Articoli di riviste: "Hele-Shaw problem"

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CROWDY, DARREN G. "Hele-Shaw flows and water waves". Journal of Fluid Mechanics 409 (25 aprile 2000): 223–42. http://dx.doi.org/10.1017/s0022112099007685.

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By adapting a new mathematical approach to the problem of steady free-surface Euler flows with surface tension recently devised by the present author, it is demonstrated that exact solutions for steady, free-surface multipole-driven Hele-Shaw flows with surface tension can be constructed using similar methods. Moreover, a (one-way) mathematical transformation between exact solutions to the two distinct free-boundary problems is identified: known exact solutions for free-surface Euler flows with surface tension are shown to automatically generate steady quadrupolar-driven Hele-Shaw flows (with non-zero surface tension) existing in exactly the same domain with the same free surface. This correspondence highlights the essential dynamical differences between the two physical problems. Using the transformation, the exact Hele-Shaw analogues of all known exact solutions for free-surface Euler flows (including Crapper's classic capillary water wave solution) are catalogued thereby producing many previously unknown exact solutions for steady Hele-Shaw flows with capillarity. In particular, this paper reports what are believed to be the first known exact solutions for Hele-Shaw flows with surface tension in a doubly-connected fluid region.

2

Kimura, Masato, Daisuke Tagami e Shigetoshi Yazaki. "Polygonal Hele–Shaw problem with surface tension". Interfaces and Free Boundaries 15, n. 1 (2013): 77–93. http://dx.doi.org/10.4171/ifb/295.

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Vasil’ev †, Alexander. "Robin's Modulus in a Hele-Shaw Problem". Complex Variables, Theory and Application: An International Journal 49, n. 7-9 (10 giugno 2004): 663–72. http://dx.doi.org/10.1080/02781070410001732188.

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Mellet, Antoine, Benoît Perthame e Fernando Quirós. "A Hele–Shaw problem for tumor growth". Journal of Functional Analysis 273, n. 10 (novembre 2017): 3061–93. http://dx.doi.org/10.1016/j.jfa.2017.08.009.

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Yadav, Dhananjay. "The effect of pulsating throughflow on the onset of magneto convection in a layer of nanofluid confined within a Hele-Shaw cell". Proceedings of the Institution of Mechanical Engineers, Part E: Journal of Process Mechanical Engineering 233, n. 5 (13 marzo 2019): 1074–85. http://dx.doi.org/10.1177/0954408919836362.

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In this article, the joint effect of pulsating throughflow and magnetic field on the onset of convective instability in a nanofluid layer, bounded in a Hele-Shaw cell is presented within the context of linear stability theory and frozen profile approach. The model utilized for nanofluid combines the impacts of Brownian motion and thermophoresis, while for Hele-Shaw cell, Hele-Shaw model is considered. The Galerkin technique is utilized to solve the eigenvalue problem. The outcome of the important parameters on the stability framework is examined analytically. It is observed that the pulsating throughflow and magnetic field have both stabilizing effects. The impact of increasing the Hele-Shaw number [Formula: see text], the modified diffusive ratio [Formula: see text] and the nanoparticle Rayleigh number [Formula: see text] is to quicken the convective motion, while the Lewis number [Formula: see text] has dual impact on the stability framework in the existence of pulsating throughflow. It is also established that the oscillatory mode of convective motion is possible only when the value of the magnetic Prandtl number [Formula: see text] is not greater than unity.

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Saffman, P. G. "Viscous fingering in Hele-Shaw cells". Journal of Fluid Mechanics 173 (dicembre 1986): 73–94. http://dx.doi.org/10.1017/s0022112086001088.

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The phenomenon of interfacial motion between two immiscible viscous fluids in the narrow gap between two parallel plates (Hele-Shaw cell) is considered. This flow is currently of interest because of its relation to pattern selection mechanisms and the formation of fractal, structures in a number of physical applications. Attention is concentrated on the fingers that result from the instability when a less-viscous fluid drives a more-viscous one. The status of the problem is reviewed and progress with the thirty-year-old problem of explaining the shape and stability of the fingers is described. The paradoxes and controversies are both mathematical and physical. Theoretical results on the structure and stability of steady shapes are presented for a particular formulation of the boundary conditions at the interface and compared with the experimental phenomenon. Alternative boundary conditions and future approaches are discussed.

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Moog, Mathias, Rainer Keck e Aivars Zemitis. "SOME NUMERICAL ASPECTS OF THE LEVEL SET METHOD". Mathematical Modelling and Analysis 3, n. 1 (15 dicembre 1998): 140–51. http://dx.doi.org/10.3846/13926292.1998.9637097.

