The motion of long bubbles through viscoelastic fluids in capillary tubes

The penetration of long gas bubble through a viscoelastic fluid in a capillary tube has been studied in order to investigate the influence of viscoelastic material properties on the hydrodynamic coating thickness and local flow kinematics. Experiments are conducted for three tailored ideal elastic (Boger) fluids, designed to exhibit similar steady shear properties but substantially different elastic material functions. This allows for the isolation of elastic and extensional material effects on the bubble penetration process. The shear and extensional rheology of the fluid is characterized using rotational and filament stretching rheometers (FSR). The fluids are designed such that the steady-state extensional viscosity measured by the FSR at a Deborah number (De) greater than 1 differs over three orders of magnitude (Trouton ratio = 10³–10⁶). The experiment set up to measure the hydrodynamic coating thickness is designed to provide accurate data over a wide range of capillary numbers (0.01 < Ca < 100). The results indicate that the coating thickness in this process increases with an increase in the extensionally thickening nature of the fluid. Experiments are also conducted using several different capillary tube diameters (0.1 < D < 1 cm), in order to compare responses at similar Ca but different flow De. Suitable scaling methods and nonlinear viscoelastic constitutive equations are explored to characterize the displacement process for polymeric fluids. Bubble tip shapes at different De are recorded using a CCD camera, and measured using an edge detection algorithm. The influence of the mixed flow field on the bubble tip shape is examined. Particle tracking velocimetry experiments are conducted to compare the influence of viscoelastic properties on the velocity field in the vicinity of the bubble tip. Local shear and extension rates are calculated in the vicinity of the bubble tip from the velocity data. The results provide quantitative information on the influence of elastic and extensional properties on the bubble penetration process in gas-assisted injection molding. The bubble shape and velocity field information provides a basis for evaluating the performance of constitutive equations in mixed flow. Received: 19 January 1999 Accepted: 30 June 1999     

Flow in a Porous Matrix with Anisotropic Structure

The flow of a viscous fluid through a porous matrix undergoing only infinitesimal deformation is described in terms of intrinsic variables, namely, the density, velocity and stress occurring in coherent elements of each material. This formulation arises naturally when macroscopic interfaces are conceptually partitioned into area fractions of fluid–fluid, fluid–solid, and solid–solid contact. Such theory has been shown to yield consistent jump conditions of mass, momentum and energy across discontinuities, either internal or an external boundary, unlike the standard mixture theory jump conditions. In the previous formulation, the matrix structure has been considered isotropic; that is, the area fractions are independent of the interface orientation. Here, that is not assumed, so in particular, the cross-section area of a continuous fluid tube depends on its orientation, which influences the directional fluxes, and in turn the directional permeability, anisotropy of the structure. The simplifications for slow viscous flow are examined, and particularly for an isotropic linearly elastic matrix in which area partitioning induces anisotropic elastic response of the mixture. A final specialization to an incompressible fluid and stationary matrix leads to potential flow, and a simple plane flow solution is presented to illustrate the effects of anisotropic permeability.     

A rising bubble in a polymer solution

We propose a boundary integral method to study the shape of a bubble rising under gravity in a dilute polymer solution. Constitutive properties are modelled using a FENE model [M.D. Chilcott, J.M. Rallison, J. Non-Newtonian Fluid Mech. 29 (1988) 381] with a pure surface tension interface. We employ a birefringent strand representation [O.G. Harlen, J.M. Rallison, M.D. Chilcott, High-Deborah-number flows of dilute polymer, J. Non-Newtonian Fluid Mech.34 (1990) 319–349] of the wake to simulate the shape and the time-dependent motion of the bubble. Steady and non-steady solutions reproduce qualitatively the bubble deformation seen in experiment with a small region of very high curvature near the rear stagnation point of the bubble. We find a limit point for steady axisymmetric solutions if the polymer concentration is increased or the surface tension is decreased. Rise speed jump discontinuities were not found.     

