Adhesion and fingering in the lifting Hele-Shaw cell: Role of the substrate

The lifting Hele-Shaw cell (LHSC) is used to study adhesion as well as viscous fingering. In the present paper we report a series of observations of development of the interface for different viscous fluids, both Newtonian and non-Newtonian, in a LHSC operated at a constant lifting force. Glass and perspex are used as the plates in two different sets of experiments. The objectives are 1) to measure the time required to separate the plates as a function of the lifting force and 2) to note the force above which viscous fingering appears. We find that for the Newtonian fluids, the plate separation time follows a universal power law with the lifting force, irrespective of fluid and substrate. The non-Newtonian fluids too, with proper scaling obey the same power law. The appearance of fingering, however, depends on the properties of the fluid as well as the substrate. We suggest a modified form of the capillary number which controls the onset of fingering; this new quantity, termed the “fingering parameter” involves the dielectric constants of the substrate and fluid in addition to the viscosity and surface tension.     

Viscoelastic theory for nematic interfaces

A complete macroscopic theory for compressible nematic-viscous fluid interfaces is developed and used to characterize the interfacial elastic, viscous, and viscoelastic material properties. The derived expression for the interfacial stress tensor includes elastic and viscous components. Surface gradients of the interfacial elastic stress tensor generates tangential Marangoni forces as well as normal forces. The latter may be present even in planar surfaces, implying that in principle static planar interfaces may accommodate pressure jumps. The asymmetric interfacial viscous stress tensor takes into account the surface nematic ordering and is given in terms of the interfacial rate of deformation and interfacial Jaumann derivative. The material function that describes the anisotropic viscoelasticity is the dynamic interfacial tension, which includes the interfacial tension and dilational viscosities. Viscous dissipation due to interfacial compressibility is described by the anisotropic dilational viscosity, and it is shown to describe the Boussinesq surface fluid appropriate for Newtonian interfaces when the director is homeotropic. Three characteristic interfacial shear viscosities are defined according to whether the surface orientation is along the velocity direction, the velocity gradient, or the unit normal. In the last case the expression reduces to the interfacial shear viscosity of the Boussinesq surface fluid. The theory provides a theoretical framework to study interfacial stability, thin liquid film stability and hydrodynamics, and any other interfacial rheology phenomena.     

A lattice Boltzmann method for immiscible multiphase flow simulations using the level set method

We consider the lattice Boltzmann method for immiscible multiphase flow simulations. Classical lattice Boltzmann methods for this problem, e.g. the colour gradient method or the free energy approach, can only be applied when density and viscosity ratios are small. Moreover, they use additional fields defined on the whole domain to describe the different phases and model phase separation by special interactions at each node. In contrast, our approach simulates the flow using a single field and separates the fluid phases by a free moving interface. The scheme is based on the lattice Boltzmann method and uses the level set method to compute the evolution of the interface. To couple the fluid phases, we develop new boundary conditions which realise the macroscopic jump conditions at the interface and incorporate surface tension in the lattice Boltzmann framework. Various simulations are presented to validate the numerical scheme, e.g. two-phase channel flows, the Young–Laplace law for a bubble and viscous fingering in a Hele-Shaw cell. The results show that the method is feasible over a wide range of density and viscosity differences.     

Temperature effects on capillary instabilities in a thin nematic liquid crystalline fiber embedded in a viscous matrix

