Showing papers on "Navier–Stokes equations published in 2011"

An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Steady-State Problems

Abstract: Steady-State Solutions of the Navier-Stokes Equations: Statement of the Problem and Open Questions.- Basic Function Spaces and Related Inequalities.- The Function Spaces of Hydrodynamics.- Steady Stokes Flow in Bounded Domains.- Steady Stokes Flow in Exterior Domains.- Steady Stokes Flow in Domains with Unbounded Boundaries.- Steady Oseen Flow in Exterior Domains.- Steady Generalized Oseen Flow in Exterior Domains.- Steady Navier-Stokes Flow in Bounded Domains.- Steady Navier-Stokes Flow in Three-Dimensional Exterior Domains. Irrotational Case.- Steady Navier-Stokes Flow in Three-Dimensional Exterior Domains. Rotational Case.- Steady Navier-Stokes Flow in Two-Dimensional Exterior Domains.- Steady Navier-Stokes Flow in Domains with Unbounded Boundaries.- Bibliography.- Index.

On the Convergence of Discrete Approximations to the Navier-Stokes Equations

Abstract: A class of useful difference approximations to the full nonlinear Navier-Stokes equations is analyzed; the convergence of these approximations to the solutions of the corresponding differential equations is established and the rate of convergence is estimated.

The Navier-Stokes Equations: An Elementary Functional Analytic Approach

Abstract: Preface.- I Introduction.- II Preliminary Results.- III The Stationary Navier-Stokes Equations.- IV The Linearized Nonstationary Theory.- V The Full Nonlinear Navier-Stokes Equations.- Bibliography.- Index.ai

Mathematical Problems of Statistical Hydromechanics

Abstract: Table Contents- 1: Functional-Analytic Expansions of Solution of Evolution Equations- 2: Elements of Measure Theory- 3: Moment Theory for Small Reynolds Numbers- 4: Space-Time Statistical Solutions of the Navier-Stokes Equations for Arbitrary Reynolds Numbers- 5: The Hopf Equation- 6: Moment Theory for Arbitrary Reynolds Numbers- 7: Homogeneous Space-Time Statistical Solutions of Navier-Stokes Equations- 8: Individual Solutions with Unbounded Energy for Navier-Stokes Equations and Other Problems- 9: Analytic First Integrals and Asymptotic Behaviour as t ? ? of Fourier Coefficients of Solutions of Two-Dimensional Navier Stokes Equations- 10: Navier-Stokes System With White Noise In A Bounded Domain- 11: The Direct and Inverse Kolmogorov Equations Corresponding to a Stochastic Navier-Stokes System- 12: Homogeneous In x Solutions of the Stochastic Navier-Stokes System With White Noise- Appendix 1: Unique Solvability "In Large" of the Three-Dimensional Navier-Stokes System and Moment Equations for a Dense Set of Data- Appendix 2: Periodic Approximations of Homogeneous Measures- Comments- References

The granular column collapse as a continuum: validity of a two-dimensional Navier–Stokes model with a μ(I)-rheology

Abstract: There is a large amount of experimental and numerical work dealing with dry granular flows (such as sand, glass beads, etc.) that supports the so-called -rheology. The reliability of the -rheology in the case of complex transient flows is not fully ascertained, however. From this perspective, the granular column collapse experiment provides an interesting benchmark. In this paper we implement the -rheology in a Navier–Stokes solver (Gerris) and compare the resulting solutions with both analytical solutions and two-dimensional contact dynamics discrete simulations. In a first series of simulations, we check the numerical model in the case of a steady infinite two-dimensional granular layer avalanching on an inclined plane. A second layer of Newtonian fluid is then added over the granular layer in order to recover a close approximation of a free-surface condition. Comparisons with analytical and semi-analytical solutions provide conclusive validation of the numerical implementation of the -rheology. In a second part, we simulate the unsteady two-dimensional collapse of granular columns over a wide range of aspect ratios. Systematic comparisons with discrete two-dimensional contact dynamics simulations show good agreement between the two methods for the inner deformations and the time evolution of the shape during most of the flow, while a systematic underestimation of the final run-out is observed. The experimental scalings of spreading of the column as a function of the aspect ratio available from the literature are also recovered. A discussion follows on the performances of other rheologies, and on the sensitivity of the simulations to the parameters of the -rheology.

