Experimental classification of nonlinear dynamics in microfluidic bubbles’ flow

A nonlinear analysis method based on the evaluation of d-infinite and largest Lyapunov exponent was used to study the complex dynamics of air bubbles carried by water and flowing in a microfluidic snake channel. A rich variety of nonlinear dynamics and flow patterns was found through the experimental observation of bubbles’ motion. The results and their graphical representation show the capability of the proposed set of dimensionless parameters to classify the nonlinearity of the process showing also its sensitivity to input flow variations.     

Effects of Bubbles on the Hydraulic Conductivity of Porous Materials – Theoretical Results

In a porous material, both the pressure drop across a bubble and its speed are nonlinear functions of the fluid velocity. Nonlinear dynamics of bubbles in turn affect the macroscopic hydraulic conductivity, and thus the fluid velocity. We treat a porous medium as a network of tubes and combine critical path analysis with pore-scale results to predict the effects of bubble dynamics on the macroscopic hydraulic conductivity and bubble density. Critical path analysis uses percolation theory to find the dominant (approximately) one-dimensional flow paths. We find that in steady state, along percolating pathways, bubble density decreases with increasing fluid velocity, and bubble density is thus smallest in the smallest (critical) tubes. We find that the hydraulic conductivity increases monotonically with increasing capillary number up to Ca 10^–2, but may decrease for larger capillary numbers due to the relative decrease of bubble density in the critical pores. We also identify processes that can provide a positive feedback between bubble density and fluid flow along the critical paths. The feedback amplifies statistical fluctuations in the density of bubbles, producing fluctuations in the hydraulic conductivity.     

The effect of the VOF–CSF static contact angle boundary condition on the dynamics of sliding and bouncing ellipsoidal bubbles

The static contact angle is the only empiricism introduced in a Volume of Fluid–Continuum Surface Force (VOF–CSF) model of bubbly flow. Although it has previously been shown to have a relatively limited effect on the accuracy of velocity and shape predictions in the case of large gas bubbles sliding under inclined walls (e.g. Cook and Behnia, 2001), it may have a more determining influence on the numerical prediction of the dynamics of smaller ellipsoidal bubbles which were shown by Tsao and Koch (1997) to bounce repeatedly when sliding under inclined walls at certain wall inclinations. The present paper reports on the influence of surface tension and the static contact angle on the dynamics of an ellipsoidal air bubble of equivalent diameter D_e = 3.4 mm. The bubble Eötvös and Morton numbers are Eo = 1.56 and Mo = 2 × 10⁻¹¹ respectively. The computational results are achieved with a Piecewise Linear Construction (PLIC) of the interface and are reviewed with reference to experimental measurements of bubble velocity and interface shape oscillations recorded using a high speed digital camera. Tests are performed at plate inclination angles θ ∈ {10°, 20°, 30°, 45°} to the horizontal and computational models consider three static contact angles θ_c ∈ {10°, 20°, 30°}. The static contact angle has been found to have a significant effect on the bubble dynamics but to varying degree depending on the plate inclination. It is shown to promote lift off and bouncing when the plate inclination angle reaches 30° in a way that does not necessarily reflect experimental observations.     

Bubble dynamics: a review and some recent results

Several aspects of small-amplitude oscillations of bubbles containing gas, vapor, or a gas-vapor mixture are discussed. An application to pressure-wave propagation in a bubbly liquid is described. Nonlinear forced oscillations are considered in the light of recent research on forced oscillations of nonlinear systems. The growth of vapor bubbles, an extension of the Rayleigh-Plesset equation to non-Newtonian liquids and appreciable mass transfer at the interface, and a boundary integral numerical method for nonspherical cavitation bubble dynamics are also briefly discussed.     

Nonlinear dynamics of self-organising bubble departures from twin nozzles

Meccanica - The nonlinear dynamics of self-organising bubble departures from twin nozzles in engine oils was analysed. Air bubbles were generated from twin brass nozzles with an inner diameter...     

Spatial distribution of the gas phase in an axisymmetric submerged impact jet

Experimental data on the spatial distribution of the gas phase in an axisymmetric impact jet are obtained by the particle image velocimetry/laser-induced fluorescence (PIV/LIF) method. It is shown that the distribution of bubbles in the flow is determined by the dynamics of vortex structures. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 4, pp. 33–38, July–August, 2009.     

