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Instability of plane Couette flow of viscoelastic liquids
Journal of Non -Newtonian Fluid Mechanics, 18 (1985) 123-141 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
INSTABILITY LIQUIDS
U. AKBAY,
123
OF PLANE COUETTE FLOW OF VISCOELASTIC
E. BECKER * and S. SPONAGEL
Institut fiu Mechanik, (F.R.G.)
Technische Hochschule Darmstadt,
(Received January 25, 1984; in revised form November,
Hochschulstr. I, D -6100 Darmstadt 8, 1984)
summaly A review of our work on the stability of plane Couette flow of a viscoelastic liquid is given. The fist part of the review is based on the assumption of a “short memory” of the fluid. The Reynolds-Orr energy criterion intimates the possibility of instability at very low Reynolds numbers. A linear stability analysis for disturbances in the flow plane shows that beyond the stability limit given by the energy criterion there are always disturbances which grow with time. A critical assessment of the short memory theory shows the severe limitations of its applicability. In the second part of the paper, the assumption of short memory is dropped. The stability of plane Couette flow with respect to special disturbances perpendicular to the flow plane is investigated for a Maxwell fluid. The flow is unstable if the product of Reynolds number and Weissenberg number is higher than a certain limit, which has the value one for a simple Maxwell fluid. This result can also be interpreted as follows: The flow becomes unstable if the velocity at the boundary walls is higher than the shear wave velocity of the fluid.
1. Introduction As is well known, flow processes of non-Newtonian, viscoelastic liquids are prone to instabilities which are absent in the flow of Newtonian liquids. In particular, slow flow, i.e. flow at very low Reynolds number, may become
* Deceased on November 14.1984.
0377-0257/85/$03.30
Q 1985 Elsevier Science Publishers B.V.
124 unstable if the liquid is non-Newtonian, whereas slow flow of a Newtonian liquid is always stable. Low Reynolds number instabilities could be the explanation for a number of phenomena which are of considerable technological import. For example, they could be the mechanism behind the observed instabilities of extrusion processes, and they might be responsible for instabilities observed in rheometers. There exists already a vast literature on such instabilities. Reviews have been presented by Petrie and Denn [14] and by Goddard [ll]. All theoretical investigations known to us which lead to explicit results concerning flow stability are based on particular constitutive equations of the liquid (see, e.g., the table, p. 185 [ll]). Their results are therefore dependent on the assumption that a particular constitutive equation describes the liquid accurately for flow with superimposed disturbances in that domain of the parameter space of the flow where instabilities are possible. If stability of instability is predicted by such a theory this need not exclude instabilities or stable flow in reality, because the constitutive equation might be inadequate. Therefore, we began, some years ago, to study stability from a different point of view, which avoids narrowing the approach by restriction to a particular constitutive equation at too early a stage. Usually the flows whose stability is to be investigated are kinematically simple; they are either viscometric flows or simple elongational flows. Our starting point is therefore the theory of nearly viscometric flow and nearly elongational flow, respectively. The present paper is a report on some of our ideas concerning the stability of viscometric shear flow at low Reynolds number. Our original ideal was to perform a stability analysis without a priori specification of a particular constitutive equation. The analysis should lead to stability criteria which contain various material functions of the shear rate of the basic flow field, part of them consisting of the viscometric functions, and the rest of non-viscometric functions. This ideal has been realized, at least to a considerable extent, only for situations to which a “short memory” assumption applies, as explained in Section 2. However, the short memory assumption is not very realistic (see Section 2.3) and the results obtained with the aid of this assumption can provide no more than a tentative indication of possible instabilities. During the course of our work we have become increasingly doubtful about the attainability of our ideal in the more general case of fluids with finite memory. Perhaps only the more limited aim of establishing stability criteria which are valid for sub-classes of the general class of “simple fluids” is a realistic one. Section 2 is essentially a brief review of that part of our work which is based on the short memory assumption, together with a critique of that assumption (Section 2.3). Section 3 contains a linear stability analysis for a particular type of disturbance which simplifies the analysis considerably
125 without sacrifice of physical content. One of the results is the detection of an instability which, to the best of our knowledge, has not yet been mentioned in the published literature. 2. Short memory theory As a suitable starting point, we introduced a simplifying assumption. This consists in postulating that the fluid has a “short memory” with respect to disturbances of the basic flow, in a sense which will be explained below. The results thereby obtained are correct in a certain asymptotic limit, as will become clear later on. 2.1 Constitutive
equation for the disturbances
The basic flow field is assumed to be a steady simple shear flow as generated, for example, by moving walls (Fig. 1). The velocity field is given by VlO=
KX2,
v;=v,o=o.
Superscript completely
(2.1)
0 denotes determined
the basic, undisturbed flow field. The extra stress is by the three viscometric functions:
0 711 -
720 =
N,(K)
=
P,(K)K2,
(2.2)
0 722
7&
N,(K)
=
V2(K)K2,
(2.3)
‘Tp2 =
U(K)
=
=
7j(K)K.
and v2 denote viscosity. Y,
x2
(2.4)
the first and the second normal
t
I-
nh
Xl
2h
~//,I//////////////////-
L
xh
’
Fig. 1. Simple shear flow between two parallel walls.
stress coefficient,
q is the
126 Now, a small disturbance
shall be superimposed
on the basic flow:
u=u”+u*.
(2.5)
The extra stress tensor is then given by
(2.6)
T= To + T’,
where To is the viscometric stress in the basic flow field (determined by (2.2)-(2.4) and T' is the additional stress due to the disturbance. The relative right Cauchy-Green tensor, which is the relevant deformation quantity, is likewise split into two parts: c,(s)
= CO(S) + c;(S).
The viscometric by
(2.7)
part C,‘(s) is contributed
by the basic flow field and is given
(2.8
0 0 11 0
Co(s)
=
‘,,
;:“,2,2
0
.
[
The symbol s denotes the time variable extending from the present (S = 0) into the past (S > 0). The additional part C,‘(S) is due to the disturbance; it can be expanded with respect to S: C,‘(S) = -2SD’ Here, field:
+ O(S2).
D’ is the symmetric
(2.9) part of the gradient
of the disturbance
velocity
(2.10) (cf. [6]). For sufficiently small disturbances, (the term “small” could be made precise by using a suitable norm of fading memory), the disturbance stress tensor T' for a “simple fluid” is a linear functional of C:(S), which can be written in the form (2.11) The kernel functions Mijk, depend on the basic flow through the argument K, and on the time variable S. The assumption of “short “memory” of the fluid with respect to disturbances implies that the kernel functions decrease so rapidly with s that it suffices to insert into (2.11) instead of C, the first term of its expansion (2.9). The integration over s can then be performed, leading to the following constitutive equation: (2.12)
127 with m ijkl
=
2/wsMij,,ds.
-
(2.13)
0
A large number of the mijk,( K) can be reduced to the viscometric functions a,( K), uz( K), T(K) (eqns. (2.2)-(2.4)). This is done by exploiting the compatibility conditions of nearly viscometric flow. The procedure is explained in [5,6). Because the liquid is considered to be density preserving, only five of the six elements D,\ are independent of each other: the equation of continuity yields Dir + L& + II& = 0. Furtermore, only the normal stress differences are of significance for the dynamics of the fluid. Therefore, (2.12) reduces to a relation between five independent stress quantitites and five independent elements Dik’ . The final result of the analysis can be written as follows [6,15]:
ml1
m21
dN1
-2v
dK
17
dN,
dK =
m31
-4
qd
2K
0
0
0
0
0
0
0
0
w3
0
0
w2
0
0
2D:2
4
9
+ N2 K
-N2 K
9
(2.14)
w3
a3
ec zxception of the three nonvanishing elements mik of the first column all elements of the constitutive matrix are determined by viscometric functions. The symbol qd denotes the “differential viscosity”: q, = dT(rc)/dK. Although the three mik are not given by viscometric functions, their behaviour in the limit K + 0 can be determined by making use of the fact that in this limit the fluid is approximated by a Rivlin-Ericksen fluid:
limm,,
= liliom2* = -q(O)
~~?TZ,,
=
lC+O
-(&(o)
2.2 The Reynolds-Orr
+0(x2),
+ 2’2(o))K
+ O(K3).
