Numerical simulations of interfacial instabilities on a rotating miscible droplet in a time‐dependent gap Hele–Shaw cell with significant Coriolis effects

Interfacial instability of a rotating miscible droplet with significant Coriolis force in a Hele–Shaw cell is simulated numerically. The influences of the relevant control parameters are first discussed qualitatively by fingering patterns. More vigorous fingerings are found at higher rotational effects, a lower viscosity contrast and a weaker effective surface tension (Korteweg constant). For a time‐dependent gap Hele–Shaw cell, a higher cell lifting rate makes the rotating droplet bear an inward straining flow, which leads to fingering enhancement. On the contrary, a higher pressing rate provides more stable effects by additional squeezing outward flow. A quantitative analysis between the Coriolis effects and tilting angles of fingers is addressed. For arbitrary combinations of all relevant control parameters, the values of tilting angles follow a nearly linear relationship with the Coriolis effects. We estimate the correlation between the relevant control parameters (dimensionless Coriolis factor Re, viscosity parameter R, cell lifting rate a) and tilting angles (θ) of fingers that can be approximated as for significant Korteweg stresses. Copyright © 2005 John Wiley & Sons, Ltd.     

Miscible displacements in Hele–Shaw cells: Nonmonotonic viscosity profiles

The influence of nonmonotonic viscosity–concentration relationships on viscous fingering of neutrally buoyant, miscible fluids in a Hele–Shaw cell has been investigated. In a first step, quasisteady base states are obtained by means of nonlinear Stokes simulations. The properties of these base states are analyzed as a function of the Péclet number, the viscosity ratio, and the profile parameters. Subsequently, the stability of these base states is investigated by means of a linear stability analysis. Overall, the nonmonotonicity of the viscosity–concentration relationship is seen to have a much smaller influence on Hele–Shaw displacements than on corresponding Darcy flows. The reason for this difference lies in the nature of the respective base states. For Darcy flows, the base state is characterized by constant velocity and a diffusively decaying concentration (and hence viscosity) profile. This base viscosity profile is strongly affected by the nonmonotonicity. On the other hand, for Hele–Shaw displacements the quasisteady base states are convectively dominated and characterized by sharp fronts, so that their shape depends only weakly on the details of the viscosity–concentration relationship. Hence, for Hele–Shaw displacements both the eigenfunctions and the associated growth rates are quite similar for monotonic and nonmonotonic profiles, in contrast to the findings by [O. Manickam, G.M. Homsy, Stability of miscible displacements in porous media with nonmonotonic viscosity profiles, Phys. Fluids A 5 (1993) 1356–1367] for Darcy flows.     

An efficient and accurate fully discrete finite element method for unsteady incompressible Oldroyd fluids with large time step

This paper proposes a second‐order accuracy in time fully discrete finite element method for the Oldroyd fluids of order one. This new approach is based on a finite element approximation for the space discretization, the Crank–Nicolson/Adams–Bashforth scheme for the time discretization and the trapezoid rule for the integral term discretization. It reduces the nonlinear equations to almost unconditionally stable and convergent systems of linear equations that can be solved efficiently and accurately. Here, the numerical simulations for L², H¹ error estimates of the velocity and L² error estimates of the pressure at different values of viscoelastic viscosities α, different values of relaxation time λ₁, different values of null viscosity coefficient μ₀ are shown. In addition, two benchmark problems of Oldroyd fluids with different solvent viscosity μ and different relaxation time λ₁ are simulated. All numerical results perfectly match with the theoretical analysis and show that the developed approach gives a high accuracy to simulate the Oldroyd fluids under a large time step. Furthermore, the difference and the connection between the Newton fluids and the viscoelastic Oldroyd fluids are displayed. Copyright © 2015 John Wiley & Sons, Ltd.     

Exact solution of the Muskat–Leibenzon problem for a growing elliptic bubble

The exact solution of the two-phase time-dependent Hele–Shaw problem (in other words, the plane Muskat–Leibenzon problem) in which a fluid occupied an unbounded channel is displaced by another fluid incoming through a slitted cut in the channel. In this case the interface between the phases, namely, fluids of different viscosity, evolves as an ellipse whose area and eccentricity vary continuously.     

