Optical properties of polarizable linear micropolar fluids

Int. J. Engng Sci. Vol. 8, pp. 31-37.

OPTICAL

Pergamon Press 1970.

Printed in Great Britain

PROPERTIES OF POLARIZABLE MICROPOLAR FLUIDS TATSUO

Department

LINEAR

TOKUOKA

ofAeronautical Engineering, Kyoto University,

Kyoto,

Japan

Ahstract - General optical behaviors of the polariz.able linear micropolar fluids are investigated. The index

tensor is reduced to a linear combination of deformation-rate tensor and gyration tensor. When the optical effect of the gyration motion are negligible with respect to that of the macro motion, the fluids show the same birefringent phenomena with the Stokesian fluids, and when the former is dominant, the fluids show the double circular refraction. The fluids are compared with Einstein’s colhd suspensions. Unless the suspended molecules have optical activity, a light propagating through the linear micropolar fluids separates into two elliptically polarized lights along the secondary principal axes of the deformation-rate quadric. The microrotatory motion affects the birefringence and the magnitude of perturbation is the second order of the concentration of the suspension. 1. INTRODUCTION

of linear micropolar fluids, introduced by Eringen[ 11, deals with a class of fluids having micro-rotational flow. The typical flow behaviors of the fluids have been investigated by Eringen[ 1 and 21, Ariman et al. [3 and 41, and Hudimoto and Tokuoka [5]. Based on the investigation of the apparent viscosity coefficients in the shear flows of the fluids, Hudimoto and Tokuoka[5] concluded that the fluids may represent Einstein’s dilute suspensions [6]. The aim of this paper is to specify the photo-constitutive equations of the polarizable linear micropolar fluids and to investigate the general optical birefringent properties of the fluids, and then to apply the obtained formulae to the two-dimensional shear flows of the fluids.

THE THEORY

2. FUNDAMENTAL

EQUATIONS

The mechanical field equations of a linear micropolar fluid[2] are

and flk -

Slk +

bkl,r

+P(1kl--&kl)

=

0,

(2)

where p is the density, vk is the velocity vector of center of gravity of macro-volume element, tkt is the stress tensor, s kI = slk is the micro-stress average, A,&,,is the first stress moment,fk is the body force per unit mass, i,, is the first stress body moment per unit mass, and &k, is the inertial spin, and throughout this paper we adopt rectangular coordinate system, an index followed by a comma represents partical differentiation, a superposed dot indicates material differentiation and repeated indices denote summation over the range (l-3) unless the particular comment appears. The electromagnetic field equations for a non-conducting, polarizable, nonmagnetic medium are given by 31

32

T. TOKUOKA

VxH=---,

1 aD c at

V.B=O,

in the system of Gaussian units, where E and H are electric and magnetic field respectively and D and B denote their fluxes, and we have H=B,

E=qD,

(4)

where r) is the index tensor, which is the inverse matrix of the dielectric tensor. Eringen [7] proposed the simple microfluids, whose constitutive equations are tensor & = defined by that tkz, ski and Altlm are functions of the deformation-rate (u~,~+ 2)&/2, the gyration tensor vkl and the gradient of it I+~,~. The micropolar fluids belong to a subclass of the microfluids and are restricted by the skew-symmetricities[2]: hklrn = - Akml.

ukl I= -u/k,

For an incompressible are given by

micropolar fluid the mechanical linear constitutive

(3 equations

t=-pI+2pd-K#,

(6)

s=-pI+2pd,

(7)

where p is the undetermined pressure, p is the shear viscosity coefficient of the linear Stokesian fluid and K,, a”, & and ye are viscosity coefficients with respect to the local deformation. Here we assume that the light propagating into the fluids is so weak that the influence on the mechanical behaviors of the fluids due to the light may be neglected. According to the principle of equipresence, we assume that the index tensor q is a function of the same independent state variables with the mechanical constitutive equations. And then we have the linear photo-constitutive equation: r) = qJ+

where q,, constants When Hermitian

Md+ iKv,

(9)

is the index coefficient at rest state, and M and K are the phenomenological specified by a given fluid. no absorption of light occurs in the fluids, the index tensor must be [8] p =

vt*

( 10)

with respect to the complex representation of the fields and fluxes, which depend on the form exp[i(k*x --ot) 1, where signs of cross and asterisk mean the transposed matrix

