Motion of inviscid fluid tongues and bubbles in a Hele-Shaw cell occupied by a viscoelastic fluid

Problems of steady-state motion of an inviscid fluid tongue or bubble in a Hele-Shaw cell occupied by a viscoelastic fluid are considered. The problems generalize the well-known Saffman-Taylor problems [1, 2] in the approximation in which the pressure jump on the interface between the two fluids depends on the normal velocity of the interface [4]. In the zero approximation the results obtained by means of an iteration technique are in good agreement with the analytic solutions obtained by Saffman and Taylor [1, 2].__________Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, 2005, pp. 117–123. Original Russian Text Copyright © 2005 by Entov, Kolganov, and Kolganova.     

Bubble growth in a Hele-Shaw cell

The general form of the solution is obtained for the problem of the development of a bubble in a Hele-Shaw channel. This solution belongs to the class of parameterized solutions characterized by the property that the partial derivatives of the mapping function are rational in the auxiliary plane. A generalized problem of the system dynamics is written for the free parameters and its complete integrability is proved. An example of the development of an asymmetric bubble with the formation of a single fiord is given.     

Variability of a typical flow in a Hele-Shaw cell

The influence of different factors on a four-eddy convective flow, typical of a Hele-Shaw cell, with reconnection of the diagonal eddies is studied experimentally and theoretically. It is shown that this self-oscillating flow develops in the Hele-Shaw cell over a wide range of values of the super-criticality in very different situations and may be both regular and stochastic. This flow is observed in cavities of different dimensions, for homogeneous fluids and mixtures, for heat-conducting and thermally insulated wide sides of the cavity, and also in the presence of vibration and other external actions.     

Unsteady motion of a bubble in a Hele-Shaw cell

New exact solutions of an idealized unsteady single-phase Hele-Shaw problem of air-bubble motion in a slot-type channel are constructed under the assumption of bubble symmetry relative to the central axis of the channel. Qualitative features of the interface evolution, which distinguish this case from the earlier considered cases of Hele-Shaw flow with different geometry, are detected.     

Diffusion stability of bubbles in a cluster

The diffusion stability of gas bubbles in one-fraction and two-fraction clusters subjected to an acoustic field is studied. For a one-fraction cluster, numerical values were obtained for the initial gas concentrations in the liquid at which the bubble tends to one of two equilibrium states because of diffusion processes between the bubble and the ambient liquid. It is found that a two-fraction cluster tends to become a one-fraction cluster. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 4, pp. 40–48, July–August, 2007.     

Convection in a Hele-Shaw cell with allowance for heat exchange on the wide faces

Convection in a Hele-Shaw cell is studied analytically and numerically with allowance for heat exchange on the wide faces according to Newton's law. It is shown that this heat exchange has a considerable effect on the amplitude in the neighborhood of the threshold and explains laboratory experiments in which convection was observed even in the case of heating from above.Perm. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 24–32, March–April, 1995.     

Thermal convection in a Hele-Shaw cell under the action of centrifugal forces

The action of centrifugal forces on convective flows in a Hele-Shaw cell is studied experimentally and theoretically for a pulsing point heat supply from below. The dependence of convection on certain factors associated with imperfection of the experimental set-up is considered. The prospects of using experimental data and numerical simulation for the design of a sensor of inertial accelerations are discussed.     

Effect of initial disturbances on the Rayleigh-Taylor instability in a Hele-Shaw cell

The effect of the initial state on the development of the Rayleigh-Taylor instability in a Hele-Shaw cell is studied. The initial disturbances were initiated using two techniques, namely, by exciting the standing Faraday waves by means of a vibrator and by periodically distorting the interface between the fluids by means of thin connecting needles. The Rayleigh-Taylor instability was realized by inverting the cell. The process of wave development at the interface between the fluids was filmed with subsequent computer processing of the videofilm obtained. It was found that the initial state is the decisive factor in the development of the process up to large times.     

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.     

Experimental and theoretical investigation of transient convective regimes in a Hele-Shaw cell

Supercritical regimes of thermal convection in a Hele-Shaw cell heated from below are investigated experimentally and theoretically. A stability map of convective regimes is plotted. Novel stable and transient pulsatory convective flows are revealed. The numerical calculations performed for a cell with thermally insulated vertical faces are in agreement with the experimental data.     

