Slip Boundary Condition Effects on the Rate of Heat Transfer in A Micro channel Including Viscous Dissipation

Accepted 19/Aug/2018, Online 30/Aug/2018 AbstractThis study analyzes laminar forced convection flow in parallel plate micro-channel with slip velocity and temperature jump. Closed-form solutions are obtained for temperature, bulk temperature and rate of heat transfer when both walls of the channel are kept at unequal temperatures. The effects of various controlling parameters such as rarefaction parameter, fluid-wall interaction parameter and Brinkman number on the thermal behavior and rate of heat transfer are discussed with the aid of line graphs. Interesting result from the present work is that increase in rarefaction parameter leads to enhancement in fluid temperature while increase in fluid-wall interaction parameter leads to increase in temperature jump on the walls of the channel. In addition, the rate of heat transfer represented as the Nusselt number at the both walls of the channel displays an unbounded swing which varies with increase in fluid-wall interaction and rarefaction parameters.

dimensionless temperature dimensionless mean temperature dimensionless variables molecular mean free path dynamic viscosity kinematic viscosity fluid density

I. INTRODUCTION
In the last few decades, research in micro-coolers, micro-fuel cells, micro-biochips and micro-reactors has been growing at a tremendous pace due to rapid developments in micro-electronics and biotechnologies. In micro-fluidic systems, micro-channel has been noticed to be one of the important elements in fluid flow within a miniature area. In addition to been a reactant delivery, the micro-channels are also used to connect different chemical chambers. The design and process controls of microfluidic and microelectromechanical systems (MEMS) involve the effect of geometrical configurations on the pressure, velocity and thermal distributions of the fluid on the micrometer ( ) scale. Knudsen number ( ) is an important factor in the flow of fluids through micro-channels and it is defined as the ratio of molecular free path ( ) to characteristic length ( ). In the slip flow regime, the gas at the surface has a tangential velocity which causes it to slip along the surface and, the temperature of the gas at the surface is finitely different from the surface temperature resulting to a jump in temperature between the surface and the adjacent gas. However, Beskok and Karniadakis [1] proposed four flow regimes in gases as shown below

II. RELATED WORK
Extensive research has been carried out recently in the field of micro geometric flows. Weng and Cheng [2] studied analytically the fully developed natural convection in open ended vertical parallel plate micro-channel, taking into account the effects of rarefaction and fluid wall interaction parameters on the volume flow rate and heat transfer. Fully developed thermocreep-driven gas micro-flow has been investigated by Weng et al [3] Laminar convection in a vertical channel with viscous dissipation and buoyancy effects have been examined by Barletta [4]. Recently, Khadrawi [5] studied the transient hydrodynamic and thermal behaviors of fluid flow in a vertical porous micro-channel under the effect of hyperbolic heat conduction model. Using semi-implicit method, Biswal et al. [6] considered the hydrodynamic and heat transfer behavior in the developing region of a micro-channel. It was deduced that slip velocity and jump temperatures enhances the rate of heat transfer.
Similarly, the effect of viscous dissipation has received considerable attention owing to its importance in the heat transfer point of view. Citing few works, Mondal and Sanchayan [7] investigated viscous dissipation effect on the limiting value of Nusselt numbers for shear driven flow through asymmetrically heated parallel plates. Tso et al. [8] and Aydin and Avci [9] studied viscous dissipation effects of power law fluid flow within parallel plates with constant heat fluxes and its effects on the rate of heat transfer in a Poiseuille flow respectively. Barletta [10] investigated laminar mixed convection with viscous dissipation in a vertical channel. Recently, Ramjee and Satyamurty [11] studied the Nusselt number for viscous dissipation flow between parallel plates kept at unequal temperatures. In another related work, Hatton and Turton [12] investigated Heat transfer in the thermal entry length with laminar flow between parallel walls at unequal temperatures. The Nusselt number for laminar forced convection in asymmetrically heated annuli with viscous dissipation was presented by Kumar and Satyamurty [13].
Free molecular flow Transition flow regime Slip flow regime Continuum regime Therefore, the objective of the present work is to analyze the influence of fluid-wall interaction parameter ( ), rarefaction parameter ( ) and viscous dissipation on the steady forced convective flow through a micro-channel. The present study extends the work of Ramjee and Satymurty 11 by taking into account the effect of velocity slip and temperature jump boundary conditions on the thermal characteristics and rate of heat transfer at both walls of the channel.

