dripping faucet originalpaper

8/13/2019 Dripping Faucet OriginalPaper

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International Journal of Theoretical Physics Vol. 35. No. 5 1996

ripping FaucetA . D I n n o c e n z o t a n d L . R e n n a t

Recei ved July 6 1995

A model for the simulation of the chaotic behavior of a leaky faucet is analyzed.It is fou nd that the mechanism Of simulating the breaking away of the drop iscrucial in order to ob tain the transition to chaos. Return maps and dripping spectraas functions of flow rate and critical parameters are obtained.

1 . I N T R O D U C T I O NDuring the l as t deca de , fo l low ing R0ss le r s (1977) sugges t ion , ma ny

au t ho r s have dem ons t r a t ed expe r i m e n t a l l y t ha t a d r ipp i ng w a t e r f auce t m i gh texhibi t a chaot ic t ransi t ion as the f low rate i s varied (Marten e t a l . 1985;Ntlfiez Y6pez e t a l . 1989; Wu e t a l . 1989; Wu and Schei ly , 1989; Cahalane t a l . 1990; Dre yer and Hicke y , 1991) . As fa r as we know, mathem at i ca lmodel s tha t s imula te care fu l ly th i s behavior have not been repor t ed in thel i t e ra ture . However , severa l years ago , Mar t i en e t a l . (1985) were ab le tos imula te , by us ing a model of a s imple one-d imensiona l nonl inear osc i l l a tor ,som e o f t he s i m p l e r beha v i o r o f the l eaky f auce t, w i t h goo d qua l i t at i veag reem en t . Thes e au t ho r s d i d no t m ake a sys t em a t ica l exhaus t i ve exp l o ra t iono f t he dependence o f t he m ode l on t he pa ram e t e r s and c l a i m ed t o ob t a i nreturn maps that only in a qual i ta t ive sense, and in l imi ted regions of theparameter space , a re s imi l a r to those produced exper imenta l ly by the fauce t .Fur the rmor e , t he mech ani sm by which the in it ia l condi t ions for the dropform at ion were res tore d was no t suf f i c ien t ly expla ined . Mo re recent ly , Bern-hard t (1991) descr ibed a s imple e l ec t ron ic c i rcu i t t ha t reproduces the typeo f ape r i od i c behav i o r t ha t m ay be found i n m any phys i ca l sy s t em s , such a sdr ipp ing fauce t s , magnetospher i c subs torms, e t c . In th i s paper i t i s demon-s t ra ted tha t t he on ly non l inear i ty req ui red to y i e ld chaos in re l axa t ion osc i l l a tor~Departimento di Fisica dell Universith, 73100 Lecce, Italy, and 1NFN Sezione di Lecce,Lecce, Italy.

9410020 774819610500 0941509.5010 996 Plenum Publishing Corporation


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9 4 2 D I n n o c e n z o a n d R e n n a

systems is a sudden change in behavior when a threshold is reached. Bern-hardt s result seems to suggest that the mechanism of formation of the dropat the threshold is crucial in order to yield chaos in mathematical models ofa dripping faucet. In order to clarify this point, we have considered the modelproposed in Martien e t a l . 1 9 8 5 ) and we analyzed the effects introduced bydifferent mechanisms of release of the drop at threshold. The amount ofwater which is released into the falling drop can depend on several physicalquantities at the threshold, such as the speed, the mass, or the momentumof the forming drop, the time of formation of the drop, etc. Moreover, as thedrop leaves the faucet, it creates oscillations in the residue (which set theinitial conditions for the following drop), affecting the time of release of thenext drop, so that the successive drops are causally related. This correlationbetween successive drops can be built up into a mathematical model in severalways: one can, for example, set the initial conditions for the following dropat the nozzle, or simulate a sudden (nonzero) reduction of the residue position.Finally, a change of the threshold can lead to further significant modificationsin the results of the model.

In this paper we analyze only some of the various modalities and shapesof the formation of a drop. Our aim is to investigate a path that should leadto the establishment of the deterministic equations describing the system. Indoing this we limit ourselves mainly to the study of the dependence of themodel on the p a r a m e t e r s o f th e t h r es h o ld : we think in fact that, owing tothe large variety of the parameters and the richness of the chaotic patternsthat can be obtained, the theoretical model can be improved by exploringmostly the mechanism of threshold that, in a certain sense, is with lessevidence connected to the physical constraints of the process. We believethat an investigation of the influence of the cr i t i ca l parameters of the modeland the structure of the drop formation can furnish suggestions for the estab-lishment of the mathematical equations describing the formation and succes-sive breaking of the drops.

