Hall effects on unsteady flow of a viscous fluid due to non-coaxial rotation of a porous disk and a fluid at infinity

An exact solution of the unsteady hydromagnetic flow due to non-coaxial rotations of a porous disk and a fluid at infinity is obtained on taking Hall currents into account. An analytical solution of the problem is obtained for small and large times after the start by the Laplace transform method. It is found that for small values of time there is no inertial oscillations while for large time the steady state is reached through inertial oscillations. The frequency of these oscillations first increases, reaches a maximum and then decreases with increase in Hall parameter.     

Oscillatory flow due to eccentrically rotating porous disk and a fluid at infinity

An analytical solution of the unsteady Navier–Stokes equations is obtained for the flow due to non-coaxial rotations of an oscillating porous disk and a fluid at infinity, rotating about an axis parallel to the axes of rotation of the disk through a fixed point. The velocity distributions and the shear stresses at the disk are obtained for three different cases when the frequency parameter is greater than, equal to or less than the rotation parameter. The flow has a boundary layer structure even in the case of blowing at the disk.     

Torsional stiffness of a layer bonded to an elastic half space

The static solution to the problem of a layer bonded to an elastic half-space, where the layer is driven by the torsional rotation of a bonded rigid circular disk, is considered here. An iterative solution, perturbing on that given for the elastic half-space, is obtained as a convergent power series, provided the ratio of the stratum depth to the radius of the disk is large. An equation for the applied static torque at the surface of the rigid disk is also calculated and compared, under limiting cases, with known results.     

Flow of a third grade fluid due to an accelerated disk

The magnetohydrodynamic (MHD) flow induced by non‐coaxial rotation of porous disk and a third grade fluid at infinity is investigated. The disk is moving with uniform acceleration and rotating with a uniform angular velocity. Numerical solution of the governing nonlinear initial and boundary value problem is obtained. The effects of physical parameters on the velocity profiles are examined in detail. The present study shows that the constant acceleration part has a greater influence than the time part of the assumed variable velocity of the disk. Copyright © 2009 John Wiley & Sons, Ltd.     

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.     

Unsteady boundary layer flow due to a stretching surface in a rotating fluid

The induced unsteady flow due to a stretching surface in a rotating fluid, where the unsteadiness is caused by the suddenly stretched surface is studied in this paper. After a similarity transformation, the unsteady Navier–Stokes equations have been solved numerically using the Keller-box method. Also, the perturbation solution for small times as well as the asymptotic solution for large times, when the flow becomes steady, has been obtained. It is found that there is a smooth transition from the small time solution to the large time or steady state solution.     

Impact of a circular disk and washer on a fluid in a cylindrical vessel

The solution of the problem of the axisymmetric motion of an ideal incompressible fluid in a cylindrical vessel of finite depth is obtained for small vibrations of a flexible circular disk and washer (disk with centered hole) on the surface of the fluid. On the basis of this solution the virtual mass is determined as a function of the dimensions of the vessel, disk and washer for the special case of a rigid nondeformable disk and washer.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 103–111, January–February, 1995.     

Rotating disk heat transfer in a fluid swirling as a forced vortex

The paper represents results of an exact solution of a laminar heat transfer problem for a rotating disk in a fluid co-rotating with the disk as a solid body. The angular speed of the fluid is less than the angular speed of the disk. Disks surface temperature varies radially accordingly to a power law. Results for the laminar regime are compared with computations for turbulent heat transfer obtained using an integral method developed earlier. On the basis of the exact solution for laminar flow and basic ideas of the integral methods solution for turbulent flow, an integral method for laminar regime is designed and an approximate analytical solution of the considered problem is derived. Inaccuracies of the laminar approximate solution over the main range of variation of the influencing parameters and Prandtl numbers from 0.71 to 1 do not exceed 2.5%. It is shown that the dependence of the Nusselt number on the ratio of the angular speeds of disk and fluid varying from 0 to 0.3 is weak and has a point of maximum within this region for laminar flow. The obtained results are important in predictions of fluid flow and heat transfer in different types of rotating machinery.     

Analysis of vibrations in a nonhomogeneous thermoelastic thin annular disk under dynamic pressure

The forced vibration analysis of nonhomogeneous thermoelastic, isotropic, thin annular disk under periodic and exponential types of axisymmetric dynamic pressures applied on its inner boundary has been performed and analytical benchmark solution has been obtained. The material has been assumed to have inhomogeneity according to a simple power law in the radial coordinate. The present analysis has been worked out in the context of generalized theory of thermoelasticity with one relaxation time. The two coupled partial differential equations have been clubbed and solved by employing Laplace transform technique to obtain the solution for radial displacement and temperature change in the space domain. In order to obtain the solution in physical domain, the inversion of the transform has been carried out by using residue calculus. The radial displacement, radial stress, circumferential stress, and temperature change have been computed numerically for copper material annular disk. The numerically computed results have been presented graphically to demonstrate the effect of two different types of dynamic pressure in reference to homogeneous and nonhomogeneous material disk. The results for homogeneous material disk have been deduced and validated with that available in literature. The closed-form solution obtained here is interesting and allows further parametric studies of nonhomogeneous structures.     

