Questo sito utilizza cookie di terze parti per inviarti pubblicità in linea con le tue preferenze. Se vuoi saperne di più clicca QUI 
Chiudendo questo banner, scorrendo questa pagina, cliccando su un link o proseguendo la navigazione in altra maniera, acconsenti all'uso dei cookie. OK

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.