Simulating fluid flows in micro and nano devices : the challenge of non-equilibrium behaviour

We review some recent developments in the modelling of non-equilibrium (rarefied) gas flows at the micro- and nano-scale, concentrating on two different but promising approaches: extended hydrodynamic models, and lattice Boltzmann methods. Following a brief exposition of the challenges that non-equilibrium poses in micro- and nano-scale gas flows, we turn first to extended hydrodynamics, outlining the effective abandonment of Burnett-type models in favour of high-order regularised moment equations. We show that the latter models, with properly-constituted boundary conditions, can capture critical non-equilibrium flow phenomena quite well. We then review the boundary conditions required if the conventional Navier-Stokes-Fourier (NSF) fluid dynamic model is applied at the micro scale, describing how 2nd-order Maxwell-type conditions can be used to compensate for some of the non-equilibrium flow behaviour near solid surfaces. While extended hydrodynamics is not yet widely-used for real flow problems because of its inherent complexity, we finish this section with an outline of recent 'phenomenological extended hydrodynamics' (PEH) techniques-essentially the NSF equations scaled to incorporate non-equilibrium behaviour close to solid surfaces-which offer promise as engineering models. Understanding non-equilibrium within lattice Boltzmann (LB) framework is not as advanced as in the hydrodynamic framework, although LB can borrow some of the techniques which are being developed in the latter-in particular, the near-wall scaling of certain fluid properties that has proven effective in PEH. We describe how, with this modification, the standard 2nd-order LB method is showing promise in predicting some rarefaction phenomena, indicating that instead of developing higher-order off-lattice LB methods with a large number of discrete velocities, a simplified high-order LB method with near-wall scaling may prove to be just as effective as a simulation tool

Molecular dynamics for fluid mechanics in arbitrary geometries

Simulations of nanoscale systems where fluid mechanics plays an important role are required to help design and understand nano-devices and biological systems. A simulation method which hybridises molecular dynamics (MD) and continuum computational fluid dynamics (CFD) models is able to accurately represent the relevant physical phenomena and be computationally tractable. An MD code has been written to perform MD simulations in systems where the geometry is described by a mesh of unstructured arbitrary polyhedral cells that have been spatially decomposed into irregular portions for parallel processing. The MD code that has been developed may be used for simulations on its own, or may serve as the MD component of a hybrid method. The code has been implemented using OpenFOAM, an open source C++ CFD toolbox (www.openfoam.org). The requirements for two key enabling components are described. 1) Parallel generation of initial configurations of molecules in arbitrary geometries. 2) Calculation of intermolecular pair forces, including between molecules that lie on mesh portions assigned to different, and possibly non-neighbouring processors. A case study of flow in a realistic nanoscale mixing channel, where the geometry is drawn and meshed in engineering CAD tools is simulated to demonstrate the capabilities of the code

On the modelling of isothermal gas flows at the microscale

This paper makes two new propositions regarding the modelling of rarefied (non-equilibrium) isothermal gas flows at the microscale. The first is a new test case for benchmarking high-order, or extended, hydrodynamic models for these flows. This standing time-varying shear-wave problem does not require boundary conditions to be specified at a solid surface, so is useful for assessing whether fluid models can capture rarefaction effects in the bulk flow. We assess a number of different proposed extended hydrodynamic models, and we find the R13 equations perform the best in this case. Our second proposition is a simple technique for introducing non-equilibrium effects caused by the presence of solid surfaces into the computational fluid dynamics framework. By combining a new model for slip boundary conditions with a near-wall scaling of the Navier--Stokes constitutive relations, we obtain a model that is much more accurate at higher Knudsen numbers than the conventional second-order slip model. We show that this provides good results for combined Couette/Poiseuille flow, and that the model can predict the stress/strain-rate inversion that is evident from molecular simulations. The model's generality to non-planar geometries is demonstrated by examining low-speed flow around a micro-sphere. It shows a marked improvement over conventional predictions of the drag on the sphere, although there are some questions regarding its stability at the highest Knudsen numbers

The structure of hypersonic shock waves using Navier-Stokes equations modified to include mass diffusion

