Presence Of Surface Tension Research Articles

Well-posedness of 3D vortex sheets with surface tension

We prove well-posedness for the initial value problem for a vortex sheet in 3D fluids, in the presence of surface tension. We first reformulate the problem by making a favorable choice of variables and parameterizations. We then perform energy estimates for the evolution equations. It is important to note that the Kelvin-Helmholtz instability is present for the vortex sheet in the absence of surface tension. Accordingly, we must construct the energy functional carefully with an eye toward the regularization of this instability. Well-posedness follows from the estimates.

Interfacial capillary waves in the presence of electric fields

Large amplitude capillary waves on an inviscid, incompressible fluid layer of density ρ 1 bounded by a second inviscid, incompressible fluid of density ρ 2 are computed in the presence of a uniform electric field acting in a direction parallel to the undisturbed configuration. Periodic travelling waves of arbitrary amplitudes and wavelengths are calculated and the effect of the electric field is studied. The solutions extend the results of Papageorgiou and Vanden-Broeck [D.T. Papageorgiou, J.-M. Vanden-Broeck, Large-amplitude capillary waves in electrified fluid sheets, J. Fluid Mech. 508 (2004) 71–88; D.T. Papageorgiou, J.-M. Vanden-Broeck, Antisymmetric capillary waves in electrified fluid sheets, Eur. J. Appl. Math. 16 (2004) 609–623] where the case of ρ 2 = 0 was treated. Fully nonlinear solutions are computed using boundary integral equation methods. It is shown that there are both symmetric and antisymmetric waves and their characteristics are explored and compared. When there is a jump in the undisturbed horizontal velocities, the flow is susceptible to Kelvin–Helmholtz instability. It is shown analytically in the linear regime, that even in the absence of surface tension, the flow is stabilized by sufficiently large electric fields. In such situations two wave speeds are possible for given electric fields and we construct these branches numerically when the amplitudes are not infinitesimal, both in the absence and presence of surface tension.

A computational framework for free surface fluid flows accounting for surface tension

This work is concerned with the simulation of incompressible Newtonian fluid flow problems with free surfaces in the presence of surface tension. The computational framework presented is based on the stabilised velocity–pressure finite element method. The movement and deformation of the computational domain are accounted for by employing the arbitrary Lagrangian–Eulerian (ALE) description of the fluid kinematics. The numerical model is extended to incorporate surface tension effects. A novel partitioned solution procedure is developed based on the Newton–Raphson methodology which incorporates full linearisation of the overall incremental problem. Several numerical examples are provided to demonstrate the efficiency of the methodology.

Adaptive method of lines solutions for the extended fifth-order Korteweg–de Vries equation

In this paper, we investigate the dynamics and interaction properties of recently discovered “embedded solitons” in an extended fifth-order KdV model inspired by water waves in the presence of surface tension. The dynamical behaviour of the solitons can be efficiently followed by using a moving or an adaptive finite difference mesh in combination with a suitable time-integrator. We will demonstrate this numerically for different types of wave solutions, such as solitary waves, multihumped waves, and interacting waves.

A coupled quadrilateral grid level set projection method applied to ink jet simulation

A coupled level set-projection method on quadrilateral grids is developed for piezoelectric ink jet simulations. The model is based on the Navier–Stokes equations for incompressible two-phase flows in the presence of surface tension and density jump across the interface separating ink and air, coupled to an electric circuit model which describes the driving mechanism behind the process, and a macroscopic contact model which describes the air–ink–wall dynamics. We simulate the axisymmetric flow on quadrilateral grids using a combination of second-order finite difference projection methods to solve the fluid equations and level set methods to track the air/ink interface. To improve the mass conservation performance of the coupled level set method, a bicubic interpolation is combined with the Fast Marching Method for level set re-initialization on quadrilateral grids. The numerical method is used to analyze the motion of the interface, droplet pinch off, formation of satellites, effect of nozzle geometry on droplet size and motion, and the dynamics for droplet landing. The simulations are faithful to the dimensions and physics of a particular class of inkjet devices.

ON THE NONLINEAR STABILITY OF MARANGONI–BÉNARD PROBLEM WITH FREE SURFACE IN THE BOUSSINESQ APPROXIMATION

We study the Bénard problem in the Boussinesq approximation with upper free surface in the presence of surface tension. Both buoyancy and Marangoni effect are taken into account. Defining a suitable energy functional, we obtain sufficient conditions for nonlinear exponential stability of the rest state.

Computerized Symbolic Computation on a Sixth-order Model for Liquid Waves in the Presence of Surface Tension or a Floating Ice

Computerized symbolic computation reflects the rapid expansion of computer sciences in various fields of science and engineering, while the studies on the liquid surfaces for rivers, oceans, aviation kerosene, liquid propellant for rockets, etc., are of current interest. In the presence of surface tension or sea ice, and with symbolic computation, the Hărăgus-Courcelle-Il’ichev model for surface liquid waves is hereby investigated. Several similarity reductions are presented, some of which are explicitly written out as exact analytic solutions having their rational expressions with respect to the dimensionless spatial variables of the model.

SYMBOLIC COMPUTATION AND POSSIBLY OBSERVABLE EFFECTS FOR THE SOLITON-LIKE LIQUID WAVE PROPAGATION IN THE PRESENCE OF SURFACE TENSION

Phenomena on the liquid surfaces are of current importance, such as those for rivers, oceans, liquid propellant for rockets and aviation kerosene. Wave propagation on a liquid surface in the presence of surface tension is hereby investigated. Computerized symbolic computation, generalized hyperbolic-function method and Wu method are combined. Some soliton-like, exact analytic solutions are obtained for a recently-proposed (2+1)-dimensional 6th-order model. Possibly observable effects are also proposed, which certain future experiments may hopefully discover.

A coupled level set projection method applied to ink jet simulation

A finite difference level set-projection method on rectangular grid is developed for piezoelectric ink jet simulation. The model is based on the Navier-Stokes equations for incompressible two-phase flows in the presence of surface tension and density jump across the interface separating ink and air, coupled to an electric circuit model which describes the driving mechanism behind the process, and a macroscopic contact model which describes the air-ink-wall dynamics. We simulate the axisymmetric flow using a combination of second order projection methods to solve the fluid equations and level set methods to track the air/ink interface. The numerical method can be used to analyze the motion of the interface, breakoff and formation of satellites, and effect of nozzle geometry on droplet size and motion. We focus on close comparison of our numerical ink jet simulation with experimental data. The corresponding author. This author was supported in part by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Department of Energy, under Contract Number DE-AC03-76SF00098, and the Division of Mathematical Sciences of the National Science Foundation.

Compressibility effects on the Rayleigh–Taylor instability growth between immiscible fluids

The linearized Navier–Stokes equations for a system of superposed immiscible compressible ideal fluids are analyzed. The results of the analysis reconcile the stabilizing and destabilizing effects of compressibility reported in the literature. It is shown that the growth rate n obtained for an inviscid, compressible flow in an infinite domain is bounded by the growth rates obtained for the corresponding incompressible flows with uniform and exponentially varying density. As the equilibrium pressure at the interface p∞ increases (less compressible flow), n increases towards the uniform density result, while as the ratio of specific heats γ increases (less compressible fluid), n decreases towards the exponentially varying density incompressible flow result. This remains valid in the presence of surface tension or for viscous fluids and the validity of the results is also discussed for finite size domains. The critical wavenumber imposed by the presence of surface tension is unaffected by compressibility. However, the results show that the surface tension modifies the sensitivity of the growth rate to a differential change in γ for the lower and upper fluids. For the viscous case, the linearized equations are solved numerically for different values of p∞ and γ. It is found that the largest differences compared with the incompressible cases are obtained at small Atwood numbers. The most unstable mode for the compressible case is also bounded by the most unstable modes corresponding to the two limiting incompressible cases.

Statistical analysis of multimode weakly nonlinear Rayleigh-Taylor instability in the presence of surface tension.

A weakly nonlinear model is proposed for the Rayleigh-Taylor instability in the presence of surface tension. The dynamics of a multimode perturbation of the interface between two incompressible, inviscid, irrotational, and immiscible fluids is analyzed. The quadratic and cubic nonlinear effects are taken into account. They include the nonlinear corrections to the exponential growths of the fundamental modulations. The role of the initial modulation spectrum is discussed. A saturation criterion in terms of the product of a local rms and a particular wave number is exhibited. It gives theoretical foundations for numerical conjectures and allows one to analyze the effects of fundamental parameters of the problem such as the dimension or the Atwood number.

On a finite element formulation for incompressible Newtonian fluid flows on moving domains in the presence of surface tension

AbstractThis work is concerned with the numerical modelling of incompressible Newtonian fluid flows on moving domains in the presence of surface tension. The solution procedure presented is based on the stabilized equal order mixed velocity‐pressure finite element formulation of the incompressible Navier–Stokes equations, which is adapted to a moving domain by means of an arbitrary Lagrangian–Eulerian (ALE) technique. The accurate and very robust integration in time is achieved by employing the generalized‐αmethod. The surface tension boundary condition is rephrased appropriately within the framework of linear finite elements. The solution procedure is verified by comparing numerical solutions with the corresponding analytical solutions and experimental data. The overall solution procedure proves to be accurate, robust and efficient. It allows the simulation of extensive deformation of the fluid domain without remeshing. Copyright © 2003 John Wiley & Sons, Ltd.

