Variational Approach to Plane Poiseuille Flow with General
Boundary Conditions
Carlo Cercignani, Maria Lampis and Silvia Lorenzani
Dipartimento di Matematica, Politecnico di Milano,
Milano, Italy 20133
Abstract. We consider the plane Poiseuille problem in the transition regime, in order to calculate the flow rate versus an
inverse Knudsen number in the case of different boundary conditions for the Boltzmann equation. The adopted technique is
a variational one, [2], [3], [4], which applies directly to the integrodifferential form of the Boltzmann equation and allows to
take into account general models of boundary conditions for the B.E., formulated in the frame of the scattering kernel theory.
THE POISEUILLE PROBLEM
Let us consider two plates separated by a distance d, and a gas flowing parallel to them in the z direction, the
flow resulting from a pressure gradient (the temperature of the wall, Tw , being supposed constant). Assuming that
the pressure gradient is small, the problem is linearized, by putting f
f0 1 h , where f0 is the Maxwellian in
equilibrium with the walls at x
d 2 and x is the coordinate normal to the walls; h is the perturbation which
satisfies the linearized B.E. The molecular velocity c is given in 2RTw 1 2 units (R is the gas constant). We assume
the linearized BGK model for the collision operator. Then we define [1], [2], [3]
1
Z x cx
c2y c2z
e
cz h x c dcy dcz
(1)
We put also
1
2
qx
e
c2x
1
Z x cx1 dcx1
(2)
q x is the mass velocity of the gas. The flow rate (in unit time through unit thickness) is given by
d 2
F
q x dx
(3)
S
(4)
d 2
The B.E. may be written
D
where DZ
cx
Z
x,
S
k 2, k
1
p
LZ
p
z
LZ
and
1
1
2
e
c2x
1
Z x cx1 dcx1
Z x cx
is the mean-free time.
To the Boltzmann equation we must add the appropriate boundary conditions on the two plates at x
CP663, Rarefied Gas Dynamics: 23rd International Symposium, edited by A. D. Ketsdever and E. P. Muntz
© 2003 American Institute of Physics 0-7354-0124-1/03/$20.00
141
(5)
d 2.
BOUNDARY CONDITIONS
In the case of thermal diffusion the velocity distribution function of the gas reemitted by a wall is a Maxwellian
corresponding to the temperature of the wall. In the case of our problem we have [1]
f
d 2 sgncx z c
f0 z c
1
kz
cx
cx
0
0
3
2
0
exp
c2
(6)
where
1
1
sgncx
We obtain
h
d 2 sgncx c
0
(7)
d 2 sgncx cx
0
(8)
and thus
Z
More general boundary conditions may be considered. According to Maxwell the reemitted molecules are partly
reflected by the wall in a specular fashion and partly diffused in a completely accommodated fashion, in conformity
with a Maxwellian distribution corresponding to the wall temperature.
In the late 1960s the problem of writing the boundary conditions for the Boltzmann equation was formulated
rigorously in terms of the so called "scattering kernel" theory [2], [3]. Under some assumptions, the distribution
functions of the reemitted and of the impinging molecules are related by
cn f c
cn 0
cn f c
c
c dc
cn
0
(9)
where cn c n and n is the unit vector normal to the wall pointing into the gas. The kernel
c
c satisfies the well
known properties of positivity, normalization and reciprocity. In the frame of the scattering kernel theory, Maxwell’s
model is equivalent to choosing
c
c
1
c
cR
f0 c c n
c n
0; c n
0
(10)
where cR c 2n n c , is the fraction of molecules diffusely scattered, is "Dirac’s delta" and f0 is the Maxwellian
in equilibrium with the wall. Another kernel with two adjustable parameters (CL model) was proposed [5]
c
c
n t
2
2 t 1
cn exp
RTw 2
cn
0
cn
1 n cn 2
2RTw n
c2n
0;
0
2;
t
ct
t
1
2
0
t
n
2
t ct
2RTw
I0
1
1
n cn cn
n RTw
(11)
In (11) I0 denotes the modified Bessel function of first kind and zeroth order; the two parameters n and t depend
on the physical nature of the gas and the wall as well as on the temperature of the latter: n and t are the
accommodation coefficients for normal energy and for tangential momentum, respectively. The accommodation
coefficient for tangential energy turns out to be t 2
t . The CL model recovers, as limiting cases, the specular
reemission (for n
0)
and
the
diffuse
reemission
(for
1 . Depending on the models for the scattering
n
t
t
kernel, we can obtain the boundary conditions for Z
d 2 sgncx cx .