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Many practical applications imply the solution of free boundary value problems. If the free boundary is complex and can change its topology, it will be hard to solve such problems numerically. In recent years a new method has been developed, which can handle boundaries with complex geometries. This new method is called the level set method. However, the level set method also has some drawbacks, which are mainly concerning conservation of mass or numerical instabilities of the boundaries. Our aim is to analyze some aspects of the level set method on the basis of two‐phase flow in a Hele‐Shaw cell. We investigate instabilities of two‐phase flow between two parallel plates. A solution of the linearized problem is obtained analytically in order to check whether the numerical schemes compute reasonable results. The developed numerical scheme is based on finite difference approximations and the level set method. The equations of two‐phase Hele‐Shaw flow are written in a modified formulation using the one‐dimensional Dirac delta‐function. Since the level set function is not smooth enough after re‐initialization, special attention during the computation of curvature is needed. We propose a method that can solve the problems for two‐phase Hele‐Shaw flow with changing topology. The numerical solution shows good agreement with the analytical solution of the linearized problem. We describe the method below and analyze the results.

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Rogosin, Sergei, e Tatsyana Vaitekhovich. "Hele-Shaw Model for Melting/Freezing with Two Dendrits". Materials Science Forum 553 (agosto 2007): 143–51. http://dx.doi.org/10.4028/www.scientific.net/msf.553.143.

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Melting/freezing process with two dendrits (or freeze “pipes”) is modelled by the complex Hele-Shaw moving boundary value problem in a doubly connected domain. The later is equivalently reduced to a couple of problems, namely, to the linear Riemann-Hilbert boundary value problem in a doubly connected domain and to evolution problem, which can be written in a form of an abstract Cauchy-Kovalevsky problem. The later is studied on the base of Nirenberg-Nishida theorem, and for the former a generalization of the Schwarz Alternation Method is proposed. By using composition of these two approaches we get the local in time solvability of this couple of problems in appropriate Banach space setting.

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Rogosin, S. "Real variable Hele-Shaw problem with kinetic undercooling". Lobachevskii Journal of Mathematics 38, n. 3 (maggio 2017): 510–19. http://dx.doi.org/10.1134/s1995080217030210.

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Jerison, David, e Inwon Kim. "The one-phase Hele-Shaw problem with singularities". Journal of Geometric Analysis 15, n. 4 (dicembre 2005): 641–67. http://dx.doi.org/10.1007/bf02922248.

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Rogosin, S. V. "ON CLASSICAL FORMULATION OF HELE‐SHAW MOVING BOUNDARY PROBLEM FOR POWER‐LAW FLUID". Mathematical Modelling and Analysis 7, n. 1 (30 giugno 2002): 159–68. http://dx.doi.org/10.3846/13926292.2002.9637188.

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Malaikah, K. R. "The Two-Phase Hell-Shaw Flow: Construction of an Exact Solution". International Journal of Applied Mechanics and Engineering 18, n. 1 (1 marzo 2013): 249–57. http://dx.doi.org/10.2478/ijame-2013-0016.

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We consider a two-phase Hele-Shaw cell whether or not the gap thickness is time-dependent. We construct an exact solution in terms of the Schwarz function of the interface for the two-phase Hele-Shaw flow. The derivation is based upon the single-valued complex velocity potential instead of the multiple-valued complex potential. As a result, the construction is applicable to the case of the time-dependent gap. In addition, there is no need to introduce branch cuts in the computational domain. Furthermore, the interface evolution in a two-phase problem is closely linked to its counterpart in a one-phase problem

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HOWISON, SAM D. "A note on the two-phase Hele-Shaw problem". Journal of Fluid Mechanics 409 (25 aprile 2000): 243–49. http://dx.doi.org/10.1017/s0022112099007740.

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We discuss some techniques for finding explicit solutions to immiscible two-phase flow in a Hele-Shaw cell, exploiting properties of the Schwartz function of the interface between the fluids. We also discuss the question of the well-posedness of this problem.

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ALEXANDROU, ANDREAS N., e VLADIMIR ENTOV. "On the steady-state advancement of fingers and bubbles in a Hele–Shaw cell filled by a non-Newtonian fluid". European Journal of Applied Mathematics 8, n. 1 (febbraio 1997): 73–87. http://dx.doi.org/10.1017/s0956792596002963.

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The problem of steady-state propagation of a finger or a bubble of inviscid fluid through a Hele–Shaw cell filled by a viscous non-Newtonian, including visco-plastic (Bingham) fluid is addressed. Only flows symmetric relative to the cell axis are considered. It is shown that, using a hodograph transform, this non-linear free boundary problem can be reduced to the solution of an elliptic system of linear partial differential equations in a fixed domain with part of the boundary being curvilinear. The resulting boundary-value problem is solved numerically using the Finite Element Method. Finger shapes are calculated, and the approach is verified for one-parameter family of solutions which correspond to the well-known Saffman–Taylor solutions for the case of a Hele–Shaw cell filled by a Newtonian fluid. Results are also shown for fingers with non-Newtonian fluids. In the case of a cell filled by visco-plastic (Bingham) fluid, it is shown that stagnant zones propagate with the finger, and that the rear part of the finger has constant width. The same approach is applied to finding a two-parametric family of solutions for steady propagating bubbles. Results are shown for bubbles in Hele–Shaw cell filled by power-law and Bingham fluids.

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Morrow, Liam C., Timothy J. Moroney, Michael C. Dallaston e Scott W. McCue. "A review of one-phase Hele-Shaw flows and a level-set method for nonstandard configurations". ANZIAM Journal 63 (16 novembre 2021): 269–307. http://dx.doi.org/10.21914/anziamj.v63.16689.

