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Journal articles on the topic 'Hele-Shaw flow'

Author: Grafiati

Published: 4 June 2021

Last updated: 26 January 2023

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1

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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2

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 a long and narrow Hele-Shaw cell appear to validate the theory in both the symmetric and the non-symmetric cases.

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3

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 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.

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4

Aronsson, Gunnar, and Ulf Janfalk. "On Hele–Shaw flow of power-law fluids." European Journal of Applied Mathematics 3, no. 4 (December 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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5

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 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.

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6

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 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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7

McDONALD, N. R. "Generalised Hele-Shaw flow: A Schwarz function approach." European Journal of Applied Mathematics 22, no. 6 (May 16, 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 those with singular potential fields, uniform gravity and centrifugal forces. Some new solutions are also constructed that variously describe equilibrium flows with higher order hydrodynamic singularities in the presence of electric point sources and an unsteady solution describing bubbles under the combined influence of strain and centrifugal potential.

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8

Mishuris, Gennady, Sergei Rogosin, and Michal Wrobel. "MOVING STONE IN THE HELE‐SHAW FLOW." Mathematika 61, no. 2 (April 8, 2015): 457–74. http://dx.doi.org/10.1112/s0025579314000461.

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9

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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10

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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11

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

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12

Kondic, Ljubinko, Peter Palffy-Muhoray, and Michael J. Shelley. "Models of non-Newtonian Hele-Shaw flow." Physical Review E 54, no. 5 (November 1, 1996): R4536—R4539. http://dx.doi.org/10.1103/physreve.54.r4536.

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13

Almgren, Robert, Wei-Shen Dai, and Vincent Hakim. "Scaling behavior in anisotropic Hele-Shaw flow." Physical Review Letters 71, no. 21 (November 22, 1993): 3461–64. http://dx.doi.org/10.1103/physrevlett.71.3461.

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14

Mishuris, Gennady, Sergei Rogosin, and Michal Wrobel. "Hele-Shaw flow with a small obstacle." Meccanica 49, no. 9 (May 15, 2014): 2037–47. http://dx.doi.org/10.1007/s11012-014-9919-8.

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15

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

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16

ENTOV, V. M., and P. ETINGOF. "On a generalized two-fluid Hele-Shaw flow." European Journal of Applied Mathematics 18, no. 1 (February 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 theory of fluids flow through porous media and microfluidic devices controlled by electric field.

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17

Ross, Julius, and David Witt Nyström. "The Hele-Shaw flow and moduli of holomorphic discs." Compositio Mathematica 151, no. 12 (August 18, 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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18

Zantop, Arne W., and Holger Stark. "Squirmer rods as elongated microswimmers: flow fields and confinement." Soft Matter 16, no. 27 (2020): 6400–6412. http://dx.doi.org/10.1039/d0sm00616e.

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19

Malaikah, K. R. "The Two-Phase Hell-Shaw Flow: Construction of an Exact Solution." International Journal of Applied Mechanics and Engineering 18, no. 1 (March 1, 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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20

Fiorucci, Giulia, Johan T. Padding, and Marjolein Dijkstra. "Small asymmetric Brownian objects self-align in nanofluidic channels." Soft Matter 15, no. 2 (2019): 321–30. http://dx.doi.org/10.1039/c8sm02384k.

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21

Inoue, Kai, and Susumu Inasawa. "Drying-induced back flow of colloidal suspensions confined in thin unidirectional drying cells." RSC Advances 10, no. 27 (2020): 15763–68. http://dx.doi.org/10.1039/d0ra02837a.

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22

Bere, Katalin Viktória, Emilie Nez, Edina Balog, László Janovák, Dániel Sebők, Ákos Kukovecz, Clément Roux, Veronique Pimienta, and Gábor Schuszter. "Enhancing the yield of calcium carbonate precipitation by obstacles in laminar flow in a confined geometry." Physical Chemistry Chemical Physics 23, no. 29 (2021): 15515–21. http://dx.doi.org/10.1039/d1cp01334c.

