On the asymptotic expansion of the solution of the plane Stokes problem in a perforated domain

2006 ◽  
Vol 135 (1) ◽  
pp. 2603-2615
Author(s):  
A. N. Averyanov
Keyword(s):  
Asymptotic Expansion ◽  
Stokes Problem ◽  
Perforated Domain

Convection in a horizontally heated sphere

2001 ◽  
Vol 438 ◽  
pp. 1-10 ◽  
Author(s):  
G. D. McBAIN
Keyword(s):  
Asymptotic Expansion ◽  
Body Force ◽  
Stokes Problem ◽  
Creeping Flow ◽  
Zeroth Order ◽  
First Order ◽  
Smooth Body ◽  
Spherical Fluid ◽  
Rayleigh Numbers ◽  
Spanwise Flow

Natural convection in horizontally heated spherical fluid-filled cavities is considered in the low Grashof number limit. The equations governing the asymptotic expansion are derived for all orders. At each order a Stokes problem must be solved for the momentum correction. The general solution of the Stokes problem in a sphere with arbitrary smooth body force is derived and used to obtain the zeroth-order (creeping) flow and the first-order corrections due to inertia and buoyancy. The solutions illustrate the two mechanisms adduced by Mallinson & de Vahl Davis (1973, 1977) for spanwise flow in horizontally heated cuboids. Further, the analytical derivations and expressions clarify these mechanisms and the conditions under which they do not operate. The inertia and buoyancy effects vanish with the Grashof and Rayleigh numbers, respectively, and both vanish if the flow is purely vertical, as in a very tall and narrow cuboid.

THE STOKES PROBLEM IN ℝn: AN APPROACH IN WEIGHTED SOBOLEV SPACES

1999 ◽  
Vol 09 (05) ◽  
pp. 723-754 ◽  
Author(s):  
F. ALLIOT ◽  
C. AMROUCHE
Keyword(s):  
Asymptotic Expansion ◽  
Sobolev Spaces ◽  
Vector Fields ◽  
Stokes Problem ◽  
Weighted Sobolev Spaces ◽  
Explicit Representation ◽  
Solenoidal Vector ◽  
Regularity Results

We prove some existence, uniqueness and regularity results for the solutions to the Stokes problem in ℝn, n≥2 in weighted Sobolev spaces [Formula: see text]. This framework enables us to characterise for which data the problem has solutions with prescribed decay or growth at infinity. Moreover, we obtain an explicit representation as well as an asymptotic expansion of the solution for non-smooth decaying data. We also establish the density of smooth solenoidal vector fields in the subspace of [Formula: see text] such that div v=0.

scholarly journals Topological asymptotic formula for the 3D non-stationary Stokes problem and application

2020 ◽  
Vol Volume 32 - 2019 - 2020 ◽  
Author(s):  
Maatoug Hassine ◽  
Rakia Malek
Keyword(s):  
Asymptotic Expansion ◽  
Stokes System ◽  
Stokes Problem ◽  
Three Dimensional ◽  
Detection Algorithm ◽  
International Audience ◽  
Flow Domain ◽  
Topological Asymptotic Expansion ◽  
Small Obstacle ◽  
Theoretical Results

International audience This paper is concerned with a topological asymptotic expansion for a parabolic operator. We consider the three dimensional non-stationary Stokes system as a model problem and we derive a sensitivity analysis with respect to the creation of a small Dirich-let geometric perturbation. The established asymptotic expansion valid for a large class of shape functions. The proposed analysis is based on a preliminary estimate describing the velocity field perturbation caused by the presence of a small obstacle in the fluid flow domain. The obtained theoretical results are used to built a fast and accurate detection algorithm. Some numerical examples issued from a lake oxygenation problem show the efficiency of the proposed approach. Ce papier porte sur l'analyse de sensibilité topologique pour un opérateur parabolique. On considère le problème de Stokes instationnaire comme un exemple de modèle et on donne une étude de sensibilité décrivant le comportement asymptotique de l'opérateur relativement à une petite perturbation géométrique du domaine. L'analyse présentée est basée sur une estimation du champ de vitesse calculée dans le domaine perturbé. Les résultats de cette étude ont servi de base pour développer un algorithme d'identification géométrique. Pour la validation de notre approche, on donne une étude numérique pour un problème d'optimisation d'emplacement des injecteurs dans un lac eutrophe. Des exemples numériques montrent l'efficacité de la méthode proposée

