pipe and channel have been studied (Pinho and Oliveira, 2000) for the flow of a Phan- Thien- Tanner fluid. The study has revealed that the viscous dissipation enhances the fluid elasticity. Performing an analytical study, using a functional analysis method, the effects of viscous dissipation on the heat transfer have been investigated (Lahjomri et al., 2003) for a thermally-developing laminar Hartman flow through a parallel plate channel with the aid of a magnetic field. In a study of thermal development of forced convection in a parallel plate channel filled by porous medium, an investigation of the effects of viscous dissipation has been done (Nield et al., 2003) with the thermal boundary condition of uniform wall temperature including axial conduction effects. Frontiers in Heat and Mass Transfer Available at www.ThermalFluidsCentral.org