Presence Of Surface Tension Research Articles

IRROTATIONAL FLOW DUE TO FORCED OSCILLATIONS OF A BUBBLE

Abstract The behaviour of an axisymmetric bubble in a pure liquid forced by an acoustic pressure field is analysed. The bubble is assumed to have a sharp deformable interface, which is subject both to surface tension and to Rayleigh viscosity damping. Two modelling regimes are considered. The first is a linearized solution, based on the assumption of small axisymmetric deformations to an otherwise spherical bubble. The second involves a semi-numerical solution of the fully nonlinear problem, using a novel spectral method of high accuracy. For large-amplitude nonspherical bubble oscillations, the fully nonlinear solutions show that a complicated resonance structure is possible and that curvature singularities may occur at the interface, even in the presence of surface tension. Rayleigh viscosity at the interface prevents singularity formation, but eventually causes the bubble to become purely spherical unless shape-mode resonances occur. An extended analysis is also presented for purely spherical bubbles, which allows for a more detailed study of the effects of resonance and the Rayleigh viscosity at the bubble surface.

Revealing the dynamics of stagnant rings of third-grade fluid film with heat transfer in the presence of surface tension

In many engineering applications, including coating and lubrication operations, analyzing the temperature behavior of thin film flows on a vertically upward-moving tube is crucial to improving predictive models. This paper examines a steady third-grade fluid film flow with a surface tension gradient on a vertical tube. The mechanisms responsible for the fluid motion are upward tube motion, gravity, and surface tension gradient. This analysis focuses on heat transfer and stagnant ring dynamics. The formulated highly nonlinear ordinary differential equations are solved using the Adomian decomposition method. The conditions for stagnant rings and uniform film thickness are attained and discussed. The inverse capillary number C, Stokes number St, Deborah number De, and Brinkman number Br emerged as flow control parameters. The temperature of the fluid film rises with an increase in the C, St, De, and Br, whereas it decreases with an increase in thermal diffusion rate. The radius of stagnant rings tends to shrink by the increase in C, St, and De. When the value of De is high, third-grade fluid behaves like solids; only free drainage happens with smaller radius stagnant rings and high temperatures. A comparison between Newtonian and third-grade fluids regarding surface tension, velocity, temperature, stationary rings, and fluid film thickness is also provided.

Effects of surface tension on the collapse time of an empty bubble

The collapse of an empty spherical bubble in an ideal liquid, in the absence of viscosity and surface tension, was studied by Lord Rayleigh. Using energy conservation, he derived an exact expression for the total collapse time as a function of the initial radius of the bubble, the density of the liquid, and the far-field pressure. In the present work, we extend Rayleigh's expression to include surface tension effects. Results are found to depend on a dimensionless parameter ϵ that measures the ratio between the work done by surface tension and that done by pressure during the collapse. This parameter is small for large bubbles but can be of order unity or larger for bubbles of small radius and, eventually, small pressure. We show that the ratio between the collapse time in the presence of surface tension and Rayleigh's collapse time is proportional to a definite integral that is a smooth, monotonically decreasing function of ϵ. This function can be easily bounded analytically for any value of ϵ, yielding a simple and accurate approximation for the collapse time that, for all practical purposes, provides a complete analytical solution to the problem at hand. We finally extend results to the case of a hyperspherical collapsing empty bubble.

Lump, Breather, Ma-Breather, Kuznetsov–Ma-Breather, Periodic Cross-Kink and Multi-Waves Soliton Solutions for Benney–Luke Equation

The goal of this research is to utilize some ansatz forms of solutions to obtain novel forms of soliton solutions for the Benney–Luke equation. It is a mathematically valid approximation that describes the propagation of two-way water waves in the presence of surface tension. By using ansatz forms of solutions, with an appropriate set of parameters, the lump soliton, periodic cross-kink waves, multi-waves, breather waves, Ma-breather, Kuznetsov–Ma-breather, periodic waves and rogue waves solutions can be obtained. Breather waves are confined, periodic, nonlinear wave solutions that preserve their amplitude and shape despite alternating between compression and expansion. For some integrable nonlinear partial differential equations, a lump soliton is a confined, stable solitary wave solution. Rogue waves are unusually powerful and sharp ocean surface waves that deviate significantly from the surrounding wave pattern. They pose a threat to maritime safety. They typically show up in solitary, seemingly random circumstances. Periodic cross-kink waves are a particular type of wave pattern that has frequent bends or oscillations that cross at right angles. These waves provide insights into complicated wave dynamics and arise spontaneously in a variety of settings. In order to predict the wave dynamics, certain 2D, 3D and contour profiles are also analyzed. Since these recently discovered solutions contain certain arbitrary constants, they can be used to describe the variation in the qualitative characteristics of wave phenomena.

