and liquids go through multiple collisions at a given instant, making the treatment of the intermolecular collision process more difficult. The dilute gas approximations, along with molecular chaos and equipar- tition of energy principles, lead us to the well established kinetic theory of gases and formulation of the Boltzmann transport equation starting from the Liouville equation. The assumptions and simplifications of this derivation are given in Vincenti and Kruger (1977) and Bird (1994). Momentum and energy transport in the bulk of the fluid happen with intermolecular collisions, as does settling to a thermodynamic equilibrium state. Hence, the time and length scales associated with the intermolecular collisions are important parameters for many applications. The distance traveled by the molecules between the intermolecular collisions is known as the mean free path. For a simple gas of hard- sphere molecules in thermodynamic equilibrium, the mean free path is given in the following form [Bird, 1994]: λ ϭ (2 1/2 π d 2 m n) Ϫ1 (6.6) The gas molecules are traveling with high speeds proportional to the speed of sound. By simple consid- erations, the mean-square molecular speed of the gas molecules is given by [Vincenti and Kruger, 1977]: ͙ C ෆ ෆ 2 ෆ ϭ Ί ϭ ͙ 3R ෆ T ෆ (6.7) where R is the specific gas constant. For air under standard conditions, this corresponds to 486m/sec. This value is about 3 to 5 orders of magnitude larger than the typical average speeds obtained in gas microflows. (The importance of this discrepancy will be discussed in Section 6.2.3.) In regard to the time scales of intermolecular collisions, we can obtain an average value by taking the ratio of the mean free path to the mean-square molecular speed. This results in t c ഡ 10 Ϫ10 for air under standard conditions. This time scale should be compared to a typical microscale process time scale to determine the validity of the thermodynamic equilibrium assumption. So far we have identified the vast number of molecules and the associated time and length scales for gas flows. That it is possible to lump all of the microscopic quantities into time- and/or space-averaged macroscopic quantities, such as fluid density, temperature, and velocity. It is crucial to determine the limitations of these continuum-based descriptions; in other words: ● How small should a sample size be so that we can still talk about the macroscopic properties and their spatial variations? ● At what length scales do the statistical fluctuations become significant? It turns out that a sampling volume that contains 10,000 molecules typically results in 1% statistical fluc- tuations in the averaged quantities [Bird, 1994]. This corresponds to a volume of 3.7 ϫ 10 −22 m 3 for air at standard conditions. If we try to measure an “instantaneous” macroscopic quantity such as velocity in a three-dimensional space, one side of our sampling cube will typically be about 72 nm. This length scale is slightly larger than the mean free path of air λ under standard conditions. Therefore, in complex micro- geometries where three-dimensional spatial gradients are expected, the definition of instantaneous macro- scopic values may become problematic for Kn Ͼ 1. If we would like to subdivide this domain further to obtain an instantaneous velocity distribution, the statistical fluctuations will be increased significantly as the sample volume is decreased. Hence, we may not be able to define instantaneous velocity distribution in a 72nm 3 volume. On the other hand, it is always possible to perform time or ensemble averaging of the data at such small scales. Hence, we can still talk about a velocity profile in an averaged sense. To describe the statistical fluctuation issues further, we present in Figure 6.2 the flow regimes and the limit of the onset of statistical fluctuations as a function of the characteristic dimension L and the nor- malized number density n/n o . The 1% statistical scatterline is defined in a cubic volume of side L, which con- tains approximately 10,000 molecules. Using Equation (6.5), we find that L/ δ Ϸ 20 satisfies this condition approximately, and the 1% fluctuation line varies as (n/n o ) Ϫ1/3 . Under standard conditions, 1% fluctuation is observed at L ϭ 72 nm, and the Knudsen number based on this value is Kn Ϸ 1. Figure 6.2 also shows the continuum, slip, transitional, and free molecular flow regimes for air at 3P ᎏ ρ Molecular-Based Microfluidic Simulation Models 6-5 © 2006 by Taylor & Francis Group, LLC 273 K and at various