A global attractor for a fluid–plate interaction model accounting only for longitudinal deformations of the plate

We study asymptotic dynamics of a coupled system consisting of linearized 3D Navier–Stokes equations in a bounded domain and the classical (nonlinear) elastic plate equation for in‐plane motions on a flexible flat part of the boundary. The main novelty of the model is the assumption that the transversal displacements of the plate are negligible relative to in‐plane displacements. These kinds of models arise in the study of blood flows in large arteries. Our main result states the existence of a compact global attractor of finite dimension. Under some conditions this attractor is an exponentially attracting single point. We also show that the corresponding linearized system generates an exponentially stable C₀‐semigroup. We do not assume any kind of mechanical damping in the plate component. Thus our results mean that dissipation of the energy in the fluid because of viscosity is sufficient to stabilize the system. Copyright © 2011 John Wiley & Sons, Ltd.     

A linear,decoupled fractional time‐stepping method for the nonlinear fluid–fluid interaction

In this paper, a linear decoupled fractional time stepping method is proposed and developed for the nonlinear fluid–fluid interaction governed by the two Navier–Stokes equations. Partitioned time stepping method is applied to two‐physics problems with stiffness of the coupling terms being treated explicitly and is also unconditionally stable. As for each fluid, the velocity and pressure are respectively determined by just solving one vector‐valued quasi‐elliptic equation and the Possion equation with homogeneous Neumann boundary condition per time step. Therefore, the cost of the fluid–fluid interaction is dominant to solve four simple linear equations, which greatly reduces the computational cost of the whole system. The method exploits properties of the fluid–fluid system to establish its stability and convergence with the same results as the standard scheme. Finally, numerical experiments are presented to show the performance of the proposed method.     

On the longtime behavior of a 2D hydrodynamic model for chemically reacting binary fluid mixtures

Two dimensional diffuse interface model for a chemically reacting incompressible binary fluid in a bounded domain is considered. The corresponding evolution system consists of the Navier–Stokes equations for the (averaged) fluid velocity that are nonlinearly coupled with a convective Cahn–Hilliard–Oono type equation for the difference ψ of two fluid concentrations. The effects of a (reversible) chemical reaction is represented in the latter equation by an additional term of the form ε(ψ ? c₀), ε > 0. Here, c₀ is the stationary spatial average of ψ, provided that, for example, no‐slip and no‐flux boundary conditions are considered. The mass is not necessarily conserved unless the spatial average of the initial datum for ψ coincides with c₀. When ε = 0 (i.e., no chemical reaction), the model reduces to the well‐known Cahn–Hilliard–Navier–Stokes system, which has been investigated by several authors. Here, we want to show that the global dynamic behavior of the system is robust with respect to ε. More precisely, we construct a family of exponential attractors, which is continuous with respect to ε. Copyright © 2013 John Wiley & Sons, Ltd.     

On the global existence of weak solution for a multiphasic incompressible fluid model with Korteweg stress

In this paper, we study a multiphasic incompressible fluid model, called the Kazhikhov–Smagulov model, with a particular viscous stress tensor, introduced by Bresch and co‐authors, and a specific diffusive interface term introduced for the first time by Korteweg in 1901. We prove that this model is globally well posed in a 3D bounded domain. Copyright © 2016 John Wiley & Sons, Ltd.     

Asymptotic behaviour of viscoelastic composites with almost periodic microstructures

In this paper, we study the acoustic properties of porous media saturated by an incompressible viscoelastic fluid. The model considered here consists of a linear deformable porous skeleton having memory that is surrounded by a viscoelastic Oldroyd fluid. Assuming the microstructures to be almost periodically distributed and under the almost periodicity hypothesis on the coefficients of the governing equations, we determine the macroscopic equivalent medium. To achieve our goal, we use some very recent tools about the sigma convergence of convolution sequences. Copyright © 2017 John Wiley & Sons, Ltd.     

