Numerical solution to the Saffman--Taylor finger problem with kinetic undercooling regularisation

The mathematical model of a steadily propagating Saffman-Taylor finger in a Hele-Shaw channel has applications to two-dimensional interacting streamer discharges which are aligned in a periodic array. In the streamer context, the relevant regularization on the interface is not provided by surface tension but instead has been postulated to involve a mechanism equivalent to kinetic undercooling, which acts to penalize high velocities and prevent blow-up of the unregularized solution. Previous asymptotic results for the Hele-Shaw finger problem with kinetic undercooling suggest that for a given value of the kinetic undercooling parameter, there is a discrete set of possible finger shapes, each analytic at the nose and occupying a different fraction of the channel width. In the limit in which the kinetic undercooling parameter vanishes, the fraction for each family approaches 1/2, suggesting that this "selection" of 1/2 by kinetic undercooling is qualitatively similar to the well-known analog with surface tension. We treat the numerical problem of computing these Saffman-Taylor fingers with kinetic undercooling, which turns out to be more subtle than the analog with surface tension, since kinetic undercooling permits finger shapes which are corner-free but not analytic. We provide numerical evidence for the selection mechanism by setting up a problem with both kinetic undercooling and surface tension and numerically taking the limit that the surface tension vanishes.

I In part I of this thesis similarity solutions to the equations of three phase flow through porous media are examined. The three phases are water, steam, and a noncondensing phase, most likely oil. The main purpose of analyzing such flows is to gain understanding of the steam flooding of oil fields. Provided steam is being injected at a higher pressure than the initial field pressure, it is shown that there will always be at least two saturation shocks. As one increases the pressure of the injected steam several regimes are encountered; first the flow develops a region where all the steam is completely condensed, then the position of two of the shocks are interchanged, and finally one of the shocks grows weaker and is eventually replaced by an expansion fan. In sections 12 and 13 the stability of steadily moving condensation fronts in porous media is analyzed. For one special problem it is found that the sign of the jump in pressure gradient at the interface determines whether the interfaces are stable or unstable. This result is applied with some caution to the similarity solutions found in the earlier sections. II Recently McLean analyzed the shapes of fingers in a Hele-Shaw cell, including the effects of surface tension. His work resolved the question of the uniqueness of the shapes first brought up by Saffman and Taylor in their analysis that did not include surface tension. It is however felt that there are still unresolved problems. In determining the pressure jump across an interface there are two principal radii of curvature. McLean only took into account the effect of the larger of these, assuming that the other was constant along the outline of the finger. Unless the smaller radius is very nearly constant, it should in fact give a larger contribution to the jump in pressure. In this thesis the effect of this smaller radius of curvature is modelled by assuming that it is a function of the normal velocity of the mean two dimensional surface of the finger. It is found that if one only takes into account the smaller radius of curvature, the problem is not uniquely determined, as in the case with no surface tension at all. When both radii of curvature are taken into account, the effect of the smaller radius of curvature is to modify the finger shapes in a way that is qualitatively in agreement with experimental data. Also, McLean's results are checked by an independent numerical scheme, and the results are found to be in excellent agreement. Using both methods of solution a second solution branch other than that described by McLean was also found.

Effects of surface tension on the dynamics of a single micro bubble near a rigid wall in an ultrasonic field

The effect of surface tension and kinetic undercooling on a radially-symmetric melting problem