Seismic Reflection from an Interface Between an Elastic Solid and a Fractured Porous Medium with Partial Saturation

The theory of Tuncay and Corapcioglu (Transp Porous Media 23:237–258, 1996a) has been employed to investigate the possibility of plane wave propagation in a fractured porous medium containing two immiscible fluids. Solid phase of the porous medium is assumed to be linearly elastic, isotropic and the fractures are assumed to be distributed isotropically throughout the medium. It has been shown that there can exist four compressional waves and one rotational wave. The phase speeds of these waves are found to be affected by the presence of fractures, in general. Of the four compressional waves, one arises due to the presence of fractures in the medium and the remaining three are those encountered by Tuncay and Corapcioglu (J Appl Mech 64:313–319, 1997). Reflection and transmission phenomena at a plane interface between a uniform elastic half-space and a fractured porous half-space containing two immiscible fluids, are analyzed due to incidence of plane longitudinal/transverse wave from uniform elastic half-space. Variation of modulus of amplitude and energy ratios with the angle of incidence are computed numerically by taking the elastic half-space as granite and the fractured porous half-space as sandstone material containing non-viscous wetting and non-wetting fluid phases. The results obtained in case of porous half-space with fractures, are compared graphically with those in case of porous half-space without fractures. It is found that the presence of fractures in the porous half-space do affect the reflection/transmission of waves, which is responsible for raising the reflection and lowering the transmission coefficients.     

Bubble jet flow interaction during pool boiling of subcooled water on horizontal thin wires

Both visual experiments and numerical analyses were conducted to investigate the interaction between bubble jet flows during pool boiling of subcooled water on horizontal thin wires. The bubble jet flows nearby attracted each other, and they can combine into one jet flow under strong interaction. As the adjacent bubble departs, the bubble jet flow would experience an unsteady evolution process with the jet flow interaction weakening. Since the unsymmetrical thermocapillary force at the bubble interface was induced by the adjacent bubble as a cold source, the bubble jet flow would trend to the adjacent bubble, and the mechanism based on thermocapillary force and cold source can explain the bubble jet flow interaction very well. The steady bubble jet flow interaction phenomena were further simulated by laminar model, and the calculated jet flow interaction phenomena were in a good agreement with the experimental results.     

Green’s formula and singularity at a triple contact line. Example of finite-displacement solution

The various equations at the surfaces and triple contact lines of a deformable body are obtained from a variational condition, by applying Green’s formula in the whole space and on the Riemannian surfaces. The surface equations are similar to the Cauchy’s equations for the volume, but involve a special definition of the ‘divergence’ (tensorial product of the covariant derivatives on the surface and the whole space). The normal component of the divergence equation generalizes the Laplace’s equation for a fluid–fluid interface. Assuming that Green’s formula remains valid at the contact line (despite the singularity), two equations are obtained at this line. The first one expresses that the fluid–fluid surface tension is equilibrated by the two surface stresses (and not by the volume stresses of the body) and suggests a finite displacement at this line (contrary to the infinite-displacement solution of classical elasticity, in which the surface properties are not taken into account). The second equation represents a strong modification of Young’s capillary equation. The validity of Green’s formula and the existence of a finite-displacement solution are justified with an explicit example of finite-displacement solution in the simple case of a half-space elastic solid bounded by a plane. The solution satisfies the contact line equations and its elastic energy is finite (whereas it is infinite for the classical elastic solution). The strain tensor components generally have different limits when approaching the contact line under different directions. Although Green’s formula cannot be directly applied, because the stress tensor components do not belong to the Sobolev space H¹(V)$H^1(\text{V})$, it is shown that this formula remains valid. As a consequence, there is no contribution of the volume stresses at the contact line. The validity of Green’s formula plays a central role in the theory.     

Numerical simulations of bouncing jets

Bouncing jets are fascinating phenomenon occurring under certain conditions when a jet impinges on a free surface. This effect is observed when the fluid is Newtonian and the jet falls in a bath undergoing a solid motion. It occurs also for non‐Newtonian fluids when the jets fall in a vessel at rest containing the same fluid. We investigate numerically the impact of the experimental setting and the rheological properties of the fluid on the onset of the bouncing phenomenon. Our investigations show that the occurrence of a thin lubricating layer of air separating the jet and the rest of the liquid is a key factor for the bouncing of the jet to happen. The numerical technique that is used consists of a projection method for the Navier–Stokes system coupled with a level set formulation for the representation of the interface. The space approximation is carried out with adaptive finite elements. Adaptive refinement is shown to be very important to capture the thin layer of air that is responsible for the bouncing. Copyright © 2015 John Wiley & Sons, Ltd.     