pressures. The mean free path varies inversely with the pressure. Hence, at isother- mal conditions, the Knudsen number varies as (n/n o ) Ϫ1 . The fundamental question of dynamic similar- ity of low-pressure gas flows to gas microflows under geometrically similar and identical Knudsen, Mach, and Reynolds number conditions can be answered to some degree by Figure 6.2. Provided that there are no unforeseen microscale-specific effects, the two flow cases should be dynamically similar. However, a distinction between the low-pressure and gas microflows is the difference in the length scales for which the statistical fluctuations become important. It is interesting to note that for low-pressure rarefied gas flows the length scales for the onset of signifi- cant statistical scatter correspond to much larger Knudsen values than do the gas microflows. For example, Kn ϭ 1.0 flow obtained at standard conditions in a 72 nm cube volume permits us to perform one instan- taneous measurement in the entire volume with 1% scatter. However, at 100 pascal pressure and 273 K temperature, Kn ϭ 1.0 flow corresponds to a length scale of 65 mm. For this case, 1% statistical scatter in the macroscopic quantities is observed in a cubic volume of side 0.72 µm, allowing about 90 pointwise instantaneous measurements. This is valid for instantaneous measurements of macroscopic properties in complex three-dimensional conduits. In large-aspect-ratio microdevices, one can always perform spanwise averaging to define an averaged velocity profile. Also, for practical reasons one can also define averaged macroscopic properties either by time or ensemble averaging (such examples are presented in Section 6.2.4). 6.2.2 An Overview of the Direct Simulation Monte Carlo Method In this section, we present the algorithmic details, advantages, and disadvantages of using the direct sim- ulation Monte Carlo algorithm for microfluidic applications. The DSMC method was invented by Graeme 6-6 MEMS: Introduction and Fundamentals L (microns) Kn = 1.0 Kn = 10 L/ = 20 Dilute gas Dense gas Kn = 0.1 Kn = 0.01 Slip flow Transitional flow Free molecular flow Continuum flow Navier−Stokes equations 10 3 10 2 10 1 10 0 10 -1 10 -2 10 -3 10 -4 10 -3 10 -2 10 -1 n/n 0 10 0 10 1 10 2 FIGURE 6.2 Limit of approximations in modeling gas microflows. L is the characteristic length, n/n o is the number density normalized with the corresponding standard conditions. The lines that define the various Knudsen regimes are based on air at isothermal conditions (T ϭ 273 K). The L/ δ ϭ 20 line corresponds to the 1% statistical scatter in the macroscopic properties. The area below this line experiences increased statistical fluctuations. © 2006 by Taylor & Francis Group, LLC A. Bird (1976, 1994). Several review articles about the DSMC method are currently available [Bird 1978, 1998; Muntz, 1989; Oran et al., 1998]. Most of these articles present an extended review of the DSMC method for low-pressure rarefied gas flow applications, with the exception of Oran et al. (1998), who also address microfluidic applications. The previous section describes molecular magnitudes and associated time and length scales. Under standard conditions in a volume of 10 µm 3 , there are about 2.69 ϫ 10 10 molecules. A molecular-based simulation model that can compute the motion and interactions of all these molecules is not possible. The typical DSMC method uses hundreds of thousands or even millions of simulated molecules or par- ticles that mimic the motion of real molecules. The DSMC method is based on splitting the molecular motion and intermolecular collisions by choos- ing a time step less than the mean collision time and tracking the evolution of this molecular process in space and time. For efficient numerical implementation, the space is divided into cells similar to the finite-volume method. The DSMC cells are chosen proportional to the mean free path λ .In order to resolve large gradients in flow with realistic (physical) viscosity values, the average cell size should be ∆x c ഡ λ /3 [Oran et al., 1998]. The time- and cell-averaged molecular quantities are obtained as the macroscopic values at the cell centers. The DSMC involves four main processes: motion of the particles, indexing and cross-referencing of particles, simulation of collisions, and sampling of the flow field. The basic steps of a DSMC algorithm are given in Figure 6.3. The first process involves motion of the simulated molecules during a time interval of ∆t. Because the molecules will