Unique solvability of equations of motion for ferrofluids

We study the differential system governing the flow of an incompressible ferrofluid under the action of a magnetic field. The system is a combination of the Navier-Stokes equations, the angular momentum equation, the magnetization equation and the magnetostatic equations. No regularizing term is added to the magnetization equation. We prove the local-in-time existence of the unique strong solution to the system posed in a bounded domain of R³ and equipped with initial and boundary conditions.     

A boundary integral equation procedure for the mixed boundary value problem of the vector helmholtz equation

A boundary integral method is developed for the mixed boundary value problem for the vector Helmholtz equation in R³. The obtained boundary integral equations for the unknown Cauchy data build a strong elliptic system of pseudodifferential equations which can therefore be used for numerical computations using Galerkin's procedure. We show existence, uniqueness and regularity of the solution of the integral equations. Especially we give the local "edge" behavior of the solution near the submanifold which divides the Dirichlet boundary from the Neumann boundary     

Global well-posedness and existence of uniform attractor for magnetohydrodynamic equations

We study the global well-posedness and existence of uniform attractor for magnetohydrodynamic (MHD) equations. The hydrodynamic system consists of the Navier–Stokes equations for the fluid velocity and pressure coupled with a reduced from of the Maxwell equations for the magnetic field. The fluid velocity is assumed to satisfy a no-slip boundary condition, while the magnetic field is subject to a time-dependent Dirichlet boundary condition. We first establish the global existence of weak and strong solutions to Equations (1.1)-(1.4). And at this stage, we further derive the existence of a uniform attractor for Equations (1.1)-(1.4).     

Numerical simulation of thermal problems coupled with magnetohydrodynamic effects in aluminium cell

A system of partial differential equations describing the thermal behavior of aluminium cell coupled with magnetohydrodynamic effects is numerically solved. The thermal model is considered as a two-phases Stefan problem which consists of a non-linear convection–diffusion heat equation with Joule effect as a source. The magnetohydrodynamic fields are governed by Navier–Stokes and by static Maxwell equations. A pseudo-evolutionary scheme (Chernoff) is used to obtain the stationary solution giving the temperature and the frozen layer profile for the simulation of the ledges in the cell. A numerical approximation using a finite element method is formulated to obtain the fluid velocity, electrical potential, magnetic induction and temperature. An iterative algorithm and 3-D numerical results are presented.     

Stability of solutions of an equation arising in hydromechanics

We study the strong stability of the equation describing small oscillations of an elastic pipe conveying an ideal fluid. We prove the existence and uniqueness theorem for the generalized solution and find sufficient conditions for strong stability in the case of a pulsing fluid. Translated fromMatematicheskie Zametki, Vol. 67, No. 1, pp. 15–24, January, 2000.     

Existence of local strong solutions to fluid–beam and fluid–rod interaction systems

We study an unsteady nonlinear fluid–structure interaction problem. We consider a Newtonian incompressible two-dimensional flow described by the Navier–Stokes equations set in an unknown domain depending on the displacement of a structure, which itself satisfies a linear wave equation or a linear beam equation. The fluid and the structure systems are coupled via interface conditions prescribing the continuity of the velocities at the fluid–structure interface and the action-reaction principle. Considering three different structure models, we prove existence of a unique local-in-time strong solution, for which there is no gap between the regularity of the initial data and the regularity of the solution enabling to obtain a blow up alternative. In the case of a damped beam this is an alternative proof (and a generalization to non zero initial displacement) of the result that can be found in [20]. In the case of the wave equation or a beam equation with inertia of rotation, this is, to our knowledge the first result of existence of strong solutions for which no viscosity is added. The key points consist in studying the coupled system without decoupling the fluid from the structure and to use the fluid dissipation to control, in appropriate function spaces, the structure velocity.     

The effect of magnetic field on dynamics of gas bubbles in visco-elastic fluids

The effect of magnetic field on nonlinear oscillations of a spherical, acoustically forced gas bubble in nonlinear visco-elastic media is studied. The constitutive equation UCM used for modeling the rheological behaviors of the fluid. By starting from the momentum equations for bubbles considering the magnetic force and considering some simplifying assumptions, the modified bubble dynamics equation (the modified Rayleigh–Plesset equation) has been achieved. Assumptions concerning the trace of the stress tensor are addressed in light of the incorporation of visco-elastic constitutive equations into modified bubble dynamics equations. The governing equations are non-dimesionalized and numerically solved by using 4th order Runge–Kutta method. The accuracy of the calculations and the formulation is compared with the previous works done for models without the presence of magnetic field. Furthermore, the bubble size variations due to acoustic motivations and stress tensor components variations in presence of different magnitudes of magnetic fields are studied. Also, the bubble size dependence on fluid conductivity variations is declared. The relevance and importance of this approach to biomedical ultrasound applications are highlighted. Preliminary results indicate that magnetic field may be an important consideration for the risk assessment of potential cavitations and also it could be possible to damp the bubble oscillations by using magnetic fields or in opposite case amplify the oscillations which could result in higher level light emissions in sonoluminescence approach.     

