Literature review

Micro- and nano-channels are widely used nowadays, examples can be easily found in medicine, biotechnologies, telecommunication and etc. The most obvious application for nano-channels is gas separation[10-13]. The specific examples of devices related to micro flow are inkjet printheads, microchannel-based chemical

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reactors, pressure sensors, micro total analysis system(μTAS) or "lab on a chip", micropumps, and Knudsen pumps [17].

Investigation of rarefied gas caused by temperature and pressure gradient as basis of a vacuum pump has been a long history since the early 1900s. The phenomenon of temperature driven flow was first explained by Reynolds[18] and contemporaneously investigated theoretically by Maxwell[19]. In early 1909 Martin Knudsen has published paper[20] dealing with molecular flow and effusion through orifices. This study introduces the idea of treating flow as impedance in the electrical sense so that one could derive a combination of a tube and an orifice in terms of their flow resistance summation. This idea was supported by Dushman [21] who stated that the flow resistance of a long tube is the sum of the resistances of the entrance opening and that of a short tube. In 1932 Clausing has introduced the concept of transmission probability on preference to flow resistance (or conductance)[22]. Further evaluation of transmission probabilities was conducted by De Marcus and Hopper to obtain approximate solutions to the Clausing-type integral equations[23]. Development of computers has allowed one to implement another useful approach, particularly well-suited for complex shapes and systems of baffles, traps, etc, is the statistical one using Monte Carlo techniques. This was described by Davis[24] and many publications since then.

One of the earliest experimental studies of gas flow through copper membrane has been conducted by Warrick and Mack [25] in 1933. Hanks and Weissberg [26] have proposed semi-empirical equation for the pressure driven flow through a circular channel. This relation was recently tested by Shinagawa et al.[27] and it was found to be valid in the range of the continuum to the upper limit of transition regime.

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In 1973 Borisov et al. [28] have studied dependence of thermomolecular pressure difference on the average pressure. Experimental studies have been conducted for different gases (He, Ne, H2 and D2) in the continuum flow regime. Researches have experimentally proven that in the continuum flow regime the value of factor γ in eq.

(1.2-13) tends toward zero. Results are in line with theoretical studies by Sone et al.

[29].

Fujimoto and Usami [30] have reported the experimental studies of gas flow through orifices or short channels over the entire flow regime from continuum to free molecular. Unfortunately all experiments have been carried out for the case of high pressure difference and most results are shown in figures that allow only quantitative comparison. Sreekanth [31] has conducted experimental studies of rarefied gas flow through short metal nano-channels in a wide range of pressure ratios. That paper proposes semi-empirical equation for estimation of mass flow rate:

( )

where <V> is the average molecular velocity; μ – viscosity of the gas; P and T are the pressure and temperature of the gas, respectively. Variables with subscripts 1 and 2 correspond to the parameters of gas in the upstream and downstream reservoirs. This relation was obtained for the case of transition flow through the extremely short tubes (Lch/dch<1 length to diameter ratio).

Works by Johnsen and Chatterjee [32] deal with experimental studies of binary gas mixtures flow through a small thin orifice (thickness ~10% of diameter); a limited range of the transition regime from molecular to continuum flow has been covered. The observed density changes of gas species are found to depend strongly on the mass ratio

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of the two flowing gases. Authors have derived a semi-empirical formula that estimates the density changes.

Recently Marino [33] has carried out experiments in order to evaluate the conductance of tubes of circular cross section as a function of Knudsen number. The main impact of this study is tabulated data of conductance of long tubes. The results discussed in this manuscript have good correlations with theoretical [34] and experimental [31] studies by other researches.

Setup of good experiments related to rarefied gas flow through micro- or nano-channels is very expensive and time consuming issue, but sometimes experiments has problem with no appropriate solution. The complicity is caused by spatial resolution since measurement in a nano-channel with lengthscale 10^-9m requires a sensor with 10

-10m scale at least. Another problem is that sensor measurements are sensitive to disturbances. Therefore, it is impossible to put a probe inside a micro-channel to measure velocity, temperature or pressure field directly. All this issues create obstacles on the way of obtaining reliable information about micro- and nano-flows. Fortunately, recent achievements of mathematics, numerical simulation methods and computer science allowed us to investigate all these tiny effects in a wide range of parameters of interest.

It is obvious that flows through membranes are more interesting from the practical point of view, but theoretical study of such complicated systems is very sophisticated.

Commonly numerical or analytical investigation of gas flows through the membrane are performed considering single channel or small set of parallel channels with further extrapolation of results to the case of membrane using correction factors [35].

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There are a lot of studies using both temperature and pressure driven flows through long channels, for example works by Sharipov [36]. This paper is related to calculations of the mass flow rate of a rarefied gas through a long capillary caused by small pressure and temperature gradients based on the s-model (a generalization of the Bhatnagar-Gross-Krook model [37], see for example [38]) for the diffuse specular gas-surface interaction in the range of the rarefaction parameter δ (δ~1/Kn) from 0.005 to 50. Author of Ref.[36] has determined that the nonlinear thermomolecular pressure difference does not depend on the shape of temperature distribution along the capillary in case of free molecular flow regime. The nonlinear thermal creep has been calculated for two different temperature distributions (linear and nonlinear) along the capillary. It has been shown that the application of the linear theory (this theory is based on the assumption of constant values of flow coefficients along the channel) to the gas flow at a large temperature ratio gives a significant error. Another work by Sharipov [39] was concentrated on research of rarefied gas flow through a long rectangular channel caused by both pressure and temperature differences. Author has proposed equations that allow one calculate a mass flow rate through a long rectangular channel. It was pointed out that the numerical results on pressure driven flow can be used for any type of gas, including polyatomic, while the data on the thermal creep can only be used for monoatomic gases. The thermal creep must be recalculated for any specific polyatomic gas by applying an appropriate kinetic equation. Unfortunately author of the last two works proposed relations for a long channel without specification of the “long” term, thus it is to be determined. Shen et al. [40] have simulated rarefied gas flows through micro-channels using information preservation method (IP) and the direct simulation Monte Carlo (DSMC) method. This study deals with stream-wise pressure distributions and mass fluxes through micro-channels given by IP method agree well with

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experimental data measure in long micro-channels by Arkilic et al. [41]. It was found that IP and DSMC calculations are able to represent flow behaviors through short micro-channel over the entire flow regime from continuum to free molecular, whereas the slip Navier-Stokes solution fails to predict it.

Paper [15] represents comprehensive numerical study of a gas flow through a long pipe with elliptic cross section. The flow considered in the work is non-isothermal, and all computations have been performed for the cases of small temperature and pressure gradients along the channel. All parameters of interest are studied on the basis of S-model kinetic equation. The main result on this work is proposed a data set which relates dependence between rarefaction parameter and non-dimensional flow rate.

Additionally this work proposes the relation allows one to compute both temperature and pressure driven flow for the cases with large gradients.

Titarev and Shakhov [42] have developed a conservative numerical method for solving the linearized S model kinetic equation. The method is intended for channels with an arbitrary cross section in the entire range of Knudsen numbers. The gas flow rate for a channel cross section specified in the form of a regular polygon was numerically computed in a wide range of rarefaction parameters. Investigations have been performed for the case of infinite channel. It was shown that the solution of the problem converges quadratically in the number of cross section sides to the limiting case of a circular pipe. The main result is that for triangular and quadrilateral cross sections, the mass flow rates and the heat fluxes were found to differ substantially from the case of a circular pipe.

On the other hand the gas flows through extremely short channels (slits and orifices) have been intensively studied as well, see for example work of Lilly et al. [43].

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Efforts of the authors were mainly concentrated on determination of propulsion properties of orifices and short channels (thick orifices). It has been found that the effect of surface secularity on a thick orifice specific impulse was found to be relatively small.

Authors of works [44-46] have done analytical and numerical studies of rarefied gas flow through slits and orifices of different shapes. Non-isothermal gas flow is discussed [44]; They proposed equations allow one to estimate flow rate of temperature driven flow through slit over a wide range of Knudsen number.