The motion and shape of a liquid drop through another continuous liquid phase (conveying phase) in a vertical Hele- Shaw cell with different two different apertures were investigated experimentally. Two different liquid-liquid systems were tested. In all cases, the continuous phase was more viscous and wetted the bounding walls. In the capillarity-dominated region, the irregular shape of the discontinuous phase changed with time and distance, with much lower velocity than that of the conveying phase. In contrast to gas-liquid systems, the velocity of these stabilized elongated drops was 2.5 to almost 5 times higher than that of conveying liquid. Despite the similarities between flow in vertical and horizontal Hele-Shaw cell, the velocity of droplets in vertical fracture is different from that of horizontal fracture. A new correlation is derived from dimensionless analysis and the experimental data to predict the elongated drop velocity as a function of the dimensionless parameters governing the system. Introduction Two-phase flow in micro-fractures is fundamental to many different fields of advanced science and technology, such as chemical process engineering, bioengineering, medical and genetic engineering as well as petroleum engineering. For instance understanding the flow of two-phase fluids in near parallel gaps through fractured rocks has a significant effect on design of different recovery methods for naturally fractured reservoir. The flow pattern of two-phase immiscible flow in a fracture depends on the flow rates of the phases, the geometry, aperture, and roughness of the fracture, the flow properties of the phases, and interfacial tension between the phases. The flow patterns in a fracture are different from that in macro size rectangular ducts or pipes due to the small aperture, which can enhance capillary effects. The flow structure in the fracture affects the flow and transport through the surrounding porous matrix blocks. The slug flow pattern in a fracture, which occurs over a wide range of parameters, is frequently encountered in oil-wet fractured reservoirs during the immiscible displacement of viscous oil. It also occurs in natural gas reservoirs during displacement of water during gas production. In the case of a smooth-walled fracture, the slug shape can be regular elongated drops (or bubbles) flowing through the conveying phase. The elongated drops (or bubbles) can have a rounded leading edge and occupy a considerable portion of the gap with length of several times to several hundred times that of the gap. The behavior of single elongated drops (or bubbles) through a continuous liquid phase problem is as rich and complex as the viscous fingering problem6. The conditions corresponding to a swarm of the elongated drops can be inferred by considering the behavior of a single drop (bubble) and their conglomeration. The interaction of many drops or bubbles across the flow cross section can also be inferred from single drop or bubble dynamics. A firm understanding of the behavior of drop or bubble flow in a single fracture is prerequisite to build predictive models for flow in complex fracture systems.

In this paper, we numerically investigate the late-time growth of high-Reynolds-number single-mode Rayleigh-Taylor instability in a long pipe by using an advanced phase-field lattice Boltzmann multiphase method. We mainly analyze the influence of surface tension on interfacial dynamic behavior and the development of the bubble front and spike front. The numerical experiments indicate that increasing surface tension can significantly reduce the complexity of formed interfacial structure and also prevents the breakup of phase interfaces. The interface patterns in the instability process cannot always preserve the symmetric property under the extremely small surface tension, but they do maintain the symmetries with respect to the middle line as the surface tension is increased. We also report that the bubble amplitude first increases then decreases with the surface tension. There are no obvious differences between the curves of spike amplitudes for low surface tensions. However, when the surface tension increases to a critical value, it can slow down the spike growth significantly. When the surface tension is lower than the critical value, the development of the high-Reynolds-number Rayleigh-Taylor instability can be divided into four different stages, i.e. the linear growth, saturated velocity growth, reacceleration, and chaotic mixing. The bubble and spike velocities at the second stage show good agreement with those from the modified potential flow theory that takes the surface tension effect into account. After that, the bubble front and spike front are accelerated due to the formation of Kelvin-Helmholtz vortices in the interfacial region. At the late time, the bubble velocity and spike velocity become unstable and slightly fluctuate over time. To determine the nature of the late-time growth, we also measure the bubble and spike normalized accelerations at various interfacial tensions and Atwood numbers. It is found that both the spike and bubble growth rates first increase then decrease with the surface tension in general. Finally, we deduce a theoretical formula for the critical surface tension, below which the Rayleigh-Taylor instability takes place and above which tension it does not occur. It is shown that the critical surface tension increases with the Atwood number and also the numerical predictions by the lattice Boltzmann method are also in accord well with the theoretical results.

We develop a time-dependent conformal method to study the effect of viscosity on steep surface waves. When the effect of surface tension is included, numerical solutions are found that contain highly oscillatory parasitic capillary ripples. These small-amplitude ripples are associated with the high curvature at the crest of the underlying viscous-gravity wave, and display asymmetry about the wave crest. Previous inviscid studies of steep surface waves have calculated intricate bifurcation structures that appear for small surface tension. We show numerically that viscosity suppresses these. While the discrete solution branches still appear, they collapse to form a single smooth branch in the limit of small surface tension. These solutions are shown to be temporally stable, both to small superharmonic perturbations in a linear stability analysis, and to some larger amplitude perturbations in different initial-value problems. Our work provides a convenient method for the numerical computation and analysis of water waves with viscosity, without evaluating the free-boundary problem for the full Navier–Stokes equations, which becomes increasingly challenging at larger Reynolds numbers.

