I
JEROME GAVE Department of Chemical Engineering, The Johns Hopkins University, Baltimore, M d .
Propagation of Transverse Waves on Viscoelastic Jets Study of transverse wave patterns imposed on jets of viscoelastic liquids may offer a new method for obtaining information about such properties of the fluids as normal stress development and relaxation
SOME
time ago the author, in collaboration with S. J. Gill, described some experiments in which a viscoelastic liquid-a solution of carboxymethylcellulose in water-was ejected from a nozzle of circular cross section which was oscillating sinusoidally transversely to its axis (2, 3). A complex but regular wave pattern was observed to occur upon the jet. T h e figure illustrates this. To the unaided eye the phenomenon appears as a set of stationary waves which are the nodal envelopes of the illustration. T h e nodal points are fixed in space for a given experiment but are located at distances apart, I, which increase as distance from the nozzle increases. Between these points, the waves of constant wave length, X, travel with a velocity, vX, where v is the nozzle vibration frequency. Gill and Gavis ( 3 )were able to account for the wave propagation properties of the jet. Their analysis showed that propagation depended upon the existence of longitudinal tension in the jet which was independent of ejection velocity, vibration frequency and amplitude, and nozzle length in the ranges of these variables studied. A dependence upon diameter, however, indicated that the tension in the jet was a function of the surface tension of the jet fluid. Calculation showed that the fluid surface tension could indeed manifest itself as a longitudinal tension but that this was not large enough to account for the total measurable tension. When the calculated surface contribution, To, was subtracted from the total tension, T, the resulting internal tension, TIM, was independent of diameter as well. The internal tension was therefore independent of the treatment the fluid received either in the nozzle or as a result of the wave disturbance. So far as the analysis of the wave patterns is concerned, the tension may be assumed simply to pre-exist in the jet. The increase in distance between nodal points could be accounted for by a progressive decrease in TMalFng the jet,
which means that TMdecreases with time and the phenomenon is one of stress relaxation. T h e origin of the internal tension can only be speculated upon at this time; it must suffice for the discussion here that the tension exists, is independent of fluid perturbations, and undergoes relaxation. Linkage with normal-stress phenomena may perhaps be expected in light of the demonstration of the development of longitudinal normal stresses in tube flow of Reiner-Rivlin fluids as reported, for example, by Truesdell ( 5 ) . Since the original experiments, wave propagation experiments have been performed with other viscoelastic systems such as methylcellulose solutions in water and polystyrene and polyisobutylene
solutions in xylene. T h e existence of a surface contribution to the tension has led to the question of whether such waves may be supported by viscous, Newtonian fluid jets. Experiments have shown that Newtonian fluid jets will support waves if their viscosities are high enough (greater than 20 c.P.) to prevent breakup under the influence of the vibration. The observations are similar to those in viscoelastic jets, except that the internodal distances are constant and distinct progressive amplitude damping occurs. These signify that there is no stress relaxation and that there is appreciable resistance to wave propagation. The propagation of transverse waves on fluid jets is thus a general occurrence and should be discussed in a more general
MOVING COORDINATE SYSTEM t p - Y l fuV vo o
- v,
-
--
-vo
VO
STATIONARY
+
-vo-
COORDINATE
-
=
-VI
SYSTEM
v,
*
VI
- v,- V , + V ,
In the moving coordinate system the nozzle moves away from the observer, and waves are propagated with velocities f vo in both directions.
In the stationary coordinate system the jet ejection velocity, VI, must b e added to the wave velocities to give two waves now traveling in the same direction but with slightly different velocities. This gives rise to interference and the pattern pictured VOL. 51, NO. 7
JULY 1959
885
manner than could be presented by Gill and Gavis. Specifically, it should be possible to generalize the equation for transverse wave propagation and to show how propagation viscous and viscoelastic jets are special cases of the same phenomenon. Alternatively, as is done here, it should be possible to start with the simplest case possible-the inviscid fluid jet with surface tension-and to $how how viscous and viscoelastic systems are obtained from this by inclusion of proper terms in the equations of motion. Inviscid Jet with Surface Tension
In this simple system the fluid issues at a uniform velocity, u l , from a nozzle vibrating sinusoidally with frequency Y in the manner described by Gavis and Gill (2). I t is convenient to follow the motion in a coordinate system moving with the jet. The equation of motion may be written for the moving coordinate system and the solutions obtained transformed back to the stationary system of the observer following the procedure of Gill and Gavis ( 3 ) . The process is illustrated pictorially in the figure. Gavis and Gill (2) have developed the wave equation for propagation of surface tension waves by equating the restoring force to the inertial force of a jet element: a211./Dt2 = n / p r D p / D X 2
(1)
where u is surface tension, p density, and r radius of the jet. The solution to this in the stationary coordinate system is identical to Equation 5 of Gill and Gavis ( 3 ) : which describes mathematically the situation illustrated in the figure, stationary coordinate system. There is no damping term; all wave lengths are constant. The propagation velocity is a function of 6 ,p, and r alone: uo = ( u / p r ) l / 2 E
vh2/21
(2)
u may be obtained from a rapid flash photograph by measurement of h and 1.
