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Discrete models for complex fluids

Scheme numerice pentru modele Lattice Boltzmann

Mostra/Nascondi contenuto.
Solving this integral is practically imposible because of the large number of degrees of freedom. According to the ensemble method [1, 2, 4] introduced by Gibbs, we may consider a large number η of systems that are macroscopically equivalent to the physical system we study. Each system of the ensemble can be represented by a single point in the phase space. If η →∞, the density of these points becomes very large and their distribution is described by a distribution function fN ({q}, {p}, t) [1, 3, 5]. This function is defined such that: fN({q}, {p}, t)dq1dp1 . . . dqNdpN (1.4) expresses the probability to find the generalized coordinates qi, pi of the particle numbered i (i = 1, . . . , N) within the intervals (qi, qi +dqi) and (pi, pi+dpi) of the phase space. According to the ergodic hypothesis [1, 4], we have: Mobs(t) = ∫ M({q}, {p})fN({q}, {p}, t)dq1dp1 . . . dqNdpN (1.5) Following the ergodic hypothesis, we may get any physical quantity as an en- semble - averaged value instead of a time average. Application of the ergodic hypothesis requires detailed study of the properties of the distribution function, which determines the evolution of the Gibbs ensemble in the phase space [1, 2, 4]. If we consider an infinitesimal variation dfN of the function fN around the point ({q}, {p}) in the phase space at time t, we have: dfN = ∂fN ∂t dt+ N∑ i=1 ∂fN ∂qi dqi + N∑ i=1 ∂fN ∂pi dpi (1.6) According to Liouville theorem [4, 5], the probability density fN remains constant along the phase space trajectory (dfN/dt = 0): dfN dt = ∂fN ∂t + N∑ i=1 [ ∂fN ∂qi q˙i + ∂fN ∂pi p˙i ] = 0 (1.7) In 1872, when Ludwig Boltzmann (1844-1906) introduced his equation [1, 3, 4, 5, 6], he started from the following hypotheses concerning the evolution of the single particle distribution function (N = 1): 1. The only possible interparticle interactions are binary collisions; 2. Initial velocities of the particles involved in the collision are not correlated (hypothesis of the molecular chaos); 3. External forces have no effect on the dynamics of local interactions. Following these hypotheses, Boltzmann considered the collision operatorQ(f1, f2) [1, 3, 4, 5, 6]: Q(f1, f2) = ∫ d~v2dΩ|~v1 − ~v2|σ(Ω)[f ′1f ′2 − f1f2] (1.8) 2

International thesis/dissertation

Facoltà: Fizică

Autore: Artur Cristea Contatta »

Composta da 116 pagine.


Questa tesi ha raggiunto 6701 click dal 19/04/2007.


Consultata integralmente una volta.

Disponibile in PDF, la consultazione è esclusivamente in formato digitale.