Oseen Operator Research Articles

On the $$C_0$$ semigroup generated by the Oseen operator around a steady flow exterior to a rotating obstacle

We consider the motion of an incompressible viscous fluid filling the whole space exterior to a moving with rotation and translation obstacle. We show that the Stokes operator around the steady flow in the exterior of this obstacle generates a $$C_0$$ -semigroup in $$L^p$$ space and then develop a series of $$L^p$$ – $$L^q$$ estimates of such semigroup. As an application, we give out the stability of such steady flow when the initial disturbance in $$L^3$$ and the steady flow are sufficiently small.

Spatial asymptotics of mild solutions to the time-dependent Oseen system

We consider mild solutions to the 3D time-dependent Oseen system with homogeneous Dirichlet boundary conditions, under weak assumptions on the data. Such solutions are defined via the semigroup generated by the Oseen operator in \begin{document}$ L^q. $\end{document} They turn out to be also \begin{document}$ L^q $\end{document} -weak solutions to the Oseen system. On the basis of known results about spatial asymptotics of the latter type of solutions, we then derive pointwise estimates of the spatial decay of mild solutions. The rate of decay depends in particular on \begin{document}$ L^p $\end{document} -integrability in time of the external force.

Pressure - velocity projection method with mixed type approximation for Oseen discrete operator

We are considering the problem of divergence - free projection in rectangular domains for the Oseen (stationary Navier-Stokes) operator. It uses nodal Finite Element method to reconstruct pressure and Finite Difference method for scalar functions and vector functions for other parts of the equations. This allows us to use simple finite difference schemes for linear part and WENO-type schemes for the the nonlinear part in the space of discrete velocities. The boundary conditions are formulated in Finite Volume conservative approach for velocity and in Finite Element approach for pressure. We illustrate that the convergence rate depends on the smoothness of the projected function and that the projection method allows one to obtain divergence-free solutions with desired tolerance. We numerically analize the projection operator that acts in the space of divergence operator image. It is shown that all eigenvalues of the operator are less than unity for Oseledets-type wall boundary conditions for velocity and zero Neumann conditions for the Poisson equation. We show that for smooth functions we obtain forth order convergence in divergence. Finally we demonstrate the method by solving the stationary lid driven cavity problem for the 2D Oseen operator.

Spectra of operators associated with flow around an infinite cylinder

Numerical approximations of the spectrum of the Oseen operator and the spectrum of the linearised Navier-Stokes operator for flow around a cylinder in two dimensions have been studied for Reynolds numbers in the range 2 to 40. By using a spectral method featuring basis functions covering the entire exterior domain, it is possible to obtain a numerical approximation to the continuous spectrum and the point spectrum. The numerical approximation of the spectra seems to agree with the expected rigorous results. Namely a parabolic tongue containing the continuous spectra for the Oseen operator and for the linearised Navier--Stokes operator a parabolic tongue containing the continuous part of the spectrum plus a finite number of isolated eigenvalues.

On 2-D Boussinesq equations for MHD convection with stratification effects

This paper is concerned with the two-dimensional magnetohydrodynamics-Boussinesq system with the temperature-dependent viscosity, thermal diffusivity and electrical conductivity. The first progress on this topic was made independently by Chae and Hou–Li [8,26] where the Boussinesq system with partial constant viscosity is obtained. Recently, Wang–Zhang [45] considered the temperature-dependent viscosity and thermal diffusivity, and Li–Xu [16] generalized the Wang–Zhang's result to the inviscid case with temperature-dependent thermal diffusivity. In this paper, we include the stratification and magnetic effects and consider the full system, in the framework of low regularity. We prove that, without any smallness assumption on the initial data, the full system is globally well-posed. Moreover, by applying the uniformly bounded generalized Oseen operator, time decay estimate of the solution is obtained.

The internal stabilization of the Stokes–Oseen equation by feedback point controllers

One designs a stabilizable feedback controller for the Stokes–Oseen equation in an open domain O⊂Rd, d≥2, with support in a finite set of points ξk∈O, j=1,…,M. More precisely, Md≥N, where N is the number of unstable eigenvalues of the Stokes–Oseen operator. The controller is expressed in terms of eigenfunctions φj∗,1≤j≤N, of the dual Stokes–Oseen operator.

On a mapping property of the Oseen operator with rotation

The Oseen problem arises as the linearization of a steady-state Navier-Stokes flow past a translating body. If the body, in addition to the translational motion, is also rotating, the corresponding linearization of the equations of motion, written in a frame attached to the body, yields the Oseen system with extra terms in the momentum equation due to the rotation. In this paper, the effect these rotation terms have on the asymptotic structure at spatial infinity of a solution to the system is studied. A mapping property of the whole space Oseen operator with rotation is identified from which asymptotic properties of a solution can be derived. As an application, an asymptotic expansion of a steady-state, linearized Navier-Stokes flow past a rotating and translating body is established.

