Rotating thermal flow in natural and industrial processes. (Q2914053)

The aim of the introductory Chapter 1 is to give basic concepts of hydrodynamics, the governing equations and dimensionless parameters describing various processes in rotating fluids, such as the dynamics of vorticity and convective phenomena.NEWLINENEWLINEIn Chapter 2, fluid systems are considered subjected to rotation, for which complications arise due to the addition of the Coriolis and centrifugal forces to the existing balance between gravitational buoyancy and viscous drag. The author provides a classification of such systems and their solutions in accordance with the increasing complexity of the studied processes. Beginning with the Rayleigh-Bénard convection with rotation in infinite layers, where the role of the centrifugal force is neglected but the Coriolis force is retained with supercritical stationary bifurcation leading to rolls, the author passes to the fluid-dynamic transition (Küppers-Lortz instability), where the spatio-temporal behavior can be characterized by straight rolls oriented at various angles and by chaotic domains.NEWLINENEWLINEAfter a variety of instabilities depending on the applied rotation rate, the author points out that there exists a range of Prandtl values for which the primary bifurcation to convection from the initial thermally diffusive state leads to oscillatory flow. The Poincaré-Hopf bifurcation occurs when the rotation rate is sufficiently large. In Section 2.5, dynamic bifurcations are studied in the forms of standing and traveling rolls or in the form of square or hexagonal lattices. Chapter 2 is finished by the consideration of centrifugal effects and turbulent rotating Rayleigh-Bénard convection.NEWLINENEWLINEChapter 3 is devoted to the geophysical Coriolis effect in the atmosphere dynamics, connected with typical flows induced by pole-equator temperature gradient in thin rotating spherical shells with centrally directed gravity. Here, the author studies the origin of the zonal winds as flows along the Earth's meridian lines and Rossby waves, or planetary waves, related to the rotation of the Earth. Separate sections are devoted to self-gravitating rotating spherical shells with convective flows induced by internal heating, and to the centrifugally driven thermal convection (convection driven by radial buoyancy in a rotating spherical annulus).NEWLINENEWLINEChapter 4 studies the baroclinic instability arising in rapidly rotating stratified fluids subjected to horizontal temperature gradients, having many geophysical and astrophysical applications. The basic results here are described in sections: linear stability analysis, fully developed nonlinear waveforms, and the route of chaos. Elementary applications to the atmospheric dynamics are presented.NEWLINENEWLINEChapter 5 ``The quasi-geostrophic theory'' explains the fundamental difference between two main fluid-dynamic instabilies in rotating fluids: one where the unstable modes involve mass and temperature redistribution (Rayleigh-Bénard or Maranyoni-Bénard convection), and the other where the stably stratified and unstratified shear instabilities -- barotropic and baroclimic instabilities -- appear due to the self-excitation of waves rather than to the direct redistribution of mass and temperature. Beginning with the concept of Rossby wave, the author introduces the modern theory of ``counter-propagating-Rossby-wave'' (CRW), and demonstrates that the CRWs should be regarded as an analysis tool which provides the baroclinic growth mechanism of the discrete solution only (which depends on CRWs amplitude ratio, phase difference, interaction coefficient and the difference in wave speeds) and cannot be applied to continuous modes.NEWLINENEWLINEChapter 6 is devoted to planetery patterns in rotating fluids exposing collective behavior. First of all, the definition of pattern is provided as follows: 1. a set of relationships that can be identified by observations of a system, or an ensemble of subsystems; 2. a simple type of emergent property of a system, where a pattern is a feature of the system as a whole but does not apply to constituent subparts of the system; 3. a property of a system by which the description of the system becomes relatively simple, allowing to detail the characteristics of its components. In subsequent sections, examples of planetery patterns and methods of investigation are considered, with applications to convective phenomena and vortex instabilities.NEWLINENEWLINEChapter 7 deals with the interplay between rotation and flows induced by surface-tension gradients, more precisely, with Marangoni-Bénard convection and hydrothermal waves as surface-tension-driven flows.NEWLINENEWLINEChapter 8 describes both buoyancy and surface-tension-driven convective instabilities arising in the crystal growth from the melt, on the examples of the Bridgman process, floating zone and Czochralski techniques. The first one is an example of the interaction between gravitational and rotational effects, in the second one, the buoyancy is replaced by Marangoni stresses, and the third one presents a situation in which coexist Rayleigh-Bénard convection, Marangoni-Bénard flow, hydrothermal and baroclimic instabilities. Chapter 8 ends with a short review of classical problems for isothermal rotating fluids with interesting applications to rotating machinery.NEWLINENEWLINEChapter 9 ``Rotating magnetic fields'' (RMF) presents technological means to suppress undesired flow instabilities. Here, the author studies the spin-up from rest of an isothermal liquid metal driven by the RMF, and discusses the influence of RMF on the effective crystal growth with higher modes of convection.NEWLINENEWLINEChapter 10 studies flows produced by angular vibrations, i.e., situations when the constant rotation rate is replaced by an angular displacement varying sinusoidally in time, and rocking motions with application to vertical Bridgman crystal growth.