单空泡与自由液面相互作用规律研究进展

空化与空泡溃灭现象普遍存在于自然界、标识码械和生物医学等领域.空泡与自由面相互作用会产生瞬态强烈耦合,涉及到空泡非球形溃灭、自由面非线性变形及失标识码象,是流体力学领域重要的前沿与基础问题. 本文围绕这一热点,从空泡非球形演化和自由面变形规律角度出发,概标识码纳近年该领域的研究进展与成果. 对于近自由面空泡的非球形演化,基于表征开尔文冲量的无量纲参数,重点关注了体积振荡、射流生成、水锤效应及溃灭标识码生成等关键过程,介绍了关键参数的理论建模方法,获得了空泡溃灭过程中能量分配机制. 针对自由液面变形演化,根据细射流和粗射流生成和发展,归纳了 4 种典型现象及特点:透明水层及水柱生成、不稳定与稳标识码水裙结构. 进一步总结了开尔文冲量理论、界面凹陷奇点概念和泰勒不稳定性等理论模型的建立和应用,讨论了气泡溃灭过程、液面标识码界面稳定性等主要机制. 此外,本文也概述了空泡脉动对球状、圆柱状等非平面液面变形行为的影响,归纳了曲率对于液面变形的影响机制. 最后,针对目前研究状况提出该领域研究中尚未解决的问题,期望对将来的空泡及空泡群与自由液面相互作用深入研究提供借鉴.      

Evaporation condensation-induced bubble motion after temperature gradient set-up

Thermocapillary (Marangoni) motion of a gas bubble (or a liquid drop) under a temperature gradient can hardly be present in a one-component fluid. Indeed, in such a pure system, the vapor–liquid interface is always isothermal (at saturation temperature). However, evaporation on the hot side and condensation on the cold side can occur and displace the bubble. We have observed such a phenomenon in two different fluids submitted to a temperature gradient under reduced gravity: hydrogen under magnetic compensation of gravity in the HYLDE facility at CEA-Grenoble and water in the DECLIC facility onboard the ISS. The experiments and the subsequent analysis are performed in the vicinity of the vapor–liquid critical point to benefit from critical universality. In order to better understand the phenomena, a 1D numerical simulation has been performed. After the temperature gradient is imposed, two regimes can be evidenced. At early times, the temperatures in the bubble and the surrounding liquid become different thanks to their different compressibility and the “piston effect” mechanism, i.e. the fast adiabatic bulk thermalization induced by the expansion of the thermal boundary layers. The difference in local temperature gradients at the vapor–liquid interface results in an unbalanced evaporation/condensation phenomenon that makes the shape of the bubble vary and provoke its motion. At long times, a steady temperature gradient progressively forms in the liquid (but not in the bubble) and induces steady bubble motion towards the hot end. We evaluate the bubble velocity and compare with existing theories.     

Wave propagation at interface of heat conducting micropolar solid and fluid media

The present investigation is concerned with the wave propagation at an interface of a micropolar generalized thermoelastic solid half space and a heat conducting micropolar fluid half space. Reflection and transmission phenomena of plane waves are investigated, which impinge obliquely at the plane interface between a micropolar generalized thermoelastic solid half space and a heat conducting micropolar fluid half space.The incident wave is assumed to be striking at the interface after propagating through the micropolar generalized thermoelastic solid. The amplitude ratios of various reflected and transmitted waves are obtained in a closed form. It is found that they are a function of the angle of incidence and frequency and are affected by the elastic properties of the media. Micropolarity and thermal relaxation effects are shown on the amplitude ratios for a specific model. The results of some earlier literatures are also deduced from the present investigation.     

Simulations of bubble growth and interaction in high viscous fluids using the lattice Boltzmann method

Phenomena of growth, coalescence and breakdown of bubbles within high viscous fluids are of great interest in the fluid dynamics of multiphase fluids because of their industrial relevance, e.g. in polymer, metal alloy and food processing fields. The dynamics of multiple bubble growth in hot viscous fluids is a complex issue governed by pressure forces, vapour diffusion, surface tension and viscous forces. Effects of water evaporation from the mixture surface are responsible for phenomena like glass transition, viscous increase and dough solidification. This article presents Lattice Boltzmann simulations of nucleating bubbles with large density ratio, that grow and interact in a hot high-viscous fluid. The work focuses on the first phases of the bubble expansion, neglecting the effects of evaporation. The simulations are performed using the Lattice Boltzmann Method (LBM). The Free Surface method is used to reduce a liquid/gas two-phase flow to a single-phase flow. The interface layer between gas and fluid is tracked using the volume of fluid (VOF) method. To avoid numerical instabilities due to the high viscosity ($\eta =100Pas$), the problem is scaled from physical to LB-units through non-dimensional quantities. The bubbles are initially punched randomly into the domain with a dimension comparable with the dimension of nucleation and are allowed to grow under an internal over-pressure. The simulated final structure of the bubbles is compared with images of a pure starch fluid, extruded under same conditions. It is shown as the final bubble distribution, matrix dimension and bubble diameters in the simulation are in good agreement with the real final conformation.     

