One-dimensional conformal metric flow

In this paper we continue our studies of the one-dimensional conformal metric flows, which were introduced in [Y. Ni, M. Zhu, Extremal one-dimensional conformal metrics, Commun. Contemp. Math., in press]. We prove the global existence and convergence of the flows, as well as the exponential convergence of the metrics under these flows.     

First‐Order system least squares for generalized‐Newtonian coupled Stokes‐Darcy flow

The coupled problem for a generalized Newtonian Stokes flow in one domain and a generalized Newtonian Darcy flow in a porous medium is studied in this work. Both flows are treated as a first‐order system in a stress‐velocity formulation for the Stokes problem and a volumetric flux‐hydraulic potential formulation for the Darcy problem. The coupling along an interface is done using the well‐known Beavers–Joseph–Saffman interface condition. A least squares finite element method is used for the numerical approximation of the solution. It is shown that under some assumptions on the viscosity the error is bounded from above and below by the least squares functional. An adaptive refinement strategy is examined in several numerical examples where boundary singularities are present. Due to the nonlinearity of the problem a Gauss–Newton method is used to iteratively solve the problem. It is shown that the linear variational problems arising in the Gauss–Newton method are well posed. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1150–1173, 2015     

Polynomial viscosity methods for multispecies kinematic flow models

Multispecies kinematic flow models are defined by systems of strongly coupled, nonlinear first‐order conservation laws. They arise in various applications including sedimentation of polydisperse suspensions and multiclass vehicular traffic. Their numerical approximation is a challenge since the eigenvalues and eigenvectors of the corresponding flux Jacobian matrix have no closed algebraic form. It is demonstrated that a recently introduced class of fast first‐order finite volume solvers, called polynomial viscosity matrix (PVM) methods [M. J. Castro Díaz and E. Fernández‐Nieto, SIAM J Sci Comput 34 (2012), A2173–A2196], can be adapted to multispecies kinematic flows. PVM methods have the advantage that they only need some information about the eigenvalues of the flux Jacobian, and no spectral decomposition of a Roe matrix is needed. In fact, the so‐called interlacing property (of eigenvalues with known velocity functions), which holds for several important multispecies kinematic flow models, provides sufficient information for the implementation of PVM methods. Several variants of PVM methods (differing in polynomial degree and the underlying quadrature formula to approximate the Roe matrix) are compared by numerical experiments. It turns out that PVM methods are competitive in accuracy and efficiency with several existing methods, including the Harten, Lax, and van Leer method and a spectral weighted essentially non‐oscillatory scheme that is based on the same interlacing property. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1265–1288, 2016     

Finite volume approximation of degenerate two‐phase flow model with unlimited air mobility

Models of two‐phase flows in porous media, used in petroleum engineering, lead to a coupled system of two equations, one elliptic and the other degenerate parabolic, with two unknowns: the saturation and the pressure. In view of applications in hydrogeology, we construct a robust finite volume scheme allowing for convergent simulations, as the ratio μ of air/liquid mobility goes to infinity. This scheme is shown to satisfy a priori estimates (the saturation is shown to remain in a fixed interval, and a discrete L2(0,T;H1(Ω)) estimate is proved for both the pressure and a function of the saturation), which are sufficient to derive the convergence of a subsequence to a weak solution of the continuous equations, as the size of the discretization tends to zero. We then show that the scheme converges to a two‐phase flow model whose limit, as the mobility of the air phase tends to infinity, is the “quasi‐Richards equation” (Eymard et al., Convergence of two phase flow to Richards model, F. Benkhaldoun, editor, Finite Volumes for Complex Applications IV, ISTE, London, 2005; Eymard et al., Discrete Cont Dynam Syst, 5 (2012) 93–113), which remains available even if the gas phase is not connected with the atmospheric pressure. Numerical examples, which show that the scheme remains robust for high values of μ, are finally given. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

Asymptotic and numerical solutions of three‐dimensional boundary‐layer flow past a moving wedge

We consider a laminar boundary‐layer flow of a viscous and incompressible fluid past a moving wedge in which the wedge is moving either in the direction of the mainstream flow or opposite to it. The mainstream flows outside the boundary layer are approximated by a power of the distance from the leading boundary layer. The variable pressure gradient is imposed on the boundary layer so that the system admits similarity solutions. The model is described using 3‐dimensional boundary‐layer equations that contains 2 physical parameters: pressure gradient (β) and shear‐to‐strain‐rate ratio parameter (α). Two methods are used: a linear asymptotic analysis in the neighborhood of the edge of the boundary layer and the Keller‐box numerical method for the full nonlinear system. The results show that the flow field is divided into near‐field region (mainly dominated by viscous forces) and far‐field region (mainstream flows); the velocity profiles form through an interaction between 2 regions. Also, all simulations show that the subsequent dynamics involving overshoot and undershoot of the solutions for varying parameter characterizing 3‐dimensional flows. The pressure gradient (favorable) has a tendency of decreasing the boundary‐layer thickness in which the velocity profiles are benign. The wall shear stresses increase unboundedly for increasing α when the wedge is moving in the x‐direction, while the case is different when it is moving in the y‐direction. Further, both analysis show that 3‐dimensional boundary‐layer solutions exist in the range −1<α<∞. These are some interesting results linked to an important class of boundary‐layer flows.     