According to this theory, the boundary conditions for the perturbation h turn out to be
h
h0
Kh
(12)
where h and h concern, respectively, the reemitted and the impinging molecules; h0 is a source term, which vanishes
in the case of our problem, and K denotes the operator
Kh
c
c h
c dc
cn 0
For thermal diffusion h
0 and Kh
cn
0
(13)
0. In the case of specular reflection
c
c
c
142
c
2n n c
(14)
we easily obtain Z cn
KZ
In the general case we have
cn
Z
cn and K is the reflection operator with respect to cn .
1
Z
c2y
exp
c2z cz Kh
c dcy dcz
(15)
and then
1
Z
cn 0
B c cn h
c dc
(16)
where
B c cn
c2y
exp
c2z cz
c
c dcy dcz
(17)
We remark that all the kernels that we have proposed [5], [6], [7] are factorized as follows
c
c
ct
t
ct
n
cn
cn
(18)
Therefore we have
Bc
cn
cn
n
cn
c2y
exp
c2z cz
ct
t
ct dcy dcz
(19)
Moreover in all the kernels above mentioned the tangential factor has the form
t
con a
ct
ct
a2
1
1
exp
1
a2
1
ct
act
2
(20)
1. Some simple calculations give
Bc
Z
cn
cn
n
cn
a
n
cn 0
cn acz exp
cy2
cn
cn dcn
cn Z
cz2
(21)
(22)
that may be written
Z
KZ
(23)
where
KZ
a
cn 0
cn
n
cn Z
cn dcn
(24)
VARIATIONAL METHOD OF SOLUTION
We introduce the following functional J of the test function Ẑ [2], [3], [4]
J Ẑ
Ẑ P DẐ
where P is the reflection operator and
and
1 2
hg
LẐ
B
2 PS Ẑ
g
d 2
exp
1 2
B
K Ẑ
PẐ
B
(25)
denote two scalar products defined as follows
d 2
h
Ẑ
cx 0
c2x h x cx g x cx dcx dx
cx exp
c2x h
cx g
cx dcx d
(26)
(27)
where
in this case denotes the sum of two terms, concerning the two plates.
The variation J of J vanishes if and only if Ẑ Z where Z is the solution of Eq.(4) with the boundary conditions
(23). Moreover
k
J Z
PS Z
1Z
(28)
2
We want to evaluate F, which is related to J Z
F
2
J Z
k
143
(29)
EVALUATION OF THE FLOW RATE
We introduce appropriate test functions; the evaluation of min J Ẑ gives an approximation of J Z and then an
evaluation of F versus the inverse Knudsen number. We remark that an important point is that, in the case of the
Poiseuille problem, the choice of the trial functions is difficult, since in the free molecular limit the flow rate tends to
an infinite value. We put
d
q̂ x
x2
Ẑ
sgncx
(30)
2
where , , , are constants to be varied in order to obtain the best value of J Ẑ . Then we consider the integral form
of the B.E.
Ẑ x cx
d
2 sgncx
x t
x
d sgnc exp
cx
x
2
exp
x
d
2 sgncx
cx Ẑ
q̂ t
k
2
cx
cx dt
(31)
We put in Eq.(31) the expressions of q̂ x and Ẑ d2 sgncx given by Eq.(30) and, by means of analytical calculations,
we obtain the expression of the trial function Ẑ x cx :
exp
x
Ẑ x cx
d
2 sgncx
k
x2
2
2
cx
k
2x cx 2 2c2x
d 2 4 d cxsgncx
2
2 c2
x
(32)
Then we put Ẑ x cx given by Eq.(32) in Eq.(25). The result of some calculations gives the expression of J Ẑ , which
is a polynomial of the second order with respect to the constants , , , that are to be determined. The derivatives of
J Ẑ with respect to , , vanish in correspondence of the optimal values of , , in the trial function we have
chosen, Eq.(30). Plugging these values into the expression of J Ẑ , we find the optimal value of the functional J Z
and arrive to the numerical results for the flow rate.
Once the pressure gradient and the distance between the plates are fixed, the dependence of the volume flow rate on
the pressure is given by the following non dimensional quantity [1]:
Q
F
2
2 kd
4
2
J Z
k 2
(33)
where
d
is the rarefaction parameter (inverse Knudsen number).
For thermal diffusion, it was shown [1], [2] that the values of Q, in the two limiting cases of free molecular flow and
hydrodynamic regime, can be approximated by:
1
2
Q
1
6
is the numerical ratio of the slip constant
log
0
(34)
Q
where
(35)
to
. For the BGK model
1 0161 [1].
