Some flow problems in noncoaxial rotating systems

Abstract

We live in an atmosphere of air and water which involve rotation to a greater or lesser extent. The atmosphere and the oceans rotate as a whole with the earth and move over its surface as winds and ocean currents which involve large-scale circulations. In a small scale, circulation due to rotatory motion is also encountered in centrifugal pumps, hydraulic turbines and many other day-to-day industrial appliances. Studies involving wing theory, meteorology, oceanography, geophysics, astrophysics and many other allied branches involve vortex motions and have largely been responsible for the growth of literature in rotating fluids. Comprehensive discussions on vortex motions have been presented by well-known authors like Goldstein, Greenspan, Lighthill, Pireusdell and Howard. The dynamics of rotating fluids has been, in the main, developed by methods special to the field, using the equations of motion of a fluid in a rotating frame of reference. It is nevertheless possible to derive all the leading results from the classical principles of fluid dynamics in non-rotating frames; especially, the rules governing the rate of change of vorticity. However, the use of rotating frame of reference has advantages in some cases, for example, in the case of nearly rigid rotations. In such cases we are principally interested in the deviations of the motion from the basic rigid rotation and it is useful to describe the motion with reference to an appropriate rotating coordinate system, developing physical concepts and mathematical approximations which appear natural from the point of view of observers fixed in the coordinate system. Such a point of view is also useful even when the motion itself is not especially close to rigid rotation, if however the boundary conditions or the driving mechanisms of the flow are naturally described in terms of a rotating coordinate system. Because of the basic rotation of the earth, most of the large-scale motions of the atmosphere and the sea are of this type. There are also other fluid motions in which the basic rotation plays an important role but in which the use of rotating coordinate system is not particularly helpful because the basic rotation is not the same at all parts of the fluid. Some instances of this type of rotational flows are the flow of a viscous fluid between differentially rotating concentric spheres, the flow between differentially rotating concentric or eccentric cylinders and the flow between parallel plates rotating coaxially or non-coaxially with different angular velocities. Problems concerning flows around rotating axially symmetric bodies arise in connection with turbine construction. Since the rotors of various turbines contain as necessary elements, plane surfaces perpendicular to the axis of rotation, the problem of finding the flow over a rotating disk remains the centre of interest of researchers. Geophysical dynamics is mainly concerned with the fundamental dynamical concepts essential to the understanding of the atmosphere and the oceans. The principles to be derived in it are largely theoretical concepts which can be applied to an understanding of the natural phenomena. Such principles are to be derived from the study of model problems whose goal is the development of conceptual comprehension rather than detailed simulation of the complex geophysical phenomena. A study sequence with a hierarchy of increasingly complex models is necessary where each stage is built on the intuition developed by the precise analysis of the simpler models. The atmosphere and the oceans are compressible fluids, but in many cases, essential features of the atmospheric or oceanic flows are not dependent on this fact and fairly satisfactory theories can be based on the mathematical models assuming an incompressible fluid. In the case of the atmosphere, density varies considerably with altitude causing a change both in pressure and temperature, but for many purposes a satisfactory model can be constructed assuming An incompressible but non-homogeneous fluid, and in some cases this can be further simplified without loss of essential physical content to an incompressible fluid in two layers, each of constant density, or even to a single homogeneous incompressible fluid. Thus rotational symmetric flow over a rotating or a stationary infinite disk, in its various forms including both the steady and unsteady versions, is an important geophysical model and the corresponding version of a conducting fluid is an important astrophysical model. The atmosphere and the oceans are located on a sphere; for motions whose length scale is relatively small compared to the earth's radius, it is usually satisfactory to model this by a plane layer of fluid. Fluid motions in rotating spherical annulus have been the object of a great deal of research in the expectation that they resemble atmospheric and oceanic circulations. As long as the annular gap width is small compared to the radius, the flows are indeed analogous in many problems with homogeneous fluids. Study of flows in rotating systems helps theoreticians to construct models for various geophysical flows. Hydromagnetic flows in rotating fluids have considerable applications in geophysics and astrophysics. Almost all geophysical and astrophysical phenomena occur where there are magnetic fields associated with materials of large conductivity and hence a strong coupling results between the motion of the matter and the magnetic field. As a result, the ordinary hydrodynamics is usually not applicable in such cases. It is now well-known that the hydromagnetic flow in the liquid core of the earth is mainly responsible for the geo-magnetic field and the theory of earth's magnetism is based on the dynamics of the core motions. An important field of technology in which flows in rotating systems are involved is the lubrication technology. When two surfaces are in contact and move relative to each other, resistance to the motion is encountered and usually the surfaces are damaged. It is possible to introduce a lubricant, which is either a Newtonian or a non-Newtonian fluid, between the surfaces to reduce the friction force and to