A finite volume method for solving Navier-Stokes problems

We develop a finite volume method for solving the Navier-Stokes equations on a triangular mesh. We prove that the unique solution of the finite volume method converges to the true solution with optimal order for velocity and for pressure in discrete H¹ norm and L² norm respectively.     

A Petrov-Galerkin method for solving the generalized equal width (GEW) equation

The generalized equal width (GEW) equation is solved numerically by the Petrov-Galerkin method using a linear hat function as the test function and a quadratic B-spline function as the trial function. Product approximation has been used in this method. A linear stability analysis of the scheme shows it to be conditionally stable. Test problems including the single soliton and the interaction of solitons are used to validate the suggested method, which is found to be accurate and efficient. Finally, the Maxwellian initial condition pulse is studied.     

Global well-posedness of the velocity–vorticity-Voigt model of the 3D Navier–Stokes equations

The velocity–vorticity formulation of the 3D Navier–Stokes equations was recently found to give excellent numerical results for flows with strong rotation. In this work, we propose a new regularization of the 3D Navier–Stokes equations, which we call the 3D velocity–vorticity-Voigt (VVV) model, with a Voigt regularization term added to momentum equation in velocity–vorticity form, but with no regularizing term in the vorticity equation. We prove global well-posedness and regularity of this model under periodic boundary conditions. We prove convergence of the model's velocity and vorticity to their counterparts in the 3D Navier–Stokes equations as the Voigt modeling parameter tends to zero. We prove that the curl of the model's velocity converges to the model vorticity (which is solved for directly), as the Voigt modeling parameter tends to zero. Finally, we provide a criterion for finite-time blow-up of the 3D Navier–Stokes equations based on this inviscid regularization.     

Parallel shear flows of fluids with a temperature dependent viscosity

We investigate whether parallel shear flows of an incompressible Newtonian fluid with a viscosity which depends linearly on temperature is possible in situations where the temperature changes along the flow direction. It is shown that parallel flow is possible only in planar or axisymmetric geometries. These two situations are investigated further. For either a plane channel or a circular pipe, we show that the temperature variation in the flow direction must be exponential.Received: December 16, 2003; revised: October 11, 2004     

Convergence of higher order upwind finite volume schemes on unstructured grids for scalar conservation laws in several space dimensions

Summary. We prove convergence of a class of higher order upwind finite volume schemes on unstructured grids for scalar conservation laws in several space dimensions. The result is applied to the discontinuous Galerkin method due to Cockburn, Hou and Shu. Received April 15, 1993 / Revised version received March 13, 1995     

On the continuation principles for the Euler equations and the quasi-geostrophic equation

We obtain new continuation principle of the local classical solutions of the 3D Euler equations, where the regularity condition of the direction field of the vorticiy and the integrability condition of the magnitude of the vorticity are incorporated simultaneously. The regularity of the vorticity direction field is most appropriately measured by the Triebel-Lizorkin type of norm. Similar result is also obtained for the inviscid 2D quasi-geostrophic equation.     

Transient film profile of thin liquid film flow on a stretching surface

In this paper we have studied a non-planar thin liquid film flow on a planar stretching surface. The stretching surface is assumed to stretch impulsively from rest and the effect of inertia of the liquid is considered. Equations describing the laminar flow on the stretching surface are solved analytically. It is observed that faster stretching causes quicker thinning of the film on the stretching surface. Velocity distribution in the liquid film and the transient film profile as functions of time are obtained. (Received: May 4, 2004; revised: February 2/August 24, 2005)     

A removable isolated singularity theorem for the stationary Navier-Stokes equations

We show that an isolated singularity at the origin 0 of a smooth solution (u,p) of the stationary Navier-Stokes equations is removable if the velocity u satisfies u∈Lⁿ or |u(x)|=o(|x|^-1) as x→0. Here n?3 denotes the dimension. As a byproduct of the proof, we also obtain a new interior regularity theorem.     

Coherent Reflection of Electromagnetic Plane Waves from Infinite Rough Slabs

We adopt the prospect of an observer interested to optimise the signal-to-noise ratio in the reception of the backward radiation coming from a surface illuminated by an electromagnetic wave with a wavelength chosen to minimize the diffuse scattering so that he has just to point his receiver in the direction of the coherent reflection. Then, to analyse the coherent reflection for harmonic plane waves impinging on a dielectric infinite film deposited on a metallic substrate we develop a formalism generalizing the customary angular spectrum representation used to tackle this kind of problem. This new approach whose efficiency is proved in the easier situation of a dielectric film endowed with an impedance, is used to get the coherent reflection from a structured 1D-dielectric film illuminated by TE and TM electromagnetic plane waves when the rough amplitude h is small enough to justify 0(h ²) approximations. The Idemen technique is used to get the boundary conditions needed to tackle these scattering problems.