A boundary element method for the solution of finite mobility ratio immiscible displacement in a Hele‐Shaw cell

In this paper, the interaction between two immiscible fluids with a finite mobility ratio is investigated numerically within a Hele‐Shaw cell. Fingering instabilities initiated at the interface between a low‐viscosity fluid and a high‐viscosity fluid are analysed at varying capillary numbers and mobility ratios using a finite mobility ratio model. The present work is motivated by the possible development of interfacial instabilities that can occur in porous media during the process of CO₂ sequestration but does not pretend to analyse this complex problem. Instead, we present a detailed study of the analogous problem occurring in a Hele‐Shaw cell, giving indications of possible plume patterns that can develop during the CO₂ injection. The numerical scheme utilises a boundary element method in which the normal velocity at the interface of the two fluids is directly computed through the evaluation of a hypersingular integral. The boundary integral equation is solved using a Neumann convergent series with cubic B‐Spline boundary discretisation, exhibiting sixth‐order spatial convergence. The convergent series allows the long‐term nonlinear dynamics of growing viscous fingers to be explored accurately and efficiently. Simulations in low‐mobility ratio regimes reveal large differences in fingering patterns compared with those predicted by previous high‐mobility ratio models. Most significantly, classical finger shielding between competing fingers is inhibited. Secondary fingers can possess significant velocity, allowing greater interaction with primary fingers compared with high‐mobility ratio flows. Eventually, this interaction can lead to base thinning and the breaking of fingers into separate bubbles. Copyright © 2015 John Wiley & Sons, Ltd.     

The Application of Nonlinear Dynamics in Nursing Research

In this article, the authors present an overview of the applications of chaos theory and nonlinear dynamics to problems of relevance not only to nurses, but to anyone dealing with human functioning and interaction. These applications have been in the areas of epidemiology, nursing management and physiological functioning. In some cases, the applications were successful in identifying information that would have been overlooked using traditional methods. In other cases, the problems of data collection and analysis, unique to nonlinear dynamics, are still being developed and tested.     

Phase Interaction in a Vapor-Liquid Medium in Transient Flow Regimes

A mathematical model and a numerical method are developed for studying nonlinear wave processes in two-phase liquids with gas or vapor bubbles under conditions of impact interaction with deformable media. On the basis of the proposed approach to the numerical modeling of the dynamics of the transient processes in the two-phase vapor-liquid and deformable media, the basic features of the phase behavior, the phase transitions, and the interphase heat and mass transfer, typical of liquids containing vapor bubbles, are analyzed. The results of solving problems of the dynamics of different vapor-liquid media are presented.__________Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, 2005, pp. 88–102.Original Russian Text Copyright © 2005 by Petushkov.     

On the modeling and simulation of a laser‐induced cavitation bubble

Single cavitation bubbles exhibit severe modeling and simulation difficulties. This is due to the small scales of time and space as well as due to the involvement of different phenomena in the dynamics of the bubble. For example, the compressibility, phase transition, and the existence of a noncondensable gas inside the bubble have strong effects on the dynamics of the bubble. Moreover, the collapse of the bubble involves the occurrence of critical conditions for the pressure and temperature. This adds extra difficulties to the choice of equations of state. Even though several models and simulations have been used to study the dynamics of the cavitation bubbles, many details are still not clearly accounted for. Here, we present a numerical investigation for the collapse and rebound of a laser‐induced cavitation bubble in liquid water. The compressibility of the liquid and vapor are involved. In addition, great focus is devoted to study the effects of phase transition and the existence of a noncondensable gas on the dynamics of the collapsing bubble. If the bubble contains vapor only, we use the six‐equation model for two‐phase flows that was modified in our previous work [A. Zein, M. Hantke, and G. Warnecke, J. Comput. Phys., 229(8):2964‐2998, 2010]. This model is an extension to the six‐equation model with a single velocity of Kapila et al. (Phys. Fluid, 13:3002‐3024, 2001) taking into account the heat and mass transfer. To study the effect of a noncondensable gas inside the bubble, we add a third phase to the original model. In this case, the phase transition is considered only at interfaces that separate the liquid and its vapor. The stiffened gas equations of state are used as closure relations. We use our own method to determine the parameters to obtain reasonable equations of state for a wide range of temperatures and make them suitable for the phase transition effects. We compare our results with experimental ones. Also our results confirm some expected physical phenomena. Copyright © 2013 John Wiley & Sons, Ltd.     

Synchronous oscillations and symmetry breaking in a model of two interacting ultrasound contrast agents

Nonlinear Dynamics - We study nonlinear dynamics in a system of two coupled oscillators, describing the motion of two interacting microbubble contrast agents. In the case of identical bubbles, the...     

OBSERVATIONS OF BUBBLE SELF-ORGANIZATION IN LASER-MATTER INTERACTIONS

Bubbles generated in laser–metal interactions (LMI) at the transition from planar-to-volume boiling behave like hard (and soft) spheres. Micrographic analysis indicates that the surface layer of vaporizing metal, which comprises small microscale bubbles, behaves like a two-dimensional granular system. Bubble collision leads to bubble self-organization which changes with laser power density Q_p, and falls into (i) the kinetic regime, (ii) the clustering regime, or (iii) the regime of inelastic collapse (when bubbles form chain-like clusters), in analogy with the results of numerical simulations of hard-sphere dynamics. At a certain power density, shock waves that travel over the surface become strong enough, so that the system of bubbles behaves like horizontally and vertically shaken granular material. Since bubbles in the vibrational process are decelerated, hydrodynamic forces deform them, causing them to stay coalesced after the collision, forming a cluster with plateau borders, which is actually metal foam. The results presented show the existence of various complexity levels in the self-organization of bubbles, their sensitivity to the variation of interaction parameters, and finally, they indicate some universal features of bubble system dynamics.     

Sorting-free Hill-based stability analysis of periodic solutions through Koopman analysis

In this paper, we aim to study nonlinear time-periodic systems using the Koopman operator, which provides a way to approximate the dynamics of a nonlinear system by a linear time-invariant system of higher order. We propose for the considered system class a specific choice of Koopman basis functions combining the Taylor and Fourier bases. This basis allows to recover all equations necessary to perform the harmonic balance method as well as the Hill analysis directly from the linear lifted dynamics. The key idea of this paper is using this lifted dynamics to formulate a new method to obtain stability information from the Hill matrix. The error-prone and computationally intense task known by sorting, which means identifying the best subset of approximate Floquet exponents from all available candidates, is circumvented in the proposed method. The Mathieu equation and an n-DOF generalization are used to exemplify these findings.

     

Numerical calculation of nonlinear normal modes in structural systems

This paper presents two methods for numerical calculation of nonlinear normal modes (NNMs) in multi-degree-of-freedom, conservative, nonlinear structural dynamics models. The approaches used are briefly described as follows. Method 1: Starting with small amplitude initial conditions determined by a selected mode of the associated linear system, a small amount of negative damping is added in order to “artificially destabilize” the system; numerical integration of the system equations of motion then produces a simulated response in which orbits spiral outward essentially in the nonlinear modal manifold of interest, approximately generating this manifold for moderate to strong nonlinearity. Method 2: Starting with moderate to large amplitude initial conditions proportional to a selected linear mode shape, perform numerical integration with the coefficient ε of the nonlinearity contrived to vary slowly from an initial value of zero; this simulation methodology gradually transforms the initially flat eigenspace for ε = 0 into the manifold existing quasi-statically for instantaneous values of ε. The two methods are efficient and reasonably accurate and are intended for use in finding NNMs, as well as interesting behavior associated with them, for moderately and strongly nonlinear systems with relatively many degrees of freedom (DOFs).     

On Nonlinear Baroclinic Waves and Adjustment of Pancake Dynamics

Three-dimensional nonhydrostatic Euler–Boussinesq equations are studied for Bu=O(1) flows as well as in the asymptotic regime of strong stratification and weak rotation. Reduced prognostic equations for ageostrophic components (divergent velocity potential and geostrophic departure/thermal wind imbalance) are analyzed. We describe classes of nonlinear anisotropic ageostrophic baroclinic waves which are generated by the strong nonlinear interactions between the quasi-geostrophic modes and inertio-gravity waves. In the asymptotic regime of strong stratification and weak rotation we show how switching on weak rotation triggers frontogenesis. The mechanism of the front formation is contraction in the horizontal dimension balanced by vertical shearing through coupling of large horizontal and small vertical scales by weak rotation. Vertical slanting of these fronts is proportional to μ^−1/2 where μ is the ratio of the Coriolis and Brunt–V?is?l? parameters. These fronts select slow baroclinic waves through nonlinear adjustment of the horizontal scale to the vertical scale by weak rotation, and are the envelope of inertio-gravity waves. Mathematically, this is generated by asymptotic hyperbolic systems describing the strong nonlinear interactions between waves and potential vorticity dynamics. This frontogenesis yields vertical “gluing” of pancake dynamics, in contrast to the independent dynamics of horizontal layers in strongly stratified turbulence without rotation. Received 8 April 1997 and accepted 29 March 1998     

