Convection in Fluids: A Rational Analysis and Asymptotic Modelling

ISBN-13: 9789048124329 / Angielski / Twarda / 2009 / 396 str.

Radyadour K. Zeytounian

Opis

This monograph, entirely devoted to Convection in Fluids, presents a unified rational approach of various convective phenomena in fluids (mainly considered as a thermally perfect gas or an expansible liquid), where the main driving mechanism is the buoyancy force (Archimedean thrust) or temperature-dependent surface tension in homogeneities (Marangoni effect). Also, the general mathematical formulation (for instance, in the Benard problem - heated from below) and the effect of free surface deformation are taken into account. In the case of atmospheric thermal convection, the Coriolis force and stratification effects are also considered. This volume gives a rational and analytical analysis of the above mentioned physical effects on the basis of the full unsteady Navier-Stokes and Fourier (NS-F) equations - for a Newtonian compressible viscous and heat-conducting fluid - coupled with the associated initials (at initial time), boundary (lower-at the solid plane) and free surface (upper-in contact with ambiant air) conditions. This, obviously, is not an easy but a necessary task if we have in mind a rational modelling process, and work within a numerically coherent simulation on a high speed computer."

Spis treści

(provisional) Preface and Acknowledgments 1 Short Preliminary Comments and Summary of Chapters 2 to 10 1 . 1 Introduction 1.2 Summary of the chapters 2 to 10 2 The Navier-Stokes and Fourier Problem 2. 1 Some relations of the thermodynamics 2.2 The case of a thermally perfect gas 2.3 The case of an expansible liquid 2.4 The NS-F starting problem 2.4.1 Equations 2.4.2 Initial and boundary conditions 2.5 SQme comments 3 The Simple Rayleigh (1916) Thermal Convection Problem 3. 1 Formulation of the exact simplified (a la Rayleigh) problem 3.2 Dimensionless dominant Rayleigh problem and the Boussinesq approximation 3.3 The Rayleigh-Bénard (RB), rigid-rigid, problem as a leading-order approximate model 3.4 Associated, to RB problem, second-order model problem 4 The Bénard (1900, 1901), Heated from Below, Convection Problem 4.1 The mathematical formulation of the full Bénard convection problem heated from below 4. 1 . 1 Bénard, basic motionless conduction, effect 4. 1 .2 Marangoni, temperature dependent surface tension, effect 4. 1 .3 Biot, heat flux through the upper surface, effect 4.4.4 Boussinesq, buoyancy, effect 4.2 Dimensionless analysis and reduced parameters 4.3 Dominant equations 4.4 Dominant boundary conditions 4.5 An ‘alternative’ 4.6 Three significant cases: 4.6. 1 Shallow convection 4.6.2 Deep convection 4.6.3 Marangoni convection 5 Thermal Shallow Convection Problem, ‘a la Rayleigh-Bénard’ 5. 1 The RB rigid-free instability problem as a significant leading-order model problem 5.2 The free surface, Marangoni and Biot effects 5.3 Second-order non Boussinesq and viscous dissipation effects 5.4 Linear classical theory 5.5 Some comments and complements 6 The Deep Thermal Convection Problem, 6. 1 The viscous dissipation effect and the ‘depth’ parameter 6.2 Thedeep thermal convection problem, ‘a la Zeytounian’ 6.3 The Charki, Errafyi and Zeytounian results: 6.3.1 Linear theory 6.3.2 Routes to chaos, 6.3.3 Lorenz system 6.3.4 Landau -Ginzburg equation 6 6.4 The Straughan and Charki rigorous mathematical results 6.5 The Hills and Roberts approach 7 The Thermocapilary, Marangoni, Convection Problem 7. 1 The free surface effects and the ‘ ‘Biot paradox' 7.2 The Bénard-Marangoni model problem 7.3 The long-wave approximation and the lubrication evolution equation for the thickness of the film 7.4 Freely falling vertical film 7.5 The KS and KS-KdV approximate equations 7 . 6 Stability results 7.7 Complements 8 Summing Up of the Three Cases Related with the Bénard, Heated from Below, Convection Problem 8. 1 The consistent conditions for the Rayleigh-Bénard model problem 8. 1 . 1 Constraints on the thickness of the fluid layer 8. 1 .2 The role of the viscous dissipation term 8. 1 .3 An equation for the free surface deformation 8.2 The particularity of the deep thermal convection model problem 8.2. 1 The role of the ‘depth’ parameter and the effect of the viscous dissipation 8.2.2 The variation of the critical Rayleigh numbers as a function of the depth parameter 8.3 The thin viscous liquid film, Marangoni, case 8.3.1 Constraints on the thickness of the liquid layer 8.3.2 The role of the free surface deformation 8.3..3 Constraints for a reduced long wave film problem 9 Atmospheric Thermal Convection Problems 9. 1 The Coriolis force and stratification effects 9.2 A rigorous formulation of the atmospheric free thermal convection 9.3 The Boussinesq approximation and the breeze problem 9.4 Infuence of a local temperature field in an atmospheric Ekman layer; a triple-deck approach 9.5 A periodic thermal free-convection problem over a curvilinear wall; the

Blurb

In the present monograph, entirely devoted to “Convection in Fluids”, the purpose is to present a unified rational approach of various convective phenomena in fluids (mainly considered as a thermally perfect gas or an expansible liquid), where the main driving mechanism is the buoyancy force (Archimedean thrust) or temperature-dependent surface tension in homogeneities (Marangoni effect). Also, the general mathematical formulation (for instance, in the Bénard problem - heated from below)and the effect of the free surface deformation are taken into account. In the case of the atmospheric thermal convection, the Coriolis force and stratification effects are also considered.

The main motivation is to give a rational, analytical, analysis of main above mentioned physical effects in each case, on the basis of the full unsteady Navier-Stokes and Fourier (NS-F) equations - for a Newtonian compressible viscous and heat-conducting fluid - coupled with the associated initiales (at initial time), boundary (lower-at the solid plane) and free surface (upper-in contact with ambiant air) conditions. This, obviously, is not an easy but a necessary task if we have in mind a rational modelling process with a view of a numerical coherent simulation on a high speed computer.