Axial eigenfunctions and inhomogeneous solutions for the three-dimensional Stokes system with an application to natural convection in a cylinder

This paper presents two contributions to the analysis of three-dimensional slow viscous flows in cylinders of circular section. First the vector axial eigenfunctions for this geometry, namely those that satisfy homogeneous boundary conditions on the flat end walls, are derived. Secondly a method is presented to find particular solutions to the inhomogeneous Stokes equations in this geometry. These new results, together with some results obtained earlier, are used to analyse slow natural convection in a vertical cylinder completely filled with a viscous liquid. The fluid motion is generated by the differential heating of the walls of the cylinder. The natural convection flow field is shown to be a superposition of an inhomogeneous field, the fields generated by the vector eigenfunctions and a Stokes flow field. A by-product of this work has been the identification of constraints on the boundary data that have to be satisfied in order for the eigenfunction expansions to work; this knowledge will be useful when attempts are made to prove the completeness of these Stokes flow eigenfunctions.     

The Cosserat eigenfunctions for the elliptic exterior problem with applications to thermoelasticity and Stokes flow

The Cosserat eigenvalue problem for the elliptic exterior is considered. It is shown that the only Cosserat eigenvalue different from the infinite multiple eigenvalues and is which also has an infinite multiplicity. The orthogonal basis of the eigenspace corresponding to is constructed. Application to thermoelasticity and Stokes flow – extensional and shear – past a rigid elliptical cylinder are presented and agreement is obtained with classical solutions for a circle. The fact that the solutions consist of only two eigenfunctions reveals the efficiency of the method.Received: February 27, 2003     

Explicit analytical solutions of the anisotropic Brinkman model for the natural convection in porous media

Some algebraically explicit analytical solutions are derived for the anisotropic Brinkman model-an improved Darcy model-describing the natural convection in porous media. Besides their important theoretical meaning (for example, to analyze the non-Darcy and anisotropic effects on the convection), such analytical solutions can be the benchmark solutions to promoting the development of computational heat and mass transfer. For instance, we can use them to check the accuracy, convergence and effectiveness of various numerical computational methods and to improve numerical calculation skills such as differential schemes and grid generation ways.     

Stokes flow solutions in infinite and semi-infinite porous channels

We derive a class of exact solutions for Stokes flow in infinite and semi-infinite channel geometries with permeable walls. These simple, explicit, series expressions for both pressure and Stokes flow are valid for all permeability values. At the channel walls, we impose a no-slip condition for the tangential fluid velocity and a condition based on Darcy's law for the normal fluid velocity. Fluid flow across the channel boundaries is driven by the pressure drop between the channel interior and exterior; we assume the exterior pressure to be constant. We show how the ground state is an exact solution in the infinite channel case. For the semi-infinite channel domain, the ground-state solutions approximate well the full exact solution in the bulk and we derive a method to improve their accuracy at the transverse wall. This study is motivated by the need to quantitatively understand the detailed fluid dynamics applicable in a variety of engineering applications including membrane-based water purification, heat and mass transfer, and fuel cells.     

On the interior regularity of weak solutions to the non-stationary Stokes system

In this paper, we prove that any weak solution to the non-stationary Stokes system in 3D with right hand side -div f satisfying (1.4) below, belongs to C( ]0, T[; C ^α (Ω)). The proof is based on Campanato-type inequalities and the existence of a local pressure introduced in Wolf [13]. Proc. Conference “Variational analysis and PDE’s”. Intern. Centre “E. Majorana”, Erice, July 5–14, 2006.     

Long‐time existence of solutions to the Navier–Stokes equations with inflow–outflow and heat convection

Long time existence of regular solutions to the Navier–Stokes equations for velocity and pressure coupled with the heat convection equation for temperature in cylindrical pipe with inflow and outflow is shown. We assume the slip boundary conditions for velocity and the Neumann conditions for temperature. First, an appropriate estimate is shown, and next the existence of solutions is proved by the Leray–Schauder fixed point theorem. The estimate is obtained for a long time, which is possible because L₂ norms of derivatives in the direction along the cylinder of the initial velocity, initial temperature and the external force are sufficiently small. Copyright © 2012 John Wiley & Sons, Ltd.     

Spatial behaviour of solutions in the plane Stokes flow

Some new methods are described in the study of the spatial behaviour of solutions in the slow flow of an incompressible viscous fluid along a semi-infinite strip subject to zero velocity on the lateral sides, a prescribed time-dependent specified velocity on the end and zero initial conditions. These methods provide an improved estimated decay rate over those previously predicted on the subject.     

On the convergence of finite element solutions to the interface problem for the Stokes system

The Stokes system with a discontinuous coefficient (Stokes interface problem) and its finite element approximations are considered. We firstly show a general error estimate. To derive explicit convergence rates, we introduce some appropriate assumptions on the regularity of exact solutions and on a geometric condition for the triangulation. We mainly deal with the MINI element approximation and then consider P1-iso-P2/P1 element approximation. Results are expected to give an instructive remark in numerical analysis for two-phase flow problems.     