Numerical simulation of liquid crystalline polymer flow into two‐dimensional thin cavity moulds

In this paper, flows of liquid crystalline polymers into two‐dimensional thin cavity moulds are simulated. The flows are modelled by Ericksen–Leslie equations of motion in the high viscosity limit. An elliptic pressure equation is derived under Hele–Shaw approximations, and the non‐isothermal natures of the flow are modelled. The equations are solved using the finite‐difference technique. A new boundary‐mapping technique is developed in this study to solve the difficulty in the finite‐difference treatment of arbitrarily shaped boundaries, which possess no natural coordinate system. This new method avoids the difficult mesh control in the body‐fitted mapping process and makes the mapping process easy to implement. It can also solve the problems caused by the uneven distribution of grid nodes in the traditional body‐fitted mapping technique. Copyright © 2008 John Wiley & Sons, Ltd.     

Buffeting in transonic flow prediction using time‐dependent turbulence model

In transonic flow conditions, the shock wave/turbulent boundary layer interaction and flow separations on wing upper surface induce flow instabilities, ‘buffet’, and then the buffeting (structure vibrations). This phenomenon can greatly influence the aerodynamic performance. These flow excitations are self‐sustained and lead to a surface effort due to pressure fluctuations. They can produce enough energy to excite the structure. The objective of the present work is to predict this unsteady phenomenon correctly by using unsteady Navier–Stokes‐averaged equations with a time‐dependent turbulence model based on the suitable (k–ε) turbulent eddy viscosity model. The model used is based on the turbulent viscosity concept where the turbulent viscosity coefficient (Cμ) is related to local deformation and rotation rates. To validate this model, flow over a flat plate at Mach number of 0.6 is first computed, then the flow around a NACA0012 airfoil. The comparison with the analytical and experimental results shows a good agreement. The ONERA OAT15A transonic airfoil was chosen to describe buffeting phenomena. Numerical simulations are done by using a Navier–Stokes SUPG (streamline upwind Petrov–Galerkin) finite‐element solver. Computational results show the ability of the present model to predict physical phenomena of the flow oscillations. The unsteady shock wave/boundary layer interaction is described. Copyright © 2005 John Wiley & Sons, Ltd.     

Stationary Boundary Points for a Laplacian Growth Problem in Higher Dimensions

It is known that corners of interior angle less than π/2 in the boundary of a plane domain are initially stationary for Hele–Shaw flow arising from an arbitrary injection point inside the domain. This paper establishes the corresponding result for Laplacian growth of domains in higher dimensions. The problem is treated in terms of evolving families of quadrature domains for subharmonic functions.     

An experimental study of miscible displacement with gravity-override and viscosity-contrast in a Hele Shaw cell

An experimental investigation of miscible displacements at constant volume flow-rate under the coupled effects of mobility contrast and gravitational segregation has been performed in a Hele Shaw cell having an aspect ratio, width to length, of 1:2. While the viscosity ratio was large (M > 180), the experiments covered both the neutrally buoyant case through to gravity-override-dominated unstable displacements. Dependence of the global displacement properties on the Gravity number (G) and the Peclet number (Pe) were quantified using a flow visualization technique. Within the experiment’s parameter range, no matter how complex the finger patterns became, and independent of G, the area grew linearly in time. As a result, the thickness of the injected less dense and less viscous fluid was almost constant at a value of 0.5–0.58 of the cell thickness with a weak dependence on Peclet number. Based on transversely averaged concentration profiles, the dependence of the average finger length was investigated and it also grew linearly in time. The displacement efficiency and breakthrough time decreased with increase of G, while the longitudinal finger growth rate increased with G. The averaged finger width followed the opposite trend and decreased as G increased. Velocity of the leading fingertip grew linearly with G at fixed Pe. The larger the value of Pe, the faster fingertips spread. As was to be expected, the larger the gravity number, the larger the global tilting of the whole finger pattern. The fractal dimension of the distorted interface at breakthrough was investigated, and it varied from 1.54 for the neutrally buoyant case to 1.08 for the gravity override dominated case.     

