Stability governing equations

The described continuous penaltyf) time-stepping scheme may yield unstable results in some problems. Therefore we consider an alternative scheme which provides better numerical stability under a wide range of conditions. This scheme is based on the U-V-P method for the slightly compressible continuity equation, described in Chapter 3, Section 1.2, in conjunction with the Taylor-Galerkin time-stepping (see Chapter 2, Section 2.5). The governing equations used in this scheme are as follows... 

Though this new algorithm still requires some time step refinement for computations with highly inelastic particles, it turns out that most computations can be carried out with acceptable time steps of 10 5 s or larger. An alternative numerical method that is also based on the compressibility of the dispersed particulate phase is presented by Laux (1998). In this so-called compressible disperse-phase method the shear stresses in the momentum equations are implicitly taken into account, which further enhances the stability of the code in the quasi-static state near minimum fluidization, especially when frictional shear is taken into account. In theory, the stability of the numerical solution method can be further enhanced by fully implicit discretization and simultaneous solution of all governing equations. This latter is however not expected to result in faster solution of the TFM equations since the numerical efforts per time step increase. 

The marginal stability envelopes are shifted when a bias voltage is applied, and recently Iwamoto et al. (1991) prepared a number of stability maps showing the effects of bias voltage. They solved the governing equations numerically. 

In the next few sections we will concentrate on the form of the governing equations (4.24) and (4.25) with the exponential approximation to f(0) as given by (4.27). We will determine the stationary-state solution and its dependence on the parameters fi and k, the changes which occur in the local stability, and the conditions for Hopf bifurcation. Then we shall go on and use the full power of the Hopf analysis, to which we alluded in the previous chapter, to obtain expressions for the growth in amplitude and period of the emerging oscillatory solutions. 

Burner-Stabilized Flame While the governing equations are the same for either the burner-stabilized or the freely propagating flames, the boundary conditions differ. For the typical burner-stabilized case, the mass flux m" is specified, as is the temperature at the burner face (z = 0). The species boundary condition is specified through the mass-... 

The aim now is twofold finding a spatially homogeneous solution of the governing equations (for a given shear rate) and investigating the stability of this solution. In this section we will describe the general procedure and give the results in Sect. 3. 

The experimental and theoretical literature on instabilities in fiber spinning has been reviewed in detail by Jung and Hyun (28). The theoretical analysis began with the work of Pearson et al. (29-32), who examined the behavior of inelastic fluids under a variety of conditions using linear stability analysis for the governing equations. For Newtonian fluids, they found a critical draw ratio of 20.2. Shear thinning and shear thickening fluids... 

To analyze a physical problem analytically, we must obtain the governing equations that model the phenomenon adequately. Additionally, if the auxiliary equations pertaining to initial and boundary conditions are prescribed those are also well-posed, then conceptually getting the solution of the problem is straightforward. Mathematicians are justifiably always concerned with the existence and uniqueness of the solution. Yet not every solution of the equation of motion, even if it is exact, is observable in nature. This is at the core of many physical phenomena where ohservahility of solution is of fundamental importance. If the solutions are not observable, then the corresponding basic flow is not stable. Here, the implication of stability is in the context of the solution with respect to infinitesimally small perturbations. 

In these equations, primes indicate differentiation with respect to y. One can rewrite these equations as a set of six first order equations and thus one would require six boundary conditions to solve them simultaneously. For the stability analysis, above equations are solved subject to homogeneous boundary conditions that are compatible with the governing equations. For example at the wall, one uses the no-slip boundary conditions,... 

The governing equations are given in the next section. The mean flow, whose stability will be studied, is given in the section 6.3. The stability equations and related numerical methods for CMM is given in section 6.4. The results and discussion follow in section 6.5. The chapter closes with some comments and outlook in section 6.6. 

