How the Reynolds equation is related to the Stokes equations

Applied Mathematics & Optimization - The Reynolds equation, which gives the pressure distribution in a thin layer of lubricating fluid, is usually motivated in a rather heuristic way. We make...     

Navier‐Stokes equations,Haar wavelets and Reynolds numbers

The paper deals with global solutions of Navier‐Stokes equations with infrared‐damped initial data in the context of Haar wavelets and function spaces of type where .     

The infinitesimal Lefschetz formulas: A heat equation proof

In this paper, the Lefschetz formulas of Atiyah and Bott (Ann. of Math.86 (1967), 374–407; 88 (1968), 451–491) and Atiyah and Singer (Ann. of Math.87 (1968), 546–604; 93 (1971), 119–138) are proved by heat equation methods in infinitesimal form, i.e., without using the usual localization on the fixed point set. The proof is obtained by a homotopic deformation of the Dirac operator and uses probabilistic methods. The main applications are potentially in noncompact situations where fixed point formulas do not always hold, while their infinitesimal analogues are still valid.     

The octahedron equation implies the cube equation: An elementary proof

It has been proved by L. Sweet that the octahedron functional equation implies the cube functional equation in all dimensionsn1. In this note we give an elementary proof of this theorem.     

Stokes equations under nonlinear slip boundary conditions coupled with the heat equation: A priori error analysis

In this work, we consider the heat equation coupled with Stokes equations under threshold type boundary condition. The conditions for existence and uniqueness of the weak solution are made clear. Next we formulate the finite element problem, recall the conditions of its solvability, and study its convergence by making use of Babuska–Brezzi's conditions for mixed problems. Third we formulate an Uzawa's type iterative algorithm that separates the fluid from heat conduction, study its feasibility, and convergence. Finally the theoretical findings are validated by numerical simulations.     

A stabilization algorithm of the Navier–Stokes equations based on algebraic Bernoulli equation

In this paper, we consider the stabilization of the nonstationary incompressible Navier–Stokes equations around a stationary solution by a boundary linear feedback control. The feedback operator is obtained from the solution of the algebraic Bernoulli equation associated with the penalized linearized Navier–Stokes equations around an unstable stationary solution and is used to locally stabilize the original nonlinear equations. We give the explicit factorized form of the stabilizing solution of the algebraic Bernoulli equation. The numerical effectiveness of this approach is demonstrated by stabilizing the vortex shedding behind a circular obstacle. Copyright © 2011 John Wiley & Sons, Ltd.     

Explicit modulation equations, Reynolds averaging and the closure problem for the Korteweg-deVries-Burgers equation

Original explicit modulation equations are determined for cnoidalwaves of the Korteweg-deVries (KdV)–Burgers equation.This formal asymptotic analysis is used to demonstrate thatthere is no single partial differential equation for the leading-ordermean velocity. The technique of Reynolds averaging is also employedto determine an equation for the mean velocity with the familiarclosure problem being encountered. The Reynolds-averaged KdV–Burgersequation is shown to be a counterexample to the existence ofa closure associated with a convective nonlinearity.     

On the asymptotic analysis of the Boltzmann equation tending towards the Stokes equations

This work deals with the analysis of the asymptotic limit for the Boltzmann equation tending towards the linearized Navier–Stokes equations when the Knudsen number $\epsilon$ tends to zero. Global existence and uniqueness theorems are proven for regular initial fluctuations. As $\epsilon$ tends to zero, the solution converges strongly to the solution of the linearized Navier–Stokes systems.     

The phase transition in random graphs: A simple proof

The classical result of Erd?s and Rényi asserts that the random graph G(n,p) experiences sharp phase transition around \begin{align*}p=\frac{1}{n}\end{align*} – for any ε > 0 and \begin{align*}p=\frac{1-\epsilon}{n}\end{align*}, all connected components of G(n,p) are typically of size O_ε(log n), while for \begin{align*}p=\frac{1+\epsilon}{n}\end{align*}, with high probability there exists a connected component of size linear in n. We provide a very simple proof of this fundamental result; in fact, we prove that in the supercritical regime \begin{align*}p=\frac{1+\epsilon}{n}\end{align*}, the random graph G(n,p) contains typically a path of linear length. We also discuss applications of our technique to other random graph models and to positional games. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013     

Bridging the gap between mathematical conjecture and proof through computer-supported cognitive conflicts

In many mathematical problems, students can feel that the universalityof a conjecture or a formula is validated by their experimentand experience. In contrast, students generally do not feelthat deductive explanations strengthen their conviction thata conjecture or a formula is true. In order to cope up withstudents’ conviction based only on empirical experienceand to create a need for deductive explanations, we developeda problem-solving activity with technology support intendedto cause cognitive conflict. In this article, we describe theprocess conducted for this activity that led students to contradictionsbetween conjectures and findings. The teacher could create familiarproblem-solving situations and use students’ naïveinductive approaches to make students think mathematically andestablish the necessity for proof via computer support.     

