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Journal articles on the topic 'Hele-Shaw flow; Stokes flow'

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Published: 4 June 2021

Last updated: 14 February 2022

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1

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, no. 3 (2000): 249–69. http://dx.doi.org/10.1017/s0956792500004149.

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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 con

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2

Heussler, F. H. C., R. M. Oliveira, M. O. John, and E. Meiburg. "Three-dimensional Navier–Stokes simulations of buoyant, vertical miscible Hele-Shaw displacements." Journal of Fluid Mechanics 752 (July 2, 2014): 157–83. http://dx.doi.org/10.1017/jfm.2014.327.

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AbstractGravitationally and viscously unstable miscible displacements in vertical Hele-Shaw cells are investigated via three-dimensional Navier–Stokes simulations. The velocity of the two-dimensional base-flow displacement fronts generally increases with the unfavourable viscosity contrast and the destabilizing density difference. Displacement fronts moving faster than the maximum velocity of the Poiseuille flow far downstream exhibit a single stagnation point in a moving reference frame, consistent with earlier observations for corresponding capillary tube flows. Gravitationally stable fronts

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3

Pozrikidis, C. "The motion of particles in the Hele-Shaw cell." Journal of Fluid Mechanics 261 (February 25, 1994): 199–222. http://dx.doi.org/10.1017/s0022112094000315.

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The force and torque on a particle that translates, rotates, or is held stationary in an incident flow within a channel with parallel-sided walls, are considered in the limit of Stokes flow. Assuming that the particle has an axisymmetric shape with axis perpendicular to the channel walls, the problem is formulated in terms of a boundary integral equation that is capable of describing arbitrary three-dimensional Stokes flow in an axisymmetric domain. The method involves: (a) representing the flow in terms of a single-layer potential that is defined over the physical boundaries of the flow as we

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4

RICHARDSON, S. "Two-dimensional Stokes flows with time-dependent free boundaries driven by surface tension." European Journal of Applied Mathematics 8, no. 4 (1997): 311–29. http://dx.doi.org/10.1017/s0956792597003057.

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We consider the two-dimensional quasi-steady Stokes flow of an incompressible Newtonian fluid occupying a time-dependent simply-connected region bounded by a free surface, the motion being driven solely by a constant surface tension acting at the free boundary. Of particular concern here are such flows that start from an initial configuration with the fluid occupying an array of touching circular disks. We show that, when there are N such disks in a general position, the evolution of the fluid region is described by a conformal map involving 2N−1 time-dependent parameters whose variation is go

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5

Oliveira, Rafael M., and Eckart Meiburg. "Miscible displacements in Hele-Shaw cells: three-dimensional Navier–Stokes simulations." Journal of Fluid Mechanics 687 (October 12, 2011): 431–60. http://dx.doi.org/10.1017/jfm.2011.367.

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AbstractThree-dimensional Navier–Stokes simulations of viscously unstable, miscible Hele-Shaw displacements are discussed. Quasisteady fingers are observed whose tip velocity increases with the Péclet number and the unfavourable viscosity ratio. These fingers are widest near the tip, and become progressively narrower towards the root. The film of resident fluid left behind on the wall decreases in thickness towards the finger tip. The simulations reveal the detailed mechanism by which the initial spanwise vorticity of the base flow, when perturbed, gives rise to the cross-gap vorticity that dr

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6

Lee, S. Y., R. Teodorescu, and P. Wiegmann. "Viscous shocks in Hele–Shaw flow and Stokes phenomena of the Painlevé I transcendent." Physica D: Nonlinear Phenomena 240, no. 13 (2011): 1080–91. http://dx.doi.org/10.1016/j.physd.2010.09.017.

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7

LIN, Y. L. "Large-time rescaling behaviours of Stokes and Hele-Shaw flows driven by injection." European Journal of Applied Mathematics 22, no. 1 (2010): 7–19. http://dx.doi.org/10.1017/s0956792510000264.

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In this paper, we give a precise description of the rescaling behaviours of global strong polynomial solutions to the reformulation of zero surface tension Hele-Shaw problem driven by injection, the Polubarinova-Galin equation, in terms of Richardson complex moments. From past results, we know that this set of solutions is large. This method can also be applied to zero surface tension Stokes flow driven by injection and a rescaling behaviour is given in terms of many conserved quantities as well.

