Royal Society Publishing

Velocity profile in the Knudsen layer according to the Boltzmann equation

Charles R Lilley, John E Sader

Abstract

Flow of a dilute gas near a solid surface exhibits non-continuum effects that are manifested in the Knudsen layer. The non-Newtonian nature of the flow in this region has been the subject of a number of recent studies suggesting that the so-called ‘effective viscosity’ at a solid surface is half that of the standard dynamic viscosity. Using the Boltzmann equation with a diffusely reflecting surface and hard sphere molecules, Lilley & Sader discovered that the flow exhibits a striking power-law dependence on distance from the solid surface where the velocity gradient is singular. Importantly, these findings (i) contradict these recent claims and (ii) are not predicted by existing high-order hydrodynamic flow models. Here, we examine the applicability of these findings to surfaces with arbitrary thermal accommodation and molecules that are more realistic than hard spheres. This study demonstrates that the velocity gradient singularity and power-law dependence arise naturally from the Boltzmann equation, regardless of the degree of thermal accommodation. These results are expected to be of particular value in the development of hydrodynamic models beyond the Boltzmann equation and in the design and characterization of nanoscale flows.

Keywords:

1. Introduction

Micro- and nanoscale systems are currently the subject of intense research and development. Gas flows in such systems often exhibit rarefied flow effects, which arise when the mean free path of gas molecules becomes significant relative to some dimension that characterizes the system. Examples include gas flows in microchannels, which are the basis for numerous micro-electro-mechanical systems flow sensing and control applications (Ho & Tai 1998), thermal force effects on microcantilevers (Gotsmann & Durig 2005) and the thermal transpiration principle upon which the solid-state Knudsen compressor (Sone 2002; McNamara & Gianchandani 2005) is based.

Accurate simulations of such rarefied flow effects are vital for the efficient design, optimization and future development of micro- and nanodevices. Conventional Navier–Stokes solutions of rarefied flows are inaccurate because rarefied flows are characterized by non-equilibrium distributions of molecular velocities, thus violating the near-equilibrium assumption that is the underlying basis of the Navier–Stokes description. Accurate rarefied flow modelling requires solutions of the more fundamental Boltzmann equation. For rarefied flows of ‘real world’ interest, the Boltzmann equation is usually solved numerically with Bird's direct simulation Monte Carlo (DSMC) method (Bird 1994). However, for many applications of current interest, DSMC calculations can impose prohibitive computational demands. This is exemplified by the work of Reese et al. (2003), who report that a two-dimensional DSMC calculation of rarefied air flow around a moving microcantilever required 24 hours on a parallel supercomputer with 3000 processors. Such intensive computational demands have motivated recent interest in the application of high-order hydrodynamic models to simulate rarefied flows (Reese et al. 2003; Guo et al. 2006; Gu & Emerson 2007; Mizzi et al. 2007; Struchtrup & Torrilhon 2007; Torrilhon & Struchtrup 2008), with the expectation that such models may be employed in computational fluid dynamics solvers to accurately capture rarefaction effects with much less computational effort than comparable DSMC calculations.

An important consideration in studying gas flows in micro- and nanoscale systems is the Knudsen layer, which is a rarefaction effect that extends to a distance the order of one mean free path from a solid surface. Within the Knudsen layer, molecules collide with the surface more frequently than they collide with each other (Gallis et al. 2006). This produces a distribution of molecular velocities that is perturbed significantly from the equilibrium Maxwellian state, and results in two important rarefaction phenomena: first, the gas at the surface has a finite velocity relative to the surface, known as the slip velocity. Second, the gas near the surface exhibits non-Newtonian behaviour. For micro- and nanoscale flows, which have characteristic dimensions of the order of several mean free paths, the Knudsen layer occupies a large portion of the flow and can therefore dominate the flow behaviour. A detailed knowledge of the Knudsen layer structure is thus essential for modelling rarefied flows in micro- and nanoscale systems.

The structure of the Knudsen layer has been studied extensively (Bardos et al. 1986; Cercignani 2000; Sone 2002; Gu & Emerson 2007; Mizzi et al. 2007; Struchtrup & Torrilhon 2007; Torrilhon & Struchtrup 2008). Recently, Lilley & Sader (2007) used existing solutions of the linearized Boltzmann equation (LBE) and precise DSMC calculations to examine the structure of the Knudsen layer in detail for shear flow past a solid wall. They discovered that the bulk gas velocity u parallel to the surface is accurately described by the remarkably simple power-law behaviourEmbedded Image(1.1)where y is the normal distance from the surface and α≈0.8. This result applies for hard sphere molecules near a diffusely reflecting surface, which corresponds to full thermal accommodation.