In Section 2, the mathematical model is explained and in Section 3bifurcation and return map drawings obtained with fixed values of criticalparameters and initial conditions are shown. In Section 4, a larger variationof parameters is performed and effects of surface tension and temperatureare analyzed. Finally, conclusions are presented in Section 5.

2. MATHEMATICAL MODELWe start from the simple mechanical model for the dripping faucetproposed in Martien e t a l . (1985). This model consists of a mass M which

grows linearly with time, pulling on a spring with stretch constant k. Thespring produces a linear restoring force equal to -k_x, where x is the position


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D r i p p i n g F a u c e t 9 4 3

o f t h e c e n t e r o f m a s s o f t h e f o r m i n g d r o p a n d t h e s p r i n g c o n s t a n t k r e p r e s e n tss u r f a c e t e n s io n . T h e d a m p i n g o f t h e re s i d u a l o s c i l l a t i o n s i s in c l u d e d in t h ef r i c t i o n f o r c e - b v , w h e r e v = dx./dt is t h e s p e e d o f th e c e n t e r o f m a s s o f t h ef o r m i n g d r o p . T h i s u n i d i m e n s i o n a l m a s s o n a s p r i n g m a y b e d e s c r i b e d b yt h e e q u a t i o n

d M v )M g k x b v 1 )d t

w h e r e g i s g r a v i t a t io n a l a c c e l e r a t i o n a n d t h e m a s s o f th e d r o p g r o w s a t ac o n s t a n t r a t e R ,

d M - R 2 )d tW h e n t h e d o w n w a r d d i s p l a c e m e n t o f t h e w a t e r r e a c h e s a c ri ti c al v a l u e x c,t h e m a s s i s s u d d e n l y r e d u c e d b y A M a n d t h e p o s i t io n o f th e r e m a i n i n g m a s so s c i l la t e s a c c o r d i n g to 1 ) a n d g r o w s a c c o r d i n g t o 2 ) . T h e d r o p m a s s A Mis ta k e n p r o p o r t i o n a l t o th e s p e e d d x / d t o f t h e m a s s a t t h e c r i ti c a l d i s t a n c e x c.

T h e r e i s, in t h is m o d e l , a c e r t a i n a r b i t r a ri n e s s i n t h e c h o i c e o f t h e v a l u e so f t h e p a ra m e t e r s , a n d s o f a r a n e x h a u s t i v e e x p l o r a t i o n o f t h e b e h a v i o r o fe q u a t i o n 1 ) in t e r m s o f t h e s e p a r a m e t e r s h a s n o t b e e n g i v e n . H o w e v e r , o n ec a n n o t i g n o r e t h e p h y s i c s , a n d w e b e l i e v e i t t o b e c o n v e n i e n t t o t a k e a ni n it ia l s e t o f th e p a r a m e t e r s w h i c h i s c l o s e t o t h e p h y s i c a l o n e s . T w o m o r ea s p e c t s o f t h e p r o b l e m s e e m i n t e r e s ti n g : th e w a y b y m e a n s o f w h i c h a ) t h em a s s o f t h e b r e a k i n g - o f f d r o p i s d e f i n e d a n d b ) th e p o s i ti o n a n d s p e e d o ft h e s u c c e s s i v e d r o p i n f o r m a t i o n a r e r e l a t e d t o t h e v a r i a b l e s M a n d v a t x,..

T h e d r o p m a s s A M c a n b e p r o d u c e d i n s e v e r al w a y s ; w e h a v e a n a l y z e dt h e f o l l o w i n g :

i ) A M = etMcv,. 3 )i i) A M = e t V c M a r t ie n e t a l . , 1 9 8 5 )