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8

Demidov, A. S., J. P. Lohéac, and V. Runge. "Stokes–Leibenson problem for Hele-Shaw flow: a critical set in the space of contours." Russian Journal of Mathematical Physics 23, no. 1 (2016): 35–55. http://dx.doi.org/10.1134/s1061920816010039.

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9

Llamoza, Johan, and Desiderio A. Vasquez. "Structures and Instabilities in Reaction Fronts Separating Fluids of Different Densities." Mathematical and Computational Applications 24, no. 2 (2019): 51. http://dx.doi.org/10.3390/mca24020051.

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Density gradients across reaction fronts propagating vertically can lead to Rayleigh–Taylor instabilities. Reaction fronts can also become unstable due to diffusive instabilities, regardless the presence of a mass density gradient. In this paper, we study the interaction between density driven convection and fronts with diffusive instabilities. We focus in fluids confined in Hele–Shaw cells or porous media, with the hydrodynamics modeled by Brinkman’s equation. The time evolution of the front is described with a Kuramoto–Sivashinsky (KS) equation coupled to the fluid velocity. A linear stabili

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10

Zhu, Lailai, and François Gallaire. "A pancake droplet translating in a Hele-Shaw cell: lubrication film and flow field." Journal of Fluid Mechanics 798 (June 15, 2016): 955–69. http://dx.doi.org/10.1017/jfm.2016.357.

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We adopt a boundary integral method to study the dynamics of a translating droplet confined in a Hele-Shaw cell in the Stokes regime. The droplet is driven by the motion of the ambient fluid with the same viscosity. We characterize the three-dimensional (3D) nature of the droplet interface and of the flow field. The interface develops an arc-shaped ridge near the rear-half rim with a protrusion in the rear and a laterally symmetric pair of higher peaks; this pair of protrusions has been identified by recent experiments (Huerre et al., Phys. Rev. Lett., vol. 115 (6), 2015, 064501) and predicted

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11

GOYAL, N., H. PICHLER, and E. MEIBURG. "Variable-density miscible displacements in a vertical Hele-Shaw cell: linear stability." Journal of Fluid Mechanics 584 (July 25, 2007): 357–72. http://dx.doi.org/10.1017/s0022112007006428.

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A computational study based on the Stokes equations is conducted to investigate the effects of gravitational forces on miscible displacements in vertical Hele-Shaw cells. Nonlinear simulations provide the quasi-steady displacement fronts in the gap of the cell, whose stability to spanwise perturbations is subsequently examined by means of a linear stability analysis. The two-dimensional simulations indicate a marked thickening (thinning) and slowing down (speeding up) of the displacement front for flows stabilized (destabilized) by gravity. For the range investigated, the tip velocity is found

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12

Kulkarni, Venkatesh M., Chu Wee Liang, C. W. Tan, P. A. Aswatha Narayana, and K. N. Seetharamu. "Simulation of Mold Filling for Non-Newtonian Fluids - Part 2." Journal of Microelectronics and Electronic Packaging 3, no. 2 (2006): 52–60. http://dx.doi.org/10.4071/1551-4897-3.2.52.

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This paper deals with the flow in the resin transfer molding process commonly used for IC chip encapsulation in the electronic packaging industry. A solution algorithm is presented for modeling the flow of a non-Newtonian fluid obeying a Power-Law model and the algorithm is used to conduct parametric studies in transfer molding. The flow model uses the Hele-Shaw approximation to solve the Navier-Stokes Equations and a pseudo-concentration algorithm for tracking the interface between the resin and the air. The Finite Element Method is employed to reduce the governing partial differential equati

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13

Demidov, A. S. "Evolution of the Perturbation of a Circle in the Stokes–Leibenson Problem for the Hele-Shaw Flow." Journal of Mathematical Sciences 123, no. 5 (2004): 4381–403. http://dx.doi.org/10.1023/b:joth.0000040301.53259.05.

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14

Dalwadi, Mohit P., S. Jonathan Chapman, Sarah L. Waters, and James M. Oliver. "On the boundary layer structure near a highly permeable porous interface." Journal of Fluid Mechanics 798 (May 31, 2016): 88–139. http://dx.doi.org/10.1017/jfm.2016.308.