A new approach to the stabilization of a fluid–structure interaction model

This article is a continuation of our work on a linear fluid–structure interaction model [Grobbelaar-Van Dalsen, On a fluid–structure model in which the dynamics of the structure involves the shear stress due to the fluid, J. Math. Fluid Mech. 10(3) (2008), pp. 388–401; Grobbelaar-Van Dalsen, Strong stability for a fluid––structure model, Math. Methods Appl. Sci., 32(2009) pp. 1452–1466]. The model describes the interaction between a 3-D incompressible fluid and a 2-D plate, the interface, which coincides with a flat flexible part of the surface of the vessel containing the fluid. The mathematical model comprises the Stokes equations and the equations for the longitudinal deflections of the plate with the inclusion of the shear stress that the fluid exerts on the plate. A dissipative damping mechanism of Kelvin–Voigt type is applied to the interior of the plate. While our earlier work shows that weak solutions in a space of finite energy are strongly asymptotically stable under no-slip transmission conditions at the interface with uniform exponential stability only attainable under an additional domination condition, the present research is directed at achieving uniform exponential stability of weak solutions without imposing the domination condition. Using energy methods we establish uniform exponential decay under a modified transmission condition at the interface. This condition entails that the fluid velocity at the interface is coupled to a linear combination of the plate velocity and displacement.     

Dynamic boundary stabilization of a Reissner–Mindlin plate with Timoshenko beam

This paper is concerned with well‐posedness results for a mathematical model for the transversal vibrations of a two‐dimensional hybrid elastic structure consisting of a rectangular Reissner–Mindlin plate with a Timoshenko beam attached to its free edge. The model incorporates linear dynamic feedback controls along the interface between the plate and the beam. Classical semigroup methods are employed to show the unique solvability of the coupled initial‐boundary‐value problem. We also show that the energy associated with the system exhibits the property of strong stability. Copyright © 2004 John Wiley & Sons, Ltd.     

Global weak solutions to the Cauchy problem of compressible Navier–Stokes–Vlasov–Fokker–Planck equations

A fluid–particles system of the compressible Navier‐Stokes equations and Vlasov‐Fokker‐Planck equation (including the case of Vlasov equation) in three‐dimensional space is considered in this paper. The coupling arises from a drag force exerted by the fluid onto the particles. We study a Cauchy problem with large data, and establish the existence of global weak solutions through an approximation scheme, energy estimates, and weak convergence. Copyright © 2016 John Wiley & Sons, Ltd.     

The role of magnetic fields in the strong stabilization of a hybrid magneto‐elastic structure

In this paper, we are concerned with a model for the magneto–elastic interactions of a three‐dimensional elastic body and a two‐dimensional flexible plate, which is attached to the flat flexible part of the surface of the body. Both the solid body and the plate are permeated by magnetic fields. The mathematical model is analyzed from the point of view of existence and uniqueness and stabilization.It turns out that, in the presence of the magnetic fields in the solid and the plate, strong stabilization can be achieved under viscous damping in the plate in one direction that is determined by the nature of the primary magnetic fields in the body and the plate. Copyright © 2011 John Wiley & Sons, Ltd.     

Automated multi‐level sub‐structuring for fluid–solid interaction problems

The automated multi‐level sub‐structuring (AMLS) method is a powerful technique to determine a large number of eigenpairs with moderate accuracy of huge symmetric and definite eigenvalue problems in structural analysis. This paper is concerned with an adapted version of AMLS for eigenfrequency analysis of fluid–solid interaction systems. Although fluid–solid vibrations are governed by an unsymmetric eigenproblem, the modified AMLS method needs approximately the same computational effort. An error bound related to the eigenvalue approximations is proved. Copyright © 2010 John Wiley & Sons, Ltd.     

