S P I N N I N G PI MOLTEN THREADLINE Stability J.
R. A.
PEARSON A N D M. A. M A T O V I C H
Department of Chemical Engineering, University of Cambridge, Pembroke Street, Cambridge, England The response of a steady, nonuniform, axially symmetric, isothermal viscous extensional flow of a Newtonian fluid to small imposed disturbances is studied. Analytic results for the amplification factor are given in terms of the frequency of the disturbance and of the over-all extension in the threadline for three sets of boundary conditions. These are compared with the amplification that is obtained in the uniform unsteady extension of a slightly deformed cylindrical filament. Both high amplifications and resonant effects, such 'as have been observed, may be expected in practice even where imposed fluctuations are very small.
HE steady-state profile of an isothermal spinning threadline extending molten fluid has been investigated (Xatovich and Pearson, 1969). We now consider the response of the system to small disturbances. There have been published reports (Bergonzoni rtnd Di Cresce, 1966; Freeman and Coplan, 1964; hliller, 1963) of resonant effects and many unpublished reports from industrial sources of overlarge denier-Le., thickness--variations in the products obtained from conventional spinning processes, overlarge in the sense that they prove to be relatively far larger than any detectable sources of fluctuation. Nost of these reports deal with rheologically complex Auibs being spun under nonisothermal conditions. Kase and hlatsuo (1967) considered part of the nonisothermal contribution to stability by showing, both numerically and experimentally, that a 1% variation in cooling air may cause a 0.2 to 0.3% variation in the ultimate fiber cross-sectional area. However, similar phenomena have been observed in spinning silicone oils under isothermal conditions. Hence there seems to be a stability problem associated with the basic mechanical process of nonuniform, unsteady elongation. What we seek therefore is some mechanism whereby small, perhaps unnoticeable, disturbances are amplified. One such mechanism has long been known in connection with the slow uniform unsteady extension of viscous fibers or sheets. The resulting linear relationship between draw ratio and amplification factor (valid for small disturbances) would lead, if relevant, to a satisfactory explanation of experimentally observed amplitudes of variation, though it cannot explain resonant phenomena. It corresponds to the approximation represented by Figure 3, c, of Matovich and Pearson (1969). This simple theory was outlined to one of us (J.R..4.P.) in 1956 by G. I. Taylor, and diligent search of the literature would probably reveal a much earlier account. A steady nonuniforin linearized analysis is then given for the restricted case of purely Newtonian isothermal viscous flow. The steady undisturbed flow is described by the very simple exponential solution given by Equation 37 of Matovich and Pearson (1969). The equations governing the disturbance are very different from those given for the former, and a resonant effect is noted in the solutions obtained. An analytical form is obtained for the amplification factor, and a few illustrative figures are shown. Further details are given by hlatovich (1966). Extension of the method of analysis to cover cases where other, more complex, steady solutions-cg., Equation 39, 40, 41, 47, or 71 of Matovich and Pearson
(1969)-are relevant is straightforward, though analytic solutions should no longer be expected: Numerical methods would have to be employed. Uniform Unsteady Extension
Growth of Disturbances. Consider a situation in which the undisturbed flow is given by (see Figure 1 for coordinate system) a = ao(t),
v(2,
(11
t ) = vo'(t)z
From continuity u = -+rvJ(t)
(21
4 = -4aOvJ
(31
and so By considering a disturbed profile of the form
+ ~ ( t sin) h ( t ) z ) l + O ( 2 )
a(r, t ) = ao(t)[1
(4)
n e study the growth of the relative disturbance amplitude, The mechanics can be represented by the constancy of filament tension-i.e., e (t).
a27; = qTa2av/az= v T F ( t )
(5 )
where q~ is the Trouton viscosity and F ( t ) is an arbitrary positive function. The relation m2/X = constant (6) follows from the convection of local maxima and minima with fluid particles, where X is taken only to 0 (E). From
(71
Da/Dt = u ( a ) = -$a(av/ax) and linearizing we obtain 2Uoe
+ iQ= 0
or, after integration, E
(t) = ~oE(t)
where E is the extension ratio. Application of this approach to the nonuniform steady flow described by Matovich and Pearson (1969), using Boundary Conditions 9 and l O i , would yield
(101
E ( 1 ) = VJVO VOL.
