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On finite amplitude shear waves in viscoelastic fluids
315
Journal of Non-Newtonian Fluid Mechanics, 5 (1979) 315-322 0 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands
ON FINITE AMPLITUDE SHEAR WAVES IN VISCOELASTIC
FLUIDS *
STEFAN ZAHORSKI Institute of Fundamental (Poland)
Technological
Research,
Polish Academy
of Sciences,
Warsaw
(Received June 22,1978)
summary It is shown that circular as well as elliptic shearing flows of incompressible simple fluids can be treated as a subclass of motions with proportional stretch history (cf. [ 31). For circular shearings, the known results on circularly polarized shear waves are derived in a different way, while for more general elliptic shearings certain new results on low frequency elliptically polarized shear waves are discussed.
1. Introduction In contrast to numerous problems of finite amplitude elastic shear waves, similar problems for viscoelastic media have attracted much less attention. A general theory of waves considered as propagating surfaces of discontinuity in dissipative materials has been presented by Coleman et al. [I]. Recently, Carroll [ 21 discussed the conditions under which motions called plane circular shearings lead to finite amplitude plane progressive or standing waves in certain fluids and solids. In the present paper, it is shown that circular, as well as elliptic, shearings in incompressible simple fluids can be treated as a subclass of motions with proportional stretch history discussed elsewhere [ 31. To this end, the flows considered must be expressed in the form of complex variable functions. In the case of circular shearings, some of Carroll’s results (cf. ref. 3) are rediscovered in a different way, while in the case of elliptic shearings similar solu* Presented at the IUTAM Symposium on Non-Newtonian Fluid Mechanics, Louvainla-Neuve, Belgium, 28 August-l September, 1978.
316
tions are obtained for low-frequency elliptically polarized plane waves by means of the procedure proposed by Niiler and Pipkin [ 41. It is also shown that the governing equations for elliptically or linearly polarized plane waves are identical to those for circularly polarized plane waves; the only differences occur in the normal stress components. 2. Plane elliptic and circular shearings as motions with proportional stretch histories Let us consider a class of plane motions in the following form: x = x + ay@) cos WT + a$(Z) sin 07, y = Y + b&Z) sin 07 - b$(Z) cos 07,
(1)
z = 2, where x, y, z denote Cartesian coordinates of a particle at an arbitrary time 7; X, Y, 2 denote Cartesian coordinates of the same particle in a reference configuration at time rR,* w denotes a constant angular frequency; and cp, $ are certain functions of 2 only. The dimensionless constant parameters a E [0, l] and b E [0, l] describe an ellipticity of motion; for a = b = 1, we have the case of circular shearing, while for either a = 0 or b = 0 we have the case of linear shearing. The corresponding velocity as well as acceleration fields can easily be calculated from eqn. (1). On introducing the auxiliary notations: 9’ = K sin 8
cp’ = K case,
(2)
where primes denote the derivatives with respect to z, we obtain (q’2 + $‘2)1/2 = K,
f3 = arctg($‘/g’),
(3)
where K is the amount of shear. It is easy to see that the planes z = con&. are material surfaces, and the paths of particles correspond to the ellipses q2 + G2 = (x2/a2) + (y2/b2); these ellipses become the circles of radii (q2 + G2)lj2 if a = b = 1 (see Fig. 1). The deformation gradient at time T ‘can be written as 1
[P(T)]
=
[0 0
0
0 1
K COS( WT -
1 K
sin(wr - 0)
or in the complex form I;(T) = Re{exp[M
8)
ei(wr-e)] 1,
1
WI =
[0
(4)
Oa 00 --ib 0
1 K
(5)
317
Fig. 1.
where i = a, and the real part of F(T) is meaningful. On writing eqn. (5) in the abbreviated form: F(r) = Re (exp(WWf)I
k(7) = eWwr--8)
,
(6)
it is immediately seen that the motion considered belongs to the class of motions with proportional stretch history [3] for which the exponent responsible for a pure stretch is proportional to a smooth function of time. The relative deformation gradient with respect to the reference configuration at present time t (T < t) Ft(r) = F(T)F’-l(t)
= exp[(k(r)
- k(t))M]
leads to the following history of right relative Cauchy-Green tensor (cf. ref. 5):
(7) deformation
C(s) = I;? (t - s) Ft ( t - s) = exp(g(s)MT ) exp(g(s)M) = 1 + g(s)(MT
+ M) + g2(s)MTM
,
(8)
where M2 = MT2 = 0, and g(s) = Iz(t -s)
- k(t) = ei(w*e)(e-iWs - 1) .