Nonlinear stability analysis of miscible displacements

A simulator for three-dimensional horizontal miscible displacements in porous media is developed. Using this simulator, we examine the initiation and development of instabilities, viscous fingers and gravity tongues.With the only perturbations to the system being truncation and round-off errors, a density ratio (the ratio of the density of the displacing fluid to that of the displaced fluid) different from one is responsible for the initiation of the instabilities, and an unfavorable mobility ratio (the ratio of the viscosity of the displaced fluid to that of the displacing fluid) is responsible for the growth of the instabilities.     

Dispersion de Taylor généralisée à un fluide à propriétés physiques variablesGeneralized Taylor dispersion for heterogeneous fluid

The investigation of non-reactive miscible solute dispersion in a vertical Hele–Shaw cell is considered. An asymptotic method is used to extend Taylor model to the case of the fluid density, the dynamic viscosity and the molecular diffusion coefficient are solute concentration-dependent. It is demonstrated that the averaged variables over the gap are governed by a convection–dispersion equation in which the dispersion tensor is concentration-dependent. To cite this article: C. Felder et al., C. R. Mecanique 332 (2004).     

Stability of vertical miscible displacements with developing density and viscosity gradients

Viscous fingering and gravity tonguing are the consequences of an unstable miscible displacement. Chang and Slattery (1986) performed a linear stability analysis for a miscible displacement considering only the effect of viscosity. Here the effect of gravity is included as well for either a step change or a graduated change in concentration at the injection face during a downward, vertical displacement. If both the mobility ratio and the density ratio are favorable (the viscosity of the displacing fluid is greater than the viscosity of the displaced fluid and, for a downward vertical displacement, the density of the displacing fluid is less than the density of the displaced fluid), the displacement will be stable. If either the mobility ratio or the density ratio is unfavorable, instabilities can form at the injection boundary as the result of infinitesimal perturbations. But if the concentration is changed sufficiently slowly with time at the entrance to the system, the displacement can be stabilized, even if both the mobility ratio and the density ratio are unfavorable. A displacement is more likely to be stable as the aspect ratio (ratio of thickness to width, which is assumed to be less than one) is increased. Commonly the laboratory tests supporting a field trial use nearly the same fluids, porous media, and displacement rates as the field trial they are intended to support. For the laboratory test, the aspect ratio may be the order of one; for the field trial, it may be two orders of magnitude smaller. This means that a laboratory test could indicate that a displacement was stable, while an unstable displacement may be observed in the field.     

Displacement of a viscous fluid from an annular Hele–Shaw cell with a sink within the framework of the Brinkman model

The stability of the radial front of viscous fluid displacement from an annular Hele–Shaw cell with a sink of finite radius is analyzed. It is shown that in the absence of the surface tension and at a minimal manifestation of molecular diffusion the role of the stabilizing factor determining the displacement front structure can be played by small viscous forces in the cell plane. The viscous fingers formed in this case turn out to be wider than those in a rectangular cell.     

Fingering patterns on an expanding miscible drop in a rotating Hele‐Shaw cell

We perform a detailed numerical study for the evolution of an expanding miscible drop in a rotating Hele‐Shaw cell. Two mathematical formulations applied to model the coating layer expansion during practical spin‐coating process, such as thinning of the layer by cell pressing and drop spreading outward due to injection, are investigated. Including miscible interfacial stresses, we focus on the investigation of dynamical and morphological influences of two different stabilizing parameters: the gap width parameter for the pressing cell and the injecting strength. In the case of a pressing cell, the fingering features of the expanding miscible drop, such as the critical radius, are distinct from those ones in the experiments of spin coating due to the different distributions of the inherent radial velocity. On the other hand, the global interfacial evolutions of an expanding drop with an additional injection bear remarkable resemblances to their immiscible counterparts. The better agreement for an injecting model suggests its appropriateness when we simulate the emerging fingering instabilities in the spin‐coating process. Moreover, we investigate the effects of Coriolis force at higher miscible Bond numbers. Coriolis force affects significantly the onset of fingering instability and the tilting angles of fingers. These stable effects are in line with the results from the previous studies for miscible and immiscible flow fields. Copyright © 2007 John Wiley & Sons, Ltd.     