(2.15) (2.16)
energy criterion
A first intimation of a possible instability of non-Newtonian flow at low Reynolds number is provided by the Reynolds-Orr energy criterion. This criterion is derived as follows: Given a basic flow field with velocity u”,
128 pressure p” and extra stress To, and a disturbed flow field with II = v” + v’, p =p” +p’, T= To + T’. The basic and the disturbed flow field satisfy the momentum balance: (2.17)
(2.18) Both v” and v satisfy equation: auf/ax;
the continuity
equation,
also v’ satisfies
= 0.
- p
JB
this
(2.19)
We assume that at the boundary i3B of a v assume the same boundary values; procedure is outlined in various books (2.17), multiplying by vf, transforming terms as possible into divergence terms (with vi = 0 on i3B) finally leads to: dK -= dt
hence
D,o,vf v:d V -
The quantity
finite spatial domain B both v” and then u’ = 0 on t3B. The following [9,12]: By subtracting (2.18) from in the resulting equation as many (using (2.19)), integrating over B
D,; T;d V.
(2.20)
JB
K can be called the kinetic energy of the disturbance
field: (2.21)
K = ; 1 v,‘vfdV. B
Di is the symmetric part of grad v’. The first term on the right-hand side of (2.20) compared with the second term is of the order of the Reynolds number Re of the basic flow field. Hence, when studying stability- for very small Reynolds numbers, the first term can be dropped: dK -=dt
JB
This neglect can be understood by introducing With the new variables t*, v,f+, Tj.lk* and K*
T:::* = T:/h,U,/h), v;*= y/u,, K* = K/pU,2, XT = Xi/h.
(2.22)
D;k q;d V. of dimensionless
variables.
129 eqn. (2.20) can be rewritten dK* -= dt*
-Re
JB
in the form:
DP,*V;.‘*V,‘*dV* -
D;zqpdV*. / B
The time scale ( h2/q,,) p is characteristic for diffusion processes. If by fixed time scale the limiting case Re --) 0 is studied, the Re term in the last equation can be dropped and the result is equivalent to eqn. (2.22). For a Newtonian fluid Tk = 2qDik; therefore: T; D,‘k= 2qD,‘, Dik > 0.
(2.23)
The integrand in (2.22) is positive definite in that case. Hence, the disturbance energy K decreases monotonically with time and vanishes asymptotically. This signifies asymptotic stability of the basic flow field (for details of the argument, see [9,12].) It is immediately clear that this argument is not applicable to the flow of a non-Newtonian liquid, because for such a liquid Di’,Ti need not be positive definite. Therefore stability is not guaranteed for such flows. In order to elaborate on this point we assume that the basic flow field is a simple shear flow according to eqn. (2.1) and that, for sufficiently small disturbances, the constitutive equation (2.14) applies. The bilinear form Di’,Ti is reduced in that case to
(2.24) In order that the quadratic form (2.24) be positive definite-which provides a sufficient condition for stability-the. coefficients of the ‘form have to satisfy various conditions. All these conditions are listed and discussed in [15]. Here, we pick out only two of them: (a) If the disturbance is a plane velocity field in the xi, x,-plane, i.e. u\ = u:(x,, x2, t), u: = ui(x,, x2, t), the three elements Di3, D& and Di3 are zero. The quadratic form (2.24) then reduces to (2.25) The condition
that (2.25) be positive
definite
is (2.26)
130 (b) If the disturbance velocity field has the form u: = u: = 0, ui = u:(x,, x2, t), the three elements &, D& and D& are zero. The quadratic form (2.24) is then reduced to (2.27) This leads to the following condition for stability: 4Y12> K’( Ni + 2N,)2.
(2.28)
Both (2.26) and (2.28) are sufficient conditions for stability to disturbances of specified form (always provided that the short memory assumption applies). The two conditions are of the same general form. Assuming for the moment that neither Ni nor q depend on K, which means also that 17= vd, condition (2.26) can be written in the form $(K)/T(K)
=
4.
(2.29)
This relation is certainly satisfied for sufficiently low shear rates. However, for high shear rates, it can be violated, indicating possible instability. The same conclusion may be drawn from (2.28). In the special case 77= q)7d = const, N2 = -NJ4 = const, eqn. (2.28) is identical with (2.29). 2.3 Linear stability analysis for plane disturbances Necessary conditions for stability can be obtained by performing a linear stability analysis. Such an analysis for simple shear flow with plane disturbances (defined as type (a) disturbances in the previous section) has been presented by us in several papers (1,3,4]. The analysis is based on the short memory assumption, i.e. on the constitutive equation (2.14) for the disturbances. Only the salient points and results will be mentioned here. Assuming that general disturbances can be decomposed into modes with periodic behaviour in x,-direction, we have studied particular disturbances of the form 4’ =+(x,/h)
exp(iax,/h)
exp(PKt).
(2.30)
+’ denotes the disturbances stream function. Inserting (2.30) into the equation for the stream function which is derived from the linearized momentum balance combined with the constitutive law (2.14) leads to the following equation for the amplitude function +(x2/h) cp”” + {201~(1 - 277,) - iaRex,/h + { a4 + ia3Rex2/h + ia&
+ j3Re) +”
+ /IRea
We(+“’ + a2+‘) = 0.
} $B
(2.31)
131 The following
abbreviations
have been used:
17”= 71/Q,
(2.32)
Re = ptch2/qd,
(2.33) (2.34)
The Reynolds number Re is defined with the differential viscosity qld; We denotes the Weissenberg number. Equation (2.31) has to be solved with the following boundary conditions: +( - 1) = cp(1) = c#L(- 1) = 9’(l) = 0.
(2.35)
By a simple argument one shows that the imaginary part pi of /3 = & + i@i is always zero (for explanation see the corresponding argument in Section 3.2). At the stability limit fi, = 0, hence #I = 0. Therefore, in order to find the stability limit, one has to put /? = 0 in (2.37). Because we are interested in the stability of flow at very low Reynolds number, we let Re + 0 and omit the terms multiplied by Re in (2.31); the influence of finite Re is discussed in [4,13]. For equation is thereby reduced to +“” + 2a2(1 - 217,)Cp” + a4+ + ia( q,)1’2We(
+“’ +\a2+‘) = 0
(2.36)
together with (2.35), (see also [7]). For fixed a and no this constitutes an eigenvalue problem for the Weissenberg number We. The first eigenvalue,
Fig. 2. Discussion of the short memory approximation.
132 for fixed v,, is shown as function of wavenumber a! in Fig. 2. (For more quantitative results see [1,3,4,7]). This “neutral curve” divides the plane into a stable and an unstable region. For values of (Yand K which correspond to a point in the unstable region & is positive. In that case the disturbance grows exponentially. An important feature of the neutral curve is the fact that it tends to the following limit for very short waves ((Y + cc): lim We=4. (2.37) lX-+* This leads to the following conclusion: If, for a given value of K, condition (2.26) is violated, one can always find a wave number (11sufficiently large such that the corresponding point in Fig. 2 lies in the unstable region. Hence, provides that the assumption of short memory is admissible, condition (2.26) is a sufficient and necessary condition for stability to plane disturbances of low Reynolds number flow. Note that, in accordane with our ideal aim as outlined in the introduction, the stability criterion contains only viscometric functions. 2.4 Range of validity of the short memory assumption The results of Section 2.3 depend critically on the short memory assumption. The following arguments show that this assumption is a rather strong one which might not be satisfied for most fluids of practical importance. Let us define a characteristic fluid time A,( K) by the definition.
