On the impulsive generation of drops at the interface of two inviscid fluids

2008 ◽  
Vol 464 (2093) ◽  
pp. 1125-1140 ◽  
Author(s):  
K.K Tjan ◽  
W.R.C Phillips
Keyword(s):  
Phase Diagram ◽  
Surface Tension ◽  
Density Ratio ◽  
Boundary Integral ◽  
Inviscid Fluid ◽  
Fluid Interface ◽  
Inviscid Fluids ◽  
Boundary Integral Formulation ◽  
Lower Fluid ◽  
Density Ratios

The numerical simulation of the deformation of an inviscid fluid–fluid interface subjected to an axisymmetric impulse in pressure is considered. Using a boundary integral formulation, the interface is evolved for a range of upper-fluid and lower-fluid density ratios under the influence of inertial, interfacial and gravitational forces. The interface is seen to evolve into axisymmetric waves or droplets depending upon the density ratio, level of surface tension and gravity. Moreover, the droplets may be spherical, tear shaped or elongated. These conclusions are expressed in a phase diagram of inverse Weber number We −1 versus Atwood number At at zero gravity, i.e. with the Froude number Fr −1 →0, and complement the earlier findings of Tjan & Phillips, who present a phase diagram of We −1 versus Fr −1 for the case in which the upper fluid has zero density. They too report tear-shaped droplets; however, while, in their paper, they form as a result of gravity, those reported here form as a result of surface tension. It is also found that the pinch-off process which effects drops remains of the power-law type with exponent 2/3 irrespective of the presence of gravity and an upper fluid. However, the constant that relates the necking radius to the time from pinch off, which is universal in the absence of gravity and an upper fluid, is affected by the presence gravity, an upper fluid and the class of drops which form.

BOUNDARY INTEGRAL FORMULATION FOR THE COLLAPSE OF AN INITIALLY SPHERICAL VAPOR CAVITY UNDER THE EFFECT OF THE CURVATURE

1995 ◽  
Vol 05 (07) ◽  
pp. 923-933
Author(s):  
A. PIACENTINI
Keyword(s):  
Surface Tension ◽  
Boundary Element Method ◽  
Mathematical Models ◽  
Spline Approximation ◽  
Boundary Integral ◽  
Small Radius ◽  
Bubble Collapse ◽  
Linear Boundary ◽  
Vapor Cavity ◽  
Boundary Integral Formulation

The effect of the curvature is usually neglected in the mathematical models of bubble collapse. For bubbles of sufficiently small radius such effect becomes of relevant interest. A linear boundary element method (B.E.M.) with a cubic spline approximation of the domain that takes the surface tension into account is presented.

Numerical simulations of sink flow in the Hele-Shaw cell with small surface tension

1997 ◽  
Vol 8 (6) ◽  
pp. 533-550 ◽  
Author(s):  
E. D. KELLY ◽  
E. J. HINCH
Keyword(s):  
Surface Tension ◽  
Integral Equation ◽  
Viscous Fluid ◽  
Numerical Simulations ◽  
Boundary Integral Equation ◽  
Numerical Algorithm ◽  
Boundary Integral ◽  
Inviscid Fluid ◽  
Small Surface ◽  
Hele Shaw Cell

The motion of an initially circular drop of viscous fluid surrounded by inviscid fluid in a Hele-Shaw cell withdrawn from an eccentric point sink is considered. Using a numerical algorithm based on a boundary integral equation, the solution for small, finite surface tension is observed. It is found that the zero-surface-tension formation of a cusp is avoided, and instead a narrow finger of inviscid fluid forms, which then rapidly propagates towards the sink. The scaling of the finger in the sink vicinity is determined.

scholarly journals Vortex ring bubbles

1991 ◽  
Vol 224 ◽  
pp. 177-196 ◽  
Author(s):  
T. S. Lundgren ◽  
N. N. Mansour
Keyword(s):  
Surface Tension ◽  
Vortex Ring ◽  
Model Equation ◽  
Boundary Integral ◽  
Boundary Integral Method ◽  
Liquid Jet ◽  
General Description ◽  
Boundary Integral Formulation ◽  
Ring Radius ◽  
Toroidal Geometry

Toroidal bubbles with circulation are studied numerically and by means of a physically motivated model equation. Two series of computations are performed by a boundary-integral method. One set shows the starting motion of an initially spherical bubble as a gravitationally driven liquid jet penetrates through the bubble from below causing a toroidal geometry to develop. The jet becomes broader as surface tension increases and fails to penetrate if surface tension is too large. The dimensionless circulation that develops is not very dependent on the surface tension. The second series of computations starts from a toroidal geometry, with circulation determined from the earlier series, and follows the motion of the rising and spreading vortex ring. Some modifications to the boundary-integral formulation were devised to handle the multiply connected geometry. The computations uncovered some unexpected rapid oscillations of the ring radius. These oscillations and the spreading of the ring are explained by the model equation which provides a more general description of vortex ring bubbles than previously available.

