Experimental investigation of nonaxisymmetric inertia waves in a rotating fluid

The properties of nonaxisymmetric waves in a fluid rotating as a rigid body in a cylindrical vessel has been studied experimentally. The method of resonance generation of the required mode is used, as this does not lead to restructuring of the basic flow. It is shown that resonance generation of the natural mode is an essentially unsteady process, whose initial stage conforms closely to the linear theory. It is established that after reaching a critical amplitude the wave ceases to grow and its original structure decays nonlinearly.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 176–180, January–February, 1987.     

Example of parametric resonance in a rotating flow

Parametric resonance is one of the common types of instability of mechanical systems [1]. A standard example of the equations describing parametric oscillations is the Mathieu equation and its generalizations. In hydrodynamics these oscillations have been closely studied in connection with the problem of the vertical oscillations of a vessel containing an incompressible fluid in a uniform gravity field [1–5]. In this paper a new example of a flow whose stability problem reduces to the Mathieu equation is given. This is a flow of special type in a rotating cylindrical channel. The direction of the angular velocity is perpendicular to the channel axis, and its magnitude varies periodically with time. Flows with this geometry are of potential interest in technical applications [6, 7].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 175–177, March–April, 1987.     

Motion of a magnetizable liquid in a rotating magnetic field

Moskowitz and Rosensweig [1] describe the drag of a magnetic liquid — a colloidal suspension of ferromagnetic single-domain particles in a liquid carrier — by a rotating magnetic field. Various hydrodynamic models have been proposed [2, 3] to describe the macroscopic behavior of magnetic suspensions. In the model constructed in [2] it was assumed that the intensity of magnetization is always directed along the field so that the body torque is zero. Therefore, this model cannot account for the phenomenon under consideration. We make a number of simplifying assumptions to discuss the steady laminar flow of an incompressible viscous magnetizable liquid with internal rotation of particles moving in an infinitely long cylindrical container in a rotating magnetic field. The physical mechanism setting the liquid in motion is discussed. The importance of unsymmetric stresses and the phenomenon of relaxation of magnetization are emphasized. The solution obtained below is also a solution of the problem of the rotation of a polarizable liquid in a rotating electric field according to the model in [3].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 40–43, July–August, 1970.     

Numerical modeling of the rotation of a solid body with a liquid-filled cavity having radial ribs

The boundary-value problem of unsteady vortex flow of a viscous incompressible fluid in a cylindrical vessel with radial ribs rotating at a variable angular velocity is solved using a finite-difference method. The results of the solution are used to calculate the motion of a system of a solid body and a cavity filled with a liquid. The results are compared with available experimental data. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 2, pp. 135–139, March–April, 2007.     

Two exact solutions of the Navier-stokes equations for the concentration of eddies in a viscous liquid

In the final analysis, vorticity in a liquid or gas is broken down by viscosity [1]; however, there are known cases of the appearance and long-term existence of three-dimensional eddies in water, air, and other media. Therefore, the conditions under which vorticity can even rise with viscosity are of interest. For example, with the flow of a liquid out of an opening in the bottom of a rotating cylindrical vessel, the total momentum with respect to the vertical axis of the vessel increases with the time [2, 3]. For some flows, there exist contradictory opinions: In [4, 5] it is asserted that an eddy around a flat sink in a viscous liquid is damped, while, in [6, 7], it is argued that, with determined Reynolds numbers, there is an increase in the vorticity around a sink. The present article gives exact solutions of the Navier—Stokes equations, demonstrating the development of eddies in a viscous liquid.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 77–81, November –December, 1977.     

