Stokes’ first problem for the rotating flow of a third grade fluid

This work is concerned with the unsteady rotating flow of the third grade fluid over a suddenly moving plate in its own plane. The non-linear problem governing the flow has been solved numerically. The influence of material parameter of third grade fluid and rotation upon the velocity has been discussed.     

On exact solutions for thin film flows of a micropolar fluid

The present paper contains the exact solutions for thin film flows of a micropolar fluid. Thin films are considered for three different geometries namely: (i) flow down an inclined plane, (ii) flow on a moving belt and (iii) flow down a vertical cylinder. The exact solution is obtained for the velocity and micro-rotation in first two cases for both weak and strong concentrations of the microelements. However, the exact expression for the micro-rotation is obtained for the third case and a numerical solution is presented for the velocity. The micro-rotation and velocity fields are sketched and discussed for different values of the material parameters. It is observed that the rotation of the microelements at the boundary increases the velocity when compared with the case when there is no rotation at the boundary.     

A pair of stream functions for three-dimensional vortex flows

Introducing a vector potential, that is based on a pair of stream functions, and a velocity potential, antisymmetric equations for the stream functions are derived with the help of a variational principle. It is found that the equations are in a suitable form to investigate flows with helical symmetry, and, for example, to connect upstream axisymmetric flows with downstream helical flows. The special case of a transition from an upstream solid-body vortex to a downstream helical flow is investigated in detail. Furthermore, the stream-function equations are particularly useful to investigate general small-amplitude inertia waves on vortex flows. Time-dependent helical flows that are time-independent in a suitably rotating frame of reference can also be discussed with the proposed method.     

On the stability of resonant rotation of a symmetric satellite in an elliptical orbit

We deal with the stability problem of resonant rotation of a symmetric rigid body about its center of mass in an elliptical orbit. The resonant rotation is a planar motion such that the body completes one rotation in absolute space during two orbital revolutions of its center of mass. In [1–3] the stability analysis of the above resonant rotation with respect to planar perturbations has been performed in detail.In this paper we study the stability of the resonant rotation in an extended formulation taking into account both planar and spatial perturbations. By analyzing linearized equations of perturbed motion, we found eccentricity intervals, where the resonant rotation is unstable. Outside of these intervals a nonlinear stability study has been performed and subintervals of formal stability and stability for most initial data have been found. In addition, the instability of the resonant rotation was established at several eccentricity values corresponding to the third and fourth order resonances.Our study has also shown that in linear approximation the spatial perturbations have no effect on the stability of the resonant rotation, whereas in a nonlinear system they can lead to its instability at some resonant values of the eccentricity.     

Variable energy blast wave through self-gravitating gas spheres

The propagation of variable energy blast wave through self-gravitating gas spheres has been studied. Similarity method has been used to obtain the solution in the form of a power series in non-dimensional shock radius. The analysis shows that the effects of self-gravitating force are of third order Therefore, the pressure, density and velocity distributions are calculated upto third approximation and shown graphically.     

Sufficient Global Stability Condition for a Model of the Synchronous Electric Motor under Nonlinear Load Moment

We study a model of the synchronous electric motor, which is described by a system of ordinary differential equations, including equations for electric currents in the windings of the rotor. The load moment is assumed to be a nonlinear function of the angular velocity of the rotor, allowing a linear estimate. The system of differential equations under consideration has a countable number of stationary solutions corresponding to the operating mode of uniform rotation of the rotor with the angular velocity equal to the angular velocity of rotation of the magnetic field in the stator. An effective sufficient condition is derived under which any motion of the rotor of the synchronous electric motor tends with time to uniform rotation.     

The effect of suction on boundary layer for rotating flows with or without magnetic field

The effect of suction on the steady laminar incompressible boundarylayer flow for a stationary infinite disc with or without magnetic field, when the fluid at a large distance from the surface of the disc undergoes a solid body rotation, has been studied. The governing coupled nonlinear equations have been solved numerically using the shooting method with least square convergence criterion. It has been found that suction tends to reduce the velocity overshoot and damp the oscillation.     

Hall Effects on Hydromagnetic Flow and Heat Transfer in a Rotating Channel

An exact solution has been obtained for the fully developedflow of a conducting fluid within a rotating channel due toaconstant pressure gradient in the presence of a uniform transversemagnetic field, taking Hall currents into account. The effectsof Hall current and rotation on the velocity and the magneticfield have been discussed. The shear stress at theplate decreasesdue to an increase in either the Hall parameter for the rotationparameter. Hall effects on the heat transfer have also beenconsidered. For small values of rotation parameter, the rateof heat transfer at the upper plate decreases while for largevalues of rotation parameter the rate of heat transfer firstdecreases, reaches a minimum and then increases with the increasein the Hall parameter. This result is reversed for the lowerplate whose temperature is below that of the upper plate.     

