Two Dimensional Microchannel Flow

stant relaxation time is used. The results using L = 200 cells for the three statistics are shown in Fig. 5.2. The main features of a typical shock tube

ow, namely, the shock wave, contact discontinuity and the expansion fan, are well represented. We can clearly delineate the dierence of three statistics. It is shown that under dierent statistics although the initial temperature, den-sity, and relaxation time are the same, the pressure, internal energy, and the temperature are dierent. It is noted that the results of MB statistics always lie between those of BE and FD statistics. Finally, we increase the tempera-ture and keep other parameters the same as (nl, u_l, θ_l) = (1.0, 0.0, 10.0) and (n_r, u_r, θ_r) = (0.7, 0.0, 12.0). The results in Fig. 5.3show that the dilute quan-tum gas in high temperature represent the quanquan-tum gas in classical limit, the dierence of three dierent statistics can not be delineated. The results are in consistent with the characters in classical limit: high temperature and low density.

5.3 Two Dimensional Microchannel Flow

The transition from molecular to viscous ow was one of the main subjects of the early experiments of Knudsen [92] in which the Knudsen minimum phenomenon was rst discovered where experimentally observed that the vol-umetric ow rate per unit pressure drop across a long capillary does not vanish as the mean pressure reduces to very small values. Since then, the capturing the Knudsen minimum phenomena in rareed gas channel ows have been a challenging and long investigating problem [97][98][99][100][101]. This phe-nomenon not only occurs in classical dilute gas but also in liquid helium at low temperature which indicates a ow minimum in phonon system [51]. The fun-damental physics of the phenomena of Knudsen minimum is associated with

rareed gas ows in the slip- and transition-ow regimes and the boundary eects of the kinetic boundary layer and so-called Knudsen layer is dominant.

This type of ow is also found in micro- and nano-scale uidic applications typ-ically involved in MEMS. The Navier-Stokes equations with no-slip boundary conditions often fail to explain important experimental observations, e.g., that the measured ow rate is higher than expected while the drag and friction fac-tor are lower than expected [102]. The main reason is that the Navier-Stokes equations can only describe ows that are close to local thermodynamic equi-librium. When the mean free path of the gas molecules approaches the length scale of the device, the ow lacks scale separation and is unable to achieve local equilibrium [103]. Gas ows in miniaturized devices are often in the slip regime (0.001 < Kn < 0.1) or the transition regime (0.1 < Kn < 10). In these regimes, the gas can no longer be described as continuous quasi-equilibrium

uid nor as a free molecular ow [104]. Also, the ows encountered in mi-cro system typically involve low Mach numbers. The LB method oers an attractive technique for micro- and nano-scale uidic applications where the microscopic and macroscopic behavior are coupled. The method retains a computational eciency comparable to Navier-Stokes solvers but is a more physically accurate method for gas ows, over a broad range of Knudsen numbers due to its original link to kinetic theory. The LB equations can be directly derived in a priori manner from the continuous Boltzmann equations [102][103]. In this section, our objective is to study the planar channel ow of gas particles of arbitrary statistics in the slip and transition regimes and in particular, we investigate the Knudsen minimum phenomena using semi-classical LB method. While considering the microchannel ow simulations, one of the two most important issues is the boundary conditions which have been discussed in Ch. 4, and another one is the relation between Knudsen

5.3. Two Dimensional Microchannel Flow 67

number Kn and relaxation time τ. In the literatures, there are many methods of calculating τ from Kn. However, generally we can write down the relation as τ = ¹₂+_c^c_∗yDim× Kn, yDim is the lattice number in the y direction, c is the sound speed in the present lattices, which will be√

3in our D2Q9 model, and c^∗ is dierent in many literatures, for example, c^∗ =√

8θ/π in [105] which is the average speed of molecular. Others chose c^∗ =√

3θ or c^∗ =√

θπ/2. Even more, Zhang et al. disclosed an idea of wall function that can provide a cor-rection to the mean free path [51]. Such that, the mean free path is variable from wall to uid, as a result, c^∗ becomes the function of Knudsen number Kn. We consider a uniform two-dimensional pressure-driven channel ow in a quantum gas. The channel length is L and height H and L/H = 25. With the given fugacity zinlet = 0.2, zoutlet = 0.09835 for the Fermi gas, zinlet = 0.2, z_outlet = 0.10191 for the Bose gas, and zinlet = 0.2,zoutlet = 0.1 for the classical limit, the temperature is θinlet = 0.5, θoutlet= 0.5, then the pressure ratio will be (Pinlet/P_outlet) = (nθ^g_g^52(z)

32(z))_inlet/(nθ^g_g^52(z)

32(z))_outlet = g⁵

2(z_inlet)/g⁵

2(z_outlet) = 2 for the three cases. Since the D2Q9 square lattice is applied, L can be written as L = (Nx− 1)δx, and H = (Ny− 1)δy where Nx and Ny are the number of lattice nodes in the x- and y-direction, respectively. The accommodation co-ecient σ = 0.6 is used in the simulation. Other values can also be calculated.

To begin with the computation, the desired Kn = λ/H is rst input, where H is the height of the channel. We also set the lattice spacing δx = δ_y = 1. In the simulations of these cases, c^∗ =√

8θ/π is chosen, such that the relaxation time τ can be expressed as

τ = Kn(Ny − 1) ×

√3π

8 + 0.5. (5.2)

Having Kn dened, appropriate Ny and τ could be chosen, which could then be used in the determination of mesh size and the collision propagation

updat-ing procedure, respectively. We used the N = 2 expansion equations set for all the cases computed. The computation domain is (0 ≤ x ≤ 500, 0 ≤ y ≤ 20) and 501 × 21 uniform lattices were used. Several Knudsen numbers covering near continuum, slip and transition ow regimes are calculated. The steady velocity proles for the three statistics, BE, MB, and FD gases for the case of z = 0.2 are shown in Fig. 5.4 (a), (b) and (c), respectively, for three dierent Knudsen numbers to represent the Knudsen, slip and Poiseuille regions. For the small Knudsen number, Kn = 0.05, the characteristic parabolic velocity prole is evident and for Kn = 0.2, the velocity slip at the walls can be clearly observed. Again, the prole for MB gas lies always in between that of the BE and FD gas and for small Knudsen number, the three proles get closer to each other. The mass ow rates for all three statistics, BE, MB, and FD gases for the case of z = 0.2 for Knudsen number covering Knudsen, slip and Poiseuille regions are shown in Fig. 5.4 (d). Seven values of Knudsen number from 0.06 to 6.0 were calculated. The Knudsen minimum can be clearly identied for all three statistics and the prole for MB gas lies always in between that of the BE and FD gas. The Knudsen minimum is found to be near Kn = 0.6.

Basically, the Knudsen minimum of a pipe of channel ow can be viewed and explained as a phenomena that appears when the ow passing through the competition between the classical Poiseuille continuum ow and the Knudsen

ow and the value of Knudsen number at this minimum should lie in the slip and transition regime. It is also found that the Knudsen number value at Knudsen minimum is very sensitive to the specularity condition (specied by σ) of the wall surface. Our value obtained here is in agreement with that reported in the literatures (See [106][107][108]). For example, in [106], rst observation of the Knudsen minimum in normal liquid ³He was reported and the position of Knudsen minimum was found to lie at Knudsen number of