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The classical model for studying one-phase Hele-Shaw flows is based on a highly nonlinear moving boundary problem with the fluid velocity related to pressure gradients via a Darcy-type law. In a standard configuration with the Hele-Shaw cell made up of two flat stationary plates, the pressure is harmonic. Therefore, conformal mapping techniques and boundary integral methods can be readily applied to study the key interfacial dynamics, including the Saffman–Taylor instability and viscous fingering patterns. As well as providing a brief review of these key issues, we present a flexible numerical scheme for studying both the standard and nonstandard Hele-Shaw flows. Our method consists of using a modified finite-difference stencil in conjunction with the level-set method to solve the governing equation for pressure on complicated domains and track the location of the moving boundary. Simulations show that our method is capable of reproducing the distinctive morphological features of the Saffman–Taylor instability on a uniform computational grid. By making straightforward adjustments, we show how our scheme can easily be adapted to solve for a wide variety of nonstandard configurations, including cases where the gap between the plates is linearly tapered, the plates are separated in time, and the entire Hele-Shaw cell is rotated at a given angular velocity. doi:10.1017/S144618112100033X

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Li, Jing, Xiaochen Li e Shijun Liao. "Stability and hysteresis of Faraday waves in Hele-Shaw cells". Journal of Fluid Mechanics 871 (24 maggio 2019): 694–716. http://dx.doi.org/10.1017/jfm.2019.335.

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The instability of Faraday waves in Hele-Shaw cells is investigated experimentally and theoretically. A novel hydrodynamic model involving capillary action is proposed to capture the variation of the dynamic contact line between two close walls of narrow containers. The amplitude equations are derived from the gap-averaged model. By means of Lyapunov’s first method, a good prediction of the onset threshold of forcing acceleration is obtained, which shows the model’s validity for addressing the stability problem for Faraday waves in Hele-Shaw cells. It is found that the effect of the dynamic contact line is much greater than that of Poiseuille assumption of velocity profile for the cases under investigation. A new dispersion relation is obtained, which agrees well with experimental data. However, we highly recommend the conventional dispersion relation for gravity–capillary waves, which can generally meet common needs. Surface tension is found to be a key factor of interface flows in Hele-Shaw cells. According to our experimental observations, a liquid film is found on the front wall of the Hele-Shaw cell when the wave is falling. As a property of the friction coefficient from molecular kinetics, wet and dry plates show different wetting procedures. Unlike some authors of previous publications, we attribute the hysteresis to the out-of-plane interface shape rather than to detuning, i.e. the difference between natural frequency and response frequency.

17

Ross, Julius, e David Witt Nyström. "The Hele-Shaw flow and moduli of holomorphic discs". Compositio Mathematica 151, n. 12 (18 agosto 2015): 2301–28. http://dx.doi.org/10.1112/s0010437x15007526.

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We present a new connection between the Hele-Shaw flow, also known as two-dimensional Laplacian growth, and the theory of holomorphic discs with boundary contained in a totally real submanifold. Using this, we prove short-time existence and uniqueness of the Hele-Shaw flow with varying permeability both when starting from a single point and also when starting from a smooth Jordan domain. Applying the same ideas, we prove that the moduli space of smooth quadrature domains is a smooth manifold whose dimension we also calculate, and we give a local existence theorem for the inverse potential problem in the plane.

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BOOS, W., e A. THESS. "Thermocapillary flow in a Hele-Shaw cell". Journal of Fluid Mechanics 352 (10 dicembre 1997): 305–30. http://dx.doi.org/10.1017/s0022112097007477.

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We formulate a simple theoretical model that permits one to investigate surface-tension-driven flows with complex interface geometry. The model consists of a Hele-Shaw cell filled with two different fluids and subjected to a unidirectional temperature gradient. The shape of the interface that separates the fluids can be arbitrarily complex. If the contact line is pinned, i.e. unable to move, the problem of calculating the flow in both fluids is governed by a linear set of equations containing the characteristic aspect ratio and the viscosity ratio as the only input parameters. Analytical solutions, derived for a linear interface and for a circular drop, demonstrate that for large aspect ratio the flow field splits into a potential core flow and a thermocapillary boundary layer which acts as a source for the core. An asymptotic theory is developed for this limit which reduces the mathematical problem to a Laplace equation with Dirichlet boundary conditions. This problem can be efficiently solved utilizing a boundary element method. It is found that the thermocapillary flow in non-circular drops has a highly non-trivial streamline topology. After releasing the assumption of a pinned interface, a linear stability analysis is carried out for the interface under both transverse and longitudinal temperature gradients. For a semi-infinite fluid bounded by a freely movable surface long-wavelength instability due to the temperature gradient across the surface is predicted. The mechanism of this instability is closely related to the long-wave instability in surface-tension-driven Bénard convection. A linear interface heated from the side is found to be linearly stable. The possibility of experimental verification of the predictions is briefly discussed.

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GRAF, F., E. MEIBURG e C. HÄRTEL. "Density-driven instabilities of miscible fluids in a Hele-Shaw cell: linear stability analysis of the three-dimensional Stokes equations". Journal of Fluid Mechanics 451 (25 gennaio 2002): 261–82. http://dx.doi.org/10.1017/s0022112001006516.

