Density-dependent incompressible fluids with non-Newtonian viscosity

We study the system of PDEs describing unsteady flows of incompressible fluids with variable density and non-constant viscosity. Indeed, one considers a stress tensor being a nonlinear function of the symmetric velocity gradient, verifying the properties of pcoercivity and (p − 1)-growth, for a given parameter p > 1. The existence of Dirichlet weak solutions was obtained in [2], in the cases p > 12/5 if d = 3 or p > 2 if d = 2, d being the dimension of the domain. In this paper, with help of some new estimates (which lead to point-wise convergence of the velocity gradient), we obtain the existence of space-periodic weak solutions for all p > 2. In addition, we obtain regularity properties of weak solutions whenever p > 20/9 (if d = 3) or p > 2 (if d = 2). Further, some extensions of these results to more general stress tensors or to Dirichlet boundary conditions (with a Newtonian tensor large enough) are obtained.Comisión Interministerial de Ciencia y Tecnologí

Approximation of Smectic-A liquid crystals

In this paper, we present energy-stable numerical schemes for a Smectic-A liquid crystal model. This model involve the hydrodynamic velocity-pressure macroscopic variables (u, p) and the microscopic order parameter of Smectic-A liquid crystals, where its molecules have a uniaxial orientational order and a positional order by layers of normal and unitary vector n. We start from the formulation given in [E’97] by using the so-called layer variable φ such that n = ∇φ and the level sets of φ describe the layer structure of the Smectic-A liquid crystal. Then, a strongly non-linear parabolic system is derived coupling velocity and pressure unknowns of the Navier-Stokes equations (u, p) with a fourth order parabolic equation for φ. We will give a reformulation as a mixed second order problem which let us to define some new energy-stable numerical schemes, by using second order finite differences in time and C 0 - finite elements in space. Finally, numerical simulations are presented for 2D-domains, showing the evolution of the system until it reachs an equilibrium configuration. Up to our knowledge, there is not any previous numerical analysis for this type of models.Ministerio de Economía y CompetitividadMinistry of Education, Youth and Sports of the Czech Republi

Numerical methods for solving the Cahn-Hilliard equation and its applicability to related Energy-based models

In this paper, we review some numerical methods presented in the literature in the last years to approximate the Cahn-Hilliard equation. Our aim is to compare the main properties of each one of the approaches to try to determine which one we should choose depending on which are the crucial aspects when we approximate the equations. Among the properties that we consider desirable to control are the time accuracy order, energy-stability, unique solvability and the linearity or nonlinearity of the resulting systems. In particular, we concern about the iterative methods used to approximate the nonlinear schemes and the constraints that may arise on the physical and computational parameters. Furthermore, we present the connections of the Cahn-Hilliard equation with other physically motivated systems (not only phase field models) and we state how the ideas of efficient numerical schemes in one topic could be extended to other frameworks in a natural way.Ministry of Education, Youth and Sports of the Czech RepublicMinisterio de Economía y Competitivida

Superconvergence in velocity and pressure for the 3D time-dependent Navier-Stokes equations

This work is devoted to the superconvergence in space approximation of a fully discrete scheme for the incompressible time-dependent Navier-Stokes Equations in three-dimensional domains. We discrete by Inf-Sup-stable Finite Element in space and by a semi-implicit backward Euler (linear) scheme in time. Using an extension of the duality argument in negative-norm for elliptic linear problems (see for instance [1] Brennet, S., Scott, L. The Mathematical Theory of Finite Element Methods, Springer, 2008) to the mixed velocity-pressure formulation of the Stokes problem, we prove some superconvergence in space results for the velocity with respect to the energy-norm, and for a weaker norm of L2 (0, T;L 2 (Ω)) type (this latter holds only for the case of Taylor-Hood approximation). On the other hand, we also obtain optimal error estimates for the pressure without imposing constraints on the time and spatial discrete parameters, arriving at superconvergence in the H1 (Ω)-norm again for Taylor-Hood approximations. These results are numerically verified by several computational experiments, where two splitting in time schemes are also considered.Ministerio de Ciencia e Innovación (España