Oscillatory flow due to eccentrically rotating porous disk and a fluid at infinity embedded in a porous medium

An oscillatory flow due to non-coaxial rotations of an oscillating porous disk and a fluid at infinity rotating about an axis parallel to the axis of rotation of the disk through a fixed point has been investigated. An analytical solution of the unsteady Navier-Stokes equations is obtained for three cases when the frequency parameter is less than, equal to or greater than the rotation parameter. The influences of the physical parameters acting on the flow are explained with the help of the figures. It is found that the depth of the penetration or the wave length of the layers decreases with an increase in porosity parameter.     

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.     

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.     

Non-linear flexure testing of a high-strength glass disk

Strength measurements of glass disk substrates were made by ring-on-ring flexural testing. The non-linear problem of a disk undergoing a large deflection in ring bending was solved approximately by a variational method. The solution has the minimum total energy and satisfies the required boundary conditions. Deflection, stress and strain distributions are evaluated as functions of the external load in terms of seven parameters related to sample material and geometry. Good agreement between experiment and theory has been obtained without any adjustable parameter for a glass disk of Corning Code 0313.     

Exact and asymptotic solutions of the Navier-Stokes equations in a fluid layer between plates approaching and moving apart from one another

Exact solutions of the Navier-Stokes equations are investigated in the layer between parallel plates the distance between which changes proportionally to the square root of time. At the boundaries of the plates the no-slip condition is assigned. For approaching plates a countable family of exact solutions each of which continuously depends on the Reynolds number is obtained. At a sufficiently large Reynolds number, near the boundary a counterflow is formed: the velocity is directed oppositely to the average velocity. On the basis of the exact solution obtained, relative errors are calculated for the asymptotic theories of Reynolds lubricating layer and Prandtl boundary layer.     

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.     

Numerical solution of a casson fluid flow in the entrance of annular tubes

The solution of the momentum equation for a Casson fluid flowing in the entrance region of an annular tube has been obtained. The results have been presented for a large range of radii ratio and dimensionless yield stress. The mathematical accuracy of the numerical procedure is demonstrated by comparing the asymptotic velocity profiles at large axial distance with fully developed solution [1]. In addition, the results of the numerical solution for the case of yield stress equal to zero are compared with the entrance flow solution for a Newtonian fluid [2].     

Wave induced thermal boundary layers in a compressible fluid: analysis and numerical simulations

The response of a semi-infinite compressible fluid to a step-wise change in temperature of its boundary is investigated analytically and numerically. Numerical results of the boundary layer structure are compared with Clarke’s analytical solution for a gas with thermal conductivity proportional to temperature. To avoid unwanted numerical dissipation in the numerical analysis, the space-time conservation element and solution element (CESE) method has been adopted to solve the unsteady 1-D Navier-Stokes equations. Good agreement between analytical and numerical results has been found for the development of the thermal boundary layer on a long time scale. Weak shock waves and expansion waves induced by the thermal boundary layer due to its compressibility, are observed in the numerical simulation. Finally, the numerical method has been applied to the reflection of a non-linear expansion wave and to a shock wave from an isothermal wall, thereby illustrating the effect of the boundary layer on the external flow field.     

Unsteady three‐dimensional boundary layer flow due to a stretching surface in a micropolar fluid

In this paper, the unsteady three‐dimensional boundary layer flow due to a stretching surface in a viscous and incompressible micropolar fluid is considered. The partial differential equations governing the unsteady laminar boundary layer flow are solved numerically using an implicit finite‐difference scheme. The numerical solutions are obtained which are uniformly valid for all dimensionless time from initial unsteady‐state flow to final steady‐state flow in the whole spatial region. The equations for the initial unsteady‐state flow are also solved analytically. It is found that there is a smooth transition from the small‐time solution to the large‐time solution. The features of the flow for different values of the governing parameters are analyzed and discussed. The solutions of interest for the skin friction coefficient with various values of the stretching parameter c and material parameter K are presented. Copyright © 2011 John Wiley & Sons, Ltd.     

Steady and modulated flow of an Ostwald fluid around a rotating disk

The solutions of the continuity equation and the equations of motion of the flow in the vicinity of a rotating disk have been established for an Ostwald fluid under steady-state conditions and in modulated flow around a mean value. Under steady-state conditions, the kinematics of the flow is scarcely dependent on the rheological parameters close to the disk, however, for n < 1 long-range effects have been put forward. For modulated flow, in the high-frequency range, a behaviour very different from that observed for a Newtonian fluid was found. In the low-frequency range an asymptotic solution has been proposed which is of special interest in mass transfer problems.Presented at the Second Conference of European Rheologists, Prague, June 17–20, 1986     

Unsteady jet flow of a viscous fluid

A solution is obtained, within the framework of the boundary layer theory, to the problem of the unsteady flow created by a two-dimensional jet source for a given momentum flux variation with time. In particular, aperiodic and periodic momentum variations are examined, and a qualitative analog with turbulent flow is noted for the latter.