Magnetohydrodynamic stability of heterogeneous dissipative conducting liquids

Summary This discussion which is restricted to the flow of heterogeneous, incompressible, inviscid and conducting liquids between two nonrotating coaxial cylinders is divided into three parts. In the first part the problem in question is investigated with an applied radial magnetic field. A sufficient condition for stability is found. In particular, if the steady flow velocity is uniform it is shown that the flow is always stable.In part two the stability of homogeneous viscous conducting fluids through two fixed coaxial cylinders with an applied radial magnetic field has been discussed. A sufficient condition for stability is derived and an upper bound for the amplification factor is given.In part three the problem in question is treated with an applied axial magnetic field. A sufficient condition for stability is given and some particular cases are investigated.     

The stability of a revolving non-newtonian liquid in the presence of an axial pressure gradient

Summary The effect of cross-viscosity on the stability in fluids has been investigated. It is found that this effect destabilizes the flow of an incompressible non-newtonian liquid between coaxial rotating cylinders when an axial pressure gradient is applied.     

Stability of Couette flow between two rotating cylinders

We investigate the stability in the large of Couette flow between two cylinders rotating in the same direction. For the case of infinitesimally small perturbations, a sufficient condition for stability of the Couette flow (the Synge condition) was obtained in [1, 2]. In [3] for an investigation of the stability in the nonlinear case, to this condition we must add certain constraints on the initial energy and the angular velocities. In the proposed study, using the second method of Lyapunov, sufficient conditions for stability in the large are obtained, which differ little from the Synge condition. In this case these conditions approach the Synge condition as the distance between the cylinders is decreased.     

Stability of liquid flow between heated rotating cylinders

The effect of a radial temperature gradient on stability of steady-state flow of a viscous liquid between two solid concentric cylinders both rotating in the same direction is considered. The linear stability problem is considered in the Boussinesq approximation. Sufficient stability and instability conditions for the flow relative to rotationally symmetric perturbation are obtained. Neutral curves are computed for a wide range of problem parameters.     

Characteristics of a viscoplastic material in the couette flow

A model governing a steady flow of a viscoplastic material between coaxial cylinders is proposed. Nonlinear velocity sensitivity typical of superplastic materials is taken into account. An algorithm of calculating the characteristics of the material is developed. The algorithm is based on the experimental data on moments and angular velocities of the rotating coaxial cylinders. The stability of the algorithm to errors in the initial data is estimated.     

Laminar unsteady magnetohydrodynamic flow in an annular channel under a radial magnetic field

Summary The paper discusses the unsteady flow of an electrically conducting viscous fluid in the region between two coaxial cylinders in the presence of a radial magnetic field emanating from the common axis in planes perpendicular to it. In the special case when the magnetic Reynolds number of the flow is the same as its Reynolds number, an exact solution in terms of Bessel functions has been obtained which after infinite time tends to the steady flow discussed by Globe.     

An asymmetrical periodic vortical structures and appearance of the self induced pressure gradient in the modified Taylor flow

An incompressible liquid flow in the gap between two coaxial cylinders, such that the inner rotating (wavy) cylinder has a periodically varying radius along the axial direction while the outer stationary cylinder has a constant radius, is studied experimentally and theoretically. Basic attention is focused on the symmetry-breaking phenomenon of the vortex flow arising from the rotation of the inner wavy cylinder. It is found that the symmetry-breaking phenomenon of the vortical flow structures in this geometry is accompanied by the occurrence of a self-induced axial pressure gradient. A theoretical formulation of the problem of periodic vortical flow prevailing in such a geometry having large axial length is presented. The comparison between the computed and the experimental results is presented and the underlying phenomena are discussed.     

Stability of flow of a thin layer of viscous magnetic liquid

The laminar flow of a thin layer of heavy viscous magnetic liquid down an inclined wall is examined. The stability and control of the flow of an ordinary liquid are affected only by alteration of the angle of inclination of the solid wall and the velocity of the adjacent gas flow. When magnetic liquids are used [1, 2], an effective method of flow control may be control of the magnetic field. By using magnetic fields of various configurations it is possible to control the flow of a thin film of viscous liquid, modify the stability of laminar film flow, and change the shape of the free surface of the laminarly flowing thin film, a factor which plays a role in mass transfer, whose rate depends on the phase contact surface area. The magnetic field significantly affects the shape of the free surface of a magnetic liquid [3, 4]. In this paper the velocity profile of a layer of viscous magnetic liquid adjoining a gas flow and flowing down an inclined solid wall in a uniform magnetic field is found. It is shown that the flow can be controlled by the magnetic field. The problem of stability of the flow is solved in a linear formulation in which perturbations of the magnetic field are taken into account. The stability condition is found. The flow stability is affected by the nonuniform nature of the field and also by its direction.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 59–65, September–October, 1977.     

