Immiscible Newtonian Liquid Research Articles

Phase-field formulated meshless simulation of Rayleigh-Taylor instability problem

The interface between two immiscible Newtonian liquids with different densities and the same viscosity, influenced by gravity, is based on the Phase-Field Method (PFM) formulation. The solution of the related governing coupled Navier-Stokes (NS) and Cahn-Hillard (CH) equations is structured by the meshless Diffuse Approximate Method (DAM) and Pressure Implicit with Splitting of Operators (PISO). The variable density is involved in the inertial and buoyancy terms (non-Boussinesq formulation). The related moving boundary problem is handled through single-domain, irregular, fixed node arrangement in two-dimensional Cartesian coordinates. The meshless DAM uses weighted least squares approximation on overlapping subdomains, polynomial shape functions of second-order and Gaussian weights. Implicit time discretisation is performed for the NS and CH equations in the momentum predictor and Phase-Field (PF) variable corrector steps of PISO, while the momentum corrector steps solve the NS equation explicitly. This solution procedure has improved stability compared to Chorin’s pressure-velocity coupling, previously used in meshless solutions of related problems. The Rayleigh-Taylor instability problem simulations are performed for an Atwood number of 0.76. The DAM parameters (shape parameter of the Gaussian weight function and number of nodes in a local subdomain) are the same as in the author’s previous studies on single-phase flows. The simulations did not need any upwinding in the range of the simulations. The results compare well with the mesh-based finite volume method studies performed with the open-source code Gerris.

Contact angle hysteresis can modulate the Newtonian rod-climbing effect

The present work investigates the role of contact angle hysteresis at the liquid–liquid–solid interface (LLS) on the rod-climbing effect of two immiscible Newtonian liquids using experimental and numerical approaches. Experiments revealed that the final steady-state contact angle, θw, at the LLS interface varies with the rod rotation speed, ω. For the present system, θw changes from ∼69° to ∼83° when the state of the rod is changed from static condition to rotating at 3.3 Hz. With further increase in ω, the θw exceeds 90°, which cannot be observed experimentally. It is inferred from the simulations that the input value of θw saturates and attains a constant value of ∼120° for ω> 5 Hz. Using numerical simulations, we demonstrate that this contact angle hysteresis must be considered for the correct prediction of the Newtonian rod-climbing effect. Using the appropriate values of the contact angle in the boundary condition, an excellent quantitative match between the experiments and simulations is obtained in terms of the climbing height, the threshold rod rotation speed for the onset of climbing, and the shape of the liquid–liquid interface. This resolves the discrepancy between the experiments and simulations in the existing literature where a constant value of the contact angle has been used for all speeds of rod rotation.

Numerical simulation of gas-bubble formation through two submerged orifices

In this work, two-dimensional numerical simulations are carried out to analyse the dynamics of gas-bubble formation at two adjacent orifices submerged in an immiscible Newtonian liquid under the condition of constant gas flow rate using finite-volume-based commercial CFD solver Ansys Fluent. Two conditions for the ambient liquid are considered: firstly the liquid is in quiet condition and secondly, it is also co-flowing with the gas stream. The full cycle from the bubble formation to its detachment and the corresponding dynamics are simulated using the volume of fluid (VOF) method, which is one of the efficient front capturing techniques of Eulerian family. The study includes: (i) time sequence profile of bubble formation to clearly represent the bubble growth, neck formation and bubble break-up at a given Weber number (We), Reynolds number (Re), Bond number (Bo) and liquid to gas mean velocity ratio (Vr); (ii) bubble growth history for different Vr and at constant Re, We and Bo; (iii) effect of orifice spacing on bubbling synchronicity and frequency at given Vr, We, Re and Bo and (iv) synchronous bubbling regime, bubble size and formation time determination, bubble coalescence phenomenon and technique for its inhibition.