A, = we/K = hd)
-1/2
d(xK)
dK
(2.38)
*
The variable on the ordinate of Fig. 2 can then be written as X,K. Furthermore, we define a second characteristics fluid time h2( K) as the time scale on which the kernel functions Mijkl( K, s) in (2.11) decay to zero. The time X2 characterizes the length of the memory of the fluid with respect to small disturbances superimposed on the basic flow with shear rate K. For the moment, we assume that the two times X, and X2 are independent of each other and that they can differ, in principle, by orders of magnitude. It can be shown that for (Y+ 0 the stability limit of Fig. 2 behaves asymptotically like (2.39)
x, = C,/a. C, is a constant of the order of 10 (depending on 77,). For the short memory approximation the additional condition is necessary. This condition can be understood by the following The oscillation frequency of a material particle is given by w = 27~/T = 2rU,/L
h,~a
GE
argument:
1
133 with the period T, average velocity U, and wavelength L. U, is of the order of the wall velocity U, (Fig. 1). Then w is of order
=
O(K(Y).
The short memory approximation is valid if A, w -z 1. That means: h,KCl
-=E 1.
Let us now define the Weissenberg number We = bola h2K
=
X,K
and a second hyper(2.40)
c,/(ll
with the condition C, -=K1 corresponding to the short memory approximation. Equation (2.40) can be rearranged by the introducing of the We-number.
(2.41) The stability criterion gives with, We =
fi( a) = C,/a,
f*(a).
The results of the short memory approximation on the marginal stability are compatible with its postulates if a crossing of curves fi(a) and fi(ar) is possible. If A, and A, are the same order, then fi(a) lies below fi(a) because C,X,/C,A, = C,/C, < 1. For usual fluid models, e.g. for the simple Maxwell fluid, A, and A, are of the same order. The short memory theory of Section 2.3 is unsuitable for predicting instability of such fluids because points on the stability limit are outside the region where the short memory theory is valid. Therefore, it is quite compatible with the results of Section 2.3 that previous stability investigations for special Maxwell fluids have shown stability for low Reynold number flow (see, e.g. [S]). If A, =K A,, a crossing of f,(a) and f2( a) is possible such that, at least for a range of sufficiently small wave numbers (Y, instability is predicted by the short memory theory. (This situation corresponds to the dash-dotted line in Fig. 2). Unfortunately, common fluid models do not satisfy A, -K A,. Therefore, the results of the short memory theory can provide no more than a tentative indication of possible instabilities in low Reynolds number flow. The fact that measurements of shear flow in rheometers indicate in many cases the onset of instabilities if the first normal stress is about four to five
134
times the shear stress [lo], (which is in surprisingly good agreement with (2.29)), must therefore be taken as incidental. It is clear at this stage that, in order to obtain clarity about the stability of non-Newtonian shear flow, the short memory theory, which becomes valid in the limit h/A, + 0, has to be dropped. A theory for finite memory has to be developed. For plane disturbances of the form (2.30) this has been done for the limit of very low Reynolds numbers, in [16]. In the meantime, we have found that it is easier to study another type of disturbance (eqn. (3.1)), which leads to similar results and which allows us to drop some restrictive assumptions made in [16]. Only a brief, qualitative interpretation of the main result of [16] will be given here. It is shown in [16] that, as in the case of vanishing memory, disturbances with wave number (Y-B cc become unstable at the lowest value of We. This lowest value, which is 4 in the short memory theory, now depends on X2~. In a h,K-X,K-plane (Fig. 3) this relation determines a stability limit. The stability limit found in [16] is shown, qualitatively, in Fig. 3. The marked point on the X, ~-axis is the result of the short memory theory. For each particular fluid model one can calculate, from the constitutive equation, &(K) and A*(K). Each particular fluid therefore determines a “constitutive curve”. X*(K)K =f(hl(~)~) in the A,K-X,K-plane, which starts which K = 0 at the origin, i.e. in the stable region. The flow becomes unstable if this curve leaves the stability domain; examples are the two fictitious dashed lines in Fig. 3. It is possible that for particular fluids the constitutive curve enters a domain of instability and leaves it again for a higher value of K (see the upper dotted curve in Fig. 3). This could indicate restabilization of an unstable flow with increasing shear rate, a phenomenon which is sometimes observed. For a simple Maxwell fluid the constitutive curve is a straight line
Fig. 3.
135 (dash-dotted in Fig. 3) which does not leave the domain of stability. This explains the failure of previous investigations to find instability for such a fluid. 3. Finite memory theory of special disturbances 3.1 Basic equations Let us assume disturbances of the special form which lead to condition (2.29): u; = u; = 0, u; = w(.q, x2, t).
(3.1)
For this disturbance field, the linear stability analysis for finite memory of the fluid is simpler than for the plane disturbance field studied in [1,3,4]. In order to find the disturbance stress field we have to determine the disturbance part C,‘(S) of the relative right Cauchy-green tensor. Let us denote by xi the coordinates of a fluid particle at time t and by & the coordinates of the same particle at time t - S. With the basic velocity field (2.1) and the disturbance velocity (3.1) the relation between & and xi becomes 51 =
X1 -
X2KS,
52 =x2,
53 =
X3 -
(3.2)
s0
w(X1
The elements &
-
X2Kr,
X2,
t -
r)dr.
of the relative deformation gradient I;;( S) are defined by (3.3)
4j.k = X/ax,, and the elements Ci, of the relative right Cauchy-Green Ci, = F/iF/k7
tensor C,(S) by (3.4)
Using (3.2) together with (3.3) and (3.4) we obtain
C,(s)=C,O(s)+
0
0
a53/ax,
0
0
x3/ax2
x,/ax,
0
[ aE,/ax,
,
(3.5)
I
with the viscometric part C,‘(s) given by (2.8). The second term on the right-hand side of (3.5) is the disturbance C:(s). Only the components Tt3, Ti3 of the disturbance stress are different from zero (which is easily shown by symmetry arguments). According to (2.11) these stress components can be
136 written
as
T;3 =
at3 x3 s),x+ b&, “)a~ ds,
K,
1
1
at3
Ti3 =
(34
2 I
+
b22(K,
(3.7)
ds,
“)jy 2
(The only four relevant kernel functions Mijkl have been denoted The momentum balance of the flow field is in linearized form:
by bik).
p3T+KX2G1-aw
aw
aT,:
aT:,
ax, + ax,.
[
(3.8)
With the aid of (3.6) and (3.7) for Ti3 and Ti3 and of (3.2~) for .$3 eqn. (3.8) is transformed into: = I
-
This is an integro-differential the homogeneous boundary
X2JW,
X2,
t -
equation conditions
(3-g)
r)drds.
for w, which has to be solved with (3.10)
W(Xi, &h, t) = 0, We now separate w(xi,
the time-dependency
x2, t) = W(x,,
Inserting
in w by assuming (3.11)
x2) e@.
(3.11) into (3.9) yields
(3.12)
xi - KX~~, x2) em8’drds. This equation is non-dimensionalized mensional variables: 2; = Xi/h, t
=
(
bi, = gBik( Here
fi = ( Re/P)“2pX
2, ~2~7),
X denotes
the following
non-di-
l?=Kii,
S=s/h,
ReL)“2r/h,
by introducing
a suitable
Re =
,,
pKh2/q.
characteristic
memory
time of the fluid.
The
non-dimensional
The boundary
form of (3.12) is
conditions
for W are
w(z 1, *1)=0. 3.2 Separable kernel functions For further study, eqn. (3.14) is simplified functions Bik are of the following form:
by assuming
that the kernel
fiik(i2, i) = Bik(lZ.) e-“.
(3.16)
One of the compatibility that
conditions
of nearly
viscometric
flow then shows (3.17)
B,, = 1. We define the following E =
( ReiZ)1’2,
abbreviations:
8 = Re/u,
Furthermore, we again assume that the disturbance modes with periodic behaviour in x,-direction: W(&
(3.18)
A = B,, + a( B,, + B22)2. can be decomposed
z2) = (8 + p + iear) 4 ( Z2) exp(i&:,).
into (3.19)
Inserting (3.16) and (3.19) into (3.14) and performing both integrations on the right-hand side yields the following equation for #(y) (with a2 = y): +“-{a2A+p2+p8+ The boundary
iae(2j3 + S)y + a2e2y2} I/J= 0. conditions
(3.20)
(3.15) carry over to (3.21)
$(-1)=+(+1)=0.