INSTABILITY OF COUPLED THERMO-SOLUTE CAPILLARY CONVECTION IN LIQUID BRIDGE AND CONTROL BY ROTATING MAGNETIC FIELD

浮区法因具有无坩埚接触污染的生长优点而成为生长高完整性和高均匀性单晶材料的重要技术.但熔体中存在的毛细对流会给浮区法晶体生长带来极大挑战,这是由于对流的不稳定会导致晶体微观瑕疵的产生和宏观条纹等缺陷的形成.为了提高浮区法生长单晶材料的品质,研究浮区法晶体生长中毛细对流特性及如何控制其不稳定性显得尤为重要.本文采用数值模拟的方法对半浮区液桥内Si_xGe_1-x体系中存在的热质毛细对流展开研究并施加旋转磁场对其进行控制.结果表明：纯溶质毛细对流表现为二维轴对称模式,温度场主要由热扩散作用决定,而浓度场则由对流和溶质扩散共同支配;纯热毛细对流呈现三维稳态非轴对称流动,浓度分布与熔体内热毛细对流的流向密切相关,等温线在对流较大的区域发生弯曲;耦合溶质与热毛细对流则为三维周期性旋转振荡流.施加旋转磁场后,熔体周向速度沿径向向外增大,熔体内浓度场和流场均呈现二维轴对称分布.     

Steady disturbed viscous incompressible gas flows in a long cylinder with one closed end

There have been many studies of steady disturbed flows in rotating channels. In [1] the steady disturbed flow between two coaxial rotating cylinders was investigated. In [2, 3] the disturbances of Hagen-Poiseuille flow due to rotation of the pipe were considered. In this article other effects: the propagation of disturbances in a long rotating pipe and their interaction with the end face are examined.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 104–112, September–October, 1993.     

Order in chaotic pseudoplastic flow between coaxial cylinders

Order is found within the chaotic nonlinear flow between rotating coaxial cylinders. The flow stability analysis is carried out for a pseudoplastic fluid through bifurcation diagram and Lyapunov exponent histogram. The fluid is assumed to follow the Carreau–Bird model, and mixed boundary conditions are imposed. The low-order dynamical system, resulted from Galerkin projection of the conservation of mass and momentum equations, includes additional nonlinear terms in the velocity components originated from the shear-dependent viscosity. It is observed that the base flow loses its radial flow stability to the vortex structure at a lower critical Taylor number, as the shear-thinning effects increase. The emergence of the vortices corresponds to the onset of a supercritical bifurcation, which is also seen in the flow of a linear fluid. However, unlike the Newtonian case, shear-thinning Taylor vortices lose their stability as the Taylor number reaches a second critical number corresponding to the onset of a Hopf bifurcation. Complete flow field together with viscosity maps are given for different scenarios in the bifurcation diagram.     

液封液桥内热毛细对流的数值模拟

建立了液封液桥(不相溶混的双层同轴液柱)内热毛细对流的物理数学模型，采用涡量-流 函数法、利用有限差分格式对微重力条件下液封液桥内热毛细对流进行了数值模拟，得到了 双层液柱主流区的温度场和流场，证实了液封技术能削弱液桥主流区的热毛细对流，从而提 高浮区晶体生长质量，找出了液封厚度以及内、外层流体物性参数比对液桥内热毛细对流的 影响规律.     

A three-dimensional non-linear constitutive law for magnetorheological fluids, with applications

In this paper we first summarize the magnetic and mechanical balance equations for magnetorheological fluids undergoing steady motion in the presence of a magnetic field. A general three-dimensional non-linear constitutive law for such a fluid is given for the case in which the magnetic induction vector is used as the independent magnetic variable. The equations are needed for the analysis of boundary-value problems involving fluids with dispersed micron-sized ferrous particles subjected to a time-independent magnetic field. For illustration, the equations are applied, in the case of an incompressible fluid, to the solution of some basic problems. We consider unidirectional flow in a region confined by two infinite parallel plates with a magnetic field applied perpendicular to the plates. Next, we examine two problems involving a circular cylindrical geometry with the fluid occupying the region between two concentric cylinders: axial flow subjected to an axial magnetic field and circumferential flow with a circumferential field. After making some simplifying assumptions on the constitutive law and choosing material parameters, numerical solutions for the velocity profiles are illustrated.     

