Related papers: Weak solution of the Hele-Shaw problem: shocks and…
We study a model for a fluid showing viscoelastic and viscoplastic behavior, which describes the flow in terms of the fluid velocity and an internal stress. This stress tensor is transported via the Zaremba--Jaumann rate, and it is subject…
When a viscous fluid partially fills a Hele--Shaw channel, and is pushed by a pressure difference, the fluid interface is unstable due to the Saffman--Taylor instability. We consider the evolution of a fluid region of finite extent, bounded…
A new class of solutions to Laplacian growth with zero surface tension is presented and shown to contain all other known solutions as special or limiting cases. These solutions, which are time-dependent conformal maps with branch cuts…
We investigate the flow of two immiscible, viscous fluids in a rotating Hele-Shaw cell, when one of the fluids is a ferrofluid and an external magnetic field is applied. The interplay between centrifugal and magnetic forces in determining…
Exact solutions are reported for a stream of asymmetric bubbles steadily moving in a Hele-Shaw channel. From the periodicity along the streamwise direction, the flow region is reduced to a rectangular unit cell containing one bubble, which…
We introduce a new sharp interface model for the flow of two immiscible, viscous, incompressible fluids. In contrast to classical models for two-phase flows we prescribe an evolution law for the interfaces that takes diffusional effects…
The dynamics of singularity formation on the interface between two ideal fluids is studied for the Kelvin-Helmholtz instability development within the Hamiltonian formalism. It is shown that the equations of motion derived in the small…
Results of investigation of the asymptotic behavior of solutions to the Cauchy problems for a quasi-linear parabolic equation with a small parameter at a higher derivative near singular points of limit solutions are presented. Interest to…
The kinetic roughening of a stable oil--air interface, moving in a Hele--Shaw cell which contains a quenched columnar disorder (tracks) has been studied. A capillary effect is responsible for the dynamic evolution of the resulting rough…
We discuss conjectural scaling limits of discrete 2-dimensional aggregation models conditioned on a semi-axis considered by Levine and Peres in arXiv:0712.3378. These are certain problems about Hele-Show flows. We study moment properties of…
In this paper, we study the dynamics of a two-dimensional viscous fluid evolving through a porous medium or a Hele-Shaw cell, driven by gravity and surface tension. A key feature of this study is that the fluid is confined within a vessel…
The Stokes wave problem in a constant vorticity flow is formulated via a conformal mapping as a modified Babenko equation. The associated linearized operator is self-adjoint, whereby efficiently solved by the Newton-conjugate gradient…
The displacement of a fluid by another less viscous one in a quasi-two dimensional geometry typically leads to complex fingering patterns. In an isotropic system, dense-branching growth arises, which is characterized by repeated…
We provide a connection between weak solution concepts of mean curvature flow. On the one side we have the viscosity solution which is based on the comparison principle. On the other, variational solutions, which are combined Brakke flows…
We are concerned with the structural stability of conical shocks in the three-dimensional steady supersonic flows past Lipschitz perturbed cones whose vertex angles are less than the critical angle. The flows under consideration are…
We consider the relationship between Hele-Shaw evolution with drift, the porous medium equation with superharmonic drift, and a congested crowd motion model originally proposed by [MRS]- [MRSV]. We first use viscosity solutions to show that…
Viscous fingering and wormhole growth are complex nonlinear unstable phenomena. We view both as the result of competition for water in which the capacity of an instability to grow depends on its ability to carry water. We derive empirical…
We investigate the shallow flow of viscous fluid into and out of a channel whose gap width increases as a power-law ($x^n$), where $x$ is the downstream axis. The fluid flows slowly, while injected at a rate in the form of $t^\alpha$, where…
We study exact solutions for the slow viscous flow of an infinite liquid caused by two rigid spheres approaching each either along or parallel to their line of centres, valid at all separations. This goes beyond the applicable range of…
This research has found a novel computational efficient method of modelling flow at low Reynolds number through fracture networks. The numerical analysis was performed by connecting Hele-Shaw cells to investigate the effect of the…