Droplet formation and its mechanism in a microchannel in the presence of an electric field

The instability of the interface between two immiscible Newtonian liquids in a microchannel in the presence of an electric field and the resulting micro droplets are experimentally investigated. In the experiments, ethylene glycol and silicone oil of different viscosities are pumped into a PMMA microchannel and a DC field is applied normal to the flat interface between the liquids. The effect of the viscosity ratio, the thickness ratio and the total flow rate, on the critical voltage, Vc at which the interface starts to deflect is investigated. An empirical formula is developed to relate the dimensionless critical electric number, Eb,c, to the thickness ratio, Hr and the viscosity ratio, μr. It is found that the thickness ratio is more important than the viscosity ratio. In the second part of the work, the evolution of the interface is studied and the droplet formation mechanism due to the electric field is presented in detail. The droplet size is investigated in terms of the thickness and viscosity ratios, and the total flow rate. An empirical formula for the droplet size is obtained as a function of the electric number, Eb the total flow rate, Q, the thickness ratio, Hr and the viscosity ratio, μr. It is possible to get smaller droplets with higher electric number, viscosity ratio, and total flow rate and smaller thickness ratio.

Exchange flows of two immiscible Newtonian liquids in a vertical tube: From falling drops to falling slugs

We present an experimental study of buoyancy-driven flows of two immiscible Newtonian liquids in a vertical tube where initially the heavier and more viscous one is placed on the top of the lighter one. Flow visualization was performed using a digital camera, and inversion velocities were determined through image analysis. The influence of the governing parameters on the speed and flow regime was examined for pairs of liquids with small density differences. Two different flow regimes were observed, namely, falling drops and falling slugs. In the first regime, spherical and ellipsoidal drops are obtained, depending on the ratio between the drop and tube diameters. The falling slug regime is a core-annular flow pattern above a critical value of interfacial tension, while no flow is noticed below this critical value. The experimental results are in good agreement with model predictions from the literature. Indeed, the results showed that terminal velocity can be estimated by empirical correlations for falling spheres.

Capillary Filling at the Microscale: Control of Fluid Front Using Geometry.

We propose an experimental and theoretical framework for the study of capillary filling at the micro-scale. Our methodology enables us to control the fluid flow regime so that we can characterise properties of Newtonian fluids such as their viscosity. In particular, we study a viscous, non-inertial, non-Washburn regime in which the position of the fluid front increases linearly with time for the whole duration of the experiment. The operating shear-rate range of our apparatus extends over nearly two orders of magnitude. Further, we analyse the advancement of a fluid front within a microcapillary in a system of two immiscible Newtonian liquids. We observe a non-Washburn regime in which the front can accelerate or decelerate depending on the viscosity contrast between the two liquids. We then propose a theoretical model which enables us to study and explain both non-Washburn regimes. Furthermore, our theoretical model allows us to put forward ways to control the emergence of these regimes by means of geometrical parameters of the experimental set-up. Our methodology allows us to design and calibrate a micro-viscosimetre which works at constant pressure.

Bubble generation in quiescent and co-flowing liquids

A numerical simulation has been accomplished to analyze the problem of dynamic bubble formation from a submerged orifice in an immiscible Newtonian liquid under the condition of constant gas inflow. We have considered two cases for the surrounding liquid, namely the liquid in a quiescent condition and the liquid as a co-flowing stream with the gas. The full cycle, from formation to detachment of the bubbles and the corresponding bubble dynamics, was simulated numerically by using a coupled level-set and volume-of-fluid (CLSVOF) method. The role of the liquid to gas mean velocity ratio, the Bond number and the Weber number in the bubble formation process was studied and the order of magnitude of forces involved in bubble dynamics are presented. Our simulation results show that the minimum radius of the neck decreases with a power law behavior and the power law exponent in a co-flowing liquid is less than 1/2 as predicted by the Rayleigh–Plesset theory for quiescent inviscid liquids. Single periodic and double periodic bubbling (with pairing and coalescence) regimes are observed in the present investigation. It is identified that a moderate co-flowing liquid may inhibit the bubble coalescence. The volume of the bubble and the bubble formation time decrease with increasing liquid to gas mean velocity ratios. For small Bond numbers, significant differences pertaining to bubble dynamics are observed between the co-flowing liquid and the quiescent liquid. Furthermore, the generation and breakup of the Worthington jet after bubble pinch-off and formation of tiny drops inside the detached bubbles are observed.

