Stokes Type Problem Research Articles

Regularity Results for a Navier–Stokes Type Problem Related to Oceanography

The problem studied in this paper derives from the modelization of an oceanic phenomenon occurring in tropical zones: the presence of zonal waves superimposed on the mean currents. Our assumptions are justified by the physical features: we have studied the perturbation of a given mean flow that verifies Navier–Stokes type equations and mixed boundary conditions of Dirichlet, Neumann or periodic types. The existence and uniqueness of the solution are studied for the problem linearized around the mean flow and for the complete nonlinear problem. In the second case, we obtain sufficient conditions, which are physically sensible: the space gradient of the mean velocity, the perturbation of the windstress and its time derivative cannot be too large. The value of the pressure pon the surface of the ocean is of great interest for physical interpretation. To define that quantity, it is necessary to have the regularity p e H1Ω. We have proved that the perturbation (u,p) of the mean circulation is such that u e L2(0,T,H2Ω)), p e L2(0,T,H1Ω)), provided the perturbation of the windstress is sufficiently regular and satisfies compatibility relations. This result is valid for the linearized and the nonlinear cases, in a domain Ω with corners. It is proved by means of an extension method, with even-odd reflection.

A spectral algorithm for the Stokes problem in vorticity-vector potential formulation and cylindrical geometry

Abstract In the numerical computation of incompressible flows using the Navier-Stokes equations, the main task generally lies in the solution of a Stokes type problem. In this paper, we focus on this key point, when using (i) a spectral method, (ii) the vorticity-vector potential formulation of the 3D Navier-Stokes equations and (iii) a cylindrical geometry. At the end we conclude with a numerical test of our Stokes solver in the framework of Rayleigh-Benard natural convection.

Spectral multigrid techniques for the Navier-Stokes equations

We present a spectral multigrid method for the effective solution of the Navier-Stokes equations for incompressible fluids. Here we consider the vorticity-streamfunction formulation. Discretization in time is performed by a second order semi-implicit scheme where the diffusive term is treated in an implicit manner while the convective term is evaluated explicitly. For the discretization of the resulting Stokes-type problem, we introduce a new spectral collocation method. This scheme is highly accurate and allows efficient solution by spectral multigrid methods. We propose an efficient finite difference preconditioner. Numerical results are presented which substantiate the usefulness of our concept.

The spectrum of a Chebyshev-Fourier approximation for the Stokes equations

A Chebyshev-Fourier approximation to the solution of the two-dimensional Stokes equations in the vorticity-streamfunction formulation is considered. The expansion in a Fourier series in the direction of periodicity leads to a family of one-dimensional Stokes-type problems being approximated by a Chebyshevcollocation method. First some results about the spectrum of the corresponding operators are derived. Then we consider a discretization in time by means of a class of semi-implicit finite differences schemes and we describe the influence matrix technique used to solve the resulting system at every time step. The properties of the spectrum of the Chebyshev-Stokes operator are used to derive some results about the stability of the resulting time marching algorithm.

A regular perturbation approach to surface tension driven flows

Surface tension induced convection in a liquid bridge held between two parallel, coaxial, solid disks is considered. The surface tension gradient is produced by a small temperature gradient parallel to the undisturbed surface. The study is performed by using a mathematical regular perturbation approach based on a small parameter, ϵ, which measures the deviation of the imposed temperature field from its mean value. The first order velocity field is given by a Stokes-type problem (viscous terms are dominant) with relatively simple boundary conditions. The first order temperature field is that imposed from the end disks on a liquid bridge immersed in a non-conductive fluid. Radiative effects are supposed to be negligible. The second order temperature field, which accounts for convective effects, is split into three components, one due to the bulk motion, and the other two to the distortion of the free surface. The relative importance of these components in terms of the heat transfer to or from the end disks is assessed.