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
Our Model A represents the velocity distribution with least error compared to other slip-boundary
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
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
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:
- If b=0, first-order accurate Maxwell's slip conditions are achieved.
- For high Kn flows (Kn -> oo), this slip formula results in a finite slip correction.
- For the slip flow regime (Kn < 0.1), this formula can be expanded by a geometric-series expansion,
and the value of b can be determined in order to give our second-order slip formula.
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
Velocity scaling at wall and center-line of the channels for slip and transition flows.
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
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 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
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
Comparisons with the proposed model are also presented.
Conclusions for the Unified Model
- A universal scaling for the velocity distribution in channel and pipe flows is
established by using the general slip boundary condition.
- The value of parameter b in the general slip boundary condition is independent of the gas type (ie. monatomic and
diatomic molecular simulations did not show any differences).
- Unified flow model predicts the velocity distribution, mass and volumetric flowrate as well as pressure distribution
in the channel and pipe flows for the entire Knudsen number regime.
- The unified model is a single parameter model. For the channel flow the value of alpha and for the pipe flow the value
of alpha_1 should be determined by comparisons to either experimental or numerical data.
- The new model predicts Knudsen's minimum for the channel flow, and use of appropriate value of alpha_1 for
pipe flows is necessary to predict Knudsen's minimum.