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In some important engineering applications, gas flows are often found to be hydrodynamic in one part of device and highly rarefied in the others. To solve this kind of multi-scale flow problems efficiently and accurately, hybrid methods coupling hydrodynamic and kinetic methods are attractive. The successful implementation of hybrid methods relies on the accurate assessment of the level of non-equilibrium (rarefaction) in the local flowfield. Currently available criteria, such as Knudsen and Mach numbers, are based on macroscopic parameters and have been shown to be restrictive. Here, we propose a new kinetic criterion that utilises the fundamental molecular distribution function to assess the local flow field. Through numerical evaluation we show that our criterion provides a reasonable assessment and, in particular, it behaves consistently for both high-speed and low-speed flows, which is not the case for the other criteria based on macroscopic parameters. As our criterion fully utilises the accurate information provided by the molecular distribution function, it is particularly suitable for recently developed multi-scale kinetic methods. © Published under licence by IOP Publishing Ltd.

World University root node

Department of Mechanical and Aerospace Engineering

Journal of Physics: Conference Series

A kinetic switching criterion for hybrid modelling of multiscale gas flows

International Journal of Heat and Mass Transfer

Erratum: Modeling of Navier - Stokes equations for high Knudsen number gas flows (International Journal of Heat and Mass Transfer (2012) 55 (4352-4358)

The possibility of modeling the Navier-Stokes equations and together with the conventional second order slip boundary condition at high Knudsen numbers is explored in this paper by incorporating the Knudsen diffusion phenomenon in rarefied gases. An effective mean free path (MFP) model is augmented to the governing equation and the slip boundary condition, as gas transport properties can be related to the MFP. This simple modification is shown to implicitly take care of the complexities associated in the transitional flow regime, without necessitating dependency of the slip coefficients on the Knudsen number. Unique analytical model with fixed values of slip coefficients is proposed and rigorous comparisons with the experimental and simulation data for pressure driven and thermally driven rarefied gas flows support this conjecture. First and second order slip coefficients have been proposed as 1.1466 and 0.9756 for rectangular channels and 1.1466 and 0.14 for the capillaries, from the continuum to the transition flow regime. The current work is significant from the numerical simulation point of view because simulation tools are better developed for Navier-Stokes equations. © 2012 Elsevier Ltd. All rights reserved.

Modeling of Navier-Stokes equations for high Knudsen number gas flows

We report molecular dynamics (MD) simulations of the dynamic wetting of nanoscale droplets on moving surfaces. The dynamic water contact angle and contact angle hysteresis are measured as a function of capillary number on smooth silicon and graphite surfaces. The hydrogen bonding and density profile variations are also reported, and the width of the water depletion layer is evaluated for droplets on three different static surfaces: silicon, graphite and a fictitious super-hydrophobic surface. Our results show that molecular displacements at the contact line are mostly influenced by interactions with the solid surface, while the viscous dissipation effects induced through the movement of surfaces are found to be negligible, especially for hydrophobic surfaces. This finding is in contrast with the wetting dynamics of macroscale droplets, which show significant dependence on the capillary number. This study may yield new insight into surfacewettability characteristics of nano droplets, in particular, developing new boundary conditions for continuum solvers for liquid flows in micro- And nanoscale devices. Copyright © 2012 by ASME.

ASME 2012 10th Int. Conf. on Nanochannels, Microchannels, and Minichannels Collocated with the ASME 2012 Heat Transfer Summer Conf. and the ASME 2012 Fluids Engineering Division Sum, ICNMM 2012

Dynamic wetting on moving surfaces: A molecular dynamics study

We present a molecular dynamics (MD) methodology to investigate temperature-driven gas flow through a nanochannel connecting two reservoirs at different temperatures. The temperature gradient along the channel drives the macroscopic movement of gas molecules from the cold to the hot regions, in the phenomenon known as thermal transpiration. Temperature, density and pressure profile variations along the axial direction are measured at various rarefaction conditions, from the slip flow to the free molecular regimes. Pressure and density evolutions are monitored as a function of time in both the hot and cold reservoirs. The normalized thermo molecular pressure difference (TMPD) values are calculated and compared with kinetic theory results. Comparisons show good agreement up to the early transition regime, although beyond this our numerical experiments over-predict the theoretical results. These numerical experiments are useful because experimental data is scarce for nano devices and these molecular simulations can mimic realistic conditions. © 2012 American Institute of Physics.

AIP Conference Proceedings

Thermal transpiration of nanoscale gas flow

We present an analytical model predicting the experimentally observed gas mass flow rate in rectangular micro channels over slip and transition regimes without the use of any fitting parameter. Previously, Sone reported a class of pure continuum regime flows that requires terms of Burnett order in constitutive equations of shear stress to be predicted appropriately. The corrective terms to the conventional Navier-Stokes equation were named the ghost effect. We demonstrate in this paper similarity between Sone ghost effect model and newly so-called 'volume diffusion hydrodynamic model'. A generic analytical solution to gas mass flow rate in a rectangular micro channel is then obtained. It is shown that the volume diffusion hydrodynamics allows to accurately predict the gas mass flow rate up to Knudsen number of 5. This can be achieved without necessitating the use of adjustable parameters in boundary conditions or parametric scaling laws for constitutive relations. The present model predicts the non-linear variation of pressure profile along the axial direction and also captures the change in curvature with increase in rarefaction. © 2012 American Institute of Physics.