NUMERICAL RESULTS
The numerical results, showing the flow rate (Q) versus the inverse of the Knudsen number ( ), are presented in Tables
1 to 4. We compare our findings with those published by other authors.
The outcomes of our method, obtained assuming the complete accommodation of the molecules on the walls, are
shown in Table 1. Since an estimate of J Z is given through the evaluation of min J Ẑ , the variational method
approximates Q
from above. This is said in order to gain a better comparison with [1]. The ability of our analytical
approach to reach very small values of the rarefaction parameter has been used to test the validity of the free
molecular limit of the flow rate given by (34) (see Figure 1).
In Tables 2 and 3 we list the results obtained in the case of Maxwell’s boundary conditions, according to which
part of the gas is reemitted specularly. In going from the complete thermal diffusion (
1) to the entirely specular
reflection (
0) the Poiseuille flow rate increases significantly.
144
TABLE 1. Poiseuille flow rate Q vs for thermal diffusion. Comparison between our
results
, Cercignani & Daneri’s results (1963) [1] and Barichello et al.’s results
(2001) [10].
From above
1
1
1
1
10
10
10
10
0 01
0 02
0 03
0 04
0 05
0 06
0 07
0 08
0 09
01
02
03
04
05
06
07
08
09
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
90
95
10
10 5
6
5
4
3
8 1650
6 8550
5 5574
4 2736
3 0497
2 7111
2 5239
2 3969
2 3022
2 2276
2 1664
2 1151
2 0710
2 0327
1 8082
1 7025
1 6409
1 6019
1 5762
1 5592
1 5483
1 5418
1 5387
1 5534
1 5948
1 6489
1 7102
1 7763
1 8454
1 9169
1 9901
2 0646
2 1401
2 2166
2 2937
2 3715
2 4497
2 5285
2 6075
2 6870
2 7667
2 8467
3 0499
2 7114
2 5242
2 3973
2 3026
2 2228
2 1668
2 1154
2 0716
2 0331
1 8087
1 7030
1 6415
1 6025
1 5769
1 5599
1 5491
1 5427
1 5396
1 5546
1 5963
1 6497
1 7117
1 7775
1 8468
1 9184
1 9928
2 0664
2 1421
2 2185
2 2957
2 3734
2 4516
2 5302
2 6091
2 6880
2 7669
2 8458
1
From below
10
Mean value
2 3023
2 0326
1 8080
1 7021
1 6403
1 6010
1 5751
1 5578
1 5465
1 5367
1 5362
1 5484
1 5862
1 6481
1 7091
1 7746
1 8432
1 9140
1 9863
2 0598
2 1343
2 2094
2 2851
2 3611
2 4375
2 5111
2 5909
2 6678
2 7447
2 8216
2 0328
1 8083
1 7025
1 6409
1 6017
1 5760
1 5588
1 5478
1 5397
1 5379
1 5515
1 5912
1 6489
1 7104
1 7760
1 8450
1 9162
1 9895
2 0631
2 1382
2 2135
2 2904
2 3672
2 4445
2 5221
2 6000
2 6779
2 7558
2 8337
2 0327
1 7025
1 6019
1 5592
1 5418
1 5387
1 5949
1 9908
2 2949
2 6093
Experiments on gas flow in capillaries with highly polished walls already showed that the observed gas volume flow
rates are greater than those calculated theoretically assuming complete diffuse molecular scattering by the walls [8].
The discrepancy between our results and those reported in [10] is negligible for
1 but it increases decreasing
the accommodation coefficient , in the free molecular regime. The source of this discrepancy should be related,
according to us, to the different ways in which the methods used (the variational approach in our work and the discrete
ordinates method in [10], [11] ) treat the double divergence met for
0 in the entirely specular reflection (
0).
The variational approach faces the problem (shared by all the boundary conditions considered) of a nearly singular
matrix in solving the linear system formed putting to zero the derivatives of J Ẑ with respect to the parameters , ,
. This feature is peculiar only to a narrow free molecular flow region (
0 1). Due to the numerical source of such
singularity, we replaced the full expression of the vanishing elements of the matrix with their limit for
0 in the
145
TABLE 2.
results
Poiseuille flow rate Q vs for Maxwell’s boundary conditions. Comparison between our
and Barichello et al.’s results (2001) [10].
