Development of a Consistent Slip-Boundary Condition at the Inlet of Microchannels

Abstract

The fast progress in Microelectromechanical systems has drawn the attention of many numerical workers to provide sufficient numerical tools to analyze the micro flows accurately. Besides many ambiguities behind the slip boundary condition research which are normally encountered in the vicinity of the solid walls (and are necessary to provide the suitable bridge between the macro and micro behaviors), there are serious uncertainties at the inlet of channels where the microflow enters into the conduit. Indeed, the macro inlet flow have been long investigated by the past workers; however, there is little effort to clarify the issue in micro inlet flow cases. In this research, we utilize the outcomes of the past research behind the macro entry flow problem and suitably extend them to micro flow applications. To avoid the singularities at the inlet and nearby the walls, we implement suitable boundary conditions far upstream of the channel. However, to achieve correct boundary conditions at the inlet, suitable symmetric boundary conditions are implemented at the fictitious walls extended upstream of the channel’s inlet. This strategy eliminates the singularities at the inlet of channel which is normally observed in numerical treatment of macro-entry flow problems. I. Introduction ROM macro-flow perspective, the velocity distribution at the inlet of a duct will undergo a development from some initial profile at the entrance to a fully developed profile at locations far downstream. The region of duct in which this velocity development arises is called the entrance region. There has been considerable interest in determining the fluid behavior within the entrance region because of its general technical importance in engineering applications, e.g. in predicting the pressure drop in a conduit with short length. The significant features of the solution in this region lie in the transport of the vorticity upstream of the duct entrance flow, in the way the entrance length varies with Reynolds number, in the finite entrance length at zero Reynolds number, and in the fact that maxima are predicted in the velocity profile at locations other than the centerline. A variety of approximate analytical and numerical methods has been employed for the determination of the flow characteristics in this region. A majority of older methods, in 1960's, employ the boundary-layer assumption which neglects the streamwise diffusion and the transversal pressure gradient. These methods are consequently valid for high Reynolds numbers, e.g. Van Dyke 1 . The numerical solution of the complete Navier-Stokes equations with no-slip boundary conditions for entrance flow problem has led to the results which were not expected from boundary-layer theory. Particularly, a flat velocity distribution at the inlet results in two overshoots in the velocity profile beside the walls near the inlet region. Generally speaking, numerical methods could be categorized based on their applied boundary conditions. Wang and Longwell 2 were the first to solve the entrance flow problem employing full Navier-Stokes equations. However, similar to Van Dyke 1 , they specified zero vorticity at the entrance instead of uniform entry condition. Mcdonald, et al. 3 solved the ψ-ω form of equations for the entrance flow in the inlet region of both a tube and a channel using either uniform or irrotational inlet boundary conditions. They concluded that irrotational entry generates lower