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23

Fabricius, John, Salvador Manjate, and Peter Wall. "Error estimates for pressure-driven Hele-Shaw flow." Quarterly of Applied Mathematics 80, no. 3 (March 30, 2022): 575–95. http://dx.doi.org/10.1090/qam/1619.

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24

Miranda, José A., and Michael Widom. "Parallel flow in Hele-Shaw cells with ferrofluids." Physical Review E 61, no. 2 (February 1, 2000): 2114–17. http://dx.doi.org/10.1103/physreve.61.2114.

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25

Glasner, Karl. "A diffuse interface approach to Hele Shaw flow." Nonlinearity 16, no. 1 (October 28, 2002): 49–66. http://dx.doi.org/10.1088/0951-7715/16/1/304.

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26

Ceniceros, Hector D., Thomas Y. Hou, and Helen Si. "Numerical study of Hele-Shaw flow with suction." Physics of Fluids 11, no. 9 (September 1999): 2471–86. http://dx.doi.org/10.1063/1.870112.

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27

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 (October 10, 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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28

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-fingering’ regime can be rationalized.

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29

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

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30

Zhang, Qi. "Some complex equations arising in hele shaw flow." Applied Mathematics Letters 6, no. 5 (September 1993): 45–47. http://dx.doi.org/10.1016/0893-9659(93)90098-8.

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31

Fast, Petri, L. Kondic, Michael J. Shelley, and Peter Palffy-Muhoray. "Pattern formation in non-Newtonian Hele–Shaw flow." Physics of Fluids 13, no. 5 (May 2001): 1191–212. http://dx.doi.org/10.1063/1.1359417.

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32

Takaki, Ryuji. "Hele Shaw Flow between Flexible and Rigid Walls." Journal of the Physical Society of Japan 54, no. 1 (January 15, 1985): 8–10. http://dx.doi.org/10.1143/jpsj.54.8.

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33

Elezgaray, J., P. Petit, and B. Bonnier. "Detecting complex singularities of a Hele-Shaw flow." Europhysics Letters (EPL) 37, no. 4 (February 1, 1997): 263–68. http://dx.doi.org/10.1209/epl/i1997-00141-0.

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34

Chesnokov, Alexander, and Valery Liapidevskii. "Viscosity-stratified flow in a Hele–Shaw cell." International Journal of Non-Linear Mechanics 89 (March 2017): 168–76. http://dx.doi.org/10.1016/j.ijnonlinmec.2016.12.016.

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35

GROSFILS, PATRICK, and JEAN PIERRE BOON. "VISCOUS FINGERING IN MISCIBLE, IMMISCIBLE AND REACTIVE FLUIDS." International Journal of Modern Physics B 17, no. 01n02 (January 20, 2003): 15–20. http://dx.doi.org/10.1142/s0217979203017023.

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With the Lattice Boltzmann method (using the BGK approximation) we investigate the dynamics of Hele-Shaw flow under conditions corresponding to various experimental systems. We discuss the onset of the instability (dispersion relation), the static properties (characterization of the interface) and the dynamic properties (growth of the mixing zone) of simulated Hele-Shaw systems. We examine the role of reactive processes (between the two fluids) and we show that they have a sharpening effect on the interface similar to the effect of surface tension.

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36

CURT, PAULA. "On some invariant geometric properties in Hele-Shaw flows with small surface tension." Carpathian Journal of Mathematics 31, no. 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.

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37

LU, H. W., K. GLASNER, A. L. BERTOZZI, and C. J. KIM. "A diffuse-interface model for electrowetting drops in a Hele-Shaw cell." Journal of Fluid Mechanics 590 (October 15, 2007): 411–35. http://dx.doi.org/10.1017/s0022112007008154.

