Solution Of Navier Research Articles

Global dynamics of large solution for the compressible Navier–Stokes–Korteweg equations

Global dynamics of large solution for the compressible Navier–Stokes–Korteweg equations

Global Spherically Symmetric Solutions for a Coupled Compressible Navier–Stokes/Allen–Cahn System

Global Spherically Symmetric Solutions for a Coupled Compressible Navier–Stokes/Allen–Cahn System

On the stability of stationary compressible Navier–Stokes flows in 3D

AbstractThis paper studies the stability of the stationary solution of the compressible Navier–Stokes equation in the 3D whole space with an external force which decays at spatial infinity. We obtain the global existence result of the non-stationary problem under the smallness assumptions on the initial perturbation around the small stationary solution. We also derive the time decay rates of the perturbations under the smallness assumption on the initial perturbations, and show the optimality of the time decay rates. The proofs are based on the combination of the spectral analysis and energy method in the framework of Besov spaces. The time-space integral estimate for the linearized semigroup around the constant state in some Besov spaces plays a crucial role in the proofs.

Existence of global weak solutions of inhomogeneous incompressible Navier–Stokes system with mass diffusion

Existence of global weak solutions of inhomogeneous incompressible Navier–Stokes system with mass diffusion

Phase transitions in the fractional three-dimensional Navier–Stokes equations

The fractional Navier–Stokes equations on a periodic domain [0,L]3 differ from their conventional counterpart by the replacement of the −νΔu Laplacian term by νsAsu , where A=−Δ is the Stokes operator and νs=νL2(s−1) is the viscosity parameter. Four critical values of the exponent s⩾0 have been identified where functional properties of solutions of the fractional Navier–Stokes equations change. These values are: s=13 ; s=34 ; s=56 and s=54 . In particular: (i) for s>13 we prove an analogue of one of the Prodi–Serrin regularity criteria; (ii) for s⩾34 we find an equation of local energy balance and; (iii) for s>56 we find an infinite hierarchy of weak solution time averages. The existence of our analogue of the Prodi–Serrin criterion for s>13 suggests the sharpness of the construction using convex integration of Hölder continuous solutions with epochs of regularity in the range 0<s<13 .

Approximate inner solvers for block preconditioning of the incompressible Navier–Stokes problems discretized by isogeometric analysis

AbstractWe deal with efficient numerical solution of the steady incompressible Navier–Stokes equations (NSE) using our in‐house solver based on the isogeometric analysis (IgA) approach. We are interested in the solution of the arising saddle‐point linear systems using preconditioned Krylov subspace methods. Based on our comparison of ideal versions of several state‐of‐the‐art block preconditioners for linear systems arising from the IgA discretization of the incompressible NSE, suitable candidates have been selected. In the present paper, we focus on selecting efficient approximate solvers for solving subsystems within these preconditioning methods. We investigate the impact on the convergence of the outer solver and aim to identify an effective combination. For this purpose, we compare convergence properties of the selected solution approaches for problems with different viscosity values, mesh refinement levels and discretization bases.

Blow-up of solutions for relaxed compressible Navier–Stokes equations

In this paper, we present a blow-up result for large data for relaxed compressible Navier–Stokes models avoiding the possibility of reaching the boundary of hyperbolicity. Thus, a previous result is improved and further examples are given illustrating possible effects of a relaxation and contrasting the classical compressible Navier–Stokes equations without relaxation where solutions for large data exist globally.

Global Solutions of 3D Isentropic Compressible Navier–Stokes Equations with Two Slow Variables

Global Solutions of 3D Isentropic Compressible Navier–Stokes Equations with Two Slow Variables

A KAM Approach to the Inviscid Limit for the 2D Navier–Stokes Equations

In this paper, we investigate the inviscid limit ν→0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ u \\rightarrow 0$$\\end{document} for time-quasi-periodic solutions of the incompressible Navier–Stokes equations on the two-dimensional torus T2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {T}}^2$$\\end{document}, with a small time-quasi-periodic external force. More precisely, we construct solutions of the forced Navier–Stokes equation, bifurcating from a given time quasi-periodic solution of the incompressible Euler equations and admitting vanishing viscosity limit to the latter, uniformly for all times and independently of the size of the external perturbation. Our proof is based on the construction of an approximate solution, up to an error of order O(ν2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(\ u ^2)$$\\end{document} and on a fixed point argument starting with this new approximate solution. A fundamental step is to prove the invertibility of the linearized Navier–Stokes operator at a quasi-periodic solution of the Euler equation, with smallness conditions and estimates which are uniform with respect to the viscosity parameter. To the best of our knowledge, this is the first positive result for the inviscid limit problem that is global and uniform in time and it is the first KAM result in the framework of the singular limit problems.

Ship Performances CFD Analysis of Hydrofoil-Supported High-Speed Catamaran Hull Form

Fast ferries with efficient fuel use are becoming increasingly in demand as fossil fuels become more in short supply and metropolises expand all over the globe in coastal and estuarine environments. These criteria need to be fulfilled by the Hydrofoil Supported Catamaran (HYSUCAT). A catamaran with wings between its demi hulls is known as an HYSUCAT, and it combines the positive traits of both the catamaran (manoeuvrability, big deck area) and the wing in-ground vehicle (excellent lift-to-drag ratio). In this research, a concept design Delft-372 catamaran is refitted with a high Reynolds number foil system to lower its overall resistance. The numerical solution of the Reynolds-averaged Navier–Stokes (RANS) equations employing the k-w SST turbulence model is used to simulate the flow around a high-speed catamaran with a hydrofoil. According to the simulation findings, the hydrofoil-assisted catamaran was determined to be a viable alternative to the current catamaran. This is due to a significant decrease in the overall resistance of the ship, reaching up to 42%, as well as notable improvements in pitch angle, reaching up to 26%.

