Both of the authors of this book are disciples and collaborators of the Brussels school of thermodynamics. Their particular domain of competence is the application of numerical methods to the many highly nonlinear problems which have arisen in the context of recent developments in the thermodynamics of irreversi- ble processes: stability of states far from equilibrium, search for marginal critical states, bifwrcation phenomena, multiple stationnary states, dissipative structures, etc. These problems cannot in general be handled using only the clas- sical and mathematically rigorous methods of the theory of differential, partial differential, and int~grodifferential equations. The present authors demonstrate how approximate methods, re lyi ng usually on powerful computers, lead to significant progress in these areas, if one is prepa- red to accept a certain lack of rigor, such as, for example, the lack of proof for the convergence of the series used in the context of problems which are not self adjoint, nor even linear. The results thus obtained must consequently be submit- ted to an exacting confrontation with experimental observations. - Even though, the '1 imited information obtained concerning the, often unsuspec- ted, mechanisms underlying the observed phenomena is both precious and frequently sufficient. This information results from the properties of the trial functions best suited to the constraints of the problem such as the initial, boundary, and "feedback" conditions, and the analysis of their behavior in the course of the evolution of the system.

Table of Contents:
A : Introduction.- I - Fundamental Laws and Basic Concepts.- 1. Balance equations for incompressible fluids.- A. Conservation of mass.- B. Conservation of momentum.- C. Conservation of energy.- 2. Fundamental thermodynamic relations; entropy balance equation and second law.- A. Alternative forms of the energy balance equation.- B. The entropy balance equation and the second law of thermodynamics.- 3. Kinetic and constitutive equations.- 4. Systems of coordinates.- A. Rectangular coordinates.- B. Cylindrical coordinates.- C. Special two-dimentional case : the stream function.- 5. Equations for the fluctuations around a steady state.- 6. Definition of stability.- 7. Normal modes.- 8. Dimensionless numbers in fluid dynamics and heat transfer problems.- Exercices.- Bibliographical notes.- II - Mathematical Background and Computational Techniques.- 1. Use of variational principles and/or stationary properties of integrals.- A. Elements of variational calculus. The Euler-Lagrange equations.- B. Variational approach to the conservations laws based on nonequilibrium thermodynamics : the theory of the local potential.- C. The numerical methods associated with the local potential theory.- D. Relation between the local potential and the Galerkin techniques.- 2. Applications to stability problems.- A. The excess local potential.- B. Variational methods for linear eigenvalue problems.- C. Stability criterion based on Lyapounov function.- 3. Purely numerical techniques.- A. Finite differences methods.- B. Conversion of a boundary value problem into an initial value problem.- Exercices.- Bibliographical notes.- B : Fluids at Constant Density, Isothermal Forced Convection.- III - Planar Flows of Newtonian Fluids.- 1. Poiseuille and Couette flow.- A. Plane Poiseuille flow and Poiseuille flow in rectangular channels.- B. Plane Couette flow.- 2. General statements of linear hydrodynamic stability of forced convection.- A. The Orr-Sommerfeld equation.- B. Variational or stationary presentations of the Orr-Sommerfeld equation. Its relation with the Galerkin technique.- C. The Chock-Schechter integration scheme.- D. The Orr and the Prigogine-Glansdorff criterion.- 3. Numerical solutions of the Orr-Sommerfeld equation.- A. Selection of trial functions.- B. Solution for U = constant.- C. Solution for plane Poiseuille flow.- a. Effect of trial functions.- b. High Reynolds numbers.- c. Two and three dimensional perturbations without elimination of variables. Relation to Squire's theorem.- d. Finite difference methods.- e. Solution using the Chock-Schechter method.- f. General discussion, comparison with experiments.- D. Solution for Couette flow.- 4. Nonlinear stability of Poiseuille flow.- A. Introduction.- B. A restricted variational approach to the nonlinear equations.