Steady solution

Figure 10. Adsorbed cation coverage as a function of electrode potential, assuming a cation interaction parameter / = 6.18 The solid line is the steady-state solution, whereas the broken line is the quasi-steady solution. Open circles indicate the unstable area. (From G. L. Griffin, J. Electrochettu Soc. 131, 18, 1984, Fig. 1. Reproduced by permission of The Electrochemical Society, Inc.)...

The solution of Eq. (10.50) determines the steady states of the liquid velocity, as well as the position of the meniscus in a heated micro-channel. Equation (10.50) can have one, two or three steady solutions. This depends on the value of the parameter (in the generic case parameter B), which takes into account the effect of the capillary forces. 

The research on the flow regimes in packed tubes suggests that laminar flow CFD simulations should be reasonable for Re <100 approximately, and turbulent simulations for Re >600, also approximately. Just as RANS models provide steady solutions that are regarded as time averages of the real time-dependent turbulent flow, it may be suggested that CFD simulations in the unsteady laminar inertial range 100 time-averaged picture of the flow field. As with wall functions, comparisons with experimental data and an improved assessment of what information is really needed from the simulations will inform us as to how to proceed in these areas. 

Once ignition has occurred in a mixture of fuel and oxidizer, propagation will continue, provided the concentrations are sufficient and no disturbance results in excessive cooling. The zeroth-order rate model is assumed to represent the lean case. Substituting the selected properties into Equation (4.43), the net release and loss curves are plotted in Figure 4.22 as a function of the flame temperature. The initial temperature of the mixture is 25 °C and fuel mass fractions are 0.05 and 0.03, representative of stoichiometric and the lower limit respectively. At this lower limit, we should see that a steady solution is not possible, and the calculations should bear this out. The burning rate is evaluated at the flame temperature, and 6K is found from Equation (4.44) with Su at the flame temperature... 

We follow the analysis of Frank-Kamenetskii [3] of a slab of half-thickness, rG, heated by convection with a constant convective heat transfer coefficient, h, from an ambient of Too. The initial temperature is 7j < 7 ,XJ however, we consider no solution over time. We only examine the steady state solution, and look for conditions where it is not valid. If we return to the analysis for autoignition, under a uniform temperature state (see the Semenov model in Section 4.3) we saw that a critical state exists that was just on the fringe of valid steady solutions. Physically, this means that as the self-heating proceeds, there is a state of relatively low temperature where a steady condition is sustained. This is like the warm bag of mulch where the interior is a slightly higher temperature than the ambient. The exothermiscity is exactly balanced by the heat conducted away from the interior. However, under some critical condition of size (rG) or ambient heating (h and Too), we might leave the content world of steady state and a dynamic condition will... 

The approach is to formulate the entire burning problem using conservation laws for a control volume. The condensed phase will use control volumes that move with the vaporization front. This front is the surface of a regressing liquid or solid without char, or it is the char front as it extends into the virgin material. The original thickness, l, does not change. While the condensed phase is unsteady, the gas phase, because of its lower density, is steady or quasi-steady in that its steady solution adjusts to the instantaneous input of the condensed phase. 

We will avoid the kinetics subsequently in solving for the burning rate and the flame temperature, but it is important in understanding extinction. The solutions for the burning rate and flame temperature are quasi-steady solutions for the gas phase. Where steady conditions do not exist, we have extinction ensuing. Extinction will be addressed later in Section 9.10. 

The right-hand side (RHS) of Equations (9.116) and (9.119) represent the net heat loss and the left-hand side (LHS) represents the energy gain. The gain and the loss terms can be plotted as a function of the flame temperature for both the diffusion and premixed flames as Semenov combustion diagrams. Intersection of the gain and loss curves indicates a steady solution, while a tangency indicates extinction. 

One approach is to approximate the unsteady solution by a quasi-steady solution where Q(t) is a function of time and time is adjusted. This can be approximated by determining the transport time (fD) from the origin to position z by... 

As will be shown, the steady solution for the detonation velocity does not involve any knowledge of the structure of the wave. The Hugoniot plot discussed in Chapter 4 established that detonation is a large Mach number phenomenon. It is apparent, then, that the integrated momentum equation is included in obtaining a solution for the detonation velocity. However, it was also noted that there are four integrated conservation and state equations and five unknowns. Thus, other... 

The Vlasov-Newton equation has many steady solutions describing a self-gravitating cluster. This is easy to show in the spherically symmetric case (the situation we shall restrict in this work, except for a few remarks at the end of this section). If one assumes a given r(r) in the steady state, the general steady solution of Eq. (4) is a somewhat arbitrary function of the constants of the motion of a single mass in this given external held, namely a funchon/(E, I ) where niE is the total energy of a star in a potenhal (r) such that r(r) = —(r/r) [d r)/dr] and where — (r.v) is the square of the... 