ESTIMATES FOR THE HOMOGENIZATION OF STOKES PROBLEM IN A PERFORATED DOMAIN

2018 ◽  
Vol 19 (1) ◽  
pp. 231-258 ◽  
Author(s):  
Amina Mecherbet ◽  
Matthieu Hillairet
Keyword(s):  
Stokes Equations ◽  
Stokes Problem ◽  
Brinkman Equation ◽  
Perforated Domain ◽  
Stat Phys ◽  
Quantitative Estimates

In this paper, we consider the Stokes equations in a perforated domain. When the number of holes increases while their radius tends to 0, it is proven in Desvillettes et al. [J. Stat. Phys. 131 (2008) 941–967], under suitable dilution assumptions, that the solution is well approximated asymptotically by solving a Stokes–Brinkman equation. We provide here quantitative estimates in $L^{p}$-norms of this convergence.

scholarly journals The geometric invariants for the spectrum of the Stokes operator

2021 ◽  
Author(s):  
Genqian Liu
Keyword(s):  
Asymptotic Expansion ◽  
Surface Area ◽  
Smooth Boundary ◽  
Stokes Problem ◽  
Stokes Operator ◽  
Bounded Domains ◽  
Spectral Invariants ◽  
Precise Information ◽  
Geometric Invariants ◽  
Dimensional Ball

AbstractFor a bounded domain $$\Omega \subset {\mathbb {R}}^n$$ Ω ⊂ R n with smooth boundary, we explicitly calculate the first two coefficients of the asymptotic expansion for the integral of the trace of the Stokes semigroup $$e^{-t S}$$ e - t S as $$t\rightarrow 0^+$$ t → 0 + . These coefficients (i.e., spectral invariants) provide precise information for the volume of the domain $$\Omega $$ Ω and the surface area of the boundary $$\partial \Omega $$ ∂ Ω by the spectrum of the Stokes problem. As an application, we show that an n-dimensional ball is uniquely determined by its Stokes spectrum among all Euclidean bounded domains with smooth boundary.

ASYMPTOTIC ANALYSIS AND PARTIAL ASYMPTOTIC DECOMPOSITION OF DOMAIN FOR STOKES EQUATION IN TUBE STRUCTURE

1999 ◽  
Vol 09 (09) ◽  
pp. 1351-1378 ◽  
Author(s):  
F. BLANC ◽  
O. GIPOULOUX ◽  
G. PANASENKO ◽  
A. M. ZINE
Keyword(s):  
Asymptotic Expansion ◽  
Asymptotic Analysis ◽  
Boundary Layers ◽  
Stokes Problem ◽  
Navier Stokes ◽  
Thin Domain ◽  
Asymptotic Decomposition ◽  
Small Parts ◽  
Rod Structures ◽  
Tube Structure

The Stokes problem posed in tube structures (or finite rod structures (see Panasenko10)), i.e. in connected finite unions of the thin cylinders with the ratio of the diameter to the height of the order [Formula: see text], is considered. The asymptotic expansion of the solution is built and justified. Boundary layers are studied. Earlier the Navier–Stokes problem in one thin domain was considered by Nazarov.8 The method of asymptotic partial decomposition of the domain (MAPDD) (see Panasenko11) is applied and justified for the Stokes problem posed in a tube structures. This method reduces the initial Stokes problem to the Stokes problem in some small parts of the domain (where the boundary layers are "concentrated".)

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