A boundary-integral model for drop deformation between two parallel plates with non-unit viscosity ratio drops

2008 ◽  
Vol 227 (20) ◽  
pp. 8807-8819 ◽  
Author(s):  
P.J.A. Janssen ◽  
P.D. Anderson
Keyword(s):  
Viscosity Ratio ◽  
Boundary Integral ◽  
Integral Model ◽  
Drop Deformation ◽  
Parallel Plates

Breakup of fluid droplets in electric and magnetic fields

1988 ◽  
Vol 188 ◽  
pp. 133-146 ◽  
Author(s):  
J. D. Sherwood
Keyword(s):  
Numerical Scheme ◽  
Viscosity Ratio ◽  
Boundary Integral ◽  
Drop Deformation ◽  
Full Time ◽  
Two Fluids ◽  
Electric And Magnetic Fields ◽  
Number Problem ◽  
Thin Thread ◽  
Surrounding Fluid

A drop of fluid, initially held spherical by surface tension, will deform when an electric or magnetic field is applied. The deformation will depend on the electric/magnetic properties (permittivity/permeability and conductivity) of the drop and of the surrounding fluid. The full time-dependent low-Reynolds-number problem for the drop deformation is studied by means of a numerical boundary-integral technique. Fluids with arbitrary electrical properties are considered, but the viscosities of the drop and of the surrounding fluid are assumed to be equal.Two modes of breakup have been observed experimentally: (i) tip-streaming from drops with pointed ends, and (ii) division of the drop into two blobs connected by a thin thread. Pointed ends are predicted by the numerical scheme when the permittivity of the drop is high compared with that of the surrounding fluid. Division into blobs is predicted when the conductivity of the drop is higher than that of the surrounding fluid. Some experiments have been reported in which the drop deformation exhibits hysteresis. This behaviour has not in general been reproduced in the numerical simulations, suggesting that the viscosity ratio of the two fluids can play an important role.

Viscous Flows Involving a Liquid-Vapor Compound Droplet

2001 ◽  
Author(s):  
D. Palaniappan
Keyword(s):  
Exact Solutions ◽  
Viscosity Ratio ◽  
Numerical Algorithms ◽  
Flow Fields ◽  
Continuous Phase ◽  
Boundary Integral ◽  
Liquid Volume ◽  
Drag Forces ◽  
Compound Droplet ◽  
Fluid Interactions

Abstract Exact analytical solutions for steady-state axisymmetric creeping flows in and around a compound multiphase droplet are presented. The solutions given here explain the droplet fluid interactions in uniform and nonuniform flow fields. The compound droplet has a two-sphere geometry with the two spherical surfaces (of unequal radii) intersecting orthogonally. The surface tension forces are assumed to be sufficiently large so that the interfaces have uniform curvature. The singularity solutions for the uniform and paraboloidal flows in the presence of a compound droplet are derived using the method of reflections. The exact solutions for the velocity and pressure fields in the continuous and dispersed phases are given in terms of the fundamental singularities (Green’s functions) and their derivatives. It is found that flow fields and the drag forces depend on two parameters namely, the viscosity ratio and the radii ratio. In the case of paraboloidal flows, a single or a pair of eddies is noticed in the continuous phase for various values of these parameters. The eddies changes their size and shape if the size of the droplet is altered. These observations may be useful in the study of hydrodynamic interactions of compound droplets in complex situations. It is found that the Stokes resistance is greater when the liquid volume is large compared to the vapor volume in uniform flow. It is also noticed that the maximum value of the drag in paraboloidal flow depends on the viscosity ratio and significantly on the liquid volume in the dispersed phase. The exact solutions presented here may be useful for boundary integral formulations that are based on special kernels and also in validating numerical algorithms and codes on multiphase flow and droplet-fluid interactions.

