Mass and momentum conservation of the least-squares spectral collocation method for the Navier–Stokes equations

From the literature, it is known that the Least-Squares Spectral Element Method (LSSEM) for the stationary Stokes equations performs poorly with respect to mass conservation but compensates this lack by a superior conservation of momentum. Furthermore, it is known that the Least-Squares Spectral Collocation Method (LSSCM) leads to superior conservation of mass and momentum for the stationary Stokes equations. In the present paper, we consider mass and momentum conservation of the LSSCM for time-dependent Stokes and Navier–Stokes equations. We observe that the LSSCM leads to improved conservation of mass (and momentum) for these problems. Furthermore, the LSSCM leads to the well-known time-dependent profiles for the velocity and the pressure profiles. To obtain these results, we use only a few elements, each with high polynomial degree, avoid normal equations for solving the overdetermined linear systems of equations and introduce the Clenshaw–Curtis quadrature rule for imposing the average pressure to be zero. Furthermore, we combined the transformation of Gordon and Hall (transfinite mapping) with the least-squares spectral collocation scheme to discretize the internal flow problems.     

Modulation equations and Reynolds averaging for finite-amplitude non-linear waves in an incompressible fluid

A formal perturbation scheme is developed to determine originalmodulation equations for laminar finite-amplitude non-linearwaves in an incompressible fluid. Three idealized problems areanalysed. The modulation equations comprise conservation ofwaves, averaged conditions for conservation of mass, momentum,kinetic energy and angular momentum and the averaged projectionof the Navier–Stokes equations onto the vorticity vector.The last of these modulation equations, which is related tovortex stretching, only appears in 3D problems. The techniqueof Reynolds averaging is also employed to obtain equations forthe mean velocities and pressure. The Reynolds-averaged Navier–Stokesequations correspond to the modulation equations for conservationof mass and momentum. However, the Reynolds stress transportequations are shown to be inconsistent with the other necessarymodulation equations. In two further idealized problems, exactsolutions of the Navier–Stokes equations are obtainedby employing the modulation equations.     

Explicit solution of the incompressible Navier–Stokes equations with linear finite elements

A three-field finite element scheme for the explicit iterative solution of the stationary incompressible Navier–Stokes equations is studied. In linearized form the scheme is associated with a generalized time-dependent Stokes system discretized in time. The resulting system of equations allows for a stable approximation of velocity, pressure and stress deviator tensor, by means of continuous piecewise linear finite elements, in both two- and three-dimensional space. Convergence in an appropriate sense applying to this finite element discretization is demonstrated, for the stationary Stokes system.     

Hydrodynamical instability initiation in two-phase stratified flow using Spectral Method

In this article, the hydrodynamical instability initiation criterion in two-phase stratified flow in a horizontal duct is examined. The nonlinear two mass and two momentum conservation equations are used for numerical simulation using the two-phase two-fluid model. The model is solved using the Finite Volume and Spectral Methods, respectively. This paper is the first to utilize the Spectral Method for the simulation of two-phase flow problems. Using the Spectral Method, we show that the numerical error and CPU time decreases noticeably relative to the Finite Volume Method. The well established Kelvin–Helmholtz (K–H) instability is selected for the test case and comparison. The results taken from each set of computer codes developed in this paper are highly compatible with the theoretical and experimental results of previous researchers who used alternative numerical methods. The results obtained from the Spectral Method in comparison with the results of other well known codes exhibit greater consistency with prior analytical results, but with much smaller computer calculation time. The step taken in the present study shows a positive progress in two-phase two-fluid model numerical solution with hydrostatic assumption. It is recommended the research to be continued with two-phase two-fluid model but with hydrodynamical assumption.     

Blow up of classical solutions to the isentropic compressible Navier–Stokes equations

In this work we consider the isentropic compressible Navier–Stokes equations in three space dimensions. Blow up result will be established, assuming the gradient of the velocity satisfies some decay constraint and the initial total momentum does not vanish. We prove the main result by a contradiction argument, based on the conservation of the total mass and the total momentum.     