(9)
For the Rivlin-Ericksen kinematic tensors defined as follows: A,(t)
= (-1)”
FI
s=o
,
n = 1, 2, . . .
(10)
318
we arrive at
Al = ioeiE(MT +M),
A2 = -a2
eif(MT + M) - 2 ~2 ez%MTM
(11)
where t: = wt - 8. Thus, eqn. (8) after taking into account eqn. (9) gives C(s) = 1 - t (e-iws - 1)Ai + &
(eViws - 1)2(Ai -i
A,) .
(12)
Substituting the above relation into the constitutive equation of an incompressible simple fluid (cf. ref. 5) :
s= ;
s=o
det C(s) = 1 ,
(C(s)),
(13)
where S is the stress deviator for suitably normalized isotropic constitutive functional, we have S(t) = ,io (k (emiws- l), k (eAiws - 1)2; A,(t), = f(w
A,(t),
A,(t)) (14)
A2(0),
where f is a tensor function of w, isotropic with respect to Al(t) and A,(t). The above result is in accordance with the representation theorem previously proved for the motions with proportional stretch history (cf. ref. 3). It is worthwhile to note that for plane circular shearings, i.e. for a = b = 1, we have MTM = 0 and Al = iweiE(MT + M) ,
A2 = -u2 eiE(MT + M) ,
implying that A2 = hAI.
In this case, we obtain instead of eqn. (12):
C(s) = 1 - $ A, sin ws + +2 A2(l - cos US) .
(15)
(16)
The above result was also proved by Carroll [ 21. The constitutive equation (13) leads to S(t) = ,!. = hW2;
(i
sin US, 5 A,(t),
(1- cos as); Al(t), A2(t))
(17)
A,(t))
where h is a tensor function even in w and isotropic with respect to the remaining tensor arguments. * Rearing in mind the widely known Rivlin-Ericksen representation of an * Although formally A2 = iwAl, it is more useful for further considerations to treat asa function of two tensor arguments.
b
319
isotropic tensor function of two tensor arguments (cf. ref. 6), we can write for both eqns. (14) and (17): S = a,Al + azA, + asA; + cw,A: + a5(A1Az + A,A,) + ix6(A;A2 + A,Af)
+ cx,(AIA~ + A;A,)
+
+ a,(AfA;
+ A;Af)
,
(W
where ci (i = 1, .. .. 8) are material functions depending on w and all joint invariants of tensors Al and AZ. Further particular forms of eqn. (18) valid for circular and elliptic shearings are discussed in the next sections. 3. Circularly polarized plane shear waves For cricular shearings (a = b = l), after taking into account the properties of tensorM (cf. eqn. (5)) and the fact that AZ = &A,, we obtain from eqn. (18) the following simplified relation: S = (a1 + ica2)Al
+ (aa - w2a4 + 2 iwa6)Af,
(19)
where ei (i = 1, . . .. 5) are even functions of o and invariants of tensor Al. Since tensor Al is complex, its invariants must be composed of Al and A;, where A; denotes the Hermitian conjugate of Al. It can be proved in a straightforward manner that in the case considered the total number of ten invariants reduces to the following non-vanishing two expressions: tr AlAi
=4
02~2
trATA1*2
,
=
4 ‘,4,‘4
(26)
Thus, the material coefficients occurring in eqn. (19) are real functions of real arguments w2 and K 2. Since the real part of eqn. (19) can be written in the form: ReS = czlReAl - ocu21mAi~+
((~3 -
u2a4)
ReA; - 2wol&nA~ ,
(21)
where Al is defined by the first of eqns. (15), we finally arrive at the following deviatoric stress components :
Sf3 = -(YIWK S23
= (Y~OK
Sill
t - (Y202KCOSt ,
cos t -
sin t ,
cx2w2~
&cp2= -(a3
-
W2Q4)W2K2
Sill
p1
= -qa3
-
Ld2cY4)02K2
COS
p
=
s33
= (-J
-
(a3
02Ct4)02K2
COS
2t- 2
~Y@~K~
COS
2[ +2
(Y5W3K2
Sin
2t- 2
(Y5W3K2
Sin
2 t, 2 t,
(22)
2 5,
,
where t = ot - 8. It is easy to see that S ’ 3 S23 are odd functions of K, while S12, S1’ and S22 are even. On the other hand, only S23 and S12 are odd functions of w, the remaining components are even. All the stress components (22) depend on variable z only, through function K(Z) (cf. eqn. (3)).
