Because of this latter instability there will be a tendency on the part of the fluid to redistribute itself and remedy the weakness in its arrangement. We also have the equation cf. Making use of this fact and remembering that the dissipation term Φ is of the second order in u in any case, we obtain for θ the equation this is the same equation as 56. The basic hydrodynamic equations For a general treatment of the problem of thermal instability described in § 6, we need the equations governing the hydrodynamical flow of a viscous fluid of varying density and temperature. The theory derived on the basis of such linearized equations is called the linear stability theory in contrast to non-linear theories which attempt to allow for the finite amplitudes of the perturbations. By supposing that the various physical variables describing the flow suffer small infinitesimal increments, we first obtain the equations governing these increments. Chandrasekhar considers most of the typical problems in hydromagnetic stability, with the exception of viscous shear flow; a specialized domain deserving a book unto itself.
As we shall presently see, this leads to the equation of heat conduction in the form required for our present purposes. Specifically, we ask: if the system is disturbed, will the disturbance gradually die down, or will the disturbance grow in amplitude in such a way that the system progressively departs from the initial state and never reverts to it? Similarly, if the initial state has spherical symmetry, we may expand the disturbance in spherical harmonics, Yml ϑ, φ , in the manner where r denotes the distance from the origin, and ϑ and φ are the spherical polar angles , and investigate the stability of the system with respect to all modes distinguished by m and l. Contents include: Rotation; Stability of More General Flows; Bénard Problem; Gravitational Equilibrium and Instability; Stability of a Magnetic Field; Thermal Instability of a Layer of Fluid heated from Below; Rayleigh-Taylor Instability; Kelvin-Helmholtz Instability; and Onset of Thermal Instability in Fluid Spheres and Spherical Shells. In thinking of the stability of a hydrodynamic system, it is often convenient to suppose that all parameters of the system, save one, are kept constant while the chosen one is continuously varied. Each chapter is accompanied by a section of bibliographical notes: indexes locating definitions and theorems are also provided. Among other books on related subjects, we may refer to: 3.
It is in the differentiation of the stable from the unstable patterns of permissible flows that the problems of hydrodynamic stability originate. In obtaining these equations from the relevant equations of motion, we neglect all products and powers higher than the first of the increments and retain only terms which are linear in them. This first lower-priced, sturdy paperback edition will be welcomed by graduate physics students and scientists familiar with Dr. This book presents a most lucid introduction to the Rayleigh-Bénard problem: it has also been applauded for its thorough, clear coverage of the theory of instabilities causing convection. The problem is thus reduced to determining p k for the various modes. Thus, if the stationary state has symmetry about an axis, we may expand the disturbance in the manner where z is the distance from the axis, and θ is the azimuthal angle , and investigate the stability of the system with respect to all modes distinguished by k and m.
Born in Lahore Punjab , he was the nephew of Sir Chandrasekhara Venkata Raman 1888-1970 , winner of the Nobel Prize for Physics in 1930. By eliminating Θ between the equations 99 and 100 , we obtain is the Rayleigh number. In astrophysical terms, our sun will at that point be a stable white dwarf. The quotation from Eddington is from : 12. In this book we shall be concerned only with the linear stability theory which presupposes that the perturbations are infinitesimal; and this must be understood even if the qualification is not explicitly made every time. The manner in which the solution for the stability problem is to be carried out is clear.
Thus, in the problem of the stability of viscous flow between two rotating coaxial cylinders, the parameters are the radii R1 and R2 of the two cylinders, the angular velocities Ω1 and Ω2 of their rotations, and the kinematic viscosity v. His main interests were in astrophysics and hydrodynamics: he studied the stellar structure and evolution and developed a theoretical model to explain the physical nature of white dwarf starts. The Boussinesq approximation In deriving the hydrodynamical equations in the preceding section, no assumptions were made regarding the constancy, or otherwise, of the various coefficients μ, cV, α, and k which were introduced. Indeed, the requirement that the equations allow non-trivial solutions satisfying the various boundary conditions leads directly to a characteristic value problem for p k. He worked at the Univers Also spelled as Subrahmanyan Chandrasekhar. People on Earth need not be bothered anticipating such drama: for us, when the sun dies, the lights will just go out.
