Mode locking for an externally excited droplet

Computers Math. Applic. Vol. 33, No. 11, pp. 21-33, 1997

Pergamon

Copyright(~)1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0898-1221/97 $17.00 -{-0.00 PII: S0898-1221 (97)00085-0

M o d e Locking for an Externally Excited Droplet R . RAVINDRAN Department of Mathematics, Indian Institute of Science Bangalore 560 012, India renrav©math, iisc. ernet, in

S. SUNDAR Department of Mathematics, Indian Institute of Technology Kharagpur 721 302, India (Received January 1996; revised and accepted March 1997)

A b s t r a c t - - W h e n subject to external acoustic excitation, a droplet exhibits behaviour depending crucially on its initial shape and the period of excitation. For shapes close to the spherical droplet, approximate expressions can be derived for the Hamiltonian and the resulting motion is almost periodic. Of special significance is the sudden change in the form of the time-dependent Hamiltonian, when "mode locking" takes place. Due to "nonlinear resonance" the droplet exhibits behaviour completely unrelated to its perturbed state. Keywords--Nonlinear oscillation, Droplet, Hamiltonian function, Resonance.

1. I N T R O D U C T I O N T h e nonlinear oscillation of a droplet subject to an external oscillation has been studied by Mbhring and Ravindran [1]. When subject to external acoustic excitation, the drop displays "strange" behaviour, depending on the initial shape of the drop (see [2]) and the period of excitation. For initial droplet shape close to the spherical one, approximate expressions can be derived for the Hamiltonian. Earlier authors have taken the drop shape to be close to the spherical. Suryanarayana and Bayazitoglu [3] have studied droplets, which are oscillations about a spherical shape. They have developed a general theory for the dynamics of a perturbed spherical droplet, subject to external forces. Wilson [4], while studying the steady thermo-capillary driven motion of a large droplet in a closed tube, has assumed the drop shape to be a large cylinder with hemispherical ends. Uijtewaal, Nijhof and Heethaar [5] have investigated droplet migration, deformation, and orientation in the presence of a plane wall, where the deformation is taken to be D = (L - B ) / ( L + B), where L is the long and B is the short axis of the deformed particle in the plane of symmetry. In this investigation, we have taken the drop shape to be a perturbation of the spherical shape, namely spheroidal. Here it is shown how subharmonic periodic solutions corresponding to the imposed period of excitation can be found using the Melnikov method (see [1]). The Melnikov function is related to the Hamiltonian for droplets, which are almost spherical. Of special significance is the sudden dramatic change in the form of the time-dependent Hamiltonian, as the initial shape of the droplet passes through a critical value. Typeset by ~ 4 ~ T E X 21

22

R. RAVINDRAN AND S. S U N D A R

2. V A R I A T I O N A L

FORMULATION--UNPERTURBED

The Lagrangian was derived for the droplet problem by MShring and Knipfer [6] and modified for the case of a droplet in an external fieldby MShring and Ravindran [l]. Using the translational invariance property for variational problems (Noether's Theorem), we find that if the center of' gravity of the drop is at rest at the origin, then to the order of accuracy considered, namely the first three terms in a series expansion (for details, see [1]).

1. The drop shape is spheroidal (see [2]) r2 z2 a~ 1 + a-~ = 1,

(2.~)

in cylindrical coordinates with a2(t) as unknown. 2. The velocity potential within the drop is given by (up to a constant) ¢ = bs

4

'

(2.2)

with b2(t) as unknown (see [1]). 3. The Lagrangian L for the unperturbed motion is given by L = Asb'2 + g (a2, b2),

(2.3)

where " denotes time derivative 2 s(

1)

(2.4)

A2= T~a2 1-'~2 and

K = a]b~Fl(as) + F2(a2),

(2.5)

where

and __arcsin ,

Fs(a~)

=

2 + -~

as>l,

2,

as = 1,

2vZ_~ arcsinh ,

(2.6)

a2
To fix the units of length and time, the volume of the droplet is taken to be ~ and the ratio of surface tension to density (alp) is taken to be I. A factor r has been absorbed in L, As, and K. The equations resulting from the variational principle are

dA2 dt dbs dt

OK Obs OK OA2 '

(2.7)

with As, b2 as canonical variables and K as the Hamiltonian. In the absence of an external excitation

dK dt

= 0.

