AD A132 852 ON A VISCOMETRIC FOW OF A SIMPLE FLUID(U) WISCONSIN /UNIV MADISON MATHEMATICS RESEARCH CENTER K R RAJAGOPALAUG 83 MRC-TSR-2547 DAA029 80 C 0041

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MtC Technical Suzoary Report #2547

C ON A VISCORSTRIC FLOW OP A SIMPLE FLUID

n K. R. Rajagopal

Mathematics Research CenterUniversity of Wisconsin-Madison610 Walnut Street

Madison, Wisconsin 53705

August 1983

>" (Received July 1, 1983)Q.C.3

LLJ

Approve for public rensLi..D bu unlimited

E LCTE

Sponsored by

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Research Triangle Parkworth Carolina 27709 83 09 22 119

UIVERSITY OF WISCONS IN-MADISONMATHEMATICS RESEARCH CENTER

ON A VISCONETRIC FLOW OF A SIMPLE FLUID

K. R. Rajagopal*

Technical Si ary Report #2547August 1983

ABSTRACT

A Class of globally viscometric flows which has relevance to slow flown

occurring between tvo infinite parallel plate. rotating with differing angular

velocities about a common axis, is studied.,__

Ame (NOS) Subject Classification: 76A05

My Words: viscometric flow, simple fluid, motions with constant stretch

history, Rivlin-gricksen fluid

Work Unit Number 6 - Miscellaneous To~pics

*Department of mechanical Engineering, University of Pittsburgh, Pittsburgh,PA 15261

Sponsored by the U. S. Army under Contract No. DAA29-B-C-0041.

SIGNIFICANCZ AND EXPLANATION

Viscometric flows are locally equivalent to steady simple shear flows and

in such flows the behavior of a simple fluid can be completely characterized

by three scalar functions of a single variable, namely the shear. Most of the

familiar flows in the literature, namely Couette flow, Poiseuille flow, etc.,

belong to the above class. In this paper we investigate a class of

viscometric flows which has relevance to the flows occurring between infinite

parallel plates rotating about a common axis with different angular

velocities.

Accession For

NTTF C'A&I

! '4 Cocles

sumary lies with N, and not with the author of this report.

am a VISCOKETIC FLOW OF A SIMPLE FLUID

K. Re Isiagopal'

1. Introduction

in one of his several pioneering papers in the fifites, Rivlin (1)

studied the torsional flow between two parallel disks. Hs considered a

velocity field of the forms

ui- -*zy. vin*zx and w-0,. (I)

a* Vo and w being the velocities in the x, y, and a directions,

respectively. The above motion in viscometric (cf. Pipkin 121) and has

relevance to the low Reynolds nusber flow between rotating disks%* The form

(1) corresponds to a flow in which each plane parallel to the plates is

rotating as though it were rigid, the angular velocity of these plates varying

linearly. Noweer, such a linear variation is by no means the only possible

one in the case of a simple fluid.

in this paper, I shall consider a generalization of (1) which is

applicable for the slow flow of a simple fluid between parallel plates

rotating with differing angular velocities about a common axis (see Pig. 1).

The assumed form for the velocity field falls into the category of pseudo-

plane motions which were studied by Derker [3).

*fepartment of Mechanical Ingineering, University of Pittsburgh, Pittsburgh,

PA 15261

Sponsored by the g. &. Army under Constract go. DA2-ft-C-OO4le

I 7

Figure 1.

We shall aseme a flow fluid of the form

u - -Q(z)y, v - 0(z)x, v - 0, (2)

where O(z) is an arbitrary function z which needs to be determined from

the equations of motion for the specific fluid under consideration.

After a brief discussion of the basic definitions and notations that we

will need, in the next section, we proceed to show that a motion of the form

(2) is viscometric. We conclude with an example of a specific fluid model

wherein Q(z) need not be linear.