Optical properties of pohrizable

linear micropdar

fluids

33

and the complex conjugate of a matrix respectively, and R is the wave-vector and OJis the circular frequency of a monochromatic light. The generalised principle of the symmetry of the kinetic coefficients [8] demands the symmetry relation

rlM VI = qtw,--P),

(11)

which is satisfied by the condition of no absorption (10). The skew symmetric tensor Yis expressed by an axial vector G such as vkl

=

eklm

G mp

(12)

where ekzmis the complete skew symmetrical unit tensor. Therefore the second relation of (4) is reduced to E=q@+MdD+iKDXG. 3. OPTICAL

(13)

PROPERTIES

The electromagnetic field equations (3) are reduced to nXE=B,

n*D=O, (14)

nXH=-D,

n-B=O,

where II = (o/c)R is the refractive index vector. Relations (14) and (4) indicate that n, D and B are mutually perpendicular, and taking x,-axis along n and eliminating E we have

s,-fs, >h =

0

(a,P=

1 and2).

(15)

Consider a conic section constructed by the quadric

dtc&%= constant

(16)

and a plane perpendicular to R through the center of the quadric, and take the coordinate axes x1 and x, along the principal axes of the conic section. Then we have

- iKGsD, +

(17)

where q. ,’

iJ.RS.Vd.SNof--c

1 ----$+Md, n2,’ &x

(a=lmd2)

(18)

T. TOKUOKA

34

and C&are the secondary principal values of the deformation-rate tensor. Eliminating D1 and D, from (17) the refractive index must have the values n, such as $=

;+;{M(d,+d,)

and the electric displacement

St [MZ(d,-d,)2+44K2G~]1’2)

(19)

i) has the components D, = ipD,

for

D2 = -bD1

n= n+,

(20)

for n = n-,

(21)

where P = &$-M(c&-d”)

+ ~M2(d,-~d,)2+4K2Gt;l”2~.

(22)

The purely imaginary value of the ratio Dg/Dl signifies that the waves are elliptically polarized, whose principal axes coincide with the secondary principal axes of the quad& (16). When KG, > 0 the directions of rotation of (20) and (21) are left- and right-handed screw direction respectively around the positive x,-axis. Now we consider two extreme cases. In the case of IM(d, - dZ) 1 s IKG3/, we have 1 -=

n$

L+Mdl or 4

L+M~2 4

(23)

and p=

0.

(24)

For this case the elliptic~ly polarized lights are reduced to the plane polarized lights along the secondary principal axes of the deformation-rate quad& and have the same birefringent formulae proposed by Tokuoka[9] with respect to the stokesian fluids. On the contrary in the case of IM(d, - d2) 1 6 IKG31, we have the double circular refraction:

(2.5) and p=

1.

f26)

In this case the two circularly polarized lights propagate with the different velocities

for IKG,J 4 1, where v. = c/no is the light velocity at rest fluids. The linearly polarized wave is rotated like so-called Faraday effect by the angle

Optical properties of polarizable linear micropolar fluids 0

In+-n-l

x=;

on%G =2ccose

2

35

(28)

per unit length in the direction of the wave vector, where 8 is the angle between n and G. 4. EXAMPLE

Hudimoto and Tokuoka[S] investigated the two-dimensional shear flows of the linear micropolar fluids. The channel has the width D (y-axis), and the wall y = 0 is at rest while the wall y = D moves with velocity V along the channel direction (x-axis). We consider that a light propagates along the normal of the two-dimensional flow (along z-axis). From their results we have sech cx{cllcosh 2a ( 5‘-$-sinha)],

~~

,

(29)