Mathematical analysis of sideband instabilities with application to rayleigh-bénard convection

Summary We introduce a new method for the analysis of sideband instabilities which are important for periodic patterns appearing in systems close to the instability threshold. The method relies on a two-fold application of the Liapunov-Schmidt reduction procedure, a first application to the nonlinear bifurcation problem and a second application to the linear spectral problem. We obtain rigorous results on the spectrum of the associated linearization in spaces allowing for general sideband perturbations by treating the sideband vector and the spectral parameter as small bifurcation parameters. We apply the theory to the small roll solutions in the Rayleigh-Bénard convection and derive domains in Rayleigh, Prandtl, and wave number space where the rolls are unstable. We recover the Eckhaus, zigzag, and skew-varicose instabilities obtained earlier by formal methods. This paper is dedicated to the memory of Juan C. Simo This paper was solicited by the editors to be part of a volume dedicated to the memory of Juan C. Simo.     

Uniform preconditioners for a parameter dependent saddle point problem with application to generalized Stokes interface equations

We consider an abstract parameter dependent saddle-point problem and present a general framework for analyzing robust Schur complement preconditioners. The abstract analysis is applied to a generalized Stokes problem, which yields robustness of the Cahouet-Chabard preconditioner. Motivated by models for two-phase incompressible flows we consider a generalized Stokes interface problem. Application of the general theory results in a new Schur complement preconditioner for this class of problems. The robustness of this preconditioner with respect to several parameters is treated. Results of numerical experiments are given that illustrate robustness properties of the preconditioner.     

Lp estimates of solutions tomixed boundary value problems for the Stokes system in polyhedral domains

A mixed boundary value problem for the Stokes system in a polyhedral domain is considered. Here different boundary conditions (in particular, Dirichlet, Neumann, free surface conditions) are prescribed on the faces of the polyhedron. The authors prove the existence of solutions in (weighted and non‐weighted) L_p Sobolev spaces and obtain regularity assertions for weak solutions. The results are based on point estimates of Green's matrix. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Weighted L~p-estimates for Stokes flow in R_+~n with applications to the non-stationary Navier-Stokes flow

We study the time-decay properties of weighted norms of solutions to the Stokes equations and the Navier-Stokes equations in the half-space Rn+ (n 2). Three kinds of the weighted Lp-Lr estimates are established for the Stokes semigroup generated by the Stokes operator in the half-space R+n (n 2). As an application of the weighted estimates of the Stokes semigroup, a class of local and global strong solutions in weighted Lp (R+n) are constructed, following the approach given by Kato.     

Global existence of weak solutions for the anisotropic compressible Stokes system

In this paper, we study the problem of global existence of weak solutions for the quasi-stationary compressible Stokes equations with an anisotropic viscous tensor. The key idea is a new identity that we obtain by comparing the limit of the equations of the energies associated to a sequence of weak-solutions with the energy equation associated to the system verified by the limit of the sequence of weak-solutions. In the context of stability of weak solutions, this allows us to construct a defect measure which is used to prove compactness for the density and therefore allowing us to identify the pressure in the limiting model. By doing so we avoid the use of the so-called effective flux. Using this new tool, we solve an open problem namely global existence of solutions à la Leray for such a system without assuming any restriction on the anisotropy amplitude. This provides a flexible and natural method to treat compressible quasilinear Stokes systems which are important for instance in biology, porous media, supra-conductivity or other applications in the low Reynolds number regime.     

On the time-periodic problem for the Stokes system in domains with cylindrical outlets to infinity

We study the time-periodic Stokes problem in the domain with cylindrical outlets to infinity in weighted function spaces. We prove that there exists a unique solution with prescribed fluxes over the sections of outlets to infinity and that, in each outlet, this solution tends to the corresponding time-periodic Poiseuille flow. Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 2, pp. 177–195, April–June, 2007.     

A fast finite difference method for biharmonic equations on irregular domains and its application to an incompressible Stokes flow

Biharmonic equations have many applications, especially in fluid and solid mechanics, but is difficult to solve due to the fourth order derivatives in the differential equation. In this paper a fast second order accurate algorithm based on a finite difference discretization and a Cartesian grid is developed for two dimensional biharmonic equations on irregular domains with essential boundary conditions. The irregular domain is embedded into a rectangular region and the biharmonic equation is decoupled to two Poisson equations. An auxiliary unknown quantity Δu along the boundary is introduced so that fast Poisson solvers on irregular domains can be used. Non-trivial numerical examples show the efficiency of the proposed method. The number of iterations of the method is independent of the mesh size. Another key to the method is a new interpolation scheme to evaluate the residual of the Schur complement system. The new biharmonic solver has been applied to solve the incompressible Stokes flow on an irregular domain.