Finally, we note that the following condition has to be satisfied in order that the s-integral on the right-hand side of (3.14) converges: p,>
(3.22)
-6.
Where j?, denotes the real part of p. Because at the stability put p = ipi in eqn. (3.20): +” - { a2A - /3; - 2ae&y
limit /3, = 0 we
- a2e2y2} 4
-i{&3,+~sSy}~=O.
(3.23)
138 For every solution I/Jof (3.23) which satisfies (3.21) pi must be zero (which is also true for every solution of (3.20)). This can be proved by multiplying (3.23) with the conjugate complex solution and integrating over y from - 1 to +1:
-2ae&j+‘l$12ydy
+ a2E2]_;‘l$12y2dy
-1
-iSpi j-I11 4 12dy - iae6/_:11 # 12ydy = 0.
(3.24)
Because of the symmetry of the problem with respect to y = 0 the function I X) I2 is an even function of y; therefore /Z :I $ I*ydy = 0. Hence, the underlined terms in (3.24) vanish and, since the real and the imaginary part of (3.24) vanish separately, pi = 0. With pi = 0, eqn. (3.23) is further reduced to $” - a2( A - e2y2 - i$y)
# = 0.
(3.25)
This equation contains the four parameters (Y, E, 6, A. For any three of these given, eqn. (3.25) together with (3.21) constitutes an eigenvalue problem with the fourth parameter as eigenvalue. The real part of (3.24) yields the following expression for A: 1 (y2
I_;‘IV12dy * / _+)d2d~
(3.26)
Taking account of the inequalities (3.27) (3.28) one derives from (3.26): (3.29)
6 < E2 - r2/4ci*. A model liquid, which admits the separation assumption (3.16) example, the simple Maxwell fluid with the constitutive equation
T= - ?JomC,(s)
e-s/xds,
is, for (3.30)
139 with q0 and X constant. For this liquid A = 1. Therefore, at the stability limit the parameter E has to satisfy the condition E2 > 1+ n2/4a2.
(3.31)
Because the wave number can be arbitrarily large, the inequality (3.31) shows that the flow is certainly stable if E -C 1 (note that the rest state K = 0, i.e. E = 0, is stable!). Instability is to be expected only if E > 1. Numerical solution of (3.25) (3.21) at our institutive (by G. Frischmann) as well as analytical investigation of the behaviour of the eigenfunctions for (Y+ cc has shown that, for 8 fixed, E decreases monotonically with increasing (Y. The eigenfunctions assume boundary layer character insofar as they are different from zero only in the immediate vicinity of the boundaries y = + 1. Figure 4 shows numerical results for the lowest order eigenfunction. The thickness of the boundary layer vanishes (like ae213) for (r + 00. Hence:
(3.32)
Therefore, from eqn. (3.26) one concludes that the lowest value of E is given by E = 1 (note that A = 1 and let (Y+ cc in eqn. (3.26)!). This means that the simple Maxwell fluid, characterized by the constitutive equation (3.30) becomes unstable with respect to the disturbances (3.1) if the parameter E increases beyond the value 1. According to the definition (3.18) the parame-
6 = 0.23 E= 1.3
Fig. 4.
140 ter E can be interpreted
in the following
PXK2A21/2 _ Kh us* i i u, = t170/PW2is the shear
&=
way (for the Maxwell fluid): (3.33)
r)O
wave velocity of the Maxwell fluid, Kh is the velocity of the bounding walls (Fig. 1). Our result can therefore be phrased as follows: If the wall velocity in the simple shear flow experiment sketched in Fig. 1 surpasses the shear wave velocity of the fluid, the flow becomes unstable. The most unstable disturbances are those with the highest wave number; in this respect, the situation is the same as for plane disturbances in a short memory fluid (Section 2). Returning to the stability of a general viscoelastic liquid, we note that r? = KX has the meaning of a Weissenberg number. In a realistic experiment, the Weissenberg number is limited and of the order of, say, 1 to 10. Because e2 = Re ii, the instability of the simple Maxwell fluid is not possible for extremely small values of the Reynolds number Re. For Re --, 0, both E + 0 and S + 0. In that case (3.25) is reduced to #” - a2A# = 0.
(3.35)
The eigenfunctions are cos nmy/2. The stability limit is determined by the lowest eigenfunction (n = 1). For this eigenfunction, cos 7ry/2, eqn. (3.35) yields A = YT~~/~cY~,
(3.36)
which is a special case of (3.29). Because, again, (Y may be arbitrarily large the stability limit for low Reynolds number flow (Re --) 0) is given by A = 0. Instability is possible only if A -C0. For each particular fluid which allows a separation of the form (3.16) (more general forms are possible, e.g. such that instead of ems one has a sum of exponentials), A is a function of K. From the constitutive equation of the fluid the function A( K) can be determined. For K + 0 each simple fluid approaches a Newtonian fluid with A = 1. Therefore low Reynolds number instability is possible only if A(K) decreases with increasing K and finally becomes negative. It is clear at this stage that theprogram of establishing general stability criteria, as outlined in the introduction, seems possible in the limit Re + 0 and for fluids which allow the separation (3.16). For more general fluids, no definite pronouncements are possible at the present time. References 1 U. Akbay, E. Becker, S. Krozer and S. Sponagel, Instability of slow viscometric flow. Mech. Res. Commun., 7 (4) (1980) 199-204.
141 2 U. Akbay, Uber den Einfluss der verlnderlichen Viskositat auf die Stabilitat der ebenen Kanalstrdmung. Rheol. Acta, 19 (1980) 196-202. 3 U. Akbay, E. Becker, S. Krozer and S. Sponagel, About a possible cause of viscoelastic turbulence, Proc. 8th Int. Congr. on Rheology, Naples, Vol. 2, 1980, pp. 79-84. 4 U. Akbay and S. Sponagel, Uber die Stabilitat viskometrischer striimungen. Rheol. Acta, 20 (1981) 579-590. 5 E. Becker, Simple non-Newtonian fluid flows. Adv. Appl. Mech., 20 (1980) 177-226. 6 E. Becker, Bemerkungen zur Instabilitat der Stromung nicht-Newtonscher Flttssigkeiten. ZAMM, 63 (1983) 43-48. 7 G. Bohme, Stromungsmechanik nicht-Newtonscher Fluide. B.G. Teubner, Stuttgart, 1981. 8 Teh Chung Ho and H.M. Denn, Stability of plane Poiseuille flow of a highly elastic liquid. J. Non-Newtonian Fluid Mech., 3 (1977/78) 179-195. 9 P.G. Drazin and W.H. Reid, Hydrodynamic Stability. Cambridge University press, 1981. 10 W. Gleissle, Spannungszustand in Kunststoffschmelzen bei Beginn von Stromungsinstabilitlten in zylindrischen Diisen. Lecture at GVC-Committee Meeting “Rheology”, Stuttgart, 1982. 11 J.D. Goddard, Polymer fluid mechanics. Adv. Appl. Mech., 19 (1979) 143-219. 12 D.D. Joseph, Stability of Fluid Motions I+II. Springer-Verlag, Berlin, Heidelberg, New York, 1976. 13 D.D. Joseph, M. Ahrens, M. Renardy and Y. Renardy, Remarks on the stability of viscometric flow. Rheol. Acta, 23 (1984) 345-354. 14 Ch.J.S. Petrie and M.M. Denn, Instabilities in polymer processing. AIChE J., 22 (1976) 209-236. 15 H. Ramkissoon, E. Becker and U. Akbay, Three-dimensional disturbances of planar viscometric flow. Rheol. Acta, 22 (1983) 284-290. 16 S. Sponagel, Uber die Stabilitat einer einfachen viskometrischen Bewegung. Dissertation, Darmstadt, 1982.
