Oscillating annular liquid membranes

Archive of Applied Mechanics 62 (1992) 43-52 Archive of Applied Mechanics �9 Springer-Verlag 1992

Oscillating annular liquid membranes

J. I. Ramos , P i t tbu rgh and J. Falgueras , M a l a g a

Summary: The response of annular liquid membranes to sinusoidal mass flow rate fluctuations at the nozzle exit is analyzed as a function of the amplitude and frequency of the axial velocity fluctuations at the nozzle exit and thermodynamic compression of the gas enclosed by the membrane. It is shown that both the pressure of the gases enclosed by the annular membrane and the axial distance at which the annular membrane merges on the symmetry axis are periodic functions of time which have the same period as that of the mass flow rate fluctuations at the nozzle exit. They are also nearly sinusoidal functions of time for small amplitudes of the mass flow rate fluctuations at the nozzle exit, and exhibit delay and lag times with respect to the sinusoidal axial velocity fluctuations at the nozzle exit. Both the delay and the lag times are functions of the amplitude and frequency of the mass flow rate fluctuations at the nozzle exit and the polytropic exponent. The amplitudes of both the pressure of the gases enclosed by the annular liquid membrane and the convergence length increase and decrease, resp., as the amplitude and frequency of the mass flow rate fluctuations at the nozzle exit, resp., are increased. They also increase as the polytropic exponent is increased.

Oszillierende rotationsschalenfiirmige Fliissigkeitsmembranen

Ubersicht: Untersucht wird das Verhalten rotationsschalenf6rmiger Flfissigkeitsmembranen in Abhfingigkeit vonder Amplitude und Frequenz der axialen Geschwindigkeitsschwankungen an der D/isenmfindung und der thermodynamischen Verdichtung des eingeschlossenen Gases, wenn sich der Massestrom an der D/isenmfindung sinusf6rmig findert. Es wird gezeigt, dab der Druck des eingeschlossenen Gases und der axiale M/indungsabstand des Scheitels der geschlossenen Membran periodische Zeitfunktionen mit der Frequenz der Masseflul3schwankung am Dtisenaustritt sind. Ffir kleine Amplituden des Massestroms ist ihr Zeitverhalten ebenfalls fast-sinusf6rmig, wobei sie beziiglich der sinusf6rmigen axialen Geschwindigkeitsschwankungen an der Dfisenmiindung eine Ansprechzeit und Phasenverschiebung aufweisen. Ansprechzeit und Phasenverschiebung sind Funktionen von Amplitude und Frequenz der Massestromschwankung sowie des polytropen Exponenten. Die Amplitude von Gasdruck und Abstand des Membranscheitels yon der Dfise w~ichst bzw. ffillt mit wachsender Amplitude und Frequenz des Massestroms. Beide nehmen aul3erdem mit dem polytropen Exponenten zu.

1 Introduction

Annular liquid jets and annu la r liquid membranes , i.e., annu la r jets of zero thickness, can be used as p ro tec t ion systems in laser fusion reactors, chemical reactors, etc. [1, 2]. Under certain condit ions, annu la r liquid jets can fo rm enclosed vo lumes which can be used to burn haza rdous toxic wastes [3, 4].

The equat ions govern ing the fluid dynamics of inviscid, incompressible , i so thermal liquid jets have been derived by the first au t ho r who in tegra ted the Nav ie r -S tokes equat ions across the annu la r jet, appl ied k inemat ic and stress condi t ions at the annu la r liquid je t ' s inner and outer interfaces, and assumed that the annu la r je t ' s thickness is m u c h smaller than the annu la r je t ' s mean radius at the nozzle exit [3, 4]. Analytical solut ions of the s teady s tate fluid dynamics equat ions were also ob ta ined for annu la r liquid jets of small slope and curva ture in grav i ta t iona l [1, 3, 4] and nongrav i t a t iona l [5] envi ronments . Both the analyt ical and the numer ica l solut ions of the s teady state fluid dynamics equat ions were found to agree very well with three sets of exper imenta l da ta [6, 7]. Exper imenta l d a t a [7], however , indicate that, under certain condit ions, annu la r liquid jets m a y exhibit an uns teady behav io r character ized by oscillations in the convergence length, i.e., the axial distance at which the annu la r liquid jet merges on the s ym m et ry axis to become a solid jet.

