Dissipative measure-valued solutions to the magnetohydrodynamic equations

We introduce the concept of a dissipative measure-valued solution to the Euler alignment system. This approach incorporates a modified total energy balance, utilizing a binary tensor Young measure. The central finding is a weak (measure-valued)-strong uniqueness principle: if both a dissipative measure-valued solution and a classical smooth solution originate from the same initial data, they will be identical as long as the classical solution exists.

We study convergence of a mixed finite element–finite volume numerical scheme for the isentropic Navier–Stokes system under the full range of the adiabatic exponent. We establish suitable stability and consistency estimates and show that the Young measure generated by numerical solutions represents a dissipative measure-valued solutions of the limit system. In particular, using the recently established weak–strong uniqueness principle in the class of dissipative measure-valued solutions we show that the numerical solutions converge strongly to a strong solutions of the limit system as long as the latter exists.

We introduce the concept of dissipative measure-valued solution to the complete Euler system describing the motion of an inviscid compressible fluid. These solutions are characterized by a parameterized (Young) measure and a dissipation defect in the total energy balance. The dissipation defect dominates the concentration errors in the equations satisfied by the Young measure. A dissipative measure-valued solution can be seen as the most general concept of solution to the Euler system retaining its structural stability. In particular, we show that a dissipative measure-valued solution necessarily coincides with a classical one on its life span provided they share the same initial data.

In this paper, we investigate the limits of Riemann solutions to the Euler equations of compressible fluid flow with a source term as the adiabatic exponent tends to one. The source term can represent friction or gravity or both in Engineering. For instance, a concrete physical model is a model of gas dynamics in a gravitational field with entropy assumed to be a constant. The body force source term is presented if there is some external force acting on the fluid. The force assumed here is the gravity. Different from the homogeneous equations, the Riemann solutions of the inhomogeneous system are non self-similar. We rigorously proved that, as the adiabatic exponent tends to one, any two-shock Riemann solution tends to a delta shock solution of the pressureless Euler system with a Coulomb-like friction term, and the intermediate density between the two shocks tends to a weighted $$\delta $$ -mesaure which forms the delta shock; while any two-rarefaction-wave Riemann solution tends to a two-contact-discontinuity solution of the pressureless Euler system with a Coulomb-like friction term, whose intermediate state between the two contact discontinuities is a vacuum state. Moreover, we also give some numerical simulations to confirm the theoretical analysis.

5R6. Theory and Applications of Viscous Fluid Flows. - RK Zeytounian (12 Rue Saint-Fiacre, Paris, 75002, France). Springer-Verlag, Berlin. 2004. 488 pp. ISBN 3-540-44013-5. $109.00.Reviewed by MF Platzer (Dept of Aeronaut and Astronaut, Naval Postgraduate Sch, Code AA/PL, Monterey CA 93943-5000).Starting with the derivation of the Navier-Stokes equations for viscous heat-conducting fluids the author proceeds to discuss various forms of these equations, including the special cases of compressible isentropic viscous flow of polytropic gases and viscous incompressible fluid flow. He then discusses the Orr-Sommerfeld theory for the plane Poiseuille flow as well as other basic flow cases, such as steady flow through an arbitrary cylinder, annular flow between concentric cylinders, Benard thermal convection flow, Benard-Marangoni flow induced by tangential gradients of variable surface tension, flow due to a rotating disc, and Rayleigh flow caused by an impulsively started flat plate. The next three chapters are devoted to the very large and very low Reynolds number limits and to the low Mach number incompressible limit. In the chapter on very large Reynolds number flow the author discusses the application of the method of matched asymptotic expansions to the two-dimensional steady flat-plate flow problem and delineates the relationship of the unsteady Navier-Stokes equations to the inviscid Euler, the Prandtl boundary layer, the one-dimensional gas dynamics and the Rayleigh compressible flow equations. He also discusses the triple deck concept, laminar flow separation on a circular cylinder, and the three-dimensional boundary layer equations. In the chapter on very low Reynolds numbers, the unsteady-state matched Stokes-Oseen solution for the flow past a sphere and the flow over an impulsively started circular cylinder are discussed, followed by a consideration of the Stokes and Oseen steady-state compressible flow equations and the asymptotic analysis for small Reynolds number flows on a rotating disc. In the next chapter on low Mach number incompressible limit, the author discusses subtleties involved in analyzing unsteady weakly compressible flows; flow in a bounded cavity and through large aspect ratio channels. He then provides further examples by analyzing the acoustic streaming effect caused by an oscillating circular cylinder, the incompressible flow past a rotating and translating cylinder, the Ekman and Stewartson layers on rotating cylinders, and the Benard-Marangoni thermo-capillary instability problem due to heating of a horizontal viscous liquid from below. Also presented are some aspects of non-adiabatic viscous atmospheric flows and a few other topics, such as the entrainment of a viscous fluid in a two-dimensional cavity and the laminar boundary layer separation phenomenon near the leading-edge region of an airfoil and on an impulsively started cylinder. In this regard, he emphasizes the need for the simultaneous solution of the boundary layer and inviscid flow equations in order to remove the singularity at the separation point, as implemented in the viscous-inviscid interaction procedures. The next two chapters are devoted to a discussion of the existence, regularity and uniqueness of solutions for the viscous incompressible and compressible flow equations and the stability theory of fluid motion. In particular, the Guiraud-Zeytounian asymptotic approach to nonlinear hydrodynamic stability is elucidated and applied to the Rayleigh-Benard convection problem, followed by an analysis of the Benard-Marangoni thermo-capillary instability problem and the Couette-Taylor viscous flow between two rotating cylinders. The final chapter of Theory and Applications of Viscous Fluid Flows presents the finite-dimensional dynamical systems approach to turbulence by reviewing the Landau-Hopf, Ruelle-Takens-Newhouse, Feigenbaum and Pomeau-Manneville transition scenarios to turbulence. The book is ended by giving examples of strange attractors occurring in various fluid flows, such as in viscous isobaric wave motions or in the flow of an incompressible but thermally conducting liquid down a vertical plane (the Benard-Marangoni problem for a free-falling vertical film). It is evident from this brief summary that the author’s emphasis is on the mathematical aspects of the viscous flow equations and their various asymptotic limit cases and analytical solution methods. His choice of topics and flow problems is meant to provide young researchers in fluid mechanics, applied mathematics and theoretical physics with an up-to-date presentation of some key problems in the analysis of viscous fluid flows. Although the author intentionally limited himself to a select few topics, teachers of advanced viscous flow courses and researchers in this field will welcome this book for its thorough review of current work and the listing of 1156 relevant papers. In my judgment, it meets the stated objective of bridging the gap between standard undergraduate texts in fluid mechanics and specialized monographs.

