Settling of an axially symmetric body in a viscous stratified fluid

The uniform, upwards flow of a continously vertically stratified fluid past an axially symmetric body is considered. The fluid is assumed to be Newtonian, incompressible, and diffusive. A matched asymptotic expansion procedure is used to calculate a correction to Stokes drag on the body. The results are valid provided that l, Re ^1/3, Fr^{2 −1/3}, Pe ^2/3, where is a stratification parameter. The results are applied to determine the quasi-steady motion of a body settling in a vertically stratified fluid.     

Finite element analysis of the axially symmetric motion of an incompressible viscous fluid in a spherical annulus

This paper presents results obtained by employing a modified Galerkin finite element method to analyse the steady state flow of a fluid contained between two concentric, rotating spheres. The spheres are assumed to be rigid and the cavity region between the spheres is filled with an incompressible, viscous, Newtonian fluid. The inner sphere is constrained to rotate about a vertical axis with a prescribed angular velocity, while the outer sphere is fixed. Results for the circumferential function Ω, streamfunction ψ, vorticity function ζ and inner boundary torque T₁ are presented for Reynolds numbers Re ? 2000 and radius ratios 0.1 ? α ? 0.9. The method proved effective for obtaining results for a wide range of radius ratios (0.1 ? α ? 0.9) and Reynolds numbers (0 ? Re ? 2000). Previous investigators who employed the finite difference method experienced difficulties in obtaining results for cases with radius ratios α ? 0.2, except for small Reynolds numbers (Re ? 100). Results for Ω, Ψ, ζ and T₁ obtained in this study for radius ratios 0.8 ≤ α ≤ 0.9 verified the development of Taylor vortices reported by other investigators. The research indicates that the method may be useful for analysing other non-linear fluid flow problems.     

Slow steady flows of a conducting fluid at large Hartmann numbers

We consider slow steady flows of a conducting fluid at large values of the Hartmann number and small values of the magnetic Reynolds number in an inhomogeneous magnetic field. The general solution is obtained in explicit form for the basic portion (core) of the flow, where the inertia and viscous forces may be neglected. The boundary conditions which this solution must satisfy at the outer edges of the boundary layers which develop at the walls are considered. Possible types of discontinuity surfaces and other singularities in the flow core are examined. An exact solution is obtained for the problem of conducting fluid flow in a tube of arbitrary section in an inhomogeneous magnetic field.The content of this paper is a generalization of some results on flows in a homogeneous magnetic field, obtained in [1–8], to the case of arbitrary flows in an inhomogeneous magnetic field. The author's interest in the problems considered in this study was attracted by a report presented by Professor Shercliff at the Institute of Mechanics, Moscow State University, in May 1967, on the studies of English scientists on conducting fluid flows in a strong uniform magnetic field.     

Branching of an axially symmetric convective flow

The stability of an axisymmetric convective gas flow with respect to finite disturbances of the bottom temperature is studied numerically using a finite-difference method. The convective flow considered approximately describes the free convection developing in the atmosphere due to heating of the substrate surface. The temperature disturbance used increases the intensity of one of two possible flows and suppresses the other flow, with the opposite signs of the vortices. Using the methods of numerical experiment, the corresponding problem of branching of the solution is examined. It is shown that the transformation of one flow into the other far from the threshold of the onset of convection requires substantial disturbances of the temperature.     

Unsteady two-dimensional rotation of a viscous fluid for steady source flow

We obtain the solution of the Navier-Stokes equations for one class of unsteady axisymmetric two-dimensional rotational flows for the case of a line source or sink of constant intensity in the fluid.     

On an axially symmetric elastic-plastic torsion problem

I.IntroductionTheelastic-plastictorsionofashaftisaclassicalmechanicsproblem.Inthemiddleofthel96os,theinventionofvariationalinequalitiesmadetheresearchofthisdirectionquiteactiveanddeep.Theresearchaboutbarswithconstantcross'sectionsisrelativelycompleteandin…     

An approximate solution of the problem of the axially symmetric flow of a viscous incompressible fluid in an angular region

An approximate solution of the axially symmetric problem of the flow of a viscous incompressible fluid in the vicinity of the point of contact between a uniformly moving plunger and a channel wall is obtained. Tomsk (Tomsk Branch of the Institute of Structural Macrokinetics, Russian Academy of Sciences). Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 157–160, January–February, 1999.     

Debye-Hückel solution for steady electro-osmotic flow of micropolar fluid in cylindrical microcapillary

Analytic expressions for speed, flux, microrotation, stress, and couple stress in a micropolar fluid exhibiting a steady, symmetric, and one-dimensional electro-osmotic flow in a uniform cylindrical microcapillary were derived under the constraint of the Debye-Hiickel approximation, which is applicable when the cross-sectional radius of the microcapillary exceeds the Debye length, provided that the zeta potential is sufficiently small in magnitude. Since the aciculate particles in a micropolar fluid can rotate without translation, micropolarity affects the fluid speed, fluid flux, and one of the two non-zero components of the stress tensor. The axial speed in a micropolar fluid intensifies when the radius increases. The stress tensor is confined to the region near the wall of the mi- crocapillary, while the couple stress tensor is uniform across the cross-section.     