44 Archive of Applied Mechanics 62 (1992)

The oscillations in the convergence length may be due to mass flow rate fluctuations at the nozzle exit [8], pressure oscillations in the gases that surround the jet [9], and fluctuations in the mass flow rate of the gas which may be injected into the volume enclosed by the annular liquid jet [10]. It must be noted that the liquid absorbs the gases that it encloses, and that, in order to maintain steady state conditions, mass must be injected into the volume enclosed by the annular liquid jet at a rate equal to the mass absorption rate by the liquid [2, 5]. However, since the diffusion coefficient of gases in liquids is very small and the Peclet number for the concentration is consequently very large, mass absorption can be neglected except at low Peclet numbers.

In this paper, the dynamics of annular liquid membranes subject to mass flow rate fluctuations at the nozzle exit is analyzed as a function of both the amplitude and the frequency of the liquid's axial velocity fluctuations at the nozzle exit and as a function of the thermodynamic compression of the gas enclosed by the membrane by means of an adaptive numerical technique which maps the curvilinear, t ime-dependent geometry of the annular liquid membrane into a unit interval.

2 Fluid dynamics equations

The dimensional equations governing the fluid dynamics of inviscid, isothermal, incompressible, annular liquid membranes can be written as [3, 4] (Fig. 1)

Om* O 0t + (m'u*) = 0, (1)

8 8J R* 8R* 8 (m'u*) + (m*u .2) = m*g + 20 ~ + (P,* -- Pi*), (2)

at*

8 - - O - - 8J*/Sz* Ot* (m'v*) + ~ (m*u*v*) = R*(pi* -- Pe*) - 2a OR,/Oz ~ , (3)

OR* OR* v* - + u* - - (4)

Ot* 8z*

where the stars denote dimensional quantities: t* is time, z* is the axial coordinate measured from the nozzle exit, m* is the annular liquid membrane's mass per unit length and per radian, u* is the liquid axial velocity, v* is the radial velocity, g is the gravitational acceleration, o- is the surface tension, R* is the annular membrane's radius, Pi* and Pe* denote the pressure of the gases enclosed by and surrounding the annular membrane, resp., and /E ''' J * = R * 1 + \ 0 z , j j " (5)

Equations (1)-(5) are applicable to both annular liquid jets of finite thickness and annular liquid membranes, i.e., annular jets of zero thickness, and were obtained by averaging the Navier-Stokes equations across the annular liquid jet and by including the kinematic and stress conditions at the jet's

I

Fig. 1. Schematic of an annular liquid jet

J. I. Ramos and J. Faigueras: Oscillating annular liquid membranes 45

inner and outer interfaces which are material surfaces. At each gas-liquid interface, the shear stress is continuous, whereas the jump in normal stresses is balanced by surface tension [3, 4].

Equations (1)-(3) represent the conservation of mass and the conservation of axial and radial momentum, resp., whereas (4) corresponds to the kinematic condition at the membrane's radius.

In this paper, the boundary conditions

m*(t*, O) = too*, u*(t*, O) = Uo* + a* sin (co*t*), (6)

v*(t*, O) = u*(t*, 0) tan 00, R*(t*, 0) = R0* (7)

were employed at the nozzle exit where the subscript 0 denotes conditions at the nozzle exit, too*, R0*, Uo*, a*, co* and 0o are time-independent, 00 denotes the angle between the symmetry axis and the velocity vector at the nozzle exit, and a* and co* are the amplitude and angular frequency, resp., of the axial velocity fluctuations at the nozzle exit. Note that since m*(t*, 0) is constant, whereas u*(t*, 0) is a periodic function of time, the mass flux m*(t*, O) u*(t*, 0) at the nozzle exit is a periodic function of time.