On the development of a wake vortex in inviscid flow

The evolution of an initial perturbation in an axisymmetric subsonic normal inviscid gas flow through a pipe is directly simulated. The basic (unperturbed) flow has a zero radial velocity component, while its axial velocity component (along the axis of symmetry) increases or decreases linearly with the radius. The perturbation is specified as a swirl (rotation about the axis) with a positive or negative velocity vanishing on the central axis and the lateral surface. Irrespective of its direction, the swirl gives rise to a steady-state vortex carried by the flow. It shape is spherical (contiguous to the rotation axis) or circular (sliding along the impermeable lateral surface).     

Flow and heat transfer of a non-Newtonian fluid past a stretching sheet with partial slip

The entrained flow and heat transfer of a non-Newtonian third grade fluid due to a linearly stretching surface with partial slip is considered. The partial slip is controlled by a dimensionless slip factor, which varies between zero (total adhesion) and infinity (full slip). Suitable similarity transformations are used to reduce the resulting highly nonlinear partial differential equations into ordinary differential equations. The issue of paucity of boundary conditions is addressed and an effective second order numerical scheme has been adopted to solve the obtained differential equations even without augmenting any extra boundary conditions. The important finding in this communication is the combined effects of the partial slip and the third grade fluid parameter on the velocity, skin-friction coefficient and the temperature field. It is interesting to find that the slip and the third grade fluid parameter have opposite effects on the velocity and the thermal boundary layers.     

Hiemenz flow and heat transfer of a third grade fluid

The laminar flow and heat transfer of an incompressible, third grade, electrically conducting fluid impinging normal to a plane in the presence of a uniform magnetic field is investigated. The heat transfer analysis has been carried out for two heating processes, namely, (i) with prescribed surface temperature (PST-case) and (ii) prescribed surface heat flux (PHF-case). By means of the similarity transformation, the governing non-linear partial differential equations are reduced to a system of non-linear ordinary differential equations and are solved by a second-order numerical technique. Effects of various non-Newtonian fluid parameters, magnetic parameter, Prandtl number on the velocity and temperature fields have been investigated in detail and shown graphically. It is found that the velocity gradient at the wall decreases as the third grade fluid parameter increases.     

Unsteady MHD Couette flow in a rotating system

The unsteady hydromagnetic Couette flow of a viscous incompressible electrically conducting fluid in a rotating system has been considered. An exact solution of the governing equations has been obtained by using a Laplace transform. Solutions for the velocity distributions as well as shear stresses have been obtained for small times as well as for large times. It is found that for large times the primary velocity decreases with increase in the rotation parameter K² while it increases with increase in the magnetic parameter M². It is also found that with increase in K², the secondary velocity v₁ decreases near the stationary plate while it increases near the moving plate. On the other hand, the secondary velocity decreases with increase in the magnetic parameter.     

The vibrations of a rotating thin piezoceramic cylindrical shell

G. H. Bryan [8] discovered and studied the dynamic phenomena that arise during the vibrations of thinwalled shells in natural modes while rotating uniformly about an axis of symmetry. In the present article a more detailed mathematical analysis has been conducted for the solution of the problem of the vibrations of a ring rotating with constant angular velocity under feasible conditions of perturbation of the vibrations and realistic mechanical properties of ceramic materials. It is established that two slowly precessing waves arise during the vibration of the ring, one of which (the one discovered by Bryan) is an inverse precession wave having an angular velocity of precession less than the angular velocity of rotation of the ring, while the second is a direct precession wave having angular velocity larger than the angular velocity of rotation of the ring. Under the actual working conditions for piezoceramic resonators the amplitudes of the direct and inverse wave become comparable. Consequently the vibrations present as a modulated standing wave, that is, the phenomenon of beats in the mode of steady-state normal vibration occurs. Translated fromMatematichni Metodi ta Fiziko-mekhanichni Polya, Vol. 39, No. 1, 1996, pp. 7–18.     

Some Aspects of Designing Multirim Composite Flywheels

Approximate solutions are given for stresses in a flexible cylindrical interlayer connecting concentric, rigid, cylindrical rims subjected to three loading cases: (i) rotation about the axis of symmetry; (ii) in-plane translation of the rims relative to each other; (iii) out-of-plane rotation of the rims relative to each other. The solutions are important for the multiple filament-wound composite rims used in energy storage flywheels, where the elastomeric interlayer idea has been proposed as a means of preventing high radial tensile stresses, which would otherwise break down the rims at less than optimal speeds. The compliances associated with the second and third loading cases are also given, establishing a simple means of analysis of the critical vibration frequencies of multirim flywheel rotors.     