DYNAMICS AND STABILITY OF PINNED–CLAMPED AND CLAMPED–PINNED CYLINDRICAL SHELLS CONVEYING FLUID

The paper examines the dynamics and stability of fluid-conveying cylindrical shells having pinned–clamped or clamped–pinned boundary conditions, where “pinned” is an abbreviation for “simply supported”. Flügge's equations are used to describe the shell motion, while the fluid-dynamic perturbation pressure is obtained utilizing the linearized potential flow theory. The solution is obtained using two methods — the travelling wave method and the Fourier-transform approach. The results obtained by both methods suggest that the negative damping of the clamped–pinned systems and positive damping of the pinned–clamped systems, observed by previous investigators for any arbitrarily small flow velocity, are simply numerical artefacts; this is reinforced by energy considerations, in which the work done by the fluid on the shell is shown to be zero. Hence, it is concluded that both systems are conservative.     

A cylindrical analog of trochoidal gerstner waves

This paper investigates isobaric motions for which the values of the pressure are conserved in fluid particles. In it, a new analytic exact particular solution of nonlinear multidimensional hydrodynamic equations is obtained; it describes a trochoidal wave in cylindrical geometry. It is also proved that trochoidal waves in cylindrical and plane geometry exhaust the class of nonlinear isobaric motions. Here and below by a wave in plane geometry we mean a wave in a uniform gravitational field which is characterized by the wave vector k. It is obvious that waves in both plane and cylindrical geometry are two-dimensional motions, since the fluid particles in motion are fixed in the plane and the motions in parallel planes are the same. The trochoidal wave in cylindrical geometry is of interest, since it describes a nonlinear wave on the surface of a cavity in a rotating fluid, a situation which is frequently encountered in applications.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 145–150, September–October, 1985.     

Plane vortex flow induced by a mass source (sink) in a rotating viscous fluid

A solution of the self-similar type, describing the development with time of a plane vortex flow excited by an axisymmetric mass source (sink) in a rotating viscous fluid, is obtained. Sources of two kinds — impulsive and of constant strength — are considered. The solutions for the velocity and vorticity fields are expressed in the form of functions similar to incomplete gamma functions and are presented in the form of graphs for various flow Reynolds numbers.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 172–175, July–August, 1991.     

The motion of a closely-fitting sphere in a fluid-filled tube

Singular perturbation techniques are used to investigate the slow, asymmetric flow around a sphere positioned eccentrically within a long, circular, cylindrical tube filled with viscous fluid. The results apply to situations in which the sphere occupies virtually the entire cross section of the cylinder, so that the clearance between the particle and tube wall is everywhere small compared with both the sphere and tube radii. The technique is an improvement over conventional “lubrication-theory” analyses.Asymptotic expansions, valid for small dimensionless clearances, are obtained for the hydrodynamic force, torque and pressure drop for flow past a stationary sphere, as well as for the case of a sphere translating or rotating in an otherwise quiescent fluid. These expansions are employed to predict the macroscopic behavior of both a neutrally-buoyant sphere suspended in a Poiseuille flow, and a sedimenting sphere in a vertical tube.The results find application in capillary blood flow, pipeline transport of encapsulated materials, and falling-ball viscometers.     

Oscillatory rotationally symmetric loss of stability of nonisothermal couette flow

A study is made of the stability of nonisothermal Couette flow — steady flow of a viscous heat conducting fluid between two rotating concentric cylinders heated to different temperatures. The methods of perturbation theory are used to establish conditions sufficient for bifurcation of a neutral curve of oscillatory instability from the neutral curve of monotonic instability. Computer calculations show that for certain values of the parameters of the problem these conditions are realized and there is an oscillatory loss of stability of the nonisothermal Couette flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 76–80, January–February, 1984.I thank V. I. Yudovich for constant interest in the work.     

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.     

