Nomenclature
final value of independent variable t
a
=
C
= final value of x
e
=
J k
=
q
=
optimal estirnate of c = function to be minimized
R
= ~ ( c a) , = t = t, =
At
=
X
=
Y
= =
t
reaction rate: constant weighting function, defined by Equation 22 random numbers with Gaussian distribution missing final condition independent variable final value o f t integration step size dependent v,ariable defined by Equation 5 or 45 observed state
literature Cited
Bellman, R . , Kagiwada, H., Kalaba, R . , Sridhar, R., “Invariant Imbedding and Nonlinear Filtering Theory,” RAND Gorp., Santa Monica, Calif., RM-4374 (December 1964). Bellman, R., Kalaba, R, J . M a t h . Phys. 1, 280 (1960). Detchmendy, D. M., Sridhar, R., “Sequential Estimation of States and Parameters in Noisy Nonlinear Dynamic Systems,” Joint Automatic Control Conference, pp. 56-63, Troy, N. Y., 1965. Kalman, R . E., Bucy, R. S., J. Basic Eng. 83, 95 (1961). Lee. E. S.. IND.END.CHEM.FUNDAMENTALS 7. 152 11968a). Lee; E. S . , “Quasilinearization and Invariant ‘ Imbedding,” Academic Press, New York, 1968b. Wing, G. M., “Introduction to Transport Theory,” Wiley, New York, 1962. RECEIVED for review December 5, 1966 ACCEPTEDAugust 8, 1967
EXPERIMENTAL TECHNIQUE
MEASUREMENT OF TENSILE STRESSES IN RAPIDLY DEFORMING VISCOELASTIC MATER IA LS G l A N N l ASTARITA, GIUSEPPE M A R R U C C I , AND D O M E N I C O ACIERNO Istituto di Eli.ttrochimica, Unioersitci di -Vafoli, Piazzale Tecchio, h7afles, Italy
The geometry of a jet of viscoelastic material issuing abruptly from a nozzle under the influence of an impact is measured by a frame-by-frame analysis of high-speed film. Application of the momentum balance equation, and a few simplifying assumptions, allow evaluation of the tensile stresses within the jet during the rapid deceleration phenomenon following the end of flow through the nozzle.
analyses of viscoelastic fluid mechanics (Metzner et al., 1966; Astaritfa, 1967) have led to the conclusion that, in high Deborah number flows, solidlike behavior is to be expected, This implies that in a suddenly accelerated flow, internal stresses are developed which do not depend on the deformation rates but rather on the total deformation of the body. Very large stresses are thus predicted over a time interval of the same order of magnitude as the natural time of the body; in the case of polymer solutions, the latter is typically in the range to second (Shertzer, 1965). Experimental confirination of this behavior has so far been only qualitative, because of the difficulty of measuring internal stresses during the first 0.001 second of a flow phenomenon. This note presents the basic idea of a n experiment where such a measurement can be made. RECENT
experiments a 1% by weight aqueous solution of ET 591, a high molecular weight additive (Dow Chemical Co.). A weight of about 3 kg. is suspended by a rod from a pivot and is held a t a height of about 3 cm. above the syringe axis by a n electromagnetic brake (Figure 1). O n release of the magnet, the weight swings and hits the piston shaft of the syringe, thus expelling an amount of liquid from the syringe nozzle. The release of the
field view
Experiment Technique
A large gas syringe (5.0-cm. I.D.) with a bell-mouthed entrance short nozzle is filled with the liquid to be tested, in these
of
I
test \ 1 m a t e r i a l L----
Figure 1.
Diagram of the apparatus
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magnet also triggers a high speed camera, which has reached full speed a t time zero-Le., the instant of impact of the weight. The camera (in the present experiment, a Pentazet type B running a t 2000 frames per second) is focused on a vertical plane containing the nozzle axis; the field of view is about 10 cm. in the horizontal direction, the nozzle tip being a t one extreme. Frame-by-frame analysis of the film strip allows measurement of the geometry of the jet of liquid issuing from the nozzle as a function of time. I n particular, the total jet length, [ ( t ) , and the vertical diameter of the jet, D(x, t ) , can be measured, x being the distance from the jet tip, and t the time. If the cross section of the jet is assumed to be circular, its area S(x, t ) is also immediately calculated. These data, as shown below, allow calculation of the internal stresses prevailing during the phenomenon.
