Efficient boundary element methods for three-dimensional exterior viscous flows

An efficient numerical algorithm for partial differential equations in complicated three-dimensional (3-D) geometries is developed in case of exterior viscous flows. The algorithm essentially consists of a boundary element method, where the resulting algebraic system is solved with a multigrid procedure. Our investigation covers and exact mathematical foundation, a detailed analysis of the discretization error, and some numerical tests with reasonable physical examples.     

Singular Cauchy problem for the equation of flow of thin viscous films with nonlinear convection

For multidimensional equations of flow of thin capillary films with nonlinear diffusion and convection, we prove the existence of a strong nonnegative generalized solution of the Cauchy problem with initial function in the form of a nonnegative Radon measure with compact support. We determine the exact upper estimate (global in time) for the rate of propagation of the support of this solution. The cases where the degeneracy of the equation corresponds to the conditions of “strong” and “weak” slip are analyzed separately. In particular, in the case of “ weak” slip, we establish the exact estimate of decrease in the L ²-norm of the gradient of solution. It is well known that this estimate is not true for the initial functions with noncompact supports. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 2, pp. 250–271, February, 2006.     

Investigation of boundary-value problem for slow flow of a sphere by viscous non-isothermal gas

We obtain a solution to a boundary-value problem of a flow of spherical form particle for stationary system of equations of viscous non-isothermal gaseous medium including the Stokes equation, heat conductivity equation, and state equation with account taken of dependence of viscosity, heat conductivity, and density of gaseous medium on temperature.     

Singular perturbations for third-order nonlinear multi-point boundary value problem

This paper is devoted to study the following third-order multi-point singularly perturbed boundary value problem

     

About steady flows of viscous compressible heat-conductive fluids on three-dimensional exterior domains

We are concerned with the general case of compressible heat-conducting Navier-Stokes equations in three-dimensional exterior domains: It is shown that for small external data and both zero or nonzero velocity at infinity, there exists a unique (steady) solution inL ^p-spaces,p > 3. We explain the approach and state the main results together with some a-priori estimates.     

A method for solving an exterior three-dimensional boundary value problem for the Laplace equation

We develop and experimentally study the algorithms for solving three-dimensionalmixed boundary value problems for the Laplace equation in unbounded domains. These algorithms are based on the combined use of the finite elementmethod and an integral representation of the solution in a homogeneous space. The proposed approach consists in the use of the Schwarz alternating method with consecutive solution of the interior and exterior boundary value problems in the intersecting subdomains on whose adjoining boundaries the iterated interface conditions are imposed. The convergence of the iterative method is proved. The convergence rate of the iterative process is studied analytically in the case when the subdomains are spherical layers with the known exact representations of all consecutive approximations. In this model case, the influence of the algorithm parameters on the method efficiency is analyzed. The approach under study is implemented for solving a problem with a sophisticated configuration of boundaries while using a high precision finite elementmethod to solve the interior boundary value problems. The convergence rate of the iterations and the achieved accuracy of the computations are illustrated with some numerical experiments.     

Analysis of singular perturbations on the Brinkman problem for fictitious domain models of viscous flows

We show the justification of a formulation by fictitious domain to study incompressible viscous flows inside fluid–porous–solid systems by using the Brinkman model in a heterogeneous fictitious porous medium covering the whole auxiliary domain. The singular perturbations of this problem are analysed when the permeability of the medium tends to infinity in the fluid domain and/or to zero in the solid domain. In addition, the viscosity inside the solid body may possibly tend to infinity. The strong convergence of the solutions is established. Some error estimates on the solution are derived as a function of the penalty parameter. The error bound on the resulting applied force for the flow around a bluff obstacle is also given. Copyright © 1999 John Wiley & Sons, Ltd.     

Solutions to a nonlinear Neumann problem in three-dimensional exterior domains

We prove the existence of multipeak solutions to a nonlinear elliptic Neumann problem involving nearly critical Sobolev exponent, in three-dimensional exterior domains.     

On the existence of free surface problem for viscous incompressible flow

In this paper we are concerned with the flow of a viscous, incompressible fluid in a bounded, three-dimensional region Ω with free surface boundary conditions. Using a method introduced by the author, that consider a two-fluid system in which the atmosphere or the vacuum is considered as a second fluid, separated from the first one by a free interface Γ(t), we prove existence of a kind of weak solution that we call quasi-weak solution.

