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 study of the flow and heat transfer in a Reiner-Rivlin fluid on a rotating porous disk

An unsteady flow and heat transfer to an infinite porous disk rotating in a Reiner—Rivlin non-Newtonian fluid are considered. The effect of the non-Newtonian fluid characteristics and injection (suction) through the disk surface on velocity and temperature distributions and heat transfer is considered. Numerical solutions are obtained over the entire range of the governing parameters.Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 1, pp. 85–95, January–February, 2005.     

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.     

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.     

Motion of a fluid between rotating cones

A study is made of the steady axisymmetric flow of a viscous fluid between two cones rotating in opposite ways round a common axis. It is shown that as in the case of the flow of fluid swirled by plane disks rotating at different speeds [1], there can be two regimes of motion in the system: a Batchelor regime with quasirigid rotation of the fluid outside the boundary layers [2] and a Stewartson regime in which the azimuthal flow is concentrated only in the boundary layers [3]. In the Stewartson regime, a boundary layer analogous to that in the single disk problem (see, for example, [4–6]) forms in the region of each cone far from the apex. For the flows outside the boundary layers, simple expressions are found which make it possible to obtain a conception of the circulation of the fluid as a whole. With minor alterations, the results can be applied to the case of the rotation of other curved surfaces.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 58–64, March–April, 1985.The author thanks A. M. Obukhov for suggesting the subject and for his interest in the work, and A. V. Danilov and S. V. Nesterov for useful discussions.     

Stability of stationary convective flow of a mixture in a vertical porous layer

In contrast to the corresponding viscous flow, the convective flow of a homogeneous liquid in a planar vertical layer whose boundaries are maintained at different temperatures is stable [1]. When a porous layer is saturated with a binary mixture, in the presence of potentially stable stratification one must expect an instability of thermal-concentration nature to be manifested. This instability mechanism is associated with the difference between the temperature and concentration relaxation times, which leads to a buoyancy force when an element of the fluid is displaced horizontally. In viscous binary mixtures, the thermal-concentration instability is the origin of the formation of layered flows, which have been studied in detail in recent years [2–4]. The convective instability of the equilibrium of a binary mixture in a porous medium was considered earlier by the present authors in [5]. In the present paper, the stability of stationary convective flow of a binary mixture in a planar vertical porous layer is studied. It is shown that in the presence of sufficient longitudinal stratification the flow becomes unstable against thermal-concentration perturbations; the stability boundary is determined as a function of the parameters of the problem.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 150–157, January–February, 1980.     

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.     

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.     

On the flow between a rotating and a porous fixed disk with suction

Numerical self-similar solutions are reported for the laminar, incompressible flow between a rotating disk and a porous, fixed one with suction. Validation of the method is obtained through the numerical integration of the full Navier-Stokes equations applied to a reference radially confined geometry, and also with hot-wire measurements of the tangential velocity component. The flow structure is analysed for different values of the rotational and suction Reynolds numbers. It is shown that suction causes an important angular acceleration of the rotating core, whose velocity may thus considerably exceed that of the rotating disk. The physical reason for this unusual behavior is discussed in detail.     

Steady plane-parallel flow of a magnetic fluid

In the hydrodynamics of a Newtonian fluid, nonlinear effects are connected only with the presence of convective derivatives in the equations and therefore disappear when plane-parallel flows are considered. Non-Newtonian effects are usually taken into account either phenomenologically in the expression for the stress tensor or by explicitly considering additional degrees of freedom. A theory of the effective viscosity of a magnetic fluid is constructed in [1] by regarding a magnetic fluid as a medium with internal rotation. It was shown that the flow of fluid in a magnetic field is non-Newtonian. Later, many authors (see, for example, [2, 3]) studied one- and two-dimensional flows under the influence of a pressure difference. However, the study was usually limited to continuous and smooth solutions. In the present work, we study the plane-parallel flow of a magnetic fluid in a homogeneous magnetic field under the influence of a longitudinal pressure gradient. We also consider discontinuous solutions. It is shown that for large longitudinal pressure gradients and sufficiently great intensities of the magnetic field, the problem has an infinite number of steady solutions which differ in the number and position of discontinuities of the magnetization and the associated abrupt changes in the velocity profile. Steady regimes and their stability are studied numerically with allowance for weak diffusion of magnetization and internal angular momentum. It is shown that the degeneracy is then lifted; however, in a certain region of parameters several stable steady regimes still exist.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 57–64, November–December, 1984.The authors would like to thank M. I. Shliomis for his constant interest in this work.     