4"
k
80.000
60,000
40.000 20,000
Analysis of Data The flow phenomenon which takes place during the experiment described above should be analyzed in some detail. Consider a mass, p W , of material, the first which issues from the nozzle starting a t time zero, the instant of impact. The details of the deformation which it undergoes in exiting from the nozzle are extremely complex and depend on the geometry of the nozzle as well as on the dynamic details of the flow within the syringe. Thus, there is little hope of obtaining any meaningful information from whatever geometrical data are obtained from the film strip concerning the time interval over which the mass under examination is expelled from the nozzle. Once the mass considered has been expelled, it will travel at some speed while the internal stresses within it relax. This phenomenon takes place in an interval of time which is of the order of the natural time of the material considered, about 0.001 second. During this time, new material will be emerging from the nozzle, so that the jet tip is expected to travel a t essentially constant speed, while the total jet volume is increasing, because of the finite flow rate through the nozzle. After some time, the syringe piston will come to a stop, because of internal friction and other dissipating causes. At this point, the flow rate through the nozzle becomes zero, but the jet tip is still traveling a t a high velocity. Thus, the jet tends to become longer and thinner, a process which, in the case of nonelastic liquids, eventually results in breakage into minute drops, owing to instability of the jet surface. T h e time interval over which the flow rate through the nozzle is finite can be approximately evaluated from Figure 2 as being of the order of 0.01 second when Uo, the jet tip velocity, is constant. At the startup of the jet elongation phenomenon, the region of the jet tip is presumably free of stresses, because stresses due to passage through the nozzle relax in about 0.001 second. Thus, elongation of the jet tip can be regarded as a suddenly accelerated flow, with reasonably simple kinematics-i.e., one-dimensional elongation along the jet axis. Consider a collection of material points extending backward from the jet tip, the total mass of which is p W (Figure 3) ; if the fluid is regarded as incompressible, volume H/ is constant with time. With the additional hypothesis that the axial-velocity distribution along any cross section of the jet is flat, volume W is contained b-tween the jet tip and a section located a t a distance Z(t) from the tip: if S(x, t ) is the cross section a t time t a t a distance x from the jet tip,
w = ~ ' ' ' ' S ( X , t ) dx and
172
dU7 - = dt
o
I&EC FUNDAMENTALS
0
0.01
0.02
t , sec Figure 2.
Experimental results
A 1Figure 3.
'(t)
--
Definition of variables
Equation 2 is the condition which makes the right side of Equation 1 a variational integral and thus determines the Z ( t )function. Substitution of Equation 1 into 2 yields
(3) The velocity U(x,t ) is obtained by similar reasoning as
where Uo is the velocity of the jet tip, which is an easily measurable quantity. The total momentum of the mass p W is
M(t) =
s,"'"
U ( X t)S(x, , t)dx
(5)
Substitution of Equation 4 into 5 gives
(11
+
A momentum balance over the time interval from t to t dt yields, if the drag of the surrounding medium on the jet surface is neglected,
r ( t ) S ( Z ,t ) =
M - ddt
(7)
where ~ ( t is) the tensile stress in the axial direction a t the cross section located a t x = 2. No convective momentum flux term appears in Equation 7, because the balance is made on a collection of material points,. T h e 7 ( t ) function is the tensile stress acting across a surface embedded in the material consideredLe., a convected surfacle. Substitution of Equation 6 into 7, and algebraic manipulation gives
Equation 8 can, in prhciple, be used for the evaluation of ~ ( t ) from the photographic data; in fact, the evaluation of the right side is a very tedious operation. Fortunately, a considerable simplification arises if the jet is cylindrical in the tip region during the high deceleration region of Figure 2. In fact, if cross section S is not a function of x a t x < Z , Equation 8 reduces to
last two terms on the right side of Equation 9 amount to no more than 1% of the first term and may be neglected. With this simplification, the deceleration curve in Figure 2 may be read directly on the stress axis; normal stresses as large as 70,400 dynes per sq. cm. are observed. Consideration of Equation 10 shows that, if tensile stresses were attributed to the prevailing shear rates, an elongational viscosity of a t least 7000 poises would be required. This is an exceedingly large value, particularly a t rather small shear rates; the elongational viscosity of elastic materials increases with increasing shear rates (Astarita, 1967), but is substantially equal to the viscometric viscosity a t shear rates much lower than the inverse of the natural time. Therefore, as could be expected for viscoelastic materials, the stresses observed should not be attributed to the deformation rate, but to the total deformation reached in a very short time interval. Accordingly, this technique could be used to infer the rheological behavior of viscoelastic materials subjected to sudden deformations. The technique described here needs refinement, but the results so far obtained encourage work in this direction as likely to yield important information on the behavior of elastic materials in suddenly accelerated flows. Ac know ledgrnent
where D ( t ) is the diameter of the cylindrical region of volume W. T h e right side of Equation 9 is easily evaluated from a frame-by-frame analysis of the high deceleration region of Figure 2. A detailed analysis has been carried out for the frames from time 0.008 to 0.022 second. T h e volume, W ,chosen was 0.131 cc., corresponding to a length Z of 0.64 cm. a t 0.008 second. During the 0.014 second of detailed analysis, no appreciable change of D was observed; this means that the change does not exceed 10% (a 10% variation in jet diameter is easily observed). Thus, the following inequality can be established.
T h e second derivative tF In Dldtz is more difficult to evaluate; yet, taking into account Equation 10, and considering that d In D/dt presumably does not change sign in the interval of time considered, (11)
Measured values of dC'o/dt are plotted in Figure 2 ; values as large as 110,000 cm. per sec.l have been observed. Thus, the
The authors acknowledge the help of G. Greco in setting u p the photographic technique. Nomenclature
D = jet diameter, cm. M = total momentum of mass p W , dyne sec.
s = jet
cross-section area, sq. cm. = time, measured from startup of flow, sec. u = axial velocity, cm. sec.-l uo = jet tip velocity, cm. set.-' PV = volume of mass p W , cc. x = distance from tip, cm. distance from tip enclosing mass pW, cm. E = auxiliary integration variable, cm. P = density, grams/cc. r = tensile stress, dyne cm.-2 t
z =
literature Cited
Astarita, G., IND.ENG.CHEWFUNDAMENTALS 6, 257 (1967). Metzner, A. B., White, J. L., Denn, M. M., A.I.Ch.E. J. 12, 863 (1966). Shertzer, C. R., Ph.D. thesis, University of Delaware, Newark, Del., 1965. for review May 15, 1367 RECEIVED ACCEPTED September 8, 1967
Work supported by the Consiglio Nazionale delle Ricerche.
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