III. MATHEMATICAL FORMULATION
Consider a fully developed flow of viscous incompressible fluid in a micro-channel formed by two parallel plates. The coordinates coincides with the centerline of the channel and the coordinate is normal to it and the distance between the two plates is as shown in figure 1. The plates are positioned at ⁄ and the average and inlet fluid temperatures are given as ̅ and respectively. The plates are kept at unequal temperatures of ⁄ and ⁄ The mathematical model extended here is a generalization of Ramjee and Satymurty 11 in the presence of slip boundary conditions. Under the usual Boussinesq approximation and neglecting convective terms, the governing equations in dimensional form are Solving ( ) ( ) using ( ) gives dimensionless velocity and temperature solutions respectively The definition for the heat coefficient based on bulk temperature as the reference fluid temperature at the wall ⁄ is given by Where the bulk temperature is defined by The Nusselt number at the lower plate ( ⁄ ) and upper plate ( ⁄ ) are given by Employing equation ( ) for and ( ) for in equations ( ) ( ), are obtained respectively as

IV. RESULTS AND DISCUSSION
The basic parameters that govern this flow are Brinkman number ( ), degree of asymmetry heating ( ), rarefaction parameter ( ) and fluid-wall interaction parameter ( ). To examine the effects of these basic parameters, the variations in velocity, temperature and Nusselt number at both walls of the channel are graphically presented and discussed. The present parametric study has been performed in continuum and slip flow regimes ( ). Also, for air and various surfaces, the values of range from near to more than and from near to respectively. Therefore this study has been performed over the reasonable ranges of .   ) at respectively. It is observed from both figures that increase in rarefaction parameter ( ) leads to increase in temperature jump on the walls of the channel. This is attributed to the weak interaction between the fluid molecules and the heated wall of the channel. Also, an increase in rarefaction parameter ( ) leads to enhancement in fluid temperature. Viscous dissipation converts the kinetic energy of the moving fluid into internal energy thereby increasing the fluid temperature. It also accounts for the distortion on the temperature profile.      respectively. It is observed from both figures that increase in fluid-wall interaction parameter ( ) leads to increase in rate of heat transfer at wall for positive dissipation. In addition, it is evident that the rate of heat transfer is significantly influenced by high value of fluid-wall interaction parameter ( ) for , while small value of fluid-wall interaction parameter ( ) has significant effect on the rate of heat transfer for .
Finally, it is worthy to note from equations ( ) ( ) that the rate of heat transfer on both walls vary with degree of asymmetric heating ( ) when dissipation is not neglected while the rate of heat transfer vary with dissipation when degree of asymmetric heating ( ).

V. CONCLUSION
This study considered fully developed laminar forced convection flow through parallel plate micro-channel under slip and jump boundary conditions. Exact solutions for the mathematical model governing the present physical situation and the expressions for the rate of heat transfer at both walls of the channel have been obtained. This study conforms to the findings of Ramjee and Satyamurty 11 when The following deductions were made from the present work: i. The increase in rarefaction parameter ( ) and fluid-wall interaction parameter ( ) leads to increase in temperature jump. ii.
At positive dissipation, increase in fluid-wall interaction parameter ( ) leads to increase in temperature while at negative dissipation, fluid temperature decreases with increase in fluid-wall interaction parameter ( ). iii.
The rate of heat transfer at both walls of the channel depends strongly on the degree of asymmetric heating ( ) and viscous dissipation ( ). iv.
The rate of heat transfer at the wall increases with an increase in rarefaction parameter ( ) for negative dissipation while for positive dissipation, increase in rarefaction parameter ( ) has an insignificant effect on the rate of heat transfer.

Funding
This research received no grant from any funding public agency, commercial, institution or any organization.

Appendix A
Constants used in the present work are as follows: