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- Cauchy problem for the Navier–Stokes–Voigt model governing nonhomogeneous flowsPublication . Antontsev, S. N.; Oliveira, H. B. deThe Navier-Stokes-Voigt model that governs flows with non-constant density of incompressible fluids with elastic properties is considered in the whole space domain R-d and in the entire time interval. If d is an element of{2,3,4}, we prove the existence of weak solutions (velocity, density and pressure) to the associated Cauchy problem. We also analyse some issues of regularity of the weak solutions to the considered problem and the large time behavior in special unbounded domains.
- The classical Kelvin-Voigt problem for incompressible fluids with unknown non-constant density: existence, uniqueness and regularityPublication . Antontsev, S. N.; Oliveira, H. B. de; Khompysh, KhThe classical Kelvin-Voigt equations for incompressible fluids with non-constant density are investigated in this work. To the associated initial-value problem endowed with zero Dirichlet conditions on the assumed Lipschitz-continuous boundary, we prove the existence of weak solutions: velocity and density. We also prove the existence of a unique pressure. These results are valid for d is an element of {2, 3, 4}. In particular, if d is an element of {2, 3}, the regularity of the velocity and density is improved so that their uniqueness can be shown. In particular, the dependence of the regularity of the solutions on the smoothness of the given data of the problem is established.
- Kelvin-Voigt equations with anisotropic diffusion, relaxation and damping: Blow-up and large time behaviorPublication . Antontsev, S.; Oliveira, H. B. de; Khompysh, KhA nonlinear initial and boundary-value problem for the Kelvin-Voigt equations with anisotropic diffusion, relaxation and absorption/damping terms is considered in this work. The global and local unique solvability of the problem was established in (J. Math. Anal. Appl. 473(2) (2019) 1122-1154). In the present work, we show how all the anisotropic exponents of nonlinearity and all anisotropic coefficients should interact with the problem data for the solutions of this problem display exponential and polynomial time-decays. We also establish the conditions for the solutions of this problem to blow-up in a finite time in three different cases: problem without convection, full anisotropic problem, and the problem with isotropic relaxation.
- Continuous/discontinuous Galerkin approximations for a fourth-order nonlinear problemPublication . Lopes, N.D.; de Oliveira, H. B.We introduce a Continuous/Discontinuous Galerkin Finite Element Method (CDFEM) with interior penalty terms to solve a nonlinear fourth-order problem that appears in the analysis of the confinement of fluid flows governed by the Stokes system. For the associated continuous and discrete problems, we prove the existence and uniqueness of weak solutions. Consistency, stability and convergence of the method are shown analytically. To show the applicability and robustness of the numerical model, several test cases are performed.
- Kelvin-Voigt equations for incompressible and nonhomogeneous fluids with anisotropic viscosity, relaxation and dampingPublication . Antontsev, S. N.; de Oliveira, H. B.; Khompysh, Kh.In this work, we consider the nonlinear initial-boundary value problem posed by the Kelvin-Voigt equations for non-homogeneous and incompressible fluid flows with fully anisotropic diffusion, relaxation and damping. Moreover, we assume that the momentum equation is perturbed by a damping term which, depending on whether its signal is positive or negative, may account for the presence of a source or a sink within the system. In the particular case of considering this problem with a linear and isotropic relaxation term, we prove the existence of global and local weak solutions for the associated initial-boundary value problem supplemented with no-slip boundary conditions. When the damping term describes a sink, we establish the conditions for the polynomial time decay or for the exponential time decay of these solutions.
- Parabolic turbulence k-epsilon model with applications in fluid flows through permeable mediaPublication . de Oliveira, H.B.In this work, we study a one-equation turbulence k-epsilon model that governs fluid flows through permeable media. The model problem under consideration here is derived from the incompressible Navier-Stokes equations by the application of a time-averaging operator used in the k-epsilon modeling and a volume-averaging operator that is characteristic of modeling unsteady porous media flows. For the associated initial- and boundary-value problem, we prove the existence of suitable weak solutions (average velocity field and turbulent kinetic energy) in the space dimensions of physics interest.
- Continuous/discontinuous finite element approximation of a 2d navier-stokes problem arising in fluid confinementPublication . Borges de Oliveira, Hermenegildo; Lopes, Nuno DavidIn this work, a stationary 2d Navier-Stokes problem with nonlinear feedback forces field is considered in the stream -function formulation. We use the Continuous/Discontinuous Finite Element Method (CD-FEM), with interior penalty terms, to numerically solve the associated boundary -value problem. For the associated continuous and discrete problems, we prove the existence of weak solutions and establish the conditions for their uniqueness. Consistency, stability and convergence of the method are also shown analytically. To validate the numerical model regarding its applicability and robustness, several test cases are carried out.