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.     

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.     

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.     

A class of exact solutions for the incompressible viscous magnetohydrodynamic flow over a porous rotating disk

The present paper is concerned with a class of exact solutions to the steady Navier-Stokes equations for the incompressible Newtonian viscous electrically conducting fluid flow due to a porous disk rotating with a constant angular speed.The three-dimensional hydromagnetic equations of motion are treated analytically to obtained exact solutions with the inclusion of suction and injection.The well-known thinning/thickening flow field effect of the suction/injection is better understood from the constructed closed form velocity equations.Making use of this solution,analytical formulas for the angular velocity components as well as for the permeable wall shear stresses are derived.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.As a result,exact formulas are obtained for the temperature field which take different forms corresponding to the condition of suction or injection imposed on the wall.     

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.     

Spontaneous rotation in the exact solution of magnetohydrodynamic equations for flow between two stationary impermeable disks

Magnetohydrodynamic (MHD) flow of a viscous electrically conducting incompressible fluid between two stationary impermeable disks is considered. A homogeneous electric current density vector normal to the surface is specified on the upper disk, and the lower disk is nonconducting. The exact von Karman solution of the complete system of MHD equations is studied in which the axial velocity and the magnetic field depend only on the axial coordinate. The problem contains two dimensionless parameters: the electric current density on the upper plate Y and the Batchelor number (magnetic Prandtl number). It is assumed that there is no external source that produces an axial magnetic field. The problem is solved for a Batchelor number of 0–2. Fluid flow is caused by the electric current. It is shown that for small values of Y, the fluid velocity vector has only axial and radial components. The velocity of motion increases with increasing Y, and at a critical value of Y, there is a bifurcation of the new steady flow regime with fluid rotation, while the flow without rotation becomes unstable. A feature of the obtained new exact solution is the absence of an axial magnetic field necessary for the occurrence of an azimuthal component of the ponderomotive force, as is the case in the MHD dynamo. A new mechanism for the bifurcation of rotation in MHD flow is found.     

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.     

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.     

Several exact solutions of the simplified Navier-Stokes equations (SNSE)

The Simplified Navier-Stokes equations (SNSE) and their exact solutions for the flow near a rotating disk and the flow in the vicinity of a stagnation point for both two- and three-dimensional flows are presented in this paper. The analysis shows that in the aforementioned cases the exact solutions of the inner-outer-layer-matched SNSE^[4] are completely consistent with those of the complete Navier-Stokes equations (CNSE) and that the exact velocity solutions of D-SNSE^[1,3] agree with those of CNSE, however, the exact pressure solutions of D-SNSE do not agree with those of CNSE. The maximum relative pressure errors between the exact solutions of D-SNSE and CNSE can be as high as a hundred per cent.     

Laminar Heat Transfer of a Swirled Flow in a Conical Diffuser. Self-similar Solution

Heat transfer in a laminar swirled air flow in the divergent channel between a disk and a cone whose vertex touches the disk is studied. A self-similar solution of the Navier-Stokes and energy equations is derived using group analysis. An exact numerical solution of the problem is obtained for different radial-to-tangential velocity ratios at the channel inlet.     

Some Steady MHD Flows of the Second Order Fluid

The exact analytic solutions of two problems of a second order fluid in presence of a uniform transverse magnetic field are investigated. The governing equation is of fourth order ordinary differential equation and is solved using perturbation method. In the first problem we discuss the flow of a second order fluid due to non-coaxial rotations of a porous disk and a fluid at infinity. In second problem the flow of a second order conducting fluid between two infinite plates rotating about the same axis is investigated, with suction or blowing along the axial direction. For second order conducting fluid it is observed that asymptotic solution exists for the velocity both in the case of suction and blowing.     

Stokes flow of a rotating sphere around the axis of a circular orifice or a circular disk

This paper presents an infinite series solution to the creeping flow equations for the axisymmetric motion of a sphere of arbitrary size rotating in a quiescent fluid around the axis of a circular orifice or a circular disk whose diameters are either larger or smaller than that of the sphere. Numerical tests of the convergence are passed and the comparison with the exact solution and other computational results shows an agreement to five significant figures for the torque coefficients in both cases. The torque coefficients are obtained for the sphere located up to a position tangent to the wall plane containing either the orifice or the disk. It is concluded that the torque coefficients of the sphere and the disk are monotonically increasing with the decrease of the distance from the disk or the orifice plane in both cases.     

