Flow Asymptotics of a Viscous Compressible Fluid with Discontinuous Initial Data

Asymptotics of a continuous solution to a plane problem on the motion of a viscous incompressible fluid with discontinuous initial velocity and pressure fields is studied by the boundarylayer method with simultaneous stretching of space and time coordinates.     

Surface Waves for a Compressible Viscous Fluid

In this paper, we are concerned with free boundary problem for compressible viscous isotropic Newtonian fluid. Our problem is to find the three-dimensional domain occupied by the fluid which is bounded below by the fixed bottom and above by the free surface together with the density, the velocity vector field and the absolute temperature of the fluid satisfying the system of Navier-Stokes equations and the initial-boundary conditions. The Navier-Stokes equations consist of the conservations of mass, momentum under the gravitational field in a downward direction and energy. The effect of the surface tension on the free surface is taken into account. The purpose of this paper is to establish two existence theorems to the problem mentioned above: the first concerns with the temporary local solvability in anisotropic Sobolev-Slobodetskiĭ spaces and the second the global solvability near the equilibrium rest state. Here the equilibrium rest state (heat conductive state) means that the temperature distribution is a linear function with respect to a vertical direction and the density is determined by an ordinary differential equation which involves equation of state. For the proof, we rely on the methods due to Solonnikov in the case of incompressible fluid with some modifications, since our problem is hyperbolic-parabolic coupled system. Dedicated to Professors Takaaki Nishida and Masayasu Mimura on their sixtieth birthdays     

On the Motion of Rigid Bodies in a Viscous Compressible Fluid

We prove the existence of global-in-time weak solutions to a model describing the motion of several rigid bodies in a viscous compressible fluid. Unlike most recent results of similar type, there is no restriction on the existence time, regardless of possible collisions of two or more rigid bodies and/or a contact of the bodies with the boundary. (Accepted September 23, 2002) Published online February 4, 2003 Communicated by Y. Brenier     

Pore-Scale Modeling of Viscous Flow and Induced Forces in Dense Sphere Packings

We propose a method for effectively upscaling incompressible viscous flow in large random polydispersed sphere packings: the emphasis of this method is on the determination of the forces applied on the solid particles by the fluid. Pore bodies and their connections are defined locally through a regular Delaunay triangulation of the packings. Viscous flow equations are upscaled at the pore level, and approximated with a finite volume numerical scheme. We compare numerical simulations of the proposed method to detailed finite element simulations of the Stokes equations for assemblies of 8–200 spheres. A good agreement is found both in terms of forces exerted on the solid particles and effective permeability coefficients.     

On Spherically Symmetric Motions Of a Viscous Compressible Barotropic and Selfgravitating Gas

We consider the Cauchy problem for the equations of spherically symmetric motions in \mathbb R³{\mathbb {R}^3}, of a selfgravitating barotropic gas, with possibly non monotone pressure law, in two different situations: in the first one we suppose that the viscosities μ(ρ), and λ(ρ) are density-dependent and satisfy the Bresch–Desjardins condition, in the second one we consider constant densities. In the two cases, we prove that the problem admits a global weak solution, provided that the polytropic index γ satisfy γ > 1.     

Heat Convection of Compressible Viscous Fluids: I

The stationary problem for the heat convection of compressible fluid is considered around the equilibrium solution with the external forces in the horizontal strip domain z ₀ < z < z ₀ + 1 and it is proved that the solution exists uniformly with respect to z ₀ ≥ Z ₀. The limit system as z ₀ → + ∞ is the Oberbeck–Boussinesq equations.     

Heat Convection of Compressible Viscous Fluids. II

We consider steady heat convections of compressible viscous fluids in the horizontal strip domain ${z_0 < z < z_0 + 1}$ under the gravity. Pattern formations are shown uniformly for ${z_0 \geq Z_0}$ . The limit of them as ${Z_0 \rightarrow + \infty}$ is that of Oberbeck-Boussinesq equations.     

On the Exponential Stability of the Rest State of a Viscous Compressible Fluid

We prove that the rest state of a viscous isothermal gas filling a bounded rigid vessel, is exponentially stable with respect a large class of "weak" perturbations that, in particular, allow for supersonic flow and discontinuous densities. In the inviscid limit, marginal stability is recovered.     

On Slightly Compressible Ideal Flow¶in the Half-Plane

We consider the Euler equations of barotropic inviscid compressible fluids in the half-plane. It is well known that, as the Mach number goes to zero, the compressible flows approximate the solution of the equations of motion of inviscid, incompressible fluids. In 2D (two dimensions) such limit solution exists on any arbitrary time interval, with no restriction on the size of the initial data. It is then natural to expect the same for the compressible solution, if the Mach number is sufficiently small. We decompose the solution as the sum of the irrotational part, the incompressible part and the remainder, which describes the interaction between the first two components. First we study the life span of smooth irrotational solutions, i.e., the largest time interval T(?) of existence of classical solutions, when the initial data are a small perturbation of size ? from a constant state. Related to this is a decay property for the irrotational part. Then, we study the interaction between the two components and show the existence on any arbitrary time interval, for any Mach number sufficiently small. This yields the existence of smooth compressible flow on any arbitrary time interval. For the proofs we use a combination of energy and decay estimates.     

Asymptotic Solutions of the Equations for a Viscous Heat-Conducting Compressible Medium

Small nonstationary perturbations in a viscous heat-conducting compressible medium are analyzed on the basis of the linearization of the complete system of hydrodynamic equations for small Knudsen numbers (Kn ≪ 1). It is shown that the density and temperature perturbations (elastic perturbations) satisfy the same wave equation which is an asymptotic limit of the hydrodynamic equations far from the inhomogeneity regions of the medium (rigid, elastic or fluid boundaries) as M _a = v/a → 0, where v is the perturbed velocity and a is the adiabatic speed of sound. The solutions of the new equation satisfy the first and second laws of thermodynamics and are valid up to the frequencies determined by the applicability limits of continuum models. Fundamental solutions of the equation are obtained and analyzed. The boundary conditions are formulated and the problem of the interaction of a spherical elastic harmonic wave with an infinite flat surface is solved. Important physical effects which cannot be described within the framework of the ideal fluid model are discussed.__________Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, 2005, pp. 76–87.Original Russian Text Copyright © 2005 by Stolyarov.     

Self-Similar Unsteady Viscous Flow

Four examples of self-similar flows of a viscous fluid are considered: separated flow over an expanding plate immersed in an unbounded unsteady viscous flow, the evolution of the velocity field induced by a vortex-source, the flow near an unsteadily moving permeable flat plate, and the flow near an unsteadily rotating disc. For the first example, a numerical solution is constructed. For the next two examples, an analytical solution is found, while the solution of the last problem is reduced to a system of ordinary differential equations.     

Incompressible Limit for the Compressible Flow of Liquid Crystals

The connection between the compressible flow of liquid crystals with low Mach number and the incompressible flow of liquid crystals is studied in a bounded domain. In particular, the convergence of weak solutions of the compressible flow of liquid crystals to the weak solutions of the incompressible flow of liquid crystals is proved when the Mach number approaches zero; that is, the incompressible limit is justified for weak solutions in a bounded domain.     

Viscous Standing Asymptotic States of Isentropic Compressible Flows Through a Nozzle

In this work we consider a viscous regularization of a well-known one-dimensional model for isentropic viscous compressible flows through a nozzle. For the existence and multiplicity of standing asymptotic states for a certain type of ducts, a complete analysis in a framework of dynamical systems is provided. As an application of the geometric singular perturbation theory, we show that all standing asymptotic states admit viscous profiles.     

An Analytical Solution for Capillary Gravity Drainage with Dominant Viscous Forces

The development of the capillary fringe during gravity drainage has a significant influence on saturation and pressure distributions in porous formations (Sarkarfarshi et al. in Int J Greenh Gas Control 23:61–71, 2014). This paper introduces an analytical solution for gravity drainage in an axisymmetric geometry with significant capillary pressure. The drainage process results from the injection of a lighter and less viscous injectant into a porous medium saturated with a heavier and more viscous pore fluid. If the viscous force dominates the capillary and the buoyancy forces, then the flow regime is approximated by differential equations and the admissible solution comprises a front shock wave and a trailing simple wave. In contrast to existing analytical solutions for capillary gravity drainage problems (e.g., Nordbotten and Dahle in 47(2) 2011; Golding et al. in J Fluid Mech 678:248–270 2011), this solution targets the saturation distribution during injection at an earlier point in time. Another contribution of this analytical solution is the incorporation of a completely drained flow regime close to the injection well. The analytical solution demonstrates the strong dependency of the saturation distribution upon relative permeability functions, gas entry capillary pressure, and residual saturation. The analytical results are compared to results from a commercial reservoir engineering software package (\(\hbox {CMG } \hbox {STARS}^{\mathrm{TM}}\)).     

On Time-Periodic Flow of a Viscous Liquid past a Moving Cylinder

We show existence, uniqueness and spatial asymptotic behavior of a two-dimensional time-periodic flow around a cylinder that moves orthogonal to its axis, with a time-periodic velocity, v. The result is proved if the size of the data is sufficiently small, and the average of v over a period is not zero.     

Hopf Bifurcation of Viscous Shock Waves in Compressible Gas Dynamics and MHD

Extending our previous results for artificial viscosity systems, we show, under suitable spectral hypotheses, that shock wave solutions of compressible Navier–Stokes and magnetohydrodynamics equations undergo Hopf bifurcation to nearby time-periodic solutions. The main new difficulty associated with physical viscosity and the corresponding absence of parabolic smoothing is the need to show that the difference between nonlinear and linearized solution operators is quadratically small in H ^s for data in H ^s . We accomplish this by a novel energy estimate carried out in Lagrangian coordinates; interestingly, this estimate is false in Eulerian coordinates. At the same time, we greatly sharpen and simplify the analysis of the previous work. Research of B.T. was partially supported under NSF grant number DMS-0505780. Research of K.Z. was partially supported under NSF grant number DMS-0300487.     

On the Capillary Problem for Compressible Fluids

Classical capillarity theory is based on a hypothesis that virtual motions of fluid particles distinct from those on a surface interface have no effect on the form of the interface. That hypothesis cannot be supported for a compressible fluid. A heuristic reasoning suggests that even small amounts of compressibility could have significant effect on surface behavior. In an earlier work, Finn took a partial account of compressibility, and formulated a variant of the classical capillarity equation for fluid surface height in a vertical capillary tube; he was led to a necessary condition for existence of a solution with prescribed mass in a tube closed at the bottom. For a circular tube, he proved that the condition also suffices, and that solutions are uniquely determined for any contact angle γ. Later Finn took more complete account of compressibility and obtained a new equation of highly nonlinear character but for which the same necessary condition holds. In the present work we consider that equation for circular tubes. We prove that the necessary condition again suffices for existence when 0 ≤ γ < π, and we establish uniqueness when 0 ≤ γ ≤ π/2. Our result is put into relief by the observation that for the unconstrained problem of a tube dipped into an infinite liquid bath, solutions do not in general exist when γ > π/2. Presumably an actual fluid would in that case descend to the bottom of the tube. This kind of singular behavior does not occur for the equation previously considered, nor does it occur in the present case under the presence of a mass constraint.