Navier-Stokes Type Research Articles

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.

Travel time prediction with viscoelastic traffic model

Travel time through a ring road with a total length of 80 km has been predicted by a viscoelastic traffic model (VEM), which is developed in analogous to the non-Newtonian fluid flow. The VEM expresses a traffic pressure for the unfree flow case by space headway, ensuring that the pressure can be determined by the assumption that the relevant second critical sound speed is exactly equal to the disturbance propagation speed determined by the free flow speed and the braking distance measured by the average vehicular length. The VEM assumes that the sound speed for the free flow case depends on the traffic density in some specific aspects, which ensures that it is exactly identical to the free flow speed on an empty road. To make a comparison, the open Navier-Stokes type model developed by Zhang (ZHANG, H. M. Driver memory, traffic viscosity and a viscous vehicular traffic flow model. Transp. Res. Part B, 37, 27–41 (2003)) is adopted to predict the travel time through the ring road for providing the counterpart results. When the traffic free flow speed is 80 km/h, the braking distance is supposed to be 45 m, with the jam density uniquely determined by the average length of vehicles l ≈ 5.8 m. To avoid possible singular points in travel time prediction, a distinguishing period for time averaging is pre-assigned to be 7.5 minutes. It is found that the travel time increases monotonically with the initial traffic density on the ring road. Without ramp effects, for the ring road with the initial density less than the second critical density, the travel time can be simply predicted by using the equilibrium speed. However, this simpler approach is unavailable for scenarios over the second critical.

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.

Hydrodynamic Diffusion in Integrable Systems.

We show that hydrodynamic diffusion is generically present in many-body, one-dimensional interacting quantum and classical integrable models. We extend the recently developed generalized hydrodynamic (GHD) to include terms of Navier-Stokes type, which leads to positive entropy production and diffusive relaxation mechanisms. These terms provide the subleading diffusive corrections to Euler-scale GHD for the large-scale nonequilibrium dynamics of integrable systems, and arise due to two-body scatterings among quasiparticles. We give exact expressions for the diffusion coefficients. Our results apply to a large class of integrable models, including quantum and classical, Galilean and relativistic field theories, chains, and gases in one dimension, such as the Lieb-Liniger model describing cold atom gases and the Heisenberg quantum spin chain. We provide numerical evaluations in the Heisenberg XXZ spin chain, both for the spin diffusion constant, and for the diffusive effects during the melting of a small domain wall of spins, finding excellent agreement with time-dependent density matrix renormalization group numerical simulations.

On some stationary Navier–Stokes type problems

The goal of this note is to present a simple proof of existence of solution to the stationary Navier–Stokes problem using a singular perturbation technique and to address some nonlocal issues related to it.

Modeling of the unsteady flow through a channel with an artificial outflow condition by the Navier–Stokes variational inequality

AbstractWe prove the global in time existence of a weak solution to the variational inequality of the Navier–Stokes type, simulating the unsteady flow of a viscous fluid through the channel, with the so‐called “do nothing” boundary condition on the outflow. The condition that the solution lies in a certain given, however arbitrarily large, convex set and the use of the variational inequality enables us to derive an energy‐type estimate of the solution. We also discuss the use of a series of other possible outflow “do nothing” boundary conditions.

Mathematical modeling of biofilm development

We perform mathematical analysis of the biofilm development process. A model describing biomass growth is proposed: It arises from coupling three parabolic nonlinear equations: a biomass equation with degenerate and singular diffusion, a nutrient transport equation with a biomass-density dependent diffusion, and an equation of the Navier–Stokes type, describing the fluid flow in which the biofilm develops. This flow is subject to a biomass’ density-dependent obstacle. The model is treated as a system of three inclusions, or variational inequalities; the equation of the Navier–Stokes type causes major difficulties for the system’s solvability. Our approach is based on the recent development of the theory on Navier–Stokes variational inequalities.