A semi-implicit in time, entropy stable finite volume scheme for the compressible barotropic Euler system is designed and analyzed and its weak convergence to a dissipative measure-valued (DMV) solution [Feireisl et al., Calc. Var. Part. Differ. Equ. 55 (2016) 141] of the Euler system is shown. The entropy stability is achieved by introducing a shifted velocity in the convective fluxes of the mass and momentum balances, provided some CFL-like condition is satisfied to ensure stability. A consistency analysis is performed in the spirit of the Lax’s equivalence theorem under some physically reasonable boundedness assumptions. The concept of Ƙ-convergence [Feireisl et al., IMA J. Numer. Anal. 40 (2020) 2227–2255] is used in order to obtain some strong convergence results, which are then illustrated via rigorous numerical case studies. The convergence of the scheme to a DMV solution, a weak solution and a strong solution of the Euler system using the weak–strong uniqueness principle and relative entropy are presented.

In this paper, we investigate the limiting behavior of Riemann solutions to the Euler equations of compressible fluid flow for modified Chaplygin gas with the body force as the two parameters tend to zero. The formation of delta shock waves and the vacuum states is identified and analyzed during the process of vanishing pressure in the Riemann solutions. The concentration and cavitation are fundamental and physical phenomena in fluid dynamics, which can be mathematically described by delta shock waves and vacuums, respectively. In this paper, our main objective is to rigorously investigate the formation of delta shock waves and vacuums and observe the concentration and cavitation phenomena. First, the Riemann problem of the Euler equations of compressible fluid flow for the modified Chaplygin gas with the body force is solved. Second, we rigorously confirm that, as the pressure vanishes, any two shock Riemann solution to the Euler equations of compressible fluid flow for the modified Chaplygin gas with the body force tends to a δ-shock solution to the pressureless gas dynamics model with a body force, and the intermediate density between the two shocks tends to a weighted δ-measure that forms the δ-shock; any two-rarefaction-wave Riemann solution to the Euler equations of compressible fluid flow for the modified Chaplygin gas with the body force tends to a solution consisting of four contact discontinuities together with vacuum states with three different virtual velocities in the limiting situation.

Compressible micropolar equations model a class of fluids with microstructure. In this paper we establish the dissipative measure-valued solution to the micropolar fluids. We also give the weak-strong uniqueness principle to this system which means its dissipative measure-valued solution is the same as the classical solution, provided they emanate from the same initial data.

For the equations of elastodynamics with polyconvex stored energy, and some related simpler systems, we define a notion of dissipative measure-valued solution and show that such a solution agrees with a classical solution with the same initial data when such a classical solution exists. As an application of the method we give a short proof of strong convergence in the continuum limit of a lattice approximation of one dimensional elastodynamics in the presence of a classical solution. Also, for a system of conservation laws endowed with a positive and convex entropy, we show that dissipative measure-valued solutions attain their initial data in a strong sense after time averaging.

