Time-invariant uniqueness for reacting viscous fluids near chemical equilibrium

A system of equations is investigated which models a reacting mixture of viscous fluids acted upon by an external body force. Through logarithmic convexity arguments the solution is shown to be unique when either the initial or the final data are given.     

Macroscopic finite element analyses of discretematerials based on directly evaluated micro‐macro transitions

We consider a homogenized macro‐continuum with locally attached microstructure of granules and derive specific micromacro transitions by a consistent transfer of discrete micro‐variables to field variables on a continuous macrostructure. Displacements and rotational constraints are imposed on the granules on the defined boundary frame of the microstructure. The constraints for linear displacements and uniform tractions on the surface yield upper and lower bound characteristics for periodic boundary conditions with regard to the aggregate stiffness. Secondly, we perform two‐scale analyses where we link simulations on the macro‐ and the microscales. Therein, coupled boundary‐value problems are solved on both scales. The macroscopic homogeneous problem is solved by a finite element method where the material model is implemented using the directly evaluated micro‐macro transitions on the basis of the discrete microstructures. Finally, a model problem is investigated to clarify the proposed method. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The FENE dumbbell model near equilibrium

We study the global existence of smooth solutions near the equilibrium to a coupled microscopic-macroscopic FENE dumbbell model which arises from the kinetic theory of diluted solutions of polymeric liquids with noninteracting polymer chains.     

On a stochastic model for particle sedimentation in fluids

The engineering sciences have long been interested in models describing the settling (or sedimentation) of particle ensembles in viscous fluids. From a theoretical point of view, sedimentation constitutes a two phase many-body phenomenon with complex interaction and as such has proved to be quite inaccessible to modelling attempts with relevance in applications. A stochastic model was introduced by Pickard and Tory in 1977 and was subsequently revised. In this paper we expand and refine the stochastic model and present a satisfactory model-fitting procedure and parameter estimators based on transit times of sedimenting particles.     

On the cauchy problem for a one‐dimensional full compressible non‐Newtonian fluids

This paper is concerned with global existence and asymptotic behavior of H¹ solutions to the Cauchy problem of one‐dimensional full non‐Newtonian fluids with the weighted small initial data. We then obtain the global existence of Hⁱ(i = 2,4) solutions and their asymptotic behavior for the system. Copyright © 2015 John Wiley & Sons, Ltd.     

A finite difference scheme for solving parabolic two‐step micro‐heat transport equations in a double‐layered micro‐sphere heated by ultrashort‐pulsed lasers

Ultrashort‐pulsed lasers with pulse durations of the order of sub‐picosecond to femtosecond domain possess exclusive capabilities in limiting the undesirable spread of the thermal process zone in the heated sample. Parabolic two‐step micro heat transport equations have been widely applied for thermal analysis of thin metal films exposed to picosecond thermal pulses. In this study, we develop a three‐level finite difference scheme for solving the micro heat transport equations in a double‐layered micro sphere. It is shown by the discrete energy method that the scheme is unconditionally stable. Numerical results for thermal analysis of a gold layer coated on a chromium padding layer are obtained. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

On the mathematical modelling of dynamic problems in a micro‐periodic composite material

The aim of this paper is to propose a certain new approach to the formulation of both discrete and continuum models for the analysis of dynamic problems in elastic composite solids with a periodic microstructure. The proposed approach is based on a periodic simplicial division of the unit cell and on the assumption of a uniform strain in every simplex. The main feature of the obtained discrete model is the finite‐difference form of governing equations. By applying smoothing operation the continuum models are derived directly from the discrete ones.     

Well‐posedness for the FENE dumbbell model of polymeric flows

We prove local and global well‐posedness for the FENE dumbbell model for a very general class of potentials. Indeed, in prior local or global well‐posedness results, conditions on the strength of the singularity (or on the parameter b) were made. Here we give a proof in the general case. We also prove global existence results if the data is small or if we restrict to the co‐rotational model in dimension 2. © 2008 Wiley Periodicals, Inc.     

Simulation of flow and electrokinetic transport in a micro‐analysis‐system

The aim of this work is the investigation of the flow and electrokinetic transport in a micro‐analysis‐system. A mathematical model for electroosmotic and electrophoretic phenomena is introduced in order to perform two‐dimensional, time‐dependent Finite Element simulations for an existing device. The model includes the feature of the flow field, the mass transfer, the external applied electric field and the involved chemistry. The results of the simulation are validated against experimental data and show good agreement. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Global solutions for micro-macro models of polymeric fluids

We provide a new proof for the global well-posedness of systems coupling fluids and polymers in two space dimensions. Compared to the well-known existing method based on the losing a priori estimates, our method is more direct and much simpler. The co-rotational FENE dumbbell model and the coupling Smoluchowski and Navier-Stokes equations are studied as examples to illustrate our main ideas.     