Scaling law behaviour of the retraction of a Newtonian droplet after a strain jump in a Newtonian matrix

We performed 3D visualisation of a Newtonian droplet embedded in an immiscible Newtonian liquid after a strain jump with a home-built counter-rotating shear device. The use of different linear polyurethanes for the droplet liquid allowed us to cover almost three decades of viscosity ratios (K) and to obtain a distinct interface with PDMS matrices with the same interfacial tension for all droplet/matrix pairs. During the droplet retraction, the major axis (L) showed universal time dependence. The apparent Hencky strain of L decayed linearly at large deformations and exponentially at small deformations. After large strain steps, the droplet axis along the vorticity direction (W) deflated and then inflated and the time dependence could be well described by a log normal function. The full width at half maximum was proportional to the droplet relaxation time for all K. The amplitude and the position of the minimum of W were proportional to the affine deformation. The results revealed interesting scaling law behaviour of the droplet retraction after large strain jumps.

Droplet dynamics in sub-critical complex flows

A newly designed eccentric cylinder device has been used to study the deformation and orientation of single Newtonian droplets immersed in an immiscible Newtonian liquid in a controlled complex flow field. Optical microscopy coupled with image acquisition analysis allows monitoring the dynamics of droplets flowing in the gap between the eccentric cylinders. Throughout the experiments, the flow intensity was kept below the critical conditions for droplet break-up. The experimental results are compared with predictions which are obtained using the transient form of the phenomenological model of Maffettone and Minale (J Non-Newtonian Fluid Mech 78:227–241, 1998; J Non-Newtonian Fluid Mech 84:105–106, 1999), incorporating a flow type parameter that accounts for the relative amount of elongational effects in the flow field and adapting the capillary number to mixed flows. For all the sub-critical flows studied here, good agreement was found between model predictions and experimental data, providing, for the first time, a quantitative assessment of drop shape predictions in complex flows.

Instability of a leaky dielectric coaxial jet in both axial and radial electric fields

The temporal linear instability of a coaxial jet with two immiscible Newtonian liquids in both the axial and radial electric fields is studied. The outer liquid is supposed to be a leaky dielectric and the inner liquid a perfect dielectric. The eigenvalue problem for both axisymmetric instability and helical instability is formulated and solved using the spectral method. Different from axisymmetric instability, for helical instability there is only one unstable mode, i.e., the helical mode, located in the long wave region. The axial electric field is found to have a strong stabilization effect on both the axisymmetric and helical modes, and the radial electric field has a great destabilization effect on them. The competition between the axisymmetric and helical instability under the action of the axial and radial electric fields is calculated. The boundary curve separating the stabilization and destabilization regions of the parasinuous mode, the neutral stability curve of the helical mode, and the boundary curve between the dominant regions of the axisymmetric and helical instability are plotted on the Q0-Eu plane and Pi-Eu plane, respectively (Q0 is the dimensionless surface charge density; Eu is the electrical Euler number representing the characteristic tangential electrostatic force; and Pi=Q0;2Eu is the dimensionless parameter representing the characteristic normal electrostatic force). In general, when surface charge density is small, the helical mode is stable, and the parasinuous mode is dominant; however, when surface charge density is sufficiently large, the helical mode is destabilized and becomes dominant in jet instability. Liquid viscosity influences the predominance of the helical mode significantly. Although liquid viscosity decreases the growth rates of both the axisymmetric and helical modes, it suppresses the axisymmetric instability much more than the helical instability, and therefore favors the realization of the helical instability in experiments.