Existence and zero‐electron‐mass limit of strong solutions to the stationary compressible Navier‐Stokes‐Poisson equation with large external force

In this study, we consider a viscous compressible model of plasma and semiconductors, which is expressed as a compressible Navier‐Stokes‐Poisson equation. We prove that there exists a strong solution to the boundary value problem of the steady compressible Navier‐Stokes‐Poisson equation with large external forces in bounded domain, provided that the ratio of the electron/ions mass is appropriately small. Moreover, the zero‐electron‐mass limit of the strong solutions is rigorously verified. The main idea in the proof is to split the original equation into 4 parts, a system of stationary incompressible Navier‐Stokes equations with large forces, a system of stationary compressible Navier‐Stokes equations with small forces, coupled with 2 Poisson equations. Based on the known results about linear incompressible Navier‐Stokes equation, linear compressible Navier‐Stokes, linear transport, and Poisson equations, we try to establish uniform in the ratio of the electron/ions mass a priori estimates. Further, using Schauder fixed point theorem, we can show the existence of a strong solution to the boundary value problem of the steady compressible Navier‐Stokes‐Poisson equation with large external forces. At the same time, from the uniform a priori estimates, we present the zero‐electron‐mass limit of the strong solutions, which converge to the solutions of the corresponding incompressible Navier‐Stokes‐Poisson equations.     

通过有漏孔管道时的层流解析解

研究了通过有漏孔管道时的层流,并解析地求解了动量方程和能量守恒方程．由Hagen-Poiseuille的速度分布,同时定义轴向和径向速度分量的未知函数,得到了压力和质量输运方程,并根据不同的参数,画出其分布图．结果表明,管道中的轴向速度、径向速度、质量输运参数和压力,随着流体沿管道的流动而减小．     

Review and complements on mixed-hybrid finite element methods for fluid flows

Mixed and hybrid finite element methods for the resolution of a wide range of linear and nonlinear boundary value problems (linear elasticity, Stokes problem, Navier–Stokes equations, Boussinesq equations, etc.) have known a great development in the last few years. These methods allow simultaneous computation of the original variable and its gradient, both of them being equally accurate. Moreover, they have local conservation properties (conservation of the mass and the momentum) as in the finite volume methods.The purpose of this paper is to give a review on some mixed finite elements developed recently for the resolution of Stokes and Navier–Stokes equations, and the linear elasticity problem. Further developments for a quasi-Newtonian flow obeying the power law are presented.     

Numerical investigation of blood flow. Part I: In microvessel bifurcations

In some diseases there is a focal pattern of velocity in regions of bifurcation, and thus the dynamics of bifurcation has been investigated in this work. A computational model of blood flow through branching geometries has been used to investigate the influence of bifurcation on blood flow distribution. The flow analysis applies the time-dependent, three-dimensional, incompressible Navier–Stokes equations for Newtonian fluids. The governing equations of mass and momentum conservation were solved to calculate the pressure and velocity fields. Movement of blood flow from an arteriole to a venule via a capillary has been simulated using the volume of fluid (VOF) method. The proposed simulation method would be a useful tool in understanding the hydrodynamics of blood flow where the interaction between the RBC deformation and blood flow movement is important. Discrete particle simulation has been used to simulate the blood flow in a bifurcation with solid and fluid particles. The fluid particle method allows for modeling the plasma as a particle ensemble, where each particle represents a collective unit of fluid, which is defined by its mass, moment of inertia, and translational and angular momenta. These kinds of simulations open a new way for modeling the dynamics of complex, viscoelastic fluids at the micro-scale, where both liquid and solid phases are treated with discrete particles.     

Continuous-time accelerated block successive overrelaxation methods for time-dependent Stokes equations

In order to solve the time-dependent Stokes equation, we follow the “Method of Lines” to obtain structured linear constant-coefficient differential–algebraic equations (DAEs). By taking advantage of the structure, we propose a class of waveform relaxation methods, called continuous-time accelerated block SOR (CABSOR) methods, for solving the obtained DAEs. The new methods are theoretically analyzed. The theory is applied to a two-dimensional time-dependent Stokes equation and verified by numerical experiments.     

