Theoretical and experimental study of the vibration of axisymmetric viscous liquid bridges

Perales, José Manuel Perales; Meseguer, José

doi:10.1063/1.858230

Cited by 72 publications

(26 citation statements)

References 40 publications

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“…In this figure, we observe that both the symmetric part and the anti-symmetric part of the signal exhibit peaks for particular values of T o . This phenomenon is not new and has been observed in several works (see for instance Perales & Meseguer 1992). These peaks can be attributed to resonance with free eigenmodes of the bridge; the vertical dash-dotted lines correspond to the periods of the linear inviscid eigenmodes of a cylindrical bridge.…”

Section: Numerical Studymentioning

confidence: 82%

“…Numerous works have also considered, for its link with the problem encountered in micro-gravity (e.g. Fowle, Wang & Strong 1979;Zhang & Alexander 1990), an inviscid liquid bridge subjected to an axially oscillating forcing (for a complete list of references, see Perales & Meseguer 1992). In the more general viscous case, as shown by Perales & Meseguer (1992), when the boundaries are oscillating axially, the bridge may in particular exhibit resonant frequencies leading to large deformations.…”

Section: Introductionmentioning

confidence: 99%

“…Fowle, Wang & Strong 1979;Zhang & Alexander 1990), an inviscid liquid bridge subjected to an axially oscillating forcing (for a complete list of references, see Perales & Meseguer 1992). In the more general viscous case, as shown by Perales & Meseguer (1992), when the boundaries are oscillating axially, the bridge may in particular exhibit resonant frequencies leading to large deformations. Dynamics of oscillating viscous liquid bridges were studied in detail by Borkar & Tsamopoulos (1991), Tsamopoulos, Chen & Borkar (1992), , Mollot et al (1993).…”

Section: Introductionmentioning

confidence: 99%

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Forced dynamics of a short viscous liquid bridge

2014

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The dynamics of an axisymmetric liquid bridge of a fluid of density ρ, viscosity µ and surface tension σ held between two co-axial disks of equal radius e is studied when one disk is slowly moved with a velocity U(t). The analysis is performed using a one-dimensional model for thin bridges. We consider attached boundary conditions (the contact line is fixed to the boundary of the disk), neglect gravity and limit our analysis to short bridges such that there exists a stable equilibrium shape. This equilibrium is a Delaunay curve characterized by the two geometric parameters 0 = V 0 /(πe 3 ) and S = (πe 2 )/V 0 , where V 0 is the volume of fluid and the length of the bridge. Our objective is to analyse the departure of the dynamical solution from the static Delaunay shape as a function of 0 , S, the Ohnesorge number Oh = µ/ √ ρσ e, and the instantaneous velocity U(t) and acceleration ∂ t U (non-dimensionalized using e and σ /ρ) of the disk. Using a perturbation theory for small velocity and acceleration, we show that (i) a non-homogeneous velocity field proportional to U(t) and independent of Oh is present within the bridge; (ii) the area correction to the equilibrium shape can be written as U 2 A i + U Oh A v + ∂ t UA a where A i , A v and A a are functions of 0 and S only. The characteristics of the velocity field and the shape corrections are analysed in detail. For the case of a cylinder (S = 1), explicit expressions are derived and used to provide some insight into the break-up that the deformation would induce. The asymptotic results are validated and tested by direct numerical simulations when the velocity is constant and when it oscillates. For the constant velocity case, we demonstrate that the theory provides a very good estimate of the dynamics for a large range of parameters. However, a systematic departure is observed for very small Oh due to the persistence of free eigenmodes excited during the transient. These same eigenmodes also limit the applicability of the theory to oscillating bridges with large oscillating periods. Finally, the perturbation theory is applied to the cylindrical solution of Frankel & Weihs (J. Fluid Mech., vol. 155 (1985), pp. 289-307) obtained for constant velocity U when the contact lines are allowed to move. We show that it can be used to compute the correction associated with acceleration. Finally, the effect of gravity is discussed and shown to modify the equilibrium shape but not the main results obtained from the perturbation theory.

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Section: Numerical Studymentioning

confidence: 82%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Forced dynamics of a short viscous liquid bridge

2014

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“…If, instead, y and the detuning parameter / [which is a scaled measure of how cióse to the 2:1 resonance we are, and depends only on the slenderness A once the liquid has been chosen, according to (13), (38), and (43)] are kept fixed, and the detuning parameter ¿o [a scaled measure of how cióse the forcing frequency is to either ftj or D, 2 = 2D, U according to (10), (38), (43), and either (45) or (60)] is varied, then we move along a horizontal straight line on the diagrams in Figs. 3(a), 8(a), and 9(a).…”

Section: Discussionmentioning

confidence: 99%

Chaotic oscillations in a nearly inviscid, axisymmetric capillary bridge at 2:1 parametric resonance

