High-order approximations for an incompressible viscous flow on a rough boundary

In this paper, we propose approximations of fluid flow that could be used for obtaining wall laws of higher order. We consider the two-dimensional laminar fluid flow, modeled by the incompressible Stokes system in a straight channel with a rough side. The roughness is periodic and the ratio of the amplitude of the rough part and the size of the flow domain is denoted by ?, being a small number. We impose periodic boundary conditions on the flow. We generalize the boundary layers needed for the construction of flow approximations of higher order with respect to ?. The existence of the layers and their features are discussed. Finally we give the error estimates for the approximations and establish an explicit wall law.     

Formulation of boundary conditions for vorticity in viscous incompressible flow problems

For the stream function-vorticity formulation of the Navier-Stokes equations, vorticity boundary conditions are required on the body surface and the far-field boundary. A two-parameter approximating formula is derived that relates the velocity and vorticity on the outer boundary of the computational domain. The formula is used to construct an algorithm for correcting the conventional far-field boundary conditions. Specifically, a soft boundary condition is set for the vorticity and a uniform flux is specified for the transversal velocity. A third-order accurate three-parameter formula for the vorticity on the wall is derived. The use of the formula does not degrade the convergence of the iterative process of finding the vorticity as compared with a previously derived and tested two-parameter formula.     

Solvability of a problem on the plane motion of a viscous incompressible fluid with a free noncompact boundary

One considers the problem of the plane motion of a viscous incompressible fluid which fills partially a container V, bounded by the straight line ₁ = {x:x ₂ = 0} and the contour (V₁), consisting of two semilines ⁽¹⁾ = {x:x ₁<–1,x ₂ = h₀} ⁽²⁾ = {x:x ₁ = 0,x ₂h₀+1} joined by a smooth curvel ⁽³⁾. One assumes that the motion is due to a nonzero flow and by the motion of the lower wall ₁ with a constant velocity R0. The unique solvability of this problem is proved for small R and .Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 110, pp. 174–179, 1981.In conclusion, the author expresses his deep gratitude to V. A. Solonnikov for his guidance.     

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.     

Estimation of the error of approximation methods for calculating flows of a viscous incompressible fluid with a free boundary

Under the assumption of a sufficient smoothness of the solutions, one investigates the error produced by the approximation methods in the computation of Navier-Stokes equations and in the restoration of the surface from its mean curvature in the course of the method of successive approximations, used for obtaining the solution of the problem on the motion of a viscous fluid with a free boundary.     

Two-dimensional incompressible viscous flow around a small obstacle

In this work we study the asymptotic behavior of viscous incompressible 2D flow in the exterior of a small material obstacle. We fix the initial vorticity ω₀ and the circulation γ of the initial flow around the obstacle. We prove that, if γ is sufficiently small, the limit flow satisfies the full-plane Navier–Stokes system, with initial vorticity ω₀ + γδ, where δ is the standard Dirac measure. The result should be contrasted with the corresponding inviscid result obtained by the authors in Iftimie et al. (Comm. Part. Differ. Eqn. 28, 349–379 (2003)), where the effect of the small obstacle appears in the coefficients of the PDE and not only in the initial data. The main ingredients of the proof are L ^p − L ^q estimates for the Stokes operator in an exterior domain, a priori estimates inspired on Kato’s fixed point method, energy estimates, renormalization and interpolation.     

