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On a Hele–Shaw Type Domain Evolution with Convected Surface Energy Density: The Third‐Order Problem

Matthias Günther, Georg Prokert
2006 SIAM Journal on Mathematical Analysis  
We investigate a moving boundary problem with a gradient flow structure which generalizes Hele-Shaw flow driven solely by surface tension to the case of nonconstant surface tension coefficient taken along  ...  The resulting evolution problem is first order in time, contains a third-order nonlinear pseudodifferential operator and is degenerate parabolic.  ...  It is the aim of the present paper to consider the generalization of the well-investigated Hele-Shaw flow problem to the case of nonconstant surface tension coefficient (or surface energy density).  ... 
doi:10.1137/050626995 fatcat:kmy65ot7kvepjakc5yywu6vblq

On a Hele--Shaw-Type Domain Evolution with Convected Surface Energy Density

Matthias Günther, Georg Prokert
2005 SIAM Journal on Mathematical Analysis  
Interest is directed to a moving boundary problem with a gradient flow structure which generalizes surface-tension driven Hele-Shaw flow to the case of nonconstant surface tension coefficient taken along  ...  In the case with kinetic undercooling regularization well-posedness of the resulting evolution problem in Sobolev scales is proved, including cases in which the surface tension coefficient degenerates.  ...  Introduction Various experimental studies investigate the influence of spatial variations of the surface energy density (corresponding to the surface tension coefficient γ) on surface tension driven Hele-Shaw  ... 
doi:10.1137/s0036141004444846 fatcat:p3he53jtfrcvnnncbiqkuy2es4

Density-driven instabilities of miscible fluids in a capillary tube: linear stability analysis

SURYA HARITH VANAPARTHY, E. MEIBURG, D. WILHELM
2003 Journal of Fluid Mechanics  
In order to analyse the experimental observations by Kuang et al. (2002) , which show that a small amount of net flow can stabilize the azimuthal instability mode and maintain an axisymmetric evolution  ...  , a base flow of Poiseuille type is included in the linear stability analysis.  ...  This exponent is very close to the value of 10/33 found by Graf et al. (2002) for the corresponding Hele-Shaw problem.  ... 
doi:10.1017/s0022112003006499 fatcat:mmqythdg5nc6tg2tsaxniwyb3e

Plume dynamics in quasi-2D turbulent convection

C. Bizon, J. Werne, A. A. Predtechensky, K. Julien, W. D. McCormick, J. B. Swift, Harry L. Swinney
1997 Chaos  
We have studied turbulent convection in a vertical thin ͑Hele-Shaw͒ cell at very high Rayleigh numbers ͑up to 7ϫ10 4 times the value for convective onset͒ through experiment, simulation, and analysis.  ...  We have conducted laboratory experiments and numerical simulations on a quasi-two-dimensional version of a canonical problem in nonlinear dynamics: buoyancy driven convection, the best known example of  ...  surrounding fluid, a Rayleigh-Taylor type instability may occur at a plume boundary, especially when two plumes of opposite density-contrast collide head-on.  ... 
doi:10.1063/1.166243 pmid:12779641 fatcat:bcktyopv3ffbpolajantscvp4u

The formation and evolution of layered structures in porous media: effects of porosity and mechanical dispersion

Stan Schoofs, Ron A. Trompert, Ulrich Hansen
2000 Physics of the Earth and Planetary Interiors  
Growth of a convective layer through convective entrainment, the formation of a stable density interface on top of the layer and destabilization of the next layer are intimately Ž . Ž . linked.  ...  This force balance also indicates whether a density interface keeps intact, migrates upwards or breaks down during the further evolution of the layered sequence.  ...  Acknowledgements The authors thank Frank Spera, Gualbert Oude Essink and an anonymous referee for constructive  ... 
doi:10.1016/s0031-9201(99)00148-x fatcat:n7yel5fafrhghoxlyu2ypb26mq

A decoupled numerical method for two-phase flows of different densities and viscosities in superposed fluid and porous layers [article]

Yali Gao, Daozhi Han, Xiaoming He, Ulrich Rüde
2021 arXiv   pre-print
The energy stability of the scheme effected with the finite element spatial discretization is rigorously established. We verify numerically that our schemes are convergent and energy-law preserving.  ...  We show that the coupled model satisfies an energy law.  ...  There are mainly two types of approach on developing diffuse interface models for two-phase flow of different densities in a single domain.  ... 
arXiv:2112.04353v1 fatcat:iriaqkas3fcwfnmtbc64454ury

Density-Driven Instabilities of Variable-Viscosity Miscible Fluids in a Capillary Tube

ECKART MEIBURG, SURYA H. VANAPARTHY, MATTHIAS D. PAYR, DIRK WILHELM
2004 Annals of the New York Academy of Sciences  
The initial interface thickness is treated as a parameter to the problem.  ...  The analysis is based on the three-dimensional Stokes equations, coupled to a convection-diffusion equation for the concentration field, in cylindrical coordinates.  ...  Support for this research was received from the NASA Microgravity and NSF/ITR programs, as well as from the Department of Energy, the UC Energy Institute, and through an NSF equipment grant.  ... 
doi:10.1196/annals.1324.032 pmid:15644370 fatcat:hxfxoj62xrfmbfn5ereehkwz3e