INSTABILITY LIQUIDS
U. AKBAY,
123
OF PLANE COUETTE FLOW OF VISCOELASTIC
E. BECKER * and S. SPONAGEL
Institut fiu Mechanik, (F.R.G.)
Technische Hochschule Darmstadt,
(Received January 25, 1984; in revised form November,
Hochschulstr. I, D -6100 Darmstadt 8, 1984)
summaly A review of our work on the stability of plane Couette flow of a viscoelastic liquid is given. The fist part of the review is based on the assumption of a “short memory” of the fluid. The Reynolds-Orr energy criterion intimates the possibility of instability at very low Reynolds numbers. A linear stability analysis for disturbances in the flow plane shows that beyond the stability limit given by the energy criterion there are always disturbances which grow with time. A critical assessment of the short memory theory shows the severe limitations of its applicability. In the second part of the paper, the assumption of short memory is dropped. The stability of plane Couette flow with respect to special disturbances perpendicular to the flow plane is investigated for a Maxwell fluid. The flow is unstable if the product of Reynolds number and Weissenberg number is higher than a certain limit, which has the value one for a simple Maxwell fluid. This result can also be interpreted as follows: The flow becomes unstable if the velocity at the boundary walls is higher than the shear wave velocity of the fluid.
1. Introduction As is well known, flow processes of non-Newtonian, viscoelastic liquids are prone to instabilities which are absent in the flow of Newtonian liquids. In particular, slow flow, i.e. flow at very low Reynolds number, may become
* Deceased on November 14.1984.
0377-0257/85/$03.30
Q 1985 Elsevier Science Publishers B.V.
124 unstable if the liquid is non-Newtonian, whereas slow flow of a Newtonian liquid is always stable. Low Reynolds number instabilities could be the explanation for a number of phenomena which are of considerable technological import. For example, they could be the mechanism behind the observed instabilities of extrusion processes, and they might be responsible for instabilities observed in rheometers. There exists already a vast literature on such instabilities. Reviews have been presented by Petrie and Denn [14] and by Goddard [ll]. All theoretical investigations known to us which lead to explicit results concerning flow stability are based on particular constitutive equations of the liquid (see, e.g., the table, p. 185 [ll]). Their results are therefore dependent on the assumption that a particular constitutive equation describes the liquid accurately for flow with superimposed disturbances in that domain of the parameter space of the flow where instabilities are possible. If stability of instability is predicted by such a theory this need not exclude instabilities or stable flow in reality, because the constitutive equation might be inadequate. Therefore, we began, some years ago, to study stability from a different point of view, which avoids narrowing the approach by restriction to a particular constitutive equation at too early a stage. Usually the flows whose stability is to be investigated are kinematically simple; they are either viscometric flows or simple elongational flows. Our starting point is therefore the theory of nearly viscometric flow and nearly elongational flow, respectively. The present paper is a report on some of our ideas concerning the stability of viscometric shear flow at low Reynolds number. Our original ideal was to perform a stability analysis without a priori specification of a particular constitutive equation. The analysis should lead to stability criteria which contain various material functions of the shear rate of the basic flow field, part of them consisting of the viscometric functions, and the rest of non-viscometric functions. This ideal has been realized, at least to a considerable extent, only for situations to which a “short memory” assumption applies, as explained in Section 2. However, the short memory assumption is not very realistic (see Section 2.3) and the results obtained with the aid of this assumption can provide no more than a tentative indication of possible instabilities. During the course of our work we have become increasingly doubtful about the attainability of our ideal in the more general case of fluids with finite memory. Perhaps only the more limited aim of establishing stability criteria which are valid for sub-classes of the general class of “simple fluids” is a realistic one. Section 2 is essentially a brief review of that part of our work which is based on the short memory assumption, together with a critique of that assumption (Section 2.3). Section 3 contains a linear stability analysis for a particular type of disturbance which simplifies the analysis considerably
125 without sacrifice of physical content. One of the results is the detection of an instability which, to the best of our knowledge, has not yet been mentioned in the published literature. 2. Short memory theory As a suitable starting point, we introduced a simplifying assumption. This consists in postulating that the fluid has a “short memory” with respect to disturbances of the basic flow, in a sense which will be explained below. The results thereby obtained are correct in a certain asymptotic limit, as will become clear later on. 2.1 Constitutive
equation for the disturbances
The basic flow field is assumed to be a steady simple shear flow as generated, for example, by moving walls (Fig. 1). The velocity field is given by VlO=
KX2,
v;=v,o=o.
Superscript completely
(2.1)
0 denotes determined
the basic, undisturbed flow field. The extra stress is by the three viscometric functions:
0 711 -
720 =
N,(K)
=
P,(K)K2,
(2.2)
0 722
7&
N,(K)
=
V2(K)K2,
(2.3)
‘Tp2 =
U(K)
=
=
7j(K)K.
and v2 denote viscosity. Y,
x2
(2.4)
the first and the second normal
t
I-
nh
Xl
2h
~//,I//////////////////-
L
xh
’
Fig. 1. Simple shear flow between two parallel walls.
stress coefficient,
q is the
126 Now, a small disturbance
shall be superimposed
on the basic flow:
u=u”+u*.
(2.5)
The extra stress tensor is then given by
(2.6)
T= To + T’,
where To is the viscometric stress in the basic flow field (determined by (2.2)-(2.4) and T' is the additional stress due to the disturbance. The relative right Cauchy-Green tensor, which is the relevant deformation quantity, is likewise split into two parts: c,(s)
= CO(S) + c;(S).
The viscometric by
(2.7)
part C,‘(s) is contributed
by the basic flow field and is given
(2.8
0 0 11 0
Co(s)
=
‘,,
;:“,2,2
0
.
[
The symbol s denotes the time variable extending from the present (S = 0) into the past (S > 0). The additional part C,‘(S) is due to the disturbance; it can be expanded with respect to S: C,‘(S) = -2SD’ Here, field:
+ O(S2).
D’ is the symmetric
(2.9) part of the gradient
of the disturbance
velocity
(2.10) (cf. [6]). For sufficiently small disturbances, (the term “small” could be made precise by using a suitable norm of fading memory), the disturbance stress tensor T' for a “simple fluid” is a linear functional of C:(S), which can be written in the form (2.11) The kernel functions Mijk, depend on the basic flow through the argument K, and on the time variable S. The assumption of “short “memory” of the fluid with respect to disturbances implies that the kernel functions decrease so rapidly with s that it suffices to insert into (2.11) instead of C, the first term of its expansion (2.9). The integration over s can then be performed, leading to the following constitutive equation: (2.12)
127 with m ijkl
=
2/wsMij,,ds.
-
(2.13)
0
A large number of the mijk,( K) can be reduced to the viscometric functions a,( K), uz( K), T(K) (eqns. (2.2)-(2.4)). This is done by exploiting the compatibility conditions of nearly viscometric flow. The procedure is explained in [5,6). Because the liquid is considered to be density preserving, only five of the six elements D,\ are independent of each other: the equation of continuity yields Dir + L& + II& = 0. Furtermore, only the normal stress differences are of significance for the dynamics of the fluid. Therefore, (2.12) reduces to a relation between five independent stress quantitites and five independent elements Dik’ . The final result of the analysis can be written as follows [6,15]:
ml1
m21
dN1
-2v
dK
17
dN,
dK =
m31
-4
qd
2K
0
0
0
0
0
0
0
0
w3
0
0
w2
0
0
2D:2
4
9
+ N2 K
-N2 K
9
(2.14)
w3
a3
ec zxception of the three nonvanishing elements mik of the first column all elements of the constitutive matrix are determined by viscometric functions. The symbol qd denotes the “differential viscosity”: q, = dT(rc)/dK. Although the three mik are not given by viscometric functions, their behaviour in the limit K + 0 can be determined by making use of the fact that in this limit the fluid is approximated by a Rivlin-Ericksen fluid:
limm,,
= liliom2* = -q(O)
~~?TZ,,
=
lC+O
-(&(o)
2.2 The Reynolds-Orr
+0(x2),
+ 2’2(o))K
+ O(K3).