0 80
0 88
0 96
10
1
1
1
1
10
10
10
10
0 01
0 05
0 10
0 30
0 50
0 70
0 90
1 00
2 00
5 00
7 00
9 00
6
11 2015
9 3720
7 5546
5 7516
4 0201
2 9586
2 5853
2 1993
2 0874
2 0320
2 0057
1 9992
2 0410
2 4369
2 7437
3 0601
5
4
3
3 0897
2 7077
2 2448
2 1023
2 0388
2 0092
2 0019
2 0414
2 4382
2 7461
3 0635
10
9 9627
8 3411
6 7317
5 1370
3 6096
2 6722
2 3388
1 9824
1 8710
1 8195
1 7963
1 7911
1 8383
2 2339
2 5394
2 8546
1 00
10
2 7383
2 4060
2 0011
1 8766
1 8220
1 7976
1 7921
1 8386
2 2351
2 5414
2 8576
8 7574
7 3436
5 9422
4 5549
3 2296
2 4154
2 1224
1 7920
1 6857
1 6396
1 6201
1 6162
1 6692
2 0645
2 3686
2 6828
2 4374
2 1482
1 7945
1 6863
1 6399
1 6202
1 6163
1 6694
2 0655
2 3704
2 6853
10
8 1648
6 8550
5 5574
4 2736
3 0485
2 2956
2 0229
1 7025
1 6019
1 5592
1 5418
1 5387
1 5947
1 9899
2 2933
2 6070
2 3023
2 0327
1 7025
1 6019
1 5592
1 5418
1 5387
1 5949
1 9908
2 2949
2 6093
TABLE 3. Poiseuille flow rate Q vs for Maxwell’s boundary conditions. Comparison between our results
, Barichello et al.’s results (2001) [10] and Siewert’s results (2002) [12].
0 10
0 30
0 50
0 70
10
1
1
1
1
10
10
10
10
0 01
0 05
0 10
0 30
0 50
0 70
0 90
1 00
2 00
3 00
5 00
7 00
9 00
6
5
4
3
35 7878
32 1378
28 4926
24 8348
21 2359
19 0675
18 3751
17 7585
17 6709
17 6079
17 5683
17 5554
17 5605
17 6681
17 9605
18 2798
18 6058
21 3780
18 2484
15 1267
12 0048
8 9536
7 1033
6 4962
5 9344
5 8397
5 7764
5 7390
5 7273
5 7412
5 8504
6 1398
6 4554
6 7786
16 4212
13 8118
11 2126
8 6219
6 1073
4 5744
4 0575
3 5600
3 4576
3 3953
3 3610
3 3510
3 3750
3 4859
3 7724
4 0844
4 4048
5 2233
4 5564
3 7785
3 5444
3 4377
3 3839
3 3682
3 3766
3 7744
4 0881
4 4102
12
7 2100
5 2428
4 5801
3 8061
3 5718
3 4640
3 4090
3 3928
3 5037
3 7884
4 1005
4 4215
12 8148
10 7254
8 6477
6 5833
4 5938
3 3761
2 9538
2 5296
2 4198
2 3610
2 3315
2 3237
2 3591
2 4717
2 7554
3 0640
3 3816
region
0 1, in order to eliminate numerically computed integrals. This trick enabled us to reach very small values
of the accommodation coefficient and of the rarefaction parameter .
A further progress in understanding the source of discrepancy between the two methods could be done comparing
the results for
0 5. Unfortunately, in [10] these results are not reported and in [13], where some points relative to
0 25 are calculated, the critical region
0 1 is completely missing.
In Table 4 we list the results for the Cercignani-Lampis scattering kernel, with two model parameters, t , n , and
compare our outputs with those reported in [13] where the S model of the Boltzmann equation was numerically
solved by the discrete velocity method. The discrepancy between the two methods near the free molecular regime is
remarkably reduced in comparison with the previous case even at small values of t and n .
From Table 4 and Figure 2, one can see that the Poiseuille flow rate significantly depends on the accommodation
146
TABLE 4. Poiseuille flow rate Q vs for Cercignani-Lampis’ scattering kernel. Comparison between our results
and Sharipov’s results (2002) [13].
