scholarly journals Exact solutions for restricted incompressible Navier–Stokes equations with Dirichlet boundary conditions

2019 ◽  
Vol 3 (8) ◽  
pp. 085008
Author(s):  
Manuel García-Casado
Keyword(s):  
Boundary Conditions ◽  
Exact Solutions ◽  
Dirichlet Boundary ◽  
Stokes Equations ◽  
Dirichlet Boundary Conditions ◽  
Navier Stokes ◽  
Navier Stokes Equations

Large-frequency global regularity for the incompressible Navier–Stokes equation

1999 ◽  
Vol 129 (5) ◽  
pp. 903-912
Author(s):  
Joel D. Avrin
Keyword(s):  
Boundary Conditions ◽  
Global Regularity ◽  
Stokes Equations ◽  
Strong Solutions ◽  
Dirichlet Boundary Conditions ◽  
Navier Stokes ◽  
Thin Domains ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations ◽  
Incompressible Navier Stokes Equation

We obtain global existence and regularity of strong solutions to the incompressible Navier–Stokes equations for a variety of boundary conditions in such a way that the initial and forcing data can be large in the high-frequency eigenspaces of the Stokes operator. We do not require that the domain be thin as in previous analyses. But in the case of thin domains (and zero Dirichlet boundary conditions) our results represent a further improvement and refinement of previous results obtained.

A mixed virtual element method for the Navier–Stokes equations

2018 ◽  
Vol 28 (14) ◽  
pp. 2719-2762 ◽  
Author(s):  
Gabriel N. Gatica ◽  
Mauricio Munar ◽  
Filánder A. Sequeira
Keyword(s):  
Fixed Point ◽  
Dirichlet Boundary ◽  
Stokes Equations ◽  
Dirichlet Boundary Conditions ◽  
Navier Stokes ◽  
Equilibrium Equations ◽  
Virtual Element Method ◽  
Navier Stokes Equations ◽  
Virtual Element ◽  
Element Method

A mixed virtual element method (mixed-VEM) for a pseudostress-velocity formulation of the two-dimensional Navier–Stokes equations with Dirichlet boundary conditions is proposed and analyzed in this work. More precisely, we employ a dual-mixed approach based on the introduction of a nonlinear pseudostress linking the usual linear one for the Stokes equations and the convective term. In this way, the aforementioned new tensor together with the velocity constitute the only unknowns of the problem, whereas the pressure is computed via a postprocessing formula. In addition, the resulting continuous scheme is augmented with Galerkin type terms arising from the constitutive and equilibrium equations, and the Dirichlet boundary condition, all them multiplied by suitable stabilization parameters, so that the Banach fixed-point and Lax–Milgram theorems are applied to conclude the well-posedness of the continuous and discrete formulations. Next, we describe the main VEM ingredients that are required for our discrete analysis, which, besides projectors commonly utilized for related models, include, as the main novelty, the simultaneous use of virtual element subspaces for [Formula: see text] and [Formula: see text] in order to approximate the velocity and the pseudostress, respectively. Then, the discrete bilinear and trilinear forms involved, their main properties and the associated mixed virtual scheme are defined, and the corresponding solvability analysis is performed using again appropriate fixed-point arguments. Moreover, Strang-type estimates are applied to derive the a priori error estimates for the two components of the virtual element solution as well as for the fully computable projections of them and the postprocessed pressure. As a consequence, the corresponding rates of convergence are also established. Finally, we follow the same approach employed in previous works by some of the authors and introduce an element-by-element postprocessing formula for the fully computable pseudostress, thus yielding an optimally convergent approximation of this unknown with respect to the broken [Formula: see text]-norm.

Vertical Layers

2006 ◽  
Author(s):  
Jean-Yves Chemin ◽  
Benoit Desjardins ◽  
Isabelle Gallagher ◽  
Emmanuel Grenier
Keyword(s):  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Stokes Equations ◽  
Point Of View ◽  
Dirichlet Boundary Conditions ◽  
Careful Study ◽  
Navier Stokes Equations ◽  
Mathematical Point ◽  
Open Questions ◽  
Ekman Layers

From a physical point of view, as well as from a mathematical point of view, horizontal layers (Ekman layers) are now well understood. This is not the casefor vertical layers which are much more complicated, from a physical, analytical and mathematical point of view, and many open questions in all these directions remain open. Let us, in this section, consider a domain Ω with vertical boundaries. Namely, let Ωh be a domain of R2 and let Ω=Ωh × [0, 1]. This domain has two types of boundaries: • horizontal boundaries Ωh × {0} (bottom) and Ωh ×{1} (top) where Ekman layers are designed to enforce Dirichlet boundary conditions; • vertical boundaries ∂Ωh × [0, 1] where again a boundary layer is needed to ensure Dirichlet boundary conditions. These layers, however, are not of Ekman type, since r is now parallel to the boundary. Vertical layers are quite complicated. They in fact split into two sublayers: one of size E1/3 and another of size E1/4 where E =νε denotes the Ekman number. This was discovered and studied analytically by Stewartson and Proudman. Vertical layers can be easily observed in experiments (at least the E1/4 layer, the second one being too thin) but do not seem to be relevant in meteorology or oceanography, where near continents, effects of shores, density stratification, temperature, salinity, or simply topography are overwhelming and completely mistreated by rotating Navier–Stokes equations. In MHD, however, and in particular in the case of rotating concentric spheres, they are much more important. Numerically, they are easily observed, at large Ekman numbers E (small Ekman numbers being much more difficult to obtain). The aim of this section is to provide an introduction to the study of these layers, a study mainly open from a mathematical point of view. First we will derive the equation of the E1/3 layer. Second we will investigate the E1/4 layer and underline its similarity with Prandtl’s equations. In particular, we conjecture that E1/4 is always linearly and nonlinearly unstable. We will not prove this latter fact, which would require careful study of what happens at the corners of the domain, a widely open problem.