This power-law behaviour prevails to a distance of about one mean free path from the surface, and thus describes the inner portion of the Knudsen layer. Importantly, equation (1.1) establishes the existence of a velocity gradient singularity at the surface. This prediction of a singularity is consistent with the work of Willis (1962) and Sone (2002), who rigorously examined the linearized (approximate) Bhatnagar–Gross–Krook (BGK) model equation (Bhatnagar et al. 1954) and discovered a singularity of logarithmic form du/dy∝ln y at the surface. The work of Lilley & Sader (2007) establishes that a singularity also exists in the full Boltzmann equation for hard spheres. This singularity is not captured by existing hydrodynamic models and contradicts recent work (Fichman & Hetsroni 2005; Lockerby et al. 2005a) that predicts a finite velocity gradient at the surface.

In this paper, we first examine the structure of the Knudsen layer for a hard sphere gas near a surface with partial thermal accommodation using the existing LBE solutions supported by precise DSMC calculations. These investigations establish that both the above power-law description and velocity gradient singularity are present in the Knudsen layer under these more general conditions. We then harness the versatility of the DSMC method to investigate the Knudsen layer for a more realistic gas model with partial thermal accommodation, and again show that the power-law description is accurate and that the singularity exists. These results corroborate and extend the applicability of the power-law description (Lilley & Sader 2007) of the Knudsen layer.

2. Background

In this paper, we study the Knudsen layer for Kramers' problem (Kramers 1949), which is illustrated in figure 1. This problem considers the unidirectional isothermal motion of a gas filling a half-space bounded by a stationary planar solid surface. The Kramers' problem is often considered in fundamental studies of the Knudsen layer, and has been researched extensively (e.g. Cercignani (2000) and Sone (2002) and references therein). The only bulk flow gradient in the Kramers' problem is du/dy, where u is the velocity component parallel to the surface and y is the normal distance from the surface. As y→∞, du/dy tends to the constant value a. Consistent with existing work on the shear-thinning nature of gases (Montanero et al. 2000; Garzó & Santos 2003), we adopt the notion of an ‘effective viscosity’ to describe the non-Newtonian behaviour inherent in the Kramers' problem. This effective viscosity μ is defined byEmbedded Imagewhere the shear stress τ is constant in the Kramers' problem. We emphasize that the effective viscosity is a mathematical construct with no connection to real gas properties, and its value will change with flow geometry (Hadjiconstantinou 2006). It is adopted here for convenience, ease of discussion and consistency with previous works.

Figure 1

Schematic of Kramers' problem. Here, y is the normal distance from the stationary solid surface at y=0 and u(y) is the bulk gas velocity in the direction parallel to the surface. The so-called macroscopic slip velocity is extrapolated from the linear velocity profile outside the Knudsen layer where y→∞, and differs from the true slip velocity u(0). ξ is called the slip length.

We normalize the flow speed u according toEmbedded ImageThe nominal mean free path λnom is given byEmbedded Imagewhere n and m are the molecular number density and mass; k is Boltzmann's constant; and T is the temperature. Here, μ[1] is the first viscosity approximation from the Chapman–Enskog solution of the Boltzmann equation (Chapman & Cowling 1970), which depends upon the molecular model. For hard spheres Embedded Image, where A is the hard sphere cross section. We normalize y according toEmbedded Imageand define the non-dimensional slip coefficient Embedded Image byEmbedded Image

Solutions of the Kramers' problem must consider the non-equilibrium distribution of molecular velocities in the Knudsen layer and hence must solve the Boltzmann equation, as noted in §1. The Boltzmann equation provides a rigorous description of a dilute gas and describes the gas behaviour in terms of the temporal evolution and spatial variation of a general molecular velocity distribution function f. For a steady flow in the absence of body forces, the Boltzmann equation for a monatomic gas isEmbedded ImageHere, f(x, v) is the distribution of molecular velocities v that depends upon the position vector x. The collision term [∂f/∂t]coll is a nonlinear integral expression that describes the change in f due to intermolecular collisions.

The nonlinear integro-differential form of the Boltzmann equation poses formidable challenges to solution by analytical methods. Indeed, complete closed-form solutions have not been found, even for the simple flows like Kramers' problem. The major difficulties arise from the collision term. For weakly non-equilibrium flows like Kramers' problem, the collision term can be linearized, yielding the LBE. The LBE retains the most important physical characteristics of the full Boltzmann equation, yet is tractable for many problems (Cercignani 1988). Computational tools offer the only practical means of solving the full nonlinear Boltzmann equation (see Cercignani 2000, p. 114), the most common being Bird's DSMC method (Bird 1994). The DSMC method captures macroscopic gas behaviour by modelling a set of simulator particles, which represent the real gas molecules, as they undergo simulated intermolecular collisions, interact with solid surfaces and move through physical space. Proofs that the DSMC method solves the Boltzmann equation have been provided by Wagner (1992) and Pulvirenti et al. (1994). Here, we consider the existing LBE solutions of the Kramers' problem in §3, and precise DSMC solutions in §4.

An important aspect of the Kramers' problem is the interaction between gas molecules and the surface. Because the physical details of such gas–surface interactions are complex, Maxwell's simple boundary condition (Maxwell 1879) is often used. Under this condition, a fraction σ of molecules are reflected diffusely from the surface, meaning that the reflected molecules have velocities distributed according to the equilibrium distribution at the surface temperature Twall. The remaining fraction 1−σ of molecules is reflected specularly, meaning that their velocity components normal to the surface are simply reversed upon reflection. Here, the parameter σ is called the thermal accommodation coefficient and ‘full’ and ‘partial’ thermal accommodation refer to cases with σ=1 and σ<1, respectively.

3. LBE solutions for hard sphere molecules

The LBE for Kramers' problem can be written (Cercignani 1988) asEmbedded Imagewhere Embedded Image denotes the linearized collision operator. The molecular velocity Embedded Image has components vx parallel to u, vy oriented in the y-direction, and vz orthogonal to vx and vy. The perturbation function h is given byEmbedded Imagewhere f* is the absolute Maxwellian distribution.

Numerical solutions of the LBE for Kramers' problem with hard sphere molecules have been published by Loyalka & Hickey (1989, 1990), Ohwada et al. (1989) and Siewert (2003). These solutions all consider full thermal accommodation (σ=1). Loyalka & Hickey (1990) and Siewert (2003) also provided solutions for partial accommodation (σ<1).

Lilley & Sader (2007) examined the LBE velocity solutions for σ=1. Specifically, they studied these solutions in the asymptotic limit as Embedded Image, by analysing the behaviour of Embedded Image versus Embedded Image on a double logarithmic scale. This plotting scheme reveals striking linearity, as shown in figure 2, and immediately leads to the power-law velocity description of equation (1.1). In terms of the normalized quantities Embedded Image and Embedded Image, this power law can be written asEmbedded Image(3.1)The corresponding effective viscosity is thenEmbedded Imagewhere μ is the standard dynamic viscosity.

Figure 2

Published LBE solutions of Kramers' problem for σ=1 compared with the power-law description of equation (3.1). This power-law description is fitted to the LBE solution of Ohwada et al. (1989), and is indistinguishable from fits to the other three LBE solutions shown here. Circles, Ohwada et al. data; solid line, power-law fit to Ohwada et al. data; triangles, Loyalka & Hickey (1989) data; squares, Loyalka & Hickey (1990) data; asterisks, Siewert data.

We emphasize that the Knudsen layer extends beyond Embedded Image, and there does not exist a strict demarcation between the Knudsen layer and the outer (external) flow region. Indeed, the Knudsen layer decays asymptotically into the outer flow region. As such, the regions marked as ‘Knudsen layer’ and ‘external flow’ around Embedded Image in figures 2, 6 and 7 are given as a guide to the eye only. Importantly, the power-law description is only valid in the region Embedded Image, as is clear from figure 2, which coincides with the inner part of the Knudsen layer.

For the hard sphere LBE solutions with σ=1, Lilley & Sader (2007) calculated the fit parameters C and α by linear regression analysis of the log data for Embedded Image. These parameters are shown in table 1, together with Embedded Image, the slip coefficients Embedded Image and the sample correlation coefficients r. In every case, r is very close to unity, demonstrating the accuracy of the power-law description within the Knudsen layer. Importantly, all LBE solutions have α distinctly less than unity. Since λ is the only natural length scale in a dilute gas flow, this power law establishes that the velocity gradient is singular at the wall Embedded Image where the corresponding effective viscosity of zero.

View this table:
Table 1

Power-law parameters and slip coefficients for the LBE solutions. (Data for σ=1 differ slightly from those of Lilley & Sader (2007) due to the different mean free paths used in data reduction. Here r is the sample correlation coefficient for the linear regression analysis of the Embedded Image versus Embedded Image data. Solutions: O, Ohwada et al. (1989); LH1, Loyalka & Hickey (1989); LH2, Loyalka & Hickey (1990); S, Siewert (2003).)

We repeated this analysis for the LBE solutions with partial thermal accommodation (0.1≤σ≤0.9) by Loyalka & Hickey (1990) and Siewert (2003), and observed a similar power-law velocity behaviour in all cases. For these solutions, values of Embedded Image, C, α, Embedded Image and r are included in table 1. Again, the correlation coefficients are small, demonstrating the accuracy of the power-law description in the Knudsen layer. In all the cases, the power-law parameter α is distinctly less than unity, as for the solutions with σ=1. Therefore, for these LBE solutions with σ<1, the power law also establishes the existence of a velocity gradient singularity at the surface where the effective viscosity is zero.

Figure 3 shows all the LBE solutions, plotted together with their corresponding power-law velocity profiles, which further demonstrates the accuracy of the power-law description within the Knudsen layer. To explore the dependence of Embedded Image, C, α and Embedded Image on the accommodation coefficient σ, these parameters are plotted versus σ in figure 4. We discuss these figures in §5.

Figure 3

LBE solutions of Kramers' problem for hard spheres at various σ, published by Loyalka & Hickey (1989, 1990), Ohwada et al. (1989) and Siewert (2003). All solutions include velocity profiles at σ=1. Only Loyalka & Hickey (1990) and Siewert (2003) give profiles for σ<1. Triangles, Ohwada et al.; crosses, Loyalka & Hickey (1989); circles, Loyalka & Hickey (1990); squares, Siewert; solid lines, power-law fits.

Figure 4

Comparison of power-law parameters (a) Embedded Image, (b) C and (c) α, and (d) slip coefficient Embedded Image. Error bars for C and α are shown for the DSMC solutions. These error bars represent 99% CIs from the nonlinear regression analysis. Errors in Embedded Image and Embedded Image are not visible on these axes. Curve fits described in equation (5.3) of §5 are shown for Embedded Image and Embedded Image. Open circles, Loyalka & Hickey; squares, Siewert; filled circles, DSMC (hard spheres); diamonds, DSMC (VSS); dot-dashed line, DSMC curve fit; solid line, curve fit.

4. DSMC solutions for hard sphere and variable soft sphere molecules

We have two aims in simulating the Knudsen layer with the DSMC method. First, we use precise DSMC calculations to confirm the accuracy of the hard sphere LBE solutions and the ensuing power-law velocity description established in §3. Since LBE is an approximation to the full nonlinear Boltzmann equation, the resulting solutions must be validated against those of the full Boltzmann equation, as afforded by precise DSMC calculations.

Second, we use DSMC calculations to probe the Knudsen layer structure for a molecular model that is more realistic than the simple hard sphere, and has not been studied with LBE. We use a major advantage of the DSMC method, in that it can, in principle, incorporate the molecular models with any level of complexity. Importantly, this provides a means of solving the Boltzmann equation for realistic gas models that cannot be studied with LBE. Here, we investigate the Knudsen layer structure for the variable soft sphere (VSS) model (Koura & Matsumoto 1991). The VSS model approximates the molecules that interact according to an intermolecular force that is inversely proportional to a power of the molecular separation distance. Details of our hard sphere and VSS models appear in appendix A.

A number of reports have studied the Knudsen layer with DSMC calculations (Bird 1977; Lockerby et al. 2005b), with several appearing in the past year (Gu & Emerson 2007; Lilley & Sader 2007; Mizzi et al. 2007; Struchtrup & Torrilhon 2007; Torrilhon & Struchtrup 2008). These studies employed Couette flow solutions to capture Kramers' problem, and we have followed suit using the geometry illustrated in figure 5. Our DSMC code solved the full Couette flow domain, and provided the mean flow velocities calculated by Embedded Image. These mean velocities were transformed into the reference frame of the Kramers' problem for subsequent analysis (see appendix A).

