Global weak solutions for a viscous liquid–gas model with transition to single-phase gas flow and vacuum

This work deals with a viscous two-phase liquid–gas model relevant to the flow in wells and pipelines. The liquid is treated as an incompressible fluid whereas the gas is assumed to be polytropic. The model is rewritten in terms of Lagrangian coordinates and is studied in a free boundary setting where the liquid and gas masses are of compact support initially, and continuous at the boundary. Consequently, the initial masses involve a transition to single-phase gas flow and vacuum at the boundary. An appropriate balance between pressure and viscous forces is identified which allows obtaining pointwise upper and lower estimates of masses. These estimates rely on the assumption of a certain relation between the rate of degeneracy of the viscosity coefficient and the rate that determines how fast the initial masses are vanishing at the boundary. By combining these estimates with basic energy type of estimates, higher order regularity estimates are obtained. The existence of global weak solutions is then proved by showing compactness for a class of semi-discrete approximations.     

Free boundary value problem for a viscous two-phase model with mass-dependent viscosity

In this paper, we study a free boundary value problem for two-phase liquid-gas model with mass-dependent viscosity coefficient when both the initial liquid and gas masses connect to vacuum with a discontinuity. This is an extension of the paper [S. Evje, K.H. Karlsen, Global weak solutions for a viscous liquid-gas model with singular pressure law, http://www.irisresearch.no/docsent/emp.nsf/wvAnsatte/SEV]. Just as in [S. Evje, K.H. Karlsen, Global weak solutions for a viscous liquid-gas model with singular pressure law, http://www.irisresearch.no/docsent/emp.nsf/wvAnsatte/SEV], the gas is assumed to be polytropic whereas the liquid is treated as an incompressible fluid. We give the proof of the global existence and uniqueness of weak solutions when β∈(0,1], which have improved the previous result of Evje and Karlsen, and get the asymptotic behavior result, also we obtain the regularity of the solutions by energy method.     

Global weak solutions for a compressible gas-liquid model with well-formation interaction

The objective of this work is to explore a compressible gas-liquid model designed for modeling of well flow processes. We build into the model well-reservoir interaction by allowing flow of gas between well and formation (surrounding reservoir). Inflow of gas and subsequent expansion of gas as it ascends towards the top of the well (a so-called gas kick) represents a major concern for various well operations in the context of petroleum engineering. We obtain a global existence result under suitable assumptions on the regularity of initial data and the rate function that controls the flow of gas between well and formation. Uniqueness is also obtained by imposing more regularity on the initial data. The key estimates are to obtain appropriate lower and upper bounds on the gas and liquid masses. For that purpose we introduce a transformed version of the original model that is highly convenient for analysis of the original model. In particular, in the analysis of the transformed model additional terms, representing well-formation interaction, can be treated by natural extensions of arguments that previously have been employed for the single-phase Navier-Stokes model. The analysis ensures that transition to single-phase regions do not appear when the initial state is a true gas-liquid mixture.     

A blow-up criterion for a 2D viscous liquid-gas two-phase flow model

In this paper, we prove a blow-up criterion in terms of the upper bound of the liquid mass for the strong solution to the two-dimensional (2D) viscous liquid-gas two-phase flow model in a smooth bounded domain. The result also applies to three-dimensional (3D) case.     

Degenerate two-phase incompressible flow II: regularity, stability and stabilization

In this paper, we analyze a coupled system of highly degenerate elliptic-parabolic partial differential equations for two-phase incompressible flow in porous media. This system involves a saturation and a global pressure (or a total flow velocity). First, we show that the saturation is Hölder continuous both in space and time and the total velocity is Hölder continuous in space (uniformly in time). Applying this regularity result, we then establish the stability of the saturation and pressure with respect to initial and boundary data, from which uniqueness of the solution to the system follows. Finally, we establish a stabilization result on the asymptotic behavior of the saturation and pressure; we prove that the solution to the present system converges (in appropriate norms) to the solution of a stationary system as time goes to infinity. An example is given to show typical regularity of the saturation.     