w h e r e v c a n d M c a r e t h e s p e e d a n d t h e m a s s a t th e t h r e s h o l d a n d e t i s ap a r a m e t e r o f p r o p o r t i o n a l i ty t o b e s u i t a b ly a d j u s t e d . T h e b e h a v i o r o f t h es o l u ti o n s o f e q u a t i o n s t ) a n d 2 ) d e p e n d s o n a t l e as t f o u r in d e p e n d e n tp a r a m e t e r s g , k, b, a n d R . In a d d i t i o n t o t h e se p a r a m e t e r s w e h a v e c o n s i d e r e d ,w i t h i n t h e m e c h a n i s m s o f d r o p f o r m a t i o n s h o w n i n r e l a ti o n s 3 ) , t w o a d d i-t i o n a l p a r a m e t e r s : t h e c o e f f i c i e n t o f p r o p o r t i o n a l i t y et a n d t h e c r it ic a l d i s t a n c exc. I n t h e f o l l o w i n g w e w i l l s h o w t h a t t h e v a l u e s o f t h e la s t tw o p a r a m e t e r sa r e c r u c i a l i n o r d e r to o b t a i n t r a n s i t i o n t o c h a o s . I n d o i n g t h i s w e w i l l m a i n t a i nc o n s t a n t , a l m o s t e v e r y w h e r e , a ll o t h e r p a r a m e t e r s e x c e p t , e v e n tu a l l y , R ) .

A s r e g a r d s t h e r e l a t i o n w i t h t h e i n i t i a l c o n d i t i o n s f o r t h e r e s i d u e m a s sm = M c - A M , w h e n a d r o p f a ll s , o n e c a n a t t e m p t to p l a c e t he re s i d u e a t


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the nozzle, putting x0 = 0, with the speed v0 = vc, of the previous dropat the threshold. Instead, we have considered two different model systemsof the breaking drop at the critical point xc:(a) A spherical drop and a residue point mass o n e - s p h e r e ) .

(b) Two spherical drops, one falling off, the other forming a residuefor the successive drop t w o - s p h e r e ) .

Figure la shows the system of point mass and sphere; the system centerof mass is at x c ; the initial position for the residue mass is given by

AMx0 = x ~ , - r (4)Mwhere r = (3AM/4rrp) u3 and p is the liquid density. Figure lb shows thetwo-sphere model; with the center of mass at xc we obtain for the positionof the residue the relation

MXo = x,. - (rl + r 2 ) - - (5)M .where r l.2 = 3 A M i . 2 1 4 ~ rp ) u3 a n d M I = A M , M 2 =- m = M c - A M .

By setting in both models v0 = Vc the total momentum P is unaffectedby the breaking. In the following section, by choosing values of theparameters g, k, and b near the physical ones, corresponding to a standardexperimental apparatus, we give an exhaustive study of the two modalitiesof the drop formation (3) with the previously suggested breaking shapeof Fig. la. Work is in progress in order to extend the analysis to thebreaking shape of Fig. tb (D'Innocenzo and Renna, n.d.); a preliminaryresult is given in the following.

3. NUMERICAL SIMULATIONSWe have set the initial values of the parameters with reference to the

physical properties of the drops.As the surface tension represents a force per length, we have, for a

w a t e r drop, a value of k roughly equal to 500 dynes/cm, while the frictionforce depends on the liquid viscosity Xl and on the eyedropper characteristics.

A preliminary test was performed, on the basis of which we have setthroughout k = 475 dynes/cm, b = 1 sec - I , m = 0.01 g, and v0 = 0.10crn/sec. This choice is justified by the necessity of limiting the number ofvariables of the model: however, one must keep in mind that there exist othervalues of these parameters for which a sensible variation of the solutions canbe obtained. For example, intermittencies with period doubling of a period-


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Dripping Faucet 945

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Fig . 1. M echanism o f drop breaking at the threshold: a) one-sphere and b) two-sphere models.

3 a tt r a c t o r o r c h a o t i c p a t t e r n s a r e o b t a i n e d f o r m = 0 . 1 0 g . I n S e c t i o n 4 al a r g e r v a r ia t i o n o f p a r a m e t e r v a l u e s i s c a r r i e d o u t .

S u b s t i tu t i n g t h e s o l u t i o nM t) = m + Rt 6 )


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of equa t ion (2 ) into equa t ion (1 ) and in t roduc ing the sp eed v = d x / d t , w et ra n s fo rm th e se e q u a t io n s i n to t h e e q u iv a l e n t sy s t e m

d r- - = v 7 )d tdUd t = M g - k x - R + b ) v

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F i g . 4 . D r i p p i n g p a t t e r n s (7 ,+ 1 v e r s u s T , ) f o r A M ~ M v < . . T h e r a n g e s f o r o r d i n a t e s a n da b s c i s s a e a r e t h e s a m e ( u n i t s, s e c o n d s ) . F l o w r a te ( m l / s e c ) : ( a) 0 . 8 8 5 , ( b ) 0 . 9 5 , ( c ) 1 .0 5.


