Existence and regularity of solutions of a phase field model for solidification with convection of pure materials in two dimensions

CNPQ - CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICOWe study the existence and regularity of weak solutions of a phase field type model for pure material solidification in presence of natural convection. We assume that the non-stationary solidification process occurs in a two dimensional bounded domain. The governing equations of the model are the phase field equation coupled with a nonlinear heat equation and a modified Navier-Stokes equation. These equations include buoyancy forces modelled by Boussinesq approximation and a Carman-Koseny term to model the flow in mushy regions. Since these modified Navier-Stokes equations only hold in the non-solid regions, which are not known a priori, we have a free boundary-value problem.We study the existence and regularity of weak solutions of a phase field type model for pure material solidification in presence of natural convection. We assume that the non-stationary solidification process occurs in a two dimensional bounded domain. The governing equations of the model are the phase field equation coupled with a nonlinear heat equation and a modified Navier-Stokes equation. These equations include buoyancy forces modelled by Boussinesq approximation and a Carman-Koseny term to model the flow in mushy regions. Since these modified Navier-Stokes equations only hold in the non-solid regions, which are not known a priori, we have a free boundary-value problem2003109125CNPQ - CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICOCNPQ - CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICO300513/87-9Anderson, D.M., McFadden, G.B., Wheeler, A.A., Diffusive-interface methods in fluid mechanics (1998) Annual Review of Fluid Mechanics, 30, pp. 139-165. , Palo Alto, CABeckermann, C., Diepers, H.-J., Steinbach, I., Karma, A., Tong, X., Modeling melt convection in phase-field simulations of solidification (1999) Journal of Computational Physics, 154, pp. 468-496Blanc, Ph., Gasser, L., Rappaz, J., Existence for a stationary model of binary alloy solidification (1995) Math. Modelling. Num. Anal., 29 (6), pp. 687-699Caginalp, G., Jones, J., A derivation and analysis of phase field models of thermal alloys (1995) Annal. 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Ações exoticas de grupos ciclicos sobre espaços euclidianos

Orientador: Hugo H. TorrianiDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica,Estatistica e Ciencia da ComputaçãoResumo: Não informado.Abstract: Not informed.MestradoMestre em Matemátic

Semi-Galerkin approximation and strong solutions to the equations of the nonhomogeneous asymmetric fluids

This paper analyzes an initial/boundary value problem for a system of equations modelling the nonstationary flow of a nonhomogeneous incompressible asymmetric (polar) fluid. Under conditions similar to those usually imposed to the nonhomogeneous 3D Navier-Stokes equations, by using a spectral semi-Galerkin method, we prove the existence of a local in time strong solution. We also prove the uniqueness of the strong solution and some global existence results. Several estimates for the solutions and their approximations are given. These can be used to find useful error bounds of the Galerkin approximations.Dans ce papier, on analyse un problème de valeurs initiales et valeurs aux limites pour un système d’équations aux dérivées partielles qui modélise le flux instationnaire d’un fluide asymmétrique incompressible non homogène. Sous des conditions similaires aux conditions usuellement imposées aux équations tridimensionelles de Navier-Stokes non homogènes, à l’aide d’une méthode de type semi-Galerkin, nous démontrons l’éxistence d’une solution forte locale en temps. On établit aussi l’unicité de solution forte et quelques résultats d’éxistence globale. Tous ces résultats reposent sur des estimations appropriées pour les solutions et leurs approximations qui permettent d’ailleurs d´eduire des estimations de l’erreur.Conselho Nacional de Desenvolvimento Científico e Tecnológico (Brasil)Fundação de Amparo à Pesquisa do Estado de São PauloDirección General de Enseñanza Superio

An optimal control problem for a generalized Boussinesq model: The time dependent case

We consider an optimal control problem governed by a system of nonlinear partial differential equations modelling viscous incompressible flows submitted to variations of temperature. We use a generalized Boussinesq approximation. We obtain the existence of the optimal control as well as first order optimality conditions of Pontryagin type by using the Dubovitskii-Milyutin formalism.Ministerio de Educación y CienciaConselho Nacional de Desenvolvimento Científico e TecnológicoFundação de Amparo à Pesquisa do Estado de São Paul

Existence and uniqueness of strong solutions for the incompressible micropolar fluid equations in domains of R3

We consider the initial boundary value problem for the system of equations describing the nonstationary flow of an incompressible micropolar fluid in a domain Ω of R3.Under hypotheses that are similar to the Navier-Stokes equations ones, by using an iterative scheme, we prove the existence and uniqueness of strong solution in Lp(Ω), for p > 3

Incompressible flow in granulated porous media with null initial velocity

In this work we study a Navier-Stokes type equation which models the flow of a viscous, homogeneous and incompressible fluid in a isotropic granular (non consolidated) porous media. Using point fixed type arguments we obtain conditions for existence of solution for the equation in Hölder's spaces.Neste trabalho nós estudamos uma equação do tipo Navier-Stokes que modela o fluxo de um fluido viscoso, homogêneo e incompressível num meio poroso granular (não consolidado). Usando argumentos do tipo ponto fixo obtemos condições para a existência de solução em espaços de Hölder

Some controllability results for linear viscoelastic fluids

We analyze the controllability properties of systems which provide a description, at first approximation, of a kind of viscoelastic fluid. We consider linear Maxwell fluids. First, we establish the large time approximate-finite dimensional controllability of the system, with distributed or boundary controls supported by arbitrary small sets. Then, we prove the large time exact controllability of fluids of the same kind with controls supported by suitable large sets. The proofs of these results rely on classical arguments. In particular, the approximate controllability result is implied by appropriate unique continuation properties, while exact controllability is a consequence of observability (inverse) inequalities. We also discuss questions concerning the controllability of viscoelastic fluids and some related open problems.Ministerio de Ciencia e Innovació

On an iterative method for approximate solutions of a generalized Boussinesq model

An iterative method is proposed for nding approximate solutions of an initial and boundary value problem for a nonstationary generalized Boussinesq model for thermally driven convection of fluids with temperature dependent viscosity and thermal conductivity. Under certain conditions, it is proved that such approximate solutions converge to a solution of the original problem; moreover, convergence-rate bounds for the constructed approximate solutions are also obtained

Efficient crack length measurement using A* shortest path methodology for a phase-field fracture framework

Accurately measuring a crack length is a crucial aspect of experimental fracture tests. In this work, we present an innovative application of the A* (A-star) shortest path methodology to track different shapes of cracks from numerical simulations. This approach is highly efficient, significantly improving the speed and accuracy of crack length measurements. Furthermore, we introduce a modified weight cost function that follows the crack path in the damage field, enhancing the accuracy of our method. The effectiveness of the proposed procedure is shown by fabricating damage fields with different geometry and good agreement when compared to the exact values. In addition, we evaluate a time-dependent crack propagation case, achieving high accuracy. We present all features and steps of the procedure to showcase its efficacy in accurately measuring the length of a crack path. Finally, we validate our method using a phase-field fracture framework and compare it with the compliance technique. The results show that the proposed method is applicable in finite element analyses with recovering accurate resultsThe authors would like to acknowledge CAPES (Coordenação de Aperfeiçoamento de Pessoal de Nível Superior) for the scholarship under grant 88882.435204/2019-01. J.A. Avila is a Serra Hunter Fellow and a CNPq fellow.Postprint (published version