The gases surrounding and enclosed by the liquid membrane are assumed to be ideal. It is also assumed that the gases surrounding the liquid are infinite in extent so that Pe* can be assumed to be constant. For ideal gases

pi* = (8)

where/~ is the specific gas constant, and ~* and T~* denote the density and temperature, resp., of the gases enclosed by the annular liquid membrane. Furthermore, for low Mach numbers, p/* is nearly uniform and (8) can be integrated to yield

pi*V* = m~*/~T~* (9)

where

L* L*

V * = S ~R*2 dz*, m , * = j" Oi*rcR .2 dz*. (10) 0 0

Here V* and m~* denote the volume and the mass of the gases enclosed by the annular membrane, resp., and L* is the convergence length, i.e., the axial distance at which the annular membrane's radius vanishes:

R*(t*, L*(t*)) = 0, (11)

In the absence of heat and mass transfer, the gases enclosed by the annular liquid membrane follow a polytropic process, i.e.,

p~*(t*) V*k(t *) = p~*(0) V'k(0) (12)

where k is the polytropic exponent. Equations (1)-(12) can be written in nondimensional form by introducing the variables

U* U* t 'b /O* Z* u- - 15-- t - - z - -

u0 Uo* Ro* Ro*'

Substitution of (13) into (1)-(5) yields

0m 0 + (mu) = 0 ,

(9 m 1 ( 0 J OR'], O (mu) + (muu) = + - - Cp,,R o-t We 7z

o 1 ( oJ/O ) O (raft) + (mu~) = Cp,,R

m, m - (13)

/T/O, '

(14)

(i5)

(16)


8 R 8 R

J = R 1 + \~-z / J

where

(17)

(18)

Fr = Uo*2/(gRo*) , W e = rno*uo*Z/(2crRo*), (19)

Cp, = C p W e , Cp = (Pi* - Pe*) Ro*Z/(mo*Uo*2), (20)

and F r and W e are the Froude and Weber numbers, resp., and Cp, denotes the pressure coefficient.

(21)

(22)

(23)

the axial velocity fluctuations at the nozzle is the ratio of the frequency of the axial characteristic residence time. Substitution of

(24)

(25)

L

L = L * / R o * , V = 5 R 2 dz. (26) 0

(27)

Substitution of (13) into (6) and (7) yields

re(t, O) = 1, u(t, O) = Uo = 1 + a sin (2reSt 0

iS(t, 0) -- u0 tan 0o, R(t, 0) = 1

where

a = a* /uo* , S t = co*Ro*/(2~Uo*).

Here a is the nondimensional amplitude of exit, and St is the Strouhal number which velocity fluctuations at the nozzle exit to a (13) into (9), (10) and (12) yields

Pi* - rceo,3 mi Pe* -- V '

where

M i * = rcRo*3pe*/(RTi*) , mi = m i * / M i * ,

Substitution of Eq. (13) into Eq. (20) yields

where

C , max = pe*Ro*/ (2a) . (28)

3 Numerical method

The annular liquid membrane's geometry is curvilinear and time-dependent and has an unknown, time-dependent, downstream boundary, i.e., the convergence point. This geometry can be trans- formed into a unit interval by means of the mapping [8]

(t, z) ~ (% 1"1), z = t, r I = z / L , 0 < t/__< 1 (29)

where L( t ) = L * ( t * ) / R o * is the Jacobian of the mapping. Substitution of (29) into (14)-(17) yields

8U 0U + u W : G (30)

where

U = [m, m R , mu, my] T, F = [mu, m R u , muu, muv] T, (31)

J. I. Ramos and J. Falgueras: Oscillating annular liquid membranes 47

G = O, me, Fr + W e N - C "R az ) Wee C ,R

I I O F dL ]

u = o u '

O R l a V J '

(33)

the superscript Tdenotes transpose, and (14) and (17) have been used to obtain the equation for the fourth component of the vector U. Equation (30) is subject to the boundary conditions at the nozzle exit (cf. (21) and (22))

U(r, 0) = [1, 1, Uo, u0 tan 0o] r (34)

and represents a system of coupled partial differential equations for U which requires the values of L and dL/dr for the evalution of U. Therefore, an equation for L is needed. Such an equation can be obtained from (11) as follows. At the convergence point (cf. (11), (13) and (29))

R(t, z = L(0 ) = 0. (35)

Differentiation of (35) with respect to t and use of (14) and (17) imply that

dL dL 4R(u (aR/&l) - eL) - - a t r / = 1 ( 3 6 )

dt dT 4R (SR/&l)

which is an ordinary differential equation for L. The spatial derivatives of (30) can be discretized in an equally spaced grid such that t/t = 0 and

r/, = 1 where I denotes the number of grid points, by means of backward differences in time, upwind differences for the advection term and central differences for G, and the resulting O(A ~, 3 t0-accurate finite difference equation can be written as

~ , + l l T , + l Ci,+l) U , + I Ui n = ---, _ --'~--'i " J i - - 1 + (I + = A-cG~ "+~ + (i 2, 3, I 1) (37)

where

c = (38)

and where the subscript i denotes the i-th grid point, i.e., r/i = iAtl, Arl is the spatial step size, 3z is the time step, and I is a 4 x 4 unit matrix.