1998

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We consider the 2:1 internal resonances (such that íljX) and £l 2 -2£l l are natural frequencies) that appear in a nearly inviscid, axisymmetric capillary bridge when the slenderness A is such that 0< A<77 (to avoid the Rayleigh instability) and only the first eight capillary modes are considered. A normal form is derived that gives the slow evolution (in the viscous time scale) of the complex amplitudes of the eigenmodes associated with £l l and Cl 2 , an d consists of two complex ODEs that are balances of terms accounting for inertia, damping, detuning from resonance, quadratic nonlinearity, and forcing. In order to obtain quantitatively good results, a two-term approximation is used for the damping rate. The coefficients of quadratic terms are seen to be nonzero if and only if the eigenmode associated with Cl 2 is even. In that case the quadratic normal form possesses steady states (which correspond to mono-or bichromatic oscillations of the liquid bridge) and more complex periodic or chaotic attractors (corresponding to periodically or chaotically modulated oscillations). For illustration, several bifurcation diagrams are analyzed in some detail for an internal resonance that appears at A -2.23 and involves the fifth and eighth eigenmodes. If, instead, the eigenmode associated with Cl 2 is °dd, and only one of the eigenmodes associated with ílj and Cl 2 is directly excited, then quadratic terms are absent in the normal form and the associated dynamics is seen to be fairly simple. O 1998 American Institute ofPhysics. [S1070-6631(98)

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“…When the desired configuration ͑slenderness and eccentricity͒ is reached, manipulation ceases and the experiment runs alone: because of evaporation the liquid bridge volume continuously decreases and the liquid bridge breaks when the stability limit is reached. Such a volume reduction process is recorded by the CCD cameras, so that from the recorded images just before the breaking, the liquid bridge contours are determined by using standard interface detection techniques already used in liquid bridge problems, 6,7 and from these contours the liquid bridge volume, V ͑the minimum volume stability limit͒, as well as the volume of liquid between the smaller disk and the liquid bridge neck, V d , are calculated. To calculate such volumes it is assumed that liquid bridge cross sections are ellipses, which agrees with published analytical approximations for the shape of liquid bridge interfaces 3,8 ͑additional details can be obtained upon request from the authors͒.…”

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confidence: 99%

Minimum volume of long liquid bridges between noncoaxial, nonequal diameter circular disks under lateral acceleration

Meseguer¹,

Espino²,

Cuerva³

et al. 2005

Physics of Fluids

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The stability limit of minimum volume and the breaking dynamics of liquid bridges between nonequal, noncoaxial, circular supporting disks subject to a lateral acceleration were experimentally analyzed by working with liquid bridges of very small dimensions. Experimental results are compared with asymptotic theoretical predictions, with the agreement between experimental results and asymptotic ones being satisfactory. © 2005 American Institute of Physics. ͓DOI: 10.1063/1.2107747͔In the simplest configuration a liquid bridge consists of an isothermal drop of liquid held by surface tension forces between two parallel, solid disks as shown in Fig. 1. Disregarding electric and magnetic fields effects, the equilibrium interface shapes and hydrostatic stability limits of liquid bridges are determined by the slenderness, ⌳ = L / ͑2R 0 ͒, where L is the distance between the supporting disks and the characteristic length R 0 is the mean radius, R 0 = ͑R 1 + R 2 ͒ /2; the ratio of the radius of the smaller disk, R 1 , to the radius of the larger one, R 2 , that is K = R 1 / R 2 , or the equivalent parameter h = ͑1−K͒ / ͑1+K͒ = ͑R 2 − R 1 ͒ / ͑R 2 + R 1 ͒; the dimensionless eccentricity, e = E / R 0 , 2E being the distance between the disks axes; the dimensionless volume, defined as the ratio of the actual volume V to the volume of a cylinder of the same length L and diameter 2R 0 : V = V / ͑R 0 2 L͒; and the lateral Bond number, B = ⌬gR 0 2 / , where ⌬ is the difference between the density of the liquid and the density of the surrounding medium, g is the lateral acceleration acting on the liquid drop, and is the surface tension.The stability limit of the minimum volume of long axisymmetric liquid bridges held between unequal, noncoaxial parallel circular supporting disks subject to lateral acceleration can be theoretically analyzed by using an analytical approximation based on the standard bifurcation theory ͑Lyapunov-Schmidt technique 1 ͒. This problem was first analyzed a decade ago, both analytically and experimentally, 2 with the following asymptotic expression for the stability limit of minimum volume obtained:where the eccentricity e is assumed to be positive when the relative position of the disks is as indicated in Fig. 1 ͑the smaller disk axis over the larger disk one͒. According to expression ͑1͒ the combined effect of lateral acceleration and eccentricity becomes maximum when the angle ␤ vanishes; this case is the only one considered in this Brief Communication. The variation of the minimum volume with the eccentricity for a lateral Bond number B = 0.05 and different values of the parameter h ͑assuming ␤ =0͒ of liquid bridges with slenderness ⌳ = 2.7 is shown in Fig. 2. Expression ͑1͒ is only valid when the liquid bridge configuration is close enough to the reference one ͑defined by the Rayleigh stability limit, ⌳ = , V =1, B = h = e =0͒, and far from this configuration of reference, Eq. ͑1͒ gives only a rough approximation of the influence on the stability limit of the different parameters considered. However, ...

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Theoretical and experimental study of the vibration of axisymmetric viscous liquid bridges

Cited by 72 publications

References 40 publications

Forced dynamics of a short viscous liquid bridge

Forced dynamics of a short viscous liquid bridge

Chaotic oscillations in a nearly inviscid, axisymmetric capillary bridge at 2:1 parametric resonance

Minimum volume of long liquid bridges between noncoaxial, nonequal diameter circular disks under lateral acceleration

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