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Differentiation with respect to 5 and resubstitution into Equation 17 yields as the governing partial differential equation
+ e-brc
all
(20 1
=0
Writing Equation 20 in the form
I
we obtain the general integrals Figure 1.
CY[ = f ( 7
Coordinate system for extending filament
+ e+)
(21 1
l
=J f(7+e+)d[+r(7)
(221
QR,~)
The local amplification factor, A , for a small disturbance introduced a t x = 0 would be simply
A (2)
=
v/vo
with a maximum of
A ( I ) = VZ/VO
(11)
However, this approach is quantitatively incorrect in its predictions. The final result for E is independenb of the wavenumber X and so is valid for an arbitrary disturbance. Steady Nonuniform Extension
Growth of Disturbances. We now consider the situation represented by an undisturbed flow E(2)
=
0
wheref and I’ are arbitrary functions. The form of Equation 21 indicates the general wave-like nature of the phenomenon, and shows how the time variation in wavenumber X in Equation 4 for the unsteady uniform case is paralleled by a coordinate stretching (e-{) in the steady nonuniform situation. We turn now to specifying boundary conditions. This is precisely the question discussed by hlatovich and Pearson (1969) and the same arguments may be used, though we have a choice as to where we put in the disturbance. It will prove convenient to decompose disturbances into their Fourier components-Le., to study the amplitude ratio A as a function of frequency w, for disturbances of the form eiwt. This is permissible because the governing equations are linear. CASE1. Input a t the die, [ = 0 A. Radius variation alone
V&’X
(z(x) = (J&-x/zx =
( & / ? r ~ o ) ~ / ~ exp
(-x/2X)
/? (0, T )
=0
(23. l a )
a(0,r ) = 0, p(0, T ) = sinwr
(23. l b )
CY
( 0 , ~ )= sin UT,
B. Velocity variation alone
We write for the disturbed flow
C. Constant output, mixed radius and velocity variation a(0,7 )= sinwr, p ( 0 , ~ = ) -2 sinwr
where M and /? we to be thought of as small. The continuity equation becomes, assuming incompressibility,
a at
- a2
+ ax-a (a%) = 0
If we suppose that the other boundary condition is to be prescribed a t x = l-i.e., 5 = A = l/X-then Equation 1Oi or IOiii of Matovich and Pearson (1969) yields either P ( 4 7 )
(15)
as before. This states that the force along the threadline is constant. If we write 7 = vot/x, 5 = x / x (16) substitute Equation 13 into Equations 14 and 15 and linearize, we obtain CYr e ( ( c q $Pc) = 0 (17) and 2a+P+Pc=4(7) (18)
+
+
respectively, where subscripts denote differentiation with respect to the arguments 7 and t-i.e., cyr = aa/ar. Substitution of Equation 17 for PE into Equation 18 yields
P = 4 (7) 606
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CY
+ 2 e - k ~+~ CY:
FUNDAMENTALS
(24. l a )
=0
(constant velocity) or
+
and the equation of motion, neglecting inertia, acceleration, gravity, and surface tension terms, a%’ = G ( t )
(23.1~)
(19)
Pc (A, 7 ) P (A, 7)
+ CY (4 = 0 7)
(24. I b )
(constant force). CASE2. Input a t the wind-up, 5 = A A. Velocity variation
P(A,7) = s i n w
(25.2a)
B. Tension fluctuation
+
4 ( T ) = &(A, 7 ) @(A,7 )
+ 2a(A,
7 )=
sin wt
(25.2b)
The relevant boundary condition a t 5 = 0 could then be CY
or, if C$
(7)is
(0, 7 ) = P (0, 7 ) = 0
(24.2a)
prescribed, a constant Q requirement in the form 2 C Y ( o 1 7 ) + @ ( 0 1 7 )=
0
(24.2b)