The exterior Dirichlet and the interior Neumann boundary value problems for the scalar Oseen equation

AbstractThe scalar Oseen equation represents a linearized form of the Navier Stokes equations. We present an explicit potential theory for this equation and solve the exterior Dirichlet and interior Neumann boundary value problems via a boundary integral equations method in spaces of continuous functions on a C2‐boundary, extending the classical approach for the isotropic selfadjoint Laplace operator to the anisotropic non‐selfadjoint scalar Oseen operator. (© 2012 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

The boundary value problems for the scalar Oseen equation

AbstractThe scalar Oseen equation represents a linearized form of the Navier Stokes equations, well‐known in hydrodynamics. In the present paper we develop an explicit potential theory for this equation and solve the interior and exterior Oseen Dirichlet and Oseen Neumann boundary value problems via a boundary integral equation method in spaces of continuous functions on a C2‐boundary, extending the classical approach for the isotropic selfadjoint Laplace operator to the anisotropic non‐selfadjoint scalar Oseen operator. It turns out that the solution to all boundary value problems can be presented by boundary potentials with source densities constructed as uniquely determined solutions of boundary integral equations.

Boundary Optimal Control of Time-Periodic Stokes-Oseen Flows

This work deals with the existence of optimal solution and the maximum principle for the optimal control problem governed by time-periodic Stokes–Oseen equations with boundary control. An example of a laminar flow is given, and the general unique continuation hypothesis for the Stokes–Oseen operator is checked in this case.

The unique continuation property of eigenfunctions to Stokes–Oseen operator is generic with respect to the coefficients

The unique continuation property of eigenfunctions φ to a Stokes–Oseen operator in Ω⊂R2 with boundary conditions {φ=0,∂∂νφ=0 on ∂Ω} is generic with respect to C2-coefficients. This implies that generically each C2 equilibrium solution to a 2-D Navier–Stokes equation is exponentially stabilizable by a tangential boundary controller.

Spectral properties in $L^q$ of an Oseen operator modelling fluid flow past a rotating body

We study the spectrum of a linear Oseen-type operator which arises from equations of motion of a viscous incompressible fluid in the exterior of a rotating compact body. We prove that the essential spectrum consists of an infinite set of overlapping parabolic regions in the left half-plane of the complex plane. The full spectrum coincides with the essential and continuous spectrum if the operator is considered in the whole 3D space. Our approach is based on the Fourier transform in the whole space and the transfer of the results to the exterior domain.

Partial differential equations with non-Euclidean geometries

Coifman and Weiss initiated the study of singular integral operators on spaces of homogeneous type. We study partial differential equations with a non-Euclidean geometry and we obtain weighted estimates using the theory of Muckenhoupt weights on related spaces of homogeneous type. Examples include squares of vector fields, elliptic operators with a drift and the Oseen operator.

A Preconditioner for the Steady-State Navier--Stokes Equations

We present a new method for solving the sparse linear system of equations arising from the discretization of the linearized steady-state Navier--Stokes equations (also known as the Oseen equations). The solver is an iterative method of Krylov subspace type for which we devise a preconditioner through a heuristic argument based on the fundamental solution tensor for the Oseen operator. The preconditioner may also be conceived through a weaker heuristic argument involving differential operators. Computations indicate that convergence for the preconditioned discrete Oseen problem is only mildly dependent on the viscosity (inverse Reynolds number) and, most importantly, that the number of iterations does not grow as the mesh size is reduced. Indeed, since the preconditioner is motivated through analysis of continuous operators, the number of iterations decreases for smaller mesh size which accords with better approximation of these operators.

JUSTIFICATION OF THE LINEARIZATION METHOD IN THE FLOW PROBLEM

The linearization method is justified in the problem of flow of viscous incompressible fluid past a body. As a preliminarily the Oseen operator is investigated, and estimates are established for its resolvent and the semigroup generated by it.

Steady Gasdynamic Flows

The steady Oseen equations for an arbitrary gas are considered. The inverse of the Oseen operator is obtained in closed form uniformly through transonic speeds in a certain limit. This limit is the same as the limit under which the Navier-Stokes equations are derivable from kinetic theory. The far field flow past an arbitrary body in terms of lift, drag, and energy dissipation of the body is obtained. As another application, the supersonic flow field past a semi-infinite flat plate is explicity obtained. The latter problem serves as the basis for an extension of the Oseen theory which gives an approximation of skin friction for arbitrary bodies.