(2.15) (2.16)
energy criterion
A first intimation of a possible instability of non-Newtonian flow at low Reynolds number is provided by the Reynolds-Orr energy criterion. This criterion is derived as follows: Given a basic flow field with velocity u”,
128 pressure p” and extra stress To, and a disturbed flow field with II = v” + v’, p =p” +p’, T= To + T’. The basic and the disturbed flow field satisfy the momentum balance: (2.17)
(2.18) Both v” and v satisfy equation: auf/ax;
the continuity
equation,
also v’ satisfies
= 0.
- p
JB
this
(2.19)
We assume that at the boundary i3B of a v assume the same boundary values; procedure is outlined in various books (2.17), multiplying by vf, transforming terms as possible into divergence terms (with vi = 0 on i3B) finally leads to: dK -= dt
hence
D,o,vf v:d V -
The quantity
finite spatial domain B both v” and then u’ = 0 on t3B. The following [9,12]: By subtracting (2.18) from in the resulting equation as many (using (2.19)), integrating over B
D,; T;d V.
(2.20)
JB
K can be called the kinetic energy of the disturbance
field: (2.21)
K = ; 1 v,‘vfdV. B
Di is the symmetric part of grad v’. The first term on the right-hand side of (2.20) compared with the second term is of the order of the Reynolds number Re of the basic flow field. Hence, when studying stability- for very small Reynolds numbers, the first term can be dropped: dK -=dt
JB
This neglect can be understood by introducing With the new variables t*, v,f+, Tj.lk* and K*
T:::* = T:/h,U,/h), v;*= y/u,, K* = K/pU,2, XT = Xi/h.
(2.22)
D;k q;d V. of dimensionless
variables.
129 eqn. (2.20) can be rewritten dK* -= dt*
-Re
JB
in the form:
DP,*V;.‘*V,‘*dV* -
D;zqpdV*. / B
The time scale ( h2/q,,) p is characteristic for diffusion processes. If by fixed time scale the limiting case Re --) 0 is studied, the Re term in the last equation can be dropped and the result is equivalent to eqn. (2.22). For a Newtonian fluid Tk = 2qDik; therefore: T; D,‘k= 2qD,‘, Dik > 0.
(2.23)
The integrand in (2.22) is positive definite in that case. Hence, the disturbance energy K decreases monotonically with time and vanishes asymptotically. This signifies asymptotic stability of the basic flow field (for details of the argument, see [9,12].) It is immediately clear that this argument is not applicable to the flow of a non-Newtonian liquid, because for such a liquid Di’,Ti need not be positive definite. Therefore stability is not guaranteed for such flows. In order to elaborate on this point we assume that the basic flow field is a simple shear flow according to eqn. (2.1) and that, for sufficiently small disturbances, the constitutive equation (2.14) applies. The bilinear form Di’,Ti is reduced in that case to
(2.24) In order that the quadratic form (2.24) be positive definite-which provides a sufficient condition for stability-the. coefficients of the ‘form have to satisfy various conditions. All these conditions are listed and discussed in [15]. Here, we pick out only two of them: (a) If the disturbance is a plane velocity field in the xi, x,-plane, i.e. u\ = u:(x,, x2, t), u: = ui(x,, x2, t), the three elements Di3, D& and Di3 are zero. The quadratic form (2.24) then reduces to (2.25) The condition
that (2.25) be positive
definite
is (2.26)
130 (b) If the disturbance velocity field has the form u: = u: = 0, ui = u:(x,, x2, t), the three elements &, D& and D& are zero. The quadratic form (2.24) is then reduced to (2.27) This leads to the following condition for stability: 4Y12> K’( Ni + 2N,)2.
(2.28)
Both (2.26) and (2.28) are sufficient conditions for stability to disturbances of specified form (always provided that the short memory assumption applies). The two conditions are of the same general form. Assuming for the moment that neither Ni nor q depend on K, which means also that 17= vd, condition (2.26) can be written in the form $(K)/T(K)
=
4.
(2.29)
This relation is certainly satisfied for sufficiently low shear rates. However, for high shear rates, it can be violated, indicating possible instability. The same conclusion may be drawn from (2.28). In the special case 77= q)7d = const, N2 = -NJ4 = const, eqn. (2.28) is identical with (2.29). 2.3 Linear stability analysis for plane disturbances Necessary conditions for stability can be obtained by performing a linear stability analysis. Such an analysis for simple shear flow with plane disturbances (defined as type (a) disturbances in the previous section) has been presented by us in several papers (1,3,4]. The analysis is based on the short memory assumption, i.e. on the constitutive equation (2.14) for the disturbances. Only the salient points and results will be mentioned here. Assuming that general disturbances can be decomposed into modes with periodic behaviour in x,-direction, we have studied particular disturbances of the form 4’ =+(x,/h)
exp(iax,/h)
exp(PKt).
(2.30)
+’ denotes the disturbances stream function. Inserting (2.30) into the equation for the stream function which is derived from the linearized momentum balance combined with the constitutive law (2.14) leads to the following equation for the amplitude function +(x2/h) cp”” + {201~(1 - 277,) - iaRex,/h + { a4 + ia3Rex2/h + ia&
+ j3Re) +”
+ /IRea
We(+“’ + a2+‘) = 0.
} $B
(2.31)
131 The following
abbreviations
have been used:
17”= 71/Q,
(2.32)
Re = ptch2/qd,
(2.33) (2.34)
The Reynolds number Re is defined with the differential viscosity qld; We denotes the Weissenberg number. Equation (2.31) has to be solved with the following boundary conditions: +( - 1) = cp(1) = c#L(- 1) = 9’(l) = 0.
(2.35)
By a simple argument one shows that the imaginary part pi of /3 = & + i@i is always zero (for explanation see the corresponding argument in Section 3.2). At the stability limit fi, = 0, hence #I = 0. Therefore, in order to find the stability limit, one has to put /? = 0 in (2.37). Because we are interested in the stability of flow at very low Reynolds number, we let Re + 0 and omit the terms multiplied by Re in (2.31); the influence of finite Re is discussed in [4,13]. For equation is thereby reduced to +“” + 2a2(1 - 217,)Cp” + a4+ + ia( q,)1’2We(
+“’ +\a2+‘) = 0
(2.36)
together with (2.35), (see also [7]). For fixed a and no this constitutes an eigenvalue problem for the Weissenberg number We. The first eigenvalue,
Fig. 2. Discussion of the short memory approximation.
132 for fixed v,, is shown as function of wavenumber a! in Fig. 2. (For more quantitative results see [1,3,4,7]). This “neutral curve” divides the plane into a stable and an unstable region. For values of (Yand K which correspond to a point in the unstable region & is positive. In that case the disturbance grows exponentially. An important feature of the neutral curve is the fact that it tends to the following limit for very short waves ((Y + cc): lim We=4. (2.37) lX-+* This leads to the following conclusion: If, for a given value of K, condition (2.26) is violated, one can always find a wave number (11sufficiently large such that the corresponding point in Fig. 2 lies in the unstable region. Hence, provides that the assumption of short memory is admissible, condition (2.26) is a sufficient and necessary condition for stability to plane disturbances of low Reynolds number flow. Note that, in accordane with our ideal aim as outlined in the introduction, the stability criterion contains only viscometric functions. 2.4 Range of validity of the short memory assumption The results of Section 2.3 depend critically on the short memory assumption. The following arguments show that this assumption is a rather strong one which might not be satisfied for most fluids of practical importance. Let us define a characteristic fluid time A,( K) by the definition.