Entropy dissipative one‐leg multistep time approximations of nonlinear diffusive equations

New one‐leg multistep time discretizations of nonlinear evolution equations are investigated. The main features of the scheme are the preservation of the non‐negativity and the entropy dissipation structure of the diffusive equations. The key ideas are to combine Dahlquist's G‐stability theory with entropy dissipation methods and to introduce a nonlinear transformation of variables, which provides a quadratic structure in the equations. It is shown that G‐stability of the one‐leg scheme is sufficient to derive discrete entropy dissipation estimates. The general result is applied to a cross‐diffusion system from population dynamics and a nonlinear fourth‐order quantum diffusion model, for which the existence of semidiscrete weak solutions is proved. Under some assumptions on the operator of the evolution equation, the second‐order convergence of solutions is shown. Moreover, some numerical experiments for the population model are presented, which underline the theoretical results. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1119–1149, 2015     

A remark on the decay of solutions to the 3‐D Navier–Stokes equations

In this paper we derive a decay rate of the L²‐norm of the solution to the 3‐D Navier–Stokes equations. Although the result which is proved by Fourier splitting method is well known, our method is new, concise and direct. Moreover, it turns out that the new method established here has a wide application on other equations. Copyright © 2007 John Wiley & Sons, Ltd.     

On the stabilizability of a structural acoustic model which incorporates shear effects in the structural component

In this paper we consider a linear three-dimensional structural acoustic model which takes account of displacement, rotational inertia and shear effects in the flat flexible structural component of the model. Thus the deflections of the structural component of the structure are governed by the Reissner–Mindlin plate equations. We show strong stabilization of the coupled model without incorporating viscous or boundary damping in the equations for the gas dynamics and without imposing geometric conditions. It turns out that damping is needed in the interior of the plate, to which end Kelvin–Voigt damping is introduced in the plate equations. As our main tool we use a resolvent criterion for strong stability due to Tomilov.     

Vibration of a Reissner–Mindlin–Timoshenko plate–beam system

In this paper, we consider a plate–beam system in which the Reissner–Mindlin plate model is combined with the Timoshenko beam model. Natural frequencies and vibration modes for the system are calculated using the finite element method. The interface conditions at the contact between the plate and beams are discussed in some detail. The impact of regularity on the enforcement of certain interface conditions is an important feature of the paper.     

Time relaxation algorithm for flow ensembles

Modeling for fast calculation of fluid flow ensembles based on time relaxation regularization is studied herein. At each time step, the proposed model requires the storage of a single coefficient matrix with multiple right‐hands sides, corresponding to each ensemble member. The time relaxation regularization penalizes the deviation of the fluctuations from the ensemble average. The algorithm is shown to be stable under a time step restriction, which holds provided fluctuations are small enough. Also, the numerical tests show that a grad‐div stabilization weakened the condition considerably. Finite element convergence results for the time relaxation ensemble algorithm are studied too. Then, 2D and 3D numerical experiments that support the theoretical results are presented. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 757–777, 2016     

Finite element method to fluid–solid interaction problems with unbounded periodic interfaces

Consider a time‐harmonic acoustic plane wave incident onto a doubly periodic (biperiodic) surface from above. The medium above the surface is supposed to be filled with a homogeneous compressible inviscid fluid of constant mass density, whereas the region below is occupied by an isotropic and linearly elastic solid body characterized by its Lamé constants. This article is concerned with a variational approach to the fluid–solid interaction problems with unbounded biperiodic Lipschitz interfaces between the domains of the acoustic and elastic waves. The existence of quasiperiodic solutions in Sobolev spaces is established at arbitrary frequency of incidence, while uniqueness is proved only for small frequencies or for all frequencies excluding a discrete set. A finite element scheme coupled with Dirichlet‐to‐Neumann mappings is proposed and the convergence analysis is performed. The Dirichlet‐to‐Neumann mappings are approximated by truncated Rayleigh series expansions. Finally, numerical tests in 2D are presented to confirm the convergence of solutions and the energy balance formula. In particular, the frequency spectrum of normally reflected signals is plotted for water–brass and water–brass–water interfaces. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 5–35, 2016     