320
Substituting eqns. (22) into the dynamical equations of equilibrium: div(S + pl) - p grad q(z) = pjc’
(23)
where p is a mean hydrostatic pressure, p is the density of a fluid and q(z) is a potential of conservative body forces, we obtain the system of linear differential equations: (a,p’ + QW$‘)
- poJ/ = 0 ,
(arl J/’ - Q1.+p’)
(24)
+ pfdq = 0,
(P + q)’ = 0 f where eqns. (2) are also used. The first two eqns. (24) can be solved for appropriate boundary conditions and a priori known functions (pi, (~z. The third eqn. (24) determines the function of mean pressure p. Equations (24) are fully equivalent to those derived in a different way by Carroll [ 21; they become identical for e1 = r4, (r2 = -ra/w2. If the solutions p(z) and G(z) of eqns. (24) are periodic or, in particular, sinusoidal, the flow considered corresponds to the case of circularly polarized plane progressive or standing waves. For fluids with linear shear response for which 0~~and c2 are independent of K, e.g. for second-order or Newtonian fluids, the general solution of eqns. (24) has the following form (cf. ref. 2): p(z) = A e-O2 COS(/~Z+ A) + B eaz cos@z + p), $(z) = A ewaz sin@ + X) -B
(25)
8” sir@z + p),
where A, B, X and /J are constants depending on the appropriate boundary conditions, and (a + i/3)2 = pw(UY, * z&)-i,
(Y> 0,
p>o.
(26)
The constant a! characterizes an exponential decay or growth of the wave amplitude, while p is simply related to the wave length. The case of a semiinfinite fluid bounded by a rigid oscillating plate and the case of a fluid contained between two parallel rigid plates one of which is oscillating in its plane were discussed in greater detail in [ 21. 4. Elliptically and linearly polarized plane shear waves For elliptic shearings (a # b # l), the constitutive equations resulting from eqns. (18) and (11) are too complex for further effective analysis. To achieve more progress for the case of low frequency waves, we apply an expansion procedure proposed by Niiler and Pipkin [ 41. To this end, we introduce the dimensionless parameters defined as follows: c2 = oT,
Q = W~IT)“~
,
(27)
where T is a fluid characteristic time, and where A denotes the finite ampli-
321
tude for Newtonian solutions. On using the dimensionless quantities:
&S
Akwa,
’
;i,=A”
Ako”
n = 1, 2 . ..
’
(28)
the constitutive equations (18) for e2 << 1 and Q = const. can be presented in the dimensionless form: $ = A, + e2P2A2 + e3p3QAf +
W4),
(29)
where Pi = I,
P2 = ~2la17’,
03 = ~,la,T,
etc.,
(30)
are dimensionless material coefficients. If all the terms of order greater then e3 many be disregarded, the coefficients f12and fi3 are constants independent of w (cf. eqn. (14)). In other words, for sufficiently low angular frequencies w or, strictly speaking, for small values of e2 = UT, the constitutive equations of a second-order incompressible fluid: S = alAl
+ cr2A2 + &,A;,
trA,=O,
(31)
where el, (Y2, OL 3 are material constants independent of o and K, may be applied. Thus, taking into account eqns. (31), (11) and the second of eqns. (5), we arrive at the following deviatoric stress components: S13 = -cxloxa
sin t - c2w2Ka cos t,
S23 = olltiKb cos t - (w,w~& sin E, Si2 = --(~~o~~K~absin 2 t,
(32)
Sil = -cx3Ca.Waz cos 2 t, S22 = a302K2b2 S33 = -(2
cos 2 t,
a2 + 013)W2K2(U2
-
b2) cos 2 t,
where t = ot - 8. It is seen that S13 and S23 are odd functions of K, while the remaining components are even. The shear stresses S23 and S13 oscillate with the angular frequency o, and the shear stress S12 and the normal components with 2 o. All the stress components (32) depend on variable z only, through function K(Z). Substituting eqns. (32) into the equations of equilibrium (23), we obtain the system of differential equations similar to (24), namely criy?” + (Y20J/” - poJ/ = 0 cY1g” - CY+p” + pwyl = 0
(33)
(p + 17)’ + 02(2 01~+ e3)(a2 - b2)[((pf2 - $‘2) cos 2 wt + 2cp’$’ sin 2 wt]’ = 0 where eqns. (2) are also used. In an analogy to the case of circularly polarized