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2. Preliminaries

Let I denote the position of an element X in the reference state at

time t and let j denote the position of X at timst. The dependence of

on x, t and T can be expressed as

I - X (X0T). (3)

The relative deformation gradient F t() is then defined through

grad xtIl.(4)

The relative right Cauchy-Green tensor is defined through

t(T)

the velocity gradient tensor L(t) through

d C)

and the Rivlin-Bricksen tensors (cef. Rivlin and ricksen (41) through

T + T

. " -n + .An- k + n- 2 -,,..(

A motion Is said to be viscometric' (cf. Coleman (S)) if at that given

material point, the right relative Cauchy-green tensor can be expressed as

-3-

22 -t( t - 2) = Aj + f- ' (8)

for all t and if relative to some orthonormal basis the Uivlin-

Ericksen tensors have the following matrix representation

A, 0 0 0 62 = 0 , (9), (10)0 0) 0 2 2

where i is usually referred to as the "shear rate"

*w* choose to use the above definition for a visconetric flow since we shallfind the need to employ the kinematical tensors A and A used in theabove definition, later on. & flow is viscometric (cf. Coleman, Markovitz and"Oil [61) if

tF (t-T) -

where R(t-T) is orthogonal with R(0) - I and N is a tensor which has thefollowing matrix representation with respect to a suitable axis

0 -000

-4-. .

3. 'T. Flow Field

Consider the motion represented by (2), i.e.,

Ua - -G(z)y,

v - a(z)x, and

W - 0.

where ua, v, and w denote the x, y, and z components of the velocity,

respectively. The motion represented by (10) is isochoric. Let us denote by

jthe 3-tuple (C, in, ).Then (10) implies that

11- ()(J (11)2

C-0. (11)3

with

CUt) -x, '1(t) - Y, and C(t) -z. (12)

A straightforward computation yields

C~)-x cos(rz)(t-rlI + y sin ((Ga)(t-r)J,(1)

1C)--x sin 1(((z))(t-r)I + Y cos I((Qw)(t-r)1, (13)2

C -T z (13)3

Thus, the relative deformation gradient has the following matrix

representations

-5-

7(T) (14)

0 01

Hence, the right relative Cauchy-Green strain history takes the simple form

1 0 sy(a'(z))

t-)- 0 1 -ex(co(z)) (15)

1+C(Qc(z))BI 2 x 247 2)

we now proceed to compute the Rivlin-Brioksen tensors F irst, it

follows from (18), the velocity gradient is given by

0 -0(3) -AO'z)

Thus, the first two Rivlin-zrick**n tensors are given by

/ 0 0 -YO'(z)

A0 3(9*(Z) ).(17)30(2 0-

and

/00 0

A2 k : ) (IS)

0 0 2(Y. ()) 2+(3. (z)) 2,

We also provide the matrix representations of A2 and A which vill be

useful later on.

2*(Q 1)J2 -39y (11z))2 0

2% _yg&) 2 0 (9

0 0

0 0 -2 (9x( 3 y[x 2 +y21

hIA2 0 2[0'(u)] 3x x 47 2 ) (20)

0 0 0

It is easy to verify that the Rivlin-lricksen tensors A, and A2 an

be expressed in the form (9) and (10) where the now basis 21 (1L1,2,3) is

related to the old cartesian basis . i (11,23) through

2 10 iti 232

-2-2

with

. 2 . + (, ,(3)}2)1/2

-7-4

rI i. . . .

It then follows from equations (15), (17), (18) and the definition of a

viscometric flow that the motion (2) under consideration is indeed

viscometric. Furthermore, a simple computation yields

A 0, V n) 3."-8

A

K _______

4. Discussion

it is easy to verify by virtue of (17)-(20) that a velocity field of the

form (10) given by*

u(x,yz) -- [ ( 2h2 z + ( 1 2_2)1Y

v(xyz) - 1 z +

w(z,y,z) - 0

satisfies the equation of motion for the non inertial flow of the classical

linearly viscous fluid and the Rivlin-gricksen fluids of second and third

grade**. In the case of the linearly viscous fluid the above solution is the

unique solution to the "Stokes flow" problem. n the came of the incompressible

Rivlin-Zricksen fluids of the second and third grade, the above flow would he

the unique solution under certain conditions if the fluids are required to be

thermodynamically compatible ** (cf. Fosdick and Rajagopal [9]).