Proximity to the water surface markedly enhances the force production on underwater flapping wings.

The potential application of flapping wings in micro-aerial vehicles is gaining interest due to their ability to generate high lift even in confined spaces. Most studies in the past have investigated hovering wings as well as those flapping near solid surfaces. However, the presence of surface tension at the water-air interface and the ability of the water surface to move might differentiate its response to the proximity of wings, compared to that of solid surfaces. Motivated by underwater, amphibian robots and several underwater experimental studies on flapping wings, our study investigated the effects of the proximity of flapping wings to the water surface at low Reynolds numbers (Re = 3400). Experiments were performed on a rectangular wing in a water tank with prescribed flapping kinematics and the aerodynamic forces were measured. The effects of surface proximity on the wing in its both upright and inverted orientations were studied. Broadly, the mean lift and drag coefficients in both orientations decreased significantly (by up to 60%) as the distance from the water surface was increased. In the case of the upright orientation, the mean lift coefficient was slightly decreased very close to the water surface with its peak being observed at the normalized clearance of [Formula: see text]. Overall, the study revealed an enhancement in the aerodynamic forces closer to the water surface.

Numerical Simulation of Sloshing in a 2D Rectangular Tank in Presence of Surface Tension Using VOF Method

Numerical Simulation of Sloshing in a 2D Rectangular Tank in Presence of Surface Tension Using VOF Method

Dynamics of 2D fluid in bounded domain via conformal variables

AbstractIn the present work, we compute numerical solutions of an integro‐differential equation for traveling waves on the boundary of a 2D blob of an ideal fluid in the presence of surface tension. We find that solutions with multiple lobes tend to approach Crapper capillary waves in the limit of many lobes. Solutions with a few lobes become elongated as they become more nonlinear. It is unclear whether there is a limiting solution for small number of lobes, and what are its properties. Solutions are found from solving a nonlinear pseudodifferential equation by means of the Newton conjugate‐residual method. We use Fourier basis to approximate the solution with the number of Fourier modes up to .

A Unique Continuation Result for a 2D System of Nonlinear Equations for Surface Waves

In this paper, we establish a result of unique continuation for a special two-dimensional nonlinear system that models the evolution of long water waves with small amplitude in the presence of surface tension. More precisely, we will show that if (eta ,Phi ) = (eta (x,y, t),Phi (x,y, t)) is a solution of the nonlinear system, in a suitable function space, and (eta ,Phi ) vanishes on an open subset Omega of mathbb {R}^2 times [-T,T], then (eta ,Phi )equiv 0 in the horizontal component of Omega . To state such property, we use a Carleman-type estimate for a differential operator mathcal {L} related to the system. We prove the Carleman estimate using a particular version of the well known Treves’ inequality.

The influence of flexible bottom on wave generation by an oscillatory disturbance in the presence of surface tension

The two-dimensional problem of wave generation by a time-harmonic pressure distribution on the surface in a finite-depth ocean is studied in this article. Here, it is considered that the ocean has a flexible base, which is modelled as a thin elastic plate and is governed by the Euler–Bernoulli beam equation. The effect of surface tension at the free surface is also taken into account. Within the framework of the linearised theory of water waves, the initial boundary value problem is solved using the Laplace–Fourier transform technique and the integral form of the free surface elevation is obtained. The method of stationary phase is used to evaluate the asymptotic solutions of the free surface elevation for large time and distance. Different forms of the surface elevation have been demonstrated graphically for variation of parameters in a number of figures, and appropriate conclusions are drawn. Moreover, the dispersion relation associated with the wave motion is also derived and analysed using contour plots to understand the characteristics of the roots. Further, the phase and group velocities of the wave motion have also been investigated in this study for both shallow and deep water.

Capillary forces and capillary bridges between a three-finger microgripper and a plate

ABSTRACT The presence of surface tension in liquids, which induces a strong capillary force between wetting particles in micro/nano scale, has aroused extensive attention. In present study, a fundamental investigation on capillary forces and rupture behaviors of capillary bridges between a three-finger microgripper and a plate is conducted in quasi-static state. Theoretical analysis is performed for solutions of the capillary force. The capillary bridges between a three-finger microgripper and a plate are established based on the principle of energy minimization. An analytical approach for computing the capillary force for the three-finger/plate geometry is proposed by means of variables obtained from the simulation models. The comparison of the single-finger capillary bridge and three-finger capillary bridge is investigated based on the developed models. The effects of separation distance, capillary bridge volume, radial distance and contact angle on the capillary force of three-finger capillary bridges are analyzed in detail. The results demonstrate that the variation of capillary force with separation distance and volume changing is not monotonic, which is caused by the edge effect of the three-finger microgripper. Capillary force measurements were experimentally characterized to demonstrate the reliability of the simulation models and the capillary force solution method based on an established experimental platform.