Linear stability analysis of capillary instabilities in a thin nematic liquid crystalline cylindrical fiber embedded in an immiscible viscous matrix is performed by formulating and solving the governing nemato-capillary equations, that include the effect of temperature on the nematic ordering as well as the effect of the nematic orientation. A representative axial nematic orientation texture with the planar easy axis at the fiber surface is studied. The surface disturbance is expressed in normal modes, which include the azimuthal wave number m to take into account non-axisymmetric modes. Capillary instabilities in nematic fibers reflect the anisotropic nature of liquid crystals, such as the ordering and orientation contributions to the surface elasticity and surface normal and bending stresses. Surface gradients of normal and bending stresses provide additional anisotropic contributions to the capillary pressure that may renormalize the classical displacement and curvature forces that exist in any fluid fiber. The exact nature (stabilizing and destabilizing) and magnitude of the renormalization of the displacement and curvature forces depend on the nematic ordering and orientation, i.e. the anisotropic contribution to the surface energy, and accordingly capillary instabilities may be axisymmetric or non-axisymmetric. In addition, when the interface curvature effects are accounted for as contributions of the work of interfacial bending and torsion to the total energy of the system, the higher-order bending moment contribution to the surface stress tensor is critical in stabilizing the fiber instabilities. For the planar easy axis, the nematic ordering contribution to the surface energy, which renormalizes the effect of the fiber shape, plays a crucial role to determine the instability mechanisms. Moreover, the unstable modes, which are most likely observed, can be driven by the dependence of surface energy on the surface area. Low-ordering fibers display the classical axisymmetric mode, since the surface energy decreases by decreasing the surface area. Decreasing temperature gives rise to the encounter with a local maximum or to monotonic increase of the characteristic length of the axisymmetric mode. Meanwhile, in the presence of high surface ordering, non-axisymmetric finite wavelength instabilities emerge, with higher modes growing faster since the surface energy decreases by increasing the surface area. As temperature decreases, the pitches of the chiral microstructures become smaller. However, this non-axisymmetric instability mechanism can be regulated by taking account of the surface bending moment, which contains higher order variations in the interface curvatures. More and more non-axisymmetric modes emerge as temperature decreases, but, at constant temperature, only a finite number of non-axisymmetric modes are unstable and a single fastest growing mode emerges with lower and higher unstable modes growing slower. For nematic fibers, the classical fiber-to-droplet transformation is one of several possible instability pathways, while others include chiral microstructures. The capillary instabilities' growth rate of a thin nematic fiber in a viscous matrix is suppressed by increasing either the fiber or matrix viscosity, but the estimated droplet sizes after fiber breakup in axisymmetric instabilities decrease with increasing the matrix viscosity. Received 15 April 2002 and Received in final form 3 October 2002 RID="a" ID="a"e-mail: alejandro.rey@mcgill.ca     

The effect of velocity-dependent boundary conditions on pattern formation

We discuss the role played by velocity-dependent boundary conditions in several problems of non-local interface dynamics. Qualitative and quantitative effects are demonstrated through three examples which are: (i) quantitative verification of the scenario of ‘microscopic solvability’ in pattern selection for the case of viscous fingering in a rectangular Hele-Shaw cell, (ii) detection of the presence of this type of boundary condition through an analysis of the statistical properties of the interface for radial Hele-Shaw flow and, (iii) generation of qualitatively different interface morphologies by tuning the magnitude of this effect in the case of radial Hele-Shaw flow. We also point out the possible implications of this last example for the observation of similar morphologies in some problems of solidification.     

I. Formation and rise of a bubble stream in a viscous liquid

The continuous emission of gas bubbles from a single ejection orifice immersed in a viscous fluid is considered. We first present a semi empirical model of spherical bubble growth under constant flow conditions to predict the bubble volume at the detachment stage. In a second part, we propose a physical model to describe the rise velocity of in-line interacting bubbles and we derive an expression for the net viscous force acting on the surrounding fluid. Experimental results for air/water-glycerol systems are presented for a wide range of fluid viscosity and compared with theoretical predictions. An imagery technique was used to determine the bubble size and rise velocity. The effects of fluid viscosity, gas flow rate, orifice diameter and liquid depth on the bubble stream dynamic were analyzed. We have further studied the effect of large scale recirculation flow and the influence of a neighbouring bubble stream on the bubble growth and rising velocity. Received: 23 July 1997 / Revised: 16 December 1997 / Accepted: 11 May 1998     

Computer simulation of viscous fingering in a lifting Hele-Shaw cell with grooved plates

We simulate viscous fingering generated by separating two plates with a constant force, in a lifting Hele-Shaw cell. Variation in the patterns for different fluid viscosity and lifting force is studied. Viscous fingering is strongly affected by anisotropy. We report a computer simulation study of fingering patterns, where circular or square grooves are etched on to the lower plate. Results are compared with experiments.     