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The method of matched asymptotic expansions is used to study the canonical problem of steady laminar flow through a narrow two-dimensional channel blocked by a tight-fitting finite-length highly permeable porous obstacle. We investigate the behaviour of the local flow close to the interface between the single-phase and porous regions (governed by the incompressible Navier–Stokes and Darcy flow equations, respectively). We solve for the flow in these inner regions in the limits of low and high Reynolds number, facilitating an understanding of the nature of the transition from Poiseuille to plug

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15

MARTIN, J., N. RAKOTOMALALA, L. TALON, and D. SALIN. "Viscous lock-exchange in rectangular channels." Journal of Fluid Mechanics 673 (February 14, 2011): 132–46. http://dx.doi.org/10.1017/s0022112010006208.

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In a viscous lock-exchange gravity current, which describes the reciprocal exchange of two fluids of different densities in a horizontal channel, the front between two Newtonian fluids spreads as the square root of time. The resulting diffusion coefficient reflects the competition between the buoyancy-driving effect and the viscous damping, and depends on the geometry of the channel. This lock-exchange diffusion coefficient has already been computed for a porous medium, a two-dimensional (2D) Stokes flow between two parallel horizontal boundaries separated by a vertical height H and, recently,

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16

Pihler-Puzović, Draga, Anne Juel, Gunnar G. Peng, John R. Lister, and Matthias Heil. "Displacement flows under elastic membranes. Part 1. Experiments and direct numerical simulations." Journal of Fluid Mechanics 784 (November 6, 2015): 487–511. http://dx.doi.org/10.1017/jfm.2015.590.

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The injection of fluid into the narrow liquid-filled gap between a rigid plate and an elastic membrane drives a displacement flow that is controlled by the competition between elastic and viscous forces. We study such flows using the canonical set-up of an elastic-walled Hele-Shaw cell whose upper boundary is formed by an elastic sheet. We investigate both single- and two-phase displacement flows in which the localised injection of fluid at a constant flow rate is accommodated by the inflation of the sheet and the outward propagation of an axisymmetric front beyond which the cell remains appro

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17

Li, Huanhao, Chun-Yi Kao, and Chih-Yung Wen. "Labyrinthine and secondary wave instabilities of a miscible magnetic fluid drop in a Hele-Shaw cell." Journal of Fluid Mechanics 836 (December 11, 2017): 374–96. http://dx.doi.org/10.1017/jfm.2017.739.

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A comprehensive experimental study is presented to analyse the instabilities of a magnetic fluid drop surrounded by miscible fluid confined in a Hele-Shaw cell. The experimental conditions include different magnetic fields (by varying the maximum pre-set magnetic field strengths,$H$, and sweep rates,$SR=\text{d}H_{t}/\text{d}t$, where$H_{t}$is the instant magnetic field strength), gap spans,$h$, and magnetic fluid samples, and are further coupled into a modified Péclect number$Pe^{\prime }$to evaluate the instabilities. Two distinct instabilities are induced by the external magnetic fields wit

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18

Demidov, A. S. "Evolution of the perturbation of a circle in the Stokes-Leibenson problem for the Hele-Shaw flow. Part II." Journal of Mathematical Sciences 139, no. 6 (2006): 7064–78. http://dx.doi.org/10.1007/s10958-006-0406-1.

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19

Zhou, Y., P. Y. Lagrée, S. Popinet, P. Ruyer, and P. Aussillous. "Experiments on, and discrete and continuum simulations of, the discharge of granular media from silos with a lateral orifice." Journal of Fluid Mechanics 829 (September 21, 2017): 459–85. http://dx.doi.org/10.1017/jfm.2017.543.

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We compare laboratory experiments, contact dynamics simulations and continuum Navier–Stokes simulations with a $\unicode[STIX]{x1D707}(I)$ visco-plastic rheology, of the discharge of granular media from a silo with a lateral orifice. We consider a rectangular silo with an orifice of height $D$ which spans the silo width $W$, and we observe two regimes. For small enough aperture aspect ratio ${\mathcal{A}}=D/W$, the Hagen–Beverloo relation is obtained. For thin enough silos, ${\mathcal{A}}\gg {\mathcal{A}}_{c}$, we observe a second regime where the outlet velocity varies with $\sqrt{W}$. This n

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20

MOYERS-GONZÁLEZ, M. A., I. A. FRIGAARD, O. SCHERZER, and T. P. TSAI. "Transient effects in oilfield cementing flows: Qualitative behaviour." European Journal of Applied Mathematics 18, no. 4 (2007): 477–512. http://dx.doi.org/10.1017/s0956792507007048.