The results are presented of a study of the application of analytical methods to the solution of two phase flow into single wells. Approximate analytical expressions for the pressure distribution in two-phase flow are found for a number of conditions. The results obtained from the analytical solutions are found to be in good agreement with results obtained by finite difference techniques using a high speed digital computer. Mathematical solutions for four sets of boundary conditions are presented. All of these solutions are composed of a short-term transient plus a steady or quasi-steady state. The rates of decay of the short-lived transients are analyzed. It is found that the durations of the short-term transients may be characterized by a parameter defined as the time constant which can be determined from simple relations. It is shown also that if the outer radius is much greater than the radius of the well, the short term transients decay at rates which are proportional to the square of the exterior radius, and the rates of decay are only slightly dependent upon the radius ratio. Numerical solutions based on finite difference techniques are presented for a number of conditions. The numerical solutions are in good agreement with the predictions based on the theoretical analysis for small and moderate drawdowns. Examples involving large drawdowns indicate that the nonlinearities in the equations of flow do not appreciably alter the longevity of the short-term transients. In all cases the time required for the short-term transients to disappear is predicted satisfactorily. Introduction The mechanism by which oil and gas flow into a single well is of vital interest to the petroleum industry. The fundamental equations of two-phase flow which describe this mechanism are nonlinear partial differential equations. Numerical solutions of these equations describing pressure transients have been obtained with the aid of electronic computers. Although solutions obtained in this manner take into account a large number of effects, the reduction of this information to useful generalities is difficult. One method of obtaining generalities is the use of linearized approximations of the nonlinear equations. Since it is possible to obtain explicit solutions of the linearized equation, general properties of the role of pressure in the flow mechanism may be ascertained. Results obtained from this approach are limited to some extent by the linearizing assumptions. The severity of these limitations may be evaluated by comparing solutions of the linear equation with numerical solutions of the more exact nonlinear equations of two-phase flow. In the past considerable amount of work has been devoted to studying pressure build-up using the single-phase flow theory. Unfortunately, most pressure build-up tests involve multiphase flow. A small amount of work has been done studying pressure build-up where the flow is two-phase. The encouraging results of these studies suggest that useful results may be found from additional studies of not only pressure build-up, but also the rapid transients associated with placing a well on production. This paper presents the basic theory of the pressure transients associated with placing a well on production and with closing it in. The paper is concerned chiefly with two-phase compressible flow; however, the results also apply to single-phase flow, The results are based on analytical solutions of the flow equations, and they are verified by numerical solutions using finite difference techniques. Much of the previous work on compressible flow into wells has been confined to single-phase flow. Some work has been done on compressible two-phase steady state flow, and solutions of the equations of flow have been found by finite difference techniques using high-speed computers. Muskat presents some rather general solutions to the equations of single-phase compressible flow into wells. Much work has been done on pressure build-up in wells (see, for example, Refs. 2–6). Almost all of the work on pressure build-up concerns single-phase flow with the exception of Ref. 5 and part of Ref. 2. Some work has been done on pressure fall-off in injection wells. Muskat presents the solution of the equations for radial steady state two-phase compressible flow. SPEJ P. 116^

Ninth International Conference on Numerical Methods in Fluid Dynamics

The Cauchy problem for the complete Euler system is in general ill-posed in the class of admissible (entropy producing) weak solutions. This suggests that there might be sequences of approximate solutions that develop fine-scale oscillations. Accordingly, the concept of measure-valued solution that captures possible oscillations is more suitable for analysis. We study the convergence of a class of entropy stable finite volume schemes for the barotropic and complete compressible Euler equations in the multidimensional case. We establish suitable stability and consistency estimates and show that the Young measure generated by numerical solutions represents a dissipative measure-valued solution of the Euler system. Here dissipative means that a suitable form of the second law of thermodynamics is incorporated in the definition of the measure-valued solutions. In particular, using the recently established weak-strong uniqueness principle, we show that the numerical solutions converge pointwise to the regular solution of the limit systems at least on the lifespan of the latter.

We prove the global-in-time existence of weak solutions to the Navier-Stokes equations of compressible isentropic flow in three space dimensions with adiabatic exponent $\gamma\ge1$. Initial data and solutions are small in $L^2$ around a non-constant steady state with densities being positive and essentially bounded. No smallness assumption is imposed on the external forces when $\gamma=1$. A great deal of information about partial regularity and large-time behavior is obtained.

<p style='text-indent:20px;'>In this paper, we consider the quasineutral limit of compressible Euler-Poisson equations based on the concept of dissipative measure-valued solutions. In the case of well-prepared initial data under periodic boundary condictions, we prove that dissipative measure-valued solutions of the compressible Euler-Poisson equations converge to the smooth solution of the incompressible Euler system when the Debye length tends to zero.</p>

In this paper, we propose a new approach to singular limits of inviscid fluid flows based on the concept of dissipative measure-valued solutions. We show that dissipative measure-valued solutions of the compressible Euler equations converge to the smooth solution of the incompressible Euler system when the Mach number tends to zero. This holds both for well-prepared and ill-prepared initial data, where in the latter case the presence of acoustic waves causes difficulties. However this effect is eliminated on unbounded domains thanks to dispersion.