MHD flow of a power-law fluid over a rotating disk

Magnetohydrodynamic flow of an electrically conducting power-law fluid in the vicinity of a constantly rotating infinite disk in the presence of a uniform magnetic field is considered. The steady, laminar and axi-symmetric flow is driven solely by the rotating disk, and the incompressible fluid obeys the inelastic Ostwald de Waele power-law model. The three-dimensional boundary layer equations transform exactly into a set of ordinary differential equations in a generalized similarity variable. These ODEs are solved numerically for values of the magnetic parameter m up to 4.0. The effect of the magnetic field is to reduce, and eventually suppress, the radially directed outflow. An accompanying reduction of the axial flow towards the disk is observed, together with a thinning of the boundary layer adjacent to the disk, thereby increasing the torque required to maintain rotation of the disk at the prescribed angular velocity. The influence of the magnetic field is more pronounced for shear-thinning than for shear-thickening fluids.     

Combined effects of rotation and translation on a MHD flow due to compressible viscous fluid

Summary The modification of an axi-symmetric viscous flow due to a relative rotation of a disk or fluid by a translation of the boundary are studied. The fluid is taken to be compressible and electrically conducting. The equations governing the motion are solved iteratively through a central-difference scheme. The effect of an axial magnetic field and disk temperature on the flow and heat transfer are included in the present analysis. The translation of the disk or fluid generates a velocity field at each plane parallel to the disk (secondary flow). The cartesian components of the velocity due to the secondary flow are oscillatory in nature when a rigid body rotation of the free stream along with a translation of the disk is considered. The magnetic field damps out the velocity field, and reduces the thickness of the boundary layer. The cross component of wall shear due to secondary flow acts in a direction opposite to the rotation of the disk or fluid for all cases of the motion. The rise in disk temperature produces an increment in the magnitude of the wall shear associated with the secondary flow.     

Exact solutions for the incompressible viscous magnetohydrodynamic fluid of a rotating disk flow

The main interest of the present investigation is to generate exact solutions to the steady Navier-Stokes equations for the incompressible Newtonian viscous electrically conducting fluid flow motion due to a disk rotating with a constant angular speed. For an external uniform magnetic field applied perpendicular to the plane of the disk, the governing equations allow an exact solution to develop taking into account of the rotational non-axisymmetric stationary conducting flow.Making use of the analytic solution, exact formulas for the angular velocity components as well as for the wall shear stresses are extracted. It is proved analytically that for the specific flow the properly defined thicknesses decay as the magnetic field strength increases in magnitude. Interaction of the resolved flow field with the surrounding temperature is further analyzed via the energy equation. The temperature field is shown to accord with the dissipation and the Joule heating. According to Fourier's heat law, a constant heat transfer from the disk to the fluid occurs, though decreases for small magnetic fields because of the dominance of Joule heating, it eventually increases for growing magnetic field parameters.     

MHD UNSTEADY FLOWS DUE TO NON—COAXIAL ROTATIONS OF A DISK AND A FLUID AT INFINITY

Exact analytical solution for flows of an electrically conducting fluid over an infinite oscillatory disk in the presence of a uniform transverse magnetic field is constructed. Both the disk and the fluid are in a state of non-coaxial rotation. Such a flow model has a great significance not only due to its own theoretical interest, but also due to applications to geophysics and engineering. The resulting initial value problem has been solved analytically by applying the Laplace transform technique and the explicit expressions for the velocity for steady and unsteady cases have been established. The analysis of the obtained results shows that the flow field is appreciably influenced by the applied magnetic field, the frequency and rotation parameters.     

