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.     

Self-gravitating stability of a rotating fluid layer

The self-gravitating instability of the present model is discussed by using a simple linear theory. The problem is formulated for a rotating fluid layer, and a dispersion relation valid for all kinds of perturbations is derived and discussed. The self-gravitating force is found to be a destabilizing factor for a small range of wavenumbers, while it is stabilizing in other ranges, depending on the density ratio of the fluids. For high values of the angular velocity, the rotational force produces a stabilizing effect and can suppress the self-gravitating instability. In the absence of the self-gravitating force, the model of a rotating fluid layer is marginally stable.     

Convective stability of a horizontal rotating fluid layer with helical turbulence

The problem of the corrective stability of a horizontal layer of turbulent fluid rotating about a vertical axis with a fixed heat flow at the boundaries is investigated in the case in which the intensity of the helical background does not depend on the rate of rotation and the degree of heating.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 33–39, January–February, 1992.The author is grateful to S. S. Moiseev for proposing the subject and to G. Z. Gershuni and D. V. Lyubimov for useful discussions.     

Convective stability of a horizontal rotating fluid layer with distributed heat sources

The results of investigating the convective instability of a horizontal layer of rotating fluid, created by a temperature difference applied at the boundaries of the layer and by heat sources distributed according to various laws, are presented. It is shown that, when the other parameters of the problem are fixed, an increase in the internal heat release lowers the limits of both monotonic and oscillatory stability of the layer, increases the wave number and reduces the neutral oscillation frequency. An increase in source concentration towards the center of the layer intensifies the effect. As the strength of the internal heat sources and their concentration towards the center of the layer increase, the oscillating convection that develops at the stability limit when the Prandtl number is low and the rotation fairly fast is first replaced by monotonic convection and then ceases altogether.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 21–28, January–February, 1989.     

Flow of a Casson fluid on a rotating disk

The flow of a thin layer of a Casson fluid on a fast rotating disk is considered. The film thickness distribution at various times for various initial thickness distribution is calculated. The stability of the flow is examined.     

Flow of a Viscoplastic Fluid on a Rotating Disk

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Fluid flow in a rotating cylindrical container with a rotating disk at the fluid surface

Fluid flow in a rotating cylindrical container of radius R_w and height H with a co-axially rotating disk of radius R_d at the fluid surface is numerically investigated. The container and the disk rotate with angular velocities Ω_w and Ω_d, respectively. We solve the axisymmetric Navier-Stokes equations using a finite-volume method. The effects of the relative directions and magnitudes of the disk and container rotations are studied. The calculations are carried out with various ratios of Ω_w and Ω_d for H/R_w = 2 and R_d/R_w = 0.7. Streamlines and velocity vectors in the meridional plane and azimuthal velocities are obtained. The flow fields in the meridional plane are discussed with relation to azimuthal velocities in the interior of the container. The numerical results are also compared with experimental data.     

Boundary of monotonic and oscillatory convective stability of a horizontal fluid layer

Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, No. 6, pp. 44–50, November–December, 1991.     

Convective stability of a rotating conducting fluid layer in an external magnetic field

The stability of the rest state of a conducting incompressible fluid forming a horizontal layer with rigid dielectric boundaries heated from below and rotating about a vertical axis, with a vertical magnetic field superimposed, is studied in the Boussinesq approximation. With increase in the Rayleigh number, depending on the relationship between the problem parameters (Taylor, Chandrasekhar and kinematic and magnetic Prandtl numbers), the eigenvalue of the critical mode of the linearization operator may be zero or imaginary, so that the instability of the rest state may be monotonic or oscillatory. The effect of the parameter values on the instability mode is investigated. In particular, the parameter ranges on which the critical eigenvalue is zero or imaginary are found.     

Boundary layer stability on a wave-periodic surface

Stability under small perturbations is investigated for flows whose velocity depends periodically on the spatial coordinate in the direction of flow. Stability calculations are carried out for the case in which the velocity distribution is a solution of the boundary-layer equations.Novosibirsk. Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 6, pp. 11–16, November–December, 1972.     

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.     

The slow rotation of an axisymmetric solid submerged in a fluid with a surfactant surface layer—I The rotating disk in a semi-infinite fluid

A formulation in terms of a Fredholm integral equation of the first kind is given for the axisymmetric problem of a solid rotating in a bounded viscous fluid whose surface is contaminated with an immiscible surfactant film. The particular case of a rotating thin circular disk immersed in a semi-infinite body of fluid is studied in detail, the problem being reduced to the solution of a Fredholm integral equation of the second kind. This equation is solved both asymptotically and numerically, and the resistive torque on the disk and surface velocity profiles are computed for varying values of the ratio of the coefficient of surface shear viscosity to the coefficient of viscosity of the substrate fluid, and depth of the disk below the surface.     

Axisymmetric magnetohydrodynamic flow of Jeffrey fluid over a rotating disk

This paper examines the magnetohydrodynamic boundary layer flow of Jeffrey fluid due to a rotating disk. The governing partial differential equations are first transformed into the coupled system of ordinary differential equations and then solved by using the homotopy analysis method. The influence of various involved physical parameters on the dimensionless radial and azimuthal velocities is sketched and analyzed. The variation of skin friction coefficients in radial and azimuthal directions is studied for various values of pertinent parameters. Copyright © 2011 John Wiley & Sons, Ltd.     

Reliability of bi-orthogonal decomposition applied to a rotating disk boundary layer

An application of bi-orthogonal decomposition to an experiment on the transition of the boundary layer over a rotating disk is performed and compared with linear, wavelet and Fourier analyses. We show how this bi-orthogonal decomposition can detect the results of these three methods, the critical Reynolds number (R _c = 268) and the first transition Reynolds number (R _t=445), and a new Reynolds number (R = 365) where the entropy fluctuates significantly, before nonlinear effects appear.     

Vortex motion in a rotating barotropic fluid layer

To study vortex motion and the mechanisms of geostrophic adjustment (i.e. the equilibrium between pressure gradient and Coriolis force, which leads to the weakening of inertio-gravity waves) in large scale geophysical flows, we simulate the dynamics of a shallow-water layer in uniform rotation, without any forcing other than the initial injection of energy and potential enstrophy. Such a flow generates inertio-gravity waves which interact with the rotational eddies. We found that both inertio-gravity waves and rotation reduce the non-linear interactions between vortices, namely the condensation of the vorticity field into isolated coherent vortices, corresponding to the inverse rotational energy cascade, and the associated production of vorticity filaments, due to the direct potential enstrophy cascade. Rotation also inhibits the direct inertio-gravitational energy cascade for scales larger than the Rossby deformation radius. Therefore, if inertio-gravity waves are initially excited at large enough scales, they will remain trapped there due to rotation and there will be no geostrophic adjustment. On the contrary, if inertio-gravity waves are only present at scales smaller than the Rossby deformation radius, which are insensitive to the effect of rotation, they will non-linearly interact and cascade towards the dissipative scales, leaving the flow in geostrophic equilibrium.