Each of these possible sets of boundary conditions will lead to a different solution whose prime characteristic can be
expressed in terms of
marieed in the form
A = ~ ( A , T= ) A(w,A)
(25 )
the overbar indicating a mean square time average. We do not give an exhaustive set of solutions here, but confine ourselves to taking one illustrative example to show how the boundary conditions can be introduced into General Solution 22, presenting a few graphical results for A (w, A ) . Let us take, therefore, Case 1A with constant velocity a t 5 = A-Le., Boundary Conditions 23.la and 24.la. We assume that we can write
f(7
+
+ q)II
(26 1
7) = Re C f o e x ~{ i w ( ~
where fo = fl
+ if'
(27)
is a complex constant, and
(34%)
for constant velocity at 5
=
A,
A' = ([Cip
+ [Si]')
(34b)
for constant tension a t { = A. Because of the relation @ = -2a at constant output-i.e., Equation 23.1~-it is most reasonable to plot the quantity R, = 2A as a measure of the amplification ratio. Thus Equation 34a yields the limit R, + 1 as w + 0, as we would wish; Figure 2 shows how R, (constant velocity) varies with w for various values of E. il "resonant" effect is clearly evident. R, + 0 as w + m for large E. The result (34b) has a different limit, R,-+ln E
If we use the Ci and Si functions defined by si
(2) =
sin u
and write
+ +
B1 = [Ci]w/~W $. wE-'[si],~~~
[COS]w/~w
B2 = [si]w/~W - wE-'[C~],/E"
[ S ~ ] ~ / E ~
(30)
for w + 0; again R, + 0 as w + co with E fixed. Figure 3 shows how R, varies with w for various values of E. The resonance is almost entirely damped. Solutions 32 and 34a exhibit singular points in the R, (or A) surface above the (w, E ) plane-i.e., points where R,+ m (B1 = BB= 0). Since large values of R, indicate the significance of smaller initiating disturbances, these points must be interpreted as points a t which the system is unstable in the classical sense with respect to the mode of disturbance being studied. Whether these points are related to observed fiber
'k
where
[flw= / ~f(w) w -f(w/E),
and
E
=
(35 )
e*
IO
we find that Ra
c
E
WD
-
34.5
2.24~
I
20
40
60
80
00
Our definition of A in 29 gives finally
- E-') + 2w(1 X (Biz+ B!?) - B'[Ci] + 3w (1 - E-I) ([Si12 + [CiI2))
A2(w, E ) = 1 (Bl[Si]
(32)
where limits w and w / E have been omitted for clarity. This is the quantity of most interest. If we consider the asymptotic limit w --f 0, we discover, as we would expect, that A -+ I, irrespective of E . This is the case where fluctuations are very slow compared with the time of flight in the threadline, and so a t any moment the steady solution of Equation 12 applies with an appropriate value of Q (= a0 &,a). If we a ,we have the case where take E to be fixed and let w fluctuations are fast compared with the time of flight. We find that, for large E,
-
A-E
397 3.78d
'OI 5
+
(33 )
which is precisely the result we obtained in Equation 11, and again is to be expected. Case 1B with Conditions 24.la and 24.lb has been similarly analyzed by Matovich (1966), and the nature of the function A (w, E ) investigated. The analytic results can be sum-
Figure 2. Amplification factor, R,, as a function of w for various values of E, with constant velocity boundary condition For convenience abscissa is w ~ a , stated multiple of w
VOL.
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~
20
60
40
80
100
0
Figure 3. Amplification factor, R,, as a function of w for various values of E, with constant tension boundary condition
Table 1.