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Dripping Faucet 9 9

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0 3 0 3 5 0 4 0 4 5 0 5 0 5 5 0 6TnFig 4 Continued

wh o se n u m e r i c a l so lu t io n s a r e o b ta in e d b y m e a n s o f a s ta n d a rd v a r i a b l e - st e pfo u r th -o rd e r Ru n g e -K u t t a me th o d in th e m o d i f i c a t io n d u e to G i l l ( s e e Bu tc h er ,1987) . Equa t ions (7) are in tegra ted u nt i l x ex ce ed s xc: thus the in tegra t ion isp e r fo rme d b a c k to x c b y u s in g th e H6 n o n (1 9 8 2 ) me th o d fo r t h e n u me r i c a lc o mp u ta t io n o f P o in c a r6 ma p s . I n o u r c a l c u l a t i o n s we h a v e v e r i f i e d t h a t i nany case the f lu id f low ra te i s a lways conse rved .

Tab le I g ives the va lues o f the pa ramete rs tha t we have kep t f ixed . Theexper im en ta l beh av io r o f a d r ipp ing fauc e t dur ing the trans it ion to chaos hasb e e n e x a min e d b y s e v e ra l a u th o r s (M a r t i e n e t a l . 1985; Ndfiez Y6pez e t a l .1989; Wu e t a l . 1989 ; Wu and Sche l ly , 1989 ; Caha lan e t a l . 1 9 9 0 ; Dre y e rand Hickey , 1991) . The exper im en ts invo lve meas urem ent o f time in terva lsbe tw een suc cess ive d rips . For each cons tan t f low ra te , the da ta a re rep resen tedin a t ime de lay d iagram ( t ,+ l ve rsus t , ) o r d i sc re te map .


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T h e e x p e r i m e n t a l i n v e st ig a t io n s c a n b e su m m a r i z e d a s f o l lo w s . A t lo wf l o w r a t e , d r i p p i n g i s f o u n d t o b e p e r i o d i c ( p e r i o d - 1 a n d p e r i o d - 2 a t t r a c to r s ) .A s t h e f l o w r a t e is v a r i e d a n d s e t t o i n c r e a s i n g l y l a r g e r v a l u e s , a p e r i o d -d o u b l i n g s e q u e n c e l e a d in g , a b o v e a c r i ti c a l f l o w r a te , to c h a o s a p p e a r s ; t h es y s t e m e x h i b i t s a b r o a d r a n g e o f d y n a m i c a l b e h a v io r , w i th m a n y e x a m p l e so f st r a n g e a t t ra c to r s . E x a m p l e s o f an i n t e r m i t te n t r o u t e t o c h a o s , w i th p e r i o d -3 a n d p e r i o d - 4 a t t ra c t o rs , a r e a l s o g i v e n ( D r e y e r a n d H i c k e y , 1 9 9 1) . V a r y in gt h e s u r f a c e t e n s i o n d r a m a t i c a l l y c h a n g e s t h e d y n a m i c s ( W u a n d S c h e l l y ,1 9 8 9 ) .

A s i n b o t h p r e v i o u s t h e o r e ti c a l a n d e x p e r i m e n t a l s t u d ie s , w e h a v ef o c u s e d o u r a t t e n t io n o n t h e t im e i n t e r v a l t,, b e t w e e n s u c c e s s i v e d r i p s F i g u r e2 s h o w s t h e d r ip s p e c t r a o f w a t e r v e r s u s R , w i th A M g i v e n in F i g . 2 a b yf o r m u l a ( i) a n d F i g . 2 b f o r m u l a ( ii ) o f ( 3 ) ( th e f i r st 1 5 0 o f 2 0 0 d r o p s


11/33

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rate: a) 0.90, b) 0.94 , c) 1.0,

a r e r e m o v e d ) . T h e s e a r e c l e a r d i a g r a m s o f b i fu r c a t io n s : in F i g 2 a a tR --~ 0 . 6 1 m l / s e c t h e r e i s a b i f u r c a t i o n f r o m a n a t t r a c t o r o f p e r i o d 1 t o a na t t r a c t o r o f p e r i o d 2 . A s R i s i n c r e a s e d , s u c c e s s i v e b i f u r c a t i o n s o c c u r u n ti ls t r a n g e p e r i o d i c a t t r a c t o r s a p p e a r . A n a l o g o u s r e s u l t s a r e o b t a i n e d i n F i g 2 b ;u p t o R = 0 . 8 2 5 m l / s e c t h e b e h a v i o r i s t h e s a m e a s in F ig . 2 a , th e n a b i p e r i o d i cd r i p p i n g l e a d i n g to c h a o s a p p e a r s . W e h a v e u s e d fo r m o d e l i ) a = 0 . 2 5c m - 1 s e c , f o r m o d e l i i) c~ = 0 . 0 2 5 g c m - ~ s e c - l a n d x~ = O . 1 9 cm fo r b o t h .