Numerical Methods and CFD Techniques for Drug Application in Brain Tumor Treatment

In our work, the governing system of equations consists of a mass conservation equation, a momentum equation and an equation for the drug concentration in the brain tumor. This system describes the penetration of drugs into the brain tumor (there is a cavity after surgical removal of a cancer tumor), which fill up the cavity after a surgery. We use techniques of computational fluid dynamics (CFD) to get a solution of the derived partial differential equations (Navier–Stokes equation with additional scalar equations and force terms) and obtain a saddle point problem after discretization of the governing system of equations with finite elements such that we can use modern CFD tools and software like FEATFLOW to get numerical solutions of this problem. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Vanishing viscosity limit of the Navier‐Stokes equations to the euler equations for compressible fluid flow

We establish the vanishing viscosity limit of the Navier‐Stokes equations to the isentropic Euler equations for one‐dimensional compressible fluid flow. For the Navier‐Stokes equations, there exist no natural invariant regions for the equations with the real physical viscosity term so that the uniform sup‐norm of solutions with respect to the physical viscosity coefficient may not be directly controllable. Furthermore, convex entropy‐entropy flux pairs may not produce signed entropy dissipation measures. To overcome these difficulties, we first develop uniform energy‐type estimates with respect to the viscosity coefficient for solutions of the Navier‐Stokes equations and establish the existence of measure‐valued solutions of the isentropic Euler equations generated by the Navier‐Stokes equations. Based on the uniform energy‐type estimates and the features of the isentropic Euler equations, we establish that the entropy dissipation measures of the solutions of the Navier‐Stokes equations for weak entropy‐entropy flux pairs, generated by compactly supported C² test functions, are confined in a compact set in H^?1, which leads to the existence of measure‐valued solutions that are confined by the Tartar‐Murat commutator relation. A careful characterization of the unbounded support of the measure‐valued solution confined by the commutator relation yields the reduction of the measurevalued solution to a Dirac mass, which leads to the convergence of solutions of the Navier‐Stokes equations to a finite‐energy entropy solution of the isentropic Euler equations with finite‐energy initial data, relative to the different end‐states at infinity. © 2010 Wiley Periodicals, Inc.     

Asymptotic behavior and dynamic stability of phase mixtures for the equations of Navier–Stokes with nonmonotonic pressure

We investigate the asymptotic behavior of the solutions of the compressible Navier–Stokes equations with nonmonotonic pressure when the initial data is large and discontinuous. We provide sufficient conditions on the pressure function for different boundary-value problems that guarantee strong convergence of the volume variable as time approaches infinity and show that, typically, fairly arbitrary discontinuous static phase mixtures can be realized as time-asymptotic limits from smooth initial data. It is required in the analysis that we improve known existence theories, which typically have small data or time-dependent bounds.     

Asymptotic behavior and dynamic stability of phase mixtures for the equations of Navier–Stokes with nonmonotonic pressure

We investigate the asymptotic behavior of the solutions of the compressible Navier–Stokes equations with nonmonotonic pressure when the initial data is large and discontinuous. We provide sufficient conditions on the pressure function for different boundary-value problems that guarantee strong convergence of the volume variable as time approaches infinity and show that, typically, fairly arbitrary discontinuous static phase mixtures can be realized as time-asymptotic limits from smooth initial data. It is required in the analysis that we improve known existence theories, which typically have small data or time-dependent bounds.     

Heat transfer of a generalized stretching/shrinking wall problem with convective boundary conditions

In this paper, we investigate the heat transfer of a viscous fluid flow over a stretching/shrinking sheet with a convective boundary condition. Based on the exact solutions of the momentum equations, which are valid for the whole Navier–Stokes equations, the energy equation ignoring viscous dissipation is solved exactly and the effects of the mass transfer parameter, the Prandtl number, and the wall stretching/shrinking parameter on the temperature profiles and wall heat flux are presented and discussed. The solution is given as an incomplete Gamma function. It is found the convective boundary conditions results in temperature slip at the wall and this temperature slip is greatly affected by the mass transfer parameter, the Prandtl number, and the wall stretching/shrinking parameters. The temperature profiles in the fluid are also quite different from the prescribed wall temperature cases.     