Well-posedness for two-dimensional steady supersonic Euler flows past a Lipschitz wedge

For a supersonic Euler flow past a straight-sided wedge whose vertex angle is less than the extreme angle, there exists a shock-front emanating from the wedge vertex, and the shock-front is usually strong especially when the vertex angle of the wedge is large. In this paper, we establish the L¹ well-posedness for two-dimensional steady supersonic Euler flows past a Lipschitz wedge whose boundary slope function has small total variation, when the total variation of the incoming flow is small. In this case, the Lipschitz wedge perturbs the flow, and the waves reflect after interacting with the strong shock-front and the wedge boundary. We first obtain the existence of solutions in BV when the incoming flow has small total variation by the wave front tracking method and then establish the L¹ stability of the solutions with respect to the incoming flows. In particular, we incorporate the nonlinear waves generated from the wedge boundary to develop a Lyapunov functional between two solutions containing strong shock-fronts, which is equivalent to the L¹ norm, and prove that the functional decreases in the flow direction. Then the L¹ stability is established, so is the uniqueness of the solutions by the wave front tracking method. Finally, the uniqueness of solutions in a broader class, the class of viscosity solutions, is also obtained.     

A blow-up criterion for 3D compressible viscoelastic flow with large initial data

In this paper, we consider a Cauchy problem for the three-dimensional compressible viscoelastic flow with large initial data. We establish a blow-up criterion for the strong solutions in terms of the gradient of velocity only, which is similar to the Beale-Kato-Majda criterion for ideal incompressible flow (cf. Beale et al. (1984) [20]) and the blow-up criterion for the compressible Navier-Stokes equations (cf. Huang et al. (2011) [21]).     

An FEM identification procedure for a wing section within a class of finite Laurent series

This paper concerns a determination procedure for conformal mapping of a wing through a finite element computation of potential function associated with the flow of 2-dimensional perfect fluid around the given wing section. Through our numerical procedure a family of mappings is obtained in the forms of finite Laurent series for an initial wing section input. Each member of the family describes a wing section located in a neighboring domain of the input one. Some of them could be expected as modified versions of the original wing section input, although they could not recover completely it.Inputting the shape of wing section has ambiguity in practical cases of wing sections such as the NACA23012 wing section. We would like to postulate that our identification procedure should be employed in the determination process of numerical profiles of the wing section considered, since identified ones are significantly easier in numerical processing than the original input shape.     

Monotonicity of control volume methods

Robustness of numerical methods for multiphase flow problems in porous media is important for development of methods to be used in a wide range of applications. Here, we discuss monotonicity for a simplified problem of single-phase flow, but where the simulation grids and media are allowed to be general, posing challenges to control-volume methods. We discuss discrete formulations of the maximum principle and derive sufficient criteria for discrete monotonicity for arbitrary nine-point control-volume discretizations for conforming quadrilateral grids in 2D. These criteria are less restrictive than the M-matrix property. It is shown that it is impossible to construct nine-point methods which unconditionally satisfy the monotonicity criteria when the discretization satisfies local conservation and exact reproduction of linear potential fields. Numerical examples are presented which show the validity of the criteria for monotonicity. Further, the impact of nonmonotonicity is studied. Different behavior for different discretization methods is illuminated, and simple ideas are presented for improvement in terms of monotonicity.     

Existence and continuous dependence of large solutions for the magnetohydrodynamic equations

Global solutions of the nonlinear magnetohydrodynamic (MHD) equations with general large initial data are investigated. First the existence and uniqueness of global solutions are established with large initial data in H ¹. It is shown that neither shock waves nor vacuum and concentration are developed in a finite time, although there is a complex interaction between the hydrodynamic and magnetodynamic effects. Then the continuous dependence of solutions upon the initial data is proved. The equivalence between the well-posedness problems of the system in Euler and Lagrangian coordinates is also showed.     

Modeling of two-phase flows with surface tension by finite pointset method (FPM)

A meshfree method for two-phase immiscible incompressible flows including surface tension is presented. The continuum surface force (CSF) model is used to include the surface tension force. The incompressible Navier–Stokes equation is considered as the mathematical model. Application of implicit projection method results in linear second-order partial differential equations for velocities and pressure. These equations are then solved by the finite pointset method (FPM), which is a meshfree and Lagrangian method. The fluid is represented as finite number of particles and the immiscible fluids are distinguished by the color of each particle. The interface is tracked automatically by advecting the color functions for each particle. Two test cases, Laplace's law and the Rayleigh–Taylor instability in 2D have been presented. The results are found to be consistent with the theoretical results.     