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We consider the situation of a heavier fluid placed above a lighter one in a vertically arranged Hele-Shaw cell. The two fluids are miscible in all proportions. For this configuration, experiments and nonlinear simulations recently reported by Fernandez et al. (2002) indicate the existence of a low-Rayleigh-number (Ra) ‘Hele-Shaw’ instability mode, along with a high-Ra ‘gap’ mode whose dominant wavelength is on the order of five times the gap width. These findings are in disagreement with linear stability results based on the gap-averaged Hele-Shaw approach, which predict much smaller wavelengths. Similar observations have been made for immiscible flows as well (Maxworthy 1989).In order to resolve the above discrepancy, we perform a linear stability analysis based on the full three-dimensional Stokes equations. A generalized eigenvalue problem is formulated, whose numerical solution yields both the growth rate and the two-dimensional eigenfunctions in the cross-gap plane as functions of the spanwise wavenumber, an ‘interface’ thickness parameter, and Ra. For large Ra, the dispersion relations confirm that the optimally amplified wavelength is about five times the gap width, with the exact value depending on the interface thickness. The corresponding growth rate is in very good agreement with the experimental data as well. The eigenfunctions indicate that the predominant fluid motion occurs within the plane of the Hele-Shaw cell. However, for large Ra purely two-dimensional modes are also amplified, for which there is no motion in the spanwise direction. Scaling laws are provided for the dependence of the maximum growth rate, the corresponding wavenumber, and the cutoff wavenumber on Ra and the interface thickness. Furthermore, the present results are compared both with experimental data, as well as with linear stability results obtained from the Hele-Shaw equations and a modified Brinkman equation.

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Morrow, Liam C., Timothy J. Moroney e Scott W. McCue. "Numerical investigation of controlling interfacial instabilities in non-standard Hele-Shaw configurations". Journal of Fluid Mechanics 877 (2 settembre 2019): 1063–97. http://dx.doi.org/10.1017/jfm.2019.623.

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Viscous fingering experiments in Hele-Shaw cells lead to striking pattern formations which have been the subject of intense focus among the physics and applied mathematics community for many years. In recent times, much attention has been devoted to devising strategies for controlling such patterns and reducing the growth of the interfacial fingers. We continue this research by reporting on numerical simulations, based on the level set method, of a generalised Hele-Shaw model for which the geometry of the Hele-Shaw cell is altered. First, we investigate how imposing constant and time-dependent injection rates in a Hele-Shaw cell that is either standard, tapered or rotating can be used to reduce the development of viscous fingering when an inviscid fluid is injected into a viscous fluid over a finite time period. We perform a series of numerical experiments comparing the effectiveness of each strategy to determine how these non-standard Hele-Shaw configurations influence the morphological features of the inviscid–viscous fluid interface. Surprisingly, a converging or diverging taper of the plates leads to reduced metrics of viscous fingering at the final time when compared to the standard parallel configuration, especially with carefully chosen injection rates; for the rotating plate case, the effect is even more dramatic, with sufficiently large rotation rates completely stabilising the interface. Next, we illustrate how the number of non-splitting fingers can be controlled by injecting the inviscid fluid at a time-dependent rate while increasing the gap between the plates. Our simulations compare well with previous experimental results for various injection rates and geometric configurations. We demonstrate how the number of non-splitting fingers agrees with that predicted from linear stability theory up to some finger number; for larger values of our control parameter, the fully nonlinear dynamics of the problem leads to slightly fewer fingers than this linear prediction.

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Howison, S. D. "Complex variable methods in Hele–Shaw moving boundary problems". European Journal of Applied Mathematics 3, n. 3 (settembre 1992): 209–24. http://dx.doi.org/10.1017/s0956792500000802.

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We discuss the one-phase Hele–Shaw problem in two space dimensions. We review exact solutions in the zero-surface-tension case, giving a unified account of the Schwarz function and conformal mapping approaches. We discuss the extension of the former method to the cases in which surface tension or ‘kinetic undercooling’ terms apply on the moving boundary, and we give some conjectures on the resulting singularity structure. Finally, we give a new interpretation of the linear stability analysis of the zero-surface-tension problem, and we suggest a possible regularization of ill-posed problems by the imposition of a unilateral constraint on the moving boundary.

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KHALID, A. H., N. R. McDONALD e J. M. VANDEN-BROECK. "Hele-Shaw flow driven by an electric field". European Journal of Applied Mathematics 25, n. 4 (10 ottobre 2013): 425–47. http://dx.doi.org/10.1017/s0956792513000351.

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The behaviour of two-dimensional finite blobs of conducting viscous fluid in a Hele-Shaw cell subject to an electric field is considered. The time-dependent free boundary problem is studied both analytically using the Schwarz function of the free boundary and numerically using a boundary integral method. Various problems are considered, including (i) the behaviour of an initially circular blob of conducting fluid subject to an electric point charge located arbitrarily within the blob, (ii) the delay in cusp formation on the free boundary in sink-driven flow due to a strategically placed electric charge and (iii) the stability of exact steady solutions having both hydrodynamic and electric forcing.