(2.8)

M o d e Locking

0.2

23

0.6 0.5

0.15

,.,,..----~,,. ,.

"-"

/"

O,.I

"~ LL

~", ' . "%k

....

~.

o.1

/

/"t

~"

0.05

.,~,

~

K(0)=4.15

'

'

O.B

1

1.2

\~.

.~: ,~"

0.3

',.\..

"k.. "~. '

/

0.1

i

\ '

\... ""

.. ."

i ~

/' /

",°°.

;./.': /-

0.4

0.2

,, \

/ [

63 --~ LL

~.,:~..

•

0

/y,~,,,... ,:.. \...

K - 4.50

X;o. x.'

i \

o

1.4

I

. . . . . . . .

0.5

1

a2

"

1.5

2

a2

1

,..~-,-~ .... z..;" "'~-...

oz -%

!-

-:i'.

o.6

"~ ,~ u_

°*2

". °%

\...""

" ""

E

S:

K - 4.es

0 ................ 0.5 1

k-. ,. . V. '"\-

;

","

t

","X~

,t

"k, \ 1.5

,k.... •

t'

0.5

"~..

0.2

°°

\:.

f-

~

\'""~..

"

l~~ t

v

V. ""

E.l:

7:

"" %

x..

:(:

0.4

z;~::--:. i::\".

1

0

2

2.5

0

a2

\

K , 5.Z0

. . . ...i,. . . . . . . . . . . . . . . . . . 0.5 1 1.5

\",,

\ 2

2.5

3

a2

F i g u r e 1. U n p e r t u r b e d case. C o m p a r i s o n of K - F2(a2) w i t h its a p p r o x i m a t e expression for various K .

O n a n y orbit, K remains constant for all time and equal to its initial value. For a spherical drop K = 4, which is the m i n i m u m value of K . For spheroidal drops K > 4. For a fixed value of K corresponding to b2 = 0, the equation g - F2(a2) = 0

(2.9)

has two positive roots, a2 = a2max ( > 1) and a2 = a2min ( < 1). Substituting 52 = g - F2(a2)

a~fl(a2) in e q u a t i o n (2.7), we have

da2

/K

- F2(a2)

=~-V -~,~-G

(2.10)

24

R. RAVINDRAN AND S, S U N D A R

6

6

5

~"

5

K(0)-4.15.TI~.0.Vp-0.25

4

4

3

3

2

2

/"~

/\

..,,

'.J',./ • ' , ~./. ' \J" t] '\/: ".."- v. . 0

0

5

10

'~.

f.~

. . . . . . . . . . . . . . . . . . .

15

K{0)-4.15.Tp=3.0.Vp=0.25

.,

,.,.

0

20

20

25

30

t

5

,~

1.,.I/,/",/v.,/,; .'... ,, ".... ..'.',,"/,/ ',.' 35

.

40

t

K(0),,,4.25.Tp=,3.0,Vp-0.25 ,'~

5

4

"~

K(0),,,4.25,Tp-3.0,Vp-0.25

4

i\

t

:'~

".,,, i ,.\ i,i, l

li

2

..'", ~ ~

~1

o0 .................. 5 10

15

I "

i~/x

o,." 20

20

~'x

,!!.,

V :.30 .......35

25

t

j

., il

i/40

t Figure 2. Plot of a2(t) for 0 < t < 40, for T~ = 3.