Convective heat transfer from a rotating or nonrotating axisymmetric body

An analysis of steady laminar mixed-convection heat transfer from a rotating or nonrotating axisymmetric body is presented. A mixed-convection parameter is proposed to serve as a controlling parameter that determines the relative importance of the forced and the free convection. In addition, a rotation parameter is introduced to indicate the relative contributions of the flow forced convection and the rotational forced convection. The values of both these two parameters lie between 0 and 1. Furthermore, the coordinates and dependent variables are transformed to yield computationally efficient numerical solutions that are valid over the entire range of mixed convection from the forced-convection limit (rotating or nonrotating bodies) to the pure free-convection limit (non-rotating bodies) and the entire regime of forced convection from the pure flow forced-convection limit (nonrotating bodies) to pure rotational forced-convection limit (rotating bodies). The effects of mixed-convection intensity, body rotation, fluid suction or injection, and fluid Prandtl number on the velocity profiles, the temperature profiles, the skin-friction parameter, and heat transfer parameter are clearly illustrated for both cases of buoyancy assisting and opposing flow conditions.     

Dynamics of Instabilities in a Viscous Fluid Rotating with an Acceleration

The flow of a viscous incompressible fluid between coaxial cylinders rotating with a constant rigid-body acceleration about the axis is studied numerically. A one-dimensional time-dependent solution of the Navier-Stokes equations is constructed analytically for the motion starting from the state of rest. On the initial time interval the one-dimensional unsteady fluid motion is unstable. Small perturbations introduced into the flow initiate the formation of secondary vortex flows with a velocity component along the axis. The dynamics of the developing instabilities and their dissipation are studied numerically. A condition determining the dimensions of the unsteady secondary flow zone is formulated. The unsteady regime is transient and starting from a certain instant of time the flow becomes stable.     

The method of fundamental solutions for Stokes flow in a rectangular cavity with cylinders

Numerical solutions based on the method of fundamental solutions are discussed for Stokes flow inside a rectangular cavity in the presence of circular cylinders. The Stokeslets are used as the fundamental solutions to obtain the solution for the flow field by a linear combination of fundamental solutions. Flow results on the cellular structure of flow field resulting from the dynamics of cylinders and the horizontal walls of the cavity are reported for (i) one rotating cylinder in a rectangular cavity with two parallel horizontal sides moving in the same directions as well as in the opposite directions, (ii) two rotating cylinders kept apart in a rectangular cavity with two parallel horizontal sides moving in the same directions as well as in the opposite directions. The effect of aspect ratio of the rectangular cavity, direction of movement of the two parallel horizontal sides of the cavity and the diameter of the rotating cylinder on the flow structure are studied. The flow results obtained for the single cylinder case are in accordance with the results available in the literature. From the computational point of view, the present numerical procedure based on the method of fundamental solutions is efficient and simple to implement as compared to the mesh-dependent schemes, which needs complex mesh generation procedure for the multiply connected geometrical domains considered in this article.     

Instability of the motion of a liquid between two rotating spherical surfaces

A study has been made of the stability of the steady-state motion of a viscous incompressible liquid, arising in a thin spherical layer, when both spheres are rotating in the same direction at different angular velocities. For a ratio of the radii of the spheres r₂/r₁=1.10, 1.07, a stability curve is obtained which is analogous to the stability curve for the motion of a liquid between rotating cylinders.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 155–156, July–August, 1970.     

Inversion d'un écoulement plan de Stokes – application à l'écoulement à débit moyen nul entre deux plans parallèles

We present the interest and some characteristics of the inverse transformation of a 2D Stokes flow. This method is applied to the cellular flow between two parallel plates induced by a rotating cylinder to obtain the flow around two circular cylinders in contact placed in the centre of a rotating circular cylinder.     

Oscillatory rotationally symmetric loss of stability of nonisothermal couette flow

A study is made of the stability of nonisothermal Couette flow — steady flow of a viscous heat conducting fluid between two rotating concentric cylinders heated to different temperatures. The methods of perturbation theory are used to establish conditions sufficient for bifurcation of a neutral curve of oscillatory instability from the neutral curve of monotonic instability. Computer calculations show that for certain values of the parameters of the problem these conditions are realized and there is an oscillatory loss of stability of the nonisothermal Couette flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 76–80, January–February, 1984.I thank V. I. Yudovich for constant interest in the work.     

Nonresonant case of intersection of bifurcation curves in the Couette-Taylor problem

The nonresonant case (Res 0) of the motion of a viscous incompressible fluid between rotating coaxial cylinders in a small neighborhood of a bifurcation point of codimension 2 is considered, where the amplitude system has only essential resonant terms. Existence and stability conditions are obtained for its solutions which correspond to various periodic and quasiperiodic solutions of the Navier-Stokes equations. In a small neighborhood of some points of the resonance Res 0, the regions of existence and stability of these solutions are determined.