From the Hele-Shaw Experiment to Integrable Systems: A Historical Overview

Alexander Vasil'ev
2008 Complex Analysis and Operator Theory  
This paper is a historical overview of the development of the topic now commonly known as Laplacian Growth, from the original Hele-Shaw experiment to the modern treatment based on integrable systems.  ...  A trivial circular solution on the plane becomes an evolution of highly non-trivial Hele-Shaw disks on a surface. Hele-Shaw flows on curved surfaces were discussed also in [106] .  ...  In the Hele-Shaw context it means sweeping the Dirac measure out in evolution with the prescribed density function (see, [30; 33] ).  ... 
doi:10.1007/s11785-008-0104-8 fatcat:4rf3px3gszf75kq2mz35al22ly

Removing the stiffness from interfacial flows with surface tension

Thomas Y. Hou, John S. Lowengrub, Michael J. Shelley
1994 Journal of Computational Physics  
A second computation shows the formation of a very ramified interface through the interaction of surface tension with an unstable density stratification.  ...  Here they are applied to computing with high resolution the motion of interfaces in Hele-Shaw flows and the motion of free surfaces in inviscid flows governed by the Euler equations.  ...  All three of the authors thank the hospitality and support of the Institute for Advanced Study where they participated in the special year in fluid mechanics during the 1991 academic year, where some of  ... 
doi:10.1006/jcph.1994.1170 fatcat:vso3bx7hovep5dlyhlqnc3uqre

Solutal convection in porous media: Comparison between boundary conditions of constant concentration and constant flux

Mohammad Amin Amooie, Mohamad Reza Soltanian, Joachim Moortgat
2018 Physical review. E  
The latter model is found to involve a nonlinear evolution of CO2 composition and associated aqueous-phase density, which depend on the formation permeability.  ...  Dissolution is associated with slow diffusion of free-phase CO2 into the underlying aqueous phase followed by density-driven convective mixing of CO2 throughout the water-saturated layer.  ...  Department of Energy's (DOE) Office of Fossil Energy funding to Oak Ridge National Laboratory (ORNL) under project  ... 
doi:10.1103/physreve.98.033118 fatcat:jqwjhci5tjh7lcdn6fpbjafd4e

Variable density and viscosity, miscible displacements in capillary tubes

S.H. Vanaparthy, E. Meiburg
2008 European journal of mechanics. B, Fluids  
The displacement of a more viscous fluid by a miscible, less viscous one of lower density in a horizontal capillary tube is studied by means of Stokes flow simulations.  ...  On the other hand, for some parameter combinations the evolution of a 'dimple' is observed on the lower side of the finger, and close to its tip.  ...  Also, the input of Prof. Tony Maxworthy, Dr. Philippe Petitjeans and Dr. Jun Kuang is gratefully acknowledged, as are the discussions with Dr. Nisheet Goyal on several computational issues.  ... 
doi:10.1016/j.euromechflu.2007.06.003 fatcat:pe6qhybhejcqxnlqg7foexx5pi

Experimental and numerical study of buoyancy-driven single bubble dynamics in a vertical Hele-Shaw cell

Xue Wang, Bart Klaasen, Jan Degrève, Bart Blanpain, Frederik Verhaeghe
2014 Physics of Fluids  
Buoyancy-driven single bubble behaviour in a vertical Hele-Shaw cell with various gap Reynolds numbers Re(h/d) 2 has been studied.  ...  The bubble behaviour was also numerically simulated by taking a two-dimensional volume of fluid method coupled with a continuum surface force model and a wall friction model in the commercial computational  ...  The authors would like to acknowledge Dr. P. Yan, Dr. Z. Sun, and B. Wu for valuable inputs and inspiring discussions.  ... 
doi:10.1063/1.4903488 fatcat:u63dpwuxsjhhtkkadd4bly7x2q

Incompressible limit of a continuum model of tissue growth with segregation for two cell populations [article]

Alina Chertock, Pierre Degond, Sophie Hecht, Jean-Paul Vincent
2018 arXiv   pre-print
Following earlier works on the single population case, we show that the model approximates a free boundary Hele Shaw type model that we characterise using both analytical and numerical arguments.  ...  The dynamics is driven by pressure and cohesion forces on the one hand and proliferation on the other hand.  ...  The work of AC was supported in part by NSF Grant DMS-521051 and RNMS Grant DMS-1107444 (KI-Net).  ... 
arXiv:1804.04090v1 fatcat:4zmk34ucjbgnznjm7y7cxl5yia

Immiscible thermo-viscous fingering in Hele-Shaw cells

S.J. Jackson, H. Power, D. Giddings
2017 Computers & Fluids  
We investigate immiscible radial displacement in a Hele-Shaw cell with a temperature dependent viscosity using two coupled high resolution numerical methods.  ...  Further numerical experiments reveal the neutral stability of the thermal effects on the fingering evolution, with classical viscous fingering dynamics eventually dominating the evolution.  ...  Acknowledgements The authors would like to thank the University of Nottingham HPC team for the use of the Minerva supercomputer cluster.  ... 
doi:10.1016/j.compfluid.2017.04.004 fatcat:dtjvoluif5d27fzmyq7pae2vla

Incompressible limit of a continuum model of tissue growth with segregation for two cell populations

Alina Chertock, Pierre Degond, Sophie Hecht, Jean-Paul Vincent
2019 Mathematical Biosciences and Engineering  
Then, following earlier works on the single population case, we provide formal arguments that the model approximates a free boundary Hele Shaw type model that we characterise using both analytical and  ...  In this paper, we study the influ-ence of the model parameters thanks to one-dimensional simulations using a finite-volume method for a relaxation approximation of the fourth order diffusion.  ...  Data Statement No new data were collected in the course of this research. Conflict of interest All authors declare no conflicts of interest in this paper.  ... 
doi:10.3934/mbe.2019290 pmid:31499739 fatcat:whiyzgeamnhxzelsly3ihjpgi4
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