Finite element approximation of a two-layered liquid film in the presence of insoluble surfactants

We consider a system of degenerate parabolic equations modelling a thin film, consisting of two layers of immiscible Newtonian liquids, on a solid horizontal substrate. In addition, the model includes the presence of insoluble surfactants on both the free liquid-liquid and liquid-air interfaces, and the presence of both attractive and repulsive van der Waals forces in terms of the heights of the two layers. We show that this system formally satisfies a Lyapunov structure, and a second energy inequality controlling the Laplacian of the liquid heights. We introduce a fully practical finite element approximation of this nonlinear degenerate parabolic system, that satisfies discrete analogues of these energy inequalities. Finally, we prove convergence of this approximation, and hence existence of a solution to this nonlinear degenerate parabolic system.

Stationary coaxial electrified jet of a dielectric liquid surrounded by a conductive liquid

The electrohydrodynamic problem describing the evolution of a coaxial jet of two immiscible Newtonian liquids injected through concentric orifices into a region of uniform electric field is formulated in the framework of the leaky dielectric model, and the dimensionless parameters governing the flow are identified. A simplified model is proposed combining a quasi-uni-directional approximation for the flow and the transport of electric charge with a fully numerical evaluation of the electric field. Results of this model are compared with boundary element solutions of the full governing equations for viscosity-dominated flows. The current/flow rate characteristic of a coaxial viscosity-dominated jet is computed and the ranges of flow rates in which a stationary axisymmetric jet is realized are discussed in the case of an outer liquid of finite electrical conductivity surrounding a dielectric liquid, in which the electric shear that stretches the jet is concentrated at its outer surface. The dependence of the flow on the applied electric field and the flow rates of the liquids, as well as the effects of the viscosity and electrical conductivity of the inner liquid, and of the surface tension of the inner surface, are discussed.

Interface stabilization in two-layer channel flow by surface heating or cooling

The linear stability of plane Poiseuille flow of two immiscible Newtonian liquids in a differentially heated channel is considered. The equations of motion and energy are fully coupled via temperature-dependent fluid-viscosity coefficients. A long-wave asymptotic formulation of the stability problem is presented together with numerical results for disturbances of arbitrary wavelength. Two combinations of immiscible liquids are analyzed: silicone/water and oil/water (water at the bottom layer in both cases). It is shown that an imposed wall temperature difference can be stabilizing or destabilizing depending on the disturbance wavenumber and layer thickness ratio. Interfacial tension has a stabilizing effect on the interface. Stabilizing influence of interfacial tension is observed at intermediate and large wavenumbers. Most importantly, for certain ranges of the controlling dimensionless parameters, stable interfaces at all disturbance wavelengths can be attained.

Drop shape under slow steady shear flow and during relaxation. Experimental results and comparison with theory

The three-dimensional small deformation of a single Newtonian drop immersed in an immiscible Newtonian liquid was investigated, both in slow steady shear and during retraction after cessation of shear. The experiments were performed in a parallel plate apparatus equipped with video- enhanced microscopy. The drop was observed from two perpendicular directions, and accurate measurements were obtained in each view by image analysis. The results were compared to existing theoretical predictions from the exact fluid dynamic problem, obtained perturbatively for the case of small deformations of the drop. Excellent agreement between data and theory was found, thus providing the first assessment of drop shape predictions in steady flow and relaxation for Newtonian fluids.

Drop formation in liquid–liquid systems before and after jetting

The formation of drops resulting from the breakup of an axisymmetric Newtonian liquid jet injected vertically into another immiscible Newtonian liquid at various Reynolds numbers is investigated here. The full transient from startup to breakup into drops was simulated numerically by solving the time-dependent axisymmetric equations of motion and continuity using a combination of the volume-of-fluid (VOF) and continuous-surface-force (CSF) methods. The numerical simulation results compare well with previous experimental data and are significantly more accurate than previous simplified analyses based on drop formation before and after jetting over a wide range of conditions.