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Quirós, Fernando, e Juan Luis Vázquez. "Asymptotic convergence of the Stefan problem to Hele-Shaw". Transactions of the American Mathematical Society 353, n. 2 (23 ottobre 2000): 609–34. http://dx.doi.org/10.1090/s0002-9947-00-02739-2.

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Savina, T. V., e A. A. Nepomnyashchy. "A dynamical mother body in a Hele-Shaw problem". Physica D: Nonlinear Phenomena 240, n. 14-15 (luglio 2011): 1156–63. http://dx.doi.org/10.1016/j.physd.2011.04.002.

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FOKAS, A. S., e S. TANVEER. "A Hele-Shaw problem and the second Painlevé transcendent". Mathematical Proceedings of the Cambridge Philosophical Society 124, n. 1 (luglio 1998): 169–91. http://dx.doi.org/10.1017/s0305004197002260.

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We consider the time evolution of an unbounded interface between a viscous fluid and a displacing fluid of negligible viscosity in a Hele-Shaw geometry. Using a conformal mapping transformation z=F(ζ, t) from the lower half ζ-plane to the flow domain Ω(t) in the z-plane, the dynamics is formulated in terms of an integro-differential equation for F on the real ζ axis. This formulation is similar to earlier results for radial and channel geometries. Using a certain explicit solution it is shown that the zero surface tension problem is ill-posed in the sense of Hadamard. Furthermore, certain earlier formal asymptotic results for small but non-zero surface tension [47], are rigorously established. This is achieved by relating the similarity reduction of the Harry–Dym equation to the Painlevé II equation, and by studying this equation in the complex plane. In particular, it is shown that there exists a unique solution of Painlevé II equation which satisfies the appropriate far-field algebraic decay condition in a certain sector Ŝ of the complex plane and which in Ŝ is free of singularities. This solution asymptotes to certain elliptic functions in other sectors of the complex plane and contains a set of denumerably infinite number of singularities that approach the boundary of Ŝ from the outside at large distances from the origin (in the similarity variable). This is consistent with earlier numerical results.

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Lokutsievskiy, Lev, e Vincent Runge. "Optimal Control by Multipoles in the Hele-Shaw Problem". Journal of Mathematical Fluid Mechanics 17, n. 2 (7 aprile 2015): 261–77. http://dx.doi.org/10.1007/s00021-015-0206-9.

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GAT, A. D., I. FRANKEL e D. WEIHS. "A higher-order Hele-Shaw approximation with application to gas flows through shallow micro-channels". Journal of Fluid Mechanics 638 (14 ottobre 2009): 141–60. http://dx.doi.org/10.1017/s002211200999125x.

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The classic hydrodynamic Hele-Shaw problem is revisited in the context of evaluating the viscous resistance to low-Mach compressible viscous gas flows through shallow non-uniform micro-fluidic configurations. Our recent study of gas flows through constricted shallow micro-channels indicates that the failure of the standard Hele-Shaw approximation to satisfy the no-slip boundary condition at the sidewalls severely restricts its applicability. To overcome this we have extended the asymptotic scheme to incorporate an inner solution in the vicinity of the sidewalls (which, in turn, allows for the characterization of the effects of channel cross-section geometry) and its matching to an outer correction. We have compared the results of the present asymptotic analysis to existing exact analytic and numerical results for straight and uniform channels and to finite-element simulations for a 90° turn and a symmetric T-junction, which demonstrate a remarkably improved accuracy relative to the standard Hele-Shaw approximation. This suggests the present scheme as a viable alternative for the rapid performance estimate of micro-fluidic devices.

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Urakov, A. R., A. A. Gordeev e S. S. Porechny. "The solution of problems on an self-similar electrochemical shaping by means of hydrodynamic analogy". Proceedings of the Mavlyutov Institute of Mechanics 6 (2008): 150–55. http://dx.doi.org/10.21662/uim2008.1.021.

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Self-similar (at preservation of geometrical similarity of borders) solutions of non-stationary Hele-Shaw problems in connection to an electrochemical shaping are considered. The problem of a flow about an arch of a circle on which the border condition for Zhukovsky’s function has the form similar to a boundary condition of a self-similar problem is used for the solution.

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Entov, Vladimir M., Pavel I. Etingof e Dmitry Ya Kleinbock. "Hele–Shaw flows with a free boundary produced by multipoles". European Journal of Applied Mathematics 4, n. 2 (giugno 1993): 97–120. http://dx.doi.org/10.1017/s0956792500001029.

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We study Hele–Shaw flows with a moving boundary and multipole singularities. We find that such flows can be defined only on a finite time interval. Using a complex variable approach, we construct a family of explicit solutions for a single multipole. These solutions turn out to have the maximal possible lifetime in a certain class of solutions.We also discuss the generalized Hele-Shaw model in which surface tension at the moving boundary is considered, and develop a method of finding steady shapes. This method yields new one-parameter families of stationary solutions. In the Appendix we discuss a connection between these solutions and a variational problem of potential theory.