(30)

and the secondary principal axes direct at an angle of rr/4 with x-axis, where l-b

CL

= l-

% -cutanha @

is the apparent viscosity and

Comparing Einstein’s theory of colloid suspension[6], Hudimoto and Tokuoka concluded that the linear micropolar fluids represent the similar mechanical behaviors with the colloid suspensions with rigid spherical particles and, for small value of a, (31) means the volume fraction of the suspended particles. Expanding (29) and (30) with respect to (Yand substituting (3 1) into them we have 12*1”‘{6(

l-5)

-i}@

+ O(gs)]

(32)

and

y,,1

Gs=~

Therefore for sufficiently dilute suspensions section may be applied and we have

-()t$+o(@). the former case depicted

(33) in the above

T. TOKUOKA

36

An = ,n,-n-,

=;y

l+ c

12*y[[(1

-5)

-;+v[‘(l

7J

-o+z]+O(+J) (34)

and (35) The birefringence An is proportional to the shearing-rate V/D and the microrotational motion influences on the birefringence and the magnitude of peculation is the second order of the concentration of the suspension. Equation (35) shows that the linearly polarized lights along the secondary principal axes are converted to the elliptically polarized lights by the effect of the local deformation and the magnitude of the deviation is the first order of the concentration. The above mentioned optical behaviors are due to the solvent and not to the solute. Thus if the suspended molecules have the optical activity, the resultant optical phenomena must be superposed with those of two effects. REFERENCES [I] A. C. ERINGEN, Devefopment in Mechanics, Vol. 3, pp. 23-40. Midwestern Mechanics (1965). [2] A. C. ERINGEN,J. Math. Me& 16, 1 (1966). [3] T. ARIMAN, A. S. CAKMAK and L. R. HILL, Phys. Fluids lo,2545 (1967). f4] T. ARIMAN, In?..!. Engng Sci. 6, 1 (1968). [S] B. HUDIMOTO andT.TOKUOKA, 1nr.J. &gag&i. 7,515 (1969). [6] A.ElNSTEfN.AnnlnPhyf. 19,289f1906):~,591(191 I). [7] A. C. ERINGEN, lnt. J. Engng Sci. 2,205 (1964). [8] L. D. LANDAU and E. M. LIFSHITZ. Electrodynamics of Continuous Media, Chap. Il. Pergamon Press ( 1960). [9] T. TOKUOKA, in:. J. Engng Sci. 4,23 (1966). (Received 28 July 1969)

R&sum&- L’auteur &die fe ~orn~~ernent optique g&&xl des fluides micro~faires fin&&es polarisables. Le tenseur d’indice est rameni B une combinaison formbe du tenseur du taux de dkformation et du tenseur de rotation. Lorsque l’effet optique du mouvement de rotation est n&gfigeabfe par rapport $ cefui du mouvement macroscopique, les fluides consid&& sont fe si&ge du m&me phtnomene de birt%ingence que les fluides de Stokes et, dans le cas inverse, ils prksentent une double r&fraction circufaire. L’auteur compare ces ffuides aux suspensions colloidales d’Einstein. Si les mofCcules en suspension ne rempfissent pas de fonction optique, une fumi&requi se propage dans un fluide micropolaire se d&compose en deux faisceaux de lumi&re & pofarisation effiptique fe long des axes principal et secondaire de la quadrique du taux de dt?formation. Le mouvement microrotatoire a@ecte la birhfringence et f’amplitude de la perturbation est du second ordre de la concentration de la suspension. Zusammenfassung- Affgemeine optische Verhaftensweisen der pofarisierbaren finearen mifcropolaren Ffiissigkeiten werden untersucht. Der Indextensor wird auf eine fineare Kombination des Deformationsgeschwin~~eitstensors und des Drehungstensors reduziert. Wenn die opt&hen Wirkungen der Drehungsbewegung mit Bezug auf die Makrobewegung vemach&ssigt werden kiinnen, weisen die Fliissigkeiten dieselben Doppelbrechungserscheinungen mit Stokes Ffiissigfceiten auf; und wenn die erstere vorherrscht, zeigen die Ffiissigkeiten die doppefte Kreisbrechung. Die Ffiissigkeiten werden mit Einstein’s Kofloidsuspensionen vergfichen. Ein Lichtstrafd, der sich in den finearen mikropofaren Ffiissigkeiten fortpflanzt, trennt sich in zwei effiptisch polarisierte Lichtstrahfen entlang der sekun&ren Hauptachsen der Deformationsgeschw~~gkeitsqua~k, ausser im Fafie optisch afctivi&er suspendierter Mifefdife. Die Mikrodrehungsbewegung beeinflusst die Doppefbrechung, und die Gtissenordnung der Stijrung ist die zweite Ordnung der Suspensionskonzentration.