Addition of Viscous Resistance
As a wave progresses along a jet, periodic bending of the jet axis occurs. If the amplitude and jet diameter are small compared to the wave length, the jet cross section remains essentially circular and all motion, aside from the translational motion of the jet itself, occurs in a direction perpendicular to the original jet axis. The nonuniform velocity profile developed in the nozzle may be considered to damp out very close to the nozzle, and for all practical purposes all portions of the jet cross section move with the same horizontal velocity, V I . T h e discussion of Hansen and others ( 4 ) may be cited in justification. T h e perpendicular motion gives rise to vertical shear in the manner usually
886
considered in the problem of bending of a beam. For the inviscid jet this has no effect. But if the fluid has viscosity or elasticity, shear forces are introduced. I n the viscous, Newtonian jet a stress proportional to the viscosity and to the rate of shear opposes the vertical shearing motion. A force balance over an infinitesimal cross section yields the equation of motion in the moving coordinate svstem:
The solution, in the stationary coordinate system, shows that the effect of the viscous term is to introduce damping and wave dispersion-i.e., to make 710. now given by: c'o s [ u / p r - a?/h2p2]1/2 e vh2/21 (4) a function of wave length. T h e distances between nodes, I , are longer than in the inviscid jet but still constant. Surface tension may be determined as before. Addition of Elasticity
I n the viscoelastic jet one might expect similar viscous resistance to wave motion to occur. Here, however, the presence of elasticity alters the situation. Under periodic shear stresses viscoelastic liquids exhibit some of the properties of solids, in that they offer elastic resistance to deformation. I n addition their viscosities decrease with rate of shear or with frequency. Vertical shear will now be opposed by rigidity as well as by viscosity and if the shear modulus of the fluid is high enough and its viscosity low enough at the frequency of the imposed wave. resistance will be caused solely by elastic stresses. T h e complete wave equation in the moving coordinate system includes the viscous term, the rigidity term with radius of gyration, K , and frequency dependent modulus, E(v), the surface contribution to the tension, Tu, and the relaxing internal tension, T;u, which in the moving coordinate system is a function of distance as well as time in the form T.Ti(t x/u1):
+
p D2*/DP
+
v(v) 6 V / D X 2 D t
K ~ E (v) =
Dx'
Tdt
+
~ / D x{ [ U / I
+
+ x/vdla11./axl
11. = A(pT)' cos 2a(vt
-
./A)
x
sin 2a Y f ( p / T ) ' I 2 d x (6)
where T = c/r
+ T-.dt)
= u/r
+ T.dx/z:)
(7)
for a fluid whose tension changes slowly. T h e propagation velocity is now given by : ~ o ( t )= [ u / p r
i 7'11(l)/pl'~
(8)
1 becomes longer as distance from the nozzle increases. T h e amplitudes should increase. but this is not noticeable, since the increase depends upon the power of T which is assumed to change slowly. T and T.ir may be plotted against t in the manner of Gavis and Gill (2). When this is done. it is found that 7' often changes very rapidly close to the nozzle. so that Equation 6 is not correct and the method of plotting is invalid for about the first nodal distance. I t is possible to obtain the solution in terms of Bessel functions close to the nozzle and to join it to Equation 6, but because it describes only a small portion of the entire jet it is not discussed here. Should wave experiments be performed with such fluids as molten polymers, it is expected that the rigidity term will have to be included in Equation 5, the viscosity term may be dropped. and the solution must include rigidity damping and dispersion and will be extremely complex. Literature Cited
(5)
Recent experiments with dilute solutions of low molecular weight polymers show that waves propagate as in viscous fluids-the moduli a t the frequencies employed (400 to 600 cp.) are too low to be noticeable in the equation. As concentration and molecular weight are increased moderately, both viscous amplitude damping and decaying tension (increasing I with distance) are observed. A solution of 5 has not been obtained for this case, however, because of the extremely complicated form it would
INDUSTRIAL AND ENGINEERING CHEMISTRY
assume. Gill and Gavis ( 3 ) found no damping or dispersion in their experiments; subsequent experiments with high molecular lveight polymer solutions also reveal no damping or dispersion. For these polymers E ( Y ) is of the order of lo3 to lo4 dynes per sq. cm. ( I ) , which is high enough to overshadow the viscous resistance but below the value of about lo6dynes per sq. cm. which may be calculated as the value where rigidity effects would just become noticeable in the experiments. For these systems the third- and fourth-order terms in Equation 5 may be dropped. T h e solution of Equation 5, transformed into the stationary coordinate system, is closely approximated by:
(1) Ashworth, J. N., Ferry, J. D., J. Am. Chem. SOC.71. 622 (1949). (2) Gavis, J., Gill, S. J.,' J. Polymer Scz.
21, 353 (1956). (3) Gill, S. J., Gavis, J., Zbid., 20, 353 11956).
(4)' Hansen, R. S., Purchase, M. E., Wallace, T. C., Woodey, R. W., J. Phys. Chem. 62,210 (1958). (5) Truesdell, C., J . Rat. .tfechech.Anal. 1, 241 (1952). RECEIVED for review January 15, 1959 .4CCEPTPD April 30, 1959 Work supported by the rational Science Foundation under Grant G3675.
