Two-dimensional shear flows of linear micropolar fluids

In?. J. Engng

Sci.

Vol. 7, pp. 5 15-522.

Pergamon Press 1969.

Printed in Great Britain

TWO-DIMENSIONAL SHEAR FLOWS OF LINEAR MICROPOLAR FLUIDS BUSUKE HUDIMOTO and TATSUO TOKUOKAt Kyoto Technical University. Kyoto. Japan Abstract-The two-dimensional parallel shear flow of a linear micropolar fluid is analysed and compared with the colloid suspensions. The suspended macro-molecules are considered as nuclei of eddy motions representing micro-rotation in macro-volume element. The velocity of center of mass of the macro-volume element is assumed as a measured velocity. The obtained results agree well with Einstein’s formula for the lypohobic sol and with the reduced viscosity-concentration relations for lyophilic sol.

I. INTRODUCTION

RECENTLY. the continua whose characteristics are affected by the local deformations in their macro-volume element have been investigated. With respect to the elastic solids, Truesdell and Toupin[ 11, Toupin[2]. Mindlin and Tiersten[3]. Eringen and Suhubi[4, 51. and Eringen[6] developed these micro theory taking into account the local deformation. With respect to the stokesian fluids. Eringen[7, 8 and 91 proposed theories of simple microfluids and micropolar fluids. Although the comparisons of such fluids with some real materials have not yet been investigated, it should seem that the fluids might express the behaviors of the colloid suspensions from the introduction of the internal freedom. This is the main aim of our paper. In section 2 we summarize the basic relations of the linear micropolar fluids and in section 3 the parallel shear flow of the fluids is analysed. In section 4 the shear flows, of colloid suspensions are compared with micropolar fluids. 2. SUMMARY

OF LINEAR

MICROPOLAR

FLUIDS

Here we r&sumC the basic relations of the linear incompressible fluids referring to [7,8]. The mechanical field equations are

isotropic micropolar

fkl,k+P(h-~J = 0

(2.1)

and fml - St?lI+ hm,k

where

p = L’k= Ikl = fk = ski = ABlm=

+puht -oh)

= 0,

mass density velocity vector of center of gravity of macro-volume stress tensor body force per unit mass slk = micro-stress average first stress moment

t Department of Aeronautical Engineering, Kyoto University, Kyoto, Japan. 515

(2.2) element

516

BUSUKE

HUDIMOTO

and TATSUO

TOKUOKA

lkl = first stress body moment per unit mass & = Gkl = inertial spin ikl= itSkI = micro-inertia moment vkl = gyration tensor. In the macro-volume

element, a material particle moves with velocity ’ = vk+ Ylk[l,

(2.3)

vk

where tk is the deviation from the center of mass of the element. Throughout this paper we employ a rectangular coordinate system &, an index followed by a comma represents partial differentiation. a superposed dot indicates material differentiation and repeated indices denote summation over the range (1,2,3). A micropolar fluid is defined by [8, equation (4.1 I]: Aklm = - hkml?

ukl

=

-

(2.4)

Ylk.

For an incompressible micropolar fluid the linear constitutive tions (3.4). (3.5) and (3.6)] are reduced to t =-,>1+2pd-?(b-

equations [8, equa-

b=),

(2.5)

s =-pI+&d.

(2.6)

and Aklm -

where

-$(Cf,+&+

p = undetermined dkl

&uk.,

=

deformation-rate ukl+

ukZ,n) ++/%~(~lr,r~km - %tr,r~kl),

(2.7)

pressure

=

bkl =

+

-)‘v)~lm,k +h"(Vkm,l-

U,,k)

tensor of macro-volume

element

vk,l

and Al.is shear viscosity coefficient of the linear skokesian fluid and K,. au. PO and yc are viscosity coefficients with respect to the local deformation. The necessary and sufficient conditions for the local Clausius-Duhem inequality for all independent process in the fluids are easily obtained by the similar calculation in [S] and given by