Research described in [45] represents studies of 2D rarefied gas flow through the thin slit into the vacuum by using the BGK and S-model equations. The obtained results have good agreement with similar studies performed using DSMC method. This paper presents numerical data on the flow rate and distributions of density, bulk velocity and temperature along the symmetry axis. According to the results listed in the paper, the mass flow rate increases with the increase of rarefaction parameter, and it reaches a constant value when the flow regime becomes continuous.

Rarefied gas flow between two containers through a thin slit is studied on the basis of the direct simulation Monte Carlo method [46]. The flow rate and flow field are calculated over the whole range of gas rarefaction for various values of the pressure ratio. It is found that at all values of the pressure ratio a significant variation of the flow rate occurs in the transition regime between the free-molecular and continuum regimes.

DSMC method has been widely used for investigations of rarefied gas flow through micro- and nano-channels, for example [15, 34, 47-50]. Varoutis et al. have investigated pressure driven flow through a long circular pipe [48]. This work covers wide range of flow regimes (rarefied flow and continuum flow, flow to the vacuum and flow between two containers with different pressures) through the different types of channel (L/R was

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varied from 0.1 to 10). It has been found that the rarefaction parameter has the most significant effect on the flow field characteristics and patterns, followed by the pressure ratio drop, while the length-to-radius ratio has a rather modest impact. Several interesting findings have been reported including the behavior of the flow rate and other macroscopic quantities in terms of these three parameters. In addition, the effects of gas rarefaction on the choked flow at large pressure drops is discussed.

In contrast to the previous papers, the work [34] describes studies of rarified gas flow through short tubes into vacuum. Authors used DSMC method to investigate the phenomenon of interest. The intermolecular potential was modeled using the hard-sphere (HS) and the variable hard hard-sphere (VHS) models. Computations are performed for various L/R ratios and rarefaction parameters. It has been shown the mass flow rate is strongly sensitive to the choice of gas-surface interaction model, meanwhile the intermolecular potential does not influence on the flow rate significantly. Results described in this paper have been supported experimentally by Marino [33].

Almost all studies mentioned above have been conducted using kinetic model or DSMC method, it should be noted that there is a relatively new method simulation technique for complex fluid systems called Lattice-Boltzmann method (LBM). Instead of solving Navier–Stokes equations, the discrete Boltzmann equation is solved to simulate the flow of Newtonian fluid with collision models such as Bhatnagar-Gross-Krook (BGK). This method has been successfully applied for the investigation of micro and nano-channel flow[51].

The paper by Ghazanfarian and Abbassi [52] deals with two-dimensional numerical simulation of gaseous flow and heat transfer in planar microchannel and nanochannel with different wall temperatures in transitional regime 0.1<Kn<1. An

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atomistic molecular simulation method was used known as thermal lattice-Boltzmann method. The simulation results of thermal lattice- Boltzmann method in the described study shows that this method is capable of modeling shear-driven, pressure-driven, and mixed shear–pressure-driven rarified flows and heat transfer up to Kn=1 in the transitional regime.

Taguchi and Charrier [53] have studied steady rarefied gas flow through periodic porous media kept at an uniform temperature. It was considered on the basis of Bhatnagar–Gross–Krook equation with the diffuse reflection condition on the solid boundary. Authors have derived, by homogenization, a fluid model that describes the global pressure distribution as well as the mass-flow rate. The derived model has a simple form and can be used as a practical tool. A simple application of the derived model to an isothermal flow through an array of circular cylinders induced by a pressure difference is presented

Despite the increasing popularity of LBM in simulating complex fluid systems, this novel approach has some limitations. At present, high Mach number flows in aerodynamics are still difficult for LBM, and a consistent thermo-hydrodynamic scheme (scheme that couples heat and mass transfer processes together) is absent. However, as with Navier–Stokes based CFD, LBM methods have been successfully coupled to thermal-specific solutions to enable heat transfer (solids-based conduction, convection and radiation) simulation capability. For multiphase/multicomponent models, the interface thickness is usually large and the density ratio across the interface is small when compared with real fluids. Nevertheless, the wide applications and fast advancements of this method during the past twenty years have proven its potential in computational physics, including microfluidics: LBM demonstrates promising results in the area of high Knudsen number flows. Sbragaglia et al.[54] have specialize the

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Boltzmann kinetic equation to describe the wetting/dewetting transition of fluids in the presence of nanoscopic grooves etched on the boundaries. This approach permits one to retain the essential supramolecular (it is well defined complex of molecules held together by noncovalent bonds) details of fluid-solid interactions without surrendering—actually boosting—the computational efficiency of continuum methods.

