A Numerical Analysis of the Graetz Problem Using the Method of Moments

In this paper, we investigate the classic Graetz problem which is concerned with the thermal development length of a fluid flowing in a pipe or channel. In our particular study, we are interested in the thermal development length associated with a rarefied gas in a 2D channel. When the gas is in a rarefied state, the boundary conditions have to be modified to account for velocity-slip and temperature-jump. Although a number of previous studies have considered rarefaction effects, they have usually taken the form of modifying the boundary conditions of the Navier-Stokes equations. Our study has involved using the Method of Moments, which represents a higher-order set of equations involving transport of stress and heat flux. The results show that the moment method captures the non-equilibrium flow features and is in good agreement with kinetic data. INTRODUCTION Over the past two decades, there has been an on-going acceleration of technological developments associated with microand nano-technology. In particular, the emergence of MEMS (Micro-Electro-Mechanical Systems), where the characteristic length scale ranges from 0.1 m μ (10 m) through to millimetres, has enabled many novel ideas and concepts to be explored. It is widely accepted that major beneficiaries will be in the fields of medicine, in point-of-care medical diagnostics, biology, with the reduction and removal of animal testing, chemistry, through improving chemical yields and safe manipulation of volatile or exothermic reactions, and advanced sensors, with greatly improved sensitivity. However, there remains a general lack of progress in modelling and simulation of microand nano-systems. As the length scales diminish, properties often ignored at the macro-scale become critically important. For example, surface tension becomes a powerful force easily capable of blocking a fluid in a channel. In addition, boundaries can be readily modified to be either hydrophobic or hydrophilic. A crucially important factor for gaseous transport is the small characteristic length scale implies that rarefaction effects have to be taken into consideration. Rarefied flow is characterised by the Knudsen number, Kn, which is determined from the ratio of the molecular mean free path over the width of some characteristic dimension, such as the diameter of a pipe or height of a channel. If the Knudsen number is very small (Kn<0.001), continuum theory is considered to be valid. However, over the last two decades, fabrication of microand nano-technology systems has made significant progress and today, many micro-electro-mechanical systems operate where the Knudsen number is above 0.001. In general, flows tend to be in the slip regime ( 0.001 0.1 Kn ≤ < ) or transition regime ( 0.1 10 Kn ≤ < ). Under these conditions, rarefaction effects need to be accounted for in the analysis and design of these devices. Any flow where is described as a free-molecular flow and the stochastic/particle nature of the gas must be modelled through the Boltzmann equation or an equivalent approach, such as direct simulation Monte Carlo. 10 ≥ Kn The thermal development characteristics of a gas flowing, or entering, a pipe or channel with a different temperature, was first analysed by Graetz [1, 2]. In his study, which described the development of a laminar flow without heat conduction and viscous dissipation, the Knudsen number was small (< 0.001) and the governing fluid dynamic equations provide an accurate description of the flow. However, with the introduction of MEMS, it is essential that the Navier-Stokes-Fourier (NSF) equations take into account velocity-slip and temperature-jump. These boundary conditions were introduced into the Graetz problem by Sparrow and Lin and other researchers [3-4] for the slip-flow regime. For flows beyond the slip-flow regime, which can be encountered in MEMS devices operating under SATP conditions when the characteristic length scale is 1.0 m, μ ≤ the NSF equations are no longer a reliable method of predicting flow behaviour. Many rarefied or non-equilibrium phenomena