Nonlinear Saffman-Taylor instability

We show, both theoretically and experimentally, that the interface between two viscous fluids in a Hele-Shaw cell can be nonlinearly unstable before the Saffman-Taylor linear instability point is reached. We identify the family of exact elastica solutions [Nye et al., Eur. J. Phys. 5, 73 (1984)]] as the unstable branch of the corresponding subcritical bifurcation which ends up at a topological singularity defined by interface pinchoff. We devise an experimental procedure to prepare arbitrary initial conditions in a Hele-Shaw cell. This is used to test the proposed bifurcation scenario and quantitatively asses its practical relevance.     

Kinetic roughening in two-phase fluid flow through a random Hele-Shaw cell

A nonlocal interface equation is derived for two-phase fluid flow, with arbitrary wettability and viscosity contrast, c=(mu(1)-mu(2))/(mu(1)+mu(2)), in a model porous medium defined as a Hele-Shaw cell with random gap b(0)+delta b. Fluctuations of both capillary and viscous pressure are explicitly related to the microscopic quenched disorder, yielding conserved, nonconserved, and power-law correlated noise terms. Two length scales are identified that control the possible scaling regimes and which scale with capillary number Ca as l(1) approximately b(0)(cCa)(-1/2) and l(2) approximately b(0)Ca-1. Exponents for forced fluid invasion are obtained from numerical simulation and compared with recent experiments.     

Viscosity minimum in bimodal concentrated suspensions under shear

We study a model of concentrated suspensions under shear in two dimensions. Interactions between suspended particles are dominated by direct-contact viscoelastic forces and the particles are neutrally bouyant. The bimodal suspensions consist of a variable proportion between large and small droplets, with a fixed global suspended fraction. Going beyond the assumptions of the classical theory of Farris (R.J. Farris, Trans. Soc. Rheol. 12, 281 (1968)), we discuss a shear viscosity minimum, as a function of the small-to-large-particle ratio, in shear geometries imposed by external body forces and boundaries. Within a linear-response scheme, we find the dependence of the viscosity minimum on the imposed shear and the microscopic drop friction parameters. We also discuss the viscosity minimum under dynamically imposed shear applied by boundaries. We find a reduction of macroscopic viscosity with the increase of the microscopic friction parameters that is understood using a simple two-drop model. Our simulation results are qualitatively consistent with recent experiments in concentrated bimodal emulsions with a highly viscous or rigid suspended component. Received 28 June 2002 RID="a" ID="a"e-mail: ernesto@pion.ivic.ve     

Dynamics of immiscible radial viscous fingering: A numerical study

In this paper we undertake a numerical investigation of the dynamics of the interface in the problem of immiscible radial viscous fingering in a Hele-Shaw cell when the areal flow rate is maintained constant. Comparison is made with experimental results to check if there is a need to introduce velocity-dependent boundary conditions and to incorporate the effect of thickness of the film left behind by the moving interface. Some new scaling results are suggested by the numerical data. These data, along with those available from laboratory experiments, provide support for a mean field theory for radial immiscible viscous fingering published recently [Phys. Rev. Lett. 65, 2680 (1990)].     