3. TWO-DIMENSIONAL

We consider the macro-volume

v, =

U(Y),

PARALLEL

SHEAR

FLOW

motion given by 21, =

21, =

0,

OsySD,

where D is the width of the channel with parallel fixed flat walls. The gyration tensor may be assumed to be independent of x and z_and then

(3.1)

517

Shear flows of linear micropolar fluids

Fig. 1.Two-dimensional

p=- ;

shear flow and coordinate system.

kY)R(Y) 0 [ --h(y) -f(Y)

h(Y) 3(Y)

0

I *

(3.2)

Substituting (3.1) and (3.2) into (2.5). (2.6) and (2.7), we obtain

where the primes denote the differentiation with respect toy. For the constant pressure and vanishing body force and the body couple, the equations of motion are (2,U+Kv)U’--K~j-’

= 0,

K&g”=0

(3.4)

and 2K& - yagn = 0, K,h - (a,. +&

f yc)h” = 0.

(3.7)

Integrating (3.6), and eliminating 18 from (3.7),, we have (2@fK,)d-Kvf=

c,

(3.8)

518

BUSUKE

HUDIMOTO

and TATSUO

4p~vf-~t,(2~+

Kt.)fll=

TOKUOKA

2K,C,

(3.9)

where C is an integration constant. From (3.6), and (3.7), we have x(-v) = 0

ifK,, # 0

(3.10)

and h(y) is any function satisfying (3.7),. In the case K,. = 0, the stress tensor t is reduced to s, namely we have the classical viscous fluids, thus we assume K!. f 0. In the case yJ2~+ K,.) = 0, (3.9)and (3.7), indicate that c (3. I 1) u’ =f= - = constant, 2/*.

where we have also the classical velocity distribution. therefore here we assume yV(2/L+ Kr) f 0. By the thermodynamical

conditions (2.8). we may put WC

CY=

75.(2j_L+Kr)

112

I

D >O,

(3.12)

which corresponds to [S, equation (7.12),] and 19, equation (8.9),]. The solutions of the differential equation (3.9) and then of (3.8) are given by f= A cash 2a7) + B sinh 2ar) + &2/1’

(3.13) (3.14)

where (3.15) and A, B, C, are the integration constants. We assume that the wall y = 0 is at rest and the wall y = D V along x-axis. The boundary conditions are u=v Determining

moves

u=j’=O at q= 0, and j=O at q=l.

constant

speed

(3.16)

the four constants by the conditions, we have f=

z 1-

secha! cosh2&

-+)

(3.17)

l-- C?” tanhcw 0 sinha U=l/rl--l/-- - CuYl,secha E.LD2

coshol ( 2 -9 ) -q l_AYLta~ PD2

sinha

(3.18)

519

Shear flows of linear micropolar fluids

It is easily verified that the velocity in the middle point of the channel is 1//2 and the velocity distribution is antisymmetrical with respect to that point. Expanding (3.17) and (3.18) with respect to (Y,we have (3.19)

f=++$+O(a4), _

(3.20)

I(= I%- 2y,l/ 3PD2 a47)(1 --r))(q -&+O(afi). The boundary conditions (3.16) and the velocity T = tzl = (p + Kr/2)Z4’ at y = 0 or D acting on the walls: T=-

1

v

(3.20) give the shear stress

_ P

D

=gp

l-XYrtanha PD2 where V/D denotes the mean rate of shear. 4. COMPARISON

WITH

[

1

1+- yF cu2_tO(cX4) , pD’

COLLOID

(3.21)