A, = we/K = hd)
-1/2
d(xK)
dK
(2.38)
*
The variable on the ordinate of Fig. 2 can then be written as X,K. Furthermore, we define a second characteristics fluid time h2( K) as the time scale on which the kernel functions Mijkl( K, s) in (2.11) decay to zero. The time X2 characterizes the length of the memory of the fluid with respect to small disturbances superimposed on the basic flow with shear rate K. For the moment, we assume that the two times X, and X2 are independent of each other and that they can differ, in principle, by orders of magnitude. It can be shown that for (Y+ 0 the stability limit of Fig. 2 behaves asymptotically like (2.39)
x, = C,/a. C, is a constant of the order of 10 (depending on 77,). For the short memory approximation the additional condition is necessary. This condition can be understood by the following The oscillation frequency of a material particle is given by w = 27~/T = 2rU,/L
h,~a
GE
argument:
1
133 with the period T, average velocity U, and wavelength L. U, is of the order of the wall velocity U, (Fig. 1). Then w is of order
=
O(K(Y).
The short memory approximation is valid if A, w -z 1. That means: h,KCl
-=E 1.
Let us now define the Weissenberg number We = bola h2K
=
X,K
and a second hyper(2.40)
c,/(ll
with the condition C, -=K1 corresponding to the short memory approximation. Equation (2.40) can be rearranged by the introducing of the We-number.
(2.41) The stability criterion gives with, We =
fi( a) = C,/a,
f*(a).
The results of the short memory approximation on the marginal stability are compatible with its postulates if a crossing of curves fi(a) and fi(ar) is possible. If A, and A, are the same order, then fi(a) lies below fi(a) because C,X,/C,A, = C,/C, < 1. For usual fluid models, e.g. for the simple Maxwell fluid, A, and A, are of the same order. The short memory theory of Section 2.3 is unsuitable for predicting instability of such fluids because points on the stability limit are outside the region where the short memory theory is valid. Therefore, it is quite compatible with the results of Section 2.3 that previous stability investigations for special Maxwell fluids have shown stability for low Reynold number flow (see, e.g. [S]). If A, =K A,, a crossing of f,(a) and f2( a) is possible such that, at least for a range of sufficiently small wave numbers (Y, instability is predicted by the short memory theory. (This situation corresponds to the dash-dotted line in Fig. 2). Unfortunately, common fluid models do not satisfy A, -K A,. Therefore, the results of the short memory theory can provide no more than a tentative indication of possible instabilities in low Reynolds number flow. The fact that measurements of shear flow in rheometers indicate in many cases the onset of instabilities if the first normal stress is about four to five
134
times the shear stress [lo], (which is in surprisingly good agreement with (2.29)), must therefore be taken as incidental. It is clear at this stage that, in order to obtain clarity about the stability of non-Newtonian shear flow, the short memory theory, which becomes valid in the limit h/A, + 0, has to be dropped. A theory for finite memory has to be developed. For plane disturbances of the form (2.30) this has been done for the limit of very low Reynolds numbers, in [16]. In the meantime, we have found that it is easier to study another type of disturbance (eqn. (3.1)), which leads to similar results and which allows us to drop some restrictive assumptions made in [16]. Only a brief, qualitative interpretation of the main result of [16] will be given here. It is shown in [16] that, as in the case of vanishing memory, disturbances with wave number (Y-B cc become unstable at the lowest value of We. This lowest value, which is 4 in the short memory theory, now depends on X2~. In a h,K-X,K-plane (Fig. 3) this relation determines a stability limit. The stability limit found in [16] is shown, qualitatively, in Fig. 3. The marked point on the X, ~-axis is the result of the short memory theory. For each particular fluid model one can calculate, from the constitutive equation, &(K) and A*(K). Each particular fluid therefore determines a “constitutive curve”. X*(K)K =f(hl(~)~) in the A,K-X,K-plane, which starts which K = 0 at the origin, i.e. in the stable region. The flow becomes unstable if this curve leaves the stability domain; examples are the two fictitious dashed lines in Fig. 3. It is possible that for particular fluids the constitutive curve enters a domain of instability and leaves it again for a higher value of K (see the upper dotted curve in Fig. 3). This could indicate restabilization of an unstable flow with increasing shear rate, a phenomenon which is sometimes observed. For a simple Maxwell fluid the constitutive curve is a straight line
Fig. 3.
135 (dash-dotted in Fig. 3) which does not leave the domain of stability. This explains the failure of previous investigations to find instability for such a fluid. 3. Finite memory theory of special disturbances 3.1 Basic equations Let us assume disturbances of the special form which lead to condition (2.29): u; = u; = 0, u; = w(.q, x2, t).
(3.1)
For this disturbance field, the linear stability analysis for finite memory of the fluid is simpler than for the plane disturbance field studied in [1,3,4]. In order to find the disturbance stress field we have to determine the disturbance part C,‘(S) of the relative right Cauchy-green tensor. Let us denote by xi the coordinates of a fluid particle at time t and by & the coordinates of the same particle at time t - S. With the basic velocity field (2.1) and the disturbance velocity (3.1) the relation between & and xi becomes 51 =
X1 -
X2KS,
52 =x2,
53 =
X3 -
(3.2)
s0
w(X1
The elements &
-
X2Kr,
X2,
t -
r)dr.
of the relative deformation gradient I;;( S) are defined by (3.3)
4j.k = X/ax,, and the elements Ci, of the relative right Cauchy-Green Ci, = F/iF/k7
tensor C,(S) by (3.4)
Using (3.2) together with (3.3) and (3.4) we obtain
C,(s)=C,O(s)+
0
0
a53/ax,
0
0
x3/ax2
x,/ax,
0
[ aE,/ax,
,
(3.5)
I
with the viscometric part C,‘(s) given by (2.8). The second term on the right-hand side of (3.5) is the disturbance C:(s). Only the components Tt3, Ti3 of the disturbance stress are different from zero (which is easily shown by symmetry arguments). According to (2.11) these stress components can be
136 written
as
T;3 =
at3 x3 s),x+ b&, “)a~ ds,
K,
1
1
at3
Ti3 =
(34
2 I
+
b22(K,
(3.7)
ds,
“)jy 2
(The only four relevant kernel functions Mijkl have been denoted The momentum balance of the flow field is in linearized form:
by bik).
p3T+KX2G1-aw
aw
aT,:
aT:,
ax, + ax,.
[
(3.8)
With the aid of (3.6) and (3.7) for Ti3 and Ti3 and of (3.2~) for .$3 eqn. (3.8) is transformed into: = I
-
This is an integro-differential the homogeneous boundary
X2JW,
X2,
t -
equation conditions
(3-g)
r)drds.
for w, which has to be solved with (3.10)
W(Xi, &h, t) = 0, We now separate w(xi,
the time-dependency
x2, t) = W(x,,
Inserting
in w by assuming (3.11)
x2) e@.
(3.11) into (3.9) yields
(3.12)
xi - KX~~, x2) em8’drds. This equation is non-dimensionalized mensional variables: 2; = Xi/h, t
=
(
bi, = gBik( Here
fi = ( Re/P)“2pX
2, ~2~7),
X denotes
the following
non-di-
l?=Kii,
S=s/h,
ReL)“2r/h,
by introducing
a suitable
Re =
,,
pKh2/q.
characteristic
memory
time of the fluid.
The
non-dimensional
The boundary
form of (3.12) is
conditions
for W are
w(z 1, *1)=0. 3.2 Separable kernel functions For further study, eqn. (3.14) is simplified functions Bik are of the following form:
by assuming
that the kernel
fiik(i2, i) = Bik(lZ.) e-“.
(3.16)
One of the compatibility that
conditions
of nearly
viscometric
flow then shows (3.17)
B,, = 1. We define the following E =
( ReiZ)1’2,
abbreviations:
8 = Re/u,
Furthermore, we again assume that the disturbance modes with periodic behaviour in x,-direction: W(&
(3.18)
A = B,, + a( B,, + B22)2. can be decomposed
z2) = (8 + p + iear) 4 ( Z2) exp(i&:,).
into (3.19)
Inserting (3.16) and (3.19) into (3.14) and performing both integrations on the right-hand side yields the following equation for #(y) (with a2 = y): +“-{a2A+p2+p8+ The boundary
iae(2j3 + S)y + a2e2y2} I/J= 0. conditions
(3.20)
(3.15) carry over to (3.21)
$(-1)=+(+1)=0.
