Direct simulation of a permeable membrane

HAL Id: jpa-00212428

https://hal.archives-ouvertes.fr/jpa-00212428

Submitted on 1 Jan 1990

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-lished or not. The documents may come from

teaching and research institutions in France or

abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

publics ou privés.

Direct simulation of a permeable membrane

U. Brosa

To cite this version:

1051

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₎ Abstract. 2014 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

hydrody-namics which _represents a membrane

_by

a

_boundary

condition. With the

_scattering

centers we obtain

different, but 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-ods of theoretical

_{hydrodynamics.}

1

_Phys.

France 51

_{( 1990)}

1051-1053 1 er JUIN _1990,

Classification

Physics

A bstracts 05.50-7.SSM

LE

JOURNAL

DE

_PHYSIQUE

Tome 51 N° 11 1 er JUIN 1990

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-macro-scopic

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

equi-libration,

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

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.

Fig.

1. - Flow thru two channels connected

by

a membrane. All _parts show flows with

_Reynolds

number 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.

In classical

_{hydrodynamics[7],}

the behavior of the membrane is described

_by

the

_boundary

con-dition

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.

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.

l’m

_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.

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.