T h e e n l a r g e m e n t o f th e s p e c t r a i n F ig . 3 r e v e a l s t he f l o w r a t e s a t w h i c hc h a r a c t e r i s t i c b e h a v i o r s o c c u r : f o r in s t a n c e , i n F i g . 3 a t r a n si t io n s t o d i f f e r e n ta t t r a c t o r s a r e e a s i l y s e e n ; b i s t a b i l i t y c a n b e o b s e r v e d i n F i g . 3 b a r o u n d f l o wr a te s ra n g i n g u p t o 2 0 . 9 2 m l / s e c . F ig u r e s 4 a n d 5 s h o w t i m e d e la y d i a g r a m sa t t h r e e s e l e c t e d f l o w r a t e s ; e a c h o f t h e s e d i a g r a m s c o n t a i n s 5 X 103 p o i n t s


12/33


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T nF i g . 5 . C o n t i n u e d .

( t h e f ir s t 1 0 p o i n t s a r e r e m o v e d ) . I t c a n b e o b s e r v e d in F i g . 4 t h a t t h ec o m p l e x i t y o f t he a tt r a c to r g r o w s w i t h i n c r e a s in g R ; a s h a rp c h a n g e o c c u r sa t R = 1 .0 2 5 m l / s e c ( s e e a l s o F i g . 3 a ). A n a n a l o g o u s c h a n g e o c c u r s f o r F ig .3 b at R = 0 . 9 3 m l /s e c . A s i t c a n b e s e e n f r o m F i g s . 5 a n d 6 w i t h A Mp r o p o r t i o n a l t o t h e s p e e d , t h e a t t r a c t o r s s e e m t o c h a n g e f r o m a p e r i o d - 2 s t a t et o a c h a o t i c s t a te , b u t t h e a n a l y s i s o f t h e d a t a s h o w s t w o c u r i o u s a s p e c t s o ft h e t i m e o f r e l e a s e . F ir s t, o n e c a n o b s e r v e t h e r e s e m b l a n c e o f t h e a t tr a c t o r si n p a r t s ( b ) a n d ( c ) o f F i g s . 4 a n d 5 i n s p i t e o f t h e d i f f e r e n c e o f th e f l o wr a t e s p e c t r a : th i s c a n b e e x p l a i n e d b y o b s e r v i n g t h a t in c a s e ( a ) , ~ = 0 . 2 5c m - 1 s e c , w h i l e i n c a s e ( b ) , e~ = 0 . 0 2 5 g c m - I s e c ; t h i s s i m i l a r i t y c a n b eu n d e r s t o o d s i nc e a t th e t h r e sh o l d b o t h c a l c u l a t e d m a s s e s a r e o f th e o r d e r o fM c -~ 0 .1 g . T h e s e c o n d a s p e c t i s e v i d e n t i n F ig . 6 , w h e r e t i m e s e r i e s d i a g r a m so f F ig s . 5 a a n d 5 b a r e s h o w n : in b o t h d i a g r a m s , a f t e r a l o n g t r a n s i to r y c h a o t i c


13/33


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. d - ? % , ~ . , . . . @ , I d -~ , : . . . ~ . ~ . ~ ' i , ' ~ , . . . . t ~ . l- . , , : : . . , , ; . ~ , , ~ , ~ , : . . ' . : ~ , ; . . , 4 , ~ : -

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F i g . 5 . C o n t i n u e d ,

r h y t h m , t h e t i m e i n t e r v a l s a l t e r n a t e , e x h i b i t i n g b i p e r i o d i n g d r i p p i n g : t h i sh a p p e n s a f t e r 2 5 0 d r o p s i n c a s e ( a) , b u t a f te r 2 2 0 0 d r o p s i n c a s e ( b ) ; th u st h is e f f e c t i s n o t e v i d e n c e d b y t h e d r ip p i n g s p e c t r a o f F ig s . 2 b a n d 3 b , w h i c hc o n t a i n 5 0 d r o p s . T h i s b e h a v i o r a p p e a r s a t R = 0 . 9 5 m l / s e c . A n a n a l o g o u se f f e c t h a p p e n s f o r c a s e ( i) a t R = 0 . 9 3 m l / s e c ; h o w e v e r , in th i s c a s e a c h a o t i cp a t t e rn t r a n s f o r m s a f t e r 8 0 0 d r i p s t o a m u l t i p e r i o d i c p a t t e r n . A s f a r a s w ek n o w , t h e s e p h e n o m e n a a r e n o t r e p o r t e d i n t h e e x p e r i m e n t a l s t u d i e s .