Uncoupling evolutionary groundwater‐surface water flows using the Crank–Nicolson Leapfrog method

Consider an incompressible fluid in a region Ω_f flowing both ways across an interface into a porous media domain Ω_p saturated with the same fluid. The physical processes in each domain have been well studied and are described by the Stokes equations in the fluid region and the Darcy equations in the porous media region. Taking the interfacial conditions into account produces a system with an exactly skew symmetric coupling. Spatial discretization by finite element method and time discretization by Crank–Nicolson LeapFrog give a second‐order partitioned method requiring only one Stokes and one Darcy subphysics and subdomain solver per time step for the fully evolutionary Stokes‐Darcy problem. Analysis of this method leads to a time step condition sufficient for stability and convergence. Numerical tests verify predicted rates of convergence; however, stability tests reveal the problem of growth of numerical noise in unstable modes in some cases. In such instances, the addition of time filters adds stability. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

A priori error estimates for a coupled finite element method and mixed finite element method for a fluid-solid interaction problem

This paper presents a heterogeneous finite element method fora fluid–solid interaction problem. The method, which combinesa standard finite element discretization in the fluid regionand a mixed finite element discretization in the solid region,allows the use of different meshes in fluid and solid regions.Both semi-discrete and fully discrete approximations are formulatedand analysed. Optimal order a priori error estimates in theenergy norm are shown. The main difficulty in the analysis iscaused by the two interface conditions which describe the interactionbetween the fluid and the solid. This is overcome by explicitlybuilding one of the interface conditions into the finite elementspaces. Iterative substructuring algorithms are also proposedfor effectively solving the discrete finite element equations.     

An iterative solver for the Oseen problem and numerical solution of incompressible Navier–Stokes equations

Incompressible unsteady Navier–Stokes equations in pressure–velocity variables are considered. By use of the implicit and semi‐implicit schemes presented the resulting system of linear equations can be solved by a robust and efficient iterative method. This iterative solver is constructed for the system of linearized Navier–Stokes equations. The Schur complement technique is used. We present a new approach of building a non‐symmetric preconditioner to solve a non‐symmetric problem of convection–diffusion and saddle‐point type. It is shown that handling the differential equations properly results in constructing efficient solvers for the corresponding finite linear algebra systems. The method has good performance for various ranges of viscosity and can be used both for 2D and 3D problems. The analysis of the method is still partly heuristic, however, the mathematically rigorous results are proved for certain cases. The proof is based on energy estimates and basic properties of the underlying partial differential equations. Numerical results are provided. Additionally, a multigrid method for the auxiliary convection–diffusion problem is briefly discussed. Copyright © 1999 John Wiley & Sons, Ltd.     

Solvability,asymptotic analysis and regularity results for a mixed type interaction problem of acoustic waves and piezoelectric structures

In the paper, we investigate the mixed type transmission problem arising in the model of fluid–solid acoustic interaction when a piezoceramic elastic body (Ω⁺) is embedded in an unbounded fluid domain (Ω^?). The corresponding physical process is described by the boundary‐transmission problem for second‐order partial differential equations. In particular, in the bounded domain Ω⁺, we have a 4×4 dimensional matrix strongly elliptic second‐order partial differential equation, while in the unbounded complement domain Ω^?, we have a scalar Helmholtz equation describing acoustic wave propagation. The physical kinematic and dynamic relations mathematically are described by appropriate boundary and transmission conditions. With the help of the potential method and theory of pseudodifferential equations based on the Wiener–Hopf factorization method, the uniqueness and existence theorems are proved in Sobolev–Slobodetskii spaces. We derive asymptotic expansion of solutions, and on the basis of asymptotic analysis, we establish optimal Hölder smoothness results for solutions. Copyright © 2017 John Wiley & Sons, Ltd.