Effect of a plane wall on a sphere moving parallel to it

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Effect of a plane wall on a sphere moving parallel to it

A. Ambari, B. Gauthier Manuel, E. Guyon

To cite this version:

A. Ambari, B. Gauthier Manuel, E. Guyon.

Effect of a plane wall on a sphere

mov-ing parallel to it.

Journal de Physique Lettres, Edp sciences, 1983, 44 (4), pp.143-146.

�10.1051/jphyslet:01983004404014300�. �jpa-00232155�

Effect of

a

plane

wall

on a

_sphere

_moving

_parallel

to

it

A. Ambari

_(*),

B. Gauthier Manuel and E.

_Guyon

Laboratoire

_{d’Hydrodynamique}

et de _{Mécanique Physique (**),}

E.S.P.C.I., 10, rue

_Vauquelin,

75231 Paris Cedex 05, France

(Re~u le 22 octobre 1982, revise le 14 _{decembre, accepte} le 23 decembre ₁₉₈₂₎

Résumé. 2014 L’étude de l’effet d’une

_{paroi plane}

sur la force

_{hydrodynamique}

exercée sur une

_sphère

en mouvement de translation

_parallèle,

_{pose des}

_{problèmes expérimentaux.}

Nous

_présentons

ici une méthode _{originale permettant} la mesure de cette force tout en _imposant une valeur constante de la distance entre la

_sphère

et la _paroi. Les résultats de cette

_expérience

sont confrontés aux théories de

O’Neill, de Goldman et al., et de Faxén. Abstract. 2014 The effect of _a

plane wall on the

_hydrodynamic

force exerted on a _sphere

_{moving parallel}

to it presents

experimental

difficulties. We describe an

_original

method for the measurement of the force while

_keeping

constant the distance between the wall and the

_sphere.

The results are _compared with the theoretical models of O’Neill, of Goldman et _al., and of Faxén.

Classification

Physics Abstracts

47.10 - 47. 55K - 47. 80

The

_{study of hydrodynamic suspensions}

of

_{spherical objects}

in fluids is

_{complex [1] :}

it involves the determination of

_{interparticle}

interactions which control the

_particles

distribution;

this distribution itself determines the

_dynamical

behaviour of each individual

_particle.

A

_simpler

case,

more studied

_{theoretically,}

is that of a fixed bed where the distribution of solid

_objects

is

_rigidly

determined as is the case in the free fall of a

_pair

of

_spheres

of

_equal

radius

_[2]

or in a dense _array

of

_particles.

We consider here a

_{particular example}

of such a

_{distribution,}

the fall of a

_sphere

S

(radius

a)

parallel

to a vertical

_plane

wall

W,

in a viscous

_liquid

_(inset

of

_{Fig. 1).}

We use an

_original

rheo-meter,

developed

in our

_laboratory

_[3],

which uses a

_{magnetic sphere}

maintained at a fixed level

by

levitation in a vertical field

_Hx.

The horizontal

_position

- in

particular

the distance b to the vertical wall -L to y is

_{adjusted by}

an additional horizontal field

_Hy(y)

_given

_by

a

_pair

of vertical

Helmholtz coils in

_opposition

and

_giving

a constant

_gradient

_dHy/dy.

A vertical

_glass

tube

contain-ing

the viscous

_liquid

is limited to half a

cylinder by fitting

in it a solid lucite wall W in a diametral

plane.

The

_assembly

is moved at a constant vertical

_{velocity Ux.}

The Stokes force

_{Fx impressed}

on the

_sphere,

whose fixed

_position

within the half

_cylinder

tube is controlled

_optically,

is detected

as an additional current sent,

by

a feed-back _system, to the horizontal coil which

_produces

the

(*) Also Laboratoire de

_Physique

des

_Liquides

et Electrochimie, Universite P. et M. Curie, T 22, 4,

place

Jussieu, 75230 Paris Cedex 05.

(**) E.R.A. no 1000.

L-144 JOURNAL DE PHYSIQUE - LETTRES

Fig.

1. - The

figure gives

the value of the vertical Stokes force exerted on a

_sphere

near a wall and normaliz-ed to the usual Stokes result far away from it. The distance is measured

_by

the lubrication ratio B = _d/a.

The _geometry of the

_sphere

rheometer _experiment and of the coordinate _system is _{given schematically} in

figure

1. _(A silicone oil 47 V 100; o silicone oil 47 V 20.)

field

_Hx.

