By Rey Pastor, Julio; Pi Calleja, Pedro; Trejo, Cesar A.
Read or Download Analisis matematico. Vol. III: Analisis funcional y aplicaciones. (Spanish) Mathematical analysis. Vol. III: Functional analysis and applications PDF
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Additional info for Analisis matematico. Vol. III: Analisis funcional y aplicaciones. (Spanish) Mathematical analysis. Vol. III: Functional analysis and applications
We denote the reference configuration of the body by χ0 (B). Now we have sufficient concepts to formulate the statement of the conservation of mass of B: d m(χ(B, t)) 0. (13) dt That is, the mass of the body B is constant in time. This idea is intuitive. Here it is a postulate, that is, something we assume. Other postulates are possible, and indeed are appropriate for single constituents of the reacting media. By (12), d dt χ(B,t) ρ dv 0. (14) It is useful to render this global equation into local form.
Thus, v21 · k −v21 · k, and the components perpendicular to k are equal. Therefore, v21 − v21 2(v21 · k)k −2(v21 · k)k . (14) Then v 1 − v1 2(v21 · k)k −2(v21 · k)k , (15) 50 4. 1. Collision geometry. and v2 − v 2 −2(v21 · k)k 2(v21 · k)k . (16) Consider particle 1 to be within dx of the point x, and to have velocity within dv1 of some velocity v1 . Then, in order for particle 2, having velocity within dv2 of v2 , to collide with it in time interval dt, so that the collision point is within the element dk of the solid angle centered about the direction k, particle 2 must lie in a cylindrical volume that is the projection of the length v21 dt on the area (2a)2 dk.
Since number, momentum, and energy are the physical parameters that are conserved, we expect that α0 + α1 · v + α2 v 2 , ln f (32) where α0 , α1 , and α2 are constants. Thus, f0 exp(α1 · v + α2 v 2 ) , f where f0 exp(α0 ). By symmetry, α1 energy2 uRe is equal to 1 2n uRe (33) 0. If we note that the particle kinetic v 2 f dv , (34) then we see that f fM (v) n 3 π uRe 3/2 exp −v