Possible Unstable Operating Points Ell2
4.495 7.070 8.963 10.53 11.86 13.09 14.20 15.25 16.20 17.15 17.96 18.80
w
4.66 10.79 17.03 23.29 29.56 35.84 42.12 48.39 54.68 60.96 67.24 73.52
instabilities will depend on whether the disturbance given by Equations 26 and 27 is the form of the most unstable mode. The first few unstable points are given in Table I. Solution 34b does not exhibit this phenomenon. Discussion
The results given by Equations 32 and 34a, b for the amplification factor show that the boundary conditions imposed a t = 0 and A are crucial in determining the quantitative nature of A (a,E ) . We could have obtained a more “general” solution in the form Aj(w, E , K, L) subject to general boundary conditions (CY
+ 2 p ) = sin at J i =0 + p + 2a) + K@ + Kaa = 62isinar b(& + /3 + 2a) + L@ = 8& sin 5=A 61i
UT,
with j = 1 , 2 , or 3. The K’s and L’s would have to be chosen so as to satisfy the (physical’ boundary conditions relevant at either end of the threadline. L would be characteristic of the take-up device; K would be chosen to represent the mechanical interaction between the upper portion of the threadline and the adjustment zone near the spinneret (Xatovich and Pearson, 1969), a t present outside our scheme of analysis; the value of j would select the source of the disturbance. (The case of j # 1, 2, or 3-i.e., no imposed fluctuation-would be of some interest. w could then be regarded as an eigenvalue, and need not be real; if nonzero solutions were found with I m ( w ) < 0, then the threadline would be wholly unstable, and infinitesimal disturbances a t any point would grow in time. 608
l&EC
FUNDAMENTALS
I n practice, the threadline would either break or exhibit large disturbances stabilized by nonlinear interactions. Observed resonances could be of this type, with Re ( w ) # 0 providing the necessary periodicity in time.) Only algebraic complexity would be encountered. Until we have reasonable methods available for selecting K, L, and j to correspond to those relevant in commercial processes, there seems little point in displaying a wealth of detailed numerical results. Furthermore, the assumption implicit in the use of the undisturbed flow given by Equation 12 may not hold in many important cases. We have considered the linearized stability equations that result when other basic flows (Xatovich and Pearson, 1969) are perturbed; unfortunately, analytic solutions do not seem so readily obtainable, and immediate recourse to purely numerical means of solution seemed to be indicated. Kevertheless, we may conclude that observed variations in filament thickness of up to 10% under practical industrial conditions where ‘imposed’ fluctuations are reduced to values well under 1% should not be regarded as unusual. Filament spinning is a basically unstable process, at least for highly viscous Newtonian liquids. The converse problem is perhaps the most interesting t o examine: Why are molten threadlines so stable in many industrial situations? Part of the answer may lie in the nonlinear and elastic nature of the fluids spun: Thus a Trouton viscosity (Equation 44, hlatovich and Pearson, 1969) that increases with rate of extension, or an elastic term (Equation 55, Xatovich and Pearson, 1969), may be shown qualitatively to lead to greater stability. Temperature variations may, however, be dominant: If the fluid cools as it passes down the threadline, thinner portions cool more rapidly and so acquire a larger viscosity; this can easily be shown qualitatively to lead to greater stability. Physicochemical studies shorn that the local structural state of polymeric materials used for melt spinning into fibers is very dependent on the extension rate and temperature histories to which they are subjected. This suggests that a fully satisfactory representation of threadline mechanics will be achieved only by use of a number of structural (or state) parameters: These could be scalars or tensors of any order; in steady flow they would be functions of position only, but would exhibit time variations under unstable conditions. Continuum relations for describing the rheological behavior of such complex materials have been proposed, but usually in very general mathematical terms, and seldom with explicit reference to temperature. Usable constitutive and state equations axe likely to be obtained only when continuum behavior is interpreted in terms of the real physical entities (molecules, entangled groups of molecules, oriented layers, or strings of molecules) that exist in stressed and deforming polymer melts. Recent methods for deducing correctly invariant rheological equations of state for dilute suspensions (Takserman-Krozer, 1963, 1967; Takserman-Krozer and Ziabicki, 1963) suggest that means may be found to unify the continuum and molecular approaches. However, until this has been done, theoretical studies of the fluid mechanics of flowing polymer systems must be largely exploratory and qualitative in their nature, particularly in situetions like that of the spinning threadline. Nomenclature
A ( w , A) ab,t)
ci (2)
= perturbation amplification factor = fiber radius = steady-state fiber radius
= integration constant = Kronecker delta = perturbation amplitude
= radius of cylindrical element = functions determining A(w, A ) defined by
Equation 30
= constant
= extension ratio = O(Z)/g(O) = force on cylindrical element
Ki Li 1
perturbation function constants in perturbation function threadline force constants in general boundary condition constant in general boundary condition length of fiber volumetric rate of flow of material in fiber L2A radial coordinate time radial velocity = axial velocity = steady-state axial velocity of fiber = velocity in cylindrical element = axial coordinate = dimensionless length
= = = = = = = = = = =
GREEKLETTERS ab,t) @(’I t,
= e-b = dimensionless length
of fiber
=
Z/X
= wavenumber function =
x/x
= axial stress = dimensionless time = uot/X = dimensionless disturbance frequency
literature Cited
Bergonzoni, A., Di Cresce, A. J., Polymer Eng. Sci. 6, 45 (Pt. I), 50 (Pt. 11) (1066). Freeman, H. I., Coplan, M. J., J. Appl. Polymer Sei. 8, 2389 (1964).