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We present an unsteady Hele–Shaw model of the fluid–fluid displacements that take place during primary cementing of an oil well, focusing on the case where one Herschel–Bulkley fluid displaces another along a long uniform section of the annulus. Such unsteady models consist of an advection equation for a fluid concentration field coupled to a third-order non-linear PDE (Partial differential equation) for the stream function, with a free boundary at the boundary of regions of stagnant fluid. These models, although complex, are necessary for the study of interfacial instability and the effects o

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21

Tsay, Ruey-Yug, and Sheldon Weinbaum. "Viscous flow in a channel with periodic cross-bridging fibres: exact solutions and Brinkman approximation." Journal of Fluid Mechanics 226 (May 1991): 125–48. http://dx.doi.org/10.1017/s0022112091002318.

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A general solution of the three-dimensional Stokes equations is developed for the viscous flow past a square array of circular cylindrical fibres confined between two parallel walls. This doubly periodic solution, which is an extension of the theory developed by Lee & Fung (1969) for flow around a single fibre, successfully describes the transition in behaviour from the Hele-Shaw potential flow limit (aspect ratio B [Lt ] 1) to the viscous two-dimensional limiting case (B [Gt ] 1, Sangani & Acrivos 1982) for the hydrodynamic interaction between the fibres. These results are also compar

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22

De´nos, R., T. Arts, G. Paniagua, V. Michelassi, and F. Martelli. "Investigation of the Unsteady Rotor Aerodynamics in a Transonic Turbine Stage." Journal of Turbomachinery 123, no. 1 (2000): 81–89. http://dx.doi.org/10.1115/1.1314607.

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The paper focuses on the unsteady pressure field measured around the rotor midspan profile of the VKI Brite transonic turbine stage. The understanding of the complex unsteady flow field is supported by a quasi-three-dimensional unsteady Navier–Stokes computation using a k-ω turbulence model and a modified version of the Abu-Ghannam and Shaw correlation for the onset of transition. The agreement between computational and experimental results is satisfactory. They both reveal the dominance of the vane shock in the interaction. For this reason, it is difficult to identify the influence of vane-wa

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23

Hilgenfeld, L., P. Cardamone, and L. Fottner. "Boundary layer investigations on a highly loaded transonic compressor cascade with shock/laminar boundary layer interactions." Proceedings of the Institution of Mechanical Engineers, Part A: Journal of Power and Energy 217, no. 4 (2003): 349–56. http://dx.doi.org/10.1243/095765003322315405.

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Detailed experimental and numerical investigations of the flowfield and boundary layer on a highly loaded transonic compressor cascade were performed at various Mach and Reynolds numbers representative of real turbomachinery conditions. The emerging shock system interacts with the laminar boundary layer, causing shock-induced separation with turbulent reattachment. Steady two-dimensional calculations have been performed using the Navier—Stokes solver TRACE-U. The flow solver employs a modified version of the one-equation Spalart—Allmaras turbulence model coupled with a transition correlation b

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24

FERNANDEZ, J., P. KUROWSKI, P. PETITJEANS, and E. MEIBURG. "Density-driven unstable flows of miscible fluids in a Hele-Shaw cell." Journal of Fluid Mechanics 451 (January 25, 2002): 239–60. http://dx.doi.org/10.1017/s0022112001006504.

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Density-driven instabilities between miscible fluids in a vertical Hele-Shaw cell are investigated by means of experimental measurements, as well as two- and three-dimensional numerical simulations. The experiments focus on the early stages of the instability growth, and they provide detailed information regarding the growth rates and most amplified wavenumbers as a function of the governing Rayleigh number Ra. They identify two clearly distinct parameter regimes: a low-Ra, ‘Hele-Shaw’ regime in which the dominant wavelength scales as Ra−1, and a high-Ra ‘gap’ regime in which the length scale

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25

CUMMINGS, L. J., S. D. HOWISON, and J. R. KING. "Two-dimensional Stokes and Hele-Shaw flows with free surfaces." European Journal of Applied Mathematics 10, no. 6 (1999): 635–80. http://dx.doi.org/10.1017/s0956792599003964.