go through intermolecular collisions, the time step for simulation chosen is smaller than the mean collision time ∆t c . Once the molecules are advanced in space, some of them will have gone through wall collisions or will have left the computational domain through the inflow–outflow bound- aries. Hence, the boundary conditions must be enforced at this level, and the macroscopic properties along the solid surfaces must be sampled. This is done by modeling the surface molecule interactions by applying the conservation laws on individual molecules rather than using a velocity distribution function that is commonly utilized in the Boltzmann algorithms. This approach allows inclusion of many other physical processes, such as chemical reactions, radiation effects, three-body collisions, and ionized flow effects, without major modifications to the basic DSMC procedure [Oran et al., 1998]. However, a priori knowl- edge of the accommodation coefficients must be used in this process. Hence, this constitutes a weakness of the DSMC method similar to the Navier–Stokes-based slip and even Boltzmann equation-based sim- ulation models. The following section discusses this issue in detail. The second process is the indexing and tracking of the particles. This is necessary because the mole- cules might have moved to new cell locations during the first stage. The new cell location of the mole- cules is indexed, and thus the intermolecular collisions and flow field sampling can be handled accurately. This is a crucial step for an efficient DSMC algorithm. The indexing, molecule tracking, and data struc- turing algorithms should be carefully designed for the specific computing platforms, such as vector super computers and workstation architectures. The third step is simulation of collisions via a probabilistic process. Because only a small portion of the molecules is simulated and the motion and collision processes are decoupled, probabilistic treatment becomes necessary. A common collision model is the no-time-counter technique of Bird (1994) that is used in conjunction with the subcell technique where the collision rates are calculated within the cells and the collision pairs are selected within the subcells. This improves the accuracy of the method by main- taining the collisions of the molecules with their closest neighbors [Oran et al., 1998]. The last process is the calculation of appropriate macroscopic properties by the sampling of molecular (microscopic) properties within a cell. The macroscopic properties for unsteady flow conditions are obtained by the ensemble average of many independent calculations. For steady flows, time averaging is also used. 6.2.3 Limitations, Error Sources, and Disadvantages of the DSMC Approach Following the work of Oran et al., (1998), we identify several possible limitations and error sources within a DSMC method. Molecular-Based Microfluidic Simulation Models 6-7 © 2006 by Taylor & Francis Group, LLC 1. Finite cell size: the typical DSMC cell should be about one-third of the local mean free path. Values of cell sizes larger than this may result in enhanced diffusion coefficients. In DSMC, one cannot directly specify the dynamic viscosity and thermal conductivity of the fluid. The dynamic viscosity is calculated via diffusion of linear momentum. Breuer et al. (1995) performed one-dimensional Rayleigh flow problems in the continuum flow regime and showed that for cell sizes larger than one mean free path the apparent viscosity of the fluid was increased. Some numerical experimen- tation details for this finding are also given in Beskok (1996). More recently, the viscosity and thermal 6-8 MEMS: Introduction and Fundamentals Interval > ∆t g ? Sample flow properties Time > t L ? Unsteady flow average runs Steady flow average samples after establishing steady flow Print final results Stop Compute collisions Reset molecular indexing Move molecules with ∆t g ; compute interactions with boundaries Initialize molecules and boundaries Set constants Read data Start Unsteady flow repeat until required sample is obtained No No FIGURE 6.3 Typical steps for a DSMC method. (Reprinted with permission from Oran, E.S. et al. [1998] “Direct Simulation Monte Carlo: Recent Advances and Applications,” Ann. Rev. Fluid Mech. 30, 403–441.) © 2006 by Taylor & Francis Group, LLC conductivity dependence on cell size have been obtained more systematically by using the Green–Kubo theory [Alexander et al., 1998; Hadjiconstaninou, 2000]. 