We have found now an equation for the evolution of the density inside the cluster without any uncontrolled parameter, except for the dimensionless number C. Below we shall do two things. First, in Section V, we shall find the steady solutions for the density, that turns out to transform into a quite simple problem, mathematically equivalent to the equilibrium of self-gravitating atmosphere. Then, in Section VI we shall look at the possible existence of finite time singularities in the dynamical problem. 

A remarkable result is that y is defined self-consistently and is a free parameter for the steady solutions. To show this point, let us introduce dimensionless quantities with overlines ... 

The next (and much more difficult) question is the stability of this solution. This is a complex issue because the coefficients of the diffusion equation depend on the solution itself. To summarize the full dynamical problem, we look at the stability of steady solutions of the dynamical problem ... 

In parallel, the simplicity of handling the locally electro-neutral case provides a convenient ground for studying the far less tractable one-dimensional version of the nonreduced system (4.1.1)-(4.1.2), asymptotically for small e. For an example of such a study we refer to [14] where the difficult question of multiplicity of steady solutions of the nonreduced system was approached through studying the multiplicity of solutions in the LEN approximation for a four layer (quadrupolar) arrangement. The theory of a bistable electronic device (thyristor) which resulted from this study will be presented in 4.3. 

It is often the case that after a sufficiently long time, a transient problem approaches a steady-state solution. When this is the case, it can be useful to calculate the steady solution independently. In this way it can be readily observed if the transient solution has the correct asymptotic behavior at long time. 

This statement comes from analytical and topological studies [4], Unlike the Lotka-Volterra model where due to the dependence of the reaction rate K(t) on concentrations NA and NB, the nature of the critical point varied, in the Lotka model the concentration motion is always decaying. Autowave regimes in the Lotka model can arise under quite rigid conditions. It is easy to show that not any time dependence of K(t) emerging due to the correlation motion is able to lead to the principally new results. For example, the reaction rate of the A + B -> 0 reaction considered in Chapter 6 was also time dependent, K(t) oc t1 d/4 but its monotonous change accompanied by a strong decay in the concentration motion has resulted only in a monotonous variation of the quasi-steady solutions of (8.3.20) and (8.3.21) jVa(t) (3/K(t) and N, (t) p/f3 = const. 

The phenomena of ignition and extinction of a flame are typical examples of discontinuous change in a system under smooth variation of parameters. It is natural that they have played a substantial role in the formation of one of the branches of modern mathematics—catastrophe theory. In Ya.B. s work it is clearly shown that steady, time-independent solutions which arise asymptotically from non-steady solutions as the time goes to infinity are discontinuous. It is further shown that transition from one type of solution to the other occurs when the first ceases to exist. The interest which this set of problems stirred among mathematicians is illustrated by I. M. Gel fand s... 

Before turning to consideration of non-steady solutions of the fundamental equation (4), we will apply the information obtained to the problem of cavitation. 

Non-steady solutions of the equation for the probability of formation of a nucleus. Above we sought a steady solution of the fundamental equation (4), which gave us the number of virile (exceeding the critical size) nuclei which form in a unit volume per unit time under constant conditions. As is obvious from the form of solution (15), in a steady state the number of formations of subcritical size does not differ from the equilibrium number ... 

Consideration of the exact solution found by Omstein and Uhlenbeck [17] in one case of diffusion in a field of forces shows that on the segment from x1 to xcrit, the non-steady solution sought has a form close to... 

Let us compare this result with Semenov s [2] interpretation of thermal explosion (Fig. 2), which operates with quantities averaged over the volume. When the coefficient of heat transfer per unit volume is decreased (which may be accomplished by increasing the dimensions of the vessel) we obtain consecutively two steady solutions At and A2, the explosion limit B, and absence of steady solutions for still smaller heat transfer (line C). 

By analogy we conclude that, of the two steady solutions under the limit (see Fig. 1), only the lower one corresponds to a stable regime (At in Fig. 2) the upper curve corresponds to an unstable solution from which a small deviation leads either to explosion or to the lower stable regime (the point A2 in Fig. 2). 

After this paper was completed, a number of studies were carried out which analyzed the structure of the family of steady solutions and the conditions of ignition in spherical and cylindrical vessels. Regarding these studies, see the monograph by Ya.B. et al.2 In this same monograph one may find literature on various applications of the concept of thermal explosion in other problems of physics, in particular, in the physics of polymers where, as was first shown by A. G. Merzhanov and his colleagues, as a result of viscous heating, steady flow of the polymer becomes impossible in the motion of polymer melts. 

A hydrostatic stress can be superposed, but it is caused only by elastic volumetric strain of the composite. The result in Eqn. (39) is, perhaps, not very useful since it is rare that a steady strain rate will be kinematically imposed. When both fiber and matrix creep, the steady solutions for a fixed stress in isothermal states are quite complex but can be computed by numerical inversion of Eqn. (39). The solution can, however, be given for the isothermal case where the fibers do not creep. (For non-fiber composites, this should be... 

---