Figure 5

Geometry of DSMC calculations of Couette flow. uc is the bulk flow velocity parallel to the wall and yc is the displacement from the midplane where yc=0. The walls are at ycH/2.

While Bird (1977), Lockerby et al. (2005b), Gu & Emerson (2007), Mizzi et al. (2007), Struchtrup & Torrilhon (2007) and Torrilhon & Struchtrup (2008) do not report the power-law velocity structure of the Knudsen layer, the DSMC solution of Lockerby et al. (2005b) does exhibit the power-law dependence with α≈0.8 for VSS molecules with diffusely reflecting walls (Lilley & Sader 2007). In Bird's study, the walls of the Couette flow simulation were close together, resulting in interference between the Knudsen layers at each wall so that the Kramers' problem was not captured accurately. This highlights the fact that the wall separation H is a critical consideration in using Couette flow simulations to capture the Kramers' problem. The wall separation must be sufficiently large such that the Knudsen layers near each wall do not interfere with each other. Here, we specify H in terms of the Knudsen number Kn, usingEmbedded ImageOwing to the computational expense of the DSMC method in the near-continuum regime where Kn is small, H cannot be arbitrarily large. A suitable H value must therefore be sufficiently large to avoid interference between the Knudsen layers, and yet still small enough to permit solution with DSMC calculations within a practical time period. To determine a suitable H, we first performed a series of DSMC calculations at various Kn, using hard spheres and σ=1. This analysis, presented in appendix B, shows that Kn≈0.06 is sufficiently small to capture the Knudsen layers at each wall without interference.

As for any numerical technique, it is essential to test the numerical convergence of DSMC solutions. Accordingly, we performed a detailed convergence analysis using hard spheres with Kn=0.0589. This analysis is presented in appendix C and demonstrates that our solution is converged. Our DSMC simulation parameters are summarized in table 2. Appendix A also contains details on the simulation time step Δt and the flow sampling interval used. An important note on the pseudo-random number generator used in our DSMC calculations appears in appendix D.

View this table:
Table 2

Summary of DSMC simulation parameters. (Embedded Image is the total number of intermolecular collisions simulated during the entire calculation.)

Using both the hard sphere and VSS models, we performed DSMC calculations of Couette flow with 0.05≤σ≤1. Samples of the resulting velocity profiles for σ=0.1, 0.6 and 1 are shown in figure 6. We used nonlinear regression to calculate Embedded Image, C and α for our DSMC solutions, according to the power-law description of equation (3.1). Our method for calculating Embedded Image is given in appendix B. These parameters are plotted versus σ in figure 4.

Figure 6

DSMC solutions of Kramers' problem compared with the published LBE solutions. ((a) σ=0.1, (b) σ=0.6, and (c) σ=1). In the reference frame of Kramers' problem, the DSMC solutions each contained 500 points. For clarity, some of these were removed for Embedded Image. The LBE solutions of Loyalka & Hickey (1989, 1990), Ohwada et al. (1989) and Siewert (2003) are shown in (c). On this plot, power-law fits to these different LBE solutions are indistinguishable so only one is shown. Squares, Siewert solution; solid line, power-law fit to LBE solution and Siewert and Loyalka & Hickey LBE solutions; filled circles, DSMC (hard spheres); crosses, DSMC (VSS); open circles, Loyalka & Hickey (1990) solution; diamonds, published LBE solutions.

The close agreement between the LBE and hard sphere DSMC solutions is immediately apparent in figures 4 and 6, verifying that LBE accurately approximates the full Boltzmann equation for the Kramers' problem with hard spheres and σ<1. Additionally, the distinct linear behaviour of the DSMC velocity profiles in figure 6 demonstrates the accuracy of the power-law velocity profile in the Knudsen layer for VSS molecules at various σ. We again emphasize that λ is the only natural length scale in this dilute gas flow. Importantly, the DSMC solutions all have α distinctly less than unity, which in turn confirms the existence of a velocity gradient singularity at the surface where the effective viscosity is zero. We note that our result of α=0.83 for the VSS model with σ=1 is consistent with the value of 0.8 estimated by Lilley & Sader (2007) from the DSMC solution published by Lockerby et al. (2005b) using VSS molecules.