2. Finite time step: due to the time splitting of the molecular motion and collisions, the maximum allowable time step is smaller than the local collision time t c . Values of time steps larger than t c result in molecules traveling through several cells prior to a cell-based collision calculation. The time-step and cell-size restrictions presented in items 1 and 2 are not a Courant–Friederichs–Lewy (CFL) stability condition. The DSMC method is always stable. However, overlooking the physical restrictions stated in items 1 and 2 may result in highly diffu- sive numerical results. 3. Ratio of the simulated particles to the real molecules: due to the vast number of molecules and limited computational resources, one always has to choose a sample of molecules to simulate. If the ratio of the actual molecules to the simulated molecules gets too high, the statistical scatter of the solution is increased. The details for the statistical error sources and the corresponding reme- dies can be found in Oran et al. (1998), Bird (1994) and Chen and Boyd (1996). A relatively well- resolved DSMC calculation requires a minimum of 20 simulated particles per cell. 4. Boundary condition treatment: the inflow–outflow boundary conditions can become particularly important in a microfluidic simulation. A subsonic microchannel flow simulation may require speci- fication of inlet and exit pressures. The flow will develop under this pressure gradient and result in a certain mass flow rate. During such simulations, specification of back pressure for subsonic flows is challenging. In the DSMC studies, one can simulate the entry problem to the channels by specifying the number density, temperature, and average macroscopic velocity of the molecules at the inlet of the channel. At the outflow, the number density and temperature corresponding to the desired back pres- sure can be specified. This and similar treatments facilitate significantly reducing the spurious numer- ical boundary layers at inflow and outflow regions. For high Knudsen number flows (i.e., Kn Ͼ 1) in a channel with blockage (such as a sphere in a pipe), the location of the inflow and outflow bound- aries is important. For example, the molecules reflected from the front of the body may reach the inflow region with very few intermolecular collisions, creating a diffusing flow at the front of the bluff body [Liu et al., 1998]. (The details of this case are presented in Section 6.2.4.) 5. Uncertainties in the physical input parameters: these typically include the input for molecular col- lision cross-section models, such as the hard sphere (HS), variable hard sphere (VHS), and variable soft sphere (VSS) models [Oran et al., 1998; Vijayakumar et al., 1999]. The HS model is usually sufficient for monatomic gases or for cases with negligible vibrational and rotational nonequilib- rium effects, such as in the case of nearly isothermal flow conditions. Along with these possible error sources and limitations, some particular disadvantages of the DSMC method for simulation of gas microflows are: 1. Slow convergence: the error in the DSMC method is inversely proportional to the square root of the number of simulated molecules. Reducing the error by a factor of two requires increasing the number of simulated molecules by a factor of four. This is a very slow convergence rate compared to the continuum-based simulations with spatial accuracy of second or higher order. Therefore, continuum-based simulation models should be preferred over the DSMC method whenever possible. 2. Large statistical noise: gas microflows are usually low subsonic flows with typical speeds of 1 mm/sec to 1 m/sec (exceptions to this are the micronozzles utilized in synthetic jets and satellite thruster control applications). The macroscopic fluid velocity is obtained by time or ensemble aver- aging of the molecular velocities. This difference of five to two orders of magnitude between the molecular and average speeds results in large statistical noise and requires a very long time averaging for gas microflow simulations. The statistical fluctuations decrease with the square root of the sam- ple size. Time or ensemble averages of low-speed microflows on the order of 0.1 m/sec require about 108 samples in order to distinguish such small macroscopic velocities. Fan and Shen (1999) introduced the information preservation (IP) technique for the DSMC method, which enables efficient DSMC simulations for low-speed flows (the IP scheme is briefly covered in Section 6.2.5). Molecular-Based Microfluidic Simulation Models 