Thus the

_experiment

leads to the measurement of the variation of the Stokes force

_Fx

exerted on a

_sphere

of radius a

_moving,

at a constant

_{velocity Ux,}

at a fixed distance b from a

wall

_[4].

This last feature is an

original

characteristic of the

_present

experiment.

In the free fall of a

sphere,

the _presence of a wall

_gives

_rise,

when b becomes of the order of the radius a, to two

contri-butions :

_i)

a decrease of the

velocity

of translation and a rotation of

particles

in the case of

spherical objects,

often

_interpreted

as a

_slip

effect at the walls

[5] ;

ii)

when inertial terms become

appreciable,

a transverse force

_Fy

_away from the wall

_{[2, 6,11].}

This effect leads to a decrease of

concentration near the wall which adds _up to a

_purely

steric contribution. The resultant of these

two actions is called the 2; effect

_[7-8].

_Thus,

when the

_viscosity

is measured with two set _ups

having

different characteristic dimensions

_comparable

with a, the measurements on

_suspensions

(unlike

those on _pure Newtonian

fluids)

will lead to different results.

Lorentz

_[9]

and Stock

_[10]

were the first to solve the effect of a fixed

_plane

wall on a

_sphere

S

moving parallel

to it

_{(our experimental}

situation not

_including

the Z

effect),

the fluid

_being

at rest at

_infinity

and not

_being

submitted to _{any constraint. But it} was Faxen

_[11]

_{who gave} a

solu-tion of the

_{problem satisfying}

Oseen’s

_equation

whilst

_allowing

the

_sphere

to rotate. He used a

« method of reflections ». With a 4th order

_expansion

_(two

reflexions on P and a

single

one on

S),

he obtained an

expression

for the

hydrodynamic

force :

b = d + a is the distance of the centre of S to

_{W ; k}

=

alb.

The numerical values of

_{f (x)}

and the

_expression

for

_Fr

which is directed away from

W,

are

given

in reference

_[11].

In

_equation

_(1),

the force

_Ex

is normalized to the usual Stokes

_expression

_{obtained far away} from the wall

_(k

-+

_0).

The

_expansion

is valid when k is small and when the

_Reynolds

number associated with the

_sphere’s

radius,

Re =

Faxen also showed that the

_{sphere undergoes}

a rotation around the z axis with an

_angular

velocity

co in the direction that it would have if it were in contact with W and would roll without

shearing.

The theoretical results are

_given

in

_figure

1 in the limit where inertial terms are

_{negligible :}

f ( Ux

bl2 v) ~

1

_{(which corresponds}

to the

_{experimental condition).}

We also

_give

the numerical result of an « exact » calculation

_by

O’Neill

[12]

done

_{by solving}

the Stokes

_{equation using bipolar}

coordinates. However the solution

_corresponds

to the case where the

_sphere

moves

_parallel

to the wall without rotation. This result was confirmed

by

a

_{calculation, [ 13,14]}

in the lubrication limit

s = dla 1.

The

_experiments

were done

_using

a

sphere

of diameter 2 a = 1 mm and a variable distance to

the wall

_{(dla -}

5 x

10-2

to

₃₎

_(Fig.

_2).

The _accuracy on the measurements of the smallest values of d is ~

+ 4

gm. We used two silicone

oils,

Rhodosil 47 V 20 and 47 V

100,

of kinematic visco-sities v = 0.2 and 1

cm2/s

_at 20 ~C. The relative velocities of the

sphere

_were

respectively Ux

=

40

_mm/min.

and

_Ux

= 8

mm/min.

Thus

Ux b/2

v ~ 6 x 10- 2 and 5 x 10 - 4 in the two cases;

the condition of

_validity

of

_(1),

_{(Ux ~/v) ~}

_1,

is also fulfilled. We have checked

_{experimentally}

that the measurements were done in a

stationary

regime

(constant

_Fx).

The

_permanent

regime

establishes itself faster

_(with

at-312

saturation

_law)

in _{the presence of} a wall than in the absence

of it

_{(t - 1/2)}

_[15].

The

_experimental

results are

_presented

in

_figure

1 for the two oils used. For

_{large separations}

(4

x

10- 2

e

0.4)

the results _agree with the different theoretical models. We attribute the deviation of

_{experimental points}

from the theoretical

_predictions

to the effect of the

_cylindrical

wall of the tube

_{(D N}

10

_mm)

which causes additional friction at

_larger

distances from the walls.