Since K is a constant throughout the orbit, this leads to an expression for the period T of the u n p e r t u r b e d motion T = 2

La2~"~ a2

rain

Fl(a2)

K - F2(a2)"

(2.11)

We can obtain an approximate expression for T as follows. It is observed t h a t for a fixed value of K , the expression K - F2(a2) can be approximated in terms of a new variable y, y = in a2, by the expression

p(/3 - y)(y - oe),

(2.12)

where/3 = lna=max, a = lna2mi,, p = ( K - 4)/lal/3. This holds for a large range of values of K. Using this approximation, we get

2 [~ ~ (e')e'dy T= ~ Ja X/(/3-Y)(Y-a)"

(2.13)

M o d e Locking

10

25

10

8

K(O).5.06,Tp.,4.0,Vp-O.~

8

K{O)-5.05.Tp-4.0.vp~2.5

6

6

4

4 :'~

2'...,-,/\/ \ k:i \ i k/

0

:f " ~

i ",j!

\I

~/ " \ .

i",j/

:",,

\",j]

................... 0 5 10

['\/

2

\',.//

'~,i'

J

15

0

20

I:%\

i/

:i" "\

'\,]"

.

\.;

40

/'\

10

~0)=,5.15,Tp,,,4.0,Vp,,=0.25

8

i

/~, 6

i

K(O)=5.15,Tp=4.0,Vp=,O.25

i

•

6

~ t'M

:

I

i

,,-~

,

4

•

~" •

\

i

4

:~

I

,

;

0

35

:K :

t

10

2>.

I"\

:\j

................... 20 25 30

t

8

f "\•

r. i'~ i . - ' ,~k/ . . ~\/

,. !,

.. ~,/

f'\

~.i. li

~ '

15

0 ................... 20 25 30

;,j

................... 0 5 10

i:'!l l

" \

j

20

t

~:'.. ~, /

\i"\ ~'\/ "\j ,'~ ~' \ 35

40

t Figure 3. Plot of a2(t) for 0 < t < 40, for Tp = 4.

If we use a series expansion in y for the numerator, we can integrate to get

T=~

1+--~+

(3a2+2afl+3fl

3)

+o(aa,f13).

(2.14)

The shape parameter a2(t) is related to b2(t) by equation (2.7), b2

=

1 da2 -a2 dt

--

Substituting in the expression for K, i.e., (2.5) we get that a2(t) satisfies the nonlinear differential equation - dt - i - + 2F1(a2------~\ dt )

+ 2Fl(a2-----~ = 0,

(2.15)

where ~ denotes the differentiation with respect to a2, and a2 satisfies the initial conditions a2 = a2max,

A first integral of this equation is

~t

da2 dt = 0,

= +~/(K

-

at t = 0.

F2(a2))/Fl(a2) which is precisely (2.10).

26

R. RAVINDRAN AND S. S U N D A R

2 1.5

K(01=4.15,Vp=S.0,Vp=0.20] ~

fl 1

0.5

0

.

.

2

4

.

.

6

10

8

t •

~-~x

K(0)=4.15,Tp-4.0,Vp,,020

1.5

% 1 0.5

!

|

0

.

...|

2

4

i

w

r

6

8

10

t • appeox Figure 4. Comparison of a2(t) w i t h its approximate expression for K ( 0 ) ___K(0)crit.

The equation for a2(t) can also be written in the form

d2a2 d .(" K - F2(a2) dr2 - da2 \ 2 F i " ~ )'

(2.16)

i.e., d~ = VV1,

where Vl(a2) = K - F2(a2) 2Fz(a2) In terms of variable y = In a2, y satisfies the differential equation

~Y +(eY~rF~'(e') +1] + 5'(e') dt 2

~-dt ] [2/'1 (eY)

2e2yF1 (e tl)

=0,

(2.17)

(2.1s)

where ' denotes differentiation with respect to y. Approximating the coefficient by polynomials, we have

Mode Locking

27

4.3

4.25

K(0)..4.15,Tp-3.0,Vp-0.1 t: ..

::,.