6-9 © 2006 by Taylor & Francis Group, LLC 3. Extensive number of simulated molecules: if we discretize a rectangular domain of 1 mm ϫ 100 µm ϫ 1µm under standard conditions for Kn ϭ 0.065 flow, we will need at least 20 cells per micrometer length scale. This results in a total of 8 ϫ 10 8 c ells. Each of these cells should contain at least 20 si mulated molecules, resulting in a total of 1.6 ϫ 10 10 particles. Combined with the number of time-step restrictions, simulation of low-speed microflows with DSMC easily exceeds the capabili- ties of current computers. An alternative treatment to overcome the extensive number of simulated molecules and long integration times is utilization of the dynamic similarity of low-pressure rarefied gas flows to gas microflows under atmospheric conditions. The key parameters for the dynamic sim- ilarity are the geometric similarity and matching of the flow Knudsen, Mach, and Reynolds numbers. Performing actual experiments under dynamically similar conditions may be very difficult; however, parametric studies via numerical simulations are possible. The fundamental question to answer for such an approach is whether or not a specific, unforeseen microscale phenomenon is missed with the dynamic similarity approach. In response to this question, all numerical simulations are inherently model based. Unless microscale-specific models are implemented in the algorithm, we will not be able to obtain more physical information from a microscopic simulation than from a dynamically similar low-pressure simulation. One limitation of the dynamic similarity concept is the onset of statistical scatter in the instantaneous macroscopic flow quantities for gas microflows for Kn > 1 (see section 6.2.1 and Figure 6.2 for details). Here, we must also remember that DSMC utilizes time or ensemble averages to sample the macroscopic properties from the microscopic variables. Hence, DSMC already determines the macroscopic properties in an averaged sense. 4. Lack of deterministic surface effects: Molecule wall interactions are specified by the accommoda- tion coefficients σ ν , σ T .For diffuse reflection σ ϭ 1, and the reflected molecules lose their incom- ing tangential velocity while being reflected with the tangential wall velocity. For σ ϭ 0 the tangential velocity of the impinging molecules is not changed during the molecule/wall collisions. For any other value of σ ,acombination of these procedures can be applied. The molecule–wall interaction treatment implemented in DSMC is more flexible than the slip conditions given by Equations (6.1) and (6.2). However, it still requires specification of the accommodation coefficients, which are not known for any gas surface pair with a specified surface root mean square (rms) rough- ness. The tangential momentum accommodation coefficients for helium, nitrogen, argon and carbon dioxide on single-crystal silicon were measured by careful microchannel experiments [Arkilic, 1997]. 6.2.4 Some DSMC-Based Gas Microflow Results This section presents some DSMC results applied to gas microflows. 6.2.4.1 Microchannel Flows The DSMC simulation results for subsonic gas flows in microchannels are presented in this section. Due to the computational difficulties explained in the previous sections, a low-aspect-ratio, two-dimensional channel with relatively high inlet velocities is studied. The results presented in the figures are for microchannels with a length-to-height ratio (L/h) of 20 under various inlet-to-exit-pressure ratios. The DSMC results are performed with 24,000 cells, of which 400 cells were in the flow direction and 60 cells were across the channel. Atotal of 480,000 molecules are simulated. The results are sampled (time aver- aged) for 10 5 times, and the sampling is performed every ten time steps. In the following simulations, diffuse reflection ( σ ν ϭ 1.0) is assumed for interaction of gas molecules with the surfaces. Because the slip amount can be affected significantly by small variations in σ ν (Equation [6.1]), the apparent value of the accommodation coefficient σ ν is monitored throughout the simulations by recording the tangential momentum of the impinging ( τ i ) and reflected ( τ r ) gas molecules. Based on these values, the apparent tangential momentum accommodation coefficient, σ ν ϭ ( τ i – τ r )/( τ i – τ w ) ϭ 0.99912 with standard deviation of σ rms ϭ 0.01603, is obtained. The velocity profiles normalized with the corresponding average speed are presented in Figure 6.4 for pressure-driven microchannel