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CAGINALP, GUNDUZ, e XINFU CHEN. "Convergence of the phase field model to its sharp interface limits". European Journal of Applied Mathematics 9, n. 4 (agosto 1998): 417–45. http://dx.doi.org/10.1017/s0956792598003520.

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We consider the distinguished limits of the phase field equations and prove that the corresponding free boundary problem is attained in each case. These include the classical Stefan model, the surface tension model (with or without kinetics), the surface tension model with zero specific heat, the two phase Hele–Shaw, or quasi-static, model. The Hele–Shaw model is also a limit of the Cahn–Hilliard equation, which is itself a limit of the phase field equations. Also included in the distinguished limits is the motion by mean curvature model that is a limit of the Allen–Cahn equation, which can in turn be attained from the phase field equations.

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Bayada, G., M. Boukrouche e M. El-A. Talibi. "The Transient Lubrication Problem as a Generalized Hele-Shaw Type Problem". Zeitschrift für Analysis und ihre Anwendungen 14, n. 1 (1995): 59–87. http://dx.doi.org/10.4171/zaa/663.

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Morris, S. J. S. "Stability of thermoviscous Hele-Shaw flow". Journal of Fluid Mechanics 308 (10 febbraio 1996): 111–28. http://dx.doi.org/10.1017/s0022112096001413.

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Viscous fingering can occur as a three-dimensional disturbance to plane flow of a hot thermoviscous liquid in a Hele-Shaw cell with cold isothermal walls. This work assumes the principle of exchange of stabilities, and uses a temporal stability analysis to find the critical viscosity ratio and finger spacing as functions of channel length, Lc. Viscous heating is taken as negligible, so the liquid cools with distance (x) downstream. Because the base flow is spatially developing, the disturbance equations are not fully separable. They admit, however, an exact solution for a liquid whose viscosity and specific heats are arbitrary functions of temperature. This solution describes the neutral disturbances in terms of the base flow and an amplitude, A(x). The stability of a given (computed) base flow is determined by solving an eigenvalue problem for A(x), and the critical finger spacing. The theory is illustrated by using it to map the instability for variable-viscosity flow with constant specific heat. Two fingering modes are predicted, one being a turning-point instability. The preferred mode depends on Lc. Finger spacing is comparable with the thermal entry length in a long channel, and is even larger in short channels. When applied to magmatic systems, the results suggest that fingering will occur on geological scales only if the system is about freeze.

33

CURT, PAULA. "On some invariant geometric properties in Hele-Shaw flows with small surface tension". Carpathian Journal of Mathematics 31, n. 1 (2015): 53–60. http://dx.doi.org/10.37193/cjm.2015.01.06.

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In this paper, by applying methods from complex analysis, we analyse the time evolution of the free boundary of a viscous fluid for planar flows in Hele-Shaw cells under injection in the non-zero surface tension case. We study the invariance in time of α-convexity (for α ∈ [0, 1] this is a geometric property which provides a continuous passage from starlikeness to convexity) for bounded domains. In this case we show that the α-convexity property of the moving boundary in a Hele-Shaw flow problem with small surface tension is preserved in time for α ≤ 0. For unbounded domains (with bounded complement) we prove the invariance in time of convexity.

34

BALSA, THOMAS F. "Secondary flow in a Hele-Shaw cell". Journal of Fluid Mechanics 372 (10 ottobre 1998): 25–44. http://dx.doi.org/10.1017/s0022112098002171.

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We examine the flow in a horizontal Hele-Shaw cell in which the undisturbed unidirectional flow at infinity is required to stream around a vertical cylinder spanning the gap between the two (horizontal) plates of the cell. A combination of matched asymptotic expansions and numerical methods is employed to elucidate the structure of the boundary layer near the surface of the cylinder. The two length scales of the problem are the gap, h, and the length of the body, l; it is assumed that h/l<<1. The characteristic Reynolds number based on l is O(1). The length scales associated with the boundary layer and the classical Hele-Shaw flow pattern are O(h) and O(l), respectively.It is found that the boundary layer contains streamwise vorticity. This vorticity is generated at the three no-slip surfaces (the two plates and the cylinder wall) as a result of the cross-flow induced by the streamwise acceleration/deceleration of the flow around the curved cylinder. The strength of the secondary flow, hence the associated streamwise vorticity, is proportional to changes in body curvature. The validity of the classical Hele-Shaw flow is examined systematically, and higher-order corrections are worked out. This results in a displacement thickness that is roughly 30% of the gap. In other words, the lowest-order correction to the classical Hele-Shaw flow may be obtained by requiring the outer flow (on the scale O(l)) to satisfy the no-penetration boundary condition on a displaced cylinder surface. The boundary layer contains ‘corner’ vortices at the intersections of the horizontal plates and the vertical cylinder surface.

35

ALEXANDROU, A. N., V. M. ENTOV, S. S. KOLGANOV e N. V. KOLGANOVA. "On bubble rising in a Hele–Shaw cell filled with a non-Newtonian fluid". European Journal of Applied Mathematics 15, n. 3 (giugno 2004): 315–27. http://dx.doi.org/10.1017/s0956792504005509.