•, ,',

,~'

~C 4.15 4.1~

V

~,'~ ,

4.05 .

.

.

.

0

•

i

.

.

.

.

i

.

.

.

.

=

20 t

10

,

=

i

,

30

40

appmg

4.3

4.25

K(0)-4.15.Tp-4.0,Vp-O.1

4.2 ~-~ 4.15 4.1 4.05 4

.

.

.

.

0

'

.

.

.

.

I0

'

20

.

.

.

.

'

'

30

'

'

'

40

t ---

•

exact

approx

Figure 5. Comparison of

K(t)

with its approximate expression for K(0) < g(0)crit.

with initial conditions

y = fl,

dy --~ = 0 ,

att=O.

In terms of y, the equation can be written as ~) = VV2,

where V2(y) = e2rj ( K - F2 (eY)) 2F1 (eY)

3. V A R I A T I O N A L EXTERNAL

(2.20)

FORMULATION--WITH EXCITATION

In the presence of an external acoustic excitation, the Lagrangian is modified as

L =

b2A2+

K(a2, b2) - A 2 T ( t ) ,

(3.1)

28

R. RAVINDRAN AND S. S U N D A R

8

8 K(0)=4.15,Vp=0.25.Tp=3

7

"-"

K(0)-4.20,Vp,=0.25,Tp=3

7

6

v

5

6

5

0

10

20

30

40

0

10

t

~-

30

40

t

8

8

7

7

6

E"

5

4

20

6

5

0

10

20

30

40

4

0

. . . . . . . . . . . . . . 10 20 30

t Figure 6. Plot of

40

t

K(t)

for 0 < t < 40, as K(0) passes through K(0)¢rit for Tp = 3.

where T ( t ) = Vpwp sinwpt.

(3.2)

Vp is associated with the amplitude of the external excitation and Wp is its frequency. In terms of canonical coordinates A2, b2, we have the governing equations dA~ OK db2 OK d~- - Ob2' d---t" = -OA---'2 + T ( t ) , (3.3) so that d K = T ( t ) dA2 dt dt "

(3.4)

K is no longer constant along an orbit but varies with t. The Hamiltonian for (3.3) is K - A 2 T ( t ) . The system of equations (3.3) is equivalent to the system da2 / K - F2(a2) :E dt = V Fl(a2) '

dK dt = -t-2a2T( t) v/( K - F2(a2) )F1 (a2).

(3.5)

Mode Locking

12

29

12

K(0)-5.05.Vp-0.25.Tp-4

K(0),-5.10,Vp,=0.25,Tp.=4

10

10

8

4

~

0

10

20

30

8

4

40

0

................... 10 20

t

30

40

t

12

12

10

10

8

~

8

6

6

4

...................

0

10

4

20

30

4C,

0

.................... 10

20

t

30

40

t

Figure 7. Plot of K ( t ) for 0 < t < 40, as K(0) passes through K(0)crit for Tp = 4.

Since/~" is now a function oft, we can no longer integrate the first equation in (3.5) as in (2.10). The equations have to be solved simultaneously. The equation satisfied by a2(t) is dt 2 + ~

2Fl(a2) \ dt )

(3.6)

+ -

= a2T(t), 2Fl(a2)

which can be compared with (2.15) in the unperturbed case. In terms of y = In a2, we have the nonhomogeneous equation d2y -dt ''~

+ (dY'~2 [Fi'(ev) + I] + F2' (ey) = T(t), \ d - t ) L2Fz (eV) 2e2VF1 (eV)

(3.7)

compared to (2.18) in the unperturbed case. A good approximation for small y is given by

---~ dt

2

k,--~,)

?(

y

c~ + / 3 ) (2

2

_

2y

_

y2)

=

T(t).

(3.8)

30

R. R A V I N D R A N

A N D S. S U N D A R

40 35 ,.<.

30 t

2s

£

PI

**

° ***

•

.,,**

20 15

Vp,,,O.25.Tp,=3.0 x

10

• .

