Existence of traveling waves associated with Lax shocks which violate Oleinik’s entropy criterion

This paper answers to the question whether a shock wave in conservation laws satisfying the Lax shock inequalities but not Oleinik’s entropy criterion is admissible under the vanishing viscosity-capillarity effects. Such a shock appears in van der Waals fluids when a secant line meets the graph of the flux function at four distinct points, and the shock jumps between the two farthest points. The existence of the corresponding traveling waves would justify the admissibility of the shock. For this purpose, we will first show that the corresponding traveling waves satisfy a system of differential equations with two saddle points and two asymptotically stable points. Second, we estimate the domains of attraction of the asymptotically stable equilibrium points, relying on Lyapunov’s stability theory. Third, we investigate the circumstances when an unstable trajectory leaving the saddle point corresponding to the left-hand state of the shock will ever enter the domain of attraction of each of the two asymptotically stable equilibrium points. Finally, we establish the existence of traveling waves associated with a Lax shock but violating the Oleinik’s entropy criterion.     

Capillary fingering: Percolation and fractal dimension

We present experimental and theoretical results concerning immiscible displacements (drainage) in 2-dimensional permeable media. When capillary forces are predominant, the injected fluid presents very thin fingers and the Representative Elementary Volume concept cannot be used for describing the partial saturations. The purpose of this paper is to show how this classical concept can be replaced by a statistical approach based on fractal geometry.Communication presented at the International Symposium on the Stochastic Approach to Subsurface Flow, Montvillargenne 4–7 June 1985.     

Cusped ripples at the plane surface of a viscous liquid

Summary We study the time-evolution of periodical ripples of a viscous liquid at the plane free surface under the action of a distant pure straining flow. We neglect inertial forces (Stokes flow) and include surface tension effects. The solutions for a contracting surface and constant strain rate show that the ripples may develop near-cusps during a stage of the evolution, though later the free surface inevitably asymptotically tends to a smooth plane with vanishing ripples due to the action of capillarity. We obtain the condition for cusp formation in this intermediate stage in terms of the initial capillary number and aspect ratio. If the capillary number is kept constant, the surface tends to shrink through a succession of self-similar trochoidal shapes, whose aspect ratio is given by the capillary number. Received 23 March 1998, accepted for publication 23 July 1998     

Fronts in two-phase porous media flow problems: The effects of hysteresis and dynamic capillarity

In this work, we study the behavior of saturation fronts for two-phase flow through a long homogeneous porous column . In particular, the model includes hysteresis and dynamic effects in the capillary pressure and hysteresis in the permeabilities. The analysis uses traveling wave approximation. Entropy solutions are derived for Riemann problems that are arising in this context. These solutions belong to a much broader class compared to the standard Oleinik solutions, where hysteresis and dynamic effects are neglected. The relevant cases are examined and the corresponding solutions are categorized. They include nonmonotone profiles, multiple shocks, and self-developing stable saturation plateaus. Numerical results are presented that illustrate the mathematical analysis. Finally, we discuss the implication of our findings in the context of available experimental results.     

A Remark on Global Existence,Uniqueness and Exp onential Stability of Solutions for the 1D Navier-Stokes-Korteweg Equations

In this paper, we investigate non-isothermal one-dimensional model of capillary compressible fluids as derived by M Slemrod(1984) and J E Dunn and J Serrin(1985). We establish the existence, uniqueness and exponential stability of global solutions in H~2×H~1× H~1 for the one-dimensional Navier-Stokes-Korteweg equations by a priori estimates,which implies the existence and exponential stability of the nonlinear C_0-semigroups S(t) on H~2× H~1× H~1.     

Existence and nonexistence of nontrivial weak solution for a class of general capillarity systems

The aim of this paper is to study the nonexistence and existence of nonnegative, nontrivial weak solution for a class of general capillarity systems. The proofs rely essentially on the minimum principle combined with the mountain pass theorem.     

Dynamics and Breakup of Viscoelastic Liquids (A Review)

The phenomena of hydrodynamic breakup of liquid jets, drops, films, bridges, and filaments are reviewed for liquids with viscoelastic properties. The reasons for breakup are capillary instabilities, collisions with rigid obstacles, and other forms of dynamic action. The relationship between the properties of the liquids and the features of the breakup process is discussed.     

Steady-state countercurrent flow in one dimension

Two phase countercurrent steady-state flow through permeable media in one dimension is discussed. For steady-state countercurrent flow in water wet porous media, a saturation profile is predicted with the water saturation decreasing in the direction that the water phase is flowing. The de la Cruz and Spanos equations predict that the Muskat relative permeability curves for countercurrent flow will be less than the Muskat relative permeability curves for steady-state cocurrent flow. This result has immediate implications regarding the use of external drive techniques to determine relative permeabilities based on the Buckley-Leverett theory and Muskat's equations. These equations and current experimental evidence involving countercurrent flow indicate that Muskat's equations do not adequately describe the multiphase flow of immiscible fluids.     

Interfacial diffusion relaxation of internal stresses in solids

The relaxation of internal stresses due to interfacial diffusion in a two-phase solid is studied theoretically with the help of the onsageristic approach of irreversible thermodynamics. In this note we derive an expression for the rate at which internal stresses associated with misfit caused by bonding a flat surface of one material to a rough surface of another. The two phases are treated as isotropic clastic substances. It is assumed that the components of only of the solids are capable of leaving their positions of migrating along the interface. The driving force for this process is minimization of total energy-clastic plus interfacial energy. We show that the time constant for relaxing these stresses is proportional to the cube of the wavelength of the roughness.