Robust numerical analysis of the dynamic bubble formation process in a viscous liquid

The dynamics of bubble formation from a submerged nozzle in a highly viscous liquid with relatively fast inflow gas velocity is studied numerically. The numerical simulations are carried out using a sharp interface coupled level set/volume-of-fluid (CLSVOF) method and the governing equations are solved through a hydrodynamic scheme with formal second-order accuracy. Numerical results agree well with experimental results and it is shown that the sharp interface CLSVOF method enables one to reproduce the bubble formation process for a wide range of inflow gas velocities. From numerical results, one can improve their understanding of the mechanisms regarding the dynamics of bubble formation. For example, it is found that for some sets of parameters that the bubble formation process reaches steady state after several bubbles are released from the nozzle. At steady state, bubbles uniformly rise freely in the viscous liquid. It is observed that the fluid flow around a formed bubble has a significant role in determining the overall dynamic process of bubble formation; e.g. the effect of the fluid flow from the preceding bubble can be seen on newly formed bubbles.     

Convective mass transfer across fluid interfaces in straight angular pores

Steady convective mass transfer to or from fluid interfaces in pores of angular cross-section is theoretically investigated. This situation is relevant to a variety of mass transport process in porous media, including the fate of residual non-aqueous phase liquid ganglia and gas bubbles. The model incorporates the essential physics of capillarity and solute mass transfer by convection and diffusion in corner fluid filaments. The geometry of the corner filaments, characterized by the fluid–fluid contact angle, the corner half-angle and the interface meniscus curvature, is accounted for. Boundary conditions of zero surface shear (‘perfect-slip’) and infinite surface shear (‘no-slip’) at the fluid–fluid interface are considered. The governing equations for laminar flow within the corner filament and convective diffusion to or from the fluid–fluid interface are solved using finite-element methods. Flow computations are verified by comparing the dimensionless resistance factor and hydraulic conductance of corner filaments against recent numerical solutions by Patzek and Kristensen (J. Colloid Interface Sci 236, 305–317 2001). Novel results are obtained for the average effluent concentration as a function of flow geometry and pore-scale Peclet number. These results are correlated to a characteristic corner length and local pore-scale Peclet number using empirical equations appropriate for implementation in pore network models. Finally, a previously published “2D-slit” approximation to the problem at hand is checked and found to be in considerable error.     

Anomalous flow properties of spherical micelle surfactant solutions passing through small-sized slits

The flow properties of complex fluids, such as aqueous solutions of polymers and surfactants, have been investigated in many studies, which revealed interesting and anomalous tendencies for several types of complex fluids in abrupt contraction and expansion flows, such as flows passing through small-sized orifices and slits. In the study, the jet thrust and excess pressure drop (net differential pressure) for experimentally observing their flow properties of water and aqueous solutions of several types of surfactants with spherical micelles in slit flows were measured. Different properties were observed for various surfactant solutions depending on the charge of the solute. The resultant jet thrust and excess pressure drop of cationic and non-ionic surfactant solutions were lower than the experimental values of water. For anionic surfactant solutions, the experimental results were similar to those of water. The types can be arranged as cationic > non-ionic > anionic in order of diminishing the jet thrust and excess pressure drop. Moreover, the effect of strong strain, boundary slip, contraction ratio, size effect, concentration, and interface phenomena was discussed.     

The SPH approach to the process of container filling based on non-linear constitutive models

In this work,the transient free surface of container filling with non-linear constitutive equation’s fluids is numerically investigated by the smoothed particle hydrodynamics(SPH) method.Specifically,the filling process of a square container is considered for non-linear polymer fluids based on the Cross model.The validity of the presented SPH is first verified by solving the Newtonian fluid and OldroydB fluid jet.Various phenomena in the filling process are shown,including the jet buckling,jet thinning,splashing or spluttering,steady filling.Moreover,a new phenomenon of vortex whirling is more evidently observed for the Cross model fluid compared with the Newtonian fluid case.     