4 •

. 4.2

. . . 4.4 4.6

.

4.8

5

5.2

r~tC

a W~cox

3

A

2.8

• • A

k-M

•

70

._o~_ 2.6

,, •

{1)

,

13._

A

•

A

2.4

•

2.2 4

. 4.2

Vp,-O.25.Tp,,3.0

.

. 4.4

. 4.6

. 4.8

5

5.2

• pea • mpea Figure 8. Averaged periods of K(t) and a2(t), respectively, for various K(0) as it passes through K(0)crit for T v = 3.

In the presence of an external excitation, the droplet motion is no longer periodic. W h e n b~(t) vanishes, the shape variable a2(t) (or equivalently, A2(t)) passes through an extreme value. The change of b2 (t) from positive to negative could be used to define an average period of the perturbed motion. W e shall denote it by Tp~t, and it is obtained numerically by averaging over a number of such crossings. It is found that for small perturbation, ~ is approximately given by

dA2

d~ -

C sinWpertt,

(3.9)

where Wpe~t = 4 , ~ and C is a constant depending on the initial conditions. This leads to

dK dt = -Cwp Vv sin O.]pertt sin wpt = A sinWpert$ sinwpt, where A = -CwpVp.

(3.10)

M o d e Locking

31

40

30

20

VN.25.Tp,=4.0

10

,'" t

e

4

m ij o

at t t t

o

S

~

~

as el ~

e

e

i

l

I

I

I

4.2

4.4

4.6

4.8

5

5.2

K(o) w

4 +

3.5 ¢q

3

."

L.

*

2.5

,~ aa~aa&

,

4

i

4.2

6

A

~ ~'~ A ; ;''~ " " Vp=0.?-5,Tp-4.0

.

,

i

i

i

4.4

4.6

4.8

5

5.2

K(0) • pen A urven Figure 9.

Averaged periods of K ( t ) and a2(t), respectively, for various K ( 0 ) _<

K(0)crit for Tp

=

4.

We can now integrate to get

g(t)

A [sin (Wpert-- 0Jp) t - g(0) = ~ L 0)pert -- ¢dp

_

sin (Wpert +wp) t] ¢dpert -~- ¢dp

(3.11) '

provided 0)pert ~ 0)p. The phenomena when ¢dpert = 0)p is called "mode-locking." As long as there is no "mode-locking," (3.11) is a good approximation to K(t). The Melnikov function (see [7]) can also be expressed in terms of the Hamiltonian. Suppose that the unperturbed system is Hamiltonian and T is the period of an orbit. Let the system be perturbed by a time-periodic perturbation of period Tp. If T Tp

m n'

32

R. RAVINDRAN AND S, SUNDAR

where m and n are integers, which are relatively prime, then the subharmonic Melnikov function is given by Mm/n(to) = 4Vpwp to+rnTpa~(t)b2(t) 15 Jto

1+

sinw~(t + to)dt (3.12)

frnTp dAs sinwp(t + to) dr,

= vp p Jo

d--r

where ~dt is to the first approximation evaluated from the unperturbed system. If we were to replace ~ from the perturbed system, then Mm/n(o) =

rnTp d K

dt Jo dt = K(mTp) - K(O).

(3.13)

If K ( t ) is given by (3.11), then since mTp is a multiple of both ~s" (= Tp) and ~OJpert s r (= Tpert ~ T), we have K(mTp) = K(O). The expression (3.11) is, however, not valid in the "mode-locked" state, and the perturbations are too large to justify the use of the Melnikov function.