A two‐phase model for compaction and damage: 2. Applications to compaction, deformation, and the role of interfacial surface tension

New equations for the dynamics of a two‐phase mixture are derived in a companion paper [Bercovici et al., this issue (a)]. These equations do not invoke a bulk viscosity as most previous papers have done, and use the existence of the pressure difference between the two phases, including the possibility of surface energy at the interface between the phases. In this paper we show how a two‐phase mixture reacts to simple stress fields. As a basic example, we discuss the deformation of a porous material confined by an impermeable jacket and loaded by a porous piston and show that the fluid can never be totally extracted from the matrix. We demonstrate that an unconfined porous sample is stronger under shear deformation than under normal stress. We consider spherically symmetric compaction and show that some unphysical results obtained using a constant matrix bulk viscosity are naturally avoided in our approach. We discuss the problem of compaction of a two‐phase liquid in the presence of surface tension. In a one‐dimensional simulation the surface tension generates porosity instabilities that tend to localize the fluid into narrow sills and dikes that cannot reach the surface.

Theoretical and computational aspects of the self-induced motion of three-dimensional vortex sheets

Theoretical and computational aspects of the self-induced motion of closed and periodic three-dimensional vortex sheets situated at the interfaces between two inviscid uids with generally different densities in the presence of surface tension are considered. In the mathematical formulation, the vortex sheet is described by a continuous distribution of marker points that move with the velocity of the fluid normal to the vortex sheet while executing an arbitrary tangential motion. Evolution equations for the vectorial jump in the velocity across the vortex sheet, the vectorial strength of the vortex sheet, and the scalar circulation field or strength of the effective dipole field following the marker points are derived. The computation of the self-induced motion of the vortex sheet requires the accurate evaluation of the strongly singular Biot-Savart integral whose existence requires that the normal vector varies in a continuous fashion over the vortex sheet. Two methods of computing the principal value of the Biot-Savart integral are implemented. The first method involves computing the vector potential and the principal value of the harmonic potential over the vortex sheet, and then differentiating them in tangential directions to produce the normal or tangential component of the velocity, in the spirit of generalized vortex methods developed by Baker (1983). The second method involves subtracting off the dominant singularity of the Biot-Savart kernel and then accounting for its contribution by use of vector identities. Evaluating the strongly singular Biot-Savart integral is thus reduced to computing a weakly singular integral involving the mean curvature of the vortex sheet, and this allows the routine discretization of the vortex sheet into curved elements whose normal vector is not necessarily continuous across the edges, and the computation of the self-induced velocity without kernel desingularization. Numerical simulations of the motion of a closed or periodic vortex sheet immersed in a homogeneous fluid confirm the effectiveness of the numerical methods for a limited time of evolution. Numerical instabilities arise after a certain evolution time due to the ill-posedness of vortex sheet dynamics. The motion may be regularized by desingularizing the Biot-Savart kernel using either Krasny's (1986 b ) method or spectrum truncation. Depending, however, on the physical mechanism that drives the motion, the instabilities may persevere.

Codimension-onepersistence beyond all orders of homoclinic orbits to singularsaddle centres in reversible systems

The persistence from a singular limit is studied ofhomoclinic orbits to a saddle-centre equilibrium infour-dimensional reversible vector fields.The linearization is assumed tohave eigenvalues ±iω, ±λ, λ,ω>0,with the singular limit being λ→0.Recent work by Lombardi has shown the generic non-persistence of such homoclinic solutions under perturbations thatbreak the integrability of a normal form due to a splitting term(Melnikov function) that is exponentially small in λ/ω.A plausible argument is presented which extends Lombardi's analysisto suggest persistence in a codimension-one sense. Specifically,keeping nonlinear terms fixed but treating λ and ω asindependent parameters, then, under the correct sign of anormal-form coefficient b2, each term in the calculablepart of the Melnikov function has a countable sequence of signchanges as a function ofω. This calculable part of the Melnikov function is only theleading-order term under an additional assumption of closenessto integrability. In that case, a zero of this functionimplies anω-value at which a curve of homoclinic orbits bifurcates fromλ = 0 in the parameter plane. The large-ω asymptotics ofthis sequence of bifurcation points is determined solely by the valueof b2.This heuristic result is supported by careful numerical experiments ona class of fourth-order reversible equations with arbitrary quadraticnonlinear terms. The implications of the results are discussed for solitarywater waves of elevation in the presence of surface tension and forrecently discovered embedded solitons in a variety of(non-integrable) nonlinear optical models.