An Isoperimetric Sloshing Problem in a Shallow Container with Surface Tension

In 1965, B. A. Troesch solved the isoperimetric sloshing problem of determining the container shape that maximizes the fundamental sloshing frequency among two classes of shallow containers: symmetric canals with a given free surface width and cross-sectional area, and radially symmetric containers with a given rim radius and volume (Commun Pure Appl Math 18(1–2):319–338, 1965, https://doi.org/10.1002/cpa.3160180124 ). Here, we extend these results in two ways: (i) we consider surface tension effects on the fluid free surface, assuming a flat equilibrium free surface together with a pinned contact line, and (ii) we consider sinusoidal waves traveling along the canal with wavenumber $$\alpha \ge 0$$ and spatial period $$2\pi /\alpha $$ ; two-dimensional sloshing corresponds to the case $$\alpha = 0$$ . Generalizing our recent variational characterization of fluid sloshing with surface tension to the case of a pinned contact line, we derive the pinned-edge linear shallow sloshing problem, which is an eigenvalue problem for a generalized Sturm-Liouville system. In the case without surface tension, we show that the optimal shallow canal is a rectangular canal for any $$\alpha > 0$$ . In the presence of surface tension, we solve for the maximizing cross-section explicitly for shallow canals with any given $$\alpha \ge 0$$ and shallow radially symmetric containers with m azimuthal nodal lines, $$m = 0, 1$$ . Our results reveal that the squared maximal sloshing frequency increases considerably as surface tension increases. Interestingly, both the optimal shallow canal for $$\alpha = 0$$ and the optimal shallow radially symmetric container are not convex. As a consequence of our explicit solutions, we establish convergence of the maximizing cross-sections, as surface tension vanishes, to the maximizing cross-sections without surface tension.

Fourth-order stability analysis for capillary-gravity waves on finite-depth currents with constant vorticity

We derive a fourth-order nonlinear evolution equation (NLEE) for narrow-banded Stokes wave in finite depth in the presence of surface tension and a mean flow with constant vorticity. The two-dimensional capillary-gravity wave motion on the surface of finite depth is considered here. The analysis is limited to one horizontal dimension, parallel to the direction of wave propagation, in order to take advantage of a formulation using potential flow theory. This evolution equation is then employed to examine the effect of vorticity on the Benjamin–Feir instability (BFI) of the Stokes capillary-gravity wave trains. It is found that the vorticity modifies significantly the modulational instability and in the case of finite depth, the combined effect of vorticity and capillarity is to enhance the instability growth rate influenced by capillarity when the vorticity is negative. The key point is that the present fourth-order analysis exhibits considerable deviations in the stability properties from the third-order analysis and gives better results consistent with the exact numerical results. Furthermore, the influence of linear shear current on Peregrine breather (PB) is studied.

ON THE ORBITAL STABILITY OF A BOUSSINESQ SYSTEM

In this paper we establish the orbital stability of ground state solitary waves for a nonlinear one-dimensional Boussinesq system that models the evolution of two dimensional long water waves with small amplitude in the presence of surface tension. We also discuss the well-posedness for the Boussinesq system, using some Strichartz type estimates associated with the system and lowering the Sobolev index obtained in some previous results.

Numerical study of the relative equilibrium of a droplet with a simply connected free surface on a rotating plane

The paper investigates the shapes of relative rest of limited layers of liquid on a rotating horizontal plane in the field of gravity in the presence of surface tension. The layers under consideration have a simply connected free surface and rotational symmetry with respect to the line of action of the angular velocity. The mathematical formulation of this problem is reduced to a system of first-order ordinary differential equations with boundary and integral closing conditions. A new algorithm for the numerical solution of the resulting system is proposed, the influence of various dimensionless parameters on the characteristics of equilibrium droplet shapes is studied, and criteria for the existence of such shapes are determined. The paper is of theoretical interest, since the problem under consideration is one of the fundamental ones in the research of capillary phenomena. The developed numerical scheme can also be applied in a wider class of differential equations. The results of the article can be used in practical tasks related to coating, fiber and powder production by the centrifugal-disk method.