SUSPENSIONS

4.1. Eddy currents In a macro-volume element the material particles move relative to the center of mass of the element. Their relative velocities are expressed by the second term of the right-hand side of (2.3). The quantity - (1/2)f(y) depicted in section 3 specifies the distribution of the angular velocity about z-axis. The positive and the negative value of f(y) denote the clockwise and the anticlockwise rotations respectively. The flow containing such eddy currents might transfer of momentum at right angles to the flow and has the increased resistance. The velocity distribution (3.18) and Fig. 1 show such property. Eringen[S, 93 investigated the flows of micropolar fluids in a circular pipe and in two-dimensional channel which show that the flow speeds are slower than the laminar flow for the same pressure gradient. 4.2. Viscosities of colloid suspensions Einstein[lO] treated the following problem. A simple Newtonian liquid has suspended in it a number of rigid spherical particles. When the suspension as a whole is subjected to a shear stress, the flow of the viscous liquid itself in the neighbourhood of a suspended particle is not homogeneous. Thus the local rate of shear of the liquid itself varies from point to point and the average value is greater than the over-all rate of shear of the suspension as a whole. The result is that the over-all viscosity of the suspension is greater than the viscosity of the suspending liquid. If the above rigid spherical particles are replaced by the small eddy currents, our linear micropolar fluids may represent Einstein’s dilute suspending solutions. In a shear flow of the fluid, the velocity of the center of mass of the macro-volume element is affected by the gyration motion and may correspond to the velocity of the solution. The suspended macro-molecules being considered as nuclei of the eddy motion are entirely small in comparison with the measuring part of the apparatus. Consider the shear flow-depicted in section 3. When we put the non-dimensional parameter (Yzero i.e. K,. = 0 in (3.19) and (3.20), we have

520

BUSUKE HUDIMOTO

and TATSUO TOKUOKA

(4.1) which may be considered to indicate the pure solvent laminar flow. When (Ytakes a nonvanishing value, the flow deviates from a laminar flow. The magnitude of deviation is expressed by the second term of the right-hand side in (3.20), and proportional to the parameter yJpD2. When ya = 0, we have the linear velocity distribution (4.1) while the gyration vector does not vanish. Thus we may conclude that (Ydepends on the concentration of the suspended molecule to the solvent, i.e. the eddy density, and yJpD2 represents the intensity of the influence on the flow of the solvent by the magnitude and shape of the macro-molecule, i.e. the intensity of disturbance by the eddy motion. The viscosity coefficient of the solution pap is given by the ratio of the shear stress to the mean rate of shear. From (3.2 1) we have

Pa, =

CL

(4.2)

l- - YL’CY tanha PD2 By the physical considerations mentioned above the viscosity of the solution must be greater than that of the solvent. This imposes the same restriction yV > 0 corresponding to (2.,8),. Einstein[ lo] investigated the case of a suspension so dilute that there was no appreciable hydrodynamic interaction between different particles. His viscosity formula is (4.3) where C#Imeans the volume fraction of the suspended particles, and Q,, 12,c and p are volume of a particle, number density, concentration (g/cm3) and mass density respectively. Our formula (4.2) have the same type of (4.3) and for small value of czwe obtain (4.4) Einstein’s formula is assured for the lyophobic sol by the various experiments [11, 121. For lyophilic sol the factor 2.5 in (4.3) and (4.4) must be increased by the apparent increase of the molecule. In order to apply (4.2) to the comparatively concentrated solution or to lyophilic sol, the next approximation is necessary. We have

where (4.6) are the specific viscosity and the limiting viscosity number respectively, called as Huggins’ constant [ 131 and

k = K/2*5* is

Shear flows of linear micropolar fluids

K=6+-;

5]+0,

521

(4.7)

and where

For rigid spherical particles Einstein[ lo], Guth and Simha[l4] and Vand[lS] proposed the same types of (4.5) and their values of K are 4.38, 14.1 and 7.349 respectively. With respect to the various suspensions many investigators measured the reduced viscosity-concentration relations[l6-181. Their results show the positive values of K. Acknowledgment-The

authors are indebted to Professor A. Cemal Eringen for his valuable advice. REFERENCES