Numerical computation of solitary waves in a two-layer fluid

2011 ◽  
Vol 688 ◽  
pp. 528-550 ◽  
Author(s):  
H. C. Woolfenden ◽  
E. I. Pǎrǎu
Keyword(s):  
Surface Tension ◽  
Free Surface ◽  
Froude Number ◽  
Solitary Waves ◽  
Integral Formula ◽  
Density Ratio ◽  
Bond Number ◽  
Boundary Integral ◽  
Boundary Integral Method ◽  
Damped Oscillations

AbstractWe consider steady two-dimensional flow in a two-layer fluid under the effects of gravity and surface tension. The upper fluid is bounded above by a free surface and the lower fluid is bounded below by a rigid bottom. We assume the fluids to be inviscid and the flow to be irrotational in each layer. Solitary wave solutions are found to the fully nonlinear problem using a boundary integral method based on the Cauchy integral formula. The behaviour of the solitary waves on the interface and free surface is determined by the density ratio of the two fluids, the fluid depth ratio, the Froude number and the Bond numbers. The dispersion relation obtained for the linearized equations demonstrates the presence of two modes: a ‘slow’ mode and a ‘fast’ mode. When a sufficiently strong surface tension is present only on the free surface, there is a region, or ‘gap’, between the two modes where no linear periodic waves are found. In-phase and out-of-phase solitary waves are computed in this spectral gap. Damped oscillations appear in the tails of the solitary waves when the value of the free-surface Bond number is either sufficiently small or large. The out-of-phase waves broaden as the Froude number tends towards a critical value. When surface tension is present on both surfaces, out-of-phase solitary waves are computed. Damped oscillations occur in the tails of the waves when the interfacial Bond number is sufficiently small. Oppositely oriented solitary waves are shown to coexist for identical parameter values.

Numerical studies of two-dimensional Faraday oscillations of inviscid fluids

2000 ◽  
Vol 402 ◽  
pp. 1-32 ◽  
Author(s):  
JEFF WRIGHT ◽  
STEVE YON ◽  
C. POZRIKIDIS
Keyword(s):  
Integral Method ◽  
Density Ratio ◽  
Extended Period ◽  
Vortex Sheet ◽  
Boundary Integral ◽  
The Other ◽  
Boundary Integral Method ◽  
Two Dimensional ◽  
Inviscid Fluids ◽  
Mathieu’S Equation

The dynamics of two-dimensional standing periodic waves at the interface between two inviscid fluids with different densities, subject to monochromatic oscillations normal to the unperturbed interface, is studied under normal- and low-gravity conditions. The motion is simulated over an extended period of time, or up to the point where the interface intersects itself or the curvature becomes very large, using two numerical methods: a boundary-integral method that is applicable when the density of one fluid is negligible compared to that of the other, and a vortex-sheet method that is applicable to the more general case of arbitrary densities. The numerical procedure for the boundary-integral formulation uses a global isoparametric parametrization based on cubic splines, whereas the numerical method for the vortex-sheet formulation uses a local boundary-element parametrization based on circular arcs. Viscous dissipation is simulated by means of a phenomenological damping coefficient added to the Bernoulli equation or to the evolution equation for the strength of the vortex sheet. A comparative study reveals that the boundary-integral method is generally more accurate for simulating the motion over an extended period of time, but the vortex-sheet formulation is significantly more efficient. In the limit of small deformations, the numerical results are in excellent agreement with those predicted by the linear model expressed by Mathieu's equation, and are consistent with the predictions of the Floquet stability analysis. Nonlinear effects for non-infinitesimal amplitudes are manifested in several ways: deviation from the predictions of Mathieu's equation, especially at the extremes of the interfacial oscillation; growth of harmonic waves with wavenumbers in the unstable regimes of the Mathieu stability diagram; formation of complex interfacial structures including paired travelling waves; entrainment and mixing by ejection of droplets from one fluid into the other; and the temporal period tripling observed recently by Jiang et al. (1998). Case studies show that the surface tension, density ratio, and magnitude of forcing play a significant role in determining the dynamics of the developing interfacial patterns.