Kase, S., Matsuo, T., J . Appl. Polymer Sci. 11, 251 (1967). Matovich, &I. A , , Ph.D. thesis, University of Cambridge] 1966. Matovich, M. A., Pearson, J. R. A,, IND.ENG.CHEM.FUNDAMENTALS
8, 512 (1969).
Miller, J. C., S.P.E. Tram. 3, 134 (1963). Takserman-Krozer, R., J . Polymer Sci. A l , 2477, 2487 (1963). Takserman-Krozer, R., J . Polymer Scz. C16, 2845 (1967). Takserman-Krozer, R., Ziabicki, A., J . Polymer Sci. 8 1 , 491, 507 (1963).
= radius perturbation = velocity perturbation
RECEIVED for review January 27, 1969 ACCEPTED July 22, 1969
S T A T I S T I C A L A N A L Y S I S OF FLUID FLOW FLUCTUATIONS I N THE VISCOUS LAYER NEAR A SOLID WALL A.
T. P O P O V I C H
Universitt Laual, DCpartement de Gdnie chimique, Qudbec, Canada Recently published experimental statistical data on the viscous sublayer in turbulent pipe flow are analyzed. Statistical ‘‘law’’ for the distribution of the wall shear stress and a tentative distribution for the dirnensionless thickness of the laminar sublayer are obtained. A new phenomenon in the distribution of the velocities is shown to exist.
T HAS been known for some time (Hinze, 1959) that the
I conditions in the viscous sublayer in turbulent flow are not
as regular as the early researchers had assumed them to be. I t seems evident now that both Prandtl and von Karman could not possibly develop a complet,ely satisfactory theory of “wall turbulence,” since they did not start from the correct experimental facts. A t the present time there is conclusive evidence that the velocity fluctuations persist in some form, almost as far as the wall itself (Fage and Townend, 1932; Hettler et al., 1964; Laufer, 1954; Popovich and Hummel, 1967a, b). Recently (Kline et al., 1967) some progress has been made in visualizing qualitatively the nature of the flow near the wall, but the complete picture of the flow mechanism is still missing. The knowledge required can, of coiirse, be obtained only by experiment. Therefore, it seems fair to say that what is still needed is not anothex theory, which would only join dozens of ‘‘theories” already existing, but more meaningful experimental data , Whereas many publications have appeared through the years on the subject of the statistical nature of turbulent flow
away from the wall, the statistical nature of the flow a t the wall itself has not been stressed. An attempt is made here to analyze the recently published (Popovich and Hummel, 1967a, b ) experimental data quantitatively. Method of Treating Data
The construction of the so-called Probit diagram is used for finding the statistical “law” according to which the experimental data are distributed. Statistical experimental data for an observed variable, t , are usually presented in the form of an ordinary histogramfor example, Figure 2. The height of a column of width At on the hist,ogram equals the number of observations of the variable t having values which fall in the interval At, divided by the column width, At. The total surface under the histogram is, thus, equal to the total number of observations, n ; and the surface between &heminimum value observed and any chosen value t equals the total number of observations having the values st; this surface is designated N ( t ) . If a distribution function-for example, Figure 2-is p (z), it is VOL.
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