Flow,heat, and species transfer over a stretching plate considering coupled Stefan blowing effects from species transfer

In this paper, we investigate the flow, heat and mass transfer of a viscous fluid flow over a stretching sheet by including the blowing effects of mass transfer under high flux conditions. Mass transfer in this work means species transfer and is different from mass transpiration for permeable walls. The new contribution from this work is, for the first time, to consider the coupled blowing effects from massive species transfer on flow, heat, and species transfer for a stretching plate. Based on the exact solutions of the momentum equations, which are valid for the whole Navier–Stokes equations, the energy and mass transfer equations are solved exactly and the effects of the blowing parameter, the Schmidt number, and the Prandtl number on the flow, heat and mass transfer are presented and discussed. The solution is given in terms of an incomplete Gamma function. It is found the coupled blowing effects due to mass transfer can have significant influences on velocity profiles, drag, heat flux, as well as temperature and concentration profiles. These solutions provide rare results with closed form analytical expressions and can be used as benchmark problem for numerical code validation.     

Problem of steady motion for a second-grade fluid in the hölder classes of functions

The paper studies a boundary-value problem (with the usual adherence boundary condition) for a stationary system of equations of motion of second-grade fluids in a bounded domain. This system is not elliptic and contains third-order derivatives of the velocity vector field. This leads to obvious difficulties in the analysis of the problem. It is known that the problem is reduced to the usual Stokes problem and to the transport equations or their analogs. We present a new easier method of such a reduction which allows us to prove the solvability of a stationary boundary-value problem for the equations of motion of second-grade fluids in the Hölder classes of functions in the case of small exterior forces. Bibliography: 6 titles.     

Analysis of the parareal approach based on discontinuous Galerkin method for time-dependent Stokes equations

This paper analyzes a parareal approach based on discontinuous Galerkin (DG) method for the time-dependent Stokes equations. A class of primal discontinuous Galerkin methods, namely variations of interior penalty methods, are adopted for the spatial discretization in the parareal algorithm (we call it parareal DG algorithm). We study three discontinuous Galerkin methods for the time-dependent Stokes equations, and the optimal continuous in time error estimates for the velocities and pressure are derived. Based on these error estimates, the proposed parareal DG algorithm is proved to be unconditionally stable and bounded by the error of discontinuous Galerkin discretization after a finite number of iterations. Finally, some numerical experiments are conducted which confirm our theoretical results, meanwhile, the efficiency of the parareal DG algorithm can be seen through a parallel experiment.     

Stability and convergence of efficient Navier‐Stokes solvers via a commutator estimate

For strong solutions of the incompressible Navier‐Stokes equations in bounded domains with velocity specified at the boundary, we establish the unconditional stability and convergence of discretization schemes that decouple the updates of pressure and velocity through explicit time stepping for pressure. These schemes require no solution of stationary Stokes systems, nor any compatibility between velocity and pressure spaces to ensure an inf‐sup condition, and are representative of a class of highly efficient computational methods that have recently emerged. The proofs are simple, based upon a new, sharp estimate for the commutator of the Laplacian and Helmholtz projection operators. This allows us to treat an unconstrained formulation of the Navier‐Stokes equations as a perturbed diffusion equation. © 2007 Wiley Periodicals, Inc.     

Mechanical Balance Laws for Boussinesq Models of Surface Water Waves

Depth-integrated long-wave models, such as the shallow-water and Boussinesq equations, are standard fare in the study of small amplitude surface waves in shallow water. While the shallow-water theory features conservation of mass, momentum and energy for smooth solutions, mechanical balance equations are not widely used in Boussinesq scaling, and it appears that the expressions for many of these quantities are not known. This work presents a systematic derivation of mass, momentum and energy densities and fluxes associated with a general family of Boussinesq systems. The derivation is based on a reconstruction of the velocity field and the pressure in the fluid column below the free surface, and the derivation of differential balance equations which are of the same asymptotic validity as the evolution equations. It is shown that all these mechanical quantities can be expressed in terms of the principal dependent variables of the Boussinesq system: the surface excursion ?? and the horizontal velocity w at a given level in the fluid.