Direct simulation of a permeable membrane
U. Brosa
To cite this version:
U. Brosa. Direct simulation of a permeable membrane. Journal de Physique, 1990, 51 (11),
pp.1051-1053. <10.1051/jphys:0199000510110105100>. <jpa-00212428>
HAL Id: jpa-00212428
https://hal.archives-ouvertes.fr/jpa-00212428
Submitted on 1 Jan 1990
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Tome 51
1 er JUIN 1990
N° 11
LE JOURNAL DE PHYSIQUE
1
Phys.
France 51
( 1990)
1 er JUIN 1990,
1051-1053
1051
Classification
Physics A bstracts
05.50-7.SSM
Short Communication
Direct simulation of
a
permeable
membrane
U. Brosa
HLRZ c/o KFA, Postfach
1913, D-517 Jülich, F.R.G.
(Reçu le 27 mars 1990, accepté le 3 avnil 1990)
Cellular automata are used to compute flow thru a permeable membrane. Scattering
centers constitute the membrane. This is in marked contrast to the approach of classical hydrodynamics which represents a membrane by a boundary condition. With the scattering centers we obtain
Abstract.
2014
more plausible results indicating that simple diffusion is the dominating process in a
porous layer. We have thus a case where cellular automata show superiority over the classical meth-
different, but
ods of theoretical
hydrodynamics.
Several investigations on the applicability of cellular automata on viscous hydrodynamics have
been done meanwhile, see e.g. [1 - 4]. In these studies, solutions of the Navier-Stokes equation
available from other sources were compared with output of cellular automata. It is clear now
that cellular automata can give a truthful picture of liquid reality as long as certain limits are not
transgressed [5]. However, at present those automata are relatively inefhcient so that their main
advantage consists in being foolproof.
Probably cellular automata will find their most relevant applications in the microscopic-macroscopic research. From the viewpoint of hydrodynamics, cellular automata are based on a schematic
model of the molecular motion. Hence one can study problems such as fluctuations and local equilibration, and one can check now much these microscopic properties show up in the macroscopic
behavior of the fluid.
For this an example shall be given right now. It is flow through a porous membrane. For a
microscopic theory, a membrane is a layer of scattering centers which the liquid has to wriggle
thru. The applications of membranes, however, are macroscopic.
We became aware of the problem in a previous study [6]. The geometry is shown in figure 1:
1Bvo channels conduct Poiseuille flow, each on its own. The problem would be trivial if it were
not for a membrane that connects the channels. The maximum velocities are v"zax at the inlets
of both channels but vmax -f- Ovmax and Vm02: - wmax at the outlets of the upper and the lower
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0199000510110105100
1052
channel, respectively. A part of the fluid, therefore, must pass the membrane. It turned out in [6]
that the computed velocity field V(r) depends much on the way in which the membrane is taken
into account. The boundary-value problem displayed in figure 1 is hence a sensitive indicator of
the treatment of porosity.
Flow thru two channels connected by a membrane. All parts show flows with Reynolds number
1.
500. The truput is 50%, i.e. làvmo:a:/vmo:a:1 = 0.5. We have solid walls in the middles, indicated by the full
lines, and mirror planes above and below, depicted by the dashed lines. See the text for the meaning of the
four parts and the papers [6, 8] for details concerning the geometry and the algorithms.
Fig.
-
In classical hydrodynamics[7], the behavior of the membrane is described by the boundary condition
meaning that the membrane allows only for fiow perpendicular to its surface. Equation (1) is to
hold on the dotted line in figure 1. However, its correctness may be doubted. First, experts in
hydrodynamics notice at once that (1) is just the slip boundary condition on a vertical wall as used
in the theory of idéal liquids. The application of slip boundary conditions to problems with viscous
flow along solid bodies is known to give misleading results. Second, there is no reason why the
fluid should not cross the membrane in an oblique direction.
With cellular automata we can easily look into this problem. All we have to know is that a
cellular automaton opérâtes on a grid. Then we select some of the nodes in the vicinity of the
dotted line and let them become points at which the molecules experience backward reflections.
For cellular automata, this amounts to non-slip boundary conditions. The difference to a solid
wall is only that the scattering centers must not be dense as the liquid has still to find ways. The
realization by a computer program is actually very simple. Details can be found in [6].
1053
The essence of the results is distilled in figure 1. Part (a) shows unhampered thruput. Instead
of the membrane, there is just a hole. Due to inertia effects the fluid passes the hole preferentially
at its downstream edge. In Part (b) the boundary condition (1) is put into force. It only makes
the velocities vertical. As mentioned already above, (1) is an equation for idéal liquids. Hence
we cannot expect a real impediment of the flow. Part (c) shows what happens when every second
node on the dotted line is made a scattering center. We see now a much broader distribution of
the velocities over the membrane. This is expected since the scattering centers induce diffusion.
In addition, the velocities pass the membrane obliquely. For the results exhibited in Part (d), only
every third node was used as scattering center. The only point to be noticed here is the small
difference with Part (c). In addition 1 have modelled the porous membrane in several different
ways. For example, 1 took nodes not only from the dotted line but from a strip around that line. I
composed nodes to form small hexagons, and so forth. Invariably pictures as shown in figure 1 (c)
and (d) were obtained. This is an almost trivial result as it is always plain diffusion which features
the flow in the porous layer. It is nevertheless remarkable that it takes so few scattering centers
to get full diffusion.
Part (b) of figure 1 shows results presented first in [6]. The juxtaposition with Part (c) exhibits
that we have in fact a significant modification.
Finding the same results by computational methods of classical hydrodynamics seems to be
possible. However, one has to discard the boundary condition (1) and must model the membrane
by a layer of small but finite thickness. In this layer, inertia is damped so that the Navier-Stokes
equation can be replaced by a diffusion equation. As a boundary condition between the porous
layer and the free flow, continuity of the velocity field ought be stipulated. Such an arrangement
complicates the application of classical methods like finite elements so that cellular automata have
an
edge.
Acknowledgements.
grateful to C. Küttner for suggesting the problem to me. Encouragement by D. Stauffer is
appreciated. Part of this work was done using the new interactive facilities at the HLRZ.
l’m
References
[1] D’HUMIERES D. and LALLEMAND P., Complex Systems 1 (1987) 599.
[2] LIM H.A., Phys. Rev. A. 40 (1989) 968;
LIM H.A., Complex Systems 2 (1988) 45.
[3] HAYOT E and RAJ LAKSHMI M., Physica D 40 (1989) 415.
[4] DUARTE J.A.M.S. and BROSA U., J. Stat. Phys. 59 (1990) 501.
[4] DAHLBURG J.P., MONTGOMERY D. and DOOLEN G.D., Phys. Rev. A 36 (1987) 2471.
[6] BROSA U., KÜTTNER C. and WERNER U., Flow thru a porous membrane simulated by cellular automata
and finite elements, submitted for publication in J. Stat. Phys.
[7] BERMAN A.S., J. Appl. Phys. 24 (1953) 1232.
[8] BROSA U. and STAUFFER D., J. Stat. Phys. 57 (1989) 399.