The Numerical Solution of the Navier-Stokes Equations for an Incompressible Fluid

Abstract: A finite difference method for solving the Navier-Stokes equations for an incompressible fluid has been developed. This method uses the primitive variables, i.e. the velocities and the pressure, and is equally applicable to problems in two and three space dimensions. Essentially it constitutes an extension to time dependent problems of the artificial compressibility method introduced in [ l ] for steady flow problems. The equations to be solved can be written in the dimensionless form

Global existence of solutions of the simplified Ericksen–Leslie system in dimension two

Abstract: In the 1960s, Ericksen and Leslie established the hydrodynamic theory for modelling liquid crystal flow. In this paper, we investigate a simplified model of the Ericksen–Leslie system, which is a system of the Navier–Stokes equations coupled with the harmonic map flow. We prove global existence of solutions to the Ericksen–Leslie system in $${\mathbb{R}^{2}}$$ with initial data, where the solutions are regular except for at a finite number of singular times.

Global Regularity Criterion for the 3D Navier–Stokes Equations Involving One Entry of the Velocity Gradient Tensor

Abstract: In this paper we provide a sufficient condition, in terms of only one of the nine entries of the gradient tensor, that is, the Jacobian matrix of the velocity vector field, for the global regularity of strong solutions to the three-dimensional Navier–Stokes equations in the whole space, as well as for the case of periodic boundary conditions.

Viscous Boundary Layers for the Navier-Stokes Equations with the Navier Slip Conditions

Abstract: We tackle the issue of the inviscid limit of the incompressible Navier–Stokes equations when the Navier slip-with-friction conditions are prescribed on impermeable boundaries. We justify an asymptotic expansion which involves a weak amplitude boundary layer, with the same thickness as in Prandtl’s theory and a linear behavior. This analysis holds for general regular domains, in both dimensions two and three.

Global mild solutions of Navier-Stokes equations

Abstract: We establish a global well-posedness of mild solutions to the three-dimensional, incompressible Navier-Stokes equations if the initial data are in the space X -1 defined by X -1 = {f E D'(R 3 ): ∫ ℝ 3 |ξ| -1 |f|dξ < ∞} and if the norms of the initial data in X -1 are bounded exactly by the viscosity coefficient μ.

The fluid/gravity correspondence

Abstract: We review the fluid/gravity correspondence which relates the dynamics of Einstein's equations (with negative cosmological constant) to the dynamics of relativistic Navier-Stokes equations.

$H^\infty$ Feedback Boundary Stabilization of the Two-Dimensional Navier-Stokes Equations

Abstract: We study the robust or $H^\infty$ exponential stabilization of the linearized Navier-Stokes equations around an unstable stationary solution in a two-dimensional domain $\Omega$. The disturbance is an unknown perturbation in the boundary condition of the fluid flow. We determine a feedback boundary control law, robust with respect to boundary perturbations, by solving a max-min linear quadratic control problem. Next we show that this feedback law locally stabilizes the Navier-Stokes system. Similar problems have been studied in the literature in the case of distributed controls and disturbances. To the authors' knowledge, it is the first time that the robust stabilization of the Navier-Stokes equations is studied for boundary controls and boundary disturbances.

A Self-Adaptive Strategy for the Time Integration of Navier-Stokes Equations

Abstract: An efficient self-adaptive strategy for the explicit time integration of Navier-Stokes equations is presented. Unlike the conventional explicit integration schemes, it is not based on a standard CFL condition. Instead, the eigenvalues of the dynamical system are analytically bounded and the linear stability domain of the time-integration scheme is adapted in order to maximize the time step. The method works independently of the underlying spatial mesh; therefore, it can be easily integrated into structured or unstructured codes. The additional computational cost is minimal, and a significant increase of the time step is achieved without losing accuracy. The effectiveness and robustness of the method are demonstrated on both a Cartesian staggered and an unstructured collocated formulation. In practice, CPU cost reductions up to more than 4 with respect to the conventional approach have been measured.