scholarly journals Limits of the Stokes and Navier–Stokes equations in a punctured periodic domain

2019 ◽  
Vol 18 (02) ◽  
pp. 211-235
Author(s):  
Michel Chipot ◽  
Jérôme Droniou ◽  
Gabriela Planas ◽  
James C. Robinson ◽  
Wei Xue
Keyword(s):  
Boundary Conditions ◽  
Periodic Boundary ◽  
Stokes Equations ◽  
Periodic Boundary Conditions ◽  
Dirichlet Boundary Conditions ◽  
Navier Stokes ◽  
Periodic Domain ◽  
Navier Stokes Equations ◽  
Forcing Function ◽  
Connected Set

We treat three problems on a two-dimensional “punctured periodic domain”: we take [Formula: see text], where [Formula: see text] and [Formula: see text] is the closure of an open connected set that is star-shaped with respect to [Formula: see text] and has a [Formula: see text] boundary. We impose periodic boundary conditions on the boundary of [Formula: see text], and Dirichlet boundary conditions on [Formula: see text]. In this setting we consider the Poisson equation, the Stokes equations, and the time-dependent Navier–Stokes equations, all with a fixed forcing function [Formula: see text], and examine the behavior of solutions as [Formula: see text]. In all three cases we show convergence of the solutions to those of the limiting problem, i.e. the problem posed on all of [Formula: see text] with periodic boundary conditions.

The Pullback Asymptotic Behavior of the Solutions for 2D NonautonomousG-Navier-Stokes Equations

2012 ◽  
Vol 4 (2) ◽  
pp. 223-237 ◽  
Author(s):  
Jinping Jiang ◽  
Yanren Hou ◽  
Xiaoxia Wang
Keyword(s):  
Asymptotic Behavior ◽  
Measure Of Noncompactness ◽  
Dirichlet Boundary ◽  
Stokes Equations ◽  
Fractal Dimensions ◽  
Dirichlet Boundary Conditions ◽  
Navier Stokes ◽  
Bounded Domains ◽  
Pullback Attractors ◽  
Navier Stokes Equations

AbstractThe pullback asymptotic behavior of the solutions for 2D Nonau-tonomousG-Navier-Stokes equations is studied, and the existence of itsL2-pullback attractors on some bounded domains with Dirichlet boundary conditions is investigated by using the measure of noncompactness. Then the estimation of the fractal dimensions for the 2DG-Navier-Stokes equations is given.

ON THE EXISTENCE AND UNIQUENESS OF SOLUTIONS TO 2D G-BÉNARD PROBLEM IN UNBOUNDED DOMAINS

2020 ◽  
Vol 65 (6) ◽  
pp. 23-30
Author(s):  
Thinh Tran Quang ◽  
Thuy Le Thi
Keyword(s):  
Existence And Uniqueness ◽  
Dirichlet Boundary ◽  
Stokes Equations ◽  
Dirichlet Boundary Conditions ◽  
Navier Stokes ◽  
Uniqueness Of Solutions ◽  
Navier Stokes Equations ◽  
Bénard Problem ◽  
Benard Problem ◽  
Homogeneous Dirichlet Boundary Conditions

We consider the 2D g-Bénard problem in domains satisfying the Poincaré inequality with homogeneous Dirichlet boundary conditions. We prove the existence and uniqueness of global weak solutions. The obtained results particularly extend previous results for 2D g-Navier-Stokes equations and 2D Bénard problem.

Parallel iterative finite-element algorithms for the Navier–Stokes equations with nonlinear slip boundary conditions

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Kangrui Zhou ◽  
Yueqiang Shang
Keyword(s):  
Finite Element ◽  
Boundary Conditions ◽  
Parallel Algorithms ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Slip Boundary Conditions ◽  
Navier Stokes Equations ◽  
Slip Boundary ◽  
Parallel Iterative ◽  
Nonlinear Slip Boundary Conditions

AbstractBased on full domain partition, three parallel iterative finite-element algorithms are proposed and analyzed for the Navier–Stokes equations with nonlinear slip boundary conditions. Since the nonlinear slip boundary conditions include the subdifferential property, the variational formulation of these equations is variational inequalities of the second kind. In these parallel algorithms, each subproblem is defined on a global composite mesh that is fine with size h on its subdomain and coarse with size H (H ≫ h) far away from the subdomain, and then we can solve it in parallel with other subproblems by using an existing sequential solver without extensive recoding. All of the subproblems are nonlinear and are independently solved by three kinds of iterative methods. Compared with the corresponding serial iterative finite-element algorithms, the parallel algorithms proposed in this paper can yield an approximate solution with a comparable accuracy and a substantial decrease in computational time. Contributions of this paper are as follows: (1) new parallel algorithms based on full domain partition are proposed for the Navier–Stokes equations with nonlinear slip boundary conditions; (2) nonlinear iterative methods are studied in the parallel algorithms; (3) new theoretical results about the stability, convergence and error estimates of the developed algorithms are obtained; (4) some numerical results are given to illustrate the promise of the developed algorithms.

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