A note on the unsteady motion of a viscous conducting liquid between two porous concentric circular cylinders acted on by a radial magnetic field

In this note the author has investigated some problems of a flow of conducting liquid through two porous non-conducting infinite circular cylinders rotating with various angular velocities for some time in the presence of a radial magnetic field.It is assumed that the rate of suction at the inner cylinder is equal to the rate of injection at the outer cylinder. Furthermore the induced electric and magnetic fields are neglected.Nomenclature a, b radii of coaxial cylinders - h ₁, h ₂ constants - m, n constants - P hydrostatic pressure - t time - A, B functions of r and b - B ₀ magnetic induction vector B ₀= _e H - H magnetic field vector - H intensity of magnetic field - L, M functions of a and b - S Suction parameter - Coefficient of viscosity - _e magnetic permeability - Kinematic coefficient of viscosity - density of liquid - conductivity of liquid - Small time - Constant - ₁, ₂ angular vrlocities - , ₁, ₂ angular velocities.     

The shape of a liquid surface between rotating concentric cylinders

Summary The shape of the free surface of a second order fluid, enclosed between vertical concentric cylinders with the inner cylinder rotating with angular velocity, is calculated. The calculation uses a perturbation procedure in which, for slow flows, the boundary conditions on the free surface are transformed to boundary conditions on the fixed horizontal surface obtained when is zero. The results are tested experimentally.With 3 figures     

Unsteady flow of a second-order fluid between concentric cylinders

The unsteady motion of an incompressible second-order fluid contained between two finite coaxial cylinders is examined when the outer cylinder is held fixed while the inner cylinder is constrained to execute an arbitrary angular velocity. A solution is obtained in closed form with the aid of transforms and an expression is obtained for the couple experienced. The particular case of a periodic angular velocity is then examined and some numerical work done. There is a marked difference between the results obtained and their classical counterparts.     

Geometric shapes of the interface surface of bicomponent flows between two concentric rotating cylinders

In this paper, the shape problem of interface of bicomponent flows between two concentric rotating cylinders is investigated. With tensor analysis, the problem is reduced to an energy functional isoperimetric problem when neglecting the effects of the dissipative energy caused by viscosity. We derive the associated Euler-Lagrangian equation, which is a nonlinear elliptic boundary value problem of the second order. Moreover, by considering the effects of the dissipative energy, we propose another total energy functional to characterize the geometric shape of the interface, and obtain the corresponding Euler-Lagrangian equation, which is also a nonlinear elliptic boundary value problem of the second order. Thus, the problem of the geometric shape is converted into a nonlinear boundary value problem of the second order in both cases.     

Unsteady flow of an elastico-viscous fluid between two coaxial circular cylinders

Unsteady flow of an Oldroyd fluid between two coaxial circular cylinders is investigated, the fluid being set in motion as the inner cylinder moves from rest for a certain period with linearly growing speed and then stops suddenly. The Laplace transform technique is used to derive the solution. For the case when the gap between the cylinders is small, a simplified solution is obtained. The expression for the shear stress on the wall of the outer cylinder is obtained and particular cases are discussed.     

Interaction between two circular cylinders in slow flow of Bingham viscoplastic fluid

The interaction between two circular cylinders was studied in the slow flow of a Bingham viscoplastic fluid in an infinite medium without any inertia effects. The configuration studied is that in which the flow direction is parallel to the centre line of the cylinders. Finite-element numerical simulations were used with an approximation by Papanastasiou's regularisation method. The case of high yield stress effect was particularly examined. The convergence of the solutions was examined in detail. Changes in the rigid zones, kinematics and stresses were determined in relation to the degree of interaction, which is a function of the distance between the cylinders and the effect of yield stress. The results compared with the case of a single cylinder show that yield stress reduces interaction effects. The transition between configurations with interacting cylinders and configuration with isolated cylinders was examined as a function of the effect of yield stress. Correlations were proposed for the drag coefficient and the stability criterion when the cylinders are interacting.     

Natural convection in vertical concentric annuli under a radial magnetic field

Exact solutions for fully developed natural convection in open-ended vertical concentric annuli under a radial magnetic field are presented. Expressions for velocity field, temperature field, mass flow rate and skin-friction are given, under more general thermal boundary conditions. It is observed that both velocity as well as temperature of the fluid is more in case of isothermal condition compared with constant heat flux case when gap between cylinders is less or equal to radius of inner cylinder while reverse phenomena occur when the gap between cylinders is greater than radius of inner cylinder.     

Numerical simulation of fluid flow of two rotating side-by-side circular cylinders by Lattice Boltzmann method

A numerical investigation of the two-dimensional laminar flow around side-by-side rotating circular cylinders using Lattice Boltzmann method is conducted. The effects of variation of rotational speed ratio β and different gap spacings g* at Reynolds number of 100 are studied. A various range of rotational speed ratio 0 ≤ β ≤ 2 for four different gap spacings of 3, 1.5, 0.7 and 0.2 are investigated. Flow conditions and its characteristics, such as lift and drag coefficients and Strouhal number, is studied. The results indicated that as β increases, the flow changes its condition from periodic to steady after a critical rotational speed. Results also indicated that variation of the gap spacing and rotational speed has significant effect on wake pattern. Wake pattern in turn has significant effect on the Strouhal number. Finally, the result is compared with experimental and other numerical data.     

