A variational principle for the Navier-Stokes equation

Motivated from Arnold's variational characterization of the Euler equation in terms of geodesic families of diffeomorphisms, a variational principle for the motion of incompressible viscous fluids is presented. A volume preserving diffusion process with drift velocity field subject to the Navier-Stokes equation is shown to extremize the energy functional of the fluid under a certain class of stochastic variations.     

Locomotion based on the control of the shape of magnetic fluid surfaces and of magnetizable media

The realization of locomotion based on the deformation of a free surface of a magnetic fluid layer in a traveling magnetic field is studied. A plane flow of an incompressible viscous magnetic fluid layer on a horizontal surface in a nonuniform magnetic field and a plane two-layers flow of incompressible viscous magnetic fluids between two parallel solid planes in a magnetic field is considered. Also the flow of an incompressible viscous magnetic fluid layer on a cylinder in a nonuniform magnetic field is an object of investigation. The deformation and the motion of a body made by a magnetizable polymer in an alternating magnetic field are experimentally studied. The cylindrical body (worm) which is located in a cylindrical tube is analyzed. These effects can be used in designing autonomous mobile robots without a hard cover. Such robots can be employed in clinical practice and biological investigations. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Existence of global weak solutions for a phase–field model of a vesicle moving into a viscous incompressible fluid

We consider in this article a model of vesicle moving into a viscous incompressible fluid. The vesicle is described through a phase–field equation and through a transport equation modeling the local incompressibility of its membrane. The equations for the fluid are the classical Navier–Stokes equations with a force resulting from the presence of the vesicle. Our main result states the existence of weak solutions for the corresponding system. The proof is based on compactness/monotonicity arguments. Copyright © 2013 John Wiley & Sons, Ltd.     

Radiation effects on the MHD flow near the stagnation point of a stretching sheet

The radiation effects using the Rosseland approximation on the flow of an incompressible viscous electrically conducting fluid over a stretching sheet near the stagnation point in the presence of uniform transverse magnetic field is studied. The governing equations transform to ordinary differential equation by using suitable similarity transformation and then by a perturbation technique the numerical results for temperature distribution were obtained and discussed graphically.     

Global existence for a regularized magnetohydrodynamic-<Emphasis Type="Italic">α</Emphasis> model

We prove local and global existence results for the Cauchy problem of a regularized magnetohydrodynamic-α model with viscous velocity field and no magnetic diffusivity for an incompressible fluid. Such a model is introduced in analogy with the Navier–Stokes equation to study the turbulent behavior of fluids in presence of a magnetic field. We consider the case of two space dimension.     

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.     

MHD flow of a Newtonian fluid over a stretching sheet: an approximate solution

An approximate solution to the problem of steady laminar flow of a viscous incompressible electrically conducting fluid over a stretching sheet is presented. The approach is based on the idea of stretching the variables of the flow problem and then using least squares method to minimize the residual of a differential equation. The effects of the magnetic field on the flow characteristics are demonstrated through numerical computations with different values of the Hartman number.     

Homogenization of the Stokes Problem With Non-homogeneous Slip Boundary Conditions

We study the Stokes system with non-homogeneous Fourier boundary conditions depending on a parameter, in a domain with periodic inclusions of the size of the period. Following the values of this parameter, we obtain at the limit a Darcy's law, a Brinkmann type equation or a Stokes type equation. We also present a physical model to which the results apply. This model describes the flow of an incompressible viscous fluid through a porous medium under the action of an exterior electric field.     

Solvability of the problem of evolution of an isolated volume of viscous,incompressible capillary fluid

The initial boundary-value problem for the Navier-Stokes equation describing the flow of a viscous, incompressible capillary fluid bounded only by a free surface is considered. At the initial time the region occupied by the fluid and the velocity field of the fluid are given. A theorem is formulated regarding the unique solvability of the problem for a finite time interval, and a model linearized problem in a half space is obtained.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Sleklova AN SSSR, Vol. 140, pp. 179–186, 1984.     

On the Equation of a Rotating Film

We study positive periodic solutions to a nonautonomous nonlinear third-order ordinary differential equation of the theory of motion of a viscous incompressible fluid with free boundary. This equation describes the steady motions of a thin layer of a fluid film on the surface of a rotating horizontal cylinder in the gravity field. The linear operator on the left-hand side of the equation has a three-dimensional kernel. Moreover, the equation contains two nonnegative parameters proportional to the gravity acceleration and surface tension. Depending on these parameters the problem in question may have either two solutions or no solutions at all. We establish some qualitative properties of solutions to the problem: in particular, their asymptotic behavior at the extremal values of the parameters.     