Scattering of linear gravity-capillary waves by a completely submerged vertical porous elastic plate

The problem of water wave scattering by a vertical porous elastic plate completely submerged in water of infinite depth in the presence of surface tension at the free surface is investigated. Using the concepts of linear water wave theory and Green's Integral theorem, a second kind Fredholm integral equation is obtained which is hypersingular in nature. The obtained hypersingular integral equation is solved numerically. A new mathematical form of the energy identity has been derived which involves an energy loss term that occurs due to the combined effects of surface tension and porous nature of the plate. Numerical estimates for physical quantities like reflection coefficient, energy loss coefficient, hydrodynamic force, and plate deflection are evaluated and then depicted graphically for a certain set of parametric values. The present methodology is validated using the energy identity and the results present in the existing literature. The effect of surface tension on the propagation of gravity waves is analyzed with the help of the various hydrodynamic quantities, which portrays that it is not always insignificant.

Nonlinear Stability of MHD Contact Discontinuities with Surface Tension

We consider the motion of two inviscid, compressible, and electrically conducting fluids separated by an interface across which there is no fluid flow in the presence of surface tension. The magnetic field is supposed to be nowhere tangential to the interface. This leads to the characteristic free boundary problem for contact discontinuities with surface tension in three-dimensional ideal compressible magnetohydrodynamics (MHD). We prove the nonlinear structural stability of MHD contact discontinuities with surface tension in Sobolev spaces by a modified Nash--Moser iteration scheme. The main ingredient of our proof is deriving the resolution and tame estimate of the linearized problem in usual Sobolev spaces of sufficiently large regularity. In particular, for solving the linearized problem, we introduce a suitable regularization that preserves the transport-type structure for the linearized entropy and divergence of the magnetic field.

Scattering of water waves by rectangular thick barriers in presence of surface tension

The influence of surface tension over an oblique incident waves in presence of thick rectangular barriers present in water of uniform finite depth is discussed here. Three different structures of a bottom-standing submerged barrier, submerged rectangular block not extending down to the bottom and fully submerged block extending down to the bottom with a finite gap are considered. An appropriate multi-term Galekin approximation technique involving ultraspherical Gegenbauer polynomial is employed for solving the integral equations arising in the mathematical analysis. The reflection and transmission coefficients of the progressive waves for two-dimensional time har- monic motion are evaluated by utilizing linearized potential theory. The theoretical result is validated numerically and explained graphically in a number of figures. The present result will almost match analytically and graphically with those results already available in the literature without considering the effect of surface tension. From the graphical representation, it is clearly visible that the amplitude of reflection coefficient decreases with increasing values of surface tension. It is also seen that the presence of surface tension, the change of width, and the height of the thick barriers affect the nature of the reflection coefficients significantly

Numerical simulation of single- and multi-mode Rayleigh–Taylor instability with surface tension in two dimensions

In this paper, we study the long-time evolutions of the single- and multi-mode Rayleigh–Taylor instability with surface tension in two dimensions by using the vortex sheet model. Applying a spectrally accurate numerical method, we investigate the effects of surface tension and density jump on the instability in various regimes of parameters. Complex phenomena of pinching, capillary waves, elongation, and roll-up appear at the interfaces. For a single-mode interface, surface tension retards the growths of bubble and spike. The effect of surface tension on the bubble and spike velocity is generally small but is large for a spike of an infinite density ratio. For multi-mode interfaces, we focus on an infinite density ratio. We show that bubbles grow with the scaling law h=αAgt2 even in the presence of surface tension, while spikes follow the scaling law weakly. It is found that both the growth rates of bubbles and spikes decrease with surface tension and the growth rate of spikes decreases larger than that of bubbles. The growth rate of the bubble front is in agreements with results of previous numerical simulations and experiments.

A priori estimates of the electrohydrodynamic waves with vorticity: horizontal electric field

ABSTRACT In this paper, we are concerned with an incompressible and inviscid conducting fluid in the presence of surface tension and horizontal electric field. It is obtained the local-in-time estimates on the interface in and the velocity fields in by using the method introduced by Shatah and Zeng [A priori estimates for fluid interface problems. Commun Pure Appl Math. 2008;61(6):848–876]. The estimates of a horizontal electric field can be obtained from the estimates of the interface and elliptic estimates.

Analysis of oblique wave diffraction by rectangular thick barrier in the presence of surface tension

Analysis of oblique wave diffraction by rectangular thick barrier in the presence of surface tension