Stationary Navier-Stokes Equations Research Articles

Correction: Ill-Posedness of the Two-Dimensional Stationary Navier–Stokes Equations on the Whole Plane

Correction: Ill-Posedness of the Two-Dimensional Stationary Navier–Stokes Equations on the Whole Plane

Ill-Posedness of the Two-Dimensional Stationary Navier–Stokes Equations on the Whole Plane

Ill-Posedness of the Two-Dimensional Stationary Navier–Stokes Equations on the Whole Plane

Overlapping finite elements for the Navier-Stokes equations

The purpose of this paper is to investigate the performance of a certain family of low-order quadrilateral finite elements in the solution of the stationary Navier-Stokes equations governing incompressible fluid flows. These finite elements are derived from the ‘overlapping finite elements’, first developed for the solution of problems in solid mechanics [5,9,48] and incorporate features of both meshfree and traditional finite element methods. One of their most remarkable properties is the insensitivity to mesh distortions. Also, since the Shepard functions are replaced by a suitable interpolation, the resulting basis functions are entirely polynomial, which allows the numerical integration of the weak forms to be performed with few integration points [6,49]. We also discuss the theoretical reasons for the stability of the solutions, that is, the absence of spurious pressure modes, and show that the proposed discretization scheme passes the relevant inf-sup test.

An $$L^p$$-theory for fractional stationary Navier–Stokes equations

An $$L^p$$-theory for fractional stationary Navier–Stokes equations

A Quasi-Variational-Hemivariational Inequality for Incompressible Navier-Stokes System with Bingham Fluid

In this paper we examine a class of elliptic quasi-variational inequalities, which involve a constraint set and a set-valued map. First, we establish the existence of a solution and the compactness of the solution set. The approach is based on results for an elliptic variational inequality and the Kakutani-Ky Fan fixed point theorem. Next, we prove an existence and compactness result for a quasi-variational-hemivariational inequality. The latter involves a locally Lipschitz continuous functional and a convex potential. Finally, we present an application to the stationary incompressible Navier-Stokes equation with mixed boundary conditions which model a generalized Newtonian fluid of Bingham type.

On well-posedness of Navier–Stokes variational inequalities

In this paper, well-posedness is carried out on variational inequalities in a general form for the stationary Navier–Stokes equations modeling viscous incompressible fluid flows with a nonsmooth slip boundary condition of friction type. For a special form of the variational inequality commonly studied in existing literature, our proof technique leads to a substantial improvement on the well-posedness condition.

Multigoal-oriented a posteriori error control for heated material processing using a generalized Boussinesq model

In this work, we develop a posteriori error control for a generalized Boussinesq model in which thermal conductivity and viscosity are temperature-dependent. Therein, the stationary Navier–Stokes equations are coupled with a stationary heat equation. The coupled problem is modeled and solved in a monolithic fashion. The focus is on multigoal-oriented error estimation with the dual-weighted residual method in which an adjoint problem is utilized to obtain sensitivity measures with respect to several goal functionals. The error localization is achieved with the help of a partition-of-unity in a weak formulation, which is specifically convenient for coupled problems as we have at hand. The error indicators are used to employ adaptive algorithms, which are substantiated with several numerical tests such as one benchmark and two further experiments that are motivated from laser material processing. Therein, error reductions and effectivity indices are consulted to establish the robustness and efficiency of our framework.

Approximation of a Solution to the Stationary Navier–Stokes Equations in a Curved Thin Domain by a Solution to Thin-Film Limit Equations

Approximation of a Solution to the Stationary Navier–Stokes Equations in a Curved Thin Domain by a Solution to Thin-Film Limit Equations

Variational–hemivariational system for contaminant convection–reaction–diffusion model of recovered fracturing fluid

Abstract This work is devoted to study the convection–reaction–diffusion behavior of contaminant in the recovered fracturing fluid which flows in the wellbore from shale gas reservoir. First, we apply various constitutive laws for generalized non-Newtonian fluids, diffusion principles, and friction relations to formulate the recovered fracturing fluid model. The latter is a partial differential system composed of a nonlinear and nonsmooth stationary incompressible Navier-Stokes equation with a multivalued friction boundary condition, and a nonlinear convection–reaction–diffusion equation with mixed Neumann boundary conditions. Then, we provide the weak formulation of the fluid model which is a hemivariational inequality driven by a nonlinear variational equation. We establish existence of solutions to the recovered fracturing fluid model via a surjectivity theorem for multivalued operators combined with an alternative iterative method and elements of nonsmooth analysis.

New Liouville type theorems for the stationary Navier–Stokes, MHD, and Hall–MHD equations

We establish new Liouville-type theorems for weak solutions of the stationary Navier–Stokes equations, stationary magnetohydrodynamics (MHD) equations and stationary Hall–MHD equations under some conditions on the growth of certain Lebesgue norms of the velocity and the magnetic field.