EFFECT OF VARYING THE DOUBLE RADIAL SWIRLER CONFIGURATION ON THE FLUID DYNAMIC AND EMISSIONS PERFORMANCES IN A CAN COMBUSTOR

The main purpose of this article is to present the fluid dynamic and combustion performances of a can combustor applying double radial flow swirler. Analysis was conducted using Computational Fluid Dynamics simulation (CFD) and compared with experimental works conducted by previous researchers. Air pollution, in the form of gaseous emissions from gas turbine generator reportedly increasing every year. Design modifications was identified and used in this study where air swirler been proven able to reduce the formation rate of gaseous emission such as NOx and CO during combustion. Swirler in a burner is a device that serves to improve the air and fuel mixture in the main combustion zone. Swirlers, presently studied are from a combination of two radial swirlers with different vane angle (primary and secondary) such as 30o/40o, 30o/50o and 30o/60o. Application of Ansys Fluent 14 CFD solvers was used in this study. Turbulent flow model used was from Reynolds Average Navier-Stokes (RANS) type involving k-ϵ. The fuel used is diesel. The parameters studied were wall temperature distribution along the combustor, the volume of NOx and volume of CO in the main combustion zone. Results from this study showed that the combination of 30o/60o generates the lowest gaseous emissions production rate. The highest average percentage error obtained by simulation studies compared to the actual experimental values for NOx was 22% and CO was 17.8%. Comparison between simulation and actual study proves that the simulation method can be used for preliminary decision and able to be used as benchmark to determine which is the best swirler configuration.

Global classical solutions to the 3D Navier–Stokes–Korteweg equations with small initial energy

In this paper, we investigate the global well-posedness of classical solutions to three-dimensional Cauchy problem of the compressible Navier–Stokes type system with a Korteweg stress tensor under the condition that the initial energy is small. This result improves previous results obtained by Hattori–Li in [H. Hattori and D. Li, Solutions for two dimensional system for materials of Korteweg type, SIAM J. Math. Anal. 25 (1994) 85–98; H. Hattori and D. Li. Global solutions of a high-dimensional system for Korteweg materials. J. Math. Anal. Appl. 198 (1996) 84–97.], where the existence of the classical solution is established for initial data close to an equilibrium in some Sobolev space [Formula: see text].

Moment‐matching based model reduction for Navier–Stokes type quadratic‐bilinear descriptor systems

AbstractWe discuss a Krylov subspace projection method for model reduction of a special class of quadratic‐bilinear descriptor systems. The goal is to extend the two‐sided moment‐matching method for quadratic‐bilinear ODEs to descriptor systems in an efficient and reliable way. Recent results have shown that the direct application of interpolation based model reduction techniques to linear descriptor systems, without any modifications, may lead to poor reduced‐order systems. Therefore, for the analysis, we transform the quadratic‐bilinear descriptor system into an equivalent quadratic‐bilinear ODE system for which the moment‐matching is performed. In view of implementation, we provide algorithms that identify the required Krylov subspaces without explicitly computing the projectors used in the analysis. The benefits of our approach are illustrated for the quadratic‐bilinear descriptor systems corresponding to semi‐discretized Navier–Stokes equations.

A Mixed-FEM for Navier–Stokes type variational inequality with nonlinear damping term

In this work we consider a mixed finite element method (FEM) for incompressible Navier–Stokes equations with nonlinear damping term and friction boundary conditions. After establishing the variational formulation, we show the well-posedness of this problem. Subsequently, we focus our attention on the mixed FEM and analyze the Galerkin approximation to this system. Then some optimal error estimates are deduced. In the end, some iterative algorithms are presented and numerical results are given to verify the theoretical analysis.

Variational inequalities of Navier–Stokes type with time dependent constraints

We consider a class of parabolic variational inequalities with time dependent obstacle of the form |u(x,t)|≤p(x,t), where u is the velocity field of a fluid governed by the Navier–Stokes variational inequality. The obstacle function p=p(x,t), imposed on u, consists of three parts, which are respectively: the degenerate part p(x,t)=0, the finitely positive part 0<p(x,t)<∞ and the singular part p(x,t)=∞. In this paper, we shall propose a sequence of approximate obstacle problems with everywhere finitely positive obstacles, and prove an existence result for the original problem by discussing convergence of the approximate problems. The crucial step is to handle the nonlinear convection term. In this paper we propose a new approach to it.