Viscosity effects in foam drainage: Newtonian and non-newtonian foaming fluids

We have studied the drainage of foams made from Newtonian and non-Newtonian solutions of different viscosities. Forced-drainage experiments first show that the behavior of Newtonian solutions and of shear-thinning ones (foaming solutions containing either Carbopol or Xanthan) are identical, provided one considers the actual viscosity corresponding to the shear rate found inside the foam. Second, for these fluids, a drainage regime transition occurs as the bulk viscosity is increased, illustrating a coupling between surface and bulk flow in the channels between bubbles. The properties of this transition appear different from the ones observed in previous works in which the interfacial viscoelasticity was varied. Finally, we show that foams made of solutions containing long flexible PolyEthylene Oxide (PEO) molecules counter-intuitively drain faster than foams made with Newtonian solutions of the same viscosity. Complementary experiments made with fluids having all the same viscosity but different responses to elongational stresses (PEO-based Boger fluids) suggest an important role of the elastic properties of the PEO solutions on the faster drainage.     

On the importance of low-energy beta beams for supernova neutrino physics

We argue that isotropization and, consequently, thermalization of the system of gluons and quarks produced in an ultrarelativistic heavy-ion collision does not follow from Feynman diagram analysis to any order in the coupling constant. We conclude that the apparent thermalization of quarks and gluons, leading to success of perfect fluid hydrodynamics in describing heavy-ion collisions at RHIC, can only be attributed to the non-perturbative QCD effects not captured by Feynman diagrams. We proceed by modeling these non-pertrubative thermalization effects using viscous hydrodynamics. We point out that matching Color Glass Condensate inital conditions with viscous hydrodynamics leads to a continuous evolution of all the components of the energy-momentum tensor and, unlike the case of ideal hydrodynamics, does not give rise to a discontinuity in the longitudinal pressure. An important consequence of such a matching is a relationship between the thermalization time and shear viscosity: we observe that small viscosity leads to short thermalization time.     

A two-phase diffuse-interface model for Hele-Shaw flows with large property contrasts

A novel two-phase diffuse-interface model is used to simulate flows inside a Hele-Shaw cell. The model assumes that the two phases coexist inside the diffuse interface, with different velocities and properties. A separate equation is used to calculate the slip velocity between the two phases inside the diffuse interface. It is shown that for one-dimensional flows parallel to the diffuse interface, the results are independent of the diffuse-interface width, regardless of the magnitude of the density and viscosity contrasts between the phases. This two-phase approach is coupled with a phase-field equation for calculating the interface motion. The model is applied to a buoyancy-driven two-phase flow involving a Rayleigh-Taylor instability and validated through a comparison with available sharp-interface results. The flows and interface topology changes are investigated for large density and viscosity contrasts between the phases. The convergence of the results with respect to the interface width is examined in detail. It is shown that the two-phase model converges better than a standard diffuse-interface model that assumes the presence of a single velocity inside the diffuse interface. Remaining interface width dependencies can be attributed to the capillary stress term in the momentum equation.     

Apparent slip at a polymer-polymer interface

We consider a planar interface between strongly-segregated homopolymers subjected to steady shear in the plane of the interface. We develop a constitutive equation for stress relaxation in an inhomogeneous system for chains obeying Rouse dynamics. Using this equation, the interfacial viscosity for a symmetric blend is found to be in agreement with a scaling prediction due to de Gennes, where is the bead friction coefficient, b is the segment length, is the segment volume and is the Flory-Huggins interaction parameter driving the phase separation. We generalize our results to asymmetric blends and describe a phenomenological extension to entangled melts. Received: 18 August 1997 / Received in final form: 1 December 1997 / Accepted: 2 December 1997     

Lattice-Boltzmann study of spontaneous emulsification

We study the dynamics of spontaneous emulsification of an initially planar oil-water interface when surfactants are added. The thermodynamic properties of the ternary oil-water-surfactant system are modeled by a Ginzburg-Landau-type free energy. The lattice Boltzmann method is used to solve the dynamic equations. The dynamics is found to be governed by a complicated interplay of convection and diffusion as the two relevant transport mechanisms. As long as the interface is almost flat, we find the interfacial area to grow first exponentially and then linearly in time. Later finger-like structures form which grow with a constant velocity. The tip velocity is found to increase roughly linearly with the mobility of the amphiphile, and to decrease as with the solvent viscosity . Received 5 January 1999     