Unsteady Flow of an Oscillating Porous Disk and a Fluid at Infinity

An exact analytic solution of the unsteady Navier–Stokes equations is obtained for the flow caused by the non-coaxial rotations of a porous disk and a fluid at infinity. The porous disk is executing oscillations in its own plane with superimposed injection or suction. An increasing or decreasing velocity amplitude of the oscillating porous disk is also discussed. Further, it is shown that a combination of suction/injection and decreasing/increasing velocity amplitude is possible as well. In addition, the flow due to porous oscillating disk and a fluid at infinity rotating about an axis parallel to the z-axis is attempted as a second problem. Sommario. Si studia il flusso non stazionario prodotto dall'oscillazione di un disco poroso in un fluido e si fornisce una soluzione analitica delle equazioni di Navier–Stokes. Si discute l'effetto di una suzione/iniezione e di una variazione sull'ampiezza della velocità' di oscillazione. Infine si studia il flusso dovuto alle oscillazioni non coassiali di un disco poroso e di un fluido all'infinito.     

Numerical modelling of rotating tangential layers (jets) in shells under strong uniform magnetic field

A numerical study of tangential layers in steady‐state magnetohydrodynamic rotating flows is presented using CFD to solve the inductionless governing equations. The analysis considers two basic flow configurations. In the first, a fluid is enclosed in a cylinder with electrically perfect conducting walls and the flow is driven by a small rotating, conducting disk. In the second, a flow is considered in a spherical shell with an inner rotating sphere. The fluid in both cases is subjected to an external axial uniform magnetic field. The results show that these flows exhibit two different types of flow cores separated from each other by a tangential layer parallel to the axis of rotation. The inner core follows a solid‐body rotation while the outer is quasistagnant. A counter‐rotating jet is developed in the tangential layer between the cores. The characteristics of the tangential layer and the properties of the meridional motion are determined for a wide range of Hartmann numbers. Distributions of angular velocity of circumferential flow and electric potential are obtained and the results are compared with those of analytic methods. Copyright © 2009 John Wiley & Sons, Ltd.     

Magnetohydrodynamic transient flows of a non-Newtonian fluid

Some exact solutions of the time-dependent partial differential equations are discussed for flows of an Oldroyd-B fluid. The fluid is electrically conducting and incompressible. The flows are generated by the impulsive motion of a boundary or by application of a constant pressure gradient. The method of Laplace transform is applied to obtain exact solutions. It is observed from the analysis that the governing differential equation for steady flow in an Oldroyd-B fluid is identical to that of the viscous fluid. Several results of interest are obtained as special cases of the presented analysis.     

An exact non‐linear Navier–Stokes compressible‐flow solution for CFD code verification

An exact similarity solution of the compressible‐flow Navier–Stokes equations is presented, which embeds supersonic, transonic, and subsonic regions. Describing the viscous and heat‐conducting high‐gradient flow in a shock wave, the solution accommodates non‐linear temperature‐dependent viscosity as well as heat‐conduction coefficients and provides the variation of all the flow variables and their derivatives. Also presented are methods to obtain time‐dependent and/or multi‐dimensional solutions as well as verification benchmarks of increasing severity. Comparisons between the developed analytical solution and CFD solutions of the Navier–Stokes equations, with determination of convergence rates and orders of accuracy of these solutions, illustrate the utility of the developed exact solution for verification purposes. Copyright © 2012 John Wiley & Sons, Ltd.     

Casson Fluid Flow Due to Non-Coaxial Rotation of a Porous Disk and the Fluid at Infinity Through a Porous Medium

The flow of the Casson fluid due to non-coaxial rotation of a disk and the fluid at infinity is investigated. Partial differential equations are made dimensionless and coupled. The exact solution of the resultant nonlinear initial-boundary-value problem is solved by applying the Laplace transform. The shear stresses at the disk surface and the steady state stresses are computed. The effects of dimensionless parameters on the dimensionless primary and secondary velocities are analyzed.     

Viscous incompressible flow over the surface of a rotary disk

Viscous incompressible film flow over the surface of an impermeable rotary disk is studied. An exact self-similar solution of the complete Navier-Stokes system of equations is obtained and the velocity and pressure fields together with the radial profiles of the fluid film are determined. A physical interpretation of the results obtained is given.Volgograd. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 39–43, November–December, 1995.     

Exact Solution of the Heat Transfer Problem for a Rotating Disk under Uniform Jet Impingement

An exact solution of the heat transfer problem for a uniform air stream impinging on a rotating disk is found. By introducing self-similar radial velocity and temperature profiles, the problem is reduced to a system of ordinary differential equations which are solved numerically. The Nusselt numbers are calculated for Prandtl numbers equal to 1 and 0.71 and various ratios of the free-stream velocity to the disk rotation velocity. The limits of the flow regime in which the heat transfer is determined solely by the impact jet parameters are found. The results are compared with experimental data for the stagnation point.     

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.