flows at Kn ϭ 0.1 and Kn ϭ 2.0. The figure also presents the molecule/cell 6-10 MEMS: Introduction and Fundamentals © 2006 by Taylor & Francis Group, LLC refinement studies as well as predictions of the VHS and VSS models. The DSMC results are compared against the linearized Boltzmann solutions [Ohwada et al., 1989], and excellent agreements of the VHS and VSS models with the linearized Boltzmann solutions are observed for these nearly isothermal flows. In regard to the molecule/cell refinement study, the number of cells and the number of simulated mole- cules are identified for each case. The first VHS case utilized only 6000 cells with 80,000 simulated mole- cules, and the results are sampled about 5 ϫ 10 5 times. Sampling is performed every 20 time steps. The refined VHS and VSS cases utilized 24,000 cells and a total of 480,000 molecules. The results for these are sampled 10 5 times, every ten time steps. Although the velocity profiles for the low-resolution case (6000 cells) seem acceptable, the density and pressure profiles show large fluctuations. The DSMC and µ Flow (spectral-element-based, continuum computational fluid dynamics [CFD] solver) predictions of density and pressure variations along a pressure-driven microchannel flow are shown in Figure 6.5.For this case, the ratio of inlet to exit pressure is Π ϭ 2.28, and the Knudsen number at the channel outlet is 0.2. Deviations of the slip flow pressure distribution from the no-slip solution are also presented in the figure. Good agreements between the DSMC and µ Flow simulations are achieved. Molecular-Based Microfluidic Simulation Models 6-11 1.2 0.8 0.4 0 0 0.2 0.4 0.6 Y 0.8 100.2 0.4 0.6 Y 0.8 1 0 0.2 0.4 0.6 Y 0.8 100.2 0.4 0.6 Y 0.8 1 U* (Y,Kn) 1.2 0.8 U* (Y,Kn) 1.2 0.8 0.4 0 U* (Y,Kn) 1.2 1 0.8 0.6 U* (Y,Kn) Kn = 0.1 Linearized Boltzmann DSMC [6,000c 80,000m (VHS)] DSMC [24,000c 480,000m (VHS)] Kn = 0.1 Linearized Boltzmann DSMC [6,000c 80,000m (VHS)] DSMC [24,000c 480,000m (VSS)] Kn = 2.0 Linearized Boltzmann DSMC [6,000c 80,000m (VHS)] DSMC [24,000c 480,000m (VHS)] Kn = 2.0 Linearized Boltzmann DSMC [6,000c 80,000m (VHS)] DSMC [24,000c 480,000m (VHS)] FIGURE 6.4 Ve locity profiles normalized with the local average velocity in the slip and transitional flow regimes. The DSMC predictions with the VHS and VSS models agree well with the linearized Boltzmann solutions of Ohwada et al. (1989). The number of cells and simulated molecules are identified on each figure. © 2006 by Taylor & Francis Group, LLC The curvature in the pressure distribution is due to the compressibility effect, and the rarefaction negates this curvature, as seen in Figure 6.5. The slip velocity variation on the channel wall is shown in Figure 6.6. Overall good agreements between both methods are observed. Pan et al. (1999) used the DSMC simula- tions to determine the slip distance as a function of various physical conditions such as the number density, 6-12 MEMS: Introduction and Fundamentals 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 X/L cl / in U in =25 ms −1 , Kn o =0.20 µFlow DSMC ° 0 0.2 0.4 0.6 0.8 1 X/L 1 1.5 2 P/P o µFlow, Kn o =0.2 µFlow, No−Slip DSMC ° FIGURE 6.5 Density (left) and pressure (right) variation along a microchannel. Comparisons of the Navier–Stokes and DSMC predictions for ratio of inlet to exit pressure of Π ϭ 2.28 and Kn o ϭ 0.20. (Reprinted with permission from Beskok, A. [1996] Ph.D. thesis, Princeton University.) 0 0.4 0.6 0.8 0.2 0.4 0.6 0.8 1 X/L U s /U i U i = 25 ms −1 , Kn o = 0.20 DSMC mFlow FIGURE 6.6 Wall slip velocity variation along a microchannel predicted by Navier–Stokes and DSMC simulations. (Reprinted with permission from Beskok, A. [1996] Simulations and Models for Gas Flows in Microgeometries, Ph.D. thesis, Princeton University.) © 2006 by Taylor & Francis Group, LLC wall temperature, and the gas mass. They determined that an appropriate slip distance is 1.125 λ gw , where the subscript gw indicates the gas-wall conditions [Pan et al., 1999]. In the transitional flow regime, Beskok and Karniadakis (1999) studied the Burnett equations for low- speed isothermal flows. This analysis has shown that the velocity profiles remain parabolic even for large Kn flows. To verify this hypothesis, they performed several DSMC simulations; the velocity distribution nondimensionalized with the local average speed is shown in Figure 6.7. They also obtained an approxi- mation to this nondimensionalized velocity distribution in the following form: U* (y,Kn) ϵ U(x,y)/U ෆ (x) ϭ ΄ Ϫ 2 ϩ ϩ ΅ (6.8) ϩ where the extended slip condition given in Equation (6.3) is used. In the above relation, the value of b ϭ Ϫ1 is determined analytically for channel and pipe flows [Beskok and Karniadakis, 1999]. In Figure 6.7, the nondimensional velocity variation obtained in a series of DSMC simulations for Kn ϭ 0.1, Kn ϭ 1.0, Kn ϭ 5.0, and Kn ϭ 10.0 flows are presented along with the corresponding linearized Boltzmann solu- tions [Ohwada et al., 1989]. The DSMC velocity distribution and the linearized Boltzmann solutions Kn ᎏ 1 Ϫ bKn 1 ᎏ 6 Kn ᎏ 1 Ϫ bKn y ᎏ h y ᎏ h Molecular-Based Microfluidic Simulation Models 6-13 1.2 0.8 0.4 0 0.2 Kn = 0.1 Kn = 1.0 Kn = 5.0 Kn = 10.0 DSMC Lin. Boltzmann b = −1 b = −1.8 b = 0 0.4 0.6 Y 0.8 1 0 0.2 0.4 0.6 Y 0.8 1 0 0.2 0.4 0.6 Y 0.8 1 0 0.2 0.4 0.6 Y 0.8 1 U(Y)/U 1.2 0.8 0.4 U(Y)/U FIGURE 6.7 Comparison of the velocity profiles obtained by new slip model Equations (6.3) and (6.8) with DSMC and linearized Boltzmann solutions [Ohwada et al., 1989]. Maxwell’s first-order boundary condition is shown by the dashed lines (b ϭ 0), and the general slip boundary condition (b ϭ Ϫ1) is shown by solid lines. (Reprinted with per- mission from Beskok, A., and Karniadakis, G.E. [1999] “A Model for Flows in Channels, Pipes, and Ducts at Micro and Nano Scales,” Microscale Thermophys. Eng. 3, pp. 43–77. Reproduced with permission of Taylor & Francis, Inc.) © 2006 by Taylor & Francis Group, LLC agree quite well. One can use Equation (6.8) to compare the results with the DSMC/linearized Boltzmann data by varying the parameter b.The case b ϭ 0corresponds to Maxwell’s first-order slip model, and b ϭϪ1corresponds to Beskok’s second-order slip boundary condition. It is clear from Figure 6.7 that Equation (6.8) with b ϭ Ϫ1 results in a uniformly valid representation of the velocity distribution in the entire Knudsen regime. The nondimensionalized centerline and wall velocities for 0.01 р Kn р 30 flows are shown in Figure 6.8. The figure includes the data for the slip velocity and centerline velocity from 20 different DSMC runs, of which 15 are for nitrogen (diatomic molecules) and 5 for helium (monatomic molecules). The differ- ences between the nitrogen and helium simulations are negligible; thus, this velocity scaling is independ- ent of the gas type. The linearized Boltzmann solutions [Ohwada et al., 1989] for a monatomic gas are also indicated by triangles in Figure 6.8. The Boltzmann solutions closely match the DSMC predictions. Maxwell’s first-order boundary condition b ϭ 0 (shown by solid lines) predicts, erroneously, a uniform nondimensional velocity profile for large Knudsen numbers. The breakdown of the slip flow theory based on the first-order slip-boundary conditions is realized around Kn ϭ 0.1 and Kn ϭ 0.4 for wall and cen- terline velocities respectively. This finding is consistent with the commonly accepted limits of the slip flow regime [Schaaf and Chambre, 1961]. The prediction using b ϭ Ϫ1 is shown by small dashed lines. The corresponding centerline velocity closely follows the DSMC results, while the slip velocity of the model with b ϭ Ϫ1 deviates from DSMC in the intermediate range for 0.1 Ͻ Kn Ͻ 5. One possible reason for this is the effect of the Knudsen layer. For small Kn flows, the Knudsen layer is thin and does not affect the overall velocity distribution too much. For very large Kn flows, the Knudsen layer covers the channel entirely. For intermediate Kn values, however, both the fully developed viscous flow and the Knudsen layer coexist in the channel. At this intermediate range, approximating the velocity profile as parabolic ignores the Knudsen layers. For this reason, the model with b ϭ Ϫ1 results in 10% error in the slip velocity 6-14 MEMS: Introduction and Fundamentals Boltzmann DSMC data b = 0 b = −1 0.5 0 0.01 0.05 0.1 0.5 Kn 1 5 10 1 1.5 U* (y,Kn) FIGURE 6.8 Centerline and wall slip velocity variations in the entire Knudsen regime. The linearized Boltzmann solutions of Ohwada et al. (1989) are shown by triangles, and the DSMC simulations are shown by closed circles. Theoretical predictions of the velocity scaling obtained by Equation (6.8) are shown for different values of b. The b ϭ 0 case corresponds to Maxwell’s first-order boundary condition, and b ϭ Ϫ1 corresponds to the general slip- boundary condition. © 2006 by Taylor & Francis Group, LLC [...]