5. Discussion

As noted in §1, two recent studies by Lockerby et al. (2005a) and Fichman & Hetsroni (2005) predicted μμ/2 at the surface in the Knudsen layer. This result contrasts with the above analysis that establishes μ=0 at the surface. It is therefore important to compare these previous models to the above power-law description and investigate how well these models approximate a solution of the Boltzmann equation.

In the first study by Lockerby et al. (2005a), the velocity profile within the Knudsen layer was modelled with a so-called ‘wall function’. This wall function, given byEmbedded Imagewas based on a curve-fit approximation to an earlier LBE solution. The wall function gives the velocity gradient Embedded Image at the surface, with the corresponding effective viscosity μ≈0.59μ. An implicit assumption in formulating the wall function was that the velocity field for Kramers' problem is analytic at y=0. Indeed, such analytic behaviour is predicted by several high-order hydrodynamic models of the Knudsen layer, which have the general form (Lockerby et al. 2005b)Embedded Image(5.1)for Kramers' problem, where the constants k1,2,3 depend upon the model and a is the velocity gradient as y→∞ (figure 1). Lockerby et al. (2005b) used the velocity gradient from the wall function at Embedded Image as a boundary condition in equation (5.1) and obtainedEmbedded Image(5.2)to describe the velocity profile in the Knudsen layer and external flow. Here, the constant K depends upon the hydrodynamic model for: the BGK–Burnett equations Embedded Image, the regularized Burnett equations Embedded Image, Zhong's augmented Burnett equations Embedded Image and the R13 equations Embedded Image. Figure 7 shows the results obtained from equation (5.2) for the regularized Burnett and Zhong's augmented equations. These two solutions form an envelope within which the BGK–Burnett solution, the R13 solution and the wall function are all contained.

Figure 7

Hard sphere LBE solutions (Loyalka & Hickey 1989; Ohwada et al. 1989; Siewert 2003) at σ=1 compared with the hydrodynamic models described by equation (5.2). The solutions of the regularized Burnett equations Embedded Image and Zhong's augmented Burnett equations Embedded Image form an envelope containing the BGK–Burnett solution, the R13 solution and the wall function. Circles, LBE solutions; solid line, power-law fit to LBE data; dashed line, regularized Burnett; dot-dashed line, Zhong's augmented Burnett; dotted line, Fichman & Hetsroni.

In the second recent study, Fichman & Hetsroni (2005) proposed the effective viscosityEmbedded Imagegiving the velocity gradient Embedded Image at the surface which is finite for σ>0. The velocity profile obtained from this effective viscosity is also shown in figure 7.

Importantly, figure 7 clearly shows that the wall function, the various hydrodynamic models and the Fichman & Hetsroni model do not capture the asymptotic form of the velocity profile in the Knudsen layer near the surface. On a log–log scale, the true velocity distribution follows a distinct line with a slope significantly less than unity, whereas the hydrodynamic models give straight lines with slopes of unity. This failure of high-order hydrodynamic models and the wall function to correctly predict the power-law structure may explain why they cannot accurately capture the Knudsen layer, as concluded by Lockerby et al. (2005b). Nonetheless, it is important to emphasize that the power-law model predicts an effective viscosity μ<μ/2 at distances less than 0.03 mean free paths away from the wall, which represents a small region of the Knudsen layer. The effect of using such approximate hydrodynamic models as opposed to the true velocity distribution in the Knudsen layer (possessing a velocity gradient singularity at the wall) in full flow modelling (Zhang et al. 2006) is unclear and requires further investigation.

Our analysis of the existing LBE solutions for Kramers' problem, together with our new DSMC calculations, clearly demonstrate that the power-law description of the Knudsen layer given in equation (3.1) accurately describes the velocity profile for both hard sphere and VSS molecules with full and partial thermal accommodation, even at a distance of only ∼λnom/100 from the surface. Since the power-law description is obtained from the LBE and DSMC solutions of the Boltzmann equation, regardless of the degree of thermal accommodation at the surface, this finding indicates that the velocity gradient singularity arises naturally from the Boltzmann equation. This is supported by Willis (1962) and Sone (2002), who proved the existence of the logarithmic singularity du/dy∝ln y in the linearized (approx.) BGK equation.