Equation (36) can be discretized as

dL 4RI[uI(RI -- R1 1)/3tl - FIL] dr 4R,(RI - Rx- 1)/Arl (39)

The values of u, R, fi and m at i = I can be calculated by linear extrapolation as

UI = 2UI_ 1 - UI- 2. (40)

It must be noted that the derivatives OR/c~r 1 and 02R/&I 2 which appear in the vector G (cf. (32) and (37)) were discretized as

(c~i~i) - R i + l - R i - l + O(At/2 ) ( i = 2 , 3, . . . , I - 1), (41) i 2At/

(c~2R) R i + t - 2 i i + R i C~rl2]i = Atl2 * + O(Atl 2) (i = 2, 3, ..., I - 1), (42)

and Rr was calcuIated from (40). Equation (37) represents a (banded) block-bidiagonal matrix of dimensions (4I - 8) x(4I - 8)

which can be easily solved by forward substitution. However, since both H and C depend on dL/dz and L, and since dL/dz can be evaluated as in (39) where U~ can be calculated by linear extrapolation as in (40), substitution of (39) into (37) would yield a sparse system of nonlinear algebraic equations


because U1 depends on the value of Ui (i = 2, 3, ..., I - 1) through (37) and (40). Such a sparse system can be solved by means of the Newton-Raphson method. However, the inversion of the Jacobian matrix of the Newton method can be avoided if one uses the following iterative technique for both U and L. Assume that U k, L k and dLk/dv are known at the k-th iteration. These values can be used to evaluate C and G in (37). The solution of (37), i.e., U k+ 1, can then be used in (39) to obtain (dL/dz) k+ 1 and

L k+l = L k + Ar \ ~ j (43)

These values of U k + 1, Lk+ 1 and (dL/dz) k+l can then be employed to evaluate C and G, and this iterative procedure can be repeated until the following convergence criterion is satisfied

[ ~ (Ui k+l Uik) T (U, k+l Uik) d- (L k+l Lk)211/2 - - - < 1 0 - 4 . ( 4 4 )

i=1 In aII the caIcuIations presented in this paper, the first iteration was always performed with the vaIues of U, L and d L / d z corresponding to the n-th time level.

The fluid dynamics equations and the numerical method presented in this paper can be used to study steady state and time-dependent problems. Under steady state condition, a = 0 (cf. (21)), i.e., the boundary conditions at the nozzle exit are independent of time, and dL/dz = O.

In order to determine the steady state geometry of the annular liquid membrane, one can simply set a l l the partial derivatives with respect to z equal to zero in the equations presented in Sect. 2 and 3, and solve the resulting finite difference equations until the convergence criterion defined in (44) is satisfied. Alternatively, one can solve the time-dependent equations subject to time-independent boundary conditions at the nozzle exit until an asymptotic steady state is reached. This time-dependent approach requires that (44) be satisfied at each time step, and the asymptotic steady solution is reached whenever the steady state convergence criterion

is satisfied where e = 10 -r in the calculations presented in this paper. In this paper, the steady state fluid dynamics field was determined asymptotically from the

solution of the time-dependent equations. Once steady state was achieved, z was reset to 0, and U(0) = U~, L(0) = L,~ and pi*(0) = (Pi*)ss where the subscript ss denotes steady state. These initial conditions were then used to analyze the response of annular liquid membranes to fluctuations in the axial velocity at the nozzle exit.

4 P r e s e n t a t i o n o f r e s u l t s

Numerical experiments were performed to determine the pressure coefficient and the convergence length as functions of the values of the polytropic exponent, the Strouhal number and the amplitude of the axial velocity fluctuations at the nozzle exit for We = 5,Fr = 10,0o = 0, Cpm~x = 0, and p~*(O)/p~* = 1. Three potytropic exponents were used in the calculations: k = 1, 1.4, and 5/3 which correspond to isothermal compression, adiabatic compression of a diatomic gas, and adiabatic compression of a monatomic gas, respectively.

