A Stochastic Model of Crystallization in an Emulsion - Laboratoire de ...

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Figure 7: Deterministic case Figure 8: Stochastic case (small coef) 26
Figure 9: Stochastic case (greater coef) It is shown in Vallet [16] that in the deterministic case, the model gives nonincreasing temperature curves only. With coef = 0, the numerical curves give globally the same behavior. This behavior is generally the one observed in the experimentation. But, in one very particular case, the temperature may rise, instead of the plateau (see gure 10). This rising occurs only with water emulsions in motor oil and when P = 0, 5. Figure 10: Water emulsion with P = 0. 5 It is known that a frozen water droplet takes a place more important than a liquid one. Moreover, during the crystallization, dendrites appear. If a liquid droplet, with temperature under TF is touched, it crystallizes instantaneously. This crystallization excess induce some more heat in the cylinder, from which temperature rises. As, in our model, wn+1 wn is with mean 0, that is, in mean, there are as many crystallizations as in the deterministic case. In order to take into account the above remark, we propose two numerical propositions : 27
Page 1 and 2: A Stochastic Model of Crystallizati
Page 3 and 4: ( , Ϝ, P ) is a complete probabili
Page 5 and 6: The Stochastic energy equality : 2
Page 7 and 8: Proof. According to equation (7 ),
Page 9 and 10: 5.3 Existence N N N∈ N∈ t E
Page 11 and 12: t t 2as 2 2as 2 2 a e E|| ϕ||
Page 13 and 14: Let us set, Remark that L and that,
Page 15 and 16: Since t t N N N ∞,N N ∇( u
Page 17 and 18: 2at 2 one has : e E|| u( t) || ||
Page 19 and 20: converges and lim = Since, one has
Page 21 and 22: 6 Uniqueness Assume that ( u, ϕ) a
Page 23 and 24: As expected, the droplets near the
Page 25: Figure 5: Stochastic case (greater
Page 29: References [1] D. Bainov, P. Simeon

A Stochastic Model of Crystallization in an Emulsion - Laboratoire de ...

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