322
waves (Section 3), it can be shown that under the assumed order of approximation (low frequencies) the motion described by eqns. (1) corresponds to the case of elliptically polarized plane progressive or standing waves, if a # b # 1, and the appropriate boundary conditions are satisfied. In particular, the case of linearly polarized waves is described either by a = 0 or b = 0. It is worthwhile to note that the governing equations for low frequency elliptically and linearly polarized waves do not differ at all from those for low frequency circularly polarized shear waves. Essential differences exist in the form of mean pressure as well as in the normal stress components. The above similarity between low frequency elliptically and circularly polarized shear waves may be of importance in such cases where circularly polarized plane progressive waves are refracted on a plane interface between two ,ion-mixing viscoelastic fluids. If the direction of propagation of a primary circularly polarized wave is not perpendicular to the interface, the resulting refracted wave must be elliptically polarized. The values of parameters a and b depend on the corresponding angles of incidence and refraction, i.e. on the direction of wave propagation and the material properties. References B.D. Coleman, M.E. Gurtin, I. Herrera and C. Truesdell, Wave Propagation in Dissipative Materials, Springer, New York, 1965. M.M. Carroll, Plane circular shearing of incompressible fluids and solids, The University of California Report, Berkeley, 1977. S. Zahorski, Flows with proportional stretch history, Arch. Mech., 24 (1972) 681. P.P. Niiler and A.C. Pipkin, Finite amplitude shear waves in some non-Newtonian fluids, Int. J. Eng. Sci., 2 (1964) 305. C. Truesdell and W. Nell, The non-linear field theories of mechanics, Encycl. of Physics, Vol. 111/3, Springer, New York, 1965. R.S. Rivlin and J.L. Ericksen, Stress-deformation relations for isotropic materials, J. Ration. Mech. Anal., 4 (1955) 323.
Journal of Non-Newtonian Fluid Mechanics, 5 (1979) 315-322 0 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands
ON FINITE AMPLITUDE SHEAR WAVES IN VISCOELASTIC
FLUIDS *
STEFAN ZAHORSKI Institute of Fundamental (Poland)
Technological
Research,
Polish Academy
of Sciences,
Warsaw
(Received June 22,1978)
summary It is shown that circular as well as elliptic shearing flows of incompressible simple fluids can be treated as a subclass of motions with proportional stretch history (cf. [ 31). For circular shearings, the known results on circularly polarized shear waves are derived in a different way, while for more general elliptic shearings certain new results on low frequency elliptically polarized shear waves are discussed.
1. Introduction In contrast to numerous problems of finite amplitude elastic shear waves, similar problems for viscoelastic media have attracted much less attention. A general theory of waves considered as propagating surfaces of discontinuity in dissipative materials has been presented by Coleman et al. [I]. Recently, Carroll [ 21 discussed the conditions under which motions called plane circular shearings lead to finite amplitude plane progressive or standing waves in certain fluids and solids. In the present paper, it is shown that circular, as well as elliptic, shearings in incompressible simple fluids can be treated as a subclass of motions with proportional stretch history discussed elsewhere [ 31. To this end, the flows considered must be expressed in the form of complex variable functions. In the case of circular shearings, some of Carroll’s results (cf. ref. 3) are rediscovered in a different way, while in the case of elliptic shearings similar solu* Presented at the IUTAM Symposium on Non-Newtonian Fluid Mechanics, Louvainla-Neuve, Belgium, 28 August-l September, 1978.