Convective instability in a rapidly rotating fluid layer in the presence of a non-uniform magnetic field

Magnetoconvective instabilities in a rapidly rotating, electrically conducting fluid layer heated from below in the presence of a non-uniform, horizontal magnetic field are investigated. It was first shown by Chandrasekhar that an overall minimum of the Rayleigh number may be reached at the onset of magnetoconvection when a uniform basic magnetic field is imposed. In this paper, we show that the properties of instability can be quite different when a non-uniform basic magnetic field is applied. It is shown that there is an optimum value of the Elsasser number provided that the basic magnetic field is a monotonically decreasing or increasing function of the vertical coordinate. However, there exist no optimum values of the Elsasser number that can give rise to an overall minimum of the Rayleigh number at the onset of magnetoconvection if the imposed basic magnetic field has an inflexion point. The project supported by the National Natural Science Foundation of China (40174026 and 40074041)     

Motion of two circular cylinders in an ideal fluid

A solution of the problem concerning the motion of two circular cylinders in an ideal fluid was given earlier using approximate methods; the error made in this treatment of the problem increased with approach of the cylinders to one another [1, 2]. In this paper we give an exact solution of the problem for arbitrary motion of the cylinders. We define a velocity potential, the kinetic energy, and the forces acting on the cylinders from the fluid side.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 80–84, November–December, 1970.The author wishes to thank N. S. Storozhuk for his aid in preparing this paper.     

磁场中旋转运动圆板的分叉与混沌研究

应用数值模拟方法研究磁场中旋转运动圆板的分叉与混沌问题。首先,基于薄板理论和麦克斯韦电磁场方程组,给出了动能、应变势能、外力虚功以及电磁力的表达式,再利用哈密顿原理,得到磁场中旋转运动圆板横向振动的非轴对称非线性磁弹性振动微分方程组。其次,采用贝塞尔函数作为圆板的振型函数进行伽辽金积分,得到了轴对称情况下横向振动的常微分方程组表达式。最后,针对主共振,取周边夹支边界条件的圆板作为算例,得到了当振型函数取一阶时,将磁感应强度、外激励振幅和激励频率作为控制参数的分叉图及庞加莱映射图等计算结果,并讨论了分叉参数对系统的分叉与混沌的影响。数值计算结果表明,这些控制参数的变化影响系统稳定性,在分叉参数逐渐变化的过程中,系统经历从混沌到多倍周期运动再到混沌的往复过程。     

Motion of a deformable droplet of magnetic fluid in a rotating magnetic field

The problem of the magnetic field-driven rotation of a magnetic fluid droplet in a viscous nonmagnetic fluid is solved analytically and experimentally. The shape of the droplet and the magnetic fields and velocities of both fluids are calculated in the weak-field approximation. The droplet is flattened for any relations between the parameters of the system. The instability of the axisymmetric shape of the droplet is established experimentally. A result of the instability is a sudden change in the droplet shape from a flattened ellipsoid of revolution to a triaxial ellipsoid elongated in the equatorial plane. The critical magnetic Bond number is determined. Perm’. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 22–30, January–February, 2000. The work was carried out with the support of the Russian Foundation for Basic Research (project No. 98-01-00182).     

Magnetohydrodynamic flow past a porous rotating disk in a circular magnetic field

This paper studies the effects of a circular magnetic field on the flow of a conducting fluid about a porous rotating disk. Using modern quasi-Newton and globally convergent homotopy methods, numerical solutions are obtained for a wide range of magnetic field strengths, suction and injection velocities and Alfven and disk speeds. Results are presented graphically in terms of three non-dimensional parameters. There is excellent agreement with previous work and asymptotic formulae.     

Steady flow of a viscous incompressible fluid between two co-axial circular cylinders with suction and injection

In the present paper an attempt has been made to study the steady flow of a viscous incompressible fluid between two co-axial circular cylinders with small outward and inward normal suction on the outer and inner cylinders respectively with the assumption that the pressure is uniform over a cross-section. The expressions for axial velocity, the volume of fluid flowing per unit time across a cross-section and components of stress at any point of the fluid are derived.

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Stationäre Strömung einer zähen, inkompressiblen Flüssigkeit zwischen zwei koaxialen Kreiszylindern bei Absaugen und Ausblasung

Zusammenfassung Es wird die stationäre Strömung eines zähen, inkompressiblen Fluids zwischen zwei koaxialen Kreiszylindern mit geringer Absaugung oder Ausblasung sowohl am Innen- wie am Außenzylinder unter der Annahme gleichförmiger Druckverteilung über jeden Querschnitt untersucht. Die Beziehungen für die Axialgeschwindigkeit, der Fluidstrom über jeden Querschnitt und die Spannungskomponenten in jedem Punkt des Fluidfeldes werden hergeleitet.

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Nomenclature density of the fluid - x axial coordinate - y radial coordinate - azimuthal coordinate - u axial velocity - v radial velocity - w azimuthal velocity - p pressure - coefficient of viscosity - kinematic viscosity - a radius of the inner cylinder - b radius of the outer cylinder - v₀ suction velocity on the inner cylinder - v₀ suction velocity on the outer cylinder =suction parameter for the inner cylinder =suction parameter for the outer cylinder =dimensionlessy coordinates Q=discharge per unit time     

Non-isothermal couette flow of a rarefied gas between two rotating cylinders

An accurate numerical solution of the momentum and the heat transfer through a rarefied gas confined between two cylinders rotating with different angular velocities and having different temperatures has been obtained over a wide range of the Knudsen number on the basis of the Bhatnagar, Gross, Krook model equation. The viscous stress tensor, heat flux, and the fields of density, temperature and velocity are found. An analysis of the influence of the angular velocities and the temperature ratio on these quantities is given.