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.     

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.     

Flow due to non-coaxial rotation of a porous disk and a second grade fluid rotating at infinity

An exact solution for the three-dimensional flow due to non-coaxial rotation of a porous disk and a second grade fluid at infinity is obtained. It is shown that for uniform suction or uniform blowing at the disk, an asymptotic profile exists for the velocity distribution. The velocity depends on two parameters: one of them is the suction parameter or blowing parameter and the other is the visco-elastic parameter. Furthermore, it is found that when the value of the visco-elastic parameter is fixed, the velocity decreases with an increase in the value of the suction parameter and when the value of the suction parameter is fixed, the velocity increases with an increase in the value of the visco-elastic parameter.     

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.     

Hall effects on oscillating flow due to eccentrically rotating porous disk and a fluid at infinity

Hall effects on the viscous incompressible fluid due to non-coaxial rotations of an oscillating porous disk and a fluid at infinity are studied. The velocity field, shear stresses and temperature distribution are obtained in closed form. It is found that with increase in frequency parameter, the primary velocity increases near the disk and becomes almost stationary away from the disk. The secondary velocity also increases with increase in frequency parameter. It is seen that with increase in Hall parameter, the primary velocity increases near the disk and decreases away from the disk. The reversed effect is observed for the secondary velocity. The shear stresses at the disk are also obtained. It is found that the shear stresses due to the primary and the secondary velocities decrease with increase in Hall parameter. The heat transfer characteristic is also studied on taking viscous dissipation into account. It is found that the mean temperature at the disk 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.     

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.     

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.     

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 fluid of a porous rotating disk flow

The present paper is concerned with a class of exact solutions to the steady Navier-Stokes equations for the incompressible Newtonian viscous fluid flow motion due to a porous disk rotating with a constant angular speed. The three-dimensional equations of motion are treated analytically yielding derivation of exact solutions with suction and injection through the surface included. The well-known thinning/thickening flow field effect of the suction/injection is better understood from the exact velocity equations obtained. Making use of this solution, analytical formulas corresponding to the permeable wall shear stresses are extracted.Interaction of the resolved flow field with the surrounding temperature is further analyzed via the energy equation. As a result, exact formulas are obtained for the temperature field which take different forms depending on whether suction or injection is imposed on the wall. The impacts of several quantities are investigated on the resulting temperature field. In accordance with the Fourier‘s heat law, a constant heat transfer from the porous disk to the fluid takes place. Although the influence of dissipation varies, suction enhances the heat transfer rate as opposed to the injection.     

Effects of Hall current and wall temperature oscillation on convective flow in a rotating fluid through porous medium

The combined effects of Hall current and plate temperature oscillation on the hydromagnetic free convective flow through a porous medium bounded by an infinite vertical limiting surface in a rotating fluid are discussed. The suction velocity perpendicular to the limiting surface is constant and the temperature of the limiting surface fluctuates with time about a non-zero constant mean. The expressions for the velocity and temperature fields are derived, and the effects of various parameters on the velocity and shear-stress are determined.

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Einflüsse von Hallstrom und Wandtemperaturschwingungen auf die Konvektionsströmung in einem rotierenden Fluid durch ein poröses Medium

Zusammenfassung Hier werden alle Einflüsse von Hallstrom und Plattentemperaturschwingungen auf eine hydromagnetische freie Konvektionsströmung durch ein poröses Medium analysiert. Das Medium wird von einer unendlichen, vertikalen Begrenzungsoberfläche in einem rotierenden Fluid eingeschränkt. Die Ansauggeschwindigkeit senkrecht zur Begrenzungsoberfläche bleibt konstant. Die Temperatur der Begrenzungsoberfläche schwankt mit der Zeit um einen konstanten, von Null verschiedenen Wert. Aussagen über die Geschwindigkeit und die Temperaturfelder werden hieraus abgeleitet und die Geschwindigkeit und Scherbeanspruchung bestimmt.