The method is used to analyze the importance of conspiring effects between hydrophobicity and roughness on the global mass flow rate of the microchannel. The mesoscopic method was also validated quantitatively against the molecular dynamics results by Cottin-Bizonne et al. [55].

Molecular Dynamics method is a simulation approach based on fundamental laws of mechanics that allows one consider behavior of every single atom/molecule of system of interest. In contrast to DSMC, it allows one to describe non-equilibrium processes, moreover MD method is able to reproduce complex mixtures of gases and/or liquids, while the LBM method experiences some problems in this case. Mi et al. [56]

have applied MD simulations to study nanochannel flows at low Reynolds numbers. A simple fluid flowing through channels of different shapes at nanoscale level is investigated. Comparing velocities and other flow parameters obtained from MD simulations with those predicted by the classical Navier-Stokes equations at same Reynolds numbers, researchers found that both results agree with each other qualitatively in the central area of a nanochannel. However, large deviation of flow field usually exists in areas near the wall. For certain complex nanochannel flow geometry, MD simulations reveal the generation and development of nano-size vortices due to the large momenta of molecules in the near-wall region while the traditional Navier-Stokes equations with the non-slip boundary condition at low Reynolds numbers cannot predict

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the similar phenomena. It is shown that although Navier-Stokes equations are still partially valid, they fail to give whole details for nanochannel flows.

Okumura and Heyes [57] have compared the results of three-dimensional MD simulations of a Lennard-Jones (LJ) liquid with a hydrostatic (HS) solution of a high temperature liquid channel which is surrounded by a fluid at lower temperature.

Because the systems were in stationary non-equilibrium states with no fluid flow, both MD simulation and the HS solution gave flat velocity profiles for a normal pressure in all temperature-gradient cases. However, the other quantities showed differences between these two methods. The MD-derived density was found to oscillate over the length of 8 LJ particle diameters from the boundary plane in the system with the infinite temperature gradient, while the HS-derived density showed simply a stepwise profile.

The MD simulation also showed another anomaly near the boundary in potential energy. Authors have found systems in which the HS treatment works well and those where the HS approach breaks down, and therefore established the minimum length scale for the HS treatment to be valid.

Molecular dynamics-continuum hybrid simulation method has been proposed by Sun et al.[58] for studies of influence of roughness and thermal boundaries effect on flow in microchannel/nanochannel. The results indicate that the molecules in the wall-neighboring area can be firmly confined in the concaves due to geometric structure and strong liquid–solid interaction and cause locking boundary in the velocity profile and linear gradient in the temperature profile. The locked boundary can further lead to negative slip length, which varies in power law with channel height. Authors of the Ref.

[58] have sown that the linear temperature gradient, as well as nearly constant temperature jump, can lead to obviously increasing Kapitza length (Kapitza length

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represents the length of a material of a given thermal conductivity providing an equivalent thermal resistance as the interface[59]) versus channel height.

When using the molecular dynamics simulation method to study low speed nanoscale flow problems, a major difficulty is the extraction of the true flow velocity because of the highly nonlinear coupling of the low bulk flow velocity and the high velocity of molecules’ thermal motion. In all published papers the reported flow velocity is the average value of the sum of these two velocities over time. For high speed flow problems the conventional MD method can give satisfactory result.

However, when the flow velocity is much smaller than the thermal velocity, the conventional molecular dynamics simulation method cannot predict the true flow velocity. To overcome this difficulty, Zhang et al. [60] have developed a new linearized algorithm. The new algorithm separates the flow velocity increment caused by external

However, when the flow velocity is much smaller than the thermal velocity, the conventional molecular dynamics simulation method cannot predict the true flow velocity. To overcome this difficulty, Zhang et al. [60] have developed a new linearized algorithm. The new algorithm separates the flow velocity increment caused by external