Unified Flow Model

Before we start developing velocity and flowrate scaling models, we continue to examine the validity of continuum-based slip models for the the transitional flow regime . In the figure below we present velocity profiles obtained by DSMC and linearized Boltzmann solutions at Kn=0.6. Our models are indicated as Model A and Model B We have also included models by Cercignani, Deissler, Maxwell, Schamberg, and Hsia and Domoto. Our Model A represents the velocity distribution with least error compared to other slip-boundary conditions. Model B does very well in the center of the channel but gives slight deviations near the walls. The maximum deviation of the model is observed at the slip location, where Model A is somewhat better. However, Model A gives larger errors than Model B towards the centerline of the channel. The errors in first-order and other second-order boundary conditions are significant. The maximum error occurs near the wall with 0.32 units of overestimation using Schamberg's boundary conditions. The models by Cercignani and by Deissler are almost identical for this case and therefore only one is shown in the figure.

Non-dimensionalized velocity distribution in half of a micro-channel for Kn=0.6 (left), and error in the solution of Navier-Stokes equations subject to various slip boundary conditions (right).

It is known from Knudsen's (1909) and Gaede's (1913) experiments in the transition flow regime that there is a minimum in the flowrate in pipe and channel flows at about Kn= 3 and Kn=1, respectively. The asymptotic limit of flowrate in a pipe is independent of the Knudsen number in the free-molecular flow regime. However for a channel, the asymptotic limit of flowrate depends on the channel legth to height ratio (more specifically it is proportional to Log_e(L/h)). If the channel length L is greater than the mean-free-path, the asymptotic limit of flowrate in the channel is proportional to Log_e(Kn). This logarithmic behavior is attributed to the degenerate geometry (infinite width) of the two-dimensional channel. For a finite width channel the flowrate tends towards a finite limit proportional to Log_e(L/h)) as was found in the experiments of Gaede (1913).

The variation of flowrate in a channel as obtained by DSMC simulations in the transition and early free-molecular flow regimes is shown in the below figure. The flowrate data is presented at the average Knudsen number in the channels The Knudsen's minimum is clearly captured by the DSMC results at Kn=1.0. The DSMC solution is compared with the semi-analytic solutions of Cercignani and Daneri and Huang etal is shown where the linearized Boltzmann equations are solved using the BGK model.

Variation of normalized flowrate in a channel as a function of Knudsen number. Comparisons are made between DSMC results and solutions of the linearized Boltzmann equation

Velocity Scaling

From the DSMC results and solutions of the linearized Boltzmann equation, it is evident that velocity profiles remain approximately parabolic for a large range of Knudsen number. This is also consistent with the Navier-Stokes or Burnett equations in long channels. Based on this observation we can then start with a parabolic velocity profile and write the dimensional form for velocity distribution in a channel of thickness h

where, the magnitude of the velocity shown by F is a function of the pressure gradient, dynamic viscosity, channel dimensions. Possible variation of the velocity magnitude due to the rarefaction effects are shown by mean-free-path (lambda) dependence of F. This velocity distribution has to be corrected with an appropriate slip boundary condition. For this purpose we used the general slip boundary conditions.

The general slip boundary condition given above has the following properties: The value of coefficient b in the general slip boundary conditions is unknown. Also the magnitude of the velocity distribution F is unknown. Therefore we non-dimensionalized the velocity distribution with the average velocity at the corresponding location along the channel. With this non-dimesionalizetion we obtained the following velocity distribution:

In the next figure we plot the non-dimensional velocity variation obtained in a series of DSMC simulations for Kn=0.1; Kn=1.0; Kn=5.0; Kn=10.0. We can now use above equation and compare with the DSMC data, by varying the parameter b which for b=0 corresponds to Maxwell's first-order and for b=-1 to our second-order boundary condition in the slip flow regime. Here we find that for b=-1 above equation still results in an accurate model of the velocity distribution for a wide range of Knudsen numbers.

Velocity profile comparisons of the model with DSMC solutions for Kn=0.1; Kn=1.0; Kn=5.0; Kn=10.0.

In the next figure we show the non-dimensionalized velocity distribution along the center-line and along the wall of the channels for the entire Knudsen number regime considered here, i.e. 0.01 < Kn < 30. We have included in the plot data for the slip-velocity and center-line velocity from 19 different DSMC runs, of which 15 of them were for nitrogen (diatomic molecules), and 4 of them were for helium (monatomic molecules). The differences between the nitrogen and helium simulations are negligible and thus this velocity scaling is independent of the gas type. The linearized Boltzmann solution of Ohwada etal for a monatomic gas is also shown in the figure by triangles. This solution closely matches the DSMC predictions. For reference we have also included the prediction given by above equation for various values of b. Maxwell's first-order boundary condition (b=0) (shown by solid line) predicts erroneously a uniform non-dimensional velocity profile for large Knudsen number. The break-down of slip flow theory based on the first-order slip boundary conditions is realized around Kn=0.1 and Kn=0.4 for the wall and the center-line velocity, respectively. This finding is consistent with commonly accepted limits of the slip flow regime. The prediction using b=-1 is shown by small-dashed lines. The corresponding center-line velocity closely follows the DSMC results while the slip-velocity of the model with b=-1 deviates from DSMC in the intermediate range for 0.1 < Kn < 5. Predictions of the model corresponding to b=-2 (dashed-doted lines) overestimate the centerline velocity for the entire flow regime and it also underestimates the asymptotic slip velocity limit for large values of Kn.

For a comparison we also included similar predictions by the second-order slip boundary condition of Hsia and Domoto. The form of their boundary conditions is similar to Cercignani's, Deissler's and Schamberg's, and they all become invalid at around Kn=0.1. Only the general slip boundary condition given above preserves the scaling of the velocity profiles accurately.