Magnetohydrodynamic lubrication theory for a cylindrical bearing

Liquid metal, which is a conductor of electric current, may be used as a lubricant at high temperatures. In recent years considerable attention has been devoted to various problems on the motion of an electrically conducting liquid lubricant in magnetic and electric fields (magnetohydrodynamic theory of lubrication), Thus, for example, references [1–3] study the flow of a conducting lubricating fluid between two plane walls located in a magnetic field. An electrically conducting lubricating layer in a magnetohydrodynamic bearing with cylindrical surfaces is considered in [4–8] and elsewhere.The present work is concerned with the solution of the plane magnetohydrodynamic problem on the pressure distribution of a viscous eletrically conducting liquid in the lubricating layer of a cylindrical bearing along whose axis there is directed a constant magnetic field, while a potential difference from an external source is applied between the journal and the bearing. The radial gap in the bearing is not assumed small, and the problem reduces to two-dimensional system of magnetohydrodynamic equations.An expression is obtained for the additional pressure in the lubricating layer resulting from the electromagnetic forces. In the particular case of a very thin layer the result reported in [4–8] is obtained. SI units are used.     

Dynamic interaction of an oscillating sphere and an elastic cylindrical shell filled with a fluid and immersed in an elastic medium

The paper studies the interaction of a harmonically oscillating spherical body and a thin elastic cylindrical shell filled with a perfect compressible fluid and immersed in an infinite elastic medium. The geometrical center of the sphere is located on the cylinder axis. The acoustic approximation, the theory of thin elastic shells based on the Kirchhoff—Love hypotheses, and the Lamé equations are used to model the motion of the fluid, shell, and medium, respectively. The solution method is based on the possibility of representing partial solutions of the Helmholtz equations written in cylindrical coordinates in terms of partial solutions written in spherical coordinates, and vice versa. Satisfying the boundary conditions at the shell—medium and shell—fluid interfaces and at the spherical surface produces an infinite system of algebraic equations with coefficients in the form of improper integrals of cylindrical functions. This system is solved by the reduction method. The behavior of the hydroelastic system is analyzed against the frequency of forced oscillations.Translated from Prikladnaya Mekhanika, Vol. 40, No. 9, pp. 75–86, September 2004.     

Deformation by the external flow of the surface of a magnetic fluid covering a circular cylinder

As one of the methods of controlling separation of a flow and reducing hydrodynamic resistance in flow round the body, [1] proposes covering the surface of the body with a layer of magnetic fluid maintained by a inhombgeneous field. The effectiveness of the method is studied in the example of flow rounda circular cylinder covered with a uniform layer of a magnetic fluid the surface of which is also assumed to be cylindrical and not dependent on the external flow. The resistance of the cylinder falls (if the viscosity of the magnetic fluid is not too high) and can be greatly reduced. It is of interest to determine the nature of the deformation produced in the surface of the magnetic fluid by the external flow, since the degree to which the boundary can be deformed determines whether the proposed method can be realized in practice. Moreover, the magnitude of the deformation evidently affects the resistance of the cylinder and determines the limits of the applicability of the approximation adopted in [1] for the nondeformed state of the surface.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 28–31, January–February, 1988.     

Convective flows in a cylindrical fluid zone in a high-frequency vibration field

Convective flows of a nonuniformly heated fluid in a cylindrical fluid zone in a high-frequency longitudinal vibration field are studied. Vibration frequencies which are high as compared with dissipative decrements and capillary frequencies, but small as compared with acoustic frequencies are considered. The general method formulated earlier for describing the behavior of inhomogeneous fluids under the influence of high-frequency vibrations is used. The interaction between the vibrational flow mechanisms and thermocapillary effects on a free surface is analyzed.Perm'. Marseilles (France). Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 53–61, September–October, 1994.     

Calculation of laminar flow of a viscous fluid between rotating disks

The results are given of a numerical investigation of the laminar flow of a viscous incompressible fluid with heat transfer from the periphery to the center between two rotating disks. The system is a simplified model of one of the elements of the cooling circuit of a gas turbine. The complete Navier—Stokes equations in the vortlcity—flow function variables were solved by an explicit conservative scheme with appoximation of the convective terms of divergence type by directed differences. The calculations were made in a wide range of variation of the dimensionless determining parameters of the problem. The results agree well with the known experimental data.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 76–81, January–February, 1982.We thank V. M. Kapinos for discussion and helpful comments.     