Weakly dissipative solutions and weak–strong uniqueness for the Navier–Stokes–Smoluchowski system

This article deals with a fluid–particle interaction model for the evolution of particles dispersed in a fluid. The fluid flow is governed by the Navier–Stokes equations for a compressible fluid while the evolution of the particle densities is given by the Smoluchowski equation. The coupling between the dispersed and dense phases is obtained through the drag forces that the fluid and the particles exert mutually. The existence of weakly dissipative solutions is established under reasonable physical assumptions on the initial data, the physical domain, and the external potential. Furthermore, a weak–strong uniqueness result is established via the relative entropy method yielding that a weakly dissipative solution agrees with a classical solution with the same initial data when such a classical solution exists.     

Strong solutions to the equations of a ferrofluid flow model

We study the differential system introduced by M.I. Shliomis to describe the motion of a ferrofluid driven by an external magnetic field. The system is a combination of the Navier-Stokes equations, the magnetization equation and the magnetostatic equations. No regularizing term is added to the magnetization equation. We prove the local-in-time existence of strong solutions to the system.     

On the “Destruction” of Solutions of Nonlinear Wave Equations of Sobolev Type with Cubic Sources

We consider model three-dimensional wave nonlinear equations of Sobolev type with cubic sources, and foremost, model three-dimensional equations of Benjamin-Bona-Mahony and Rosenau types with model cubic sources. An essentially three-dimensional nonlinear equation of spin waves with cubic source is also studied. For these equations, we investigate the first initial boundary-value problem in a bounded domain with smooth boundary. We prove local solvability in the strong generalized sense and, for an equation of Benjamin-Bona-Mahony type with source, we prove the unique solvability of a “weakened” solution. We obtain sufficient conditions for the “destruction” of the solutions of the problems under consideration. These conditions have the sense of a “large” value of the initial perturbation in the norms of certain Banach spaces. Finally, for an equation of Benjamin-Bona-Mahony type, we prove the “failure” of a “weakened” solution in finite time.     

Well-posedness and finite element approximation of time dependent generalized bioconvective flow

We consider the finite element approximation of a time dependent generalized bioconvective flow. The underlying system of partial differential equations consists of incompressible Navier–Stokes type convection equations coupled with an equation describing the transport of micro-organisms. The viscosity of the fluid is assumed to be a function of the concentration of the micro-organisms. We show the existence and uniqueness of the weak solution of the system in two dimensions and construct numerical approximations based on the finite element method, for which we obtain error estimates. In addition, we conduct several numerical experiments to demonstrate the accuracy of the numerical method and perform simulations of the bioconvection pattern formations based on realistic model parameters to demonstrate the validity of the proposed numerical algorithm.     

On a decoupled linear FEM integrator for eddy-current-LLG

We propose a numerical integrator for the coupled system of the eddy-current equation with the nonlinear Landau–Lifshitz–Gilbert equation. The considered effective field contains a general field contribution, and we particularly cover exchange, anisotropy, applied field and magnetic field (stemming from the eddy-current equation). Even though the considered problem is nonlinear, our scheme requires only the solution of two linear systems per time-step. Moreover, our algorithm decouples both equations so that in each time-step, one linear system is solved for the magnetization, and afterwards one linear system is solved for the magnetic field. Unconditional convergence – at least of a subsequence – towards a weak solution is proved, and our analysis even provides existence of such weak solutions. Numerical experiments with micromagnetic benchmark problems underline the performance and the stability of the proposed algorithm.     

Vanishing viscosity limit of short wave–long wave interactions in planar magnetohydrodynamics

We study several mathematical aspects of a system of equations modelling the interaction between short waves, described by a nonlinear Schrödinger equation, and long waves, described by the equations of magnetohydrodynamics for a compressible, heat conductive fluid. The system in question models an aurora-type phenomenon, where a short wave propagates along the streamlines of a magnetohydrodynamic medium. We focus on the one dimensional (planar) version of the model and address the problem of well posedness as well as convergence of the sequence of solutions as the bulk viscosity tends to zero together with some other interaction parameters, to a solution of the limit decoupled system involving the compressible Euler equations and a nonlinear Schrödinger equation. The vanishing viscosity limit serves to justify the SW–LW interactions in the limit equations as, in this setting, the SW–LW interactions cannot be defined in a straightforward way, due to the possible occurrence of vacuum.     