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Electrowetting has recently been explored as a mechanism for moving small amounts of fluids in confined spaces. We propose a diffuse-interface model for drop motion, due to electrowetting, in a Hele-Shaw geometry. In the limit of small interface thickness, asymptotic analysis shows that the model is equivalent to Hele-Shaw flow with a voltage-modified Young–Laplace boundary condition on the free surface. We show that details of the contact angle significantly affect the time scale of motion in the model. We measure receding and advancing contact angles in the experiments and derive their influence through a reduced-order model. These measurements suggest a range of time scales in the Hele-Shaw model which include those observed in the experiment. The shape dynamics and topology changes in the model agree well with the experiment, down to the length scale of the diffuse-interface thickness.

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38

Saffman, P. G. "Viscous fingering in Hele-Shaw cells." Journal of Fluid Mechanics 173 (December 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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39

Rogosin, S. V. "ON CLASSICAL FORMULATION OF HELE‐SHAW MOVING BOUNDARY PROBLEM FOR POWER‐LAW FLUID." Mathematical Modelling and Analysis 7, no. 1 (June 30, 2002): 159–68. http://dx.doi.org/10.3846/13926292.2002.9637188.

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40

Taheri, Amir, Jan David Ytrehus, Bjørnar Lund, and Malin Torsæter. "Experimental Study of the Use of Tracing Particles for Interface Tracking in Primary Cementing in an Eccentric Hele–Shaw Cell." Energies 14, no. 7 (March 29, 2021): 1884. http://dx.doi.org/10.3390/en14071884.

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We present the results of the displacement flows of different Newtonian and Herschel–Bulkley non-Newtonian fluids in a new-developed eccentric Hele–Shaw cell with dynamic similarly to real field wellbore annulus during primary cementing. The possibility of tracking the interface between the fluids using particles with intermediate or neutral buoyancy is studied. The behaviors and movements of particles with different sizes and densities against the primary vertical flow and strong secondary azimuthal flow in the eccentric Hele–Shaw cell are investigated. The effects of fluid rheology and pumping flow rate on the efficiency of displacement and tracing particles are examined. Moreover, the behavior of pressure gradients in the cell is described and analyzed. Successful results of tracing the interface using particles give us this opportunity to carry out a primary cementing with high quality for the cases that the risk of leakage is high, e.g., primary cementing in wells penetrating a CO2 storage reservoir.

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41

Schuszter, Gábor, Fabian Brau, and A. De Wit. "Flow-driven control of calcium carbonate precipitation patterns in a confined geometry." Physical Chemistry Chemical Physics 18, no. 36 (2016): 25592–600. http://dx.doi.org/10.1039/c6cp05067k.

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42

Cueto-Felgueroso, Luis, and Ruben Juanes. "A phase-field model of two-phase Hele-Shaw flow." Journal of Fluid Mechanics 758 (October 9, 2014): 522–52. http://dx.doi.org/10.1017/jfm.2014.512.

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AbstractWe propose a continuum model of two-phase flow in a Hele-Shaw cell. The model describes the multiphase three-dimensional flow in the cell gap using gap-averaged quantities such as fluid saturation and Darcy flux. Viscous and capillary coupling between the fluids in the gap leads to a nonlinear fractional flow function. Capillarity and wetting phenomena are modelled within a phase-field framework, designing a heuristic free energy functional that induces phase segregation at equilibrium. We test the model through the simulation of bubbles and viscously unstable displacements (viscous fingering). We analyse the model’s rich behaviour as a function of capillary number, viscosity contrast and cell geometry. Including the effect of wetting films on the two-phase flow dynamics opens the door to exploring, with a simple two-dimensional model, the impact of wetting and flow rate on the performance of microfluidic devices and geological flows through fractures.

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43

Moog, Mathias, Rainer Keck, and Aivars Zemitis. "SOME NUMERICAL ASPECTS OF THE LEVEL SET METHOD." Mathematical Modelling and Analysis 3, no. 1 (December 15, 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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44

Park, Jang Min, and Seong Jin Park. "Modeling and Simulation of Fiber Orientation in Injection Molding of Polymer Composites." Mathematical Problems in Engineering 2011 (2011): 1–14. http://dx.doi.org/10.1155/2011/105637.

