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Asymptotic Modeling of Flows in Micro-Channel by Using ...

Asymptotic Modeling of Flows in Micro-Channel by Using ...

Asymptotic Modeling of Flows in Micro-Channel by Using

Asymptotic Modeling of Flows in Micro-Channel by Using Macroscopic Balance Equations Renée Gatignol and Cédric Croizet Université Pierre et Marie Curie (UPMC, Paris 6) & CNRS, Institut Jean le Rond d’Alembert 4 place Jussieu, 75005 Paris, France Abstract. The introduction of a small parameter related to the micro-channel geometry and the application of the Principle of Least Degeneracy to the dimensionless Navier-Stokes equations allow to yield models for the study of microchannel flows with small Mach number and small or moderate Knudsen number. The first approximation is calculated for a steady gas flow inside a micro-channel with a temperature gradient along the walls. On the other hand, Direct Simulation Monte Carlo method is used to analyze the same problem. Some comparisons are presented. Analytical asymptotic solutions and DSMC numerical simulations are in very good agreement. Keywords: Microfluidics, Micro-channel, Rarefied Gas, DSMC Simulation. PACS: 05.10.Ln – 47.10.ad – 47.11.Mn – 47.45.Gx – 47.61.-k INTRODUCTION The micrometric apparatus are present in various fields of technology such as process engineering, heat exchangers, etc. For the description of gas flows with heat exchanges in micro-channels which occur in these systems, the Direct Simulation Monte Carlo (DSMC) method is well adapted. However, this method is time computer expensive. Our purpose is to study flows of compressible fluids in micro-channels by using an asymptotic macroscopic approach. The flow equations are the usual Navier-Stokes equations for mass, momentum and energy. First order jump conditions for the velocity and the temperature are written along the walls of the micro-channel. A large number of works about gas flows in micro-channels through different methods are available, theoretical, numerical and experimental as well. Depending on Knudsen number, these flows are considered in continuous, slipping, transitional or free molecular regimes [1,2,3,4,5]. The DSMC methods are successful in the case of transitional regimes; consequently they are relevant to study gas flows in micro-systems [6,7]. The lattice Boltzmann methods are also an interesting approach to compute these flows, particularly for complex geometries, although very few applications have been carried out at the present time [8]. In this paper, an asymptotic modeling is proposed for thermal compressible gas flows in micro-channels with small Mach numbers M and small or moderate Knudsen numbers Kn . Slipping boundary conditions of first order for the velocity and the temperature are taken into account [1,2]. It is possible to yield steady analytical asymptotic solutions in order to describe the physical phenomena. In addition, these solutions are compared with DSMC simulations obtained with the code DS2V of Bird [5,9]. Asymptotic solutions and DSMC simulations are in a very good agreement. EXAMPLES OF GAS FLOWS IN MICRO-CHANNELS BY DSMC METHOD Let us consider the steady laminar thermal flow of a gas in a two-dimensional micro-channel of length l and width h , with no volumetric force neither heat source. The complete geometry is shown on Fig. 1: It consists in two areas of identical sizes (length l ' = 7 µm , width h ' = 5 µm ) filled up with the same gas (Nitrogen) and connected to each other by a micro-channel of length l = 10 µm and width h = 1 µm . The axis of the micro-channel is a symmetry axis. The gas is flowing from one area to the other one through the channel. We begin with some DSMC simulations. The initial values for pressure and temperature P in and T in are given on the entrance of the left area, whereas P out and T out denote the initial values at the end of the right exit area. The constant pressure boundary

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