- C. Influence of the initial amplitude of the disturbance.- 5. An oscillatory solution in planar-Poiseuille flow.- A. Introduction.- B. Existence of statistically steady states.- C. Existence of periodic flows.- D. Stability and/or instability of the new periodic flow.- 6. Remarks on the transition to turbulence.- Bibliographical notes.- IV - Cylindrical Flows of Newtonian Fluids.- 1. A. Poiseuille flow in a pipe.- B. Poiseuille flow down an annular pipe.- 2. General statements on linear stability of forced convection in cylindrical coordinates.- A. An equivalent of the Orr-Sommerfeld equation.- B. Non axisymmetric disturbances.- 3. Linear stability of pipe Poiseuille flow.- A. Stability with respect to two-dimensional axisymmetric disturbances.- B. Stability with respect to three-dimensional non axisymmetric disturbances.- Bibliographical notes.- V - Flow Stability of Non-Newtonian Fluids.- 1. Stress-Strain relations for some particular non-newtonian fluids.- A. Introduction.- B. The Coleman-Noll model.- 2. Stability of plane Poiseuille flow for a second order viscoelastic fluid.- A. The generalized Orr-Sommerfeld equation.- B. The solution of the generalized Orr-Sommerfeld equation for plane flow.- C. Plane Poiseuille flow : sufficient condition for stability.- D. Instability of plane Poiseuille flow of a second order fluid : a numerical result.- 3. Stability of pipe Poiseuille flow for a second order fluid..- Bibliographical notes.- C : Non Isothermal One Component Systems.- VI - Free Convection in One Component Fluid.- 1. Introduction.- 2. The linear theory of the Benard problem.- A. The eigenvalue problem. Its solution for simple boundary conditions.- B. Solutions based on approximate numerical calculations.- a. The local potential method.- b. The Chock-Schechter numerical integration.- C. Solution based on the thermodynamic stability criterion.- D. Experimental aspect.- E. Effect of lateral boundaries.- F. Extension of the Benard problem.- a. Surface tension effect.- b. Effect of a magnetic field.- 3. The non-linear theory of the Benard problem.- A. Approximate computational techniques.- B. Global properties of the flow.- a. Variation of the Nusselt number with the Rayleigh number (free boundary conditions).- b. Variation of the Nusselt number with the Rayleigh number (rigid boundary conditions).- c. Variation of the number of convective cells with the Rayleigh number.- C. Fine structure of the flow.- D. Behavior near threshold.- E. Behavior far from the critical point.- a. The Lorenz model.- b. The routes to turbulence.- 4. The thermogravitational process.- A. The steady state profile.- B. The stability of the steady state profile.- Bibliographical notes.- VII - Non Isothermal Forced Convection in a One-Component Fluid.- 1. General aspects of the effect of temperature gradients.- 2. Temperature gradients imposed by the boundary conditions.- 3. Temperature gradients due to viscous heating.- A. Experimental interest.- B. Cylindrical Poiseuille flow with viscous heating.- a. the steady state.- b. stability of cylindrical Poiseuille flow including viscous heating.- 4. Further discussion on the multiplicity of steady states when taking into account viscous heating.- Bibliographical notes.- VIII - Mixed Convection in a One-Component Fluid.- 1. Introduction in the Benard problem with flow.- 2. Relation between two and three dimensional disturbances; extension of Squire's theorem.- 3. Experiments on the onset of free convection with a superposed small laminar flow.- 4. Effect of lateral boundaries.- Bibliographical notes.- D : Multicomponent Systems.- IX - Free Convection in a Multicomponent Fluid.- 1. Introduction to the influence of concentration gradients on hydrodynamic stability.- 2. Formulation of the linearized problem.- A. The conservation equations.- B. The thermohaline problem.- C. The effect of thermal diffusion (or Soret effect).- 3. The thermohaline convection : linear stability analysis.- A. The role of boundary conditions.- B. Free boundaries with specified solute concentrations and temperatures.- C. Experimental observations.- 4. Free convection with thermal diffusion : linear analysis.- A. Coupled equations for temperature and mass.- B. Exact solution of the simplified problem for free and pervious boundaries.- C. Variational solution for rigid boundaries.- D. 0.- B. Results for s < 0.- 3. Postface.- Bibliographical notes.- Appendix A.- Appendix B.