316
tions are obtained for low-frequency elliptically polarized plane waves by means of the procedure proposed by Niiler and Pipkin [ 41. It is also shown that the governing equations for elliptically or linearly polarized plane waves are identical to those for circularly polarized plane waves; the only differences occur in the normal stress components. 2. Plane elliptic and circular shearings as motions with proportional stretch histories Let us consider a class of plane motions in the following form: x = x + ay@) cos WT + a$(Z) sin 07, y = Y + b&Z) sin 07 - b$(Z) cos 07,
(1)
z = 2, where x, y, z denote Cartesian coordinates of a particle at an arbitrary time 7; X, Y, 2 denote Cartesian coordinates of the same particle in a reference configuration at time rR,* w denotes a constant angular frequency; and cp, $ are certain functions of 2 only. The dimensionless constant parameters a E [0, l] and b E [0, l] describe an ellipticity of motion; for a = b = 1, we have the case of circular shearing, while for either a = 0 or b = 0 we have the case of linear shearing. The corresponding velocity as well as acceleration fields can easily be calculated from eqn. (1). On introducing the auxiliary notations: 9’ = K sin 8
cp’ = K case,
(2)
where primes denote the derivatives with respect to z, we obtain (q’2 + $‘2)1/2 = K,
f3 = arctg($‘/g’),
(3)
where K is the amount of shear. It is easy to see that the planes z = con&. are material surfaces, and the paths of particles correspond to the ellipses q2 + G2 = (x2/a2) + (y2/b2); these ellipses become the circles of radii (q2 + G2)lj2 if a = b = 1 (see Fig. 1). The deformation gradient at time T ‘can be written as 1
[P(T)]
=
[0 0
0
0 1
K COS( WT -
1 K
sin(wr - 0)
or in the complex form I;(T) = Re{exp[M
8)
ei(wr-e)] 1,
1
WI =
[0
(4)
Oa 00 --ib 0
1 K
(5)
317
Fig. 1.
where i = a, and the real part of F(T) is meaningful. On writing eqn. (5) in the abbreviated form: F(r) = Re (exp(WWf)I
k(7) = eWwr--8)
,
(6)
it is immediately seen that the motion considered belongs to the class of motions with proportional stretch history [3] for which the exponent responsible for a pure stretch is proportional to a smooth function of time. The relative deformation gradient with respect to the reference configuration at present time t (T < t) Ft(r) = F(T)F’-l(t)
= exp[(k(r)
- k(t))M]
leads to the following history of right relative Cauchy-Green tensor (cf. ref. 5):
(7) deformation
C(s) = I;? (t - s) Ft ( t - s) = exp(g(s)MT ) exp(g(s)M) = 1 + g(s)(MT
+ M) + g2(s)MTM
,
(8)
where M2 = MT2 = 0, and g(s) = Iz(t -s)
- k(t) = ei(w*e)(e-iWs - 1) .
(9)
For the Rivlin-Ericksen kinematic tensors defined as follows: A,(t)
= (-1)”
FI
s=o
,
n = 1, 2, . . .
(10)
318
we arrive at
Al = ioeiE(MT +M),
A2 = -a2
eif(MT + M) - 2 ~2 ez%MTM
(11)
where t: = wt - 8. Thus, eqn. (8) after taking into account eqn. (9) gives C(s) = 1 - t (e-iws - 1)Ai + &
(eViws - 1)2(Ai -i
A,) .
(12)
Substituting the above relation into the constitutive equation of an incompressible simple fluid (cf. ref. 5) :
s= ;
s=o
det C(s) = 1 ,
(C(s)),
(13)
where S is the stress deviator for suitably normalized isotropic constitutive functional, we have S(t) = ,io (k (emiws- l), k (eAiws - 1)2; A,(t), = f(w
A,(t),
A,(t)) (14)
A2(0),
where f is a tensor function of w, isotropic with respect to Al(t) and A,(t). The above result is in accordance with the representation theorem previously proved for the motions with proportional stretch history (cf. ref. 3). It is worthwhile to note that for plane circular shearings, i.e. for a = b = 1, we have MTM = 0 and Al = iweiE(MT + M) ,
A2 = -u2 eiE(MT + M) ,
implying that A2 = hAI.
In this case, we obtain instead of eqn. (12):
C(s) = 1 - $ A, sin ws + +2 A2(l - cos US) .