On non‐Newtonian incompressible fluids with phase transitions

A modified model for a binary fluid is analysed mathematically. The governing equations of the motion consists of a Cahn–Hilliard equation coupled with a system describing a class of non‐Newtonian incompressible fluid with p‐structure. The existence of weak solutions for the evolution problems is shown for the space dimension d=2 with p? 2 and for d=3 with p? 11/5. The existence of measure‐valued solutions is obtained for d=3 in the case 2? p< 11/5. Similar existence results are obtained for the case of nondifferentiable free energy, corresponding to the density constraint |ψ| ? 1. We also give regularity and uniqueness results for the solutions and characterize stable stationary solutions. Copyright © 2006 John Wiley & Sons, Ltd.     

Convergence to equilibrium for a fully hyperbolic phase‐field model with Cattaneo heat flux law

We consider a fully hyperbolic phase‐field model in this paper. Our model consists of a damped hyperbolic equation of second order with respect to the phase function χ(t) , which is coupled with a hyperbolic system of first order with respect to the relative temperature θ(t) and the heat flux vector q (t). We prove the well‐posedness of this system subject to homogeneous Neumann boundary condition and no‐heat flux boundary condition. Then, we show that this dynamical system is a dissipative one. Finally, using the celebrated ?ojasiewicz–Simon inequality and by constructing an auxiliary functional, we prove that the solution of this problem converges to an equilibrium as time goes to infinity. We also obtain an estimate of the decay rate to equilibrium. Copyright © 2008 John Wiley & Sons, Ltd.     

On a first‐order hyperbolic coagulation model

We consider a first‐order hyperbolic initial–boundary value problem that models coagulation of particles. We establish a comparison principle and construct monotone sequences to show the existence and uniqueness of the solution. Copyright © 2003 John Wiley & Sons, Ltd.     

Subharmonics near an equilibrium point for hamitonian systems

This work is devoted to the study of subharmonic solutions near an equilibrium for certain Hamiltonian systems. We impose a weak condition on the Floquet exponents of the linearized system, and a superquadratic condition on the higher order term. This last condition is reduced to the center manifold. This research was supported in part by the Air Force Office of Scientific Research under Grant AFOSR-87-0202.     

On the hyperbolic system of a mixture of Eulerian fluids: a comparison between single‐ and multi‐temperature models

The first rational model of homogeneous mixtures of fluids was proposed by Truesdell in the context of rational thermodynamics. Afterwards, two different theories were developed: one with a single‐temperature (ST) field of the mixture and the other one with several temperatures. The two systems are from the mathematical point of view completely different and the relationship between their solutions was not clarified. In this paper, the hyperbolic multi‐temperature (MT) system of a mixture of Eulerian fluids will be explained and it will be shown that the corresponding single‐temperature differential system is a principal subsystem of the MT one. As a consequence, the subcharacteristic conditions for characteristic speeds hold and this gives an upper‐bound esteem for pulse speeds in an ST model. Global behaviour of smooth solutions for large time for both systems will also be discussed through the application of the Shizuta–Kawashima condition. Finally, as an application, the particular case of a binary mixture is considered. Copyright © 2006 John Wiley & Sons, Ltd.     

On the blow‐up criterion of the periodic incompressible fluids

Kato, Ponce, Beale and Majda prove the existence and uniqueness of maximal solution of Euler and Navier–Stokes equations and some blow‐up criterion. In the periodic case, we establish that if the maximum time T* is finite, then the growth of is at least of the order of (T* ? t)^{?2m / 5}. Copyright © 2012 John Wiley & Sons, Ltd.     

On the small time asymptotics of stochastic non‐Newtonian fluids

In this paper, a small‐time large deviation principle for the stochastic non‐Newtonian fluids driven by multiplicative noise is proved. Copyright © 2016 John Wiley & Sons, Ltd.     

On measure‐valued solutions to a two‐dimensional gravity‐driven avalanche flow model

This paper concerns measure‐valued solutions for the two‐dimensional granular avalanche flow model introduced by Savage and Hutter. The system is similar to the isentropic compressible Euler equations, except for a Coulomb–Mohr friction law in the source term. We will partially follow the study of measure‐valued solutions given by DiPerna and Majda. However, due to the multi‐valued nature of the friction law, new more sensitive measures must be introduced. The main idea is to consider the class of x‐dependent maximal monotone graphs of non‐single‐valued operators and their relation with 1‐Lipschitz, Carathéodory functions. This relation allows to introduce generalized Young measures for x‐dependent maximal monotone graph. Copyright © 2005 John Wiley & Sons, Ltd.