W e f i n i s h t h i s s e c t i o n b y s h o w i n g i n F i g . 7 t h e d r i p s p e c t r a a t R - - 0 . 9m l / s e c a s a f u n c t i o n o f th e c r i t ic a l d i s t a n c e xc. T h e s e d i a g r a m s a r e u s e f u l i nt h a t t h e y s u g g e s t a s t ro n g d e p e n d e n c e o f t he m o d e l o n th i s p a r a m e t e r ; t h es p e c t r a c a l c u l a t e d a t d i f f e r e n t f l o w r a t e s c a n f u r n i s h u s e f u l i n d i c a t i o n s i no r d e r t o o b t a i n d i f f e r e n t f o r m s f o r t h e c h a o t i c a t t r a c t o r . O b s e r v i n g F i g . 7 b ,o n e c a n s e e a n a n o m a l o u s d e c r e a s e o f t h e d r i p p i n g t i m e i n t e rv a l s as t hec r i ti c a l d i s t a n c e x c g r o w s i n t h e c a s e w h e r e t h e d r o p m a s s A M i s t a k e n


14/33

9 5 4 D ' l n n o c e n z o a n d R e n n a

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1 IFig . 6 . T ime in te rva l versus d rop number fo r F igs 5a and 5b .


15/33

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16/33

9 56 D I n n o c e n z o and enna

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::? i..:.:?::L.::[email protected].':~ ...i i:ii.i.:i..-:.i:.::::.:::::L~I...i:i.i:.i:L".-::.~.. : . '.' . - . ..'-.....:.......'.......... -- :.?. .'..':. ..- . .. .--. ..-.',...'~...'.... .....'.~.... . ....' .. . .

i . . . . ~ " " ~" i ~: f i~" :~" '~ ; # ' ; " " '~: i - : ' i i ' i ' ( " , : , . i , , . .4" ~ . . . . . " ; ;. ' ." ~ ' :

.0 3 l , , , , l , , , , I ,.. , ~ , l ~ , I , , , , l0 1000 2000 3000 4000 5000

Fig. 8. Time series diagram for the drop interval of Fig. 4a. (b-e): Time delay diagrams ofthe four at t ractors of the t ime series in (a) . (f) Sixty successive drop intervals connected withstraight l ines for enhanced visual effect .

proportional to the drop speed v. This argument seems to privilege the modeli) over ii) of relation 3).Moreover, all these results suggest that a correct model of a dripping

faucet must contain a certain dependence among a, xc, and R.

4. ATTRACTORSFigure 8 shows the case illustrated in Fig. 4a in greater detail: the time

series diagrams show four patterns that evolve in the chaotic attractors ofFigs. 4b and 4c. In the time delay diagrams of Figs 8b-8e the four attractorsof the time series of Fig. 8a are shown separately. The shape of Fig. 8d canbe obtained from the shape of Fig. 8c by a double reflection about the timeaxes. In Fig. 8f 60 successive drop intervals from the middle of the timeseries of Fig. 8a are connected with straight lines for enhanced visual effectin order to see more clearly the alternate periodic behavior of the droptime intervals.Some interesting maps are obtained upon varying both x . and c~. Figure9 shows some attractors for case i) of 3); these attractors are very similar


17/33


b

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T nFig. 8. Continued.

I

t o e x p e r im e n t a l a t t ra c to r s . T h e d e p e n d e n c e o n t h e s e p a r a m e t e r s is e v i d e n c e do n c o m p a r i n g t h e m a p o f F i g . 9 b w i t h t h a t o f F i g . 9 d , w h o s e a t t r a c t o r s t r u c tu r ei s q u i t e d i f f e r e n t .

