Dissipative structures mathematical model

The statistical theory of open systems is not yet developed enough to be applied to physico-chemical problems. Both catastrophe and dissipative structure theories are of more general philosophic rather than practical value. So, only the classic Poincare-Andronov s bifurcatirMi theory gives real tools for the formulation and investigation of the mathematical models of the processes developing in physical and chemical systems far away from equilibrium. Some examples are presented in Chap. 5 where these tools were successfully applied to electrochemical systems. Main principles of such applications are given below. 

In order to resolve the above-mentioned problems, perforated wall structures have been introduced especially in small craft harbors. The simplest perforated wall structure may be a curtain-wall breakwater (sometimes called wave screen or skirt breakwater), which consists of a vertical wall extending from the water surface to some distance above the seabed. Recently, Isaacson et al." proposed a slotted curtain-wall breakwater. Another simple perforated wall structure may be an array of vertical piles, which is called a pile breakwater in this chapter. The closely spaced piles induce energy dissipation due to viscous eddies formed by the flow through the gaps. To examine the wave scattering by vertical piles, hydraulic model tests have been used. Efforts toward developing analytical models to calculate the reflection and transmission coefficients have also been made. Recently, Suh et introduced a curtain-wall-pile breakwater, the upper part of which is a ciu tain waU and the lower part consisting of an array of vertical piles. They developed a mathematical model that predicts various hydrodjmamic characteristics of a cmtain-wall-pile breakwater. More recently, Suh and Ji extended the model to a multiple-row breakwater. 

This part of the book moves from the chemical structure and reactions used to form polymers to their physical properties. Some of the material in this section may be familiar to one who has studied the mechanics of materials, but it is worth delving deeper into polymers and discover some of their unusual characteristics (viscoelasticity, for one). Topics that are addressed here include mechanical strength, flexibility, polymer responses to compression, stretching and shear forces, and the equipment used to test these properties. One chapter is devoted to mathematical models to describe viscoelastic behavior (using combinations of so-called springs and dashpots) to approximate the complex response of polymers to stresses that include both elastic stretch (which is particularly useful in the waistbands of underwear) and viscous deformation (which is very useful in automobile bumpers to dissipate the energy of a collision). 

---