An electromagnet with Hall excitation

In 1911 Corbino showed that if a disk with a current flowing to the axis is placed in a magnetic field parallel to the axis of the disk, then due to the Hall emf the initially straight line of electric current is turned into a spiral. This leads to an increase in the length of the current line and thus to an increase in the disk resistance. The change in the disk resistance in a magnetic field was used in [1] to switch the current in the circuit of an inductive energy store. If the electric current carriers move from the edge of the disk to the axis, the azimuthal Hall current is accompanied by an increase in the magnetic field inside the disk compared with that outside it [2]. The same processes occur in a hydromagnet [3–5], in which the radial flow of a conducting liquid in an axial magnetic field is used to amplify a magnetic field. In the papers mentioned earlier the transients which occur when the steady magnetic field is established were not considered. To produce a magnetic field, and particularly for switching, the switch-on time of the device is of considerable importance. Hence, in this paper we consider the nonstationary problem of the amplification of a magnetic field. The amplification of the field is obtained and the time taken for the stationary state to build up is found. Both quantities depend exponentially on the magnetic Reynolds number. For a hydromagnet it is shown that the steady-state magnetic field differs considerably from that obtained in [4, 5]. The disagreement between the results is due to the fact that the boundary conditions in [4, 5] were arbitrarily chosen.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 44–48, May–June, 1979.The author thanks E. I. Bichenkov and R. L. Rabinovich for useful discussions and advice.     

直流磁场下销盘摩擦接触区的电磁感应现象

为了研究直流磁场对45钢销盘摩擦副的摩擦磨损特性的影响,根据磁场销盘摩擦试验机的结构和销盘摩擦副在摩擦过程中的实际接触情况,建立了二维微凸峰接触静态磁场和瞬态磁场有限元模型,分析了销盘摩擦接触区的电磁感应现象,得出以下结论:磁感应强度B在摩擦接触区分布不均,在微凸峰接触点区域的磁感应强度B值较大;摩擦试验中,在销盘磨痕和磨屑的微凸峰接触区将产生较高频率的动态磁化,同时在微凸峰上产生较大的感应电流,这些现象促进了销盘磨痕表面和磨屑的氧化.     

Exact solutions for the incompressible viscous magnetohydrodynamic fluid of a rotating-disk flow with Hall current

The focus of the present study is to obtain exact solutions for the flow of a viscous hydromagnetic fluid due to the rotation of an infinite disk in the presence of an axial uniform steady magnetic field with the inclusion of Hall current effect. In place of the traditional von Karman's axisymmetric evolution of the flow, the rotational non-axisymmetric stationary conducting flow is taken into consideration here, whose governing equations allow an exact solution to develop bounded everywhere in the normal direction to the wall.The three-dimensional equations of motion are treated analytically yielding derivation of exact solutions, which differ from those of corresponding to the classical von Karman's conducting flow. Making use of this solution, analytical formulas for the angular velocity components, for the current density field as well as for the wall shear stresses are extracted. The critical peripheral locations at which extrema of the local skin friction occur are also determined. It is proved from the analytical results that for the specific flow the properly defined thicknesses decay as the magnetic field strength increases in magnitude, approaching their hydrodynamic value in the limit of large Hall numbers.Interaction of the resolved flow field with the surrounding temperature is further analyzed via the energy equation. The temperature field is shown to accord with the dissipation function. According to the Fourier's heat law, a constant heat transfer from the disk to the fluid occurs, though it increases by the presence of magnetic field, the increase is slowed down by the Hall effect eventually reaching its hydrodynamic limit.     

Electromagnetic diffraction by a thin conducting circular disk placed at the plane interface of two different media

In this paper, the problem of diffraction of time harmonic, electromagnetic waves by a thin ideally conducting disk lying at the plane interface of two different media is considered. In this analysis, the incident wave is a plane wave travelling in a direction perpendicular to the plane interface of the two media. A Hertz vector formulation is applied to reduce our electromagnetic diffraction problem to a system of two scalar problems which are solved by the help of two pairs of Fredholm integral equations of the second kind. Low frequency approximations to the tangential components of the magnetic intensity associated with the diffracted field at the surfaces of the disk, the induced surface current density on the disk and the scattering cross section are obtained.     

Three-dimensional rotational MHD flows in bounded volumes

Three-dimensional laminar flows of a viscous conducting gas in the neighborhood of a rotating disk are considered. The simultaneous impact of an external magnetic field, suction from the disk surface, and the axial temperature gradient as well as the action of the external axial magnetic field on three-dimensional flows in the neighborhood of rigid permeable surfaces are first studied. An exact analytic solution of the system of the boundary layer equations is obtained. It is found that the direction of the radial flow initiated in the boundary layer can be varied by changing the temperature ratio in the external flow and on the disk for various Prandtl numbers Pr. An approximate solution of the problem of flow in the rotating cylinder in the presence of a retarding cover is constructed on the basis of the approach developed for extended disks.     