minimise the damage to them. The theory and practice of lubrication is based upon the assumption that the dominating influence is that of the lubricant viscosity. When dragged into the narrow space between two inclined surfaces in relative motion, the pressure generated in the fluid film by the viscous forces tends to keep the surfaces apart, thus forming a bearing. The most favourable geometry and kinematics of such systems have been thoroughly investigated. The journal bearings which support the radial load, are extensively used in machinery used in various fields of industry and technology. In a journal bearing the shaft member is called the journal and the cylindrical body around the journal is referred to as the bearing. In most cases, the journal rotates and the bearing is stationary, although the reverse can occur or both can be rotating. In order to analyse the flow phenomena in journal bearings, the viscous flow between eccentric cylinders with a narrow gap has been studied extensively by numerous researchers. Such studies help the engineers, faced with design problems involving journal bearings or rotor shafts surrounded by a fluid annulus, to make accurate predictions of the nature of the fluid flow and the steady-state forces involved in the design. In classical lubrication theory, the inertia effect is completely neglected. The advent of low viscosity lubricants and the higher speeds of rotation occurring at the journal bearings have necessitated the study of modifications in the results which follow from taking into account the effects of inertia. The geometry of rotating parallel plates with a narrow clearance has been found to be useful in the design of bearings meant for taking axial load. Flows in rotating systems are extensively used in the theory and practice of rheometry. Among the various flows associated with the mechanics of fluids, the one which is of particular interest for the determination of the physical properties of liquids is that which is generated by a Rotor. The reason lies in the fact that relatively simple, self-contained apparatus may be used to generate the flow and permit measurement of some of the constitutive parameters. A well-known apparatus for generating such flows is the rotating-type viscometer. Some of the rotating-type viscometers which have been in use for long are the concentric cylinder viscometer, the concentric sphere viscometer, and the cone and plate viscometer. The principle employed in these viscometers is to shear the fluid by the rotation of one of the instrument members and to determine the viscosity using the measured values of the torque and the angular velocity. The use of non-Newtonian fluids has increased tremendously both in industry and in day-to-day life. Determination of the complex viscosity of elastic-viscous fluids has become very important for various industrial purposes. The advent of low-viscosity fluids has made it necessary to take the inertia effect on the flow into account. Rheologists have designed new types of rheometers exploiting new geometrical configurations and developed theory to connect the viscosity with certain measurable physical quantities such as the components of the force on one of the instrument members. The flows in non-coaxial rotating systems have found application in the new type of rheometers, like the Maxwell orthogonal rheometer, the balance rheometer, and the eccentric cylinder rheometer. In the Maxwell orthogonal rheometer, the test fluid is sheared by placing it in between two parallel disks rotating with the same angular velocity about two non-coincident parallel axes, with a small distance between them. There is an arrangement to measure the components of the force on one of the disks and with the help of relevant theory the complex viscosity is determined using the measured values of the forces. In the balance rheometer, the test fluid is taken in the annular gap between an inner sphere and a concentric outer spherical container and they rotate with the same angular velocity about two axes which are slightly inclined to each other. The components of the couple or the forces on the inner sphere are measured. In the case of the eccentric cylinder rheometer also, theory is available to connect the viscosity with the measured values of the forces on the inner cylinder. Further, there is well-developed theory and experimental procedure available for using the cone and plate rheometer. The theoretical investigation of some flow problems in non-coaxial rotating systems forms the subject matter of this thesis. These flows are important in rheometry and the theory of lubrication and also may be of significance in connection with certain geophysical and astrophysical phenomena. The flow problems in the following geometrical configurations are studied: (i) non-coaxial rotating parallel disks (ii) non-coaxial rotating cylinders (iii) concentric spheres rotating about non-coincident axes (iv) cone and plate rotating about axes slightly inclined to each other. The thesis is divided into five chapters. Each chapter begins with an introduction to the general aspects of the problem including a survey of the related works and this is followed by a brief account of the investigation carried out by the author. The symbols and notations used in different chapters or in different parts of a chapter (wherever a chapter is divided into parts) are independent of each other and are explained in that chapter or in that part of the chapter. The original papers referred to, in the study of the problem in a chapter, are given at the end of the chapter. The mathematical technique employed in the solutions of the problems taken up for investigation is mainly the perturbation procedure. It is well-known that the perturbation theory is a powerful tool in solving problems in fluid mechanics. This theory gives valuable insight into the actual behaviour of the mathematical model employed and often such information is useful in the construction of efficient algorithms useful in numerical computations. Whenever exact solution cannot be obtained, the perturbation method is employed to generate approximate solutions and these approximate solutions can be used to provide necessary.