PDF in anisotropic fluids AND FLUENT BODIES, AND INTERACTION FIELDS

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A THESIS OH

n o n-l in e a r it y i n a n iso tr o pic f l u i d s,b e s i u e h t

AND FLUENT BODIES, AND INTERACTION FIELDS

By

P. D. S. Terms Assistant Professor Department of Mathematics Indian Institute of Technology

Kharagpur

Submitted to the Indian Institute of Technology,Kharagpur for the award of

the degree of Doctor of Science (Mathematics)

October, 1964

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The author is and will ever be grateful to Prof. B.R.Seth, Prof.J.L.Bricksen and

Prof. C.Truesdell for inspiring him to do the research work.

r-1). <i • >-

' 2-1-10.

( P. D. S. ? s r u )

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CONCLUSIONS AND SUGGESTIONS FOB 180 FURTHER RESEARCH

BIBLIOGRAPHY 136

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taken by me and actually carried out by

2

^&$* I also certify that the thesis is apaposed by me*

C P .D .S . Versa )

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PHS*ACS

Linear theories of tb® behaviour of Batter h a w swayed scientific thought over two centuries. Even now any new departures come back to them for their validity and verification. They havg advanced science snd tech

nology to an unforeseen degree. They have borne rich fruits in the form of classical theories of elasticity, fluid flow and heat conduction. They have enriched all basic sciences and are responsible for the discovery and study of many new functions in mathematics.

These theories are based on the assumption that matter is continuous. With rapid growth of nuclear research some have been misled to look upon continuun theories without appreciating that there aust toe some

thing bridging the gulf between macroscopic and oicro- scoplc behaviours of matter. The non-linear theories of elasticity and fluid mechanics have played an impor

tant part.

It should not be believed that the last word has been said on the nuclear description of matter. With the discovery of new elementary particles every day sad with mutual transmutation of the unstable ones, it 1*

not unreasonable to Imagine that their birth asf have te

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be explained on the basis of ■ continuum principle.

Instability in a continaua can make it appear aa a discreet system. Perhaps Einstein's undying faith in the unified field theory was also inspired by the implicit belief in the continuua structure of the

universe. The nature and properties of thia principle aey have to be defined more precisely, and perhaps in a for® different from the classical types. In any ease, the non-linear theories dealing with both elasticity and fluidity either separately or in a combined fashion have a vital role to play. Hot only thia but the Mutual interactions of different field theories will also

become important to consider. Me shall deal with, in this thesis, several such aspects of non-linear conti

nuum mechanics. We devide the whole subject into three parts. The first part deals with a theory of aniso

tropic fluids, recently developed in a series of papers L f> *•> ^ ^ W J- Ihis theory is prinarlly aeant for describing viscoelastic behaviour of aaterlala. The fluid is visualized as a suspension of non spharleal particles, e .g . ellipsoidal particles, whose boundaries are surfaces of revolution, the preferred direction being the axis of revolution. It eaploys a single set

* of constitutive equations and no yield criterion. It has been shown that for a certain class of these

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anisotropic fluids, the predicted behaviour is like that of the Bingham materials. In the one dimensional descriptions of Bingham materials used by Buckingham

L ^ J and Heiner / _ / or in the three dimen

sional formulation introduced by Oldroyd L 1 7 D y two sets of constitutive equations are used* One set, applying at low stresses, describes an ordinary elastic material. The other, applying at high stresses, in

volves a viscosity similar to that which characterizes Newtonian fluids. K scalar yield criterion is used for determining which set applies. This theory of anisotro

pic fluids also describes / «& —/ SOffl® fluids, commonly treated as isotropic, better than comparably simple theories of isotropic fluids. After explaining its formulation and the main features in the first chapter, we analyse a few problems in the next three chapters v i z ., Couette flow / f _ / , Helical flow

L

H -J» and flow between two rotating cones £ / ° J?, obtaining their solution in a rather simple closed form*

These analyses go to show that surfaces of discontinuity, or the yield surface in the terminology of non-Bewtonian

fluids, can be predicted using only one set of consti

tutive equations and no yield criterion. &x$erimnts

* are in progress to verify these results.

t

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In the second part, we discuss the resilient and fluent bodies. It may be said that solids and fluids are only limit cases of a general type of material called

•hygrosteric material1. We divide this hygrosteric material into two classes! (a) fluent bodies,and (b) resilient bodies* A medium is called a 'fluent body*

i f it can be at ease only when the stress Is equal to ■ certain hydrostatic pressure, depending on density

(A body is said to be at ease i f the tine derivatives of all functions describing its state vanish). After giving in chapter f Noll's / IS _ / development of the genera*

U s e d theory, originally due to Maxwell Z / 9 „ / * v*

proceed to discuss in chapter 71 an exact solution L _/« Some of tha effects that we find are new »nd interesting.

In chapters H I to IX , we take up resilient bodies.

A medium i s called a resilient body i f it can be at ease for arbitrary stresses within a certain range. Hypoelas

tic materials fall under this category. An hypoelastle material Is one in which, for a given Initial stress,the stress at a final state depends only on the paths by which the material points reach the final state and net . upon the rate at which they traverse these paths.

The theory of hypoelasticity is an

attempt to extend the elassieal view of elastic response

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by formulating it in terms of rates. The theory is built up on principles of invariance. Hypoelastic materials of various grades are defined as possible models for physical materials, no idea of approxima

tion is necessarily implied. This differs from older theories In that its equations are of a fora admissible for deformations of any magnitude and speed. The stress*

strain relations are not known in an explicit form as In classical theory of elasticity. They are determined In an individual problem by integrating the differen

tial equations and therefore depend cm the manner in which load is applied. In some esses, yield is possible to predict without assuming any yield condition. Arbi*

trsry initial stresses oan also be taken Into account.

In chapter 711, we describe the resilient bodies with special reference to the theory of hypoelasticity.

'•fe give Its main features especially in reference to other theories of elasticity and plasticity. In chapter T il l , we give solutions of five problem using the hypoelastlc equations of grade zero and grade one.

While doing the problems for grade one, we make an inter*

eating observation regarding an apparent instability la . the solution. The question seems to be still open.

Applications to plasticity L D are also indies ted.

Zn chapter IX , we have shown the possibility of tfaa

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existence of do formation energy for an hypoelastlc mate

rial. In Tact, we calculated it for grade zero £

for the first time. In this chapter, an indication is also made as to how we can introduce thermal stresses in hypoelasticity - the constitutive equations as well as the heat conduction equation are derived for grades zero and one.

finally in part I I I , we discuss the interactions of different fields, for example, the affect of polari

zation on an elastic dielectric subjected to large defor

mations, the effect of a given magnetic field on the torsional vibrations of a conducting circular cylinder and that of electrical conduction through finitely deformed bodies. In chapter X, we explain the basic ideas of electroelastostatics and study the symmetrical expansion of a hollow spherical dielectric with the help of a theory i _ / recently formulated for an elastic dielectric subject to large deformations and polarisa

tions. A thick, incompressible, spherical shell, of internal radius h-i and external radius h z , carrying a uniform surface charge at the inner surface, is expan

ded symmetrically in the radial direction and held in equillbriuBi under internal and external pressures. We determine the electrostatic field and, the stress tensor.

