Solution phase models cellular model

The solution obtained as described above is valid at small values of volume content of the liquid phase. The average distance between drops is proportional to and therefore with the growth of m the average distance between drops decreases, which results in the need to take into account the mutual influence of drops, in other words, to take into account hindrance. To this end, the boundary condition in the bulk gas needs to be formulated, proceeding from the cellular model (see Section 10.7) ... 

Figure 5.3. A cellular automata model of the interface between two immiscible bquids, after the demixing process has reached an equilibrium. A solute (encircled cells) has partitioned into the two phases according to its partition coefficient...

From a mathematical point of view, the onset of sustained oscillations generally corresponds to the passage through a Hopf bifurcation point [19] For a critical value of a control parameter, the steady state becomes unstable as a focus. Before the bifurcation point, the system displays damped oscillations and eventually reaches the steady state, which is a stable focus. Beyond the bifurcation point, a stable solution arises in the form of a small-amplitude limit cycle surrounding the unstable steady state [15, 17]. By reason of their stability or regularity, most biological rhythms correspond to oscillations of the limit cycle type rather than to Lotka-Volterra oscillations. Such is the case for the periodic phenomena in biochemical and cellular systems discussed in this chapter. The phase plane analysis of two-variable models indicates that the oscillatory dynamics of neurons also corresponds to the evolution toward a limit cycle [20]. A similar evolution is predicted [21] by models for predator-prey interactions in ecology. 

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