T h e e f f e c t o f c h a n g i n g t h e l i q u id s u r f a c e t e n s i o n h a s b e e n a l s o a n a l y z e d ;r e s u l t s a r e r e p o r t e d i n F i g . 1 0 f o r a f l o w r a t e o f 0 .9 5 m l / s e c . I t c a n b e n o t e dt h a t t h e st r a n g e a t t r a c t o r o f F i g. 4 b b e c o m e s b i p e r i o d i c w i t h d e c r e a s i n gs u r f a c e t e n s io n ( k = 4 5 0 d y n e s / c m ) , w h i l e i t e v o l v e s i n to a n a t t ra c t o r o fm o r e c o m p l e x s t r u c tu r e f o r k = 5 0 0 d y n e s / c m .

F i n al ly , w e s h o w s o m e d e l a y d i a g ra m s a n d t h e c o r r e s p o n d i n g t i m e s e r ie sf o r t h e t w o - s p h e r e m o d e l o f F i g . l b . I n F i g . 11 a n a t t r a c t o r - t y p e d i n o s a u r i so b t a i n e d w i t h d r o p m a s s A M ~ M ov e w h i le in F ig . 1 2, t ak in g AM cc v c, w es h o w a c l o s e d d i s c r e t e a t t r a c t o r w i t h a s u g g e s t i v e r e g u l a r i t y i n t h e t i m es e r i e s b e h a v i o r .


18/33

9 5 8 D l n n o c e n z o a n d R e n n a

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t1 t I I I t t i I l t f I t f

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F i g 8 C o n t i n u e d

T h u s w i t h t h e p r e s e n t m o d e l s e v e r a l f e a t u r e s o f t h e d r i p p i n g f a u c e tb e h a v i o r c a n b e o b s e r v e d a s c l o s e d - l o o p p a t t e r n s a n d p e r i o d i c o r s t r a n g ea t t r a c t o r s .

5 . C O N C L U S I O N ST h e m o d e l o f a d ri p p i n g f a u c e t w e h a v e p r e s e n t e d i n th i s p a p e r r e p r o -

d u c e s m a n y o f th e e x p e r i m e n t a l b e h a v i o r s . I n a d di ti o n it c o n t a i n s a d y n a m i c sw h i c h r e v e a l s m a n y i n t e r e s t i n g f e a t u r e s . F u r t h e r i n v e s t i g a t i o n s c a n b e p e r -f o r m e d i n o r d e r t o u n d e r s t a n d t h e n a t u r e o f v a r i o u s a t t r a c t o r s a n d t h e e f f e c to f th o s e p a r a m e t e r s w h i c h a r e n o t v a r i e d in o u r a n a ly s i s . T h i s m a t h e m a t i c a lm o d e l d i s p l a y s in f a c t v a r i o u s f o r m s o f c h a o s a n d t h u s l e n d s i t s e l f w e l l t o


19/33

Dripping Faucet 959d

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T nFig. 8. Continued.

. 0 4 1 . 0 4 2

t h e re p r e s e n t a ti o n o f t y p i c a l p h e n o m e n a o f n o n l i n e a r p h y s i cs . T h e r e s u l ts o fo u r w o r k c a n b e s u m m a r i z e d a s f o l l o w s :

i ) T h e d i f f e r e n t i a l e qu a t i ons 1 ) a nd 2 ) , w i t h one p r e f i x e d m e c h a n i s mf o r the r e l e a s e o f t he d r op , s e e m t o u s una b l e t o r e p r o du c e a ll the c ha r a c t e r i s t icp a t te r n s o f t h e d r i p p i n g f a u c e t .

i i ) The relevant n o n l i n e a r i t y r e q u i r e d t o y i e l d c h a o s i s g i v e n b y t h esudden changes a t t he t h r e s ho l d .

i ii ) A c o ns i s t e n t de f i n i t i on o f threshold i s ne c e s s a r y i n o r de r t o g i veun i c i t y t o t he m ode l .i v ) T h e m e c h a n i s m o f r e le a s e o f th e d r o p w i th a m a s s p r o p o r t i o n a l t om o m e n t u m s e e m s t o b e m o r e r e a l i s t i c .

v ) link between the threshold parameters i s ne c e s s a r y i n o r de r t oo b t a i n a m o r e c o m p l e t e r e p r o d u c t i o n o f e x p e r i m e n t a l r e s u lt s .

W e t h i nk t ha t f u r t he r s t ud i e s a l ong t he a bove l i ne s c a n l e a d t o a goodr e f in e m e n t o f t h e m o d e l.


20/33

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21/33


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nF i g . 8 . C o n t i n u e d .