Optical properties of polarizable linear micropolar fluids

37

Sommarlo-Si investigano i comportamenti generali ottici dei fluidi micropolari lineari polarizzabili. 11 tensore d’indice b ridotto a una combinazione lineare de1 tensore de1 tasso di deformazione e de1 tensore di rotazione. Quando l’effetto ottico de1 movimento rotatoriob trascurabilerispettoa quello delmacromovimento,ifluidimostranoglistessifenomenibirifrangenti deifluidiStokensianiequandoiprimisonodominanti, i fluidi mostrano la doppia r&zione circolam. I fluidisono ConfTontati con le sospensiom colloidali di Einstein. A meno the le molewle in sospensione abbiano attiviti ottica, ma leggera propagazione attraverso i fluidi micropolari lineari si separa in due luci polarizzate ellitticamente lung0 gli assi principali secondari della quad&a de1 tasso di deformazione. 11movimento microrotatorio influenza la biigenza e l’ordine di grandezzadellaperturbazionebilsecondo ordinedellaconcentrazionedellasospensione. A6wpa~--Mmy~aloTcn O6mTie OnTH'IecKHe CBOfiCTBa flOJTBpH3yeMblX JIHHehbiX MHKpOnOJWHblX ~~~Te8.~OKa3aTenbHbI8TeH30pnpHBOlUtTCIlKnHHeaHO~ woM6HHamHTeH3OpaCTeneHHae@OpMaUHH H TeH3opa BpaUeHHn. Korna OnmTHsecKoe BJmIHHe BpaluaTeJIbnoro IIBHXCeHHa He3HaYHTeJtbHO no OTHOmeHHIO K MaKpOnBHTKeHTIBDo, B ~(HRKOCTIIX BbICTynaTOT Te XCe CaMble RBBeHHs aBonHoro JTYYeBOrO npenoMneHHB,KaK BXW~KOCTRX C~oy~ca,Kor~ta ~enpeO6JTaZWT BpaUlaTeJtbHOeJlBHKieHHe,BXWAKOCTaX BblCTynaeT ItBOfiHOeKpyrOBOe npenoMneHkie. ~HrurocTH CpaBHHBaIOTCRC pa3XHmKeHHMMH CYcueH3HaMH 38HmTenHa. EU~H CycneHmipoBaHHbte MoneKynbI He nOKa3bIBaIOT OnTHYeCKO~ BKTHBHOCTH, TO CBeT pacnpocTpaHmom8cr 9epe3 mHetinbre wiKpononnpme nomKom4 pacmennneTcn Ha ma ~~H~HY~CKH CTeneHH ne@opbiauTw. nonnpH30BaHHbte nywoi cBe.Ta Bnonb BTOpHYHblX rnaBHbtx ocett KeanpHKa MHKpOBpamaTeJlbHOe JJBHIKeHHe BO3BenCTByeT Ha LtBOfiHoe Jly'teBOe nt=iOMBeHHe H BeJlH’lHHa BOJM)‘Ul~HHR OCTaeTCIl B JBBHCHMOCTH OT BTO@i CTeneHH KOHUeHTpaUHH cyCneH3HH.