scholarly journals Drop deformation by laser-pulse impact

2016 ◽  
Vol 794 ◽  
pp. 676-699 ◽  
Author(s):  
Hanneke Gelderblom ◽  
Henri Lhuissier ◽  
Alexander L. Klein ◽  
Wilco Bouwhuis ◽  
Detlef Lohse ◽  
...  
Keyword(s):  
Surface Tension ◽  
Laser Pulse ◽  
Time Scale ◽  
Pulse Shape ◽  
Thin Sheet ◽  
Boundary Integral ◽  
Drop Deformation ◽  
Nanosecond Laser Pulse ◽  
Nanosecond Laser ◽  
Energy Partition

A free falling, absorbing liquid drop hit by a nanosecond laser pulse experiences a strong recoil pressure kick. As a consequence, the drop propels forward and deforms into a thin sheet which eventually fragments. We study how the drop deformation depends on the pulse shape and drop properties. We first derive the velocity field inside the drop on the time scale of the pressure pulse, when the drop is still spherical. This yields the kinetic energy partition inside the drop, which precisely measures the deformation rate with respect to the propulsion rate, before surface tension comes into play. On the time scale where surface tension is important, the drop has evolved into a thin sheet. Its expansion dynamics is described with a slender-slope model, which uses the impulsive energy partition as an initial condition. Completed with boundary integral simulations, this two-stage model explains the entire drop dynamics and its dependence on the pulse shape: for a given propulsion, a tightly focused pulse results in a thin curved sheet which maximizes the lateral expansion, while a uniform illumination yields a smaller expansion but a flat symmetric sheet, in good agreement with experimental observations.

The effects of surfactants on drop deformation and breakup

1990 ◽  
Vol 220 ◽  
pp. 161-186 ◽  
Author(s):  
H. A. Stone ◽  
L. G. Leal
Keyword(s):  
Equation Of State ◽  
Interfacial Tension ◽  
Active Material ◽  
Interfacial Area ◽  
Boundary Integral ◽  
Boundary Integral Method ◽  
Drop Deformation ◽  
Local Concentration ◽  
Interfacial Velocity ◽  
Surface Active

The effects of surface-active agents on drop deformation and breakup in extensional flows at low Reynolds numbers are described. In this free-boundary problem, determination of the interfacial velocity requires knowledge of the distribution of surfactant, which, in turn, requires knowledge of the interfacial velocity field. We account for this explicit coupling of the unknown drop shape and the evolving surfactant distribution. An analytical result valid for nearly spherical distortions is presented first. Finite drop deformation is studied numerically using the boundary-integral method in conjunction with the time-dependent convective–diffusion equation for surfactant transport. This procedure accurately follows interfacial tension variations, produced by non-uniform surfactant distribution, on the evolving interface. The numerical method allows for an arbitrary equation of state relating interfacial tension to the local concentration of surfactant, although calculations are presented only for the common linear equation of state. Also, only the case of insoluble surfactant is studied.The analytical and numerical results indicate that at low capillary numbers the presence of surfactant causes larger deformation than would occur for a drop with a constant interfacial tension equal to the initial equilibrium value. The increased deformation occurs owing to surfactant being swept to the end of the drop where it acts to locally lower the interfacial tension, which therefore requires increased deformation to satisfy the normal stress balance. However, at larger capillary numbers and finite deformations, this convective effect competes with ‘dilution’ of the surfactant due to interfacial area increases. These two different effects of surface-active material are illustrated and discussed and their influence on the critical capillary number for breakup is presented.

scholarly journals Two-phase flow in Hele-Shaw cells: numberical studies of sweep efficiency in a five-spot pattern

1988 ◽  
Vol 29 (4) ◽  
pp. 375-400 ◽  
Author(s):  
Leonard W. Sahwartiz ◽  
Anthony J. Degregoria
Keyword(s):  
Dimensionless Parameter ◽  
Viscosity Ratio ◽  
Nonlinear Behavior ◽  
Boundary Integral ◽  
Porous Media Flow ◽  
Sweep Efficiency ◽  
Two Phase ◽  
Production Wells ◽  
Range Of Values