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26

Hedenmalm, Håkan, and Sergei Shimorin. "Hele–Shaw flow on hyperbolic surfaces." Journal de Mathématiques Pures et Appliquées 81, no. 3 (2002): 187–222. http://dx.doi.org/10.1016/s0021-7824(01)01222-3.

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27

Zeybek, M., and Y. C. Yortsos. "Parallel flow in Hele-Shaw cells." Journal of Fluid Mechanics 241 (August 1992): 421–42. http://dx.doi.org/10.1017/s0022112092002106.

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We consider the parallel flow of two immiscible fluids in a Hele-Shaw cell. The evolution of disturbances on the fluid interfaces is studied both theoretically and experimentally in the large-capillary-number limit. It is shown that such interfaces support wave motion, the amplitude of which for long waves is governed by a set of KdV and Airy equations. The waves are dispersive provided that the fluids have unequal viscosities and that the space occupied by the inner fluid does not pertain to the Saffman-Taylor conditions (symmetric interfaces with half-width spacing). Experiments conducted in

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28

Morris, S. J. S. "Stability of thermoviscous Hele-Shaw flow." Journal of Fluid Mechanics 308 (February 10, 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 viscosit

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29

Nizamova, A. D., and Valiev A. A. Valiev. "Mathematical model of oil displacement by water in a plane channel." Multiphase Systems 15, no. 3-4 (2020): 208–11. http://dx.doi.org/10.21662/mfs2020.3.131.

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Unstable displacement of immiscible liquids in a plane channel is a topical research in both theoretical and practical applications. In this paper, we consider a plane channel filled with an incompressible fluid. Over time, another fluid is injected into the channel. The fluids are immiscible. The paper builds a mathematical model of the process of oil displacement by water in a plane channel, which allows further numerical studies and comparison of the results with the obtained experimental data using the example of the Hele-Show cell. The mathematical model for a multiphase, multicomponent f

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30

GRAF, F., E. MEIBURG, and 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 (January 25, 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 waveleng

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31

BOOS, W., and A. THESS. "Thermocapillary flow in a Hele-Shaw cell." Journal of Fluid Mechanics 352 (December 10, 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 solut

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32

BALSA, THOMAS F. "Secondary flow in a Hele-Shaw cell." Journal of Fluid Mechanics 372 (October 10, 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 bou

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33

Almgren, Robert, Wei-Shen Dai, and Vincent Hakim. "Scaling behavior in anisotropic Hele-Shaw flow." Physical Review Letters 71, no. 21 (1993): 3461–64. http://dx.doi.org/10.1103/physrevlett.71.3461.

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34

Ceniceros †, Hector D., and José M. Villalobos. "Topological reconfiguration in expanding Hele—Shaw flow." Journal of Turbulence 3 (January 2002): N37. http://dx.doi.org/10.1088/1468-5248/3/1/037.

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35

Kondic, Ljubinko, Peter Palffy-Muhoray, and Michael J. Shelley. "Models of non-Newtonian Hele-Shaw flow." Physical Review E 54, no. 5 (1996): R4536—R4539. http://dx.doi.org/10.1103/physreve.54.r4536.

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36

Mishuris, Gennady, Sergei Rogosin, and Michal Wrobel. "MOVING STONE IN THE HELE‐SHAW FLOW." Mathematika 61, no. 2 (2015): 457–74. http://dx.doi.org/10.1112/s0025579314000461.

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37

Hedenmalm, Haakan, and Anders Olofsson. "Hele-Shaw flow on weakly hyperbolic surfaces." Indiana University Mathematics Journal 54, no. 4 (2005): 1161–80. http://dx.doi.org/10.1512/iumj.2005.54.2651.

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38

Goldstein, Raymond E., Adriana I. Pesci, and Michael J. Shelley. "Instabilities and singularities in Hele–Shaw flow." Physics of Fluids 10, no. 11 (1998): 2701–23. http://dx.doi.org/10.1063/1.869795.

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39

Mishuris, Gennady, Sergei Rogosin, and Michal Wrobel. "Hele-Shaw flow with a small obstacle." Meccanica 49, no. 9 (2014): 2037–47. http://dx.doi.org/10.1007/s11012-014-9919-8.

