Introduction

Babskii, V. G.; Kopachevskii, N. D.; Slobozhanin, Lev A.; Tyuptsov, A. D.

doi:10.1007/978-3-642-70964-7_1

Cited by 5 publications

(29 citation statements)

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“…For the considered configuration, the static stability problem is given by (Myshkis et al. 1987, p. 129) with the boundary condition where and denote eigenfunctions and eigenvalues, respectively.…”

Section: Discussion Of Resultsmentioning

confidence: 99%

“…To establish the vibration model, we follow the theoretical framework set forth in Myshkis et al. (1987) for oscillations of capillary surfaces in an external force field. Two assumptions are made: (i) the liquid is assumed to be incompressible and ideal so that the potential flow theory can be used; (ii) the small-amplitude oscillations are considered, i.e.…”

Section: Mathematical Formulationmentioning

confidence: 99%

“…We consider the Bernoulli equation (2.3) on and then substitute the linear form of (2.4) into (2.3) to obtain the dynamic pressure balance (Myshkis et al. 1987, p. 280) where is the Laplace–Beltrami operator depending on the equilibrium shape , and , are the two principle curvatures of . It is worth noting that the sign of the two principal curvatures of the axisymmetric drop surface depends only on the direction of the normal vector of the liquid surface, and here and have negative signs (see Appendix A).…”

Section: Mathematical Formulationmentioning

confidence: 99%

“…In (2.14 e ) the boundary parameter is given by (Myshkis et al. 1987, p. 126) where is the curvature of the solid surface at the CL .…”

Section: Mathematical Formulationmentioning

confidence: 99%

“…These models are to solve a functional eigenvalue problem governing the linear dynamics of sessile drops (Myshkis et al. 1987, p. 281). The eigenvalues are the natural frequencies of vibration, and the eigenvectors are the shapes of these vibrational modes.…”

Section: Introductionmentioning

confidence: 99%

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Effects of gravity on natural oscillations of sessile drops

2023

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Natural oscillations of sessile drops with a free or pinned contact line in different gravity environments are studied based on a linear inviscid irrotational theory. The inviscid Navier–Stokes equations and boundary conditions are reduced to a functional eigenvalue problem by the normal-mode decomposition. We develop a boundary element method model to numerically solve the eigenvalue problem for predicting the natural frequencies. Emphasis is placed on the frequency shifts of modes due to gravity for a wide range of contact angles $\alpha$ and Bond numbers $Bo$ . Three types of $\alpha$ – $Bo$ diagrams reflecting how gravity shifts the frequency are identified. Specifically, the frequency of zonal modes shifts downwards (upwards) when $\alpha$ is smaller (larger) than a critical value, while the frequencies of most sectoral modes are shifted downwards regardless of $\alpha$ . As a result, gravity can transform the lowest mode from a zonal mode to a sectoral mode. The spectral degeneracy of hemispherical drops inherited from the Rayleigh–Lamb spectrum is also broken by gravity. However, we discover that gravity has no effect on the mode associated with the horizontal motion of the centre of mass, whose frequency is always zero regardless of $\alpha$ and $Bo$ . This implies that the ‘walking’ drop instability reported in previous literature does not exist.

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Section: Discussion Of Resultsmentioning

confidence: 99%

Section: Mathematical Formulationmentioning

confidence: 99%

Section: Mathematical Formulationmentioning

confidence: 99%

“…In (2.14 e ) the boundary parameter is given by (Myshkis et al. 1987, p. 126) where is the curvature of the solid surface at the CL .…”

Section: Mathematical Formulationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Effects of gravity on natural oscillations of sessile drops

2023

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Static and dynamic stability of pendant drops

Zhang

Zhou

2023

J. Fluid Mech.

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Despite the widespread occurrence of pendant drops in nature, there is still a lack of combined studies on their dynamic and static stability. This study focuses on the dynamic and static stability of elongated drops with either a free or pinned contact line on a plane. We first examine static stability for both axisymmetric and non-axisymmetric perturbations subject to volume or pressure constraints. The stability limits for volume and pressure disturbances (axisymmetric) correspond to the maximum volume and pressure of the drops, respectively. Drops with free contact lines are marginally stable to non-axisymmetric perturbations because of their horizontal translational invariance, whereas pinned drops are stable. The linear dynamic stability is then investigated numerically through a boundary element model, restricted to volume disturbances. Results show that when the stability limit is reached, the first zonal mode has a zero frequency, suggesting that the thresholds for static and dynamic stability are essentially equivalent. Furthermore, natural frequencies experience sharp changes as the stability limit is approached. Another zero frequency mode associated with the horizontal motion of the centre of mass is also revealed by the numerical results, reflecting the horizontal translational invariance of drops with free contact lines. Finally, the frequency spectrum modified by gravity is explored, resulting in the identification of five gravity-induced frequency shift patterns. The frequency shifts break the spectral degeneracy for hemispherical drops with free contact lines, leading to various spectral orderings according to polar and azimuthal wavenumbers.

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Exotic flotation theory with new method for floating stability analysis

Zhang,

Tan

et al. 2023

J. Fluid Mech.

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A macroscale floating object moving downwards will encounter an increasing buoyancy force exerted by the liquid. However, considering the surface tension and the deformed meniscus, we find an exotic floating object of specific shape that withstands a constant total force exerted by the liquid when it moves vertically and slowly. This constant total force consists of the surface tension force and the hydrostatic pressure force, from which a model to determine the shape of the exotic floating object is proposed. Results show that there exist three types of exotic floating objects in both the two-dimensional symmetric and axisymmetric cases, dependent on their concavity and convexity. To ensure that the menisci around the exotic floating objects can be sustained in practice, the stabilities of these menisci are checked. Apart from the meniscus stabilities (of liquid surfaces), the floating stabilities (of solid objects) are also studied. It is demonstrated that the exotic floating object remains in a critical state of floating stabilities no matter where this object locates vertically, from which a new method to predict the floating stabilities for general floating objects of arbitrary shape is put forward, based on the contact angle and the geometrical parameters at the contact point. With the new method, the floating stabilities can be predicted conveniently, without performing an extra force analysis.

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Introduction

Cited by 5 publications

References 0 publications

Effects of gravity on natural oscillations of sessile drops

Effects of gravity on natural oscillations of sessile drops

Static and dynamic stability of pendant drops

Exotic flotation theory with new method for floating stability analysis

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