AbstractThe unsteady Hele-Shaw problem is a model nonlinear system that, for a certain parameter ranger, exhibits the phenomenon known as viscous fingering. While not directly applicable to multiphase porous-media flow, it does prove to be an adequate mathematical model for unstable dieplacement in laboratory parallel-plate devices. We seek here to determine, by use of an accurate boundary-integral frount-tracking scheme, the extent to which the simplified system captures the canonical nonlinear behavior of displacement flows and, in particular, to ascertain the role of noise in such systems. We choose to study a particular pattern of injection and production “wells.” The pattern chosen is the isolated “five-spot,” that is a single source surrounded by four symmetrically-placed sinks in an infinite two-dimensional “reservoir.” In cases where the “pusher” fluid has negligible viscosity, sweep efficiency is calculated for a range of values of the single dimensionless parameter τ, an inverse capillary number. As this parameter is reduced, corresponding to increased flow rate or reduced interfacial tension, this efficiency decreases continuously. For small values of τ, these stable displacements change abruptly to a regime characterized by unstable competing fingers and a significant reduction in sweep efficiency. A simple stability argument appears to correctly predict the noise level required to transit from the stable to the competing-finger regimes. Published compilations of experimental results for sweep efficiency as a function of viscosity ratio showed an unexplained divergence when the pusher fluid is less viscous. Our simulations produce a similar divergence when, for a given viscosity ratio, the parameter τ is varied.

Deformation and breakup of a viscoelastic drop in a Newtonian matrix under steady shear

2007 ◽  
Vol 584 ◽  
pp. 1-21 ◽  
Author(s):  
NISHITH AGGARWAL ◽  
KAUSIK SARKAR
Keyword(s):  
Steady State ◽  
Viscosity Ratio ◽  
Equation Model ◽  
Deborah Number ◽  
Steady Shear ◽  
Drop Deformation ◽  
Ordinary Differential Equation Model ◽  
Differential Equation Model ◽  
Critical Capillary Number ◽  
Viscous Stresses

The deformation of a viscoelastic drop suspended in a Newtonian fluid subjected to a steady shear is investigated using a front-tracking finite-difference method. The viscoelasticity is modelled using the Oldroyd-B constitutive equation. The drop response with increasing relaxation time λ and varying polymeric to the total drop viscosity ratio β is studied and explained by examining the elastic and viscous stresses at the interface. Steady-state drop deformation was seen to decrease from its Newtonian value with increasing viscoelasticity. A slight non-monotonicity in steady-state deformation with increasing Deborah number is observed at high Capillary numbers. Transient drop deformation displays an overshoot before settling down to a lower value of deformation. The overshoot increases with increasing β. The drop shows slightly decreased alignment with the flow with increasing viscoelasticity. A simple ordinary differential equation model is developed to explain the various behaviours and the scalings observed numerically. The critical Capillary number for drop breakup is observed to increase with Deborah number owing to the inhibitive effects of viscoelasticity, the increase being linear for small Deborah number.

scholarly journals Boundary-integral model of permanent magnet of a tube segment as shape

2011 ◽  
Vol 60 (4) ◽  
pp. 413-432
Author(s):  
Krystyn Pawluk ◽  
Renata Sulima
Keyword(s):  
Magnetic Field ◽  
Permanent Magnet ◽  
Permanent Magnets ◽  
Boundary Integral ◽  
General Idea ◽  
Integral Approach ◽  
Integral Model ◽  
Magnetic Flux Density ◽  
The Magnetic Field ◽  
Tube Segment

Boundary-integral model of permanent magnet of a tube segment as shape The magnetic field due to a permanent magnet of a tube-side segment as shape and of radial-oriented magnetization is considered. Such a sheet modelling a single pole of the magnet is used to express the suitable contribution to magnetic quantities. A boundary-integral approach is applied that is based on a virtual scalar quantity attributed to the magnet pole. Such an approach leads to express analytically the scalar magnetic potential and the magnetic flux density by means of the elliptic integrals. Numerical examples of the computed fields are given. The general idea of the presented approach is mainly directed towards designing the magnetic field within the air gap of electric machines with permanent magnets as an excitation source. Other technical structures with permanent magnets may be a subject of this approach as well.

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