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40

Pareschi, Lorenzo, Giovanni Russo, and Giuseppe Toscani. "A kinetic approximation of Hele–Shaw flow." Comptes Rendus Mathematique 338, no. 2 (2004): 177–82. http://dx.doi.org/10.1016/j.crma.2003.11.006.

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41

Perrin, Charlotte. "A remark on memory effects in constrained fluid systems." ESAIM: Proceedings and Surveys 69 (2020): 56–69. http://dx.doi.org/10.1051/proc/202069056.

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The goal of this note is to put into perspective the recent results obtained on memory effects in partially congested fluid systems of Euler or Navier-Stokes type with former studies on free boundary obstacle problems and Hele-Shaw equations. In particular, we relate the notion of adhesion potential initially introduced in the context of dense suspension flows with the one of Baiocchi variable used in the analysis of free boundary problems.

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42

Aronsson, Gunnar, and Ulf Janfalk. "On Hele–Shaw flow of power-law fluids." European Journal of Applied Mathematics 3, no. 4 (1992): 343–66. http://dx.doi.org/10.1017/s0956792500000905.

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This paper reviews the governing equations for a plane Hele–Shaw flow of a power-law fluid. We find two closely related partial differential equations, one for the pressure and one for the stream function. Some mathematical results for these equations are presented, in particular some exact solutions and a representation theorem. The results are applied to Hele–Shaw flow. It is then possible to determine the flow near an arbitrary corner for any power-law fluid. Other examples are also given.

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43

ENTOV, V. M., and P. ETINGOF. "On a generalized two-fluid Hele-Shaw flow." European Journal of Applied Mathematics 18, no. 1 (2007): 103–28. http://dx.doi.org/10.1017/s0956792507006869.

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Generalized two-phase fluid flows in a Hele-Shaw cell are considered. It is assumed that the flow is driven by the fluid pressure gradient and an external potential field, for example, an electric field. Both the pressure field and the external field may have singularities in the flow domain. Therefore, combined action of these two fields brings into existence some new features, such as non-trivial equilibrium shapes of boundaries between the two fluids, which can be studied analytically. Some examples are presented. It is argued, that the approach and results may find some applications in the

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44

McDONALD, N. R. "Generalised Hele-Shaw flow: A Schwarz function approach." European Journal of Applied Mathematics 22, no. 6 (2011): 517–32. http://dx.doi.org/10.1017/s0956792511000210.

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An equation governing the evolution of a Hele-Shaw free boundary flow in the presence of an arbitrary external potential – generalised Hele-Shaw flow – is derived in terms of the Schwarz functiong(z,t) of the free boundary. This generalises the well-known equation ∂g/∂t= 2∂w/∂z, wherewis the complex potential, which has been successfully employed in constructing many exact solutions in the absence of external potentials. The new equation is used to re-derive some known explicit solutions for equilibrium and time-dependent free boundary flows in the presence of external potentials, including th

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KHALID, A. H., N. R. McDONALD, and J. M. VANDEN-BROECK. "Hele-Shaw flow driven by an electric field." European Journal of Applied Mathematics 25, no. 4 (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 electr

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Tryggvason, Grétar, and Hassan Aref. "Finger-interaction mechanisms in stratified Hele-Shaw flow." Journal of Fluid Mechanics 154 (May 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-fi

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Hansen, E. B., and H. Rasmussen. "A numerical study of unstable Hele-Shaw flow." Computers & Mathematics with Applications 38, no. 5-6 (1999): 217–30. http://dx.doi.org/10.1016/s0898-1221(99)00228-x.

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Miranda, José A., and Michael Widom. "Parallel flow in Hele-Shaw cells with ferrofluids." Physical Review E 61, no. 2 (2000): 2114–17. http://dx.doi.org/10.1103/physreve.61.2114.

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Glasner, Karl. "A diffuse interface approach to Hele Shaw flow." Nonlinearity 16, no. 1 (2002): 49–66. http://dx.doi.org/10.1088/0951-7715/16/1/304.

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Takaki, Ryuji. "Hele Shaw Flow between Flexible and Rigid Walls." Journal of the Physical Society of Japan 54, no. 1 (1985): 8–10. http://dx.doi.org/10.1143/jpsj.54.8.

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