Three different axial velocity fluctuations at the nozzle exit were investigated: the first fluctuation corresponds to St = 0.1 and a = 0.1, the second to St = 0.5 and a = 0.1, and, the third to St = 0.1 and a = 0.5. These fluctuations are identified as case I, II and III, resp., and are illustrated in Fig. 2 which shows the axial velocity at the nozzle exit as a function of time.

Figures 3 and 4 show the pressure coefficient and the convergence length as functions of time for the first fluctuation, and indicate that both Cv, and L' = L/L(O), where L(0) = 0.326 2, take some time to respond to the axial velocity fluctuations at the nozzle exit. This delay time decreases as the polytropic exponent is increased. Furthermore, both the pressure coefficient and the convergence


1.3

1.2

1.1

! 1.0

0.9

0.8

0.7 Fig. 2. Axial velocity at the nozzle exit as a function of time

0 10 20 30 40 50 t - - - - - - , , - -

0.1

0.2

I 0

r

:= 1.4 =

-0,1

-0.2

t - - - - - ~ 10 20 30 40 50

P

Fig. 3. Pressure coefficient as a function of time (case I)

1.12

I I.O0

J

0.88

t

j

l : = 5/3,

:= 1.4 / : = 1 - -

10 20 30 40 50 t ~

Fig. 4. Convergence length as a function of time (case I)


k = 5/3

k ~ ] . 4

c)

-0.02 , - 0 10 20 30 40

t ~,- 50

Fig. 5. Pressure coefficient as a funct ion of t ime (case II)

1.1

I 1.0

2J

0.9 0

)

0.02

k= 1, 1.4, 5/3

10 20 30 40 50 t

Fig. 6. Convergence length as a funct ion of t ime (case II)

0.6

0.5

0.4

0.3

I 0.2

o 0.1

0

-0.1

-0.2

-0.3 0 I0 2'0

/ k=5/3 ---~,~ kt14\/l

30 4'0 50

Fig. 7. Pressure coefficient as a funct ion of t ime (case III)


1.28

1.00

0.72

y 1;

t -

k = 5/3--~ k = 1 . 4 ~

3'0 40

i

f ,

/ 0

i

i

!

J 50

Fig. 8. Convergence length as a function of time (case III)

length become periodic functions of time after the delay time and their amplitudes increase as the polytropic exponent is increased. Figures 3 and 4 also show that the amplitude of the oscillations in the convergence length does not increase as much as that of the pressure coefficient as the polytropic exponent is increased.

Figures 5 and 6 correspond to the second axial velocity fluctuation at the nozzle exit, and indicate that the amplitude of the oscillations of both the pressure coefficient and the convergence length decreases whereas their frequency increases as the Strouhal number is increased.

Figures 7 and 8 correspond to the third axial velocity fluctuation at the nozzle exit, and indicate that the amplitude of the oscillations of both the pressure coefficient and the convergence length increases as the amplitude of the axial velocity fluctuations at the nozzle exit is increased.

The results presented in Figs. 3-8 indicate that the response of annular liquid membranes to sinusoidal axial velocity or sinusoidal mass flow rate fluctuations at the nozzle exit is characterized by periodic pressure coefficients and convergence lengths after an initial delay time that depends on the polytropic exponent, the Strouhal number and the amplitude of the axial velocity fluctuations at the nozzle exit. The delay time in the response of the pressure coefficient decreases as the polytropic exponent, the amplitude of the axial velocity fluctuations at the nozzle exit and the Strouhal number are increased. The delay in the response of the convergence length is nearly independent of both the polytropic exponent and the amplitude of the axial velocity fluctuations at the nozzle exit, and decreases as the Strouhal number is increased.

Figures 3, 5 and 7 and Fig. 4, 6 and 8 also indicate that the pressure coefficient and the convergence length, resp., exhibit a periodic behavior after the delay time whose deviation from the sinusoidal one increases as the amplitude of the axial velocity fluctuations at the nozzle exit is increased. In particular, the results presented in Fig. 3 -6 are nearly sinusoidal, whereas those of Figs. 7 and 8 are not.