Numerical solution of the exact equations for capillary–gravity waves

1979 ◽  
Vol 95 (1) ◽  
pp. 119-139 ◽  
Author(s):  
Leonard W. Schwartz ◽  
Jean-Marc Vanden-Broeck
Keyword(s):  
Surface Tension ◽  
Numerical Solution ◽  
Gravity Waves ◽  
High Accuracy ◽  
Boundary Integral ◽  
Two Dimensional ◽  
Boundary Equation ◽  
Boundary Integral Formulation ◽  
Wave Forms ◽  
Nonlinear Form

A numerical method is presented for the computation of two-dimensional periodic progressive surface waves propagating under the combined influence of gravity and surface tension. The dynamic boundary equation is used in its exact nonlinear form. The procedure involves a boundary-integral formulation coupled with a Newtonian iteration. Solutions of high accuracy can be achieved over much of the range of wavelengths and heights including limiting waves. A number of different continuous families of solutions have been produced, all of which ultimately exhibit closed bubbles at their troughs. The so-called critical wavelengths are less important than have been previously assumed; the number of possible wave forms does increase with increasing wavelength, however.

Surface-tension effects in the contact of liquid surfaces

1989 ◽  
Vol 203 ◽  
pp. 149-171 ◽  
Author(s):  
Hasan N. Oguz ◽  
Andrea Prosperetti
Keyword(s):  
Surface Tension ◽  
Experimental Evidence ◽  
Integral Method ◽  
Model Problem ◽  
Mathematical Formulation ◽  
Boundary Integral ◽  
Boundary Integral Method ◽  
Inviscid Fluid ◽  
Condensation Nuclei ◽  
Rain Drops

The process by which two surfaces of the same liquid establish contact, as when two drops collide or raindrops fall on water, is studied. The mathematical formulation is based on the assumption of an incompressible, inviscid fluid with surface tension. A model problem with a simplified geometry is solved numerically by means of a boundary-integral method. The results imply that a number of toroidal bubbles form and remain entrapped between the contacting surfaces. Experimental evidence for this process, which is important for boiling nucleation and the formation of condensation nuclei for rain drops, is found in the literature.

scholarly journals Geometric series expansion of the Neumann–Poincaré operator: Application to composite materials

2021 ◽  
pp. 1-26
Author(s):  
ELENA CHERKAEV ◽  
MINWOO KIM ◽  
MIKYOUNG LIM
Keyword(s):  
Composite Materials ◽  
Series Expansion ◽  
Function Theory ◽  
Effective Conductivity ◽  
Two Dimensions ◽  
Boundary Integral ◽  
Geometric Function Theory ◽  
Boundary Integral Formulation ◽  
Geometric Function ◽  
Series Expression

The Neumann–Poincaré (NP) operator, a singular integral operator on the boundary of a domain, naturally appears when one solves a conductivity transmission problem via the boundary integral formulation. Recently, a series expression of the NP operator was developed in two dimensions based on geometric function theory [34]. In this paper, we investigate geometric properties of composite materials using this series expansion. In particular, we obtain explicit formulas for the polarisation tensor and the effective conductivity for an inclusion or a periodic array of inclusions of arbitrary shape with extremal conductivity, in terms of the associated exterior conformal mapping. Also, we observe by numerical computations that the spectrum of the NP operator has a monotonic behaviour with respect to the shape deformation of the inclusion. Additionally, we derive inequality relations of the coefficients of the Riemann mapping of an arbitrary Lipschitz domain using the properties of the polarisation tensor corresponding to the domain.

scholarly journals Nonlinear two-dimensional free surface solutions of flow exiting a pipe and impacting a wedge

2021 ◽  
Vol 126 (1) ◽  
Author(s):  
Alex Doak ◽  
Jean-Marc Vanden-Broeck
Keyword(s):  
Surface Tension ◽  
Integral Equation ◽  
Free Surface ◽  
Boundary Integral ◽  
Boundary Integral Method ◽  
Two Dimensional ◽  
Numerical Schemes ◽  
Additional Advantage ◽  
Infinite Wedge ◽  
Good Agreement

AbstractThis paper concerns the flow of fluid exiting a two-dimensional pipe and impacting an infinite wedge. Where the flow leaves the pipe there is a free surface between the fluid and a passive gas. The model is a generalisation of both plane bubbles and flow impacting a flat plate. In the absence of gravity and surface tension, an exact free streamline solution is derived. We also construct two numerical schemes to compute solutions with the inclusion of surface tension and gravity. The first method involves mapping the flow to the lower half-plane, where an integral equation concerning only boundary values is derived. This integral equation is solved numerically. The second method involves conformally mapping the flow domain onto a unit disc in the s-plane. The unknowns are then expressed as a power series in s. The series is truncated, and the coefficients are solved numerically. The boundary integral method has the additional advantage that it allows for solutions with waves in the far-field, as discussed later. Good agreement between the two numerical methods and the exact free streamline solution provides a check on the numerical schemes.

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