Dense palladium-based membranes. Shown in Table 10.1 are modeling studies of packed-bed dense membrane shell-and-tube reactors. All utilized Pd or Pd-alloy membranes except one [Itoh et al., 19931 which used yttria-stabilized zirconia membranes. As mentioned earlier, the permeation term used in Ihe governing equations for the tube and shell sides of the membrane is expressed by Equation (10-51b) with n equal to 0.5 [c.g., Itoh, 1987] or 0.76 [e.g., Uemiya et al., 1991]. 

In this section, we consider these problems in some detail, although with the major simplifications of assuming that the processes are isothermal and that the liquid is incompressible. As we shall see, the governing equations for even this simplified ID problem are nonlinear, and thus most features can be exposed only by either numerical or asymptotic techniques. In fact, the problem of single-bubble motion in a time-dependent pressure field turns out to be not only practically important, but also an ideal vehicle for illustrating a number of different asymptotic techniques, as well as introducing some concepts of stability theory. It is for this reason that the problem appears in this chapter. 

Therefore, to study the stability problem, we must solve the governing equations, (6-83) and (6-85), with (6-91) as the initial condition, to determine whether h grows or decays in time. The result depends, of course, on the values of the parameters (6-87)-(6-89) and also on the form (and, generally, also the magnitude) of disturbance function. If the disturbance decays so that h - 1 as t - oo, the system is said to be stable to that particular disturbance. 

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... 

The equations (12-20)-( 12-24) are the so-called linear stability equations for this problem in the inviscid fluid limit. We wish to use these equations to investigate whether an arbitrary, infinitesimal perturbation will grow or decay in time. Although the perturbation has an arbitrary form, we expect that it must satisfy the linear stability equations. Thus, once we specify an initial form for one of the variables like the pressure p, we assume that the other variables take a form that is consistent with p by means of Eqs. (12-20)-(12-24). Now the obvious question is this How do we represent a disturbance function of arbitrary form For this, we take advantage of the fact that the governing equations and boundary conditions are now linear, so that we can represent any smooth disturbance function by means of a Fourier series representation. Instead of literally studying a disturbance function of arbitrary form, we study the dynamics of all of the possible Fourier modes. If any mode is found to grow with time, the system is unstable because, with a disturbance of infinitesimal amplitude, every possible mode will always be present. 

An arbitrary disturbance form in the x and y directions could be expressed as a sum of the Fourier modes of wave number a x and a y, but because the governing equations are linear with coefficients that are independent of x, y, it is enough to consider the stability of these disturbance quantities one mode at a time, for arbitrary values of a x and a y. The fimctions of z must be chosen to satisfy boundary conditions on the fluid interface. The stability is determined by the sign of the real part of a. The reader is reminded that the primes on all of the symbols mean that they are dimensional. 

In the inviscid limit, the general linear stability problem takes the following simpler form. First, the governing equations, (12-65a) and (12-65b), are reduced to a pair of second-order DEs ... 

The governing equations for the linear stability theory are the same as for the Rayleigh-Benard problem, namely (12-215), except that it is customary to drop the buoyancy terms because these are of secondary importance for very thin fluid layers where Marangoni instabilities are present but Ra <neutral state. Assuming that... 

The governing equations (12-307) for a 2D disturbance (ay = v = 0) can be combined to obtain a single higher-order equation that can be used to study the stability conditions. First, we note that the continuity equation (12-307d) can be satisfied by introducing a function [Pg.875]

W hen the dimensionless permeability k/d1 is equal to zero, the problem reduces to the Darcy approximation that was analyzed in the previous problem. On the other hand, for k/d1 — oo, the problem is the classical Rayleigh Benard case for a single viscous fluid. With the notation of Section H, the governing equation at the neutral stability point becomes... 

Nonlinear equations may admit no real solntions or mnltiple real solutions. For example, the quadratic equation can have no real solutions or two real solutions. Thus, it is important to know whether a given equation governing the behavior of an engineering system can admit more than one solution, since it is related to the issue of operation and performance of the system. In this subsection, criteria for the existence of multiple steady-state solutions to the governing equations of a CSTR and tubular reactors and, subsequently, the stability of these multiple steady states are presented. 

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