In these cases the stable states, when perturbed, execute undamped oscillations with certain definite characteristic frequencies; while in the unstable states small initial perturbations tend to grow exponentially with time; and the marginal states themselves are stationary. These situations occur when the variability in the density and in the various coefficients is due to variations in the temperature of only moderate amounts. As we have explained, stability means stability with respect to all possible infinitesimal disturbances. The nature of the physical problem Consider then a horizontal layer of fluid in which an adverse temperature gradient is maintained by heating the underside. An identical equation governs Θ.
Chandrasekhar's work, particularly in light of the resurgence of interest in the Rayleigh-Bénard problem. Clearly, a system must be considered as unstable even if there is only one special mode of disturbance with respect to which it is unstable. This book presents a most lucid introduction to the Rayleigh-Bénard problem: it has also been applauded for its thorough, clear coverage of the theory of instabilities causing convection. We shall distinguish two kinds of bounding surfaces: rigid surfaces on which no slip occurs and free surfaces on which no tangential stresses act. Hence Consider the integral on the right-hand side. We shall now show, further, that the lowest characteristic value of R is, indeed, a true minimum. It follows from this that the locus Σ X1,…, Xj separating the regions of complete stability from the regions of instability in the X1,…, Xj -space is the envelope of the loci Σ k; further, that when the system becomes unstable as we cross Σ at some particular point, the mode of disturbance which will be manifested at onset will be the one whose locus Σ k touches Σ at that point.
In recent years the class of such problems of stability has been enlarged by the interest in hydrodynamic flows of electrically conducting fluids in the presence of magnetic fields. Only for particular values of R for given a² will the problem allow a non-zero solution. The variations in the other coefficients consequent to variations in density of the amounts stated must be of the same order. The earliest experiments to demonstrate in a definitive manner the onset of thermal instability in fluids are those of Bénard in 1900, though the phenomenon of thermal convection itself had been recognized earlier by Count Rumford 1797 and James Thomson 1882. The first of these is due to Pellew and Southwell. Making use of equations 25 , 27 , 28 , and 29 , we can rewrite the first term on the right-hand side of equation 33 as also, alternative forms of the third and the fourth terms are Combining the foregoing equations, we have Since this equation must hold for every volume V, we must have Making use of the equation of continuity, we can simplify the foregoing equation to the form Equations 2 , 14 , 17 , 29 , and 39 are the basic hydrodynamic equations.
Problems in hydromagnetic stability are also considered in : 2. This book is devoted to a consideration of some typical problems in hydrodynamic and hydromagnetic stability; and in this introductory chapter we shall formulate the basic principles and concepts of the subject. This is the domain of hydromagnetics; and there are problems of hydromagnetic stability even as there are problems of hydrodynamic stability. The analysis into normal modes According to the general ideas described in Chapter I § 3 , we must analyse an arbitrary disturbance into a complete set of normal modes and examine the stability of each of these modes, individually. It remains to interpret the last term on the right-hand side of equation 25 ; this is an integral over the volume of the quantity Using the definitions of eij and Pij equations 7 and 14 , we can write The first term on the right-hand side is the volume integral of this quantity represents, therefore, the increase in the internal energy due to the compression which the fluid experiences. We start from an initial flow which represents a stationary state of the system. He worked at the University of Cambridge and at the University of Chicago.
The solution of equation 50 appropriate to the problem on hand is β is the adverse temperature gradient which is maintained. The foregoing arguments establish the stationary property of the characteristic values when they are regarded as given by equation 140. Chandrasekhar considers most of the typical problems in hydromagnetic stability, with the exception of viscous shear flow; a specialized domain deserving a book unto itself. The corresponding distribution of the density is given by With this expression for ρ, equation 48 can be integrated to give Let the initial state described by equations 47 , 51 , 52 , and 53 be slightly perturbed. Variations of this small amount can, in general, be ignored. This first lower-priced, sturdy paperback edition will be welcomed by graduate physics students and scientists familiar with Dr. By ignoring terms of the second and higher orders in the perturbations, equations 43 and 46 give see equation 62 below and the velocity field remains, of course, solenoidal: It is of interest to verify that we obtain the same equations 55 — 57 from the general equations without first making the Boussinesq approximation.