We find that the effect of polarisation is to increase only the radial stress. In this chapter we also give*

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the theory of electrical conduction / **?_/ through finitely deformed bodies recently developed by Bivlln L -3°_y* We show how to simplify the basic equations of this theory so as to apply to the specific problems and illustrate with the help of an example.

Chapter XI deals with th® effect of a uni fora L-51!

axial magnetic field on the torsional vibrations of an isotropic circular cylinder. It Is found that the type

of simple harmonic wave motion remains the same as given by Love / 3 £ _y , though the velocity of propagation off such waves along the cylinder is increased by the aagne- tic field. In the last chapter H I , we have analysed [3:

the propagation of a wave surface of order one into an elastic medium, at rest in its unstrained position by

following Thomas's £ 3 if J methods. Mien, unlike Thomas, we retain the quadratic terms In the strain

tensor, it is found thst for irrotational waves the velc city propagation is no longer constant and consequently

the successive positions of the Irrotational wave m r m faces H . (t) do not fora a family o f parallel sarffaee*

in space. ?or equivoluiclnal waves, however the velocity of propagation is still constant.

Mb conclude the thesis by giving a few suggestion*

for future research. A number of problems aust be solved

~ 7 -

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theories of non-linear contin\ra» mechanics. All the problems discussed in t iis thesis are published and reprints attached at the end. Almost all the solu

tions are in a simple closed form.

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PAHT I

ANISOTROPIC FLUIDS

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1 .1 . Definition of anisotropic fluids.

The quality of variation of any physical pro

perty with the direction in a body along which It is measured is called anisotropy. I f there exists no preferred direction with respect to a certain consti

tutive property of the material, the body is said to be isotropic with respect to that property, otherwise aelotropic or anisotropic. For example, the resisti

vity of certain single crystals measured with the electric field along a particular crystallographlc direction may be higher than that along direction per

pendicular to it . Such crystals are anisotropic with respect to resistivity. It is well known that a

material may be anisotropic with respect to one pro

perty and isotropic with respect to another. Sxaaples of bodies which are anisotropic In soae of their pro

perties are liquid crystals, single crystals and

* aggregate of polycrystals with a preferred orienta- tion.

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10

An anisotropic fluid may be visualised aa a suspension of non-spherlcal particles i . e . , particles having some preferred direction. In particular, the anisotropic fluid that we consider here can be regarded as composed of dumbell or ellipsoidal molecules whose boundaries are surfaces of revolution, the preferred direction being the axis of revolution. This direction may change with position and time, its motion being governed by the fluid motion.

1 .2 . Formulation of the theory.

We represent the preferred direction mentioned in the last section by a vector y v of variable magni

tude. The total linear momentum, moment of momentum sod kinetic energy respectively of a pair of mass points can then be written as

In ( 1 . 2 . 1 ) , the dot denotes material derivative,

(

1

.

2

.

2

)

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and and ape the two masses located at points with coordinates ^ and Z.^ respectively.

The square bracket stands for antisymmetric part as described later. Now i f V be any material volume with boundary

S

^and be the vector element area on

S

directed outward relative to

V

, the conser- vation laws can be put in the form

— f f <A V - o

I t ^ ’ (1 .2 .3 )

77 j M ^ <^aj +

^{[ x} ^d\!}

^ $ y P ( 1 .2 .4 )

TZ I w ij | * r U .i \ x r .n «AV

d t

V

S ^ (1 .2 .5 )

A

a t 6

)

V

Here f , ^ ; and ^ are the volume densities of total mass, linear momentum, bivector 'moment of momentum and energy respectively, Is

the stress tensor, the body force per unit volume

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12

and ^ the heat flux. The square bracket stands for 'the antisymmetric part o f e.g.

“ t y ] = * i X i h ~ x i ^j ⁱ ⁾ ^-

For the type of anisotropic fluid that we are considering, one can write

). 3.

I I r . (1 .2 .7 )

(1 .2 .3 )

(1 .2 .9 )

where £ Is the total less kLnetic energy I . e . , the internal energy density. Following Oseen / we Introduce one more equation of the form

di

(1*2*

^ V

where ^ . is the sum of an extrinsic part sod an intrln- sic part. The extrinsic part may be non zero I f there

are applied magnetic fields. We assume that there is no magnetic field so that the extrinsic part Is zero.

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The derivation of differential forms for these relations by using the divergence theorem and the identity,

cit

J

v leads to

f + f X l,i = ° « (1 .2 .1 1 )

^ ‘ ' f /“

p •/ p r

1 1 ' H 1 n ’ (1*2*12)

I «

f h ~ > (1 .2 .1 3 )

> £ * ^ - * £, i >

Tlr . - 4- tl 3jJ U vJ

(1 .2 .1 5 )

The theory developed by Oseen / appears to be con- sistend with (1 .2 .1 1 ) to ( 1 .2 .1 4 ) , but not with (1 .2 .1 S ).

In sccordance with usual practice in mechanics ha assu~

mes that i ^ =. fc-. . However, tjhere is soae pre- ?

o «

cedent for ( 1 .2 .1 5 ) . In his theory of elastic dielectric*

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Toupin I 2>i J introduced an aquation formally identi

cal with ( 1 .2 .1 5 ) , with vi i interpreted as the polari

zation vector, representing intrinsic local part of the electric field, and denoting the local part of the stress tensor. Grad / 3 discussed conserva

tion laws which are mor® general than used here. In doing so, he introduces an equation similar to (1 .2 .1 5 ).