Averaged equations in a Hele-Shaw cell: A stratification model

An averaged model of viscous fluid displacement from a Hele-Shaw cell with consideration of the transverse stratification of flow between the cell plates is proposed.     

Thermocapillary drift of an air bubble in a horizontal Hele-Shaw cell

An experimental and theoretical study has been made of the motion of an air bubble due to the thermocapillary effect in a thin horizontal layer of liquid bounded above and below by solid walls. The dependence of the speed of thermocapillary drift of the bubble on its radius has three characteristic regions, which correspond to different ratios of the bubble diameter to the thickness of the liquid layer. The results of theoretical solution of the problem for each of the three regions are compared with the experimental results.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 62–67, May–June, 1984.We thank V. A. Briskman for suggesting the subject and for constant interest in the work, and also A. F. Pshenichnikov for valuable advice.     

An experimental study of non-isothermal miscible displacements in a Hele-Shaw cell

Non-isothermal miscible displacements in a radial Hele-Shaw cell were experimentally investigated using a scheme in which room temperature liquids of relatively high viscosity were displaced by high-temperature (80 °C), less-viscous liquids. Fundamental characteristics have been presented regarding how the effect of a non-isothermal field on miscible displacement patterns varies in terms of factors such as the viscosity ratio of the more- and less-viscous liquids at 20 °C, M₂₀, the rate of an increase in the pattern’s area, R, and the gap width of the cell, b. The concept of area density was used to quantitatively evaluate the effect of the non-isothermal fields on the patterns. We have found that the effect of the non-isothermal field on the patterns does not monotonically vary with M₂₀ and b. In contrast, it increases with R in the present experimental condition. The experimental results can be explained by introducing an assumption in which heat is transferred mainly to the plates of the cell, in other words, the temperature of the more-viscous liquid remains constant, whereas that of the less-viscous liquid spatiotemporally decreases and the viscosity of it increases along with the temperature decrease. Visualization of non-isothermal field in the cell has been done by means of a thermo sheet and the results support the assumption mentioned above.     

Displacement of a viscous fluid from a Hele-Shaw cell with a sink

It is found that the radial geometry does not stabilize the evolution of instability in the displacement of a more viscous fluid by a less viscous fluid from a circular Hele-Shaw cell with a sink. A linear analysis shows the absolute instability of the radial displacement front. The appearance of isolated fingers is observed during numerical simulations.     

Inertial migration of sedimenting particles in a suspension flow through a Hele-Shaw cell

Within the framework of the model of two interpenetrating continua, a horizontal laminar dilute-suspension flow in a vertical Hele-Shaw cell is investigated. Using the method of matched asymptotic expansions, an asymptotic model of the transverse migration of sedimenting particles is constructed. The particle migration in the horizontal section of the cell is caused by an inertial lateral force induced by the particle sedimentation and the shear flow of the carrier phase. A characteristic longitudinal length scale is determined, on which the particles migrate across the slot through a distance of the order of the slot half-width. The evolution of the particle number concentration and velocity fields along the channel is studied using the full Lagrangian method. Depending on the particle inertia parameter, different particle migration regimes (with and without crossing of the channel central plane by the particles) are detected. A critical value of the particle inertia parameter corresponding to the change in migration regime is found analytically. The possibility of intersection of the particle trajectories and the formation of singularities in the particle number concentration is demonstrated.     

On the stability of gas bubbles in liquid-gas solutions

It was shown some time ago by use of diffusion theory that a gas bubble in a liquid-gas solution was unstable. This problem has been reconsidered recently in two papers both of which propose to develop a stability analysis solely from thermodynamic considerations. The first of these studies purports to find stability for a gas bubble in a liquid-gas solution. Some possible sources of error in this analysis are mentioned here. The second study considers a particular system of a bubble in a liquid drop immersed in a second liquid in which the gas is insoluble. A condition of stability is then found. This system is reconsidered here simply in terms of the ideas of diffusion theory. The stability conditions may then be stated in simple physical terms.