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.

On some practical issues concerning the implementation of Cahn–Hilliard–Navier–Stokes type models

Suspensions, solutions and colloids are physical systems that can be generally denoted as “multicomponent systems” or “mixtures”. Physical systems of this type are frequently met in many industrial applications which rises the need for numerical simulations of the behaviour of such complex systems. A particular practical example of such system is glass/tin/nitrogen system that must be studied in modelling of the float glass process (Pilkington process) that partly motivated this research. The aim of this paper is to discuss the numerical challenges along with some practical issues concerning the implementation of a chosen Cahn–Hilliard–Navier–Stokes type model that showed up to be particularly suitable for the description of three-component systems. The discretization of the system of governing partial differential equations is based on the finite element method using the FEniCS Project. Numerical experiments carried out in two and three spatial dimensions verify our straightforward implementation that uses parallel direct sparse solvers to resolve the intermediate linear systems of algebraic equations. Possible improvements of the current implementation are briefly outlined within the paper.

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.

Two-Level Defect-Correction Method for Steady Navier-Stokes Problem with Friction Boundary Conditions

AbstractIn this paper, we present two-level defect-correction finite element method for steady Navier-Stokes equations at high Reynolds number with the friction boundary conditions, which results in a variational inequality problem of the second kind. Based on Taylor-Hood element, we solve a variational inequality problem of Navier-Stokes type on the coarse mesh and solve a variational inequality problem of Navier-Stokes type corresponding to Newton linearization on the fine mesh. The error estimates for the velocity in the H1 norm and the pressure in the L2 norm are derived. Finally, the numerical results are provided to confirm our theoretical analysis.

Global existence of weak solutions for a nonlocal model for two-phase flows of incompressible fluids with unmatched densities

We consider a diffuse interface model for an incompressible isothermal mixture of two viscous Newtonian fluids with different densities in a bounded domain in two or three space dimensions. The model is the nonlocal version of the one recently derived by Abels, Garcke and Grün and consists in a Navier–Stokes type system coupled with a convective nonlocal Cahn–Hilliard equation. The density of the mixture depends on an order parameter. For this nonlocal system we prove existence of global dissipative weak solutions for the case of singular double-well potentials and non-degenerate mobilities. To this goal we devise an approach which is completely independent of the one employed by Abels, Depner and Garcke to establish existence of weak solutions for the local Abels et al. model.

On a non‐stationary fluid flow problem in an infinite periodic pipe

We study linearized, non‐stationary Navier–Stokes type equations with the given flux in an infinite pipe periodic of period length L with respect to . The existence and uniqueness of the solution is proved. Moreover, the convergence of the solution in a finite pipe of length to the L‐periodic solution as is investigated.

Stochastic PDEs and Lack of Regularity

We review results on the existence and uniqueness for a surface growth model with or without space–time white noise. If the surface is a graph, then this model has striking similarities to the three dimensional Navier-Stokes equations in terms of energy estimates and scaling properties, and in both models the question of uniqueness of global weak solutions remains open. In the physically relevant dimension $d=2$ and with the physically relevant space–time white noise driving the equation, the direct fixed-point argument for mild solutions fails, as there is not sufficient regularity for the stochastic forcing. The situation is the simplest case where the method of regularity structures introduced by Martin Hairer can be applied, although we follow here a significantly simpler approach to highlight the key problems. Using spectral Galerkin method or any other type of regularization of the noise, one can give a rigorous meaning to the stochastic PDE and show existence and uniqueness of local solutions in that setting. Moreover, several types of regularization seem to yield all the same solution. We finally comment briefly on possible blow up phenomena and show with a simple argument that many complex-valued solutions actually do blow up in finite time. This shows that energy estimates alone are not enough to verify global uniqueness of solutions. Results in this direction are known already for the 3D-Navier Stokes by Li and Sinai, treating complex valued solutions, and more recently by Tao by constructing an equation of Navier-Stokes type with blow up.