The effect of the slip condition on unsteady flow due to non-coaxial rotations of disk and a fluid at infinity

An exact solution of the Navier-Stokes equation is constructed for the magnetohydrodynamic (MHD) flow. The flow is due to non-coaxially rotations of a porous disk with slip condition and a fluid at infinity. The solutions for steady and unsteady cases are obtained by Laplace transform method. The effects of magnetic field and slip parameters are shown and discussed.     

Boundary layer on a rotating disk with account for the Hall effect

The steady rotation of a disk of infinite radius in a conducting incompressible fluid in the presence of an axial magnetic field leads to the formation on the disk of a three-dimensional axisymmetric boundary layer in which all quantities, in view of the symmetry, depend only on two coordinates. Since the characteristic dimension is missing in this problem, the problem is self-similar and, consequently, reduces to the solution of ordinary differential equations.Several studies have been made of the steady rotation of a disk in an isotropically conductive fluid. In [1] a study was made of the asymptotic behavior of the solution at a large distance from the disk. In [2] the problem is linearized under the assumption of small Alfven numbers, and the solution is constructed with the aid of the method of integral relations. In the case of small magnetic Reynolds numbers the problem has been solved by numerical methods [3,4]. In [5] the method of integral relations was used to study translational flow past a disk. The rotation of a weakly conductive fluid above a fixed base was studied in [6,7], The effect of conductivity anisotropy on a flow of a similar sort was studied approximately in [8], In the following we present a numerical solution of the boundary-layer problem on a disk with account for the Hall effect.     

EXACT SOLUTIONS FOR MAGNETOHYDRODYNAMIC FLOW IN A ROTATING FLUID

An analytical solution is obtained for the flow due to solid-body rotations an oscillating porous disk and of a fluid at infinity. Neglecting the induced magnetic field, the effects of the transversely applied magnetic field on the flow are studied. Further, the flow confined between two disks is also discussed. It is found that an infinite number of solutions exist for the flow confined between two disks.     

磁头/磁盘滑动接触下磁盘温度及热退磁临界条件的研究

采用二维轴对称有限元模型计算磁头/磁盘滑动接触下,铝质磁盘的稳态温度和热应力场以及热退磁临界条件.结果表明:磁盘温度在极短时间内升至摩擦稳态值,然后缓慢线性升高到最终稳态值;经过充分热传导和热交换后磁盘的温度梯度较小,此时磁层内的热应力集中分布于磁盘固定端边缘附近;磁盘的稳态温度和热应力均随速度增大而增大,且载荷越大其值增大越快;热应力小于1.2 GPa时所对应的速度和载荷为安全工况;温升大于373 K时所对应的工况将导致磁盘退磁.     

直流磁场下销盘摩擦过程中的电磁感应力矩分析

为了研究磁场对材料摩擦磨损性能的影响,本文分析了直流磁场下销盘摩擦过程中的涡流和电磁感应力,并通过积分得到感应力矩的计算公式.在销盘0.1 mm间隙时检测了磁吸力和在不同试验条件下盘试样所受的感应力矩,利用试验数据对感应力矩公式中的修正系数进行了最小二乘法拟合,所得感应力矩公式的计算结果与不同电流和转速下的试验数据很好地吻合.在摩擦磨损过程中,可利用感应力矩公式计算出相应试验条件下盘所受的感应力矩,分析其对摩擦系数的影响.     

RETRACTED ARTICLE: Exact solutions for the incompressible viscous magnetohydrodynamic fluid of a rotating disk flow

The main interest of the present paper is to generate exact solutions to the steady Navier-Stokes equations for the incompressible Newtonian viscous electrically conducting fluid flow motion due to a disk rotating with a constant angular speed. In place of the traditional von Karman’s axisymmetric evolution of the flow, the rotational non-axisymmetric stationary conducting flow is taken into consideration here. As a consequence, for an external uniform magnetic field applied perpendicular to the plane of the disk, the governing equations allow an exact solution to develop, which is influenced by a fixed point on the disk and also is bounded everywhere in the normal direction to the wall.     

