The interaction of thermocapillary convection and low-frequency vibration in nearly-inviscid liquid bridges

Nicolás, Josep; Rivas, Damiân; Vega, José M.

doi:10.1007/s000330050040

Cited by 36 publications

(24 citation statements)

References 28 publications

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“…The equations derived there by means of an additional multiple scale analysis led to a surprisingly simple description of the resulting system, consisting of a pair of decoupled, albeit nonlocal equations of the type already studied at length in [17], together with a set of equations governing the interaction of the spatial phase of the wave amplitudes and the viscous mean flow. Since the Reynolds number of this flow can be (indeed must be) substantial these equations must be treated numerically as already done in other circumstances [31,32]. Such solutions will be reported elsewhere.…”

Section: Discussionmentioning

confidence: 99%

Nearly inviscid Faraday waves in annular containers of moderately large aspect ratio

Vega

Knobloch

Martel

2001

Physica D: Nonlinear Phenomena

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Nearly inviscid parametrically excited surface gravity-capillary waves in two-dimensional domains of finite depth and large aspect ratio are considered. Coupled equations describing the evolution of the amplitudes of resonant left-and right-traveling waves and their interaction with a mean flow in the bulk are derived, and the conditions for their validity established. Under suitable conditions the mean flow consists of an inviscid part together with a viscous mean flow driven by a tangential stress due to an oscillatory viscous boundary layer near the free surface and a tangential velocity due to a bottom boundary layer. These forcing mechanisms are important even in the limit of vanishing viscosity, and provide boundary conditions for the Navier-Stokes equation satisfied by the mean flow in the bulk. For moderately large aspect ratio domains the amplitude equations are nonlocal but decouple from the equations describing the interaction of the slow spatial phase and the viscous mean flow. Two cases are considered in detail, gravity-capillary waves and capillary waves in a microgravity environment.

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Section: Discussionmentioning

confidence: 99%

Nearly inviscid Faraday waves in annular containers of moderately large aspect ratio

Vega

Knobloch

Martel

2001

Physica D: Nonlinear Phenomena

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“…Still, in this case, a 2l and a 22 and thus a is quite insensitive to variations of C in the realistic range 10~4:SC:S10~2. Instead, &Tand / are scaled measures of detuning from resonance that may be varied independently by choosing the forcing frequencies and the slenderness, respectively; thus they must be treated as free parameters.…”

Section: Amplitude Equations and Solvability Conditionsmentioning

confidence: 77%

“…Íiií and -rw ; e~! Íiií , respectively, and the first and second equations (27) by rU k e~' ük ' and rW k e~' ük ', respectively, add, intégrate in 0 <r<\, -A 0 <z<A 0 , intégrate by parts, and take into account the continuity equations (20) and (26), and the boundary conditions (22)- (24) and (28)- (30), to obtain…”

Section: Amplitude Equations and Solvability Conditionsmentioning

confidence: 99%

“…In this case the coefficients, a 2l and a 22 , of the quadratic, nonlinear terms vanish in the amplitude equations ing the resulting equation by A k and adding the complex conjúgate. Thus the not-directly-excited mode decays exponentially in the viscous time scale t~\a 4k \¡C+ a 5k C\ (which is the slow scale for the evolution of the modulation of the excited mode).…”

Section: Forcing With the Lower Frequency £L 1 (02=02=0)mentioning

confidence: 99%

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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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“…II we shall perform a weakly nonlinear, multiscale analysis in the limit (1), (4). The velocity and the stagnation pressure in the bulk (outside the boundary layers) and the free surface deflection will be given by [cf (22) and (23) -n (18) are the damping rate and detuning (with respect to the natural frequency ft) of the capillary waves and satisfy Q<d<\ and \S\<\.…”

Section: J -Ajomentioning

confidence: 99%

Weakly nonlinear nonaxisymmetric oscillations of capillary bridges at small viscosity

2002

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Weakly nonlinear nonaxisymmetric oscillations of a capillary bridge are considered in the limit of small viscosity. The supporting disks of the liquid bridge are subjected to small amplitude mechanical vibrations with a frequency that is cióse to a natural frequency. A set of equations is derived for accounting the slow dynamics of the capillary bridge. These equations describe the coupled evolution of two counter-rotating capillary waves and an associated streaming flow. Our derivation shows that the effect of the streaming flow on the capillary waves cannot be a priori ignored because it arises at the same order as the leading (cubic) nonlinearity. The system obtained is simplified, then analyzed both analytically and numerically to provide qualitative predictions of both the relevant large time dynamics and the role of the streaming flow. The case of parametric forcing at a frequency near twice a natural frequency is also considered.

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The interaction of thermocapillary convection and low-frequency vibration in nearly-inviscid liquid bridges

Cited by 36 publications

References 28 publications

Nearly inviscid Faraday waves in annular containers of moderately large aspect ratio

Nearly inviscid Faraday waves in annular containers of moderately large aspect ratio

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

Weakly nonlinear nonaxisymmetric oscillations of capillary bridges at small viscosity

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