4. R E S U L T S Numerical calculations have been done for a large range of initial droplet shape, where K(0) is allowed to vary from 4.05 to 5.20 (K = 4 corresponds to the spherical droplet). The deviation of K(0) from the value 4 is a measure of how much the initial spheroidal droplet differs from the spherical one. The perturbation is restricted to two cases. CASE (i). Vp = 0.25, Tp = 3, where Tp varies only slightly from the period of the unperturbed droplet. CASE (ii). Vp = 0.25, Tp = 4. In the unperturbed case, K - F2(as) is approximated for our entire range of interest by p(t3 - y)(y - ~) as in (2.12). Figure 1 shows the accuracy of the approximation. This helps obtain approximate expression for the period of oscillation of the droplet in closed form. In the perturbed case, the plot of as(t) vs. t shows quasiperiodic behaviour. In both Cases (i) and (ii), for K(0) close to 4, the amplitude of as remains small. In Case (i), as K(0) changes from 4.15 to 4.25, the amplitude increases threefold. The same dramatic increase is noticed in Case (ii) as K(0) changes from 5.05 to 5.15 (Figures 2 and 3). For larger values of K(0) in both cases, the larger amplitude persists. The value of K(0), for which this sudden increase in amplitude is first observed is called the critical value of K(0) and is denoted by K(0)crit. Numerically, the following values have been obtained: g(0)crit -- 4.25,

in Case (i),

= 5.15,

in Case (ii).

For K(0) < g(0)crit, the function a2(t) is given by the approximate equation (2.19) (Figure 4). The function K ( t ) clearly exhibits the same dramatic increase in amplitude and period as K(0) passes through g(0)crit in Cases (i) and (ii). For K(0) < K(0)crlt, K ( t ) is given by the simple expression (3.11) and the quasiperiodic nature of the function is observed (Figure 5). As K(0) passes through g(0)cri t in both Cases (i) and (ii), there is a sudden increase in both amplitude and period (Figures 6 and 7). The amplitude persists for all K(0) > g(0)crit, but the period decreases. What causes this remarkable increase in amplitude? If we compute numerically the average period of the perturbed motion, then in Case (i) Tpert locks into Tp at g ( 0 ) = 4.25 and when "mode locking" occurs, then due to "nonlinear resonance," a sudden change in period is observed when

Mode Locking

33

comparing the approximate period as obtained from (3.9),(3.11) with the calculated one, there is a striking change as K ( 0 ) passes through K(0)cri t (Figure 8). In Case (ii), Tpert is close to T and increases with K ( 0 ) gradually till "mode locking" occurs at K(0) = 5.15 (Figure 9). Thereafter, the same behaviour as in Case (i) is observed. If the period of the imposed excitation is close to the natural period T of the droplet, then "mode locking" occurs for all droplets which initially differ only slightly from the spherical shape. If the period of disturbance differs significantly from the natural period, then only droplets whose initial shape differ greatly from the spherical shape exhibit "mode locking." This study shows t h a t a true indicator of mode locking is the Hamiltonian of the system. It undergoes a dramatic change in period and amplitude as mode locking occurs.

REFERENCES 1. W. M6hring and R. Ravindran, Forced vibrations of a non-viscous drop, Max Planck lnstitut filr StrSmungsforschung, GSttingen, Bericht 24, 1-62, (1992). 2. E. Becker J.W. Hiller and T.A. Kowalewski, Experimental and theoretical investigations of large amplitude liquid droplets, J. Fluid Mech. 231, 189-210, (1991). 3. P.V.R. Suryanarayana and Y. Bayazitoglu, Effect of static deformation and external forces on the oscillations of levitated droplets, Phys. Fluids A3, 967-977, (1991). 4. S.K. Wilson, The steady thermo-capillary driven motion of a large droplet in a closed tube, Phys. Fluids AS, 2064-2066, (1993). 5. W.S.J. Uijtewaal, E.-J. Nijhof and R.M. Heethaar, Droplet migration, deformation and orientation in the presence of a plane wall: A numerical study compared with analytical theories, Phys. Fluids A5, 819-825, (1993). 6. W. MShring and A. Knipfer, A model for nonlinear axisymmetric droplet vibration, Physica D 64, 404-419, (1993). 7. J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields, Springer-Verlag, (1983).