Debye-H¨uckel solution for steady electro-osmotic flow of micropolar fluid in cylindrical microcapillary

Analytic expressions for speed, flux, microrotation, stress, and couple stress in a micropolar fluid exhibiting a steady, symmetric, and one-dimensional electro-osmotic flow in a uniform cylindrical mi...     

On the steady vibrations problem in linear theory of micropolar solid–fluid mixture

In this paper we study the steady vibrations problem in linear theory of isothermal micropolar solid–fluid mixture. With the help of fundamental solution we establish representations of Somigliana type. Then, using the potentials of single layer and double layer, we reduce the boundary value problems to singular integral equations for which the Fredholm’s theorems are valid. Existence and uniqueness results for interior and exterior problems are presented.     

Asymptotics of axially symmetric fluid flows with a free boundary in the case of dwindling viscosity

For high Reynolds numbers asymptotic expansions are constructed of the solution of the axially symmetric wave problem on the surface of a viscous incompressible fluid of infinite depth under the assumption that the tangential stresses on the free surface are of the order 0(1/Re). The principal terms of the asymptotic expansion are solutions of linear partial differential equations. The obtained result is then adapted to the case in which the fluid fills a bounded region whose boundary is a free surface. Some examples are given.     

Numerical study of the axially symmetric motion of an incompressible viscous fluid in an annulus between two concentric rotating spheres

The research reported herein involved the study of the transient motion of a system consisting of an incompressible Newtonian fluid in an annulus between two concentric, rotating, rigid spheres. The primary purpose of the research was to study the use of a numerical method for analysing the transient motion that results from the interaction between the fluid in the annulus and the spheres which are started suddenly by the action of prescribed torques. The problems considered in this research included cases where: (a) one or both spheres rotate with prescribed constant angular velocities and (b) one sphere rotates due to the action of an applied constant or impulsive t?orque. In this research the coupled solid and fluid equations were solved numerically by employing the finite difference technique. With the approach adopted in this research, only the derivatives with respect to spatial variables were approximated with the use of the finite difference formulae. The steady state problem was also solved as a separate problem (for verification purposes), and the results were compared with those obtained from the solution of the transient problem. Newton's algorithm was employed to solve the algebraic equations which resulted from the steady state problem, and the Adams fourth-order predictor–corrector method was employed to solve the ordinary differential equations for the transient problem. Results were obtained for the streamfunction, circumferential function, angular velocity of the spheres and viscous torques acting on the spheres as a function of time for various values of the system dimensionless parameters.     

Orientational phase transitions in systems of axially symmetric particles in steady elongational flow combined with potential fields

Effects of uniaxial elongational flow combined with an external potential field (electric, magnetic) on the isotropic-to-nematic phase transition are considered basing on the Onsager free energy approach and self-consistent field theory. Ellipsoidal particles of uniaxial symmetry subjected to dipole and quadruple fields are assumed. Mean-field potential of the interparticle interactions accounting for dipole and quadruple contributions is considered.The self-consistency equations include dipole and quadruple contributions of the external and mean-field potentials. The equilibrium thermodynamic potentials, critical conditions of phase instability, and phase equilibria are discussed.Example computations indicate that orientation of rigid, prolate particles in the elongational flow and/or an external potential field results in narrowing the range of phase instability, reducing the difference in orientational order between the phases in equilibrium, and shifting phase equilibria to smaller values of the interparticle interactions parameter.Uniaxial elongational flow contributes a potential with quadruple symmetry, additive to other quadruple terms of the external potential. Specific nature of the flow potential, different to the non-hydrodynamic potentials, is evident in the behaviour of internal energy and entropy. At the phase transitions, the flow contributes to the discontinuity of internal energy and entropy, a term originated from the activation energy of viscous motion.     

Adaptive finite elements for the steady free fall of a body in a Newtonian fluid

The numerical simulation of the free fall of a solid body in a viscous fluid is a challenging task since it requires computational domains which usually need to be several order of magnitude larger than the solid body in order to avoid the influence of artificial boundaries. Toward an optimal mesh design in that context, we propose a method based on the weighted a posteriori error estimation of the finite element approximation of the fluid/body motion. A key ingredient for the proposed approach is the reformulation of the conservation and kinetic equations in the solid frame as well as the implicit treatment of the hydrodynamic forces and torque acting on the solid body in the weak formulation. Information given by the solution of an adequate dual problem allows one to control the discretization error of given functionals. The analysis encompasses the control of the free fall velocity, the orientation of the body, the hydrodynamic force and torque on the body. Numerical experiments for the two dimensional sedimentation problem validate the method. To cite this article: V. Heuveline, C. R. Mecanique 333 (2005).     

Instability of axially symmetric MHD flows

The problem of linear stability of axially symmetric steady-state flows of an ideal incompressible fluid in a magnetic field is studied. A necessary and sufficient condition of stability of these flows with respect to perturbations of the same symmetry type is obtained by the direct Lyapunov method. This condition represents a generalization of the well-known Rayleigh criterion [3, 4] of centrifugal stability of rotating streams to the magnetohydrodynamic case. Two-sided exponential estimates of the perturbation growth are derived. A class of the most rapidly growing perturbations is identified and exact formulas for determining their growth rate are obtained. The corresponding exponents are calculated using the steady flow parameters and initial data for the perturbation field. From the mathematical point of view, the results of the present paper are preliminary in character, since the theorems of existence of the solutions of the problem in question have not been proved.Novosibirsk. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 19–25, November–December, 1995.