Equations linear stability continuity

The infinitely long cylinder with no motion of the interface or of the fluid within the cylinder is, of course, a possible equilibrium configuration, in the sense that it is a surface of constant curvature so that the stationary constant-radius fluid satisfies all of the conditions of the problem, including the Navier-Stokes and continuity equations (trivially), as well as all of the interface boundary conditions including especially the normal-stress balance, which simply requires that the pressure inside the cylinder exceed that outside by the factor 1//a. The question for linear stability theory is whether this stationary configuration is stable to infinitesimal perturbations of the velocity, the pressure, or the shape of the cylinder. 

To analyze the linear stability of a Couette flow, we begin with the Navier Stokes and continuity equations in a cylindrical coordinate system. The frill equations in dimensional form can be found in Appendix A. We wish to consider the fate of an arbitrary infinitesimal disturbance to the base flow and pressure distributions (12 114) and (12 116). Hence we consider a linear perturbation of the form... 

In searching for the necessary conditions under which the smooth-stratified flow configuration is stable, linear stability analysis is carried out on the transient two-fluid continuity and momentum Equations 1, 2, 7. The equations are perturbed around the smooth fully developed stratified flow pattern. Following the route of temporal stability analysis h = heKkx-m). gi(icx-[Pg.327]

The methods presented include linear algebra methods (linearization, stability analysis of the linear system, constrained linear systems, computation of nominal interaction forces), nonlinear methods (Newton and continuation methods for the computation of equilibrium states), simulation methods (solution of discontinuous ordinary differential and differential algebraic equations) and solution methods for... 

That is to say, the meaning of stability of scheme (21) is that a solution (21) depends continuously on the right-hand side and this dependence is uniform in the parameter h. This implies that a small change of the right-hand side results in a small change of the solution. If the scheme is solvable and stable, it is correct. Note that the uniqueness of the scheme (21) solution is a consequence of its solvability and stability and, hence, we might get rid of the uniqueness requirement in condition (1). Indeed, assume to the contrary that there were two solutions to equation (21), say and By the linearity property of the operator A, their difference = y — yf should satisfy the homogeneous equation... 

The stability of transport and rate systems is studied either by nonequilibrium thermodynamics or by conventional rate theory. In the latter, the analysis is based on Poincare s variational equations and Lyapunov functions. We may investigate the stability of a steady state by analyzing the response of a reaction system to small disturbances around the stationary state variables. The disturbed quantities are replaced by linear combinations of their undisturbed stationary values. In nonequilibrium thermodynamics theory, the stability of stationary states is associated with Progogine s principle of minimum entropy production. Stable states are characterized by the lowest value of the entropy production in irreversible processes. The applicability of Prigogine s principle of minimum entropy production is restricted to stationary states close to global thermodynamic equilibrium. It is not applicable to the stability of continuous reaction systems involving stable and unstable steady states far from global equilibrium. The steady-state deviation of entropy production serves as a Lyapunov function. 

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... 

To achieve a deeper understanding of the stability properties of the inhomogeneous fixed point under the influence of the control force and to obtain the general form of the characteristic equation which determines the eigenvalues of this linearized system, we perform the linearization of the original continuous system (5.30) at the spatially inhomogeneous fixed point (ao(x),uo). Introducing... 

This means that the stability of the high-symmetry form (Q= 0) can only be satis-fled if, in Eq. (B.l), the linear term is absent (A = 0) and B > 0. For B < 0, the low-symmetry form is stable, and the equilibrium values of Q, is between 0 and 1. Equations (B.2) and (B.3) can further be satisfled by setting aU odd-order terms in Eq. (B.l) to zero. The transition is then continuous between the states character-... 

Rousseau and Howell (1982) considered the merits of using different measurements for stabilizing low order cycling of CSD in a continuous crystallizer with both fines destruction and product classification. The analysis was carried out on a simulated process using population and mass balances along with kinetic equations and employed finite difference techniques to solve the system. The main advantages of using a finite difference method in comparison with a linearized form of analytical solution were cited as (a) no modifications to the models were necessary to accommodate different removal functions and (b) any form of nucleation kinetics could be used. 

Rawlings etal. (1992) analysed the stability of a continuous crystallizer based on the linearization of population and solute balance. Their model did not depend on a lumped approximation of partial difference equations and successfully predicted the occurrence of sustained oscillations. They demonstrated that simple proportional feedback control using moments of CSD as measurements can stabilize the process. It was concluded that the relatively high levels of error in these measurements require robust design for effective control. 

---