Navier Stokes Type Equations Research Articles

Strong Well-Posedness of the Q-Tensor Model for Liquid Crystals: The Case of Arbitrary Ratio of Tumbling and Aligning Effects ξ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\xi $$\\end{document}

The Beris–Edwards model of nematic liquid crystals couples an equation for the molecular orientation described by the Q-tensor with a Navier–Stokes type equation with an additional non-Newtonian stress caused by the molecular orientation. Both equations contain a parameter ξ∈R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\xi \\in \\mathbb {R}$$\\end{document} measuring the ratio of tumbling and alignment effects. Previous well-posedness results largely vary on the space dimension n and the constraints of the parameter ξ∈R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\xi \\in \\mathbb {R}$$\\end{document}. This work addresses strong well-posedness of this model, first locally and then globally for small initial data, both in the Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p$$\\end{document}-L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document}-setting for p>44-n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p > \\frac{4}{4-n}$$\\end{document}, in the general cases, i.e., for n=2,3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n = 2, 3$$\\end{document} and without any restriction on ξ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\xi $$\\end{document}. The approach is based on methods from quasilinear equations and the fact that the associated linearized operator admits maximal Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p$$\\end{document}-L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document}-regularity. The proof of the latter property relies on techniques from sectorial operators, Schur complements and J\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {J}$$\\end{document}-symmetry.

Localization of energies in Navier–Stokes models with reaction terms

We analyze a general class of coupled systems of stationary Navier–Stokes type equations with variable coefficients and non-homogeneous terms of reaction type in the incompressible case. Existence of solutions satisfying the homogeneous Dirichlet condition in a bounded domain in [Formula: see text], [Formula: see text], and localization results for the corresponding kinetic energy and enstrophy are obtained by using a variational approach and the fixed point index theory.

Research of an oblique detonation wave in a limited area

This work presents the results of two-dimensional numerical simulation of processes, arising from the interaction of an incoming flow of a well-mixed stoichiometric hydrogen-air mixture with a wedge in a bounded region at high Mach velocities using Navier-Stokes type equations taking into account chemical interactions. The Mach number of the incoming flow from 9 to 12 were considered. Various wave configurations were obtained depending on the velocity of the incoming flow. For the flow Mach number of 9, the presence of a detonation wave perpendicular to the flow was obtained. For high flow velocities, another similar wave configuration of the same type was obtained.

GPU-based molecular dynamics of fluid flows: Reaching for turbulence

Fluid dynamics is a ubiquitous problem that arises in different branches of science and industry. It is usually tackled by numerically solving continuum Navier-Stokes type equations. Molecular dynamics has been not a feasible tool to approach fluid dynamics until very recently due to its disproportional computational complexity for relevant system sizes. In this paper, we propose a new type of boundary conditions for molecular dynamics simulations of stationary fluid flows and present its possible GPU-based implementations in OpenMM and LAMMPS. We examine the performance and scalability of the proposed implementations. The benchmarking results show promising performance that makes it possible to reach turbulence in atomistic models of stationary fluid flows using modern supercomputers.

Existence of global weak solutions to an inhomogeneous Doi model for active liquid crystals

In this paper, we consider an inhomogeneous Doi model which was introduced by E and Zhang (2006) [17]. We extend their model, which couples a Smoluchowski equation to a Navier-Stokes type equation, for active particles by introducing an additional stress tensor. Exploiting the energetic and entropic structure of the system, we establish the existence of global-in-time weak solutions in two and three space dimensions for both passive and active particles. In particular, our result holds for minimal regularity assumptions on the initial data and without restrictions on the Reynolds and Deborah number.

Existence of Sobolev regular solutions for the incompressible flow of liquid crystals in three dimensions

<abstract><p>This paper considers a simplified three dimensional Ericksen-Leslie System for nematic liquid crystal flows in the unbounded domain $ \Omega: = \mathbb R^+\times \mathbb R^2 $ or the smooth bounded domain $ \Omega $. The hydrodynamic system consists of the Navier-Stokes type equations for the fluid velocity coupled with a convective Ginzburg-Landau type equation for the averaged molecular orientation. We first establish the global existence of Sobolev regular solution with finite energies in Sobolev space $ H^{s}(\Omega)\times H^{s}(\Omega) $, where the index $ s $ of the Sobolev space can be any large fixed integer, but $ s\neq+\infty $. Then we give an asymptotic expansions of a family of Sobolev regularity solutions for such system in $ \Omega $.</p></abstract>

Distribution dependent SDEs for Navier-Stokes type equations

To characterize Navier-Stokes type equations where the Laplacian is extended to a singular second order differential operator, we propose a class of SDEs depending on the distribution in future. The well-posedness and regularity estimates are derived for these SDEs.