We set

Ll

‘■i

J >

. 0 . (1 .2 .1 6 )

and

A

71 CO

The vector that while

A

Ti- plays an important role. It can b* shown remains invariant under time dependant

T

orthogonal transformations, ^ does not. Using (1 .2 .1 5 ) to ( 1 .2 .1 7 ) , we obtain

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I f we take help of ( 1 .2 .1 8 ) , energy equation (1.2.14) can be rewritten In the form

\

Brillouin's principle, as proposed by Truesdell 3$

for constructing constitutive equations states that stress and heat flux depend on the same variables*

Supposing that this principle holds good for ^ . as well, we can say that t- > ^ and ^ at a particle P

at a time t are all functions of the variables

evaluated at P at time t * "T* being the absolute

* .

temperature. Since 7lj_ and A,; : do not remain invar i'

ll

ant, the variables listed in (1 .2 .2 0 ) can be replaced by

f> T , Uj_ 9 n c , £ 7 . (i.2 « 2 i)

We farther restrict the present analysis to the fluids is which i-. P fy. and ^ are linear functions of the . variables kC.- , d . ■ and "1..- . So we write

1 V

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A ^ /

where A A , J> and C -4 are functions of f , T and rtj_ * We further assume that these relations are invariant under reflections through all planes contain

ing n- as well as the rotations. This implies that A A , B > and L A are transversely isotropic tensors with respect to direction . Smith and Rivlin £ 3^ _ / have shown that any such tensor can be represented as a linear combination of outer products formed from the tensors

■W.- and <$\• i .e . Yli VI: (1 .2 .2 3 )

i

4

?or our purposes the scalar coefficients in these combi- nations can be reduced to functions of

J , T , n "(= n c . (1 .2 .2 4 )

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where YlL — Til a15*3 ~ riL 1 *®* an<3 -■rt-i, are physically indistinguishable. From these considerations and from ( 1 .2 .2 2 ) , (1 .2 .2 3 ) and ( 1 .2 .2 4 ), we can, after necessary simplifications, obtain the

following relations:

where > , ji * and 1 A are functions of T , in".

In order to have a determinate system of equations (1 .2 .1 5 ) should be an identity. This means that

To find the equation of heat conduction, we oust combine the equation of state

(

1

.

2

.

26

)

(

1

.

2

.

86

)

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- 18

^ Cl.2.29)

with the energy equation ( 1 .2 .1 9 ) , $ being the entropy per unit ma3s. We have the invariance requirement that (1*2.29) retains its form under all rotations, which means that it reduces to the form

(1 .2.30)

fefe know that

- 6 t:

(1 .2 .3 1 )

and assume that it can be solved for S in terms of f and r\~ , We introduce the free energy function

(1 .2 .3 8 )

such that

^ _ ? j 1 ----L , , C> t | I J , | I ^ £i

>r < - - k It „ viM ' * - i * (1*2.33)

' °>* r n ;n" ®1,1 »s,j> ^ ‘Tjf

From ( 1 .2 .1 7 ) , we see that

1

a- n. = n t ^ ^.

(1 .2 .3 4 )

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From ( 1 .2 .1 1 ) , ( 1 .2 .1 4 ) , ( 1 .2 .3 0 ) , (1 .2 .3 3 ) and (1 .2 .3 4 ), the rate of production of entropy is given by

1 T - . ^ I . . _ ^ + ^ ^ ^ .

(1 .2 .3 6 ) The form of the Clausius inequality most often used in irreversible thermodynamics is

= 0-ii + t — o ,; ) A ;; - (jH + A . \ _ a -r |T

<* ^ r ^ - w - N c * * ' (1.2. w ) CP .

Equations ( 1 .2 .1 1 ) , ( 1 .2 .1 2 ) , (1 .2 .1 3 ) and (1 .2 .3 S ) yield eight equations for the eight unknowns ( f > T , a~i.

^ ) , it being understood that the body force and the relations ( 1 .2 .2 5 ) , ( 1 .2 .2 6 ) , (1 .2 .2 7 ) and (1 .2 .3 2 ) ere specified.

1 .3 . Summary of the basic equations used.

The theory is further particularised to apply only to Incompressible fluids under constant temperature. It is also taken for granted that 'molecular inertia',

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- 20

is negliglba so that » o » T and f are omitted on the grounds of simplicity. With all these simplifi

cations, we give below a summary of governing equations*

tij = -j-Jy + i /u i£j + ( M

+ i “ i l ^ W i + ^ t V j O ,

\ - “ ij i- X O l j «■ - d k<>-t » „ r,L ) > ( U M )

^ i l 5 0 ’ <1.3.3)

I

H «

®X; = t • + r'l

5 'J ' V ' (1.3.4)

n where A^> and are material parameters* These are simple functions of i ^ snd V 4 introduced in (1 .2 .2 5 ) to (1 .2 .2 8 ) as functions of f , T and

For fixed ^ > "f as we are considering isothermal and incompressible flow, these are functions of w only.

It may be noted that equations ( 1 .3 .1 ) and (1 .3 .2 ) are still non-linear in yi^ .

In the problems that we give in chapters 11,111 end IV , we take n, i)L = j *s an additional condition

*• 4

so that these material parameters become constants.

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Their values are yet to be determined from experiments that are in progress.

1 .4 . Main features of the theory.

Ericksen L & -J formulated this theory of aniso

tropic fluids to present something that could incorpo

rate some features of Oseen's / 33~_/ theory describing the behaviour of liquid crystals and at the same time be properly invariant. He has applied this theory to simple shearing flow of incompressible anisotropic

fluids and has shown that for a certain class of these, the predicted behaviour is like that of the Binghaa

material. In the one dimensional descriptions of Bingham materials used by Buckingham / ,5~_/ and Beiner / * l i j or in the three dimensional formulation introduced by Oldroyd / i ’J _ / , two sets of constitutive equation* ere used. One set, applying at low stresses, describes an ordinary elastic material. The other applying at high stresses Involves viscosity similar to that whieh cha

racterizes Newtonian materials. A scalar yield criteria is used for determining which set applies. The analyses of Veraa Zfy#/IL/ that are the subject matter of chapter

* I I , III and I? show that the single s£t of constitutive equations used in the above theory, unlike in

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22

Oldroyd's L (7 -/ theory, is enough to predict the surfaces of discontinuity that may arise in tha flow process. There is no need of using an yield condition

for this purpose. It has also been suggested / _ / that this theory Is applicable also to some fluids

commonly treated as isotropic and describes them better than simpler theories of isotropic fluids. No doubt some of the real viscoelastic fluids have been excluded by the assumptions that we make, yet, a3 said before and as we shall see in later chapters, the theory is good enough to score over other theories of viscoelas

ticity. It becomes more useful because we can solve the problems In a closed form without much difficulty and have a clear picture of what is happening. Hie theory is non-linear in and gives Interesting results. It has a large scope for future research.

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2 .1 . Statement of the problem.

We consider the steady motion of an Incompre

ssible anisotropic fluid confined between two coaxial right circular cylinders of radii and C Ai-7> ^| ) which are themselves rotating about the common axis with constant angular velocities and . We use cylindrical polar coordinates A.; & and % ,

taking the

2

-axis to be the axis of the cylinders. The velocity field is then given by

u » o , V = y y ( K ) 9 ( 2 a a )

where V"C is to be determined. Ufa assume that the material adheres to the walls. Hence i>-(A) is subject

to the conditions:

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- 24 -

2*2. Reduced form of the basic equations.

By substituting ( 2 .1 .1 ) into ( 1 .2 .1 6 ) , we have

[ 1 . 1 , l V J

^ '/t* (v -v i* j 0

'h j l =

0 - [jL (v+»ti) o

J ^ W ) 0 «

0

^{^} ^* ₀ _o _o

w

(

2

.

