Blowing up

There exists an interesting (or, perhaps annoying) phenomenon in the theory of stochastic processes called blowing up. This means that a process defined by its infinitesimal quantities grows in such a quick way that within a finite interval of time the random variable characterising the process in a fixed time point takes on infinite values with greater than zero probabilities. This phenomenon can be found in the case of the usual (polynomial) CDS model of the reaction 

It is not astonishing at all if one considers the deterministic model of the same reaction and verifies that all the solutions of the induced kinetic differential equation are such that they cannot be extended to the whole positive half line, or to use the stochastic terminology, the deterministic model blows up too (cf. Toth, 1986). 

A very natural (but so far only historical) fact is that sufficient conditions not to blow up are known for the same mechanisms in the deterministic and in the stochastic case. (It may be useful to note that this kind of blowing up has nothing to do with shock waves, but it sometimes does depend on concentration ratios.) 

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It turns out that one cannot propagate Y using standard numerical methods because Y blows up whenever g is zero. To circumvent this one must propagate Y by invariant imbedding . The basic idea here is to construct a propagator Y which satisfies... 

Dijference between exact and numerical trajectory for the approach of two argon atoms for a time stef) of he simulation blows up. ... 

Reactive Chemicals Reviews The process chemistry is reviewed for evidence of exotherms, shock sensitivity, and other insta-bihty, with emphasis on possible exothermic reactions. It is especially important to consider pressure effects— Pressure blows up people, not temperature The pumose of this review is to prevent unexpected and uncontrolled chemical reactions. Reviewers should be knowledgeable people in the field of reactive chemicals and include people from loss prevention, manufacturing, and research. 

If you blow up a balloon, energy is stored in it. There is the energy of the compressed gas in the balloon, and there is the elastic energy stored in the rubber membrane itself. As you increase the pressure, the total amount of elastic energy in the system increases. 

If we then introduce a flaw into the system, by poking a pin into the inflated balloon, the balloon will explode, and all this energy will be released. The membrane fails by fast fracture, even though well below its yield strength. But if we introduce a flaw of the same dimensions into a system with less energy in it, as when we poke our pin into a partially inflated balloon, the flaw is stable and fast fracture does not occur. Finally, if we blow up the punctured balloon progressively, we eventually reach a pressure at which it suddenly bursts. In other words, we have arrived at a critical balloon pressure at which our pin-sized flaw is just unstable, and fast fracture just occurs. Why is this ... 

In the tubular process a thin tube is extruded (usually in a vertically upward direction) and by blowing air through the die head the tube is inflated into a thin bubble. This is cooled, flattened out and wound up. The ratio of bubble diameter to die diameter is known as the blow-up ratio, the ratio of the haul-off rate to the natural extrusion rate is referred to as the draw-down ratio and the distance between the die and the frost line (when the extrudate becomes solidified and which can often be seen by the appearance of haziness), the freeze-line distance. 

The major advantage of film blowing is the ease with which biaxial orientation can be introduced into the film. The pressure of the air in the bubble determines the blow-up and this controls the circumferential orientation. In addition, axial orientation may be introduced by increasing the nip roll speed relative to the linear velocity of the bubble. This is referred to as draw-down. 

Example 4.3 A plastic shrink wrapping with a thickness of 0.05 mm is to be produced using an annular die with a die gap of 0.8 mm. Assuming that the inflation of the bubble dominates the orientation in the film, determine the blow-up ratio required to give uniform biaxial orientation. 

A plastic film, 0.1 mm thick, is required to have its orientation in the transverse direction twice that in the machine direction. If the film blowing die has an outer diameter of 100 mm and an inner diameter of 98 mm estimate the blow-up ratio which will be required and the lay flat film width. Neglect extrusion induced effects and assume there is no draw-down. 

Design a die which will produce plastic film 0.52 mm thick at a linear velocity of 20 mm/s. The lay-flat widfli of the film is to be 450 mm and it is known that a blow-up ratio of 1.91 will give the necessary orientation in the film. Assume that there is no draw-down. 

Now it makes the solution simpler to assume that the blow up ratio is given by Dft/Di (ie rather than Db/D ). Also this seems practical because the change from D to D is caused solely by inflation whereas the change from D to Db includes die swell effects. 

Wecanmodifythesolutiontoincludethisbehavior.Thecriticalradiuscanbeexpressed as amultiple of theinitial radius at time zero. We want the functiontoliterally "blow-up"... 

Table 2 Properties of the PE Resins and Their Blow Film Samples (Blow up Ratio-2) [56]...

Figure 11 S11F/S33F ratio vs. blow up ratio for different...

In their study on LLDPE resins containing 1-butene, 1-hexane, and 1-octene comonomers, Kalyon and Moy [30] found a significant variation in their film thickness when measured around the circumference of tubular bubbles processed under identical conditions. The samples blown with a blow-up ratio of two, exhibited more significant variation in thickness than those prepared with a blow-up ratio of three. However, film processed at a higher blow-up ratio has been found to have less variation in thickness. 

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