     

Effects of temperature dependent fluid properties and variable Prandtl number on the transient convective flow due to a porous rotating disk

In this paper we have studied the effects of temperature dependent fluid properties such as density, viscosity and thermal conductivity and variable Prandtl number on unsteady convective heat transfer flow over a porous rotating disk. Using similarity transformations we reduce the governing nonlinear partial differential equations for flow and heat transfer into a system of ordinary differential equations which are then solved numerically by applying Nachtsheim–Swigert shooting iteration technique along with sixth-order Runge–Kutta integration scheme. Comparison with previously published work for steady case of the problem were performed and found to be in very good agreement. The obtained numerical results show that the rate of heat transfer in a fluid of constant properties is higher than in a fluid of variable properties. The results further show that consideration of Prandtl number as constant within the boundary layer for variable fluid properties lead unrealistic results. Therefore, modeling thermal boundary layers with temperature dependent fluid properties Prandtl number must treated as variable inside the boundary layer.     

On one slip condition for the equations of a viscous fluid

By studying the joint flow of a viscous and a micropolar fluid, we obtained a new boundary condition for the equations of the viscous fluid for the case where a thin layer of a granular fluid is present on the interface with the solid. Examples of using this condition in problems of drilling mud flow in the presence of a mud cake on the borehole wall are given.     

Rotating flow of a third grade fluid in a porous space with Hall current

This study investigates the rotating magnetohydrodynamic (MHD) flow of a third-grade fluid in a porous space. Modified Darcy's law has been utilized for the flow modeling. The Hall effects are taken into consideration. The basic equations governing the flow are reduced to a highly nonlinear ordinary differential equation. This equation has been solved analytically by employing the homotopy analysis method (HAM). The effects of the various interesting parameters on the velocity distribution have been discussed.     

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.     

Hall effects on the unsteady hydromagnetic oscillatory flow of a second-grade fluid

An exact solution of an oscillatory flow is constructed in a rotating fluid under the influence of an uniform transverse magnetic field. The fluid is considered as second-grade (non-Newtonian). The influence of Hall currents and material parameters of the second-grade fluid is investigated. The hydromagnetic flow is generated in the uniformly rotating fluid bounded between two rigid non-conducting parallel plates by small amplitude oscillations of the upper plate. The exact solutions of the steady and unsteady velocity fields are constructed. It is found that the steady solution depends on the Hall parameter but is independent of the material parameter of the fluid. The unsteady part of the solution depends upon both (Hall and material) parameters. Attention is focused upon the physical nature of the solution, and the structure of the various kinds of boundary layers is examined. Several results of physical interest have been deduced in limiting cases.     

Drag of a sphere in an unsteady flow of viscous fluid at small reynolds numbers

An unsteady flow of viscous incompressible fluid past a sphere is investigated. The values of the inertial and unsteady terms in the Navier-Stokes equations are characterized by translational (R) and vibrational (R_k) Reynolds numbers, which are assumed small. The solution is constructed in the form of an expansion with respect to max(R, R _k ^1/2 ) by the method of matched asymptotic expansions. A correction to the Stokes force, correct to o[max(R, R _k ^1/2 )], is calculated. It is shown that the result depends strongly on the ratio R/R _k ^1/2 and goes over into the well-known equations for the cases R 0, R_k 0.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 11–16, January–February, 1988.     

Effects of Hall current on f lows of a Burgers’ fluid through a porous medium

This paper generalizes the analysis of four magnetohydrodynamic (MHD) flow problems of an Oldroyd-B fluid discussed by Asghar et al. [Int. J. Non-linear Mech. 40, 589–601 (2005)] into three directions: (i) to discuss the problems in a porous medium using modified Darcy’s law (ii) to see the influence of Hall current (iii) to determine the effect of rheological parameter of Burgers’ fluid. Analytical solutions of velocity distribution valid at large and small times are given in each problem. Comparison has been provided for Oldroyd-B and Burgers’ fluids through graphs. The physical interpretation is also included.