Finally, we note that the following condition has to be satisfied in order that the s-integral on the right-hand side of (3.14) converges: p,>
(3.22)
-6.
Where j?, denotes the real part of p. Because at the stability put p = ipi in eqn. (3.20): +” - { a2A - /3; - 2ae&y
limit /3, = 0 we
- a2e2y2} 4
-i{&3,+~sSy}~=O.
(3.23)
138 For every solution I/Jof (3.23) which satisfies (3.21) pi must be zero (which is also true for every solution of (3.20)). This can be proved by multiplying (3.23) with the conjugate complex solution and integrating over y from - 1 to +1:
-2ae&j+‘l$12ydy
+ a2E2]_;‘l$12y2dy
-1
-iSpi j-I11 4 12dy - iae6/_:11 # 12ydy = 0.
(3.24)
Because of the symmetry of the problem with respect to y = 0 the function I X) I2 is an even function of y; therefore /Z :I $ I*ydy = 0. Hence, the underlined terms in (3.24) vanish and, since the real and the imaginary part of (3.24) vanish separately, pi = 0. With pi = 0, eqn. (3.23) is further reduced to $” - a2( A - e2y2 - i$y)
# = 0.
(3.25)
This equation contains the four parameters (Y, E, 6, A. For any three of these given, eqn. (3.25) together with (3.21) constitutes an eigenvalue problem with the fourth parameter as eigenvalue. The real part of (3.24) yields the following expression for A: 1 (y2
I_;‘IV12dy * / _+)d2d~
(3.26)
Taking account of the inequalities (3.27) (3.28) one derives from (3.26): (3.29)
6 < E2 - r2/4ci*. A model liquid, which admits the separation assumption (3.16) example, the simple Maxwell fluid with the constitutive equation
T= - ?JomC,(s)
e-s/xds,
is, for (3.30)
139 with q0 and X constant. For this liquid A = 1. Therefore, at the stability limit the parameter E has to satisfy the condition E2 > 1+ n2/4a2.
(3.31)
Because the wave number can be arbitrarily large, the inequality (3.31) shows that the flow is certainly stable if E -C 1 (note that the rest state K = 0, i.e. E = 0, is stable!). Instability is to be expected only if E > 1. Numerical solution of (3.25) (3.21) at our institutive (by G. Frischmann) as well as analytical investigation of the behaviour of the eigenfunctions for (Y+ cc has shown that, for 8 fixed, E decreases monotonically with increasing (Y. The eigenfunctions assume boundary layer character insofar as they are different from zero only in the immediate vicinity of the boundaries y = + 1. Figure 4 shows numerical results for the lowest order eigenfunction. The thickness of the boundary layer vanishes (like ae213) for (r + 00. Hence:
(3.32)
Therefore, from eqn. (3.26) one concludes that the lowest value of E is given by E = 1 (note that A = 1 and let (Y+ cc in eqn. (3.26)!). This means that the simple Maxwell fluid, characterized by the constitutive equation (3.30) becomes unstable with respect to the disturbances (3.1) if the parameter E increases beyond the value 1. According to the definition (3.18) the parame-
6 = 0.23 E= 1.3
Fig. 4.
140 ter E can be interpreted
in the following
PXK2A21/2 _ Kh us* i i u, = t170/PW2is the shear
&=
way (for the Maxwell fluid): (3.33)
r)O
wave velocity of the Maxwell fluid, Kh is the velocity of the bounding walls (Fig. 1). Our result can therefore be phrased as follows: If the wall velocity in the simple shear flow experiment sketched in Fig. 1 surpasses the shear wave velocity of the fluid, the flow becomes unstable. The most unstable disturbances are those with the highest wave number; in this respect, the situation is the same as for plane disturbances in a short memory fluid (Section 2). Returning to the stability of a general viscoelastic liquid, we note that r? = KX has the meaning of a Weissenberg number. In a realistic experiment, the Weissenberg number is limited and of the order of, say, 1 to 10. Because e2 = Re ii, the instability of the simple Maxwell fluid is not possible for extremely small values of the Reynolds number Re. For Re --, 0, both E + 0 and S + 0. In that case (3.25) is reduced to #” - a2A# = 0.
(3.35)
The eigenfunctions are cos nmy/2. The stability limit is determined by the lowest eigenfunction (n = 1). For this eigenfunction, cos 7ry/2, eqn. (3.35) yields A = YT~~/~cY~,
(3.36)
which is a special case of (3.29). Because, again, (Y may be arbitrarily large the stability limit for low Reynolds number flow (Re --) 0) is given by A = 0. Instability is possible only if A -C0. For each particular fluid which allows a separation of the form (3.16) (more general forms are possible, e.g. such that instead of ems one has a sum of exponentials), A is a function of K. From the constitutive equation of the fluid the function A( K) can be determined. For K + 0 each simple fluid approaches a Newtonian fluid with A = 1. Therefore low Reynolds number instability is possible only if A(K) decreases with increasing K and finally becomes negative. It is clear at this stage that theprogram of establishing general stability criteria, as outlined in the introduction, seems possible in the limit Re + 0 and for fluids which allow the separation (3.16). For more general fluids, no definite pronouncements are possible at the present time. References 1 U. Akbay, E. Becker, S. Krozer and S. Sponagel, Instability of slow viscometric flow. Mech. Res. Commun., 7 (4) (1980) 199-204.
141 2 U. Akbay, Uber den Einfluss der verlnderlichen Viskositat auf die Stabilitat der ebenen Kanalstrdmung. Rheol. Acta, 19 (1980) 196-202. 3 U. Akbay, E. Becker, S. Krozer and S. Sponagel, About a possible cause of viscoelastic turbulence, Proc. 8th Int. Congr. on Rheology, Naples, Vol. 2, 1980, pp. 79-84. 4 U. Akbay and S. Sponagel, Uber die Stabilitat viskometrischer striimungen. Rheol. Acta, 20 (1981) 579-590. 5 E. Becker, Simple non-Newtonian fluid flows. Adv. Appl. Mech., 20 (1980) 177-226. 6 E. Becker, Bemerkungen zur Instabilitat der Stromung nicht-Newtonscher Flttssigkeiten. ZAMM, 63 (1983) 43-48. 7 G. Bohme, Stromungsmechanik nicht-Newtonscher Fluide. B.G. Teubner, Stuttgart, 1981. 8 Teh Chung Ho and H.M. Denn, Stability of plane Poiseuille flow of a highly elastic liquid. J. Non-Newtonian Fluid Mech., 3 (1977/78) 179-195. 9 P.G. Drazin and W.H. Reid, Hydrodynamic Stability. Cambridge University press, 1981. 10 W. Gleissle, Spannungszustand in Kunststoffschmelzen bei Beginn von Stromungsinstabilitlten in zylindrischen Diisen. Lecture at GVC-Committee Meeting “Rheology”, Stuttgart, 1982. 11 J.D. Goddard, Polymer fluid mechanics. Adv. Appl. Mech., 19 (1979) 143-219. 12 D.D. Joseph, Stability of Fluid Motions I+II. Springer-Verlag, Berlin, Heidelberg, New York, 1976. 13 D.D. Joseph, M. Ahrens, M. Renardy and Y. Renardy, Remarks on the stability of viscometric flow. Rheol. Acta, 23 (1984) 345-354. 14 Ch.J.S. Petrie and M.M. Denn, Instabilities in polymer processing. AIChE J., 22 (1976) 209-236. 15 H. Ramkissoon, E. Becker and U. Akbay, Three-dimensional disturbances of planar viscometric flow. Rheol. Acta, 22 (1983) 284-290. 16 S. Sponagel, Uber die Stabilitat einer einfachen viskometrischen Bewegung. Dissertation, Darmstadt, 1982.