Surface tension effects on interaction between two fluids near a wall

Interaction between two fluids near a fixed solid surface ismodelled, with surface tension acting as an important influenceon the assumed planar motion. The two fluids are immiscible,incompressible and have small density and viscosity ratios;the heavier more viscous body of fluid is approaching the solidsurface and the other fluid is lying as a thin layer in between.In the so-called supercritical range where, for both fluids,inviscid forces dominate over viscous ones, a pair of pressure–shaperelations is found which leads to a nonlinear integro-differentialequation for the unknown interface shape. Analysis, computationand comparisons are applied to the equation. Travelling-statesolutions are found of periodic and non-periodic form, includinginteresting cases which exhibit parabolic growth of the layerthickness in the far field.     

On the weak discontinuity surfaces and tapes of equation systems in fluid mechanics

The weak discontinuity surfaces for a system of quasi-linear differential equations of higher order are developed. The classification of equation systems in fluid mechanics is based on the propagative weak discontinuity surfaces. Types of equations for different flow models are discussed. The conclusion is as follows:(a) For incompressible nonviscous flow, incompressible viscous flow and compressible viscous flow, the types of equations are all parabolic in the unsteady case and elliptic in the steady case.(b) For compressible nonviscous flow, the type of equations is hyperbolic in the unsteady case or steady supersonic case, and the type is elliptic in the steady subsonic case.     

Motion of a Ferrofluid in a Periodic Magnetic Field due to Surface Deformation

The article considers a plane flow of an incompressible, viscous magnetizable fluid. Cause of the flow is an alternating magnetic field, which consequently affects the free surface to periodical deformation. The proof of a peristaltic material transport is desired. The STOKES approximation for creeping flows and the two-dimensional continuity equation serve for governing equations. The methods used include power series expansion with regard to the small ratio of ε perturbed and unperturbed fraction of the magnetic field. The analytical result is discussed with regard to functional parameters. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A novel type of wave behaviour in a compressible inviscid dipolar fluid and stability characteristics of generalized fluids

Summary A new type of wave behaviour is found for third order waves in a compressible inviscid dipolar fluid. Several stability-like results are presented for the theories of a viscous incompressible dipolar fluid and a mixture of two viscous incompressible fluids.     

A free boundary problem in an absorbing porous material with saturation dependent permeability

We prove a well posedness result for a free boundary problem describing the filtration of an incompressible viscous fluid in a porous medium containing water absorbing granules.?The location of the wetting front (the free boundary) is described by a Stefan like problem for a parabolic equation which is coupled with an hyperbolic equation describing the absorption kinetic of the granules. Received December 1999     

A Method for Determining Linear Solutions in Magnetohydrodynamics

Given any two-dimensional and incompressible flow describedby a set of linear partial differential equations, a methodis presented for determining the solution for the dependentvariables (velocity, pressure, etc.). The method is then usedto investigate the following magnetohydrodynamic flows. (a) The flow of an irrotational and inviscid fluid. (b) The flow of a viscous fluid. (c) The flow of an electrically conducting, inviscid fluid inthe presence of a magnetic field aligned with the flow at infinity. (d) The flow of an electrically conducting, viscous fluid inthe presence of a magnetic field having arbitrary direction.     

Stochastic Vorticity and Associated Filtering Theory

The focus of this work is on a two-dimensional stochastic vorticity equation for an incompressible homogeneous viscous fluid. We consider a signed measure-valued stochastic partial differential equation for a vorticity process based on the Skorohod--Ito evolution of a system of N randomly moving point vortices. A nonlinear filtering problem associated with the evolution of the vorticity is considered and a corresponding Fujisaki--Kallianpur--Kunita stochastic differential equation for the optimal filter is derived.     

竖板表面温度的变化对磁流体动力学流动的影响

在横向磁场作用下,不可压缩的粘性导电流体,流经一个半无限的竖板,完成了壁面温度变化对磁流体动力学流动的分析.假定由粘性耗散和感应磁场产生的热量可以忽略不计.无量纲的控制方程为二维非稳态耦合的非线性方程.结果显示,磁场参数对空气和水的速度有着抑制作用.     

Completeness of the Floquet solutions to the problem of a solid oscillating in a fluid

We prove the completeness of the Floquet solutions to the parabolic equation describing small oscillations of a fluid-solid system. The symmetry axis of the solid is fixed inside a container of an arbitrary shape which is filled with an incompressible viscous fluid. The solid oscillates torsionally under the action of an elastic force with time periodic rigidity.