2

.

1

)

where 8 dash means di fferentiation with respect to K • Ths equations (1 .3 .1 ) reduce to

t„ = - |> + /a, n % a, O - u-n.‘) £ ^ ^ ^

W ' !> + U 1 "L + *, n3, #

t3= - (> +/*,

*3

+ ^ ^ ( v - W r S 1) ( / A ^ ) ^

^ii 'A( \ + C Jr/- y- * ' ) 5t v ni 3

% , u i V V i + A V > 3) ,

(

²

.

²

.

²

)

(2*2*3)

(2 .2 *4 )

( 2 . 2 . S)

(2*2*8)

(2*2*7}

We assume that "ft =. n ( t j . The equations (1*3*2)

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<

²

.

²

.

⁸

)

dtt,

— - r 1 C‘i at h.

( 2 .2 .9 ) 3 , ,

- -f- (if ■

dt ^

— \JJ

(

²

.

²

.

¹⁰

)

where t\, +- n ^{a k} = i

(

2

.

2

.

11

)

Equations ( 1 .3 .4 ) reduce to

U 2,1.) = “ f 3*

H. (2 .2 .1 2 )

(2 .2 .1 3 )

A / j

(2 .2 .1 4 )

where we have used the fact that the stresses are ftinefcion*

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26 -

Of h. and t alone and also that there is no body fore® acting.

2*3* General solution.

Integrating ( 2 .2 .8 ) , (2 .2 .9 ) and (2 .2 .1 0 ), we h«ve

L

k. +• x A -I

A n(

(2 .3 .1 )

(2 .3 .3 )

where K, and |3 are arbitrary functions of K only, and L and Y are given by

L

In finding the above solution, we have not made use of

* any special method except the ordinary process of eliai-

e

nation. See the appendix for aore details.

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2 .4 . Steady state solution and general analysis.

life restrict our attention to fluids for which

\k\ ;> t . Than as f" -*oo , the solution, in 2 .3 , approaches the steady state solution

StXv cp , = Qxxj, , ^ (2 .4 .1 )

where <^> is the constant angle determined by the condi

tions that

i

^ A t * . ,

A-t 1 and that it lies in tha range

0 < f < 7T ^ when Y > o

,

X > I , (8.4.2)

^ /t ^ f < .7 T

jX

^when / ^ 0 , > ; (2.4.3) 7T /* < ^ when y > o p A (2.4.4) 3ftItf ^ /i' when / <5 ? X > I . (2 .4 .5 )

The theory treats A and — Yl as indistinguishable, which means that there is no loss in generality in

taking

0

< <$> < fT , as is done here. Ms consider the stable steady state solution as given by (2 .4 .1 ) through ( 2 .4 .5 ) proceeding under the assumption that

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- 28 -

V ^ o anywhere in the annulus. Than the equations ( 2 .2 .2 ) to (2 .2 .7 ) give us the stresses as followat

\ -

^{- f}

t + I

^A

1,

4 — i 2-

11 f> + /*, j> + h % { J

V

- b

*, -it

V

S . ‘ ° ,

where

+- O Y ,

(2 .4 .6 )

(2 .4 .7 )

(2 .4 .8 )

(2 .4 .9 )

(2 .4 .1 0 )

(2 .4.11 )

f ,

We observe from ( 2 .4 .1 2 ) , ( 2 .4 .1 ) and ( 2 .3 .2 ) that th*

constants A, & “Jnd £ reverse sign when Y does.

Therefore places where Y - o must receive special car®,

^or the time being, we exclude these'from the annulus and find out the corresponding restriction on the

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t* K

12. ~ --- ? (2.4.13)

where K is the torque per unit height, whether or not ( 2 .4 .9 ) holds. Substituting the value of t (1_ from (2 .4 .1 3 ) in ( 2 .4 .9 ) , we get

= _K . __ C

I T O ) (2 .4 .1 4 )

where

C

represents one of the two possible values given by ( 2 .4 .1 2 ) . Physically, we expect the stresses to do non-negative work in deforming the fluid. Therefore we require that, for all i ,

2 W =.

I {

(C-t- 2-j) /) J (2 .4 .1 5 )

i . e . , that I* > o and that C~ and 7 be of the m m

sign. From (2 .4 .1 2 ) and ( 2 .4 .2 ) to (2 .4 .P ) we find that this requires that

F ' \ ° j j * 2 . C/'4" - ! ) w i»n I ; (2 .4 .1 6 )

, / t ( < 0 , yU.t (K C" l ) ^ O when (2 .4 .1 7 )

e

It is then clear from (2 .4 .1 4 ) that )(, and V m at be

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of the same sign. From (2 .4 .1 4 ) it is also obvious that 'i will not be zero anywhere in the annulus provided that the torque per unit height, \c , supposed to be given, satisfies

- 30 -

I Ik ^ |r \ --r

2ir * i ^ i - ■ (2 .4 .1 8 )

Reversing the sign of j\ , which amounts to reversing the directi on of tho applied torque, changes the solution in a rather trivial and obvious way. We therefore take the sign of K to be given as positive and continue

with the above solution under the assumption that (2 .4 .1 3 ) holds. Integration of ( 2 .2 .1 2 ) , with the help of ( 2 . 4 . 6 ) ,

( 2 .4 .7 ) and of ( 2 .4 .1 4 ) , then gives us respectively:

and

=

Y / L

_

‘u ( 2 .4 .2 0 )

# where we regard k., h.^ , k L and say, vTLj f u given and find - Jl^ and p tro® the boundary conditions ( 2 .1 .2 ) or in other words from the following

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equations respectively:

K ziri)

A C

Lj)

u

.1, « v s i

(2 .4 .2 1 ) and

P - - - L

(2*4.22)

This will give an acceptable solution, as long as (2 .4 .1 3 ) holds, for th® entire annulus /*. ^ h.^ .