Periodic orbits,basins of attraction and chaotic beats in two coupled Kerr oscillators

Kerr oscillators are model systems which have practical applications in nonlinear optics. Optical Kerr effect, i.e., interaction of optical waves with nonlinear medium with polarizability χ ⁽³⁾ is the basic phenomenon needed to explain, for example, the process of light transmission in fibers and optical couplers. In this paper, we analyze the two Kerr oscillators coupler and we show that there is a possibility to control the dynamics of this system, especially by switching its dynamics from periodic to chaotic motion and vice versa. Moreover, the switching between two different stable periodic states is investigated. The stability of the system is described by the so-called maps of Lyapunov exponents in parametric spaces. Comparison of basins of attractions between two Kerr couplers and a single Kerr system is also presented.     

Dynamics of microbubble oscillators with delay coupling

Two vibrating bubbles submerged in a fluid influence each others’ dynamics via sound waves in the fluid. Due to finite sound speed, there is a delay between one bubble’s oscillation and the other’s. This scenario is treated in the context of coupled nonlinear oscillators with a delay coupling term. It has previously been shown that with sufficient time delay, a supercritical Hopf bifurcation may occur for motions in which the two bubbles are in phase. In this work, we further examine the bifurcation structure of the coupled microbubble equations, including analyzing the sequence of Hopf bifurcations that occur as the time delay increases, as well as the stability of this motion for initial conditions which lie off the in-phase manifold. We show that in fact the synchronized, oscillating state resulting from a supercritical Hopf is attracting for such general initial conditions.     

Mathematical modelling of bubble driven flows in metallurgical processes

The complex fluid dynamics of two-phase bubbly flows in metallurgical reactors is modelled numerically by using a k–e turbulence model for the liquid phase, with a driving force determined by considering the motion of the bubbles. The latter are affected by the buoyancy forces and the drag caused by their relative motion with the mean and turbulent motions of the liquid, the turbulent component being obtained by random sampling to give an ensemble of bubble trajectories. The two-way coupling between the two phases is resolved by an iterative procedure which converges on a stable overall solution. The results are compared with measurements carried out on an air-water model and show good overall agreement.     

A new approach to investigate the nonlinear dynamics in a (3 + 1)-dimensional nonlinear evolution equation via Wronskian condition with a free function

This paper proposes a new approach to investigate the nonlinear dynamics in a (3 + 1)-dimensional nonlinear evolution equation via Wronskian condition with a free function. Firstly, a Wronskian condition involving a free function is introduced for the equation. Secondly, by solving the Wronskian condition, some exact solutions are presented. Thirdly, the dynamical behaviors are analyzed by choosing specific functions in the Wronskian condition. In addition, some exact solutions are graphically illustrated by using Mathematica symbolic computations. The dynamical behaviors include stationary y-breather, line-soliton resonance, line-soliton-like phenomenon, parabola–soliton interaction, cubic–parabola–soliton resonance, kink behavior, and singular waves. These results not only illustrate the merits of the proposed method in deriving new exact solutions but also novel dynamical behaviors in the (3 + 1)-dimensional nonlinear evolution equation.

     

On the dynamics and instability of bubbles formed during underwater explosions

Dynamics of explosion bubbles formed during underwater detonations are studied experimentally by exploding fuel (hydrogen and/or carbon monoxide)–oxygen mixture in a laboratory water tank. Sub-scale explosions are instrumented to provide detailed histories of bubble shape and pressure. Using geometric and dynamic scaling analyses it has been shown that these sub-scale bubbles are reasonable approximations of bubbles formed during deep sea underwater explosions. The explosion bubble undergoes pulsation and loses energy in each oscillation cycle. The observed energy loss, which cannot be fully explained by acoustic losses, is shown here to be partly due to the excitation of instability at the interface between the gaseous bubble and the surrounding water. Various possible mechanisms for the dissipation of bubble energy are addressed. The analysis of the experimental data gives quantitative evidence (confirmed by recent numerical studies) that the Rayleigh–Taylor instability is excited near the bubble minimum. The dynamics of the bubble oscillation observed in these experiments are in good agreement with experimental data obtained from deep sea explosions