The Rayleigh–Taylor instability of two superposed incompressible fluids of different densities in the presence of small rotation, surface tension and suspended dust particles is investigated. The linearized equations of the problem are constructed and the general dispersion relation is obtained using normal mode analysis by applying the appropriate boundary conditions. The effects of surface tension, the Atwood number, small rotation and suspended dust particles are studied on both conditions of Rayleigh–Taylor instability and growth rate of the unstable Rayleigh–Taylor mode. The numerical calculations have been performed to see the effect of rotation, the Atwood number, relaxation frequency and mass concentration of suspended dust particles. It is found that the growth rate of Rayleigh–Taylor instability depends upon the mass concentration and relaxation frequency of suspended dust particles. The uniform small rotation, relaxation frequency and mass concentration of suspended dust particles all have stabilizing influence on the growth rate of Rayleigh–Taylor instability. It is also found that the Atwood number has destabilizing influence on the growth rate of the considered Rayleigh–Taylor configuration.

The phase split behavior of air–water slug flow at branching micro‐T‐junctions was studied by the volume of fluid model. The effects of velocity, surface tension, and viscosity on phase split results have been tested by slug units of fixed length. The split of a gas bubble at the T‐junction shows obstructing and nonobstructing modes. Through comparison, we found that the appearance of tunnels changes the driving force of bubble breakup from upstream pressure to viscous shearing force. Split features are influenced by the length of slug unit and the flow ratio in the obstructing mode. Nevertheless, the viscous force and surface tension should not be slighted for nonobstructing mode. Moreover, tendency of the flow maldistribution of nonobstructing mode is more pronounced with higher velocity and smaller surface tension. The liquid slug length has a negligible effect on the split results of gas phase. Thus, the results of two‐phase split could be obtained from the gas split ratio solely. Pattern diagram for the transition between the 2 modes was proposed. Eventually, the importance of the length information of slug unit was also verified by the predictive model of phase split that formulated with the regression analysis of numerical data.

Nonlinearities in sedimentation: a microscopic study

Formation of gas bubbles through an orifice under a liquid pool is important as a topic of fundamental research and is relevant in diverse industrial applications. This complex process is influenced by a large number of process parameters including the properties of the liquid and the gas phase. In the present work investigations have been made for bubble evolution at a submerged orifice using a computational simulation. Smoothed particle hydrodynamics is used to model the formation and detachment of the bubble from the orifice mouth. Using the advantage of this particle-based method, effects of density, viscosity, and surface tension of the liquid are investigated by noticing instantaneous bubble contour and the duration of bubble growth. The present simulation has been validated satisfactorily against published results. Finally, the process of neck formation and bubble pinch off has also been studied by varying the specific fluid properties individually. It has been noticed that the effect of surface tension and the effect of viscosity on bubble growth and necking are just the reverse of the effect of density on these processes. Moreover, compared to surface tension and density, the effect of viscosity on bubble evolution is marginal.

In the asymptotic calculations of the first order of smallness by the dimensionless amplitude of capillary waves on the surface of charged jets of polar liquid, the effect of the relaxation effect of surface tension on the regularities of their implementation is investigated. Calculations are carried out on the model of an ideal non-compressible electrically conductive fluid. It has been shown that taking into account the effect of dynamic surface tension leads to an increase in the order of the dispersion equation, which has another damping root, which is obliged to destroy the near-surface double electric layer (destruction of the ordering of polar molecules in the near-surface layer), which undergoes electrostatic instability at sufficiently large charges (pre-breakdown in the sense of ignition of corona discharge in air). In the ideal fluid mathematical model used, the relaxation motion of the jet surface disturbances that occurs when the surface tension relaxation effect is turned on and the attenuation decrements of capillary wave motions are purely of a relaxation nature.

Dynamic Surface Tension Effects in Impact of a Drop with a Solid Surface

The effects of surface tension on flooding in counter-current two-phase flow in an inclined tube

Hydrodynamic instabilities and interface dynamics of two immiscible liquids driven by a rotating disk.