A cascadic multigrid algorithm for the Stokes equations

A variant of multigrid schemes for the Stokes problem is discussed. In particular, we propose and analyse a cascadic version for the Stokes problem. The analysis of the transfer between the grids requires special care in order to establish that the complexity is the same as that for classical multigrid algorithms. Received September 10, 1997 / Revised version received February 20, 1998     

A combined BIE-FE method for the Stokes equations

A numerical algorithm for the biharmonic equation in domainswith piecewise smooth boundaries is presented. It is intendedfor problems describing the Stokes flow in the situations whereone has corners or cusps formed by parts of the domain boundaryand, due to the nature of the boundary conditions on these partsof the boundary, these regions have a global effect on the shapeof the whole domain and hence have to be resolved with sufficientaccuracy. The algorithm combines the boundary integral equationmethod for the main part of the flow domain and the finite-elementmethod which is used to resolve the corner/cusp regions. Twoparts of the solution are matched along a numerical ‘internalinterface’ or, as a variant, two interfaces, and theyare determined simultaneously by inverting a combined matrixin the course of iterations. The algorithm is illustrated byconsidering the flow configuration of ‘curtain coating’,a flow where a sheet of liquid impinges onto a moving solidsubstrate, which is particularly sensitive to what happens inthe corner region formed, physically, by the free surface andthe solid boundary. The ‘moving contact line problem’is addressed in the framework of an earlier developed interfaceformation model which treats the dynamic contact angle as partof the solution, as opposed to it being a prescribed functionof the contact line speed, as in the so-called ‘slip models’.     

A posteriori error estimators for the Stokes equations

Summary We present two a posteriori error estimators for the mini-element discretization of the Stokes equations. One is based on a suitable evaluation of the residual of the finite element solution. The other one is based on the solution of suitable local Stokes problems involving the residual of the finite element solution. Both estimators are globally upper and locally lower bounds for the error of the finite element discretization. Numerical examples show their efficiency both in estimating the error and in controlling an automatic, self-adaptive mesh-refinement process. The methods presented here can easily be generalized to the Navier-Stokes equations and to other discretization schemes.This work was accomplished at the Universität Heidelberg with the support of the Deutsche Forschungsgemeinschaft     

On a free boundary problem for the Reynolds equation derived from the Stokes system with Tresca boundary conditions

The asymptotic behaviour of a Stokes flow with Tresca free boundary friction conditions when one dimension of the fluid domain tends to zero is studied. A specific Reynolds equation associated with variational inequalities is obtained and uniqueness is proved.     

The planar Dirichlet problem for the Stokes equations

The Dirichlet problem for the Stokes equations is studied in a planar domain. We construct a solution of this problem in form of appropriate potentials and determine the unknown source densities via integral equation systems on the boundary of the domain. The solution is given explicitly in the form of a series. As a consequence we determine a solution of the Dirichlet problem for a compressible Stokes system and a solution of a boundary value problem on a domain with cracks. Copyright © 2011 John Wiley & Sons, Ltd.     

A functional equation proof of the distributive-triples theorem

A 1987 article studied the fact that some attributes can be measured in two ways: via concatenations and via decomposition in a conjoint fashion. The measures were shown to be the same provided a distribution law is satisfied. The proof was algebraic and indirect. A direct proof is provided using functional equation arguments.     

Uniqueness for the Navier–Stokes equations and multipliers between Sobolev spaces

We re-prove various uniqueness theorems for the Navier–Stokes equations, stating the assumptions in terms of multipliers between Sobolev spaces instead of Lebesgue or Lorentz spaces.     

A least-square mixed method for Stokes equations

We prove the convergence of a least-square mixed method for Stokes equations by use of an operator theoretic approach. The method does not require LBB condition on the finite dimensional subspaces. The resulting bilinear form is symmetric and positive definite, which leads to optimal convergence and the h⁻² condition number. © 1997 John Wiley & Sons, Inc.     

A separation between the boundary shape and the boundary functions in the parametric integral equation system for the 3D Stokes equation

Numerical Algorithms - The paper introduces the analytical modification of the classic boundary integral equation (BIE) for Stokes equation in 3D. The performed modification allows us to obtain...