Linear hydrodynamic stability theory

Detonation, Nonlinear Theory of Unstable One-Dimensional. J.J. Erpenbeck describes in PhysFluids 10(2), 274-89(1969) CA 66, 8180-R(1967) a method for calcg the behavior of 1-dimensional detonations whose steady solns are hydrodynamic ally unstable. This method is based on a perturbation technique that treats the nonlinear terms in the hydro-dynamic equations as perturbations to the linear equations of hydrodynamic-stability theory. Detailed calcns are presented for several ideal-gas unimol-reaction cases for which the predicted oscillations agree reasonably well with those obtd by numerical integration of the hydrodynamic equations, as reported by W. Fickett W.W. Wood, PhysFluids 9(5), 903-16(1966) CA 65,... 

The designation linear to describe the theoretical study of the fate of small initial disturbances is due to the fact that the dynamics of such small disturbances can be described by means of a linear approximation of the Navier-Stokes, continuity, and other transport equations. Because the governing equations are linear, analytic theory is often possible but this requires that the unperturbed state or flow, whose stability we wish to study, be known analytically. Furthermore, this base solution must be quite simple for even the linear approximation of the equations to be analytically tractable. In practice this reduces significantly the number of problems in which complete analytic results are possible and also explains why hydrodynamic stability theory has been particularly successful in analyzing problems... 

A set of linear equations describing the infinitesimal disturbances in the mean flow variables follow from Equations 4.1-4.3 and 5.1, 5.2 with allowance made for Equation 9.1. Analysis of this linear set and of its characteristic equation should be accomplished along the well-known standard lines of the hydrodynamic stability theory which are exemplified for similar stability problems in reference [15,34]. In addition to this general formulation of the stability problem, different simplified versions of this problem can be considered, and in particular, those corresponding to the simplified fluid dynamic models discussed in Section 6. 

This principle is very general, relating neither to the linearity nor to the symmetry of the transport laws. On the other hand, it is difficult to attribute a physical meaning to dxP- The authors later attempted to derive a local potential from this property, and they applied this concept to the study of the chemical and hydrodynamical stability (e.g., the Benard convection). The results of this approach were published in Glansdorff and Prigogine s book Thermodynamic Theory of Structure, Stability and Fluctuations (LS.IO, 10a), published in 1971. 

We will consider the cold-gas-convex surface of the flame front as a curved cell of the flame which had been formed after the plane flame lost its stability. The steady state of the convex flame is a result of the nonlinear hydrodynamic interaction with the gas flow field (see Zeldovich, 1966, 1979). In the linear approximation the flame perturbation amplitude grows in time in accordance with Landau theory, but this growth is restricted by nonlinear effects. 

Several attempts have been made to explain theoretically the effects of flow on the phase behavior of polymer solutions [112,115-118,123,124]. This has been done by modification of the mean-field free energy. The key point is to include properly the elastic energy of deformation produced by flow. A more rigorous approach originates from Helfand et al. [125, 126] and Onuki [127, 128] who proposed hydrodynamic theories for the dynamics of concentration fluctuations in the presence of flow coupled with a linear stability analysis. 

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