(15)
(16)
The above result was also proved by Carroll [ 21. The constitutive equation (13) leads to S(t) = ,!. = hW2;
(i
sin US, 5 A,(t),
(1- cos as); Al(t), A2(t))
(17)
A,(t))
where h is a tensor function even in w and isotropic with respect to the remaining tensor arguments. * Rearing in mind the widely known Rivlin-Ericksen representation of an * Although formally A2 = iwAl, it is more useful for further considerations to treat asa function of two tensor arguments.
b
319
isotropic tensor function of two tensor arguments (cf. ref. 6), we can write for both eqns. (14) and (17): S = a,Al + azA, + asA; + cw,A: + a5(A1Az + A,A,) + ix6(A;A2 + A,Af)
+ cx,(AIA~ + A;A,)
+
+ a,(AfA;
+ A;Af)
,
(W
where ci (i = 1, .. .. 8) are material functions depending on w and all joint invariants of tensors Al and AZ. Further particular forms of eqn. (18) valid for circular and elliptic shearings are discussed in the next sections. 3. Circularly polarized plane shear waves For cricular shearings (a = b = l), after taking into account the properties of tensorM (cf. eqn. (5)) and the fact that AZ = &A,, we obtain from eqn. (18) the following simplified relation: S = (a1 + ica2)Al
+ (aa - w2a4 + 2 iwa6)Af,
(19)
where ei (i = 1, . . .. 5) are even functions of o and invariants of tensor Al. Since tensor Al is complex, its invariants must be composed of Al and A;, where A; denotes the Hermitian conjugate of Al. It can be proved in a straightforward manner that in the case considered the total number of ten invariants reduces to the following non-vanishing two expressions: tr AlAi
=4
02~2
trATA1*2
,
=
4 ‘,4,‘4
(26)
Thus, the material coefficients occurring in eqn. (19) are real functions of real arguments w2 and K 2. Since the real part of eqn. (19) can be written in the form: ReS = czlReAl - ocu21mAi~+
((~3 -
u2a4)
ReA; - 2wol&nA~ ,
(21)
where Al is defined by the first of eqns. (15), we finally arrive at the following deviatoric stress components :
Sf3 = -(YIWK S23
= (Y~OK
Sill
t - (Y202KCOSt ,
cos t -
sin t ,
cx2w2~
&cp2= -(a3
-
W2Q4)W2K2
Sill
p1
= -qa3
-
Ld2cY4)02K2
COS
p
=
s33
= (-J
-
(a3
02Ct4)02K2
COS
2t- 2
~Y@~K~
COS
2[ +2
(Y5W3K2
Sin
2t- 2
(Y5W3K2
Sin
2 t, 2 t,
(22)
2 5,
,
where t = ot - 8. It is easy to see that S ’ 3 S23 are odd functions of K, while S12, S1’ and S22 are even. On the other hand, only S23 and S12 are odd functions of w, the remaining components are even. All the stress components (22) depend on variable z only, through function K(Z) (cf. eqn. (3)).
320
Substituting eqns. (22) into the dynamical equations of equilibrium: div(S + pl) - p grad q(z) = pjc’
(23)
where p is a mean hydrostatic pressure, p is the density of a fluid and q(z) is a potential of conservative body forces, we obtain the system of linear differential equations: (a,p’ + QW$‘)
- poJ/ = 0 ,
(arl J/’ - Q1.+p’)
(24)
+ pfdq = 0,
(P + q)’ = 0 f where eqns. (2) are also used. The first two eqns. (24) can be solved for appropriate boundary conditions and a priori known functions (pi, (~z. The third eqn. (24) determines the function of mean pressure p. Equations (24) are fully equivalent to those derived in a different way by Carroll [ 21; they become identical for e1 = r4, (r2 = -ra/w2. If the solutions p(z) and G(z) of eqns. (24) are periodic or, in particular, sinusoidal, the flow considered corresponds to the case of circularly polarized plane progressive or standing waves. For fluids with linear shear response for which 0~~and c2 are independent of K, e.g. for second-order or Newtonian fluids, the general solution of eqns. (24) has the following form (cf. ref. 2): p(z) = A e-O2 COS(/~Z+ A) + B eaz cos@z + p), $(z) = A ewaz sin@ + X) -B
(25)
8” sir@z + p),
where A, B, X and /J are constants depending on the appropriate boundary conditions, and (a + i/3)2 = pw(UY, * z&)-i,
(Y> 0,
p>o.