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Sunto In questo lavoro studiamo il moto di un fluido viscoso e incomprimibile in una regione limitata tridimensionale Ω, con condizioni al contorno di superficie libera. Utilizzando un metodo, dovuto all’autore, che consiste nel considerare l’atmosfera o il vuoto come un secondo fluido, separato dal primo da un’interfaccia mobile Γ(t), dimostriamo l’esistenza di una sorta di soluzione debole, denominata soluzione quasi-debole.

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Work supported by Progetto Murst n. 9801262841.     

Global regularity of three-dimensional density patches for inhomogeneous incompressible viscous flow

Toward Lions' open question in Lions(1996) concerning the propagation of regularity for density patch, we prove that the boundary regularity of the 3-D density patch persists by time evolution for inhomogeneous incompressible viscous flows, with the initial density given by(1-η)1_(?_0)+ 1_(?_0~c )for some small enough constant η and some W~(k+2p),domain ?_0, p ∈]3, ∞[, and with the initial velocity satisfying some smallness condition and appropriate conormal regularities.     

The coupling of bem and fem for the two-dimensional viscous flow problem

In this paper we propose a numerical scheme for treating the problem of sJow viscous flow past an obstacle in the plane. This scheme is a combination of boundary element and finite element methods. By introducing an auxiliary boundary curve, we divide the region under consideration into two subregions, an inner and an outer region. In the inner region, we employ a finite element method (FEM) for solving a system of simplified field equations with proper natural boundary conditions. In the outer region, the solution is expressed in the form of a simple-layer potential with density function satisfying a system of modified integral equations of the first kind. The latter are solved by a boundary element method (BEM). Both solutions are matched on the common auxiliary boundary curve. Error estimates in suitable function spaces are derived in terms of the mesh widths as well as the small parameters, the Reynolds numbers     

Singular perturbations in the interaction representation

A general method for treating highly singular perturbations V of self-adjoint operators H in Hilbert space is applied to the case of perturbations of $\text{(?}\text{id}\text{dx}\text{)}{}^{\mathrm{n}}$ in L₂ ($\text{R}$¹ by (multiplications by) distributions. A self-adjoint operator H_V that agrees with H + V in the usual sense when V is sufficiently regular, and is moreover a continuous function of V, within the class of distributions under consideration, in the strong operator topology for unbounded self-adjoint operators, is shown to exist. This operator H_V need not be semi-bounded, or determined by a sesquilinear form associated with H + V. The method proceeds by construction of the corresponding unitary propagator in the interaction representation, essentially e^?itHVe^itH, which is shown to be expressible as a uniformly convergent perturbative series for small times.     

A variational conjugate gradient method for determining the fluid velocity of a slow viscous flow

The problem considered is that of determining the fluid velocity for linear hydrostatics Stokes flow of slow viscous fluids from measured velocity and fluid stress force on a part of the boundary of a bounded domain. A variational conjugate gradient iterative procedure is proposed based on solving a series of mixed well-posed boundary value problems for the Stokes operator and its adjoint. In order to stabilize the Cauchy problem, the iterations are ceased according to an optimal order discrepancy principle stopping criterion. Numerical results obtained using the boundary element method confirm that the procedure produces a convergent and stable numerical solution.     

The slow stationary flow of a viscous liquid about porous spheroids

The slow stationary flow of a viscous liquid about porous spheroids is examined. The solution describing the Stokes stream function is found to be a one-parameter family of surfaces and for special values of the parameter we recover the flow past impermeable spheroids. The ranges of the parameter are determined and the case of the porous sphere as well as that of a porous disc are deduced as limiting case.     

The exterior Dirichlet problem for a quasilinear elliptic boundary value problem suggested by plane shear flow

We study the existence of a classical solution of the exterior Dirichlet problem for a class of quasilinear elliptic boundary value problems that are suggested by plane shear flow. In this connection only bounded solutions which tend to zero at infinity are of interest. A priori bounds on solutions and constructive existence proofs are given. Finally, we prove the existence of a unique bounded solution of the shear flow and we show, under certain hypotheses about the asymptotic behavior of the nonlinearity, that this solution tends to zero at infinity. As an example, we consider the case of the parabolic shear flow.