[1] C. TRUESDELL and R. TOUPIN, Classical Field Theories, Handbuch der Physik, III/l. Springer (1960). [2] R. TOUPIN,Arch. rat. Mech.Anal. 11,385 (1962). [3] R. D. MINDLIN and H. F. TIERSTEN,Arch. rat. Mech.Anal. 11,415 (1962). [4] A. C. ERINGEN andE.S. SUHUBI,Inf.J.EngngSci.2, 189(1964). [5] E. S. SUHUBI and A. C. ERINGEN, Int. J. Engng Sci. 2,389 (1964). [6] A. C. ERINGEN,J. Math. Mech. 15,909 (1966). [7] A. C. ERINGEN, 1nr.J. Engng Sci. 2,205 (1964). [81 A. C. ER1NGEN.J. Math. Mech. 16, 1 (1966). [9] A. C. ERINGEN, Development in Mechanics, Vol. 3, pp. 23-40. Midwestern Mechanics (1965). [lo] A.E1NSTE1N,AnnlnPhys.19,289(1906);34,591 (1911). [ll] M. BANCELIN.Kolloid.Zh.9, 154 (1911). [12] F. EIRICH, M. BUNZL and H. MARGARETHA, Kolloid. Zh. 74,276 (1936). [13] M. L. HUGGINS,J.Am. them. Sot. 64,2216 (1942), Ind. Engng Chem. 35,980 (1943). [14] E. GUTH and R. SIMHA, Kolloid. Zh. 74,266 (1936). [15] V. VAND,J.phys. toll. Chem. 52,277 (1949). [16] G. TAYLOR, J. Am. them. Sot. 69, 635 (1947); T. G. FOX and P. J. FLORY, 73, 1909 (1951); 73,1915 (1951). [17] S. N. CHINAI, P. C. SCHERER, C. W. BONDURANT and D. W. LEVI, J. Polymer Sci. 22, 527 (1951); L. H. TUNG, 24, 333 (1957); S. N. CHINA1 and R. A. GUZZI, 41,475 (1959); F. DIDOT, S. N. CHINA1 and D. W. LEVI, 43,557 (1960). [18] K. IMAI, U. MAEDA and S. MATSUMOTO, Kobunshikagaku (in Japanese) 14, 419 (1957); S. MATSUMOTO and K. IMAI, 14,425 (1957); R. NAITO, 15,597 (1958); 15,604 (1958); S. FURUYA and M. HONDA, 16,612 (1959); Y. NAKAMURA and M. SAITO, 17,718 (1960); R. ENDO, 18, 143,214 (1961); 19,39 (1962). (Received 4 January 1968) Resume- Dans la pr&ente Etude, nous analysons l’ecoulement tangentiel parallele a deux dimensions d’un fluide micropolaire lineaire et nous le comparons au comportement des suspensions colldidales. Les macromol6cules en suspension sont consider&es comme les noyaux de mouvements tourbillonnaires representant une microrotation dans l’tlement de macrovolume. La vitesse du centre de masse de l’tlement de macrovolume est conside& comme &ant une vitesse mesuree. Les r6sultats s’accordent avec laformule d’Einstein pour les sols (suspensions colldidales) lyophobes et avec les relations reduites viscositeconcentration pour les sols lyophiles. Zusammenfassong- Der zweidimensionale parallele Scherungsfluss einer linearen mikro-polaren Fhissigkeit wird untersucht und mit dem einer kolloiden Suspension verglichen. Die suspendierten M&o-Molekiile werden als Keme von Wirbelbewegungen angesehen, die Mikro-Rotation im Makro-Volumen Element darstellen. Die Geschwindigkeit des Schwerpunktes des Makro-Volumen Elementes wird als eine gemessene

522

BUSUKE

HUDIMOTO

and TATSUO TOKUOKA

Geschwindigkeit angenommen. Die erhaltenen Ergebnisse sind in guter Ubereinstimmung mit Einstein’s Formel fir das lyophobe Sol und mit den verringerten Viskositiit-Konzentration VerhHltnissen fiir das lyophile Sol. Sommario- Si analizza, e si confronta con le sospensioni colloidali, il flusso di taglio parallelo bidimensionale di un fluid0 micropolare lineare. Le macromolecole in sospensione sono considerate alla stregua di nuclei di movimenti parassitirappresentantiuna microrotazione in element0 di macrovolume. La velocita del centro della massa dell’elemento di macrovolume i? presunto come una velocita misurata. I risultati ottenuti collimano egregiamente con la formula di Einstein sul sol liofobico e con i rapporti di viscosita ridotta-concentrazione peril sol liofilico. A@rpatcr-llonaepraercn

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