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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
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Approve for public rensLi..D bu unlimited
E LCTE
Sponsored by
u: s: f lsarch office EP0.Box 12211
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
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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.
-10-
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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REPORT DOCMENTATIO PAGE ADMSRT 2
1.RPRMUMBE GOVT ACCESSION NO: S.ECIPIEN S CATAIOG NUMS ft
2547 - a_ _ _ _ _ _ _ _ _ _ _ _ _
4. TITLE (end &"ow) S. TYPE OF REPORT & PERIOD COVERED
Summary Report - no specificON A VISCOM4ETRIC FLOW OF A SIMPWLE FLUID - reporting period
6. PeRFOINGO ORG. REPORT NUMBER
7. AUTHOR(@) S. CONTRACT OR GRANT HUM11ER11(a)
K. R. Rajagopal DAAG9-OC-00 41it- PERFORMANG ORGANIZATION NAME AND ADDRESS ID RNELEMENT. PiRECT.TS
Mathematics Research Center, University of AMk UntAmbr
610 Walnut Street Wisconsin miscellaneous TopicsMadison,* Wisconsin 53706 _____________
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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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