Howard Brenner has recently proposed modifications to the Navier-Stokes equations that relate to a diffusion of fluid volume that would be significant for flows with high density gradients. In a previous paper (Greenshields & Reese, 2007), we found these modifications gave good predictions of the viscous structure of shock waves in argon in the range Mach 1.0-12.0 (while conventional Navier-Stokes equations are known to fail above about Mach 2). However, some areas of concern with this model were a somewhat arbitrary choice of modelling coefficient, and potentially unphysical and unstable solutions. In this paper, we therefore present slightly different modifications to include molecule mass diffusion fully in the Navier-Stokes equations. These modifications are shown to be stable and produce physical solutions to the shock problem of a quality broadly similar to those from the family of extended hydrodynamic models that includes the Burnett equations. The modifications primarily add a diffusion term to the mass conservation equation, so are at least as simple to solve as the Navier-Stokes equations; there are none of the numerical implementation problems of conventional extended hydrodynamics models, particularly in respect of boundary conditions. We recommend further investigation and testing on a number of different benchmark non-equilibrium flow cases.Comment: written for the 2nd European Conference on AeroSpace Sciences (EUCASS), Belgium, 200

The concept of mass-density in classical thermodynamics and the Boltzmann kinetic equation for dilute gases

In this paper we discuss the mass-density of gas media as represented in kinetic theory. It is argued that conventional representations of this variable in gas kinetic theory contradict a macroscopic field variable and thermodynamic property in classical thermodynamics. We show that in cases where mass-density variations exist throughout the medium, introducing the mass-density as a macroscopic field variable leads to a restructuring of the diffusive/convective fluxes and implies some modifications to the hydrodynamic equations describing gas flows and heat transfer. As an illustration, we consider the prediction of mass-density profiles in a simple heat conduction problem between parallel plates maintained at different temperatures

The structure of shock waves as a test of Brenner's modifications to the Navier-Stokes equations

Brenner has recently proposed modifications to the Navier-Stokes equations that are based on theoretical arguments but supported only by experiments having a fairly limited range. These modifications relate to a diffusion of fluid volume that would be significant for flows with high density gradients. So the viscous structure of shock waves in gases should provide an excellent test case for this new model. In this paper we detail the shock structure problem and propose exponents for the gas viscosity-temperature relation based on empirical viscosity data that is independent of shock experiments. We then simulate shocks in the range Mach 1.0-12.0 using the Navier-Stokes equations, both with and without Brenner's modifications. Initial simulations showed Brenner's modifications display unphysical behaviour when the coefficient of volume diffusion exceeds the kinematic viscosity. Our subsequent analyses attribute this behaviour to both an instability to temporal disturbances and a spurious phase velocity-frequency relationship. On equating the volume diffusivity to the kinematic viscosity, however, we find the results with Brenner's modifications are significantly better than those of the standard Navier-Stokes equations, and broadly similar to those from the family of extended hydrodynamic models that includes the Burnett equations. Brenner's modifications add only two terms to the Navier-Stokes equations, and the numerical implementation is much simpler than conventional extended hydrodynamic models, particularly in respect of boundary conditions. We recommend further investigation and testing on a number of different benchmark non-equilibrium flow cases

Solving the Boltzmann equation deterministically by the fast spectral method : application to gas microflows

Based on the fast spectral approximation to the Boltzmann collision operator, we present an accurate and efficient deterministic numerical method for solving the Boltzmann equation. First, the linearised Boltzmann equation is solved for Poiseuille and thermal creep flows, where the influence of different molecular models on the mass and heat flow rates is assessed, and the Onsager-Casimir relation at the microscopic level for large Knudsen numbers is demonstrated. Recent experimental measurements of mass flow rates along a rectangular tube with large aspect ratio are compared with numerical results for the linearised Boltzmann equation. Then, a number of two-dimensional micro flows in the transition and free molecular flow regimes are simulated using the nonlinear Boltzmann equation. The influence of the molecular model is discussed, as well as the applicability of the linearised Boltzmann equation. For thermally driven flows in the free molecular regime, it is found that the magnitudes of the flow velocity are inversely proportional to the Knudsen number. The streamline patterns of thermal creep flow inside a closed rectangular channel are analysed in detail: when the Knudsen number is smaller than a critical value, the flow pattern can be predicted based on a linear superposition of the velocity profiles of linearised Poiseuille and thermal creep flows between parallel plates. For large Knudsen numbers, the flow pattern can be determined using the linearised Poiseuille and thermal creep velocity profiles at the critical Knudsen number. The critical Knudsen number is found to be related to the aspect ratio of the rectangular channel