Effect of inertia on thermoelastic flow instability

Thermal effects induced by viscous heating cause thermoelastic flow instabilities in curvilinear shear flows of viscoelastic polymer solutions. These instabilities could be tracked experimentally by changing the fluid temperature T₀ to span the parameter space. In this work, the influence of T₀ on the stability boundary of the Taylor–Couette flow of an Oldroyd-B fluid is studied. The upper bound of the stability boundary in the Weissenberg number (We)–Nahme number (Na) space is given by the critical conditions corresponding to the extension of the time-dependent isothermal eigensolution. Initially, as T₀ is increased, the critical Weissenberg number, We_c, associated with this upper branch increases. Increasing T₀ beyond a certain value T^* causes the thermoelastic mode of instability to manifest. This occurs in the limit as We/Pe → 0, where Pe denotes the Péclet number. In this limit, the fluid relaxation time is much smaller than the time scale of thermal diffusion. T₀ = T^* represents a turning point in the We_c–Na_c curve. Consequently, the stability boundary is multi-valued for a wide range of Na values. Since the relaxation time and viscosity of the fluid decrease with increasing T₀, the elasticity number, defined as the ratio of the fluid relaxation time to the time scale of viscous diffusion, also decreases. Hence, O(10) values of the Reynolds number could be realized at the onset of instability if T₀ is sufficiently large. This sets limits for the temperature range that can be used in experiments if inertial effects are to be minimized.     

Similar laminar boundary layers in incompressible micropolar fluids

Summary A search for similar solutions reveals as only possible similar boundary layer flow in micropolar fluids the flow near a stagnation point. The corresponding equations have been solved numerically by means of a shooting method. Consideration is given not only to the coupling parameterC ₁ and the microdiffusivity parameterC ₂ but also to the microinertia parameterC ₃. It is shown that macroscopic properties of steady boundary layer flows are not very much affected by these parameters, while the microrotation and therefore the inner structure of the layer is very sensitive to all three parameters. These properties of the microstructure can become important in certain unsteady flow problems; then also the macroscopic behaviour may be different to the behaviour of Newtonian fluids.

---

Zusammenfassung In der vorliegenden Untersuchung wird gezeigt, daß ähnliche Grenzschichten in mikropolaren Flüssigkeiten nur in der Nähe eines Staupunkts existieren. Die zugehörigen gewöhnlichen Differentialgleichungen werden mit einem Einschießverfahren numerisch gelöst. Neben dem KopplungsparameterC ₁ und dem MikrorotationsparameterC ₂ wird dabei auch der Einfluß der Mikroträgheit im ParameterC ₃ berücksichtigt. Es zeigt sich, daß diese Parameter die makroskopischen Eigenschaften stationärer Grenzschichtströmungen relativ wenig beeinflussen, während sich die Mikrorotation und damit die innere Struktur der Grenzschicht mit diesen Parametern sehr stark ändern kann. Man kann vermuten, daß diese Eigenschaften mikropolarer Flüssigkeiten bei instationären Vorgängen durchaus auch im makroskopischen Verhalten zu größeren Abweichungen gegenüber newtonschen Flüssigkeiten führen können.