Computation of Axisymmetric Suction Flow through Porous Media in the Presence of Surface Tension

The effect of small surface tension on a class of axisymmetric flows with suction is studied numerically. The dynamic evolution of a blob of incompressible viscous fluid, surrounded by air and drawn into an interior sink, is considered. The velocity field of the viscous fluid is assumed to satisfy Darcy's law and thus the motion is that of a flow through porous media. The fluid interface motion is computed using a highly accurate boundary integral method. This method combines recent numerical techniques to achieve efficient and high-order space and time discretizations for this type of flow. Through accurate computations, it is shown that, in the presence of small surface tension, the dynamic behavior of the axisymmetric flows is very similar to that of the (two-dimensional) Hele–Shaw counterparts. A long finger develops in the fluid interface and forms a cone singularity when it reaches the sink before all the fluid is sucked out. The finger bulges and develops a neck for sufficiently small surface tension. The bulge–neck formation is enhanced by the additional component of the mean curvature in the three-dimensional flow. However, its effect is not strong enough to cause the interface to pinch off at the neck for the data considered here.

Evolution of perturbations on the surface of a viscoelastic liquid

The stability of a layer of a viscoelastic liquid on an inclined plane is studied within the framework of the model with a time-dependent “memory” in the presence of surface tension. It is shown analytically and numerically that these flows can be stable or unstable depending on the Reynolds number. Profiles of the free surface are found as functions of the Reynolds and Weber numbers.

Breaking Homoclinic Connections for a Singularly Perturbed Differential Equation and the Stokes Phenomenon

Behavior of the separatrix solution y(t)=−(3/2)/cosh2(t/2) (homoclinic connection) of the second order equation y″=y+y2 that undergoes the singular perturbation ɛ2y″″+y″=y+y2, where ɛ>0 is a small parameter, is considered. This equation arises in the theory of traveling water waves in the presence of surface tension. It has been demonstrated both rigorously [1, 2] and using formal asymptotic arguments [3, 4] that the above‐mentioned solution could not survive the perturbation.The latter papers were based on the Kruskal–Segur method (KS method), originally developed for the equation of crystal growth [5]. In fact, the key point of this method is the reduction of the original problem to the Stokes phenomenon of a certain parameterless “leading‐order” equation. The main purpose of this article is further development of the KS method to study the breaking of homoclinic connections. In particular: (1) a rigorous basis for the KS method in the case of the above‐mentioned perturbed problem is provided; and (2) it is demonstrated that breaking of a homoclinic connection is reducible to a monodromy problem for coalescing (as ɛ→0) regular singular points, where the Stokes phenomenon plays the role of the leading‐order approximation.

Dynamic generation of capillary waves

We investigate the dynamic generation of capillary waves in two-dimensional, inviscid, and irrotational water waves with surface tension. It is well known that short capillary waves appear in the forward front of steep water waves. Although various experimental and analytical studies have contributed to the understanding of this physical phenomenon, the precise mechanism that generates the dynamic formation of capillary waves is still not well understood. Using a numerically stable and spectrally accurate boundary integral method, we perform a systematic study of the time evolution of breaking waves in the presence of surface tension. We find that the capillary waves originate near the crest in a neighborhood, where both the curvature and its derivative are maximum. For fixed but small surface tension, the maximum of curvature increases in time and the interface develops an oscillatory train of capillary waves in the forward front of the crest. Our numerical experiments also show that, as time increases, the interface tends to a possible formation of trapped bubbles through self-intersection. On the other hand, for a fixed time, as the surface tension coefficient τ is reduced, both the capillary wavelength and its amplitude decrease nonlinearly. The interface solutions approach the τ=0 profile. At the onset of the capillaries, the derivative of the convection is comparable to that of the gravity term in the dynamic boundary condition and the surface tension becomes appreciable with respect to these two terms. We find that, based on the τ=0 wave, it is possible to estimate a threshold value τ0 such that if τ⩽τ0 then no capillary waves arise. On the other hand, for τ sufficiently large, breaking is inhibited and pure capillary motion is observed. The limiting behavior is very similar to that in the classical KdV equation. We also investigate the effect of viscosity on the generation of capillary waves. We find that the capillary waves still persist as long as the viscosity is not significantly greater than surface tension.

EHD kelvin-helmholtz instability in viscous porous medium permeated with suspended particles

The electrohydrodynamic Kelvin-Helmholtz instability of the plane interface between two uniform, superposed viscous and streaming dielectric fluids permeated with suspended particles through porous medium is considered under the influence of a tangential electric field. In the absence of surface tension, it is found that perturbations transverse to the direction of streaming are unaffected by the presence of both streaming and the tangential electric field, if perturbations in the direction of streaming are ignored. For perturbations in all other directions there exists instability for a certain wavenumber range. In the presence of surface tension, it is found that the instability of this system is suppressed by the presence of the tangential electric field. Both the tangential electric field and the surface tension have stabilizing effects and they are able to suppress Kelvin-Helmholtz instability for small wavelength perturbations. The medium porosity reduces the stability range given in terms of a difference in streaming velocities and the electric field effect, while the suspended particles do not affect the above results.