MHD flow of a third-grade fluid due to eccentric rotations of a porous disk and a fluid at infinity

The problem of magnetohydrodynamics (MHD) flow of a conducting, incompressible third-grade fluid due to non-coaxial rotations of a porous disk and a fluid at infinity in the presence of a uniform transverse magnetic field is considered. An exact analysis is carried out to model the governing non-linear partial differential equation. A numerical solution of the third-order non-linear partial differential equation has been obtained. Several graphs and tables have been drawn to show the influence of porosity ε, magnetic parameter N, material parameters α and β on the velocity distribution.     

中碳钢/不锈钢磁场摩擦中磨屑的行为和作用

以45钢销/302不锈钢盘摩擦副为研究对象,采用自制的销、盘摩擦磨损试验机,研究了直流磁场作用下磨屑在摩擦过程中的行为及其对摩擦磨损性能的作用. 为此分析了有、无磁场作用下磨屑在磨损面上的分布特点,利用扫描电镜观察了磨屑及45钢销磨损面的形貌,采用三维形貌仪表征了磨损面特征区域的相对高度. 与无磁场时的摩擦磨损情况相比,磁场作用下45钢销的磨损量有所增大,而摩擦系数稍有减小. 摩擦过程中出现了302不锈钢盘向45钢销的材料转移并形成了不连续的转移层,该转移层相对高度较大,承担了主要的摩擦磨损并趋于平滑. 磁场作用下45钢销磨损面吸附少量磨屑并使之细化和氧化,该吸附磨屑在一定程度上减小了摩擦副的摩擦系数,并阻碍试样之间的材料转移,从而增加了45钢销的磨损量.      

表面纹理磁盘滑动接触的温度和应力及退磁临界条件的研究

采用应力场和温度场耦合的有限元方法,计算磁头/磁盘滑动接触下铝基磁盘磁层内瞬态温度场和应力场及退磁临界条件,分析接触压力、滑动速度、摩擦系数以及磁盘表面纹理对磁层内最大摩擦温升值和最大应力值的影响.结果表明:波形纹理表面瞬间滑动接触所产生的温度分布呈波形特征,表面纹理越尖锐,磁层内的温度和应力越大;滑动速度对磁层内温度的影响大于对应力的影响;当磁层最大应力小于1.2 GPa时,所对应的速度和压力为安全工况;当温升大于180 K时所对应的工况将导致磁盘退磁.     

Rotationally symmetric flow over a rotating disk

The rotationally symmetric flow over a rotating disk in an incompressible viscous fluid is analyzed by a new method when the fluid at infinity is in a state of rigid rotation (in the same or in the opposite sense) about the same axis as that of the disk. Asymptotic expansions for the velocity field over the entire flow field are obtained for the general class of one-parameter rotationally symmetric flows. This method is further extended to the case when a uniform suction or injection is assumed at the rotating disk. Fluid motion induced by oscillatory suction of small amplitude at the rotating disk is also discussed.An initial-value analysis reveals that resonance is possible only when the angular velocity of the rotating fluid is greater than that of the rotating disk.     

Unsteady compressible flow due to relative rotation and translation of a viscous fluid on a disk

Summary The modification of the axisymmetric viscous flow due to relative rotation of the disk or fluid by a translation of the boundary is studied. The fluid is taken to be compressible, and the relative rotation and translation velocity of the disk or fluid are time-dependent. The nonlinear partial differential equations governing the motion are solved numerically using an implicit finite difference scheme and Newton's linearisation technique. Numerical solutions are obtained at various non-dimensional times and disk temperatures. The non-symmetric part of the flow (secondary flow) describing the translation effect generates a velocity field at each plane parallel to the disk. The cartesian components of velocity due to secondary flow exhibit oscillations when the motion is due to rotation of the fluid on a translating disk. Increase in translation velocity produces an increment in the radial skin friction but reduces the tangential skin friction.