On some free boundary problem for a compressible barotropic viscous fluid flow

In this paper, we prove a local in time unique existence theorem for the free boundary problem of a compressible barotropic viscous fluid flow without surface tension in the \(L_p\) in time and \(L_q\) in space framework with \(2 < p < \infty \) and \(N < q < \infty \) under the assumption that the initial domain is a uniform \(W^{2-1/q}_q\) one in \({\mathbb {R}}^{N}\, (N \ge 2\) ). After transforming a unknown time dependent domain to the initial domain by the Lagrangian transformation, we solve problem by the Banach contraction mapping principle based on the maximal \(L_p\) – \(L_q\) regularity of the generalized Stokes operator for the compressible viscous fluid flow with free boundary condition. The key issue for the linear theorem is the existence of \({\mathcal {R}}\) -bounded solution operator in a sector, which combined with Weis’s operator valued Fourier multiplier theorem implies the generation of analytic semigroup and the maximal \(L_p\) – \(L_q\) regularity theorem. The nonlinear problem we studied here was already investigated by several authors (Denisova and Solonnikov, St. Petersburg Math J 14:1–22, 2003; J Math Sci 115:2753–2765, 2003; Secchi, Commun PDE 1:185–204, 1990; Math Method Appl Sci 13:391–404, 1990; Secchi and Valli, J Reine Angew Math 341:1–31, 1983; Solonnikov and Tani, Constantin carathéodory: an international tribute, vols 1, 2, pp 1270–1303, World Scientific Publishing, Teaneck, 1991; Lecture notes in mathematics, vol 1530, Springer, Berlin, 1992; Tani, J Math Kyoto Univ 21:839–859, 1981; Zajaczkowski, SIAM J Math Anal 25:1–84, 1994) in the \(L_2\) framework and Hölder spaces, but our approach is different from them.     

Analysis of boundary conditions for approximate solution of problems of viscous incompressible flow past a body

Translated from Aktual'nye Voprosy Prikladnoi Matematiki, pp. 49–54, 1989.     

Friction boundary conditions on thermal incompressible viscous flows

This work adresses an unsteady heat flow problem involving friction and convective heat transfer behaviors on a part of the boundary. The problem is constituted by a variational motion inequality with energy dependent coefficients, and the energy equation in the framework of L ¹-theory for the dissipative term. Using the duality theory of convex analysis, it also envolves the existence of Lagrange multipliers. Weak solutions of an approximate coupled system are proven by a fixed point argument for multivalued mappings and compactness methods. Then the existence result for the initial coupled system is proven by the passage to the limit. This work was partially supported by FCT research program POCTI (Portugal/FEDER-EU).     

On a problem of viscous strip deformation with a free boundary

We consider a class of solutions to the 2D Navier-Stokes equations in a strip such that the longitudinal component of velocity is a linear function of the longitudinal coordinate while the transversal component and the pressure do not depend on this coordinate. One of the strip boundaries is free and the other boundary can be a solid wall or free too. We formulate conditions for a global solvability in time of corresponding initial boundary value problems, describe their asymptotic properties, give examples of exact solutions, study blowing up solutions in the case when the both strip boundaries are free.     

Homogenization of an incompressible viscous flow in a porous medium with double porosity

We study the homogenization of an incompressible viscous flow in a porous medium with double porosity. We derive a macroscopic model with local Navier–Stokes system in the large cavities, Darcy law in the thinner porous rock, and a contact law between the two. We use Γ-convergence methods associated with multi-scale convergence notions in order to get this limit law. We exhibit a critical ratio between the two scales of the pores.     

On viscous incompressible flows of nonsymmetric fluids with mixed boundary conditions

In this paper we are concerned with the initial boundary value problem for the micropolar fluid system in nonsmooth domains with mixed boundary conditions. The considered boundary conditions are of two types: Navier’s slip conditions on solid surfaces and Neumann-type boundary conditions on free surfaces. The Dirichlet boundary condition for the microrotation of the fluid is commonly used in practice. However, the well-posedness of problems with different types of boundary conditions for microrotation are completely unexplored. The present paper is devoted to the proof of the existence, regularity and uniqueness of the solution in distribution spaces.     

On MHD flow of an incompressible viscous fluid

In this paper, we apply Homotopy Perturbation Method (HPM) to find the analytical solutions of nonlinear MHD flow of an incompressible viscous fluid through convergent or divergent channels in presence of a high magnetic field. The flow of an incompressible electrically conducting viscous fluid in convergent or divergent channels under the influence of an externally applied homogeneous magnetic field is studied both analytically and numerically. The graphs are presented to reveal the physical characteristics of flow by changing angles of the channel, Hartmann and Reynolds numbers.     

Efficient computation of non-isothermal highly viscous incompressible flow

We consider the numerical solution of the non–isothermal incompressible Navier–Stokes equations using a discrete projection method. The computation of velocity and temperature subproblems is carried out on different meshes chosen with respect to the physical behavior of these quantities. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)