22/33


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T nF i g . 9 . E x a m p l e s o f s t r a n g e a t t r a c t o r s f o r A M ~ M ~ . v . . T h e v a l u e s o f eL ( u n i t s c m - t s c c )R ( m l / s e c ) , x ,. ( c m ) a r e , r e s p e c t i v e l y : ( a ) o~ = 0 . 2 , R = 1 . 3 , x ,. - - 0 . 1 5 ; ( b ) o~ - - 0 . 3 , R = 0 . 4 ,x . = 0 . 1 5 ; ( c ) ~ = 0 . 4 , R = 0 . 6 , x c = 0 . 2 5 ; ( d ) ~ = 0 . 5 , R = 0 . 3 , x c = 0 . 3 ; (e ) ~ t = 0 . 6 ,R = 0 . 6 , x c = 0 . 2 5 .


23/33

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24/33

964 D Innoc enzo and Renna

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25/33


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27/33

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28/33

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Dripping Faucet 969

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~ ' ~ ' IP . k . . . - . ~'~ ,.,~ '-. ~ . * . . . - ~ .ioo . . . . . . .~ ' ' . ,, .: ~ . ' . 4 ' : ' ~ = ' : ' / . ' ; - ~ , k . ' ~ ' ~. ' . . Y ,~- :: . . ' 4... ' . ~: : ' ~ . ~

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T nFig . 11 . (a) Discre te map and (b) t ime ser ies d iagr am for the two-sphere model wi th AMM v c. Value s of the pa ra me te rs are (~ = l, R = 0.9, & = 0.25. Unit s are as in the previ ousfigures for the case AM oc M .v.


30/33


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. ' . : : ' .~ . ': . : : : ' : ~ . . . - : .- , . ; - . . ' ' , . ' . . . L ' ' ' : : ' . ' , ' ' ~ ' : ' ; . . , ' l ' : . . ' : .' * * ' . , - . : : ' . ' . ' .- . . . , . ~ ' . . . . - . ': .. ': .. . ' ' : ' - , ' Y ' , . - t - , ; . . : , . - . . . . . . . . : . . . . : ' . . : - . . . . :

, . '. ' , : . , . : ' . k ~ - , . : . : : . . . . : ' : , ' . . ' : . . . . . . ' . . . , . . ' .-' ' . ' = ' - ' : , : : . ' . ' : ' ~ . ~ . . . . . ' ~ J . ' t : : : . . ~ : , ~ . . . 2 ' ~ . . . - ' . ' ~ .

I i ~ i i t ~ t I t ~ I ~ ~ t I0 1 0 0 0 ~ 0 0 0 ~ 0 0 00 0 0 3 0 0 0

nF i g . 11 . C o n t i n u e d .


31/33


I

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Fig, 12. a) Discrete map and b) time series diagram fo r the two-s phere model with AMvc. Values of the parame ters are cL = 0.008, R = 0.9, x,_ = 0.3. Units are as in the previo usfigures for the case AM o: v,..


32/33

972b0 3 5

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i I I I I ~ I ~ i ~0 1 0 00 2 0 0 0 3 0 0 0 4 0 0 0 5 0 0 0Fig. 12. Continued.

N O T E A D D E D I N P R O O F SA f t e r s u b m i s s i o n o f t h i s p a p e r f o r p u b l i c a t i o n , t h e a u t h o r s b e c a m e

a c q u a i n t e d w i t h a n a r t i c l e b y J . C . S a r t o r e l l i , W . M . G o n ~ a l v e s , a n d R . D .P i n t o P h y s i c a l R e v i e w E , 4 9 1 9 9 4 ) , 3 9 6 3 ) , i n w h i c h e x p e r im e n t a l e v i d e n c e so f s u d d e n c h a n g e s f r o m c h a o t i c t o p e ri o d ic r e g i m e s o f a d r ip p i n g f a u c et h a v eb e e n r e p o r t e d . T h e s e r e s u l t s a r e v e r y s i m i l a r t o t h o s e s h o w n i n F i g . 6 a a n d6 b o f t h i s p a p e r a n d a r e c o n s i d e r e d b y t h e a u t h o r s a s s t r o n g e n c o u r a g e m e n tt o c o n t i n u e t h e s t u d y o f t h e s e p h e n o m e n a .

R E F E R E N C E SBernhardt, P. A. (1991). Physica D 5 2 , 4 8 9 .Butcher, J . C. (1987) . The Numeri cal Analysis o f Ordinary. Differential Equations Wiley, New

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