Zn the case when (2 .4 .1 8 ) is violated, we have two possibilities:

/

(2 .4 .2 3 )

or

(2 .4 .2 4 )

I f (2 .4 .2 3 ) holds, we can adopt the previous solution for the annulus A , ^ V <C h, where 1Z =x K | tirC , and assume rigid rotation for h. ^ & <i \ L in order to get a combination of solutions physically acceptable

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32

tor all k. • On the other hand, I f (2 .4 .2 4 ) holds, we assume rigid rotation for the entire annulus

For investigating the stresses at any point in the rigidly rotating annulus when either (2 .4 .2 3 ) or (2 .4 .2 4 ) holds, we refer back to the basic equations, since the solution given before no longer applies. The equations (2 .2 .8 ) to (2 .2 .1 1 ) then reduce to

Assuming Aj and to be independent of & and X , we get

The stresses will take the form

t

(2 .4 .2 7 )

(2 .4 .2 8 )

(2 .4 .2 9 )

(42)

(2 .4 .3 0 )

(2 .4 .3 1 )

Equations (2 .4 .1 3 ) and (2 .4 .3 0 ) give

(2 .4 .3 2 )

Combining (2 .4 .3 2 ) with ( 2 .4 .2 6 ) , we obtain

We observe here that the solution (2 .4 .2 7 ) - (2 .4 .3 3 ) and (2 .4 .3 6 ) below, is valid tor h < ^ or

for A., < i_ h. according as (2 .4 .2 3 ) holds or (2 .4 .2 4 ) holds. In the first case i . e . , when (2.4*23) holds, we have Interface 4 - h. across which the above two solutions, separately valid for A., <L k. k and for /u < t 'tj_ , should match. ?or this pur

pose, we note from the equations ( 2 .4 .1 ) to ( 2 .4 .5 ) that

^ OT ^1 according as X > I or I • Thus in case (2 .4 .2 3 ) holds, continuity of Tl require*

that we have the following 3Cheme in 4 2 .4 .3 3 )

(43)

- 34

(2.4.34)

and A . b e i n g positive in either case. It follows fro« the definition of H. , (2 .4 .1 2 ) and (2 .4 .2 ) to ( 2 .4 .5 ) that U is continuous at = A. . Row wo can find j> , when (2 .4 .2 3 ) or (2 .4 .2 4 ) holds, by inte

grating (2 .2 .1 2 ) and using the solution (2 .4 .2 7 ) to ( 2 .4 .3 2 ) :

by (2 .4 .3 4 ) or (2 .4 .3 5 ) according as X > I or <i-| .

*In ( 2 .4 .3 6 ) , the constant should be ckosen to yield con

tinuity of the stresses at A. = A. . This arguaent fails

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when (2 .4 .2 4 ) holds. The solution for rigid rotation, given above,will apply. Physically, it seems plausible that this is what one would obtain i f one started with a torque satisfying (2 .4 .1 8 ) and then decreased it until

(2 .4 .1 8 ) failed. This line of thought suggests how to remove the ambiguity of sign in (2 .4 .3 3 ) when (2.4*24) holds. We use the choice given by (2 .4 .3 4 ) and (2 .4 .3 5 ).

This completes the possibilities.

2 .5 . Stress relaxation.

We consider a hypothetical case when the fluid is in a steady 3tate of motion, but ti does not take on its steady state value. Physically, of course, it is hard to attain this since the analysis then becomes Inconsistent with the equations of motion with zero body

force. In the transition period i . e . , when n star

ting from the initial value, approaches its steady state value, the stresses relax, and we can calculate from ( 2 . 4 . 9 ) , ( 2 . 4 . 3 ) , ( 2 .4 .4 ) and (2 .4 .1 3 ) as follows:

( 2 .5 .1 )

where

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33

A ,* ju, nJ > M ~ ^xJ ^ K jAV^+O/L- (2 .5 .2 )

\ = ( M k|)/[A J<jrU)-H j j + (2 .5 .3 )

In the case when there exists a lower bound on y f we can only say from above that there will be an upper bound on the relaxation time and the discrepancy tends to zero*

For further Information regarding relaxation times, however, we must know more about the functions ^ | t \ ) and It >L)

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APPENDIX (Details of 2 .3)

Rewriting ( 2 .2 . 8 ) , ( 2 .2 .9 ) and ( 2 .2 .1 0 ) , we obtain

~7> n

"bt 1 \ ( a - -a ( 2 .3 ,1 ) '

V n ,

(2.3.3)*

* %

- s — i i K n t y ) v r,

(2 .3 .2 )*

where

(2*3*2) 9 we have:

y\ -z c) n

(2.3.4)*

^vvi

sum n ^- i % n

£r{ tf.f/ (2 .3 .5 )*

Multiplying (2 .3 .4 ) by ^ on both sides, ( 2 .3 .5 ) by /- q )\ n. , on both sides, and equating the right hand sides, we gat on simplification:

'±_

at

K- £ A Ai.

(2 *3 *6 )’

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- 38 -

Integrating ( 2 .3 .6 ) / , we obtain

\ ~ k, < + L ;

where is an arbitrary function of ^ and

/ low substituting the value of n ( from (2 .3 .7 )

(2 .3 .2 ) / and Integrating the result, we have

\ ~ j j ^ U + b j |(ATi; ( y t + 5j j - L

^X

r ,. „ , , r , V 7 1

X | K ( +- l' L (Vb+S)J

0slng ( 2 .3 .7 ) , ( 2 .3 .9 ) and ( 2 .2 .1 1 ) , w« get

(2 .3 .7 )*

(2 .3 .8 )*

into

(2.3.9)*

(2 .3 .1 0 )*

where P is an aroitrary function of ^ . We, thus, have finally A , 0 L and ^ as functions of L and ^7 •

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3 .1 . Formulation of the problem.

Helical flow is a laminar flow which can occur in an annular mass of fluid contained batwean two in

finite coaxial cylinders of radii , and rota

ting about their common vertical axis with angular velocities J l( and J 1 L respectively. We take the flow under a constant pressure gradient p parallel to the axis of rotation. This flow contains concentric pipe flow and Couette flow as special cases involving, respectively, zero applied torque and zero driving force in the axial direction. We use cylindrical polar coor

dinates h. ? (§) , £ taking the z-axis to be the axis of the cylinders. The velocity field is then given by

* assume that the material adheres to the walls* Hence

e

w ( > )

and

W

(."■) are subject to conditions

> ( 3 .1 .1 )

where W (>) and

W

are to be determined. Wa

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40

2)

We suppose that the body force ^ par unit mass have a single valued potential , so that = — grad ^ . This helps us to work with a modified pressure given by

(3 .1 .3 )

3 .2 . Reduced form of the basic equations.