A New Two-grid Algorithm Based on Newton Iteration for the Stationary Navier–Stokes Equations with Damping

A New Two-grid Algorithm Based on Newton Iteration for the Stationary Navier–Stokes Equations with Damping

Analysis of a class of globally divergence-free HDG methods for stationary Navier-Stokes equations

Analysis of a class of globally divergence-free HDG methods for stationary Navier-Stokes equations

Liouville-type theorems for steady MHD and Hall-MHD equations in [formula omitted

In this paper, we study the Liouville-type theorems for three-dimensional stationary incompressible MHD and Hall-MHD systems in a slab with periodic boundary condition. We show that, under the assumptions that (uθ,bθ) or (ur,br) is axisymmetric, or (rur,rbr) is bounded, any smooth bounded solution to the MHD or Hall-MHD system with local Dirichlet integral growing as an arbitrary power function must be constant. This improves the result of Theorem 1.2 in Pan (2021) [33], where the Dirichlet integral of u is assumed to be finite. Inspired by Bang et al. (2022) [3], our proof relies on establishing Saint-Venant's estimates associated with our problem, and the result in the current paper extends that for stationary Navier-Stokes equations shown by [3] to MHD and Hall-MHD equations. To achieve this, more intricate estimates are needed to handle the terms involving b properly.

Linearized Learning with Multiscale Deep Neural Networks for Stationary Navier-Stokes Equations with Oscillatory Solutions

Linearized Learning with Multiscale Deep Neural Networks for Stationary Navier-Stokes Equations with Oscillatory Solutions

Asymptotic behavior and Liouville-type theorems for axisymmetric stationary Navier-Stokes equations outside of an infinite cylinder with a periodic boundary condition

We study the asymptotic behavior of solutions to the steady Navier-Stokes equations outside of an infinite cylinder in R3. We assume that the flow is periodic in x3-direction and has no swirl. This problem is closely related with two-dimensional exterior problem. Under a condition on the generalized finite Dirichlet integral, we give a pointwise decay estimate of the vorticity at the spatial infinity. This reveals the relation between the integrability of ∇v and the decay rate of ω near the spatial infinity. Moreover, we prove a Liouville-type theorem only from the condition of the generalized finite Dirichlet integral.

Non-isothermal non-Newtonian fluids: The stationary case

The stationary Navier–Stokes equations for a non-Newtonian incompressible fluid are coupled with the stationary heat equation and subject to Dirichlet-type boundary conditions. The viscosity is supposed to depend on the temperature and the stress depends on the strain through a suitable power law depending on [Formula: see text] (shear thinning case). For this problem we establish the existence of a weak solution as well as we prove some regularity results both for the Navier–Stokes and the Stokes cases. Then, the latter case with the Carreau power law is approximated through a FEM scheme and some error estimates are obtained. Such estimates are then validated through some two-dimensional numerical experiments.

A Liouville-type theorem for the stationary Navier–Stokes equations

In this paper, we study the Liouville-type theorem of the stationary Navier–Stokes equations on an oscillation growth condition of the potential tensors. Based on the oscillation growth condition, we prove Poincaré-type inequalities on different balls, which play a decisive role to get the result.

Anisotropic Liouville type theorem for the stationary Navier–Stokes equations in [formula omitted

In this brief note we study the Liouville type problem for the stationary Navier–Stokes equations in R3. We show that under certain anisotropic integrability conditions on the components of the velocity implies that the solution is trivial.

Existence of solutions to steady Navier-Stokes equations via a minimax approach

Our objective in this paper is to develop and utilize a minimax principle for proving the existence of symmetric solutions for the stationary Navier-Stokes equations. Notwithstanding its application to symmetric solutions in this paper, our minimax principle is broad enough to capture other types of solutions provided the equation and the external force are compatible under a family of operations including but not limited to being invariant by compact groups. The subset of functions compatible under this family of operations is not required to be a linear subspace, and being a closed convex set suffices for our purpose.

Regularity of weak solutions for the stationary Ericksen–Leslie and MHD systems

We consider two elliptic coupled systems of relevance in the fluid dynamics. These systems are posed on the whole space R3, and they consider the action of external forces. The first system deals with the simplified Ericksen–Leslie (SEL) system, which describes the dynamics of liquid crystal flows. The second system is the time-independent magneto-hydrodynamic (MHD) equations. For the SEL system, we obtain a new criterion to improve the regularity of weak solutions, provided that they belong to some homogeneous Morrey space. As a bi-product, we also obtain some new regularity criterion for the stationary Navier–Stokes equations and for a nonlinear harmonic map flow. This new regularity criterion also holds true for the MHD equations. Furthermore, for this last system, we are able to use the Gevrey class to prove that all finite energy weak solutions are analytic functions, provided that the external forces belong to some Gevrey class.