Bi-velocity gas dynamics of a micro lid-driven cavity heat transfer subject to forced convection

Abstract

We investigate heat transfer in a rarefied gas in a lid-driven cavity flow initiated by instantaneously heating and cooling opposite walls for different flow regimes. Volume diffusion model is used as an extension to the standard Navier–Stokes-Fourier set for simulating the flows. Numerical simulations are presented and compared with standard Navier–Stokes-Fourier. For higher Knudsen numbers the volume diffusion model captures non-local equilibrium effects in corners of the cavity that are missed by the Navier–Stokes-Fourier model. It is generally observed that one can use volume diffusion corrections to capture disequilibrium effects in high rarefaction regimes.

Introduction

Lid-driven cavity flow problem is often used to benchmark new continuum fluid models as several investigations are carried out using the configuration [1], [2]. The various heat transfer mechanics, namely, natural convection, forced-convection and mixed convection are all investigated using hydrodynamic models as understanding thermal behaviors of a lid-driven cavity fuels applications in electronic cooling, manufacturing processes and others. Iwatsu and Hyun provided a numerical study of three-dimensional flows in a cubical container with stable vertical temperature stratification [3]. Ghasemi and Aminossadati presented the numerical study of mixed convection in a lid-driven filled with water and nanoparticles [4]. Cheng and Liu investigated the effect of the cavity inclination on mixed convection heat transfer [5].

With advances in nano- and micro electromechnical systems (NEMS/MEMS) in fabrication technology, understanding new mass and heat transfer processes in rarefaction regimes is of another great interest. Accurate modeling of gas flows at micro/nano scale involves in principle accurate modeling of rarefaction, gas–surface interactions and inter-molecular collisions [6]. To determine the degree of gas rarefaction, the Knudsen number (Kn) is introduced, Kn = λ/L where λ is the mean free path of the gas molecules and L is a characteristic length of the flow system [7]. Existing models in dealing with these transport phenomena include continuum, molecular and hybrid methods. The standard continuum fluid approach is thought to be valid for Kn ≤ 0.001. It may be extended beyond Kn = 0.01 into the slip flow regime by introducing velocity slip and temperature jump at boundaries. For 0.01 ≤ Kn ≤ 10, in the transition regime, the continuum assumption breaks down and Boltzmann equation and other particle methods are adopted. The flow is considered as a free-molecular flow for Kn > 10 where inter-molecular collisions become rare. Gas flows in MEMS and NEMS are usually in the slip (Kn ≤ 0.01) and transition regimes (0.01 ≤ Kn ≤ 10).

Heat transfer characteristics in rarefied gas in the driven cavity problem are in paucity in the literature. Mizzi et al. provided DSMC flow features in the cavity in the slip flow regime [8]. General agreement is reported between NSF and DSMC at Kn = 0.02 and 0.05 in the velocity fields and differences for the temperature fields. Though understanding counter-gradients heat transfer mechanism exhibited in the DSMC flow contours are still indistinct and are not predictable by NSF. Boundary treatment is determinant in DSMC flow feature predictions [9]. The inappropriateness of the standard NSF model in describing temperature field in a stationary gas in the continuum limit has long been shown by Sone et al. [10]. These authors therefore provided corrections to the standard constitutive equations involving Korteweg diffuse interface type stress tensor required to acquire appropriate heat transfer description in a gas around a body at rest in a close domain.

An alternative continuum approach to the description of a fluid accounting for its non-continuum aspect has been recently introduced [11]. This theory was first initiated based on the observation in thermophoresis experiments [12]. The underlying principle is the existence of two independent velocities in any continuum fluid; the volume velocity and the mass velocity [13]. The resulting equations are completed in a thermo-mechanically consistent continuum flow equations [14]. A numerical analysis using this approach to predict gas dynamics in a micro channel Couette flow is presented in [15]. It was concluded that the Bi-velocity (volume diffusion) equations show reasonable results compared with existing Burnett equations that are traditionally derived from the Boltzmann equation. An analytical solution based on this new continuum approach to the prediction of gas mass flow rates in a pressure driven rarefied gas over a range of Knudsen numbers is presented in [16]. It was found that volume diffusion (Bi-velocity) theory concords with experimental data from low to Knudsen number of 5.

In the present paper we propose an additional lid-driven cavity flow configuration to evaluate heat transfer mechanics in non-equilibrium rarefaction regimes. We compare volume diffusion continuum approach with standard NSF. It is generally observed that the volume diffusion continuum approach may allow for better description of counter-gradient heat transfer mechanics.