Struwe-like solutions for an evolutionary model of magnetoviscoelastic fluids

In this work we investigate the existence and uniqueness of Struwe-like solutions for a system of partial differential equations modeling the dynamics of magnetoviscoelastic fluids. The considered system couples a Navier-Stokes type equation with a dissipative equation for the deformation tensor and a Landau–Lifshitz–Gilbert type equation for the magnetization field.The main purpose is to establish a well-posedness theory in a two-dimensional periodic domain under standard assumption of critical regularity for the (possibly large) initial data. We prove that the considered weak solutions are everywhere smooth, except for a discrete set of time values.The proof of the uniqueness is based on suitable energy estimates for the solutions within a functional framework which is less regular than the one of the Struwe energy level. These estimates rely on several techniques of harmonic analysis and paradifferential calculus.

An open mapping theorem for the Navier–Stokes type equations associated with the de Rham complex over $mathbb R^n$

An open mapping theorem for the Navier–Stokes type equations associated with the de Rham complex over $mathbb R^n$

Incompressible and fast rotation limit for barotropic Navier–Stokes equations at large Mach numbers

We study the incompressible and fast rotation limit for the barotropic Navier–Stokes equations with Coriolis force, in the case when the Mach number Ma is large with respect to the Rossby number Ro: namely, we focus on the regime Ro≪Ma. For this, we follow a recent approach in Danchin and Mucha (2017) and take also a large bulk viscosity coefficient. We prove that the limit dynamics is described by an incompressible Navier–Stokes type equation, recasted in the vorticity formulation, where however an additional unknown, linked to density oscillations around a fixed constant reference state, comes into play. The proof of the convergence is based on a compensated compactness argument and on the derivation of sharp decay estimates for solutions to a heat equation with fast diffusion in time.

On the stability phenomenon of the Navier-Stokes type equations for elliptic complexes

ABSTRACT Let be a Riemannian n-dimensional smooth closed manifold, , be smooth vector bundles over and be an elliptic differential complex of linear first order operators. We consider the operator equations, induced by the Navier-Stokes type equations associated with on the scale of anisotropic Hölder spaces over the layer with finite time T>0. Using the properties of the differentials and parabolic operators over this scale of spaces, we reduce the equations to a nonlinear Fredholm operator equation of the form , where K is a compact continuous operator. It appears that the Fréchet derivative is continuously invertible at every point of each Banach space under consideration and the map is open and injective in the space.

A BDF2-semismooth Newton algorithm for the numerical solution of the Bingham flow with temperature dependent parameters

This paper is devoted to the numerical solution of the non-isothermal instationary Bingham flow with temperature dependent parameters by semismooth Newton methods. We discuss the main theoretical aspects regarding this problem. Mainly, we discuss the existence of solutions for the problem, and focus on a multiplier formulation which leads us to a coupled system of PDEs involving a Navier–Stokes type equation and a parabolic energy PDE. Further, we propose a Huber regularization for this coupled system of partial differential equations, and we briefly discuss the well posedness of the regularized problem. A detailed finite element discretization, based on the so called (cross-grid P1) - Q0 elements, is proposed for the space variable, involving weighted stiffness and mass matrices. After discretization in space, a second order BDF method is used as a time advancing technique, leading, in each time iteration, to a nonsmooth system of equations, which is suitable to be solved by a semismooth Newton (SSN) algorithm. Therefore, we propose and discuss the main properties of a SSN algorithm, including the convergence properties. The paper finishes with two computational experiments that exhibit the main properties of the numerical approach.

On convergent schemes for a two-phase Oldroyd-B type model with variable polymer density

AbstractThe paper is concerned with a diffuse-interface model that describes two-phase flow of dilute polymeric solutions with a variable particle density. The additional stresses, which arise by elongations of the polymers caused by deformations of the fluid, are described by Kramers stress tensor. The evolution of Kramers stress tensor is modeled by an Oldroyd-B type equation that is coupled to a Navier–Stokes type equation, a Cahn–Hilliard type equation, and a parabolic equation for the particle density. We present a regularized finite element approximation of this model, prove that our scheme is energy stable and that there exist discrete solutions to it. Furthermore, in the case of equal mass densities and two space dimensions, we are able to pass to the limit rigorously as the regularization parameters and the spatial and temporal discretization parameters tend towards zero and prove that a subsequence of discrete solutions converges to a global-in-time weak solution to the unregularized coupled system. To the best of our knowledge, this is the first existence result for a two-phase flow model of viscoelastic fluids with an Oldroyd-B type equation. Additionally, we show that our finite element scheme is fully practical and we present numerical simulations.