Figures 2 8 show that there are phase lags between the axial velocity fluctuations at the nozzle exit and the pressure coefficient, and between the convergence length and the axial velocity fluctuations at the nozzle exit. These phase lags are functions of the polytropic exponent, the Strouhal number and the amplitude of the axial velocity fluctuations at the nozzle exit. The phase lag between the axial velocity fluctuations at the nozzle exit and the convergence length is larger than that between the pressure coefficient and the axial velocity fluctuations at the nozzle exit.

Figures 3 and 7 indicate, for St = 0.1, the amplitude of the pressure coefficient oscillations is larger than that of the axial velocity fluctuations at the nozzle exit, i.e., it is larger than 0.1 and 0,5, respectively. A similar behavior is also observed in the convergence length as illustrated in Figs. 4 and 8. However, the amplitudes of both the pressure coefficient and the convergence length are smaller than that of the axial velocity fluctuations at the nozzle exit for St = 0.5 as shown in Fig. 2, 5 and 6.

4*


5 Conclusions

The response of annular liquid membranes to sinusoidal mass flow rate fluctuations at the nozzle exit has been analyzed as a function of the amplitude and frequency of the axial velocity oscillations at the nozzle exit and as a function of the compression of the gas enclosed by the membrane by means of an iterative block implicit technique. It has been shown that both the convergence length and the pressure of the gases enclosed by the annular membrane exhibit a delay time, after which they become periodic functions of time. The amplitude and frequency of both the pressure coefficient and the convergence length increase as the amplitude and frequency of the mass flow rate fluctuations at the nozzle exit, resp., are increased. The amplitudes of both the convergence length and the pressure of the gases enclosed by the annular liquid membrane also increase as the polytropic exponent is increased.

The delay time of both the convergence length and the pressure coefficient, and the phases lags between the axial velocity at the nozzle exit and the convergence length and between the axial velocity at the nozzle exit and the pressure of the gases enclosed by the membrane are functions of the amplitude and frequency of the mass flow rate fluctuations at the nozzle exit and the polytropic exponent.

Acknowledgements

The calculations presented in this paper were performed at the Facultad de InformAtica of the Universidad de MAlaga, Spain.

References

1. Ramos, J. I.: Liquid curtains-I. Fluid mechanics. Chem. Eng. Sci. 43 (1988) 3171 3184 2. Ramos, J. I.; Pitchumani, R.: Liquid curtains-II. Gas absorption. Chem. Eng. Sci. 45 (1990) 1595-1604 3. Ramos, J. I.: Liquid membranes: Formulation and steady state analysis. Report CO/89/4, Dept. of Mech. Engineering,

Carnegie Mellon Univ. Pittsburgh, Pennsylvania 1989 4. Ramos, J. I.: Annular liquid jets: Formulation and steady state analysis (in press). Z. angew. Math. Mech. 71 (1991) 5. Ramos, J. I.: Annular liquid jets in zero gravity. Appl. Math. Modelling 14 (1990) 630-640 6. Ramos, J. I.: Analytic, asymptotic and numerical studies of liquid curtains and comparisons with experimental data. Appl.

Math. Modelling 14 (1990) 170-183 7. Chigier, N.; Ramos, J. I.; Kihm, K.: Experimental and theoretical studies of vertical annular liquid jets. In: Proc. 6th symp.

on energy engineering sciences: Flow and transport in continua, pp. 18 25. Argonne, Illinois: U.S. Department Energy, Argonne National Laboratory 1988

8. Ramos, J. I.: Adaptive finite difference methods for liquid membranes. Part I: Mass flow rate fluctuations at the nozzle exit. Report CO/91/1, Dept. of Mech. Engineering, Carnegie Mellon Univ., Pittsburgh, Pennsylvania 1991

9. Ramos. J. I.: Dynamic response of liquid curtains to time-dependent pressure fluctuations. Appl. Math. Modelling 14 (1991) 126 135

10. Ramos, J. I.: Adaptive finite difference methods for liquid membranes. Part II: Mass injection. Report CO/91/2, Dept. of Mech. Engineering, Carnegie Mellon Univ., Pittsburgh, Pennsylvania 1991

Received June 4, 1991

Prof. J. I. Ramos

Dept. of Mechanical Engineering Carnegie Mellon University Pittsburgh, Penn. 15213-3890 U.S.A.

Prof. J. Falgueras

Facultad de Informatica Universidad de Malaga Plaza E1 Ejido, s/n E-29013 Malaga Spain

Oscillating annular liquid membranes

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