By substituting ( 3 .1 .1 ) into ( 1 .2 .1 6 ) , we obtain

w -

o >/,. vj,_

• N -

oo

0 0

VJ 0 o J i 'W ' o o

( 3 .2 .1 )

where the dash denotes differentiation with respect to h, • The equations ( 1 .3 .1 ) reduce to

(3 .2 .2 )

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3)

^3 =

~f

+ ,Ml V * ^ A + W 71, ^ 2^

(3 .2 .4 )

\j=-

/*/£-© + yu j y), nt-H / ( n ^ + ^ u^ t

^ j*"i

^| ^{^CV*-}

ni.J

^r

^ 3= /a / ^ & - h ^ |n( ri3 ^ / ( » 1_ c , ^ + q ) ^ Jk± +

-f- [fc^e.viL7)3 + y ^ 0 ( n j > T ^ ) j

\ f A V

3

+ y ( ni ^ 8 + n3 W h v » 3 H

+ / ^ [ »ij f r ^ e yi, fl3j-

(3 .2 .5 )

(3*2«C)

(3 .2 .? )

where Y 0 ~ W ; ^»/ 1*9 f) “ ^ U? so that */ If the rate of shear given by J grad y j and is always

positive. ( 3 ,2 .8 )

Assuming n ~ %. t the equations ( 1 .3 .2 ) take the fora

(51)

42 •

= v (^»,g^.8) ( l+ x - iX ^ s .) - ;U W 3 ( > '" ) , (3-8-10)

= V (i ^ S w e ^ l t - X -* * »>) - » A 'A|/ii*ij(Y4>^ j ( 3iS-u)

where rl f Y) + fh, - I • Equations ( 1 .3 .4 ) , in the

I 1* 1

I I*

absence of body forces, reduce to

c H * l i ) + 4' L t \\ - f -’l ts (3 .2 .1 2 )

0 7

;

( 3 . 2 . 1 3 )

j

( 3 . 2 . 1 4 )

where we have used the fact that the stresses are func

tions of k. sn<3 "t alone.

*3 .3 . General solution.

Integrating ( 3 . 2 . 9 ) , (3 .2 .1 0 ) and (3 .2 .1 1 ) with the help of th® assumption 7\j^ v\j^ =. | i we get

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the following equations that are enough to determine _> 71^ and Yi^ in an explicit form;

V-l - j. h = k v Y * ^ b - V c~*c a ), (3>3>1)

i * i

I v c*

* + { f s j - j

SJ

TVt- ^«.

( 3 .3 .2 )

(3 .3 .3 )

where S', - ® a. 6. / W ] , K, = K j tA-0 “ » * and K ar® arbitrary functions of /L . The method of

integration is similar to one given in Appendix of

Chapter I I . Eliminating 7)^ out of ( 3 .2 .9 ) and ( 3 .2 .1 0 ) , Tij_ out of ( 3 .2 .9 ) and (3 .2 .1 1 ) we get two aquations

oui of which it is easy to get ( 3 .3 .1 ) on integration.

3 .4 . Steady state solution and general analysis.

We again restrict our attention to fluids for which [\ |> | . Then, as "t , this solution

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44

approaches the steady state solution

whore £ is th® constant angle determined by the condi

tions that "kir ^ lyj^AriyV and that i £ lie*

in the range

O < i < 'T/I) when " A X ,

^{(3 .4 .2 )}

rrk < It when K <--1,

( 3 .4 .3 )

irk y

_i

' ^ >

^>•

when

^A

<^— I

; (3 .4 .4 )

3rr(‘i i

^V"

;» < K when > i

^. ^{(3 .4 .5 )}

The theory treats Ti and - n as indistinguishable, which means that there is no los3 in generality in

taking o <L S, < i| as is done here. We consider the stable, steady state solution as given by (3*4.1) proceeding under the assumption that -s.'co and w do not vanish together anywhere in the annulus. Thus the . equations ( 3 .2 .2 ) to ( 3 .2 .7 ) give us the stress** as

under

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(3 .4 .6 )

tz~ - i + /*■ I ^ +• 5> / Cao^

( 3 .4 .7 )

t = - J +-*, + B y ^ ^

t (L= (t +

( 3 .4 .8 )

(3 .4 .9 )

"tj;" + 3 */j ,

(3 .4 .1 0 )

t = ( ^tc ^ f-&y) c«e

(3 .4 .1 1 )

where

\- A £ u * 2l | ,

+ V j )

= / 1 $U« 2. ” ^

I f -

X]) _ 1 ( ^ f ^ - ) + 1 Mj_ U ^ 2-^ .

(3 .4 .1 2 )

•Excluding the places where y =-0 .from th® anmilus for the time being, we find the corresponding restric

tion on the constants. Integration of (3 .2 .1 3 ) and

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- 46

(8 .2 .1 4 ) gives*.

(3*2*13)

i . n ,

x x.

(3 .2 .1 4 )

where H is th® torque per unit height and L is an arbitrary constant. Equations (3 .4 .1 3 ) and (3 .4 .1 4 ) hold

independently of ( 3 .4 .9 ) and (3 .4 .1 0 ) respectively.

Substituting the value of t ^ from (3 .4 .1 3 ) in (3 .4 .9 ) and of t, , from (3 .4 .1 4 ) into ( 3 .4 .1 0 ) , we get

YC*o 9 = k, 'Zo

= J L

•^TTj)

c

(3.4*15)

YSuOitt c: vy= X

3 c

where C represents one of the two values given by ( 3 .4 .1 2 ) . Physically we expect the stresses to do non- .negative work in deforming the fluid. Therefore we

require that

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(3 .4 .1 ? )

i . e . , 3) ^ o and C. > o since Y is always positive.

From (3*4*12) and (3*4.2) to (3 * 4 .5 ), we find that this requires that

/A-, j> o , ^{l ' h L ~} ^\ ^{) +} ^{l \} XL + > O when \ > l , ( 3 .4 .1 8 )

( 3 . 4 . 1 9 )

It is then clear from (3 .4 .1 S ) and (3 .4 .1 6 ) that h.v) / and W ' will not vanish together anywhere In th* ansa*

lus provided the torque per unit height ^ , supposed to be given, satisfies

(3*4 *20)

and ( L + j T

K

lj > 0 , ( 3 . 4 . 8 1 )

where L can be found in terms of M from boundary conditions.

Assuming that (3 .4 .2 0 ) and (3 .4 .2 1 ) hold, we con

tinue with the above solution. Integration of ( 3 . 2 . 1 2 ) ,

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• 48

“ “ “ ‘ h* 'MlP ° f ( 3 ' 4 ' 6 ) - __ 0 ^ . 1 3 ) m ) ( 3 . 4 . 1 4 ) ,^♦ gives us

/*. ^ | + A y + /«, S i j j t + H j i - y d i + ? * - ^ c J - g

+■ f J ' t S 1oliv + CUw4o~i,

( 3 . 4 . 2 2 )

and

f t = ? _ i L ______ Cj l _ x 1 4 * 3 ^ * - 0

* j * ‘{(m| ( l * +{P I*-)

( 3 . 4 . 2 3 ) W = 0 , + _ Ta.1,

i d L Tt4J>

_ _ c f L / t + i . ? ^ .

j — ^ ^

( 3 . 4 . 2 4 )

* « « r* * * rd ^ K ana . „ , J L , . , , l M n

and find v / L - J l T> A i

2- I ; V and L from tha boundary conditions (3. 1. 2 ) or in other words from

J!L

( L _ _ L \ _ _ M C r ^ i

^ ^ ^ * * * \ k $ l i i i t + (L t + U f £ f * k ’ (3 .4 .3 8 )

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A ( = T, - ( M j i r m t ) - ( M C |j t i) j

%

'k1 (3 .4 .2 6 )

= Q, +(L|3>) l ^ K , - ( h ^ * )

— (£ i J>) J ( i L / L t ? i J ^ | i r ) + (tL\ + -pA.3)1}

*Ri

fRfc

J

0~K+jr'f;0

{ ( ■ + ( L ,l+ i

?A>/ } ^h..