Low-dimensional representations of early transition in rotating fluid flow

Low-dimensional representations of the axisymmetric Navier-Stokes equations are generated by a Galerkin projection. Proper orthogonal decomposition (POD) techniques based on snapshots generated from a finite-difference algorithm are used. The Reynolds number range is extended by adding displacement vectors to the Galerkin basis. For the fluid flow enclosed in a cylindrical vessel with rotating end cover, the first transition from steady to oscillatory motion is detected as a supercritical Hopf bifurcation. Comparison with the full numerical solution of the Navier-Stokes equations as well as experimental results show excellent agreement.One of the authors (EAC) acknowledges financial support from the Danish Research Academy (Grant S910171), the Danish Technical Research Council (Grant 16-4967-2 OS), the Louis Dreyer Myhrwold's Fund, and the Fisker & Nielsens Fund.     

Some properties of multi-fluid stokes flows

The solution of Stokes' equations for a rotating axisymmetric body which possesses reflection symmetry about a planar interface between two infinite immiscible quiescent viscous fluids is shown to be independent of the viscosities of the fluids and identical with the solution when the fluids have the same viscosity. The result is generalized to a rotating axisymmetric system of bodies which possesses reflection symmetry about each interface of a plane stratified system of fluids. An analogous result for two-fluid systems with a nonplanar static interface is also derived. The effect on torque reduction produced by the presence of a second fluid layer adjacent to a rotating axisymmetric body is considered and explicit calculations are given for the case of a sphere. A proof of uniqueness for unbounded multi-fluid Stokes' flow is given and the asymptotic far field structure of the velocity field is determined for axisymmetric flow caused by the rotation of axisymmetric bodies.     

Numerical investigation of the hydrodynamic drag of a circular cylinder coated with a thin layer of magnetic fluid

In order to reduce the drag of bodies in a viscous flow it has been proposed to apply to the surface exposed to the flow a layer of magnetic fluid, which can be retained by means of a magnetic field and thus act as a lubricant between the external flow and the body [1, 2]. In [1] the hydrodynamic drag of a current-carrying cylindrical conductor coated with a uniform layer of magnetic fluid was theoretically investigated at small Reynolds numbers. In order to simplify the equations of motion, the Oseen approximation was introduced for the free stream and the Stokes approximation for the magnetic fluid [3]. This approach has led to the finding of an exact analytic solution from which it follows that at Reynolds numbers Re 1 the drag of the cylinder can be considerably reduced if the viscosity of its magnetic-fluid coating is much less than the viscosity of the flow. The main purpose of the present study is to investigate, with reference to the same problem, how the magnetic-fluid coating affects the hydrodynamic drag at Reynolds numbers 1 Re 10²–10³, i.e., under separated flow conditions. In this case the simplifications associated with neglecting the nonlinear inertial terms in the Navier—Stokes equation are inadmissible, so that a solution can be obtained only by numerical methods.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 11–16, May–June, 1986.     

Transonic Euler computation in streamfunction co-ordinates

Numerical simulation by a finite element method is used to examine the problem of the rotating flow of a viscoelastic fluid in a cylindrical vessel agitated with a paddle impeller. The mathematical model consists of a viscoelastic constitutive equation of Oldroyd B type coupled to the hydrodynamic equations expressed in a rotating frame. This system is solved by using an unsteady approach for velocity, pressure and stress fields. For Reynolds numbers in the range 0.1–10, viscoelastic effects are taken into account up to a Deborah number De of 1.33 and viscoelasticity and inertia cross-effects are studied. Examining the velocity and stress fields as well as the power consumption, it is found that their evolutions are significantly different for low and moderate inertia. These results confirm the trends of experimental studies and show the specific contribution of elasticity without interference of the pseudoplastic character found in actual fluids.