The discussion above clearly shows that the various high-order hydrodynamic models considered by Lockerby et al. (2005b) do not accurately capture the velocity structure of the Knudsen layer as predicted by the LBE solutions. Importantly, these models do not provide a proper treatment of the boundary conditions at the wall (Gu & Emerson 2007; Mizzi et al. 2007; Struchtrup & Torrilhon 2007; Torrilhon & Struchtrup 2008), and indeed can only be formally valid in the outer part of the Knudsen layer since they are derived from high-order hydrodynamic treatments of the Boltzmann equation (Hadjiconstantinou 2006). Thus, it is not surprising that deviations to predictions of the Boltzmann equation exist for the inner part of the Knudsen layer, as shown above. The power-law description, and the feature of a velocity gradient singularity at the surface, is expected to motivate further research into hydrodynamic models of rarefied flow.

Our results show that the entire velocity profile for Kramers' problem is accurately represented byEmbedded ImageThe corresponding effective viscosity for Embedded Image isEmbedded Imagewhich is strongly non-Newtonian. The functional dependencies of Embedded Image, C, α and Embedded Image on the accommodation coefficient σ were determined empirically using several trial functions and nonlinear regression to yieldEmbedded Image(5.3)The fits for Embedded Image and Embedded Image are included in figure 4.

A striking feature of the data shown in figure 4, is that the power-law structure of the Knudsen layer appears to be preserved in the asymptotic limit as σ→0. Indeed, the power-law exponent α(σ) varies very little while going between the limits of fully diffuse (α=1) and fully specular (α=0) reflection at the surface. Furthermore, the product Embedded Image obtained from the above formulae, increases by only approximately 6% as σ decreases from unity to zero. Given the scatter in the numerical data for both α and C (see figure 4), we then conclude that the product αC≈1.05 is a sound approximation for all σ. The reasons for this intriguing constant behaviour in αC, which appears directly in the expression for the effective viscosity (see above), and the limited variability in the power-law exponent, are unknown at present.

Interestingly, our DSMC calculations show that the structure of the Knudsen layer for VSS molecules is very similar to that for hard spheres, indicating that the power-law description is only weakly dependent on the molecular model. This strongly suggests that the power-law description is a general physical phenomenon, within the framework of the Boltzmann equation, which applies for all pure monatomic gases. However, as noted by Lilley & Sader (2007), detailed solutions of the Boltzmann equation using realistic intermolecular potentials validated by accurate experimental measurements are necessary to make a definitive general statement about the accuracy of the power-law behaviour in real gases. Any such investigations must consider gas mixtures containing molecules with rotational energy such as air, which are important in most practical applications. Since the DSMC method offers the only practical means of solving the Boltzmann equation for such mixtures, DSMC calculations will be an essential component of future investigations into the power-law behaviour in real gases.

6. Summary and conclusions

We have examined the structure of the velocity profile in the Knudsen layer using LBE solutions of Kramers' problem for hard sphere molecules with partial thermal accommodation, according to Maxwell's boundary condition (Maxwell 1879). Our study establishes that the velocity profile in the Knudsen layer, under these conditions, also follows the power-law description originally found by Lilley & Sader (2007) for hard spheres with full thermal accommodation. This in turn shows that the velocity gradient is singular at the surface, i.e. the effective viscosity is zero, under arbitrary thermal accommodation. These findings were verified using precise DSMC calculations.

We also performed DSMC calculations to probe the structure of the Knudsen layer for a gas composed of VSS molecules, which are more realistic than simple hard spheres. These simulations also revealed the power-law velocity behaviour within the Knudsen layer over a full range of accommodation coefficients. The small difference we observed between the hard sphere and the VSS solutions indicates that the power-law behaviour is only weakly dependent on the molecular model, and suggests that it arises directly from the Boltzmann equation.

These results are expected to motivate future work into understanding the origin of such behaviour by rigorous asymptotic analysis of the Boltzmann equation. Given the importance of rarefied gas dynamics in small-scale flows, our findings are thus expected to impact on the development and application of nanoscale devices.

Acknowledgments

This research was supported by the Particulate Fluids Processing Centre, a special research centre of the Australian Research Council and by the Australian Research Council Grants Scheme.

Footnotes

    • Received February 18, 2008.
    • Accepted March 14, 2008.

References

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