Systems of partial differential equations of mixed hyperbolic-parabolic type

We prove the global existence and uniqueness of solutions of certain mixed hyperbolic-parabolic systems of partial differential equations in one space dimension with initial data that is assumed to be pointwise bounded with possibly large oscillation and with small total energy. The systems we consider are general enough to include the Navier-Stokes equations of compressible flow, the equations of compressible MHD, models of chemical combustion, and others. In particular, the application of our results to the MHD system gives an existence result which is new.     

Global well-posedness of strong solutions to a two-phase model with magnetic field for large oscillations in three dimensions

In this paper, we consider Cauchy problem for a two-phase model with magnetic field in three dimensions. Under some smallness assumptions on the initial data but possibly large oscillations, we obtain the global well-posedness of strong solution as well as its large-time behavior. Compared with Wen and the first author's work [28] where global well-posedness and large time behavior of strong solutions were obtained subject to smallness of initial data in $H^2$ norm, we only need the smallness of initial energy which allows large oscillations of the initial data.     

Mixed convection boundary layer flow over a permeable vertical surface with prescribed wall heat flux

The effects of suction/injection on the laminar mixed convection boundary-layer flow on a vertical wall with a prescribed heat flux are considered. The conditions which allow the equations to be reduced to similarity form are derived and numerical solutions of the resulting equations are obtained for a range of values of the suction/injection and buoyancy parameters. Two specific cases, corresponding to a stagnation point flow and uniform wall heat flux, are treated in detail. Results are presented in terms of the skin friction and wall temperature with a selection of velocity and temperature profiles also being given. Dual solutions are found to exist for assisting flow, these are an addition to what has been reported previously for opposing flows. Solutions for some limiting values of the parameters are also derived.      

Pointwise gradient estimates of solutions to onedimensional nonlinear parabolic equations

We present here an improved version of the method introduced by the first author to derive pointwise gradient estimates for the solutions of one-dimensional parabolic problems. After considering a general qualinear equation in divergence form we apply the method to the case of a nonlinear diffusion-convection equation. The conclusions are stated first for classical solutions and then for generalized and mild solutions. In the case of unbounded initial datum we obtain several regularizing effects for t > 0. Some unilateral pointwise gradient estimates are also obtained. The case of the Dirichlet problem is also considered. Finally, we collect, in the last section, several comments showing the connections among these estimates and the study of the free boundaries associated to the solutions of the diffusion-convection equation.     

Strong solutions of the Navier-Stokes equations for isentropic compressible fluids

We study strong solutions of the isentropic compressible Navier-Stokes equations in a domain . We first prove the local existence of unique strong solutions provided that the initial data ρ₀ and u₀ satisfy a natural compatibility condition. The important point in this paper is that we allow the initial vacuum: the initial density may vanish in an open subset of Ω. We then prove a new uniqueness result and stability result. Our results are valid for unbounded domains as well as bounded ones.     

Review on mathematical analysis of some two-phase flow models

The two-phase flow models are commonly used in industrial applications, such as nuclear, power, chemical-process, oil-and-gas, cryogenics, bio-medical, micro-technology and so on. This is a survey paper on the study of compressible nonconservative two-fluid model, drift-flux model and viscous liquid-gas two-phase flow model. We give the research developments of these three two-phase flow models, respectively. In the last part, we give some open problems about the above models.     

Particle trajectories beneath small amplitude shallow water waves in constant vorticity flows

We investigate the particle trajectories in a constant vorticity shallow water flow over a flat bed as periodic waves propagate on the water’s free surface. Within the framework of small amplitude waves, we find the solutions of the nonlinear differential equations system which describes the particle motion in the considered case, and we describe the possible particle trajectories. Depending on the relation between the initial data and the constant vorticity, some particle trajectories are undulating curves to the right, or to the left, others are loops with forward drift, or with backward drift, others can follow some peculiar shapes.     

Existence and asymptotic behavior of C solutions to the multi-dimensional compressible Euler equations with damping

In this paper, the existence and asymptotic behavior of C¹$C^1$ solutions to the multi-dimensional compressible Euler equations with damping on the framework of Besov space are considered. Comparing with the well-posedness results of Sideris–Thomases–Wang [T. Sideris, B. Thomases, D.H. Wang, Long time behavior of solutions to the three-dimensional compressible Euler with damping, Comm. Partial Differential Equations 28 (2003) 953–978], we weaken the regularity assumptions on the initial data. The global existence lies on a crucial a-priori estimate which is obtained by the spectral localization method. The main analytic tools are the Littlewood–Paley decomposition and Bony’s paraproduct formula.