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The problem of a bubble rising due to buoyancy in a Hele–Shaw cell filled with a viscous fluid is a classical free-boundary problem first posed and solved by Saffman & Taylor [11]. In fact, due to linearity of the flow equations the problem is reduced to that of a bubble transported by uniform fluid flow. Saffman and Taylor provided explicit expressions for the bubble shape. Steady propagation of bubbles and fingers in a Hele–Shaw cell filled with a nonlinearly-viscous fluid was studied by Alexandrou & Entov [1]. In Alexandrou & Entov [1], it was shown that for a nonlinearly viscous fluid the problem of a rising bubble cannot be reduced to that of a steadily transported bubble, and should be treated separately. This note presents a solution of the problem following the general framework suggested in Alexandrou & Entov [1]. The hodograph transform is used in combination with finite-difference and collocation techniques to solve the problem. Results are presented for the cases of a Bingham and power-law fluids.

36

Hwang, Hyung Ju, Youngmin Oh e Marco Antonio Fontelos. "The vanishing surface tension limit for the Hele-Shaw problem". Discrete and Continuous Dynamical Systems - Series B 21, n. 10 (novembre 2016): 3479–514. http://dx.doi.org/10.3934/dcdsb.2016108.

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37

Kim, Inwon. "Long time regularity of solutions of the Hele–Shaw problem". Nonlinear Analysis: Theory, Methods & Applications 64, n. 12 (giugno 2006): 2817–31. http://dx.doi.org/10.1016/j.na.2005.09.021.

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38

Crowdy, D. G. "Exact solutions to the unsteady two-phase Hele-Shaw problem". Quarterly Journal of Mechanics and Applied Mathematics 59, n. 4 (20 ottobre 2006): 475–85. http://dx.doi.org/10.1093/qjmam/hbl012.

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39

Loutsenko, Igor. "The Variable Coefficient Hele-Shaw Problem, Integrability and Quadrature Identities". Communications in Mathematical Physics 268, n. 2 (8 settembre 2006): 465–79. http://dx.doi.org/10.1007/s00220-006-0099-9.

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40

Constantin, P., e M. Pugh. "Global solutions for small data to the Hele-Shaw problem". Nonlinearity 6, n. 3 (1 maggio 1993): 393–415. http://dx.doi.org/10.1088/0951-7715/6/3/004.

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41

Požár, Norbert. "Homogenization of the Hele-Shaw Problem in Periodic Spatiotemporal Media". Archive for Rational Mechanics and Analysis 217, n. 1 (16 dicembre 2014): 155–230. http://dx.doi.org/10.1007/s00205-014-0831-0.

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42

Chen, Xinfu. "The Hele-Shaw problem and area-preserving curve-shortening motions". Archive for Rational Mechanics and Analysis 123, n. 2 (1993): 117–51. http://dx.doi.org/10.1007/bf00695274.

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43

RICHARDSON, S. "Plane Stokes flows with time-dependent free boundaries in which the fluid occupies a doubly-connected region". European Journal of Applied Mathematics 11, n. 3 (giugno 2000): 249–69. http://dx.doi.org/10.1017/s0956792500004149.

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Abstract (sommario):

Consider the two-dimensional quasi-steady Stokes flow of an incompressible Newtonian fluid occupying a time-dependent region bounded by free surfaces, the motion being driven solely by a constant surface tension acting at the free boundaries. When the fluid region is simply-connected, it is known that this Stokes flow problem is closely related to a Hele-Shaw free boundary problem when the zero-surface-tension model is employed. Specifically, if the initial configuration for the Stokes flow problem can be produced by injection at N points into an empty Hele-Shaw cell, then so can all later configurations. Moreover, there are N invariants; while the N points at which injection must take place move, the amount to be injected at each of these points remains the same. In this paper, we consider the situation when the fluid region is doubly-connected and show that, provided the geometry has an appropriate rotational symmetry, the same results continue to hold and can be exploited to determine the solution of the Stokes flow problem.

44

Wilson, S. D. R. "The Taylor–Saffman problem for a non-Newtonian liquid". Journal of Fluid Mechanics 220 (novembre 1990): 413–25. http://dx.doi.org/10.1017/s0022112090003329.

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The Taylor–Saffman problem concerns the fingering instability which develops when one liquid displaces another, more viscous, liquid in a porous medium, or equivalently for Newtonian liquids, in a Hele-Shaw cell. Recent experiments with Hele-Shaw cells using non-Newtonian liquids have shown striking qualitative differences in the fingering pattern, which for these systems branches repeatedly in a manner resembling the growth of a fractal. This paper is an attempt to provide the beginnings of a hydrodynamical theory of this instability by repeating the analysis of Taylor & Saffman using a more general constitutive model. In fact two models are considered; the Oldroyd ‘Fluid B’ model which exhibits elasticity but not shear thinning, and the Ostwald–de Waele power-law model with the opposite combination. Of the two, only the Oldroyd model shows qualitatively new effects, in the form of a kind of resonance which can produce sharply increasing (in fact unbounded) growth rates as the relaxation time of the fluid increases. This may be a partial explanation of the observations on polymer solutions; the similar behaviour reported for clay pastes and slurries is not explained by shear-thinning and may involve a finite yield stress, which is not incorporated into either of the models considered here.