Global regularity for some classes of large solutions to the Navier-Stokes equations

Abstract: In previous works by the first two authors, classes of initial data to the three-dimensional, incompressible Navier-Stokes equations were presented, generating a global smooth solution although the norm of the initial data may be chosen arbitrarily large. The main feature of the initial data considered in one of those studies is that it varies slowly in one direction, though in some sense it is "well-prepared" (its norm is large but does not depend on the slow parameter). The aim of this article is to generalize that setting to an "ill prepared" situation (the norm blows up as the small parameter goes to zero). As in those works, the proof uses the special structure of the nonlinear term of the equation.

Entropy density of spacetime and the Navier-Stokes fluid dynamics of null surfaces

Abstract: It has been known for several decades that Einstein's field equations, when projected onto a null surface, exhibit a structure very similar to the nonrelativistic Navier-Stokes equation I show that this result arises quite naturally when gravitational dynamics is viewed as an emergent phenomenon Extremizing the spacetime entropy density associated with the null surfaces leads to a set of equations which, when viewed in the local inertial frame, becomes identical to the Navier-Stokes equation This is in contrast to the usual description of the Damour-Navier-Stokes equation in a general coordinate system, in which there appears a Lie derivative rather than a convective derivative I discuss this difference, its importance, and why it is more appropriate to view the equation in a local inertial frame The viscous force on fluid, arising from the gradient of the viscous stress-tensor, involves the second derivatives of the metric and does not vanish in the local inertial frame, while the viscous stress-tensor itself vanishes so that inertial observers detect no dissipation We thus provide an entropy extremization principle that leads to the Damour-Navier-Stokes equation, which makes the hydrodynamical analogy with gravity completely natural and obvious Several implications of these results are discussed

A Connection Between Scott-Vogelius and Grad-Div Stabilized Taylor-Hood FE Approximations of the Navier-Stokes Equations

Abstract: This article studies two methods for obtaining excellent mass conservation in finite element computations of the Navier-Stokes equations using continuous velocity fields. With a particular mesh construction, the Scott-Vogelius element pair has recently been shown to be inf-sup stable and have optimal approximation properties, while also providing pointwise mass conservation. We present herein the first numerical tests of this element pair for the time dependent Navier-Stokes equations. We also prove that the limit of the grad-div stabilized Taylor-Hood solutions to the Navier-Stokes problem converges to the Scott-Vogelius solution as the stabilization parameter tends to infinity. That is, we provide theoretical justification that choosing the grad-div parameter large does not destroy the solution. Numerical tests are provided which verify the theory and show how both Scott-Vogelius and grad-div stabilized Taylor-Hood (with large stabilization parameter) elements can provide accurate results with excellent mass conservation for Navier-Stokes approximations.

A High-Order Unifying Discontinuous Formulation for the Navier-Stokes Equations on 3D Mixed Grids

Abstract: The newly developed unifying discontinuous formulation named the correction pro- cedure via reconstruction (CPR) for conservation laws is extended to solve the Navier-Stokes equations for 3D mixed grids. In the current development, tetrahedrons and triangular prisms are considered. The CPR method can unify several popular high order methods including the dis- continuous Galerkin and the spectral volume methods into a more efficient differential form. By selecting the solution points to coincide with the flux points, solution reconstruction can be com- pletely avoided. Accuracy studies confirmed that the optimal order of accuracy can be achieved with the method. Several benchmark test cases are computed by solving the Euler and compress- ible Navier-Stokes equations to demonstrate its performance.

Experimental and numerical investigation of acoustic streaming excited by using a surface acoustic wave device on a 128° YX-LiNbO3 substrate

Abstract: This work uses a finite volume method to investigate three-dimensional acoustic streaming patterns produced by surface acoustic wave (SAW) propagation within microdroplets. A SAW microfluidic interaction has been modelled using a body force acting on elements of the fluid volume within the interaction area between the SAW and fluid. This enables the flow motion to be obtained by solving the laminar incompressible Navier–Stokes equations driven by an effective body force. The velocity of polystyrene particles within droplets during acoustic streaming has been measured and then used to calibrate the amplitudes of the SAW at different RF powers. The numerical prediction of streaming velocities was compared with the experimental results as a function of RF power and a good agreement was observed. This confirmed that the numerical model provides a basic understanding of the nature of 3D SAW/liquid droplet interaction, including SAW mixing and the concentration of particles suspended in water droplets.