Transition regime analytical solution to gas mass flow rate in a rectangular micro channel

In this paper we investigate the problem of isothermal pressure driven gas flow through a channel using a continuum equations set with a mass diffusion correction to the mass-density equation. The additional term is invoked as a way of releasing the local-equilibrium assumption in the presence of strong local gradients. Then the dissipative mass flux is proportional to local density/pressure gradient which is constant in a cross section and results in a correction to the velocity profile. Subsequently, better mass-flow rate predictions are obtained than those obtained using Maxwell-type first order slip boundary conditions. Results indicate that the mass diffusion correction captures the "Knudsen paradox" and change in curvature of the streamwise pressure profile without the need for setting higher order slip boundary conditions. © 2011 American Institute of Physics.

Isothermal micro-channel gas flow using a hydrodynamic model with dissipative mass flux

We propose a power-law based effective mean free path (MFP) model so that the Navier-Stokes-Fourier equations can be employed for the transition-regime flows typical of gas micro/nanodevices. The effective MFP model is derived for a system with planar wall confinement by taking into account the boundary limiting effects on the molecular free paths. Our model is validated against molecular dynamics simulation data and compared with other theoretical models. As gas transport properties can be related to the mean free path through kinetic theory, the Navier-Stokes-Fourier constitutive relations are then modified in order to better capture the flow behavior in the Knudsen layers close to surfaces. Our model is applied to fully developed isothermal pressure-driven (Poiseuille) and thermal creep gas flows in microchannels. The results show that our approach greatly improves the near-wall accuracy of the Navier-Stokes-Fourier equations, well beyond the slip-flow regime. © 2011 American Society of Mechanical Engineers.

Fulltext

Journal of Fluids Engineering, Transactions of the ASME

Modeling of Knudsen layer effects in micro/nanoscale gas flows

We investigate a power-law form for the probability distribution function of free paths of dilute gas molecules in a confined region. A geometry-dependent effective molecular mean free path (MFP) model is then derived for a planar wall confinement, by taking into account the boundary limiting effects on the molecular paths. The power-law based effective MFP is validated against molecular dynamics simulation data and compared with exponential effective MFP models. The Navier-Stokes constitutive relations are then modified according to the kinetic theory of gases i.e. transport properties can be described in terms of the free paths which the molecules describe between collisions. Results for isothermal pressure-driven Poiseuille gas flows in micro-channels are reported, and we compare results with conventional hydrodynamic models, solutions of the Boltzmann equation and experimental data. © 2011 American Institute of Physics.

Behaviour of microscale gas flows based on a power-law free path distribution function

Fluid slip has been observed experimentally in micro- and nanoscale liquid flow devices by several investigators. While observations of fluid slip continue to expand, the generating mechanism responsible for fluid slip is not well understood and indeed generalized mathematical formulation is not available. In the present paper, the author gave an attempt to explain the generating mechanism for the fluid slip on hydrophobic surface. The importance of the present theory lies in the fact that it obviates the need to impose the ad hoc Newtons slip at the fluid-wall interface and also the pre-assumption of thin gas layer close to the wall. Surface interactions with the liquid/fluid at molecular scale are incorporated together with the phase field theory to accurately predict the phase of the fluid close to the wall, which is imperative to accurately determine the fluid slip close to the wall. It is noticed that the incorporation of these molecule-surface interactions have significant effect on the resulting coating windows on both hydrophobic and hydrophilic substrates, however it is more predominant for the hydrophobic one. © 2010 Elsevier B.V.

Chemical Engineering and Processing: Process Intensification

Liquid slip/stick over hydrophobic/hydrophilic surfaces and their implications in coating processes

We present the results of investigations into the distribution of molecular free paths in rarefied gases using molecular dynamics simulations. Our tests on a range of different gas densities and confinements (unbounded, single bounding wall and parallel bounding walls) indicate that the molecules perform Lévy-type flights, irrespective of the presence of a bounding wall. The free paths most closely follow a power-law distribution. Simulations of gases confined by planar surfaces indicate that the local molecular mean free path varies sharply close to a solid surface. These results may yield new insight into diffusive transport in rarefied gases, in particular, the constitutive behaviour of gas flows in micro- and nanoscale devices. © 2011 IOP Publishing Ltd.