Periodic reiterated homogenization of stationary Navier–Stokes type equations

This paper is devoted to the reiterated homogenization of stationary Navier–Stokes type equations in a periodic structure. The usual Laplace operator in the classical Navier–Stokes equations is here replaced by an elliptic linear differential operator of order two, in divergence form with periodically oscillating coefficients. Our approach is the well known two-scale convergence method. One convergence theorem is proved and we derive de macroscopic homogenized model.

Lubrication and shallow-water systems Bernis-Friedman and BD entropies

This paper concerns the results recently announced by the authors, in C.R. Acad. Sciences Maths volume 357, Issue 1, 1-6 (2019), which make the link between the BD entropy introduced by D. Bresch and B. Desjardins for the viscous shallow-water equations and the Bernis-Friedman (called BF in our paper) dissipative entropy introduced to study the lubrication equations. More precisely different dissipative BF entropies are obtained from the BD entropies playing with drag terms and capillarity formula for viscous shallow water type equations. This is the main idea in the paper which makes the link between two communities. The limit processes employ the standard compactness arguments taking care of the control in the drag terms. It allows in one dimension for instance to prove global existence of nonnegative weak solutions for lubrication equations starting from the global existence of nonnegative weak solutions for appropriate viscous shallow-water equations (for which we refer to appropriate references). It also allows to prove global existence of nonnegative weak solutions for fourth-order equation including the Derrida-Lebowitz-Speer-Spohn equation starting from compressible Navier-Stokes type equations.

Smoothness with Respect to Viscosity of the Solutions of Navier–Stokes-Type Operator Equations

An evolutionary Navier–Stokes-type equation is considered. Due to the presence of a bilinear operator term in it, it is possible to introduce a small parameter and expand the solution in it. The main objective of the paper is to find the conditions for the ordinary (not asymptotic) convergence of the series obtained in this case.

An existence and uniqueness result for the Navier–Stokes type equations on the Heisenberg group

The aim of this paper is to give an existence and uniqueness of solutions for the Cauchy problem of the Navier–Stokes type equations associated to the sublaplacian provided by the left invariant vector fields on the Heisenberg group. To avoid the difficulty of the non-commutative which is intrinsic in 2-step stratified Lie groups, we construct the solenoidal space by using the right invariant vector fields on the Heisenberg group.

A CFD hybrid approach to simulate liquid-phase chemical reactors

Design and safety assessments of chemical reactors can be done using Reynolds Averaged Navier-Stokes (RANS) type equations. The averaging procedure of transport equations gives rise to unclosed terms, which must be properly modeled independently on the computational cell size. In particular, presence of chemical reactions leads to an additional source term in species equation. The averaged value of this term involves effects of both chemical kinetics and turbulence. Turbulence-kinetics interaction (TKI) models must be then developed in order to close species transport equations, so that Computational Fluid Dynamics (CFD) can be used to reduce the number of experiments required to design a chemical reactor.Many TKI models have been developed in the past, mainly for gaseous systems, while liquid-phase models have been less investigated because of demanding theoretical challenges. Therefore, the purpose of this work is the development of a new TKI model for liquid phase reactions, which combines the Laminar Rate model (for kinetic controlled systems) with the Multiple Time Scales model (for turbulence controlled systems) allowing its use also when kinetic and turbulent mixing characteristic times are comparable. An analysis of the influence of the turbulence model coupled with the proposed model was carried out to identify the most suitable turbulence model, and two different case studies were investigated to show the potentialities of the proposed approach.

Periodic homogenization of the non-stationary Navier–Stokes type equations

In this paper we study the homogenization of the Cauchy–Dirichlet initial boundary value problems for the Navier–Stokes type equations, on one hand in an open set \(\Omega \) of \(\mathbb {R}^{N}\), on the other hand in porous media \(\Omega ^{\varepsilon }\). In the second case, the equations are classical unstationary Navier–Stokes one, and the porous media are periodic.

Transport equations with integral terms: existence, uniqueness and stability

We prove some theorems on the existence, uniqueness, stability and compactness properties of solutions to inhomogeneous transport equations with Sobolev coefficients, where the inhomogeneous term depends upon the solution through an integral operator. Contrary to the usual DiPerna–Lions approach, the essential step is to formulate the problem in the Lagrangian setting. Some motivations to study the above problem arise from the description of polymeric flows, where such kind of equations are coupled with other Navier–Stokes type equations. Using the results for the transport equation we will provide, in a separate paper, a sequential stability theorem for the full problem of the flow of concentrated polymers. At the end of the note we also point out a relevant example about the strong stability of the continuity equation, which highlights the role of an important assumption in our main stability statement.