45

Logvinov, Oleg A. "Viscous fingering in poorly miscible power-law fluids". Physics of Fluids 34, n. 6 (giugno 2022): 063105. http://dx.doi.org/10.1063/5.0088487.

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Abstract (sommario):

A renowned problem of a viscous fluid displacement by a less viscous one from a Hele–Shaw cell is considered. Both fluids exhibit non-Newtonian properties: a power-law viscosity dependence on strain rates (Ostwald–de Waele rheology). A unified approach independent of particular rheology is applied to derive averaged two-dimensional equations of motion (so-called Hele–Shaw models). The equations are based on Reynolds class averaging procedure. Under these governing equations, linear stability analysis of the radial interface is conducted with a new key idea—possibility of characteristic size selection even in the absence of stabilizing factors such as surface tension and molecular diffusion. For proving this, proper boundary conditions are set on the interface, namely, the equality of full normal stresses including viscous ones, instead of the simple equality of pressures.

46

Tryggvason, Grétar, e Hassan Aref. "Finger-interaction mechanisms in stratified Hele-Shaw flow". Journal of Fluid Mechanics 154 (maggio 1985): 287–301. http://dx.doi.org/10.1017/s0022112085001537.

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Interactions between a few fingers in sharply stratified Hele-Shaw flow are investigated by numerical integration of the initial-value problem. It is shown that fingers evolving from an initial perturbation of an unstable interface consisting of a single wave are rather insensitive to variations of the control parameters governing the flow. Initial perturbations with at least two waves, on the other hand, lead to important finger-interaction and selection mechanisms at finite amplitude. On the basis of the results reported here many features of an earlier numerical study of the ‘statistical-fingering’ regime can be rationalized.

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Deckelnick, Klaus, e Charles M. Elliott. "Local and global existence results for anisotropic Hele–Shaw flows". Proceedings of the Royal Society of Edinburgh: Section A Mathematics 129, n. 2 (1999): 265–94. http://dx.doi.org/10.1017/s0308210500021351.

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Abstract (sommario):

In this paper we study a moving boundary problem for an anisotropic two-phase Hele–Shaw flow. Using a regularization technique, we prove existence of a local solution. Under suitable conditions on the initial free boundary we obtain a global solution and study its asymptotic behaviour.

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Bazaliy, B. V., e N. Vasylyeva. "The Two-Phase Hele-Shaw Problem with a Nonregular Initial Interface and Without Surface Tension". Zurnal matematiceskoj fiziki, analiza, geometrii 10, n. 1 (25 marzo 2014): 3–43. http://dx.doi.org/10.15407/mag10.01.003.

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49

CUMMINGS, L. J., e J. R. KING. "Hele–Shaw flow with a point sink: generic solution breakdown". European Journal of Applied Mathematics 15, n. 1 (febbraio 2004): 1–37. http://dx.doi.org/10.1017/s095679250400539x.

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Abstract (sommario):

Recent numerical evidence [8, 28, 33] suggests that in the Hele–Shaw suction problem with vanishingly small surface tension $\gamma$, the free boundary generically approaches the sink in a wedge-like configuration, blow-up occurring when the wedge apex reaches the sink. Sometimes two or more such wedges approach the sink simultaneously [33]. We construct a family of solutions to the zero-surface tension (ZST) problem in which fluid is injected at the (coincident) apices of an arbitrary number $N$ of identical infinite wedges, of arbitrary angle. The time reversed suction problem then models what is observed numerically with non-zero surface tension. We conjecture that (for a given value of $N$) a particular member of this family of ZST solutions, with special complex plane singularity structure, is selected in the limit $\gamma\,{\to}\,0$.

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CENICEROS, HECTOR D., e THOMAS Y. HOU. "The singular perturbation of surface tension in Hele-Shaw flows". Journal of Fluid Mechanics 409 (25 aprile 2000): 251–72. http://dx.doi.org/10.1017/s0022112099007703.

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Morphological instabilities are common to pattern formation problems such as the non-equilibrium growth of crystals and directional solidification. Very small perturbations caused by noise originate convoluted interfacial patterns when surface tension is small. The generic mechanisms in the formation of these complex patterns are present in the simpler problem of a Hele-Shaw interface. Amid this extreme noise sensitivity, what is then the role played by small surface tension in the dynamic formation and selection of these patterns? What is the asymptotic behaviour of the interface in the limit as surface tension tends to zero? The ill-posedness of the zero-surface-tension problem and the singular nature of surface tension pose challenging difficulties in the investigation of these questions. Here, we design a novel numerical method that greatly reduces the impact of noise, and allows us to accurately capture and identify the singular contributions of extremely small surface tensions. The numerical method combines the use of a compact interface parametrization, a rescaling of the governing equations, and very high precision. Our numerical results demonstrate clearly that the zero-surface-tension limit is indeed singular. The impact of a surface-tension-induced complex singularity is revealed in detail. The singular effects of surface tension are first felt at the tip of the interface and subsequently spread around it. The numerical simulations also indicate that surface tension defines a length scale in the fingers developing in a later stage of the interface evolution.

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