A unifying model for fluid flow and elastic solid deformation: A novel approach for fluid–structure interaction

Fluid–structure coupling is addressed through a unified equation for compressible Newtonian fluid flow and elastic solid deformation. This is done by introducing thermodynamics within Cauchy׳s equation through the isothermal compressibility coefficient that is experimentally measurable for both fluids and solids. The vectorial resolution of the governing equation, where every component of velocity vectors and displacement variation vectors is calculated simultaneously in the overall multi-phase system, is characteristic of a monolithic resolution involving no iterative coupling. For system equation closure, mass density and pressure are both re-actualized from velocity vector divergence, when the shear stress tensor within the solid phase is re-actualized from the displacement variation vectors. This novel approach is first validated on a two-phase system, involving a plane fluid–solid interface, through the two following test cases: (i) steady-state compression and (ii) longitudinal and transverse elastic wave propagations. Then the 3D study of compressive fluid injection towards an elastic solid is analyzed from initial time to steady-state evolution.     

Momentum evolution of ejected and entrained fluid during laminar vortex ring formation

The evolution of total circulation and entrainment of ambient fluid during laminar vortex ring formation has been addressed in a number of previous investigations. Motivated by applications involving propulsion and fluid transport, the present interest is in the momentum evolution of entrained and ejected fluid and momentum exchange among the ejected, entrained fluid and added mass during vortex ring formation. To this end, vortex rings are generated numerically by transient jet ejection for fluid slug length-to-diameter (L/D) ratios of 0.5–3.0 using three different velocity programs [trapezoidal, triangular negative slope (NS), and positive slope (PS)] at a jet Reynolds number of 1,000. Lagrangian coherent structures (LCS) were utilized to identify ejected and entrained fluid boundaries, and a Runge-Kutta fourth order scheme was used for advecting these boundaries with the numerical velocity data. By monitoring the center of mass of these fluid boundaries, momentum of each component was calculated and related to the total impulse provided by the vortex ring generator. The results demonstrate that ejected fluid exchanges its momentum mostly with added mass during jet ejection and that the momentum of the entrained fluid at jet termination was < 11% of the total ring impulse in all cases except for the triangular NS case. Following jet termination, momentum exchange was observed between ejected and entrained fluid yielding significant increase in entrained fluid’s momentum. A performance metric was defined relating the impulse from over-pressure developed at the nozzle exit plane during jet ejection to the flow evolution, which increased preferentially with L/D over the range considered. An additional benefit of this study was the identification of the initial (i.e., before jet initiation) location of the fluid to be entrained into the vortex ring.     

环形喷管喷口气泡演化的实验研究

水下气泡的发展演化及气泡动力学行为是气液两相动力学的基础理论与水下射流应用的重要基础. 环形喷管/喷口形成的气泡及气体射流具有其不同于圆孔实心射流的特殊表现与规律机制,随着同心筒破水发射等特殊应用的出现,环形喷口气体射流/泡流的基础现象观测和机制分析成为迫切的需求. 基于环形喷管的设计和水下射流条件的分析,设计建立了一套环形喷管水箱实验系统,对水下环形喷管喷口气泡发展演化过程进行了初步的实验研究. 为观测研究气体通过环形喷管气泡生长发展过程,在较低压力、较低流速下,采用高速摄影仪记录气泡生长及发展演化过程. 结合对气泡发展演化过程的图像处理与分析,研究分析了环形喷口气泡形成区制、气泡生长过程形态发展特点、以及气泡形成时间及气泡体积变化特点. 研究表明:在本实验气体流量范围内(50.8~237.3 dm3/min),环形喷口气泡发展演化过程呈现较为明显的三周期区制,前泡尾流影响是环形气泡呈三周期区制的主要原因;不同周期内的气泡形成时间具有较稳定规律,并受到流量影响;气泡生长过程中有较为明显的下沉、回升特征;气泡表面张力、液体惯性与流动的共同作用,造成了典型的气泡顶部坍塌现象.      

Three-dimensional viscoelastic numerical analysis of the encapsulation phenomena in coextrusion

In recent years coextrusion process had gained wide recognition as an approach to achieving high quality or quantity and low costs by multilayered system as well as an polymer alloy or blend. We performed a strict 3-D numerical simulation with viscoelastic model on the encapsulation phenomena which is one of the problems in the coextrusion process. The effects of material properties such as viscosity ratio, extensional viscosity and second normal stress difference and an effect of flow rate ratios on the encapsulation phenomena were examined. Numerical results showed that encapsulation phenomena were affected by not only viscous properties but also elastic properties or non-linear properties. It was found that the difference of second normal stress differences working on the interface between fluid I and fluid II (we defined it as DN ₂) had a correlation with the interface shape. Received: 31 August 1998 Accepted: 15 September 1998