n
0 25
1
10
6
1
10
4
0 01
01
02
10
20
35
10 0
0 25
0 50
0 75
10
0 25
0 50
0 75
10
0 25
0 50
0 75
10
0 25
0 50
0 75
10
0 25
0 50
0 75
10
0 25
0 50
0 75
10
0 25
0 50
0 75
10
0 25
0 50
0 75
10
0 25
0 50
0 75
10
n
05
13
t
13 979
10 269
8 922
8 166
11 371
7 660
6 313
5 557
8 766
5 100
3 781
3 048
7 380
3 887
2 672
2 023
7 155
3 608
2 422
1 803
6 898
3 340
2 145
1 539
6 920
3 372
2 188
1 595
7 096
3 550
2 368
1 776
8 113
4 558
3 366
2 766
n
0 75
13
9 230
5 262
3 822
3 052
7 918
4 109
2 754
2 039
7 536
3 808
2 501
1 817
6 944
3 370
2 164
1 554
6 940
3 391
2 205
1 611
7 110
3 566
2 384
1 792
8 125
4 572
3 380
2 780
13 723
10 099
8 837
8 166
11 116
7 490
6 228
5 557
8 562
4 958
3 709
3 048
7 342
3 840
2 643
2 023
7 105
3 605
2 418
1 808
6 884
3 330
2 139
1 539
6 917
3 371
2 187
1 595
7 094
3 549
2 367
1 776
8 101
4 550
3 363
2 766
8 699
5 014
3 726
3 052
7 596
3 952
2 693
2 039
7 319
3 699
2 457
1 817
6 913
3 352
2 157
1 554
6 963
3 389
2 204
1 611
7 110
3 565
2 383
1 792
8 114
4 564
3 377
2 780
n
10
13
13 581
10 004
8 790
8 166
10 974
7 395
6 181
5 557
8 445
4 877
3 668
3 048
7 316
3 808
2 625
2 023
7 102
3 598
2 411
1 808
6 870
3 321
2 134
1 539
6 915
3 369
2 186
1 595
7 094
3 549
2 367
1 776
8 092
4 544
3 360
2 766
8 480
4 893
3 675
3 052
7 441
3 865
2 655
2 039
7 202
3 633
2 429
1 817
6 889
3 338
2 150
1 554
6 932
3 386
2 203
1 611
7 109
3 565
2 383
1 792
8 104
4 557
3 373
2 780
13
13 484
9 939
8 757
8 166
10 877
7 331
6 149
5 557
8 364
4 820
3 639
3 048
7 296
3 785
2 611
2 023
7 101
3 571
2 399
1 808
6 856
3 311
2 129
1 539
6 912
3 367
2 186
1 595
7 094
3 549
2 367
1 776
8 083
4 538
3 357
2 766
8 369
4 824
3 642
3 052
7 356
3 812
2 630
2 039
7 134
3 590
2 408
1 817
6 871
3 326
2 144
1 554
6 928
3 384
2 202
1 611
7 109
3 565
2 383
1 792
8 095
4 551
3 370
2 780
coefficient of the tangential momentum t in the whole range of the Knudsen numbers considered: it increases
decreasing t and ddQ is not constant. The dependence on the energy accommodation coefficient n is very weak:
t
it is slightly evident only near the minimum of the curves representing Q and in the limit
0.
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1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
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Cercignani, C., Rarefied Gas Dynamics, Cambridge Texts in Applied Mathematics, Cambridge University Press (2000).
Cercignani, C., J. Stat. Phys. 1, 297-311 (1969).
Cercignani, C., Lampis, M., Transport Theory and Statistical Physics 1, 101-114 (1971).
Cercignani, C., Lampis, M., Mechanics Research Communications 26, 451-456 (1999).
Cercignani, C., Lampis, M. and Lentati, A., Transport Theory and Statistical Physics 24, 1319-1336 (1995).
Porodnov, B.T., Suetin, P.E., Borisov, S.F., Akinshin, V.D., J. Fluid Mech. 64, 417-437 (1974).
Siewert, C.E., Garcia, R.D.M. and Grandjean, P., J. Math. Phys. 21, 2760-2763 (1980).
Barichello, L.B., Camargo, M., Rodrigues, P., Siewert, C.E., Z. angew. Math. Phys. 52, 517-534 (2001).
147
FIGURE 1. Poiseuille flow rate Q vs for thermal diffusion in the limit
the asymptotic values given by (34) (dashed) is shown.
0. A comparison between our results (solid) and
FIGURE 2. Poiseuille flow rate Q vs for Cercignani-Lampis’ scattering kernel.
0 5 (dashed) and t 0 3 (dot dashed).
t
n
11. Barichello, L.B., Siewert, C.E., Z. angew. Math. Phys. 50, 972-981 (1999).
12. Siewert, C. E., J. Quantitative Spectroscopy & Radiative Transfer 72, 75-88 (2002).
13. Sharipov, F., Eur. J. Mech. B/fluids 21, 113-123 (2002).
148
0 5 and
t
0 9 (solid),
t
0 7 (dotted),