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We review the fundamental modeling and numerical simulation for a prediction of fiber orientation during injection molding process of polymer composite. In general, the simulation of fiber orientation involves coupled analysis of flow, temperature, moving free surface, and fiber kinematics. For the governing equation of the flow, Hele-Shaw flow model along with the generalized Newtonian constitutive model has been widely used. The kinematics of a group of fibers is described in terms of the second-order fiber orientation tensor. Folgar-Tucker model and recent fiber kinematics models such as a slow orientation model are discussed. Also various closure approximations are reviewed. Therefore, the coupled numerical methods are needed due to the above complex problems. We review several well-established methods such as a finite-element/finite-different hybrid scheme for Hele-Shaw flow model and a finite element method for a general three-dimensional flow model.

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45

Box, Finn, Gunnar G. Peng, Draga Pihler-Puzović, and Anne Juel. "Flow-induced choking of a compliant Hele-Shaw cell." Proceedings of the National Academy of Sciences 117, no. 48 (November 16, 2020): 30228–33. http://dx.doi.org/10.1073/pnas.2008273117.

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After centuries of striving for structural rigidity, engineers and scientists alike are increasingly looking to harness the deformation, buckling, and failure of soft materials for functionality. In fluidic devices, soft deformable components that respond to the flow have the advantage of being passive; they do not require external actuation. Harnessing flow-induced deformation for passive functionality provides a means of developing flow analogs of electronic circuit components such as fluidic diodes and capacitors. The electronic component that has so far been overlooked in the microfluidics literature—the fuse—is a passive safety device that relies on a controlled failure mechanism (melting) to protect a circuit from overcurrent. Here, we describe how a compliant Hele-Shaw cell behaves in a manner analogous to the electrical fuse; above a critical flux, the flow-induced deformation of the cell blocks the outflow, interrupting (choking) the flow. In particular, the pressure distribution within the fluid applies a spatially variant normal force to the soft boundary, which causes nonuniform deformation. As a consequence of lateral confinement and incompressibility of the soft material, this flow-induced elastic deformation manifests as bulging near the cell outflow; bulges that come into contact with the rigid cell roof interrupt the flow. We identify two nondimensional parameters that govern the central deflection and the choking of the cell, respectively. This study therefore provides the mechanical foundations for engineering passive-flow limiters into fluidic devices.

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Zeybek, M., and Y. C. Yortsos. "Long waves in parallel flow in Hele-Shaw cells." Physical Review Letters 67, no. 11 (September 9, 1991): 1430–33. http://dx.doi.org/10.1103/physrevlett.67.1430.

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Elezgaray, J. "Modeling the singularity dynamics of a Hele-Shaw flow." Physical Review E 57, no. 6 (June 1, 1998): 6884–87. http://dx.doi.org/10.1103/physreve.57.6884.

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ISHIDA, Takanori, Tsutomu TAKAHASHI, Masataka SHIRAKASHI, and Tomiichi HASHEGAWA. "Observation of Hele-Shaw flow of a viscoelastic fluid." Proceedings of the JSME annual meeting 2002.3 (2002): 1–2. http://dx.doi.org/10.1299/jsmemecjo.2002.3.0_1.

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Choi, Sunhi, David Jerison, and Inwon Kim. "Local regularization of the one-phase Hele-Shaw flow." Indiana University Mathematics Journal 58, no. 6 (2009): 2765–804. http://dx.doi.org/10.1512/iumj.2009.58.3802.

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PASA, Gelu. "Some Contradictions in the Multi-Layer Hele-Shaw Flow." International Journal of Petroleum Technology 6, no. 1 (December 22, 2019): 41–48. http://dx.doi.org/10.15377/2409-787x.2019.06.5.

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