Steady Periodic Solution

The first law applied to the transversally lumped, otherwise differential system shown in Fig. 3.16(a), and interpreted in terms of Fig. 3.16(b), yields 

Relating u to T, employing two-dimensional Fourier s law and Newton s law for upward and downward convection, Eq. (3.84) maybe rearranged to give 

For an application of this formulation, reconsider Ex. 2,9. Let the fin have a uniform initial temperature Too, and let the base temperature be suddenly raised to temperature Tq and held constant thereafter. 

We have from Eq. (3.85), after eliminating the energy generation and the y dependence of temperature, and introducing m2 = (hi + hi)/kS, 

As we learned in this chapter, the formulation of unsteady distributed problems leads to partial differential equations. The solution of these equations is much more involved than that of ordinary differential equations. Among the techniques available, the analytical and computational methods are most frequently referred to. Exact analytical methods such as separation of variables and transform calculus are beyond the scope of the text. However, the method of complex temperature and the use of charts based on exact analytical solutions, being useful for some practical problems, are respectively discussed in Sections 3.4 and 3.6. Among approximate analytical methods, the integral method, already introduced in Sections 2.4 and 3.1, is further discussed in Section 3.5. The analog solution technique is also briefly treated in Section 3.7. 

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Seeking a steady periodic solution, the right hand side suggest that the solution should be [6],... 

The imaginary part of Eq. 3.64 is the starting periodic solution of the complementary problem.) As t - oo, the last term in parentheses approaches zero, and Eq. (3.65) reduces to the steady periodic solution,... 

For linear problems, the input and response have the same harmonic variation. A steady periodic solution must then have the form... 

In Section 3.2 we focused on the unsteady solution and its steady part for periodic lumped problems. We learned then the practical importance of steady periodic solutions and, in terms of the method of complex temperature, an easy way of obtaining only the steady part of periodic solutions. In this section we apply the method of complex... 

Experimental observations of regular waves and instability of the main flow lead to search nonlinear steady regimes. The problem of bifurcations of nonlinear steady periodic solutions of (9) was formulated in paper Sh, where the first family of waves was found. This family softly bifurcates from the waveless flow on neutral curve and exists at s G (0,1). Because nonlinear waves of first family move with phase velocity c < 3 they were named as slow waves. The second family of fast waves together with some other bifurcating solutions are obtained by Shkadov et al. (1981), see also Bunov et al. (1984). The full study of intermediate bifurcations is fulfilled as well as full two-parametric manifold of bifurcations is constructed up to now (Sisoev and Shkadov, 1997a, 1999). ... 

One way to examine the validity of the steady-state approximation is to compare concentration—time curves calculated with exact solutions and with steady-state solutions. Figure 3-10 shows such a comparison for Scheme XIV and the parameters, ki = 0.01 s , k i = 1 s , 2 = 2 s . The period during which the concentration of the intermediate builds up from its initial value of zero to the quasi-steady-state when dcfjdt is vei small is called the pre-steady-state or transient stage in Fig. 3-10 this lasts for about 2 s. For the remainder of the reaction (over 500 s) the steady-state and exact solutions are in excellent agreement. Because the concen-... 

In this simplified version of the Brusselator model, the trimolecular autocatalytic step, which is a necessary condition for the existence of instabilities, is, of course, retained. However, the linear source-sink reaction steps A—>X—>E are suppressed. A continuous flow of X inside the system may still be ensured through the values maintained at the boundaries. The price of this simplification is that (36) can never lead to a homogeneous time-periodic solution. The homogeneous steady states are... 

In Fig. 21 we have drawn the bifurcation diagram of the fundamental steady-state solutions for three values of p [ Kxn is plotted versus UK) as the bifurcation parameter]. There is a subcritical region in the upper or lower branch, depending on the relative height of the peaks in Fig. 20c. The asymptotes K and K" of these branches correspond to half-period solutions of infinite length. When p 2 the asymptote K merges with the w-axis therefore situation 2 above can be viewed as a particular case of situation 3 above, in which the bifurcation point moves to infinity. 

The same equations, albeit with damping and coherent external driving field, were studied by Drummond et al. [104] as a particular case of sub/second-harmonic generation. They proved that below a critical pump intensity, the system can reach a stable state (field of constant amplitude). However, beyond the critical intensity, the steady state is unstable. They predicted the existence of various instabilities as well as both first- and second-order phase transition-like behavior. For certain sets of parameters they found an amplitude self-modula-tion of the second harmonic and of the fundamental field in the cavity as well as new bifurcation solutions. Mandel and Erneux [105] constructed explicitly and analytically new time-periodic solutions and proved their stability in the vicinity of the transition points. 

Fig. 4.15 Time-dependent, nondimensional, solution of Eq. 4.104, with the cylinder wall rotating in an oscillatory manner, w = sin(2jrt/tp), where tp is the nondimensional period of the oscillation. The fluid is initially at rest, with the inner rod beginning to oscillate suddenly at t = 0. After a few cycles, the solution comes to a repeatable steady oscillating solution. This solution was generated in a spreadsheet with 16 uniformly spaced nodes and a nondimensional timestep dt = 0.001.