... motion of the charged surface relative to the stationary liquid, by an applied electric field 4 Sedimentation potential: electric field created by the motion of charged particles relative to a stationary liquid (opposite of electrophoresis) 6.3.2 The Electro-Osmotic Flow The electro-osmotic flow is created by applying an electric field in the streamwise direction, where this electric field (E) interacts... Schematic diagram of the electric double layer next to a negatively charged solid surface Here, ψ is the electric potential, ψs is the surface electric potential, ζ is the zeta potential, and yЈ is the distance measured from the wall © 2006 by Taylor & Francis Group, LLC Molecular-Based Microfluidic Simulation Models 6-2 1 where ψ is the electric potential field, ζ is the zeta potential, ρe is the net charge...Molecular-Based Microfluidic Simulation Models 6-1 5 at Kn ϭ 1 However, the velocity distribution in the rest of the channel is described accurately for the entire flow regime Based on these results, Beskok and Karniadakis (1999) developed a unified flow model that can predict the velocity profiles, pressure distribution, and mass flow rate in channels, pipes, and arbitrary aspect-ratio rectangular... redistribution of the ions close to the wall, keeping the bulk of the liquid far away from the wall electrically neutral The distance from the wall at which the electric potential energy is equal to the thermal energy is known as the Debye length (λ), and this zone is known as the electric double layer The electric potential distribution within the fluid is described by the Poisson–Boltzmann equation:... for the recent experimental findings of [Maurer et al., 2003], who measured the second-order slip coefficient in two-dimensional channel flow What sets this model apart from all other approaches is its ability to quantitatively account for the effect of the Knudsen layers on the flow; this, in fact, holds the key to obtaining an accurate second-order slip model As we show below, the effect of the Knudsen... Taylor & Francis Group, LLC Hydrodynamics of Small-Scale Internal Gaseous Flows 7-7 7.2.2.2 A Second-Order Slip Model for the Hard-Sphere Gas Recently, Hadjiconstantinou (2003a) has shown how the above theoretical results can be used to develop a slip model for the hard-sphere gas that is a more realistic model of isothermal gaseous flows2 In one-dimensional flows (∂u/∂z ϭ ∂u/∂x ϭ 0), this new model reduces... description within the Knudsen layers, for Kn գ 0.1 the effect of these layers on the remainder of the flow field (within the Navier–Stokes approximation) can be captured by a set of effective slip/jump boundary conditions In particular, according to slip flow, the velocity of the gas at the wall, ugas|wall, differs from the velocity of the wall uw by an amount that is proportional to the normal velocity gradient... exponent of approximately T 0.7 To remedy this, collision models with variable collision cross-sections have been proposed [Bird, 1994]; one example is the variable hard-sphere (VHS) model in which the collision cross-section is a function of the relative velocity of the colliding molecules The work presented here can be easily extended to these modified collision models One of the basic geometries... describe the liquid and dense gas flows in submicron-scale conduits The effects of van der Waals forces between the fluid and the wall molecules and the presence of longer range Coulombic forces and an electrical double layer (EDL) can significantly affect the microscale transport [Ho and Tai, 1998; Gad- el- Hak, 1999] For example, the streaming potential effect present in pressure-driven flows under the. .. [Cercignani, 1988; Grad, 1969] 4 The velocity slip expression equation (7.4) does not include the thermal creep contribution in the presence of a temperature gradient along the wall A discussion of this form of velocity slip can be found in [Fukui and Kaneko, 1988] Thermal creep phenomena extend beyond the slip-flow regime; thermal creep flow for the hard-sphere model and the associated thermal creep coefficient . of the Navier–Stokes equations, as well as the molecular interaction models of the MD and the DSMC methods. Although it may seem that the molecular simulation methods are more fundamental, they require. reason, the model with b ϭ Ϫ1 results in 10% error in the slip velocity 6-1 4 MEMS: Introduction and Fundamentals Boltzmann DSMC data b = 0 b = −1 0 .5 0 0.01 0. 05 0.1 0 .5 Kn 1 5 10 1 1 .5 U* (y,Kn) FIGURE. presents the molecule/cell 6-1 0 MEMS: Introduction and Fundamentals © 2006 by Taylor & Francis Group, LLC refinement studies as well as predictions of the VHS and VSS models. The DSMC results