Global in time solutions for the Poiseuille flow of Oldroyd type in 3D domains

A Poiseuille flow in a 3D cylindrical domain is considered for a non-newtonian fluid of Oldroyd type. We prove existence (and uniqueness) of a global (in time) weak solution. Moreover, this weak solution is an strong solution when data are more regular. These results has already been obtained in the case of two concentrical cylinders . Now, we consider an extension to an unique cylinder. Then, a mixed parabolic-hyperbolic PDE's system appears but the parabolic equation is of degenerate type. The key of the proofs is to estimate in appropriate Sobolev weighted spaces (and to obtain strong convergence in weak norms by means of a Cauchy argument)

Convergence to equilibrium for smectic-A liquid crystals in 3D domains without constraints for the viscosity

In this paper, we focus on a smectic-A liquid crystal model in 3D domains, and obtain three main results: the proof of an adequate Lojasiewicz-Simon inequality by using an abstract result; the rigorous proof (via a Galerkin approach) of the existence of global intime weak solutions that become strong (and unique) in long-time; and its convergence to equilibrium of the whole trajectory as time goes to in nity. Given any regular initial data, the existence of a unique global in-time regular solution (bounded up to in nite time) and the convergence to an equilibrium have been previously proved under the constraint of a su ciently high level of viscosity. Here, all results are obtained without imposing said constraint

Global in time solution and time-periodicity for a smectic-A liquid crystal model

In this paper some results are obtained for a smectic-A liquid crystal model with time-dependent boundary Dirichlet data for the so-called layer variable φ (the level sets of φ describe the layer structure of the smectic-A liquid crystal). First, the initial-boundary problem for arbitrary initial data is considered, obtaining the existence of weak solutions which are bounded up to infinity time. Second, the existence of time-periodic weak solutions is proved. Afterwards, the problem of the global in time regularity is attacked, obtaining the existence and uniqueness of regular solutions (up to infinity time) for both problems, i.e. the initial-valued problem and the time-periodic one, but assuming a dominant viscosity coefficient in the linear part of the diffusion tensor.Ministerio de Educación y Ciencia MTM2006–07932Junta de Andalucía P06-FQM-0237

Convergence and error estimates of two iterative methods for the strong solution of the incompressible korteweg model

We show the existence of strong solutions for a fluid model with Korteweg tensor, which is obtained as limit of two iterative linear schemes. The different unknowns are sequentially decoupled in the first scheme and in parallel form in the second one. In both cases, the whole sequences are bounded in strong norms and convergent towards the strong solution of the system, by using a generalization of the Banach’s Fixed Point Theorem. Moreover, we explicit a priori and a posteriori error estimates (respect to the weak norms), which let us to compare both schemes.Dirección General de Investigación (Ministerio de Educación y Ciencia)Junta de Andalucí

Analysis of the hydrostatic Stokes problem and finite-element approximation in unstructured meshes

The stability of velocity and pressure mixed finite-element approximations in general meshes of the hydrostatic Stokes problem is studied, where two “inf-sup” conditions appear associated to the two constraints of the problem; namely incompressibility and hydrostatic pressure. Since these two constraints have different properties, it is not easy to choose finite element spaces satisfying both. From the analytical point of view, two main results are established; the stability of an anisotropic approximation of the velocity (using different spaces for horizontal and vertical velocities) with piecewise constant pressures, and the unstability of standard (isotropic) approximations which are stable for the Stokes problem, like the mini-element or the Taylor-Hood element. Moreover, we give some numerical simulations, which agree with the previous analytical results and allow us to conjecture the stability of some anisotropic approximations of the velocity with continuous piecewise linear pressure in unstructured meshes.Dirección General de Investigación (Ministerio de Educación y Ciencia)Junta de Andalucí