In Fig. 15.8 notice that during the time integration, the steady-state residuals increased for a period as the transient solution trajectory climbed over a hill and into the valley where the solution lies. This behavior is quite common in chemically reacting flow problems, especially when the initial starting estimates are poor. In fact it is not uncommon to see the transient solution path climb over many hills and valleys before coming to a point where the Newton method will begin to converge to the desired steady-state solution. 

Because the solute diffusivity in the solid is far smaller than in the liquid, any diffusion in the solid will be neglected. In most cases of interest, the transient period required to produce a quasi-steady-state solute distribution at the interface is relatively small.1 At a relatively short time after the establishment of the quasi-steady-state concentration spike, the flux relative to an origin at the interface moving at velocity v is... 

It should be said that the phase plane of itself cannot be guaranteed to give the whole behavior, for, in computing phase planes, one cannot be sure that one has started in the regions of attraction of all the steady or periodic solutions present. In 1956, the theory that is needed for this to be done systematically was 20 years into the future, and the calculating engines were just beginning to become really powerful. Fortran had not been invented, and each cell had to be addressed by its number, in octal 7... 

An obvious map to consider is that which takes the state (x(t), y(t) into the state (x(t + r), y(t + t)), where r is the period of the forcing function. If we define xn = x(n t) and y = y(nr), the sequence of points for n = 0,1,2,... functions in this so-called stroboscopic phase plane vis-a-vis periodic solutions much as the trajectories function in the ordinary phase plane vis-a-vis the steady states (Fig. 29). Thus if (x , y ) = (x +1, y +j) and this is not true for any submultiple of r, then we have a solution of period t. A sequence of points that converges on a fixed point shows that the periodic solution represented by the fixed point is stable and conversely. Thus the stability of the periodic responses corresponds to that of the stroboscopic map. A quasi-periodic solution gives a sequence of points that drift around a closed curve known as an invariant circle. The points of the sequence are often joined by a smooth curve to give them more substance, but it must always be remembered that we are dealing with point maps. 

Bailey, J. E., 1977, Periodic phenomena in chemical reactor theory. In Chemical Reactor Theory (Edited by Lapidus, L. and Amundson, N. R.). Prentice-Hall, Englewood Cliffs, NJ. Balakotaiah, V. and Luss, D., 1982, Structure of the steady state solutions of lumped parameter chemical reacting systems. Chem. Engng Sci. 37, 1611-1623. 

The Hopf bifurcation where a stable focus becomes unstable and sheds or absorbs a periodic solution is an important transition which has received a great deal of attention (for a review see Marsden McCracken 1976). Clearly the lines over which it can take place are the loci of steady-states whose eigenvalues are purely imaginary. These are shown on the sides of the fin in figure 5. Because this is a two-dimensional system we can write down the condition quite explicitly. Writing the equations ... 

All transitions associated with secondary bifurcations have now been completely classified. Moreover, several examples of tertiary or even quaternary branchings are known both for steady-state and for time-periodic solutions (see, e.g., Iooss6 and Erneux and Reiss7 for some recent results). 

A very important example of global bifurcations is the succession of period-doubling transitions, leading from a steady-state solution to time-periodic solutions of increasing period and finally to nonperiodic behavior. This phenomenon has been studied extensively for iterative equations of the form11,12... 

The previous 13 figures show several transition stages in the behavior of the solution to the given IVP from having one fixed asymptotic steady-state solution for low values of hf] through small oscillation for all times, to limit cycles, irregularity, and chaos back to repeated oscillations with ever-decreasing numbers of periods and then back to one... 

As an example, if only quasi-steady flow elements are used with volume pressure elements, a model s smallest volume size (for equal flows) will define the timescale of interest. Thus, if the modeler inserts a volume pressure element that has a timescale of one second, the modeler is implying that events which happen on this timescale are important. A set of differential equations and their solution are considered stiff or rigid when the final approach to the steady-state solution is rapid, compared to the entire transient period. In part, numerical aspects of the model will determine this, but also the size of the perturbation will have a significant impact on the stiffness of the problem. It is well known that implicit numerical methods are better suited towards solving a stiff problem. (Note, however, that The Mathwork s software for real-time hardware applications, Real-Time Workshop , requires an explicit method presumably in order to better guarantee consistent solution times.)... 

For T = 16 s, the single nephron model undergoes a supercritical Hopf bifurcation at a = 11 (outside the figure), fn this bifurcation, the equilibrium point loses its stability, and stable periodic oscillations emerge as the steady-state solution. For a = 19.5, at the point denoted PDla 2 in Fig. 12.5, this solution undergoes a period-... 

The theory of nonlinear oscillations can describe the periodic solution that appears beyond the instability of the steady state. Stable states exist before the instability. The perturbations correspond to complex values of the normal mode frequencies and spiral toward the steady state to a focus. As soon as the steady state becomes unstable, a stable periodic... 

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