A study of effects of wall-normal rotation on the turbulent channel flow

The effects of the wall–normal rotation on the turbulence channel flow have been studied. A series of direct numerical simulations have been performed with various rotation rates for Reynolds number 180 based on the friction velocity in the non–rotating case. All remarkable changes are discussed. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

计及Hall和离子滑移电流的旋转流体在与时间相关初值条件下的MHDCouette流动

考虑Hall和离子滑移电流的影响,在旋转系统中研究导电流体非稳定的MHD Couette流动．在小数值磁场Reynolds数假定下,推导出基本的控制方程,使用著名的Laplace变换技术,数值地求解该基本方程．分两种情况:磁场相对于流体或者移动平板固定时,得到速度和表面摩擦力统一的闭式表达式．用图形讨论了问题的不同参数,对速度和表面摩擦力的影响．所得结果显示,主流速度u和次生速度v随着Hall电流而增大．离子滑移电流的增大,也会导致主流速度u的增大,但会使次生速度v减小．还给出了旋转、Hall和离子滑移参数的综合影响,确定了次生运动对流体流动的贡献．     

在不均匀旋转大气中瞬时圆形轨道的人造卫星的寿命

本文用能量观点研究了人造卫星在不均匀旋转大气的阻力作用下的寿命.根据流体动力学,大气旋转角速度随高度而减小.大气密度按指数形式而减少.文中推出了卫星在上述旋转大气模型中瞬时圆形轨道上的寿命表达式,并对具体卫星的寿命做了数值估计.结果表明:在本文中所计算出的卫星寿命比前人将大气旋转同地球自转角速度一样的模型所算出的寿命要短些.     

DNS of a Turbulent Channel Flow with Streamwise Rotation - Study of the Reverse Effect of the Cross Flow

Modeling of rotating turbulent flows is a major issue in engineering applications. In this work a turbulent channel flow rotating about the streamwise direction is presented. The theory is based on the investigations of [3] employing Lie group analysis. It was found that a cross flow in spanwise direction is induced. A series of direct numerical simulations (DNS) has been conducted for both different rotation rates and different Reynolds numbers to validate the cross flow. In addition some new interesting effects were observed. The averaged profile ū₃ of the cross flow is formed like a ‘S’ that means it exhibits a triple zero-crossing which denotes regions of reverse flow. Alaso a reverse effect is seen which means that for small rotation rates up to Ro=10 the spanwise mean velocity profiles increase and at rotation number Ro=14 this effect appears to reverse. Both effects were observed at two different Reynolds numbers Re = 180 and Re = 270. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Exact analytical solutions for the Poiseuille and Couette-Poiseuille flow of third grade fluid between parallel plates

Exact analytical solutions for the velocity profiles and flow rates have been obtained in explicit forms for the Poiseuille and Couette-Poiseuille flow of a third grade fluid between two parallel plates. These exact solutions match well with their numerical counter parts and are better than the recently developed approximate analytical solutions. Besides, effects of various parameters on the velocity profile and flow rate have been studied.     

Fokker-planck equation for brownian motion of rotating spherical particles : PMM vol. 40, n 6, 1976, pp. 1127–1131

The Langevin equation to derive the Fokker-Planck equation is used for the Brownian motion of particles in translational motion. The Fokker-Planck equation for the Brownian motion of particles which have, in addition to the translational velocity also an angular velocity, has not, so far, been derived. This can apparently be explained by the fact that in the case of the rotational motion, the Langevin equation for the translational motion velocity vector must be supplemented by a corresponding equation for an angular velocity vector. The latter equation must contain, in addition to the systematic moment of reaction linearly dependent on the angular velocity of rotation itself, a random moment rapidly varying with time. Moreover, to ensure the compatibility of two differential vector equations within the system, additional relations which must be introduced, must connect not only the coefficients of the systematic reactions, but also the. random vectors varying rapidly with time.In [1],the Boltzmann's equation for a mixture of two gases was used to derive a Fokker-Planck equation for a translational motion of Brownian particles. The same method can be applied to the Brownian motion of spherical particles which have, in addition to the translational velocities, angular velocities of self-rotations. In this case there is no need to introduce additional relations connecting the random rapidly varying vectors.In the present paper we derive the Fokker-Planck equations for a new model of rotating spherical molecules which was used in [2].