Dewetting processes in a cylindrical geometry

Dewetting of liquid films of water-glycerol solutions of different viscosities has been studied experimentally in PVC cylindrical tubes. In contrast with plane surfaces, the dewetting capillary number Ca_vd increases with the film thickness h₀ over a large part of the experimental range and follows a same global trend independent of viscosity as a function of h₀. This increase is only partly explained by variations of the capillary driving force predicted in a recent theoretical work for a cylindrical geometry. An additional explanation is suggested, based on different spatial distributions of the viscous dissipation in the dewetting bump in the planar and cylindrical geometries. This mechanism is investigated for films of different thicknesses in a numerical model assuming a polynomial variation of the liquid thickness with distance in the bump region.     

Propagation of travelling waves in sub-excitable systems driven by noise and periodic forcing

It has been reported that traveling waves propagate periodically and stably in sub-excitable systems driven by noise [Phys. Rev. Lett. 88, 138301 (2002)]. As a further investigation, here we observe different types of traveling waves under different noises and periodic forces, using a simplified Oregonator model. Depending on different noises and periodic forces, we have observed different types of wave propagation (or their disappearance). Moreover, reversal phenomena are observed in this system based on the numerical experiments in the one-dimensional space. We explain this as an effect of periodic forces. Thus, we give qualitative explanations for how stable reversal phenomena appear, which seem to arise from the mixing function of the periodic force and the noise. The output period and three velocities (normal, positive and negative) of the travelling waves are defined and their relationship with the periodic forces, along with the types of waves, are also studied in sub-excitable system under a fixed noise intensity. Electronic supplementary material Supplementary Online Material     

Dynamics of viscous vesicles in shear flow

The dynamics of giant lipid vesicles under shear flow is experimentally investigated. Consistent with previous theoretical and numerical studies, two flow regimes are identified depending on the viscosity ratio between the interior and the exterior of the vesicle, and its reduced volume or excess surface. At low viscosity ratios, a tank-treading motion of the membrane takes place, the vesicle assuming a constant orientation with respect to the flow direction. At higher viscosity ratios, a tumbling motion is observed in which the whole vesicle rotates with a periodically modulated velocity. When the shear rate increases, this tumbling motion becomes increasingly sensitive to vesicle deformation due to the elongational component of the flow and significant deviations from simpler models are observed. A good characterization of these various flow regimes is essential for the validation of analytical and numerical models, and to relate microscopic dynamics to macroscopic rheology of suspensions of deformable particles, such as blood.     

Effects of symmetric and asymmetric contact angles and division of menisci during separation

Meniscus and viscous forces are sources of adhesive force when two surfaces are separated with a micro-meniscus present at the interface. The adhesive force can be one of the main reliability issues when the contacting surfaces are ultra-smooth and the normal load is small, as is common for micro/nano devices. In this paper, both meniscus and viscous forces of menisci with symmetric and asymmetric contact angles are modelled. Equations for both meniscus and viscous forces in division of menisci are analytically formulated. The role of these two forces is evaluated during the separation process. The effects of the contact angles, division of menisci, as well as liquid thicknesses, surface tension and viscosity of the liquid, and separation distance and time during separation are presented. It is found that contact angles significantly affect the break point and meniscus force, and the magnitude of meniscus force can be largely reduced by choosing proper asymmetric contact angles. ‘Force scaling’ effects are found to be true for both meniscus and viscous forces when one larger meniscus is divided into large numbers of identical micro-menisci. Meniscus force increases with the number of divisions whereas viscous force decreases by an order of inverse the number of division (1/N). Optimal configurations for low adhesion are identified. This study presents a comprehensive analysis of meniscus and viscous forces during separation of menisci under different physical configurations. It provides a fundamental understanding of the physics of the process and knowledge for control of adhesion due to liquid menisci.