Convection in an oscillating force field and microgravity

Convective flows in a plane layer of viscous fluid in the presence of an oscillating external force are investigated numerically [1 – 8]. The layer is assumed to be placed in a gravitational field. The cases in which the external field oscillations are generated by rotation about the horizontal axis or by vibration in the longitudinal direction are considered. The Navier-Stokes equations and the Boussinesq approximation are used for describing the fluid motion. The flows developing in the layer in the presence of a transverse temperature gradient are determined, the stability boundaries of these flows are found, and the supercritical motion regimes are studied. These investigations are carried out using the averaging method (in order to find the stability limits for high rotation velocities and vibration frequencies) and the Galerkin method.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 99–106, September–October, 1994.     

Micromechanics of residual saturation formation during immiscible displacement in a porous medium

The process of displacement of a nonwetting fluid has been studied experimentally on a transparent model of a porous medium for various percolation velocities in the stable front regime, when the viscosity of the displacing fluid is greater than that of the fluid displaced. The flow structures in the final displacement regime, when the nonwetting phase is distributed in the pore space in the form of individual drops or ganglia, have been visually investigated. Imbibition is numerically modeled on a two-dimensional network model with allowance for various microdisplacement mechanisms. The effect of the initial displacing phase saturation on the magnitude and structure of the residual displaced fluid saturation is demonstrated. The fractal dimensionality of the displacement boundary is measured.Novosibirsk. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.3, pp. 116–121, May–June, 1994.     

Magnetohydrodynamic flow past a porous rotating disk in a circular magnetic field

This paper studies the effects of a circular magnetic field on the flow of a conducting fluid about a porous rotating disk. Using modern quasi-Newton and globally convergent homotopy methods, numerical solutions are obtained for a wide range of magnetic field strengths, suction and injection velocities and Alfven and disk speeds. Results are presented graphically in terms of three non-dimensional parameters. There is excellent agreement with previous work and asymptotic formulae.     

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.     

Effects of Hall current and slip condition on unsteady flow of a viscous fluid due to non-coaxial rotation of a porous disk and a fluid at infinity

The unsteady hydromagnetic flow due to non-coaxial rotations of a porous disk with slip condition and a fluid at infinity has been studied on taking Hall currents into account. An exact solution of the governing equation has been obtained by the Laplace transform technique. Asymptotic solution is obtained for large time. It is found that for large time there exists a thin boundary layer near the disk. The thickness of this layer decreases with increase in either suction or magnetic parameter.     

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.     

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.     

Unsteady Motion of a Viscous Liquid between Rotating Parallel Walls in the Presence of a Crossflow

The paper considers the unsteady flow of a viscous incompressible fluid inside an infinitely long slot with uniform injection or suction of the fluid through the porous walls of the slot. The plates with the fluid are rotated rigidly with constant angular velocity. The unsteady flow is induced by nontorsional vibrations of the upper plate. The flowvelocity field and the tangential stress vectors exerted by the fluid on the upper and lower walls of the slot are determined. In this case, one can find an exact solution of the threedimensional nonstationary Navier–Stokes equations. No restrictions are imposed on the motion pattern of the plate.     

Effect of the Coriolis force on the onset of convection in a layer with a fixed heat flux on the boundaries

The effect of the Coriolis force on the onset of convection in a plane horizontal layer of viscous fluid with a fixed heat flux on the rigid lower and free upper boundaries is investigated. Expressions for the critical Rayleigh numbers and wave number are obtained analytically in the rapid rotation limit.Perm'. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 42–46, May–June, 1994.