(26)
The constant a! characterizes an exponential decay or growth of the wave amplitude, while p is simply related to the wave length. The case of a semiinfinite fluid bounded by a rigid oscillating plate and the case of a fluid contained between two parallel rigid plates one of which is oscillating in its plane were discussed in greater detail in [ 21. 4. Elliptically and linearly polarized plane shear waves For elliptic shearings (a # b # l), the constitutive equations resulting from eqns. (18) and (11) are too complex for further effective analysis. To achieve more progress for the case of low frequency waves, we apply an expansion procedure proposed by Niiler and Pipkin [ 41. To this end, we introduce the dimensionless parameters defined as follows: c2 = oT,
Q = W~IT)“~
,
(27)
where T is a fluid characteristic time, and where A denotes the finite ampli-
321
tude for Newtonian solutions. On using the dimensionless quantities:
&S
Akwa,
’
;i,=A”
Ako”
n = 1, 2 . ..
’
(28)
the constitutive equations (18) for e2 << 1 and Q = const. can be presented in the dimensionless form: $ = A, + e2P2A2 + e3p3QAf +
W4),
(29)
where Pi = I,
P2 = ~2la17’,
03 = ~,la,T,
etc.,
(30)
are dimensionless material coefficients. If all the terms of order greater then e3 many be disregarded, the coefficients f12and fi3 are constants independent of w (cf. eqn. (14)). In other words, for sufficiently low angular frequencies w or, strictly speaking, for small values of e2 = UT, the constitutive equations of a second-order incompressible fluid: S = alAl
+ cr2A2 + &,A;,
trA,=O,
(31)
where el, (Y2, OL 3 are material constants independent of o and K, may be applied. Thus, taking into account eqns. (31), (11) and the second of eqns. (5), we arrive at the following deviatoric stress components: S13 = -cxloxa
sin t - c2w2Ka cos t,
S23 = olltiKb cos t - (w,w~& sin E, Si2 = --(~~o~~K~absin 2 t,
(32)
Sil = -cx3Ca.Waz cos 2 t, S22 = a302K2b2 S33 = -(2
cos 2 t,
a2 + 013)W2K2(U2
-
b2) cos 2 t,
where t = ot - 8. It is seen that S13 and S23 are odd functions of K, while the remaining components are even. The shear stresses S23 and S13 oscillate with the angular frequency o, and the shear stress S12 and the normal components with 2 o. All the stress components (32) depend on variable z only, through function K(Z). Substituting eqns. (32) into the equations of equilibrium (23), we obtain the system of differential equations similar to (24), namely criy?” + (Y20J/” - poJ/ = 0 cY1g” - CY+p” + pwyl = 0
(33)
(p + 17)’ + 02(2 01~+ e3)(a2 - b2)[((pf2 - $‘2) cos 2 wt + 2cp’$’ sin 2 wt]’ = 0 where eqns. (2) are also used. In an analogy to the case of circularly polarized
322
waves (Section 3), it can be shown that under the assumed order of approximation (low frequencies) the motion described by eqns. (1) corresponds to the case of elliptically polarized plane progressive or standing waves, if a # b # 1, and the appropriate boundary conditions are satisfied. In particular, the case of linearly polarized waves is described either by a = 0 or b = 0. It is worthwhile to note that the governing equations for low frequency elliptically and linearly polarized waves do not differ at all from those for low frequency circularly polarized shear waves. Essential differences exist in the form of mean pressure as well as in the normal stress components. The above similarity between low frequency elliptically and circularly polarized shear waves may be of importance in such cases where circularly polarized plane progressive waves are refracted on a plane interface between two ,ion-mixing viscoelastic fluids. If the direction of propagation of a primary circularly polarized wave is not perpendicular to the interface, the resulting refracted wave must be elliptically polarized. The values of parameters a and b depend on the corresponding angles of incidence and refraction, i.e. on the direction of wave propagation and the material properties. References B.D. Coleman, M.E. Gurtin, I. Herrera and C. Truesdell, Wave Propagation in Dissipative Materials, Springer, New York, 1965. M.M. Carroll, Plane circular shearing of incompressible fluids and solids, The University of California Report, Berkeley, 1977. S. Zahorski, Flows with proportional stretch history, Arch. Mech., 24 (1972) 681. P.P. Niiler and A.C. Pipkin, Finite amplitude shear waves in some non-Newtonian fluids, Int. J. Eng. Sci., 2 (1964) 305. C. Truesdell and W. Nell, The non-linear field theories of mechanics, Encycl. of Physics, Vol. 111/3, Springer, New York, 1965. R.S. Rivlin and J.L. Ericksen, Stress-deformation relations for isotropic materials, J. Ration. Mech. Anal., 4 (1955) 323.