Dynamic breakup of liquid–liquid jets

The axisymmetric, dynamic breakup of a Newtonian liquid jet injected vertically into another immiscible Newtonian liquid at various Reynolds numbers is investigated here. The full transient from jet start-up to breakup into drops was simulated numerically by solving the time-dependent axisymmetric equations of motion and continuity using an algorithm based on the Volume of Fluid (VOF) method that was previously proven successful in simulations of steady-state liquid jets (i.e., of the jet region close to the nozzle before breakup). The algorithm has been further refined here based on its performance on transient problems such as the solution of the free liquid–liquid capillary jet breakup problem. The comparison of the simulation results with previous experimental measurements of jet length under conditions where all forces, i.e., viscous, inertial, buoyancy, and surface tension, are important, can be judged satisfactory given the sensitive dependence of the results on details of the experimental setup that are not available. The comparison involves the jet length till breakup as well as the jet and drop shapes, often far from regular. In comparison with experiment, the results of the present numerical method show a greater sensitivity of the jet length to the Reynolds number than the best predictions previously available based on the linear stability analysis of the free liquid–liquid capillary jet breakup problem.

Steady laminar flow of liquid–liquid jets at high Reynolds numbers*

The axisymmetric steady-state laminar flow of a Newtonian liquid jet injected vertically into another immiscible Newtonian liquid is investigated for various Reynolds numbers. The steady-state solution was calculated by solving the axisymmetric transient equations of motion and continuity using a numerical scheme based on the volume of fluid (VOF) method combined with the new continuum surface force (CSF) algorithm. The analysis takes into account pressure, viscous, inertial, gravitational, and surface tension forces. Comparison with previous experimental measurements, performed on a xylene/water system, under conditions where all of these forces are important, shows good agreement over the entire range of conditions studied. Comparisons of the present numerical method with the numerical results of previous boundary-layer methods help establish their range of validity.

Co-current axisymmetric flow in complex geometries: Experiments

The interface location and velocity profiles in co-current low-Reynolds number flow of immiscible Newtonian and viscoelastic liquids through a sudden expansion were measured using laser speckle photography. The results were compared with the flow of the same liquids through a sudden contraction, as well as to the expansion and contraction flow of single liquids. The flow structure is very sensitive to the viscosity ratio, but is insensitive to whether or not the fluids are viscoelastic when the viscosity ratio is large.

Laminar jet contraction and velocity distribution in immiscible liquid-liquid systems

The flow behavior of a Newtonian liquid jet injected vertically into an immiscible Newtonian liquid phase is analyzed. Boundary-layer type approximations are used to simplify the general equations, and an approximate momentum-integral type numerical solution is obtained. This solution predicts the velocity distribution in each phase and the jet radius. The effects on jet behavior of the five dimensionless groups needed to characterize the gravitational, interfacial tension and viscous forces are shown. In particular the importance of the continuous phase viscosity is demonstrated. Experimental measurements of jet radius confirm the essential features of the analysis and illustrate the shortcomings of the approximate solution.

Prediction of jet length in immiscible liquid systems

AbstractThe stability theory is used to predict jet length from jet inception to disruption for injection of one Newtonian liquid into a second immiscible Newtonian liquid. Knowledge of the length is essential for predicting the size of drops formed from jets. At low velocities jet length is controlled by the amplification of symmetrical waves which travel at the interfacial velocity of the jet. At higher velocities an abrupt lengthening of the jet may occur as a result of drop merging, and the jet length is then controlled by the growth rate of sinuous waves which are strongly velocity dependent. Jet disruption results from a geometrical limitation on the maximum amplitude of the sinuous waves. Predictions show good quantitative agreement with experimental data for thirteen mutually saturated systems over a wide range of variables and qualitative agreement with limited experimental data on the effects of initial disturbance level and mass transfer.