A stabilization algorithm of the Navier–Stokes equations based on algebraic Bernoulli equation

In this paper, we consider the stabilization of the nonstationary incompressible Navier–Stokes equations around a stationary solution by a boundary linear feedback control. The feedback operator is obtained from the solution of the algebraic Bernoulli equation associated with the penalized linearized Navier–Stokes equations around an unstable stationary solution and is used to locally stabilize the original nonlinear equations. We give the explicit factorized form of the stabilizing solution of the algebraic Bernoulli equation. The numerical effectiveness of this approach is demonstrated by stabilizing the vortex shedding behind a circular obstacle. Copyright © 2011 John Wiley & Sons, Ltd.     

Global analysis for strong solutions to the equations of a ferrofluid flow model

In this paper we are concerned with the differential system proposed by Shliomis to describe the motion of an incompressible ferrofluid submitted to an external magnetic field. The system consists of the Navier-Stokes equations, the magnetization equations and the magnetostatic equations. No regularizing term is added to the magnetization equations. We prove the local existence of unique strong solution for the Cauchy problem and establish a finite time blow-up criterion of strong solutions. Under the smallness assumption of the initial data and the external magnetic field, we prove the global existence of strong solutions and derive a decay rate of such small solutions in L²-norm.     

Conditions for the existence of a global strong solution to a class of nonlinear evolution equations in a Hilbert space

We study a nonlinear operator differential equation in a Hilbert space. This equation represents an abstract model for the system of Navier-Stokes equations. The main result consists in proving the existence of a strong solution to this equation under the condition that a certain other system of equations (related to the original equation) has only the zero solution.     

On Schwarz Alternating Methods for the Subsonic Full Potential Equation

The Schwarz alternating method can be used to solve elliptic boundary value problems on domains which consist of two or more overlapping subdomains. The solution is approximated by an infinite sequence of functions which results from solving a sequence of elliptic boundary value problems in each of the subdomains. The full potential equation is derived from the Navier–Stokes equations assuming the fluid is compressible, inviscid, irrotational and isentropic. It is being used by the aircraft industry to model flow over an airfoil or even an entire aircraft. This paper shows that the additive and multiplicative versions of the Schwarz alternating method, when applied to the full potential equation in three dimensions, converge to the true solution geometrically. The assumptions are that the initial guess and the true solution are everywhere subsonic. We use the convergence proof by Tai and Xu and modify it for certain closed convex subsets.     

Global existence of weak and strong solutions to Cucker–Smale–Navier–Stokes equations in R2

We study the existence theory for the Cucker–Smale–Navier–Stokes (in short, CS–NS) equations in two dimensions. The CS–NS equations consist of Cucker–Smale flocking particles described by a Vlasov-type equation and incompressible Navier–Stokes equations. The interaction between the particles and fluid is governed by a drag force. In this study, we show the global existence of weak solutions for this system. We also prove the global existence and uniqueness of strong solutions. In contrast with the results of Bae et al. (2014) on the CS–NS equations considered in three dimensions, we do not require any smallness assumption on the initial data.     

Fully discrete T‐ψ finite element method to solve a nonlinear induction hardening problem

We study an induction hardening model described by Maxwell's equations coupled with a heat equation. The magnetic induction field is assumed a nonlinear constitutional relation and the electric conductivity is temperature‐dependent. The T ‐ψ method is to transform Maxwell's equations to the vector–scalar potential formulations and to solve the potentials by means of the finite element method. In this article, we present a fully discrete T ‐ψ finite element scheme for this nonlinear coupled problem and discuss its solvability. We prove that the discrete solution converges to a weak solution of the continuous problem. Finally, we conclude with two numerical experiments for the coupled system.     

An iterative splitting approach for linear integro-differential equations

The motivation for this paper is to solve a model based on the dynamics of electrons in a plasma using a simplified Boltzmann equation. Such problems have arisen in active plasma resonance spectroscopy, which is used for plasma diagnostic techniques; see Braithwaite and Franklin (2009) [1]. We propose a modified iterative splitting approach to solve the Boltzmann equations as a system of integro-differential equations. To enable solution by fast and iterative computations, we first transform the integro-differential equations into second order differential equations. Second, we split each second order differential equations into two first order differential equations via a splitting approach. We carry out an error analysis of the higher order iterative approach. Numerical experiments with a simplified Boltzmann equation will be discussed, along with the benefits of computing with this splitting approach.