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.

Unconditional energy conservation and conditional regularity for the incompressible Navier-Stokes Maxwell system

In this paper, we study a hydrodynamic system modeling the evolution of a plasma subject to a self-induced electromagnetic Lorentz force in incompressible viscous fluids. The system consists of the Navier–Stokes equations coupled with a Maxwell equation. In the three dimensional case, we show that every weak solution verifies the energy equality for the incompressible Navier–Stokes–Maxwell equations with damping. We also establish some non-explosion criteria in terms of the velocity and magnetic of local strong solution for standard Navier–Stokes–Maxwell system.

The Double‐Logarithmic Regularity Criterion of Pressure for the 3D Navier–Stokes Equations in the Besov Space

In the paper, we consider the regularity of the weak solutions for the incompressible 3D Navier–Stokes (N–S) equations. Our main result is the double‐logarithmic regularity criterion of pressure for the 3D Navier–Stokes equations in the Besov space. Our results extend and generalize previous works.

Group theoretic approach and analytical solutions of the compressible Navier–Stokes equations

Group theoretic approach and analytical solutions of the compressible Navier–Stokes equations

Modeling of Wave Processes in Hydraulic Drive Systems of Technological Equipment

The article, based on the performed theoretical research, solves the essential scientific and technical problem of increasing the accuracy of identification of wave processes in a hydrodynamic system (pipeline) by developing a generalized method of mathematical designing of the dynamics of a continuous viscous and weakly compressed fluid in the hydrodynamic system pipeline based on the Navier-Stokes equation. Amplitude-frequency characteristics represent parameters of wave processes in the hydraulic drive system. A partial solution of Navier–Stokes equations, under zero initial conditions, is proposed in the form of four-pole equations, the components of which are represented in the form of the Laplace image of the corresponding relative pressure and flow coordinates and the the hydraulic line parameters determine the four-pole elements themselves It is also proposed to determine the values of the four-pole elements based on time constants and relative damping coefficients on the frequency characteristics of hydraulic lines with distribution parameters based on the condition of equality of the first resonant frequencies and amplitudes (at these frequencies). With the help of the developed methods, the primary dynamic parameters of the amplitude-frequency characteristics of continuous viscous and weakly compressed liquid in the pipeline of hydraulic systems for different flow ranges. This made it possible to achieve the following practical results: the high degree of adequacy of the developed mathematical model indicates an increase in the reliability of determining the operating characteristics when designing a hydraulic drive. The high accuracy of determining the first resonant frequencies and amplitudes allows for creating a hydraulic pump with improved operational characteristics.

Designing the edges of holes (with bias flow) to maximise acoustic damping

Circular holes with a mean bias flow passing through them can amplify or damp acoustic energy and this property is relevant for many industrial applications. In this work, we propose a methodology to design the edges of such holes so that the acoustic damping is maximised. The approach relies on a Bayesian optimisation framework and is illustrated in a short circular hole with a mean laminar bias flow. The acoustic response of the perforation is characterised numerically using a two-step approach where, first, a steady mean flow is computed as the solution of the incompressible Navier–Stokes equations. Second, small-amplitude acoustic perturbations are superimposed on this mean flow and their dynamics are obtained as the solution of the linearised compressible Navier–Stokes equations. Both the upstream and downstream edges of the hole are modified with 45°chamfers. The sizes of these two chamfers are the control parameters optimised to maximise the acoustic absorption coefficient. The results of this letter show that the careful design of the edges can dramatically increase the acoustic energy that holes can damp: the hole investigated here goes from one generating 55% more acoustic energy than incident upon it at a given frequency to one that damps 46% of this acoustic energy.

Geometric regularity criteria for the Navier–Stokes equations in terms of velocity direction

In this paper, we are inspired by a famous result by Constantin and Fefferman who proved that a simple geometrical assumption on the direction of the vorticity leads to the regularity of weak solutions of the 3D Navier–Stokes equations. We show that the same result can be achieved if the vorticity direction is replaced by the velocity direction. We further strengthen this result and prove that in fact it is not necessary to consider the velocity direction in all close space points but only in the points whose distance equals to a small positive number dependant on the data. In the second part of the paper, we extend a result by Berselli and C rdoba concerning the role of the helicity for the regularity of the weak solutions of the Navier–Stokes equations.

On the Stability of Periodic Solutions of a Model Navier–Stokes Equation in a Thin Layer

We study the existence and stability of periodic solutions of the model Navier–Stokes equation in a thin three-dimensional layer depending on the existence and stability of periodic solutions of a special limit two-dimensional equation.

On weak (measure valued)–strong uniqueness for Navier–Stokes–Fourier system with Dirichlet boundary condition

In this article, our goal is to define a measure valued solution of compressible Navier–Stokes–Fourier system for a heat conducting fluid with Dirichlet boundary condition for temperature in a bounded domain. The definition will be based on the weak formulation of entropy inequality and ballistic energy inequality. Moreover, we obtain the weak (measure valued)–strong uniqueness property of this solution with the help of relative energy.

An Eulerian finite element method for tangential Navier-Stokes equations on evolving surfaces

The paper introduces a geometrically unfitted finite element method for the numerical solution of the tangential Navier–Stokes equations posed on a passively evolving smooth closed surface embedded in R 3 \mathbb {R}^3 . The discrete formulation employs finite difference and finite elements methods to handle evolution in time and variation in space, respectively. A complete numerical analysis of the method is presented, including stability, optimal order convergence, and quantification of the geometric errors. Results of numerical experiments are also provided.