A generalized Rayleigh?Taylor condition for the Muskat problem

@article{Escher2010AGR,
  title={A generalized Rayleigh?Taylor condition for the Muskat problem},
  author={Joachim Escher and Anca-Voichita Matioc and Bogdan-Vasile Matioc},
  journal={Nonlinearity},
  year={2010},
  volume={25},
  pages={73-92}
}
In this paper we consider the evolution of two fluid phases in a porous medium. The fluids are separated from each other and also the wetting phase from air by interfaces which evolve in time. We reduce the problem to an abstract evolution equation. A generalized Rayleigh–Taylor condition characterizes the parabolicity regime of the problem and allows us to establish a general well-posedness result and to study stability properties of flat steady states. When considering surface tension effects… 

Figures from this paper

Viscous displacement in porous media: the Muskat problem in 2D
  • B. Matioc
  • Mathematics
    Transactions of the American Mathematical Society
  • 2018
We consider the Muskat problem describing the viscous displacement in a two-phase fluid system located in an unbounded two-dimensional porous medium or Hele-Shaw cell. After formulating the
Rayleigh-Taylor instability for the two-phase Navier-Stokes equations with surface tension in cylindrical domains
This article is concerned with the dynamic behaviour of two immiscible and incompressible fluids in a cylindrical domain, which are separated by a sharp interface. In case that the heavy fluid is
The Rayleigh–Taylor instability for the Verigin problem with and without phase transition
Isothermal compressible two-phase flows in a capillary are modeled with and without phase transition in the presence of gravity, employing Darcy’s law for the velocity field. It is shown that the
On the Muskat flow
Of concern is the motion of two fluids separated by a free interface in a porous medium, where the velocities are given by Darcy's law. We consider the case with and without phase transition. It is
The multiphase Muskat problem with equal viscosities in two dimensions
We study the two-dimensional multiphase Muskat problem describing the motion of three immiscible fluids with equal viscosities in a vertical homogeneous porous medium identified with R under the
Modelling and Analysis of the Muskat Problem for Thin Fluid Layers
We consider the evolution of two thin fluid films in a porous medium. Starting from the classical equations modelling the Muskat problem we pass to the limit of small layer thickness and obtain a
Rayleigh-Taylor breakdown for the Muskat problem with applications to water waves
The Muskat problem models the evolution of the interface between two different fluids in porous media. The Rayleigh-Taylor condition is natural to reach linear stability of the Muskat problem. We
The domain of parabolicity for the Muskat problem
We address the well-posedness of the Muskat problem in a periodic geometry and in a setting which allows us to consider general initial and boundary data, gravity effects, as well as surface tension
...
...

References

SHOWING 1-10 OF 31 REFERENCES
The Rayleigh-Taylor condition for the evolution of irrotational fluid interfaces
For the free boundary dynamics of the two-phase Hele-Shaw and Muskat problems, and also for the irrotational incompressible Euler equation, we prove existence locally in time when the Rayleigh–Taylor
Contour Dynamics of Incompressible 3-D Fluids in a Porous Medium with Different Densities
We consider the problem of the evolution of the interface given by two incompressible fluids through a porous medium, which is known as the Muskat problem and in two dimensions it is mathematically
Nonlinear stability of the Muskat problem with capillary pressure at the free boundary
Interface evolution: the Hele-Shaw and Muskat problems
We study the dynamics of the interface between two incompressible 2-D ows where the evolution equation is obtained from Darcy’s law. The free boundary is given by the discontinuity among the
Well-posedness of two-phase Hele–Shaw flow without surface tension
  • D. Ambrose
  • Mathematics
    European Journal of Applied Mathematics
  • 2004
We prove short-time well-posedness of a Hele–Shaw system with two fluids and no surface tension (this is also known as the Muskat problem). We restrict our attention here to the stable case of the
Existence and Stability Results for Periodic Stokesian Hele-Shaw Flows
TLDR
It is proved the existence of a unique classical solution if the initial data is near a constant, identify the equilibria of the flow, and study their stability.
A moving boundary problem for periodic Stokesian Hele–Shaw flows
This paper is concerned with the motion of an incompressible, viscous fluid in a Hele–Shaw cell. The free surface is moving under the influence of gravity and the fluid is modelled using a modified
The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid
  • P. Saffman, G. Taylor
  • Engineering
    Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences
  • 1958
When a viscous fluid filling the voids in a porous medium is driven forwards by the pressure of another driving fluid, the interface between them is liable to be unstable if the driving fluid is the
On the parabolicity of the Muskat problem: Well-posedness, fingering, and stability results
We consider in this paper the Muskat problem in a periodic geometry and incorporate capillary as well as gravity effects in the modelling. The problem re-writes as an abstract evolution equation and
Removing the stiffness from interfacial flows with surface tension
A new formulation and new methods are presented for computing the motion of fluid interfaces with surface tension in two-dimensional, irrotational, and incompressible fluids. Through the
...
...