---

---

With 6 figures and 1 table     

An experimental investigation of initial oscillations in a radial Hele-Shaw cell

Hele-Shaw cell is a laboratory device consisting of two parallel plates of glass separated by a thin gap. In this cell, in the flow of two immiscible fluids, when a fluid of higher viscosity is displaced by a fluid of lower viscosity, the less viscous fluid is observed to form “fingers” into the more viscous one due to the unstable interface. The Saffman-Taylor or viscous finger instability has been examined and modeled for over forty years for the rectilinear Hele-Shaw cell and about half as long for the radial Hele-Shaw cell. In this paper, we study, in detail, the early development of viscous instabilities in a radial Hele-Shaw cell. This source flow configuration has been chosen so that the instability can be monitored precisely. The objective of this study is to examine the onset of fingering, i.e. initial number of fingers that form, and the evolution of interface instability. Our experiments suggest that there may be some order in this formation process and one can model this aspect by considering the unsteady velocity components and predicting temporal changes in wavenumber responsible for the initial number of fingers and may be later accounting for the fingertip oscillations and splitting. We injected a water-based fluid into an oil in a radial Hele-Shaw cell at constant flow rate and recorded the movement of the less viscous droplet as it evolved. The relative curvature changes on the expanding droplet boundary was plotted with the angular positions about the interface and subtracting out the average radius, resulting in a plot of the change in amplitude with respect to time for the interface configuration. Three unstable configured tests at kinematic viscosity contrast (v ^O) of 0.34, 0.68, and 0.94 were run at approximately the same flow rate (2π cm²/s). The droplet exhibited oscillatory movement for these unstable configuration. The amplitude and the rate of oscillations were measured from digitized data. The smaller the viscosity difference, the smaller was the amplitude growth rate and resulted in a longer time to form visible finger initiation. This work was supported by National Science Foundation, grant number EID-9017555. We also like to thank Dr. Len Schwartz, Professor of Mechanical Engineering at the University of Delware for his insight and helpful suggestions.     

Miscible Thermo-Viscous Fingering Instability in Porous Media. Part 2: Numerical Simulations

The development of the thermo-viscous fingering instability of miscible displacements in homogeneous porous media is examined. In this first part of the study dealing with stability analysis, the basic equations and the parameters governing the problem in a rectilinear geometry are developed. An exponential dependence of viscosity on temperature and concentration is represented by two parameters, thermal mobility ratio β _T and a solutal mobility ratio β _C , respectively. Other parameters involved are the Lewis number Le and a thermal-lag coefficient λ. The governing equations are linearized and solved to obtain instability characteristics using either a quasi-steady-state approximation (QSSA) or initial value calculations (IVC). Exact analytical solutions are also obtained for very weakly diffusing systems. Using the QSSA approach, it was found that an increase in thermal mobility ratio β _T is seen to enhance the instability for fixed β _C , Le and λ. For fixed β _C and β _T , a decrease in the thermal-lag coefficient and/or an increase in the Lewis number always decrease the instability. Moreover, strong thermal diffusion at large Le as well as enhanced redistribution of heat between the solid and fluid phases at small λ is seen to alleviate the destabilizing effects of positive β _T . Consequently, the instability gets strictly dominated by the solutal front. The linear stability analysis using IVC approach leads to conclusions similar to the QSSA approach except for the case of large Le and unity λ flow where the instability is seen to get even less pronounced than in the case of a reference isothermal flow of the same β _C , but β _T  = 0. At practically, small value of λ, however, the instability ultimately approaches that due to β _C only.     

Numerical simulation of two‐dimensional Kelvin–Helmholtz instability using weakly compressible smoothed particle hydrodynamics

The growth of the Kelvin–Helmholtz instability generated at the interface between two ideal gases is studied by means of a Smoothed Particle Hydrodynamics (SPH) scheme suitable for multi‐fluids. The SPH scheme is based on the continuity equation approach where the densities of SPH particles are evolved during the simulation, in combination with a momentum equation previously proposed in the literature. A series of simulations were carried out to investigate the influence of viscosity, smoothing, the thickness of density and velocity transition layers. It was found that the effective viscosity of the presented results are strongly dependent on the artificial viscosity parameter α_AV, with a linear dependence of 0.15. The utilisation of a viscosity switch is found to significantly reduce the spurious viscosity dependence to 1.68 × 10^?4 and generated qualitatively improved behaviour for inviscid fluids. The linear growth rate in the numerical solutions is found to be in satisfactory agreement with analytical expectations, with an average relative error 〈η_smooth〉=13%. In addition, the role played by velocity and density transition layers is also in general agreement with the analytical theory, except for the sharp‐velocity, finite‐density gradient cases where the larger growth rate than the classical growth rate is expected. We argue the inherited smoothing properties of the velocity field during the simulations are responsible for causing this discrepancy. Finally, the SPH results are in good agreement for finite velocity and density gradient scenarios, where an average relative error of 〈η_smooth〉=11.5% is found in our work. Copyright © 2015 John Wiley & Sons, Ltd.