Velocity scaling at wall and center-line of the channels for slip and transition flows.

Flowrate Scaling

Volumetric flowrate depends on the magnitude of the velocity profile, which has been cancelled in determination of the velocity scaling in the previous section. Here we try to identify the magnitude of the velocity profile given in the following functional form: F(dP/dx, mu, h, lambda).

Using the Navier-Stokes equations the volumetric flow rate is given as:

In continuum as well as in the free-molecular flow regime the volumetric flowrate is proportional to the pressure gradient (dP/dx), and therefore we can normalize our volumetric flowrate data with (dP/dx), which results in the following variation of normalized volumetric flow rate as a function of Kn.

Volumetric flowrate (per channel width) per pressure gradient as a function of Kn for nitrogen flow.

If we use the Navier-Stokes description of the flowrate, both the first-order slip model (dashed-lines) and the velocity scaling corrected model (shown by dashed-dotted lines) gives errenous results. This indicates that a further correction to the above formula is necessary for rarefied flows. As we examine the flowrate equation given by the Navier-Stokes equations, the dynamic viscosity in the denominator seems to be a good candidate for explaining the deviations from continuum behavior as the Kn is increased.

Dynamic viscosity (mu) is a transport coefficient that is based on diffusion of momentum through intermollecular collisions. However, as the flow gets more rarefied the amont of intermollecular collisions gets significantly less, and molecule-wall collisions becomes more dominant. The kinetic theory definition of dynamic viscosity is proportional to mean-free-path*mean molecular speed*density (mu~lambda c rho). This is true for cases when lambda < channel thickness (or diameter). When mean-free-path becomes larger than the channel thickness (or diameter), use of mean-free-path in the definition of the dynamic viscosity becomes meaningless and instead a typical channel dimension should be used. With this discussion we have shown that the dynamic viscosity variation is given as:

where mu_o is the dynamic viscosity and Cr(Kn) is called "rarefaction coefficient". It is seen that the rarefaction coefficient Cr(Kn)=(1 + alpha Kn) gives reasonable answers, where the value of alpha is determined as 2.2 for nitrogen and 1.6 for helium flow. See also the performance of the model for the volumetric flowrate variation given in the previous figures.

Variation of the "rarefaction coefficient", Cr(Kn) as a function of Kn.

The volumetric flowrate is given as:

The mass flowrate is given as:

where Pi and Po shows inlet and exit pressures, L is the channel length, h is the channel thickness. The comparison of the corrected model with the DSMC data is given in the figure below. The model predicts the Knudsen's minimum obtained by the DSMC calculations quite accurately at Kn = 1.0. Consistent with the DSMC solutions the model predicts a flowrate independent of Kn for the free-molecular flow limit. However, this constant flowrate for larger Kn is slightly (13 %) lower than the DSMC predictions. The reason of this deviation is that in the channel flow the corresponding asyptotic free-molecular limit depends on the channel length to height ratio, and our current model is not able to predict such a geometry dependent variation. For comparisons with the model and the DSMC data, we also plot the flowrate obtained by the continuum and the first-order slip models in the figure. The continuum model behaves like 1/Kn and gives the wrong variation while the slip flow model results in flowrate values three times less than the DSMC calculations.

Variation of mass flowrate (per pressure drop (Pi-Po) as a function of exit Knudsen number in the channel. Knudsen's minimum is captured by the model at Kn = 1.

We also obtain the pressure distribution very accurately. This is demonstrated in the figure below, where deviations of pressure distribution from linear pressure drop for Nitrogen flow with Kn_o =0.2 and inlet to exit pressure ratio of 2.28 is shown. It is seen that the analytic model (shown by the solid line) gives very close results to the DSMC simulations.

Curvature in the pressure distribution for nitrogen flow at Kn=0.2 obtained by mu-flow, DSMC and unified model.

Unified Model for Pipe Flows

We have developed a similar model for pipe flows. Our DSMC simulation (for axisymmetric flows) and linearized Boltzmann solution in the literature once again gives b=-1 as a reasonable value for decribing the non-dimensional velocity scaling. We have also assumed that the rarefaction coefficient has the same form of Cr=1 + alpha Kn. Under these circumstances the velocity scaling is given as:

and the volumetric flowrate is given as:

The mass flowrate is given as:

The value of alpha can be determined by using the corresponding free molecular mass flowrate limit:

For our model the correct free molecular flowrate is achieved if the value of alpha is sellected as:

In the figure below we show the variation of mass flowrate (non-dimesionalized with the corresponding free molecular value) as a function of Kn. For comparison purposes we present Knudsen's two equation model as well as Loyalka and Hamoodi's linearized Boltzmann solutions. Our model with b=-1 and alpha determined by above equation is shown by dashed lines. It is seen that the model predicts the flowrate variation with reasonable accuracy. However it can not predict the Knudsen's minimum. Therefore we modified our model based on Loyalka and Hamoodi's linearized Boltzmann solutions and have obtained that the value of alpha in our model should be a function of Kn. This variation is determined as

with this modification our model is able to predict Knudsen's minimum. However, modification of alpha in this form results in coorect behaviour for transition and free-molecular flows, but it should not be used in the slip flow regime as the leading order correction to no-slip continuum flow becomes O(Kn^(1/2)) rather than O(Kn) in the slip-flow regime.

Free-molecular scaling of Knudsen's model, Loyalka and Hamoodi's linearized Boltzmann solutions, and axi-symmetric DSMC solutions. Comparisons with the proposed model are also presented.

Conclusions for the Unified Model