Fig. 2. -

Photograph of the _{magnetic sphere} next to the vertical

_reflecting

wall W. The distance d is accu-rately determined _using the reflection image of the _sphere, when the tube is

_{displaced vertically.}

L-146 JOURNAL DE PHYSIQUE - LETTRES

Despite

the limited _accuracy of the measurements of the smallest

_separations

d the results also indicate a saturation of the forces as d --+ 0. In O’Neill’s solution there is a

_divergence

of Fx

in the

lubrication limit s -~ 0 which would be due to the friction of the wall and the non

_{rotating sphere}

in contact with it. On the other hand a saturation effect is

_predicted

in Faxen’s model where the

sphere

is free to rotate near the wall. A similar saturation effect is also found with the model of Goldman et al.

_[14].

In their treatment

_they

consider

_separately

the effects of translation and

rotation,

and

_{by superposition they}

calculate

_exactly

the force in the lubrication limit. When d -+

0,

they

find that

_{~M/t/ ~ 1/4.}

This result is

_incompatible

with the no

_slip

condition at the wall which would

_imply

_{awl U}

= 1. Thus _a saturation effect is also

expected

for small

separations.

The _present

_experimental

_techniques

can

_easily

be extended to other

_{projects involving}

sus-pensions ;

like for

_{example measuring}

the force on a levitated

_magnetic

sphere placed

in the middle

of an _array of

_{spheres falling}

around it or fixed

mechanically

and in a

moving liquid.

Such

situa-tions are closer to theoretical models where the

_velocity

fields rather than force fields - as in

classical

_experiments

- are defined.

Acknowledgments.

- We are _very

_grateful

to M. Clement for his

_helpful

assistance. The authors thank C.

_Deslouis,

J.

_Dufaux,

J. P.

_Jorre,

and D. Lhuillier for

_helpful

discussions.

References

[1] A list of

_problems

to be studied

_{experimentally}

and

_{theoretically}

has been

_{proposed by}

DE _GENNES,

P. G.,

Phys.

Chem.

_Hydro.

2 ₍₁₉₈₁₎ 31.

[2] Cox, R. G. and _MASON, S. _G., Ann. Rev. Fluid Mech. 3 ₍₁₉₇₁₎ 291.

[3] GAUTHIER _{MANUEL, B., MEYER,} R. and _{PIERANSKI, P.,} Submitted to J.

_Phys.

E.

[4] It is also

_possible

to measure the

_{component Fy}

of the force

_{by measuring}

the horizontal

_displacement

of the

_sphere,

when the

_liquid

is set in _{motion, knowing}

_dHy/dy.

This was not done in the present

preliminary study.

[5] BRUNN, P., Int. J. _Multiphase

_flow

7 ₍₁₉₈₁₎ 221.

[6] GOLDSMITH, H. L. and _MASON, S. G.,

Rheology,

vol. 4 _{(Acad. Press)} 1967, p. 150.

[7] SCHULTZ-GRUNOW, F., Ing. Archiv. 28 ₍₁₉₅₉₎ 307.

[8]

BRANDT-BUGLIARELLO, Trans. Soc. Rheol. 10 ₍₁₉₆₆₎ 229.

[9] LORENTZ, H. A.,

Zittingsverlag

Akad. Wet. Amsterd. 5 ₍₁₈₉₆₎ 168.

_{Abhand-Lungen}

über theoretisch 1

(1907)

S. 23

_(Leipzig).

[10] STOCK, J., Bull. Int. Acad. Cracovie

₍₁₉₁₁₎

A 18-27.

[11] FAXÉN, H., Einwirkung der

_{Gerfäßwände auf}

den Widerstand gegen die Bewegung einer kleinen Kugel

in einer Zähen

_{Flüssigkeit (Diss) (Upsala) (1921)}

_{pp. 55-128.}

[12] O’NEILL, M. E., Mathematica 11

₍₁₉₆₄₎

67.

[13] O’NEILL, M. E. and STEWARTSON, K., J. Fluid Mech. 27

₍₁₉₆₇₎

705.

[14] GOLDMAN, A. J., Cox, R. G., BRENNER, M., Chem. _Eng. Sci. 22 ₍₁₉₆₇₎ 637.