Journal of Physics D: Applied Physics

Molecular free path distribution in rarefied gases

We investigate whether a power-law form of probability distribution function better describes the free paths of dilute gas molecules in a confined system. An effective molecular mean free path model is derived, which allows the mean free path to vary close to bounding surfaces. Our model is compared with molecular dynamics simulation data, and also other classical mean free path models. As gas transport properties can be related to the mean free path through kinetic theory, the Navier-Stokes constitutive relations are then modified and applied to various benchmark test cases. Results for isothermal pressure-driven Poiseuille flows in micro-channels are reported, and we compare our results with conventional hydrodynamic models, solutions of the Boltzmann equation, and experimental data. Our new approach provides good results for mean free path and crosssectional flow velocity profiles up to Knudsen numbers around 1, and for integral flow parameters such as flow rate and friction factor up to Knudsen number of 10. We discuss some limitations of our power-law model, and point to the way forward for further development. Copyright © 2010 by ASME.

ASME 2010 8th International Conference on Nanochannels, Microchannels, and Minichannels Collocated with 3rd Joint US-European Fluids Engineering Summer Meeting, ICNMM2010

The importance of mean free path in determining gas micro flow behaviour

We test an extended continuum-based approach for analyzing micro-scale gas flows over a wide range of Knudsen number and Mach number. In this approach, additional terms are invoked in the constitutive relations of Navier-Stokes-Fourier equations, which originate from the considerations of phoretic motion as triggered by strong local gradients of density and/or temperature. Such augmented considerations are shown to implicitly take care of the complexities in the flow physics in a thermo-physically consistent sense, so that no special boundary treatment becomes necessary to address phenomenon such as Knudsen paradox. The transition regime gas flows, which are otherwise to be addressed through computationally intensive molecular simulations, become well tractable within the extended quasi-continuum framework without necessitaing the use of any fitting parameters. Rigorous comparisons with direct simulation Monte Carlo (DSMC) computations and experimental results support this conjecture for cases of isothermal pressure driven gas flows and high Mach number shock wave flows through rectangular microchannels. © Springer-Verlag 2010.

Microfluidics and Nanofluidics

Predicting microscale gas flows and rarefaction effects through extended Navier-Stokes-Fourier equations from phoretic transport considerations

Despite the enormous scientific and technological importance of micro-channel gas flows, the understanding of these flows, by classical fluid mechanics, remains incomplete including the prediction of flow rates. In this paper, we revisit the problem of micro-channel compressible gas flows and show that the axial diffusion of mass engendered by the density (pressure) gradient becomes increasingly significant with increased Knudsen number compared to the pressure driven convection. The present theoretical treatment is based on a recently proposed modification (Durst et al. in Proceeding of the international symposium on turbulence, heat and mass transfer, Dubrovnik, 3-18 September, pp 25-29, 2006) of the Navier-Stokes equations that include the diffusion of mass caused by the density and temperature gradients. The theoretical predictions using the modified Navier-Stokes equations are found to be in good agreement with the available experimental data spanning the continuum, transition and free-molecular (Knudsen) flow regimes, without invoking the concept of Maxwellian wall-slip boundary condition. The simple theory also results in excellent agreement with the results of linearized Boltzmann equations and Direct Simulation Monte Carlo (DSMC) method. Finally, the theory explains the Knudsen minimum and suggests the design of future micro-channel flow experiments and their employment to complete the present days understanding of micro-channel flows. © Springer-Verlag 2008.

Pressure-driven diffusive gas flows in micro-channels: From the Knudsen to the continuum regimes

In this paper we provide solution of the Navier-Stokes equations for gaseous slip flow in long microchannels with a second-order accurate slip boundary condition at the walls. The obtained solution is general enough to allow evaluation of various slip models proposed in the literature. We compare our solution against the first-order accurate slip boundary condition and show that the solution has a weak dependence on Reynolds number, which was neglected in the earlier theory. It is emphasized that first-order slip models do not predict the "Knudsen paradox" (appearance of a minima in normalized volume flux at Knudsen number approximately unity), or a change in curvature of centerline pressure at Knudsen numbers of 0.16. A comparison with Boltzmann's solution suggests that the derived solution agrees reasonably well up to Knudsen number approximately 5, which shows that the validity of Navier-Stokes to rarefied gases can possibly be increased by using a high order slip boundary condition and proper choice of the slip coefficients. This result is significant from the perspective of numerical simulations of rarefied gases. © 2007 Elsevier Ltd. All rights reserved.

Analytical solution of gaseous slip flow in long microchannels

Slot coaters are widely used in the coating industry and they are employed as dies that can be operated in their bead coating mode and their curtain coating mode. The bead mode is usually used for low coating velocities and the curtain mode for high velocities. Slot coaters, operating in their bead mode, usually provide coated layers of high quality. The minimum requirement for stable coating can be expressed as a requested pressure distribution between the lip of the slot coater and the substrate. The computer program ASC-B-100 has been setup at FMP Technology GmbH to compute permissible ranges of pressure differences as a function of web speed. Shear thinning and shear thickening liquids are introduced with their rheological properties being expressed by analytical functions or by computed values obtained through experiments.