Multicomponent flow modeling

We present multicomponent flow models derived from the kinetic theory of gases and investigate the symmetric hyperbolic-parabolic structure of the resulting system of partial differential equations.We address the Cauchy problem for smooth solutions as well as the existence of deflagration waves,also termed anchored waves.We further discuss related models which have a similar hyperbolic-parabolic structure,notably the SaintVenant system with a temperature equation as well as the equations governing chemical equilibrium flows.We next investigate multicomponent ionized and magnetized flow models with anisotropic transport fluxes which have a different mathematical structure.We finally discuss numerical algorithms specifically devoted to complex chemistry flows,in particular the evaluation of multicomponent transport properties,as well as the impact of multicomponent transport.     

Two‐phase flows in karstic geometry

Multiphase flow phenomena are ubiquitous. Common examples include coupled atmosphere and ocean system (air and water), oil reservoir (water, oil, and gas), and cloud and fog (water vapor, water, and air). Multiphase flows also play an important role in many engineering and environmental science applications. In some applications such as flows in unconfined karst aquifers, karst oil reservoir, proton membrane exchange fuel cell, multiphase flows in conduits, and in porous media must be considered together. Geometric configurations that contain both conduit (or vug) and porous media are termed karstic geometry. Despite the importance of the subject, little work has been performed on multiphase flows in karstic geometry. In this paper, we present a family of phase–field (diffusive interface) models for two‐phase flow in karstic geometry. These models together with the associated interface boundary conditions are derived utilizing Onsager's extremum principle. The models derived enjoy physically important energy laws. A uniquely solvable numerical scheme that preserves the associated energy law is presented as well. Copyright © 2013 John Wiley & Sons, Ltd.     

Unified finite element discretizations of coupled Darcy–Stokes flow

In this article, we discuss some new finite element methods for flows which are governed by the linear stationary Stokes system on one part of the domain and by a second order elliptic equation derived from Darcy's law in the rest of the domain, and where the solutions in the two domains are coupled by proper interface conditions. All the methods proposed here utilize the same finite element spaces on the entire domain. In particular, we show how the coupled problem can be solved by using standard Stokes elements like the MINI element or the Taylor–Hood element in the entire domain. Furthermore, for all the methods the handling of the interface conditions are straightforward. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Nonlinear simulations of vesicle wrinkling

In this paper, we study nonlinear wrinkling dynamics of a vesicle in an extensional flow. Motivated by the recent experiments and linear theory on wrinkles of a quasi‐spherical membrane, we are interested in examining the linear theory and exploring wrinkling dynamics in a nonlinear regime. We focus on a quasi‐circular vesicle in two dimensions and show that the linear analytical results are qualitatively independent of the number of dimensions. Hence, the two‐dimensional studies can provide insights into the full three‐dimensional problem. We develop a spectral accurate boundary integral method to simulate the nonlinear evolution of surface tension and the nonlinear interactions between flow and membrane morphology. We demonstrate that for a quasi‐circular vesicle, the linear theory well predicts the characteristic wavenumber during the wrinkling dynamics. Nonlinear results of an elongated vesicle show that there exist dumbbell‐like stationary shapes in weak flows. For strong flows, wrinkles with pronounced amplitudes will form during the evolution. As far as the shape transition is concerned, our simulations are able to capture the main features of wrinkles observed in the experiments. Interestingly, numerical results reveal that, in addition to wrinkling, asymmetric rotation can occur for slightly tilted vesicles. The mathematical theory and numerical results are expected to lead to a better understanding of related problems in biology such as cell wrinkling. Copyright © 2013 John Wiley & Sons, Ltd.     

Coupling of compressible and incompressible flow regions using the multiple pressure variables approach

In many cases, multiphase flows are simulated on the basis of the incompressible Navier–Stokes equations. This assumption is valid as long as the density changes in the gas phase can be neglected. Yet, for certain technical applications such as fuel injection, this is no longer the case, and at least the gaseous phase has to be treated as a compressible fluid. In this paper, we consider the coupling of a compressible flow region to an incompressible one based on a splitting of the pressure into a thermodynamic and a hydrodynamic part. The compressible Euler equations are then connected to the Mach number zero limit equations in the other region. These limit equations can be solved analytically in one space dimension that allows to couple them to the solution of a half‐Riemann problem on the compressible side with the help of velocity and pressure jump conditions across the interface. At the interface location, the flux terms for the compressible flow solver are provided by the coupling algorithms. The coupling is demonstrated in a one‐dimensional framework by use of a discontinuous Galerkin scheme for compressible two‐phase flow with a sharp interface tracking via a ghost‐fluid type method. The coupling schemes are applied to two generic test cases. The computational results are compared with those obtained with the fully compressible two‐phase flow solver, where the Mach number zero limit is approached by a weakly compressible fluid. For all cases, we obtain a very good agreement between the coupling approaches and the fully compressible solver. Copyright © 2014 John Wiley & Sons, Ltd.