(3 .4 .2 ? )

(3 .4 .2 8 )

Ihe volume discharge per unit time through a cross section perpendicular to the z-axis is

h

S - 2ir \ /jl \J JL a.

(3 .4 .2 9 )

*1

This will give us an acceptable solution, so long as (3 .4 .2 0 ) and (3 .4 .2 1 ) hold} for the entire annulus

^ ^ <C . In case one or both of them are violated, we have two possibilities

b i

(3 .4 .3 0 )

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50

or

( u , + < c < \

( 3 .4 . a )

I f (3 .4 .3 0 ) holds, we can adapt the previous solution for the annulus 'R( ^

root of the equation

for the annulus ^ h <1 li where \ is a real

i'^-1 t ( L /t + 1 ^ = C h. L (3 .4 .3 2 )

uir; ^ ^ '

and assume rigid motion for "R A. ^ 1?^ in order to get a combination of solutions physically acceptable for all h. • On the other hand i f (3 .4 .3 1 ) holds, we assume rigid motion for the entire annulus ' R ^

In case when (3 .4 .2 1 ) is violated but (3 .4 .2 0 ) holds, we again have two possibilities:

L + i ? < > o ,

* (3 .4 .3 3 )

and L + -L <1 0 . (3 .4 .3 4 )

t

I f (3 .4 .3 3 ) holds, we adapt the previous solution for the annulus It, ^ /i <£ ^ where 'R* is a root of

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L +• (P £ j a*) r o and assume Couette flow for

He , In order to get a combination of solutions physically acceptable for all k. • On the other hand if (3 .4 .3 4 ) holds, we may assume Coutte flow for the entire annulus. ?or the stresses in rigidly moving annulus or in Couette flow, we can refer to chapter I I .

3 .« . Stress relaxation.

We consider a hypothetical case, as before, when the fluid is in a steady 3tate of motion, but u doea not take on its steady state value. Physically, of

course, It is hard to attain since the analysis then becomes Inconsistent with the equations of motion with body force present. In the transition period i . e . , when

M. starting from the Initial value, approaches its steady state value^ the stresses relax and we can calcu

late as in the previous chapter

^ 0 ) = h .'d W '

4 ( C/v )+ S. J - i /3

- y }

( 3 . S . 1 )

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52

In case when there exists a lower bound on Y , we can say from (3«5«1) that there will be an upper bound on the relaxation time and discrepancy tends to zero*

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4 .1 . Problem.

We consider the sieady motion of an Incompressi

ble anisotropic fluid confined between two coaxial right circular cones of semivertical angles 0( and

> &t) » with the common axis vertically upwards and vertex downward, which are thenss elves rotating about the common axis with constant angular velocities JL ( and J\.j_ . Vte use cylindrical polar coordinates K, Q and % , taking vertex as origin and the upward axis as z-axis. The velocity field is then

given by

( 4 .1 .1 )

where (_ ^ |

z)

is to be determined subject to the boundary conditions

(63)

defined by .

The physical components of and are given

*>y

o ?c v, o

> N ] "

6 - C ^ 'k ^ ) o

'/^ 0 V- o )i\

J> 'ji+L- °

L° -v a o

(4 .1 .3 )

where Y( = h -A^ , i L = ^ t the sufflauo ^ and X/- denoting partial differentiation with respect to ft. and with respect to Z. respectively.

4 .2 . Reduced form of governing equations.

The constitutive equations ( 1 .3 .1 ) give

V

i ⁼

{ *l + A LA *,\+ K \ n 3) n ) i - A ^ , n j ( ,

! V i \ (/, w y v ^ ) ^

(4 .2 .1 )

( 4 .2 .2 )

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^ - M

is -' V .«, M j t ,t L ^ ^, ^{^} ^{t ^} ^ ^

‘ ‘ A W . , ( 4 . 2 . 3 ) 3

- +

( 4 . 2 . 4 )

^ = 4, !', ••'!. f u,

( f m * \

( 4 . 2 . 5 )

L l ' V j-* C V a ) ^

( 4 . 2 . 6 )

w here we work w ith ^ = |- + ^ \p ^ cj =. — <j/ •

We assume t h a t Y\ - a /^ \ j x ? . Th® »<|fl»tioM ( 1 . 3 . 2 ) now become

^At i . / v i. \

H - ~ <2 I rii- A m( ) - X l^| ^ ? (4 .2 .7 )

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56 -

r|, . f x

- ~ '

(\-\-l

- X »i*j,vi3 Y t

where n, + 4* rt-^ ~ 1 • Equations (1 .3 .4 ) reduce to

£ (

⁴

' > * £ ( « * ) ^ c t irW - - r * J f ;

^{( 4} ^. ^, ^{1 0 )}

*. L t . ) r 1 - fir ) i i -t- _ .

,t (4.2.n)

~~

t K . )

+• —

■ t .

) {- J_ t =■ pa

dv ~ *z v b "is r j . <4.2.12)

where we have used the fact that the stresses are

func

tions of 4 , % and t only.

4 .3 . General solution.

Integrating ( 4 . 2 . 7 ) , ( 4 .2 .8 ) and (4 .2 .9 ) with the help of 71 < n. = i » we get

"fe. k

\

+■ I

H \ T ) t -

1<

j y j

_ (n3 j YV) J ^

(66)

- =

5

Ai y‘

V . /,|y,

(4. 3. 2 )

<

⁺ ^{= | y}

( 4 .3 .3 )

where \([ - K {(£+ lj , ^ % IJ A'F and K. and ^ ara arbitrary functions of

A.

and Z • The method of

integration remains the same as in case of Couette flow and Helical flow*

4*4* Steady state solution and general analysis*

We^agaln restrict our attention to flulda for which

JAj >1

^• ^Then ^cw^{t ~>o}, thia solution froai ( 4 .3 .1 ) to ( 4 . 3 . 2 ) , approaches the steady state aolutlon

(4 .4 .x )

where j> is the constant angle determined by the con

ditions that

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- S8 -

UsS j> = ^

i -0 / ^ 0 } , (4 .4 .2 )

and that it lies in the range

0 <

t ir /(j when ,\ > 1 ^{^} (4 .4 .3 )

« . l / t f t li / v when (4 .4 .4 )

f ^{^ r}^' ^/'

(1 1 i- j •£. M f /4 when X < - 1 , (4 .4 . n)

i i •- '■^*44* ■ X/ IT when X > 1, (4 .4 .6 )

where

j ^, 1

V ^ j % /,

" y > [^ £*>X- s. The theory treats 71 and ■- n as in distinguishable , which means that there is no loss in generality in taking

® 'If' , as is done her®. We consider the stable steady state solution as given by ( 4 .4 .1 ) through (4 .4 .6 ) proceeding under the assumption that j3 ^ o anywhere between the cones. Then we have fro® ( 4 .2 .1 ) to (4 *2 .6 ):

(4 .4 .7 )

( 4 .4 .3 )

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t -

(C + ■ J |2 j) X ,

( 4 .4 .9 )

(4 .4 .1 0 )

(4 .4.11)

(4 .4.12)

where Xft,- '

M.S**

/ '

v t itu3j ; 3- B i^}

Lf ) 4 ji ^ ( v . u ) + / V ^ *-{ ^.

(4 .4 .1 3 )

W® must take special care of the places between the two cones where |i - o or Y| =. Y^=- o . For the time being we exclude them from between 0 = ©, and (E> * 0„ and

I *-

find out the corresponding restriction on the constant so that the steady state Clow may be possible.

Integration of (4 .2 .1 1 ) with the help of ( 4 .4 .1 0 ), (4 .4 .1 1 ) and ( 4 .4 .6 ) gives

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60

where <k and lr are arbitrary constants to be found out by applying boundary conditions ( 4 .1 .2 ) ; and X.

stands for >uj% or "Iam $ . One can see from (4 .4 .1 4 ) that

C j " 1. X 4 (x S i)'lhL a * 3

(4*4#IK)

Here in (4 .4 .1 4 ) and ( 4 .4 .1 5 ) , C represents one of the two possible values given by ( 4 .4 .1 3 ). One can

suppose that & and & are known in terms of Jl( , C and !>, and . Physically we expect the stresses to do non-negative work in deforming the fluid. Therefore we require that for all p ,

n \ \

(4 .4 .1 6 )

i . e . , that % > o and C ^>o as is positive.

From (4 .4 .1 3 ) and (4 .4 .3 ) to (4 .4 .6 ), we find that this requires that

u, >°> /U2.Ca" ij +■ (f X*"( when ^ > 1, (4 .4 .1 7)

/ j ^ /*t (Av-l) + AX2*(/* +«.) ^ o ' when X<1-|, (4*4.13)

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that J t , , J l t , C , j,, y, , flj, a*« such that r‘w

5 L • (4 .4 .1 9 )

Equations (4 .2 .1 3 ) and (4 .2 .1 2 ) give H

f ) * h L* 'Itd + ( f l f ) }

“ ti v f ~

and

*1

O'/6

= ^ f

("'^ j ( ■ * ■ ) ( ^ P y ^ C ' ' 1-!kf)J :a /,

V

(4 .4.21)

respectively. The solution mentioned above, in case when (4 .4 .1 9 ) bolds, will be physically possible provi

ded (4 .4 .2 0 ) and (4 .4 .2 1 ) are consistent with each other

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62

for all X . For that, we must have the following equation satisfied for all X •

I f 3| and are both chosen sufficiently near or in other words i f X is large enough, then froa (4 .4 .2 2 ) it follows that the steady flow is possible provided

Only in the exceptional fluids that have /*(= /V

, can we satisfy all the conditions for steady flow in a vide gap between rotating cones* In case

(4 .4 .1 9 ) does not hold good, we can proceed in the same way as in the previous chapters. It will also be inter

esting to note that i f ji = o everywhere in the gap, for which we must have ( C. 1 j| < |«-| a**" 6, , the whole fluid between the cones will move like a rigid body.

(4 .4.22)

(4 .4 .2 3 )

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We consider a hypothetical case when the fluid is in a steady state of motion hut n does not take on its steady state value. Physically ofcourse it may be hard to attain. In the transition period, the stress relax and as before we have

= « e ,

U . K . l )

where

I I i

r

* h

<- i1 *

(4 .S .2 )

In ease when there exists a lower bound on ft we can only interpret from ( 4 .5 .1 ) that there will be an upper hound on the relaxation time and that the discrepancy tends to zero.

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PART II

RESILIEHT A M3 FLUENT BODIES

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5 .1 . Hygrosterlc materials.

We consider the rollowing equation proposed by Maxwell /

J

cl

S

,

-— =1 ,U 4 A __ O (5 .1 ,1 )

d ' a t t ’

where

S

denotes stress, Cl is the strain, yt i* the elastic modulus and T Is the relaxation tine. For

T >o , ( 5 .1 .1 ) becomes

^ ° ^"

^ t

(5 .1 .S )

wliich is the differentiated fora of the stress-strain relation

S - va. a , (5 .1 .3 )

J

characterizing an elastic solid. I f -we put 7j =■ T « where 'lj is a viscosity constant, we can writ* (5*1*1)

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Hm

«HCD

>4

«

4*ft

mou

£ o VI•»

O faa a

aa

£

t<«

• *0 5 ? (Se!

ca

«HH 81 3

©

*o

£ O

««

H« Oa.

c

<D

•Hm •>

• x>

o r5

£>£?

fa C

0) •) c * o

S 5*

H fa « ^{<0 C}

cr « ■H H

<D «rl ^{o h} c »a ^{• >}

3 l CO'S

fa (0

«> W C XI

d ^•Hn Htt

o 9)

«H (0 ? '

« O

cc tn

*H n >

HO

«M fa

01 0)

« Q.'QJ

H SS'H

©

O O fO

H ■H

a Q, V

O ft O -H

fa *> fa ♦>

+J 0)

0 * 0 o a

••5 to H

M rt M •

(76)

in the form

(5 .1 .4 )

For 'V -> o , >) fixed, this becomes a relation between stress and ’ rate of deformation'

which characterizes a viscous fluid.

The equation ( 5 .1 .1 ) describes a class of materials including both solids and fluids. Materials

PDF in anisotropic fluids AND FLUENT BODIES, AND INTERACTION FIELDS

A THESIS OH

CONTENTS

CONCLUSIONS AND SUGGESTIONS FOB 180 FURTHER RESEARCH

not unreasonable to Imagine that their birth asf have te

In chapters H I to IX , we take up resilient bodies.

for future research. A number of problems aust be solved

' °>* r n ;n" ®1,1 »s,j> ^ ‘Tjf

0

0

\ ~ j j ^ U + b j |(ATi; ( y t + 5j j - L

O < i < 'T/I) when " A X ,

tz~ - i + /*■ I ^ +• 5> / Cao^

and ( L + j T

' > * £ ( « * ) ^ c t irW - - r * J f ;

JAj >1

f ) * h L* *'Itd + ( f l * f ) }

PART II

f ) * h L* 'Itd + ( f l f ) }