* This is Rivlin's (4] result extended to the case when both the top andbottom plates are rotating.** The stress constitutive equations for the linearly viscous fluid and the

incompressible Rivlin-Mricksen fluids of second and third grade are given by(of. Truesdell and Holl [7]):

- -P + Uhl + a#

- pi * "a 1 + a 1 2 + Q21 A3 + 02(AA2 + AA1] + 142

*ta The fluid is said to be thermodynamically compatible If it meets the

Claasius-Duhem inequality in all its motions and if the specific Relmholtxfree energy is a minimum when the fluid is at rest under isothermalconditions. The uniqueness result is not a consequence of Tanner's therorm(101 as the flow in question is not plane.

However, the flow (2) is by no means the only one possible in a general

simple fluid. We give below an example of a simple fluid which is properly

frame invariant in which an infinity of solutions is possible for the above

problem. of course, the fluid model may not be a realistic one. It should

however be noted that one could easily construct fluid models wherein the

stress in expressible as polynomials of the gradients of velocity and the

(n-l)th accelerations, the class of models studied by Rivlin [I], where non-

unique solutions for 2)(W) are possible.

Let us consider a fluid model whose Cauchy stress T is given by

1

T -pl + - A

(tr A) -2

Such a fluid model is definitely permissible under the class of simple fluids

(cf. Wineman and Pipkin (11] ). A trivial computation, for the problem in

question, verifies that

! 2= 0 0 0 .

(tr 2)2(0 0 1

It then follows that any smooth l(z) which is such that it is B1 at the

top and 2 2 at the bottom would be permissiblet

Acknowledgements The author would like to thank Ms. K. Spear for usefuldiscussions.

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Bibliography

[ 11 Rivlin, R.., Solutions of Some Problems in the Ract Theory of Viso-Elasticity, J. National Mach. Anal., 5 179-187 (1956).

21 Pipkin, A.C., Controllable viscometric flow, Quarterly Appl. Hath., 26e87-100 (1967).

[ 3) Darker, R., Integration d6l equations du mouvement dun fluid* visquex,incompressible, Handbuch der Physik, VIUI/2, Springer, Berlin-Gottingen-Heidelberg (1963).

4] Rivlin, R.S., Ericksen, J.L., Stress deformation relations for Isotropicmaterials, J. Rational Mech. Anal., 4, 323-425 (1955).

51 Coleman, B.D., Kinematical concepts with application@ in the mechanicsand thermodynamics of incompressible viscoelastic fluids, Arch. RationalMach. Anal., 9, 273-300 (1962).

[ 6) Coleman, S.D., Markovitz, H., Noll, W., Viscometric flown of Non-Newtonian fluids, Springer-Verlag, New York (1966).

7] Truesdell, C. Noll, W., The non-linear field theories of mechanics,Fi9ges Haandbuch dew Physik, 111/3, Springer, Derlin-Naidelberg-Nov York(1965).

8 8] Fosdick, R.L., Majagopal, K.R., Uniqueness and drag for fluids of secondgrade in steady motion, Intl. J. Non-linear Nechanics, 13, 131-137(1978).

[ 9 Posdick, R.L., Pajagopal, K.R., Thermodynmics and Stability of fluidsof grade three, Proc. Roy. Soc. London, A 339, 351-377 (1980).

[10] Tanner, l.I., Plane creeping flow of incompressible second orderfluids, Physics of Fluids, 9, 1246-1247 (1966).

(11] Wineman, A.B., Ptpkin, A.C., Material symetry restriction. onconstitutive equations, Arch. Rational Kach. Anal., 17, 184-214 (1964).

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WS SUPPLEMENTARY NOTES

IS. KIEY WORDS (Cenfinu. an revereso eiitaae..gw0.gE and5 M byoo Wiumber)

viscometric flow, simple fluid, motions with constant stretch history,Rivlin-Ericksen fluid.

20. ABSTRACT (Centifte as reuerse sde Nf neees and ideatfy by bock ambesr)

A class of globally viscometric flows which has relevance to slow flowsoccurring between two infinite parallel plates rotating with differing angularvelocities about a common axis, is studied.

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