Bulletin of the American Physical Society
61st Annual Meeting of the APS Division of Fluid Dynamics
Volume 53, Number 15
Sunday–Tuesday, November 23–25, 2008; San Antonio, Texas
Session HF: Hele-Shaw Cells and Marangoni Instabilities |
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Chair: Sanjay Kumar, The University of Texas at Brownsville Room: 003B |
Monday, November 24, 2008 10:30AM - 10:43AM |
HF.00001: Hydrodynamic Stability in Hele-Shaw and Porous Media Flows Prabir Daripa We will discuss some stability problems of two-phase flows in Hele-Shaw cell and porous media and provide some stability results based on Darcy's law and saturation model. Effects of surface tension in Hele-Shaw flows and capillarity in porous media flows on slowdown of instabilities will be quantified within linear theory. Results on hydrodynamic instability in immiscible porous media flows in the presence of capillarity will be provided. We will present analysis and provide arguments that show that slowdown of instabilities due to capillarity is usually very rapid which makes the flow almost, but not entirely, stable. The profiles of the stable and unstable waves in the far-field will be characterized using a novel but very simple approach. Open problems in this direction will be discussed. Time permitting, we will briefly outline the proof of nonlinear instability for single phase flows in a Hele-Shaw cell with viscosity gradient in the direction of flow. [Preview Abstract] |
Monday, November 24, 2008 10:43AM - 10:56AM |
HF.00002: Miscible viscous fingering involving viscosity changes of the displacing fluid by chemical reactions Yuichiro Nagatsu, Kenji Matsuda, Chika Iguchi, Yoshihito Kato, Yutaka Tada We experimentally studied the effects of changes in the viscosity of the displaced more-viscous liquid by instantaneous reactions on miscible viscous fingering pattern (Nagatsu et al., J. Fluid Mech., 571, 475 (2007)). In the present study, experiments have been performed on the miscible viscous fingering involving changes in the viscosity of the displacing less-viscous liquid by instantaneous reactions in a Hele-Shaw cell. We have found that the shielding effect is suppressed and the fingers are widened when the viscosity of the displacing less-viscous liquid is increased. In contrast, the shielding effect is enhanced and the fingers are narrowed when the viscosity is decreased. These results are essentially same as those obtained by the above-mentioned previous study. This shows that the effects of changes in the viscosity due to the instantaneous reactions are independent of whether the changes occurs in the displaced more-viscous liquid or in the displacing less-viscous liquid. A reason is proposed to explain these results. [Preview Abstract] |
Monday, November 24, 2008 10:56AM - 11:09AM |
HF.00003: Spiral pattern in a radial displacement in a Hele-Shaw cell Mitsumasa Ban, Yuichiro Nagatsu, Atsushi Hayashi, Yoshihiro Kato, Yutaka Tada When a reactive and miscible less-viscous liquid displaces a more-viscous liquid in a Hele-Shaw cell, reactive miscible viscous fingering takes place. We have experimentally shown that the pattern created by the displacement of a more-viscous fluid by a less-viscous one in a radial Hele-Shaw cell develops not radially but spirally when a more-viscous sodium polyacrylate solution is displaced by a less-viscous trivalent iron ion (Fe$^{3+})$ solution with a sufficiently high concentration of Fe$^{3+}$. Another experiment in order to investigate the mechanism of spiral pattern formation revealed that an instantaneous chemical reaction takes place between the two fluids and at high Fe$^{3+}$ concentrations it produces a film of the gel at the contact plane. The gel is formed by three-dimensional network structures between the polyacrylate solution and the trivalent iron ion (Fe$^{3+})$ solution. We have proposed a physical model that the gel's film is responsible for the form of the spiral pattern. [Preview Abstract] |
Monday, November 24, 2008 11:09AM - 11:22AM |
HF.00004: Hexagonal patterns in longwave oscillatory convection in a binary liquid A.A. Nepomnyashchy, S. Shklyaev, A. Oron Oscillatory longwave Marangoni convection is studied in a heated layer of binary mixture with the concentration gradient induced by the Soret effect. Weakly-nonlinear analysis on a hexagonal lattice shows that the cubic-order truncation of the set of amplitude equations is degenerate, so that a three-parameter family of Asynchronous Hexagons (AH) is stable in this framework. Each of AH represents a superposition of three standing waves (SW) with the amplitudes depending on their phase shifts. Twisted Rectangles (TwR) and Wavy Rolls 2 (WR2) are the particular cases of AH corresponding to equal amplitudes of SW. Another limiting case of AH is Alternating Rolls (AR) corresponding to superposition of two equal SW, while the third one vanishes. For selection of the stable patterns from AH, the fifth-order terms are accounted for. The stability analysis demonstrates that either WR2 is selected or no stable patterns exist near the stability threshold. In the latter case, a heteroclinic cycle connecting WR2, TwR, and AR emerges. [Preview Abstract] |
Monday, November 24, 2008 11:22AM - 11:35AM |
HF.00005: Viscous potential flow analysis of radial fingering in a Hele-Shaw cell Daniel D. Joseph, Hyungjun Kim, Toshio Funada The problem of radial fingering in two-phase gas/liquid flow in a Hele-Shaw cell under injection of gas is studied here. The fingers arise as an instability of a time dependent flow which is rigorously resolved in the linearized approximation for the first time. One consequence of the unsteady basic flow is continuous nucleation of new finger as the radius $R(t)$ of the unperturbed interface increases. Another consequence is that the evolution of the perturbed interface $f(R(t))$ is not governed by the local (in time) value of the unperturbed interface $R(t)$ but depends globally on all past values of $R(t)$ up to the present. The problem is analyzed as a viscous potential flow VPF in which the potential flow analysis of Paterson (1981) [J. Fluid Mech. 113, 513] and others is augmented to account for the effects of viscosity on the normal stress at the gas/liquid interface. The unstable cases in which gas is injected into liquid or liquid is withdrawn from gas lead to fingers. Here we show that the viscous normal stress should not be neglected. [Preview Abstract] |
Monday, November 24, 2008 11:35AM - 11:48AM |
HF.00006: Photochemical Marangoni Convection Alexander Golovin, Vladimir Volpert Marangoni convection caused by a photochemical reaction is studied. Two cases are considered: convection in a thin liquid film and in a deep liquid layer. In the first case a system of strongly nonlinear evolution equations is derived and solved numerically. It is shown that Marangoni flow caused by a photochemical reaction can result in either film dry-out or sustained wavy patterns. In the case of a deep layer the conditions for Marangoni instability to occur are found and their dependence on the reaction kinetic parameters is analyzed. [Preview Abstract] |
Monday, November 24, 2008 11:48AM - 12:01PM |
HF.00007: Marangoni instabilities in two-layer systems due to concentration dependent transfer properties P.M.J. Trevelyan, V. Pimienta, K. Eckert, A. De Wit We consider a Hele-Shaw cell containing two immiscible liquids. A chemical species initially dissolved in an organic phase crosses the interface into the aqueous phase. In the aqueous phase this chemical reactant is involved in a reaction producing a surfactant which undergoes micellisation when the critical micelle concentration is reached. These micelles increase solubility which in turn increases the transfer rate and hence favours additional formation of micelles. To model such an autocatalytic increase of solubility, we consider here that the partition coefficient is a function of the surfactant concentration. Through the solutal Marangoni effect, this surfactant can induce tangential stresses leading to interfacial motion. The aim of our study is to theoretically examine the conditions for an instability in such a system. In particular, we seek to understand whether Marangoni effects can be observed because of a concentration dependent partition coefficient in a system that would be stable in the case of a constant partition coefficient according to the classical stability conditions of Sternling and Scriven (AIChE J., 5, p.514, 1959). [Preview Abstract] |
Monday, November 24, 2008 12:01PM - 12:14PM |
HF.00008: Experimental study of viscous fingering of miscible circular samples displaced linearly in a Hele-Shaw cell Renaud Maes, Anne De Wit Viscous fingering is a hydrodynamical instability occuring when a less viscous fluid displaces a more viscous one in a porous medium : their interface is unstable and develops ``fingers'' in the course of time. In the case of a finite width sample of fluid displaced linearly by a miscible carrying fluid of different viscosity, such a viscous fingering instability contributes to the spreading and distortion of the sample [1]. The goal of our work is to quantify experimentally the contribution of viscous fingering to the spreading of initially circular samples displaced linearly by another miscible carrying fluid using a Hele-Shaw cell as a 2D model for porous media. In a first stage, we analyze dispersion in the cell for stable interfaces using samples and displacing fluid of same viscosity, identifying a transition between a diffusive-like (dispersive) regime and a purely advective one when the injection speed is increased. We next characterize the dynamics of viscously unstable interfaces. By measuring the area and perimeter of the viscous samples as function of time, we have quantified the effects of viscous fingering on the spreading of the sample as a function of the injection speed and of the viscosity contrast between the two fluids. [1] De Wit, A., Bertho, Y., and Martin, M. Viscous fingering of miscible slices, Phys. Fluids, 17 (2005), 054114. [Preview Abstract] |
Monday, November 24, 2008 12:14PM - 12:27PM |
HF.00009: Stability analysis of an evaporating binary mixture Hatim Machrafi, Alexey Rednikov, Pierre Colinet, Pierre Dauby Rayleigh-B\'{e}nard-Marangoni instabilities in an evaporating binary mixture, consisting of a solvent and a solute of weak concentration, are studied theoretically. Local thermodynamic equilibrium is assumed at the flat gas-liquid interface. Solvent evaporation and air absorption in the liquid are neglected. At a certain height above the interface, the temperature and the concentration are fixed. One of the goals of the study is to track down the effects of this artifact on the results. Non-linear quasi-stationary basic profiles (due to evaporation) of the temperature and the solute concentration in the gas phase are considered, while the temperature distribution in the liquid is assumed to be linear and quasi-stationary. For the solute concentration in the liquid phase, two variants of the reference solution are studied, one just linear and quasi-stationary, whereas the other involves a fully transient non-linear profile. The latter is a more realistic option, given the relatively slow diffusion time in the liquid. A linear stability analysis is then carried out numerically, and illustrated for an aqueous solution of ethyl alcohol. [Preview Abstract] |
Monday, November 24, 2008 12:27PM - 12:40PM |
HF.00010: Ferrofluid patterns in a radial magnetic field: Linear stability, nonlinear dynamics, and exact solutions Jose Miranda, Rafael Oliveira, Eduardo Leandro The response of a ferrofluid droplet to a radial magnetic field is investigated, when the droplet is confined in a Hele-Shaw cell. We study how the stability properties of the interface and the shape of the emerging patterns react to the action of the magnetic field. At early linear stages it is found that the radial field is destabilizing and determines the growth of fingering structures at the interface. In the weakly nonlinear regime we have verified that the magnetic field favors the formation of peaked patterned structures that tend to become sharper and sharper as the magnitude of the magnetic effects is increased. A more detailed account of the patterns' morphology is provided by the determination of nontrivial exact stationary solutions for the problem with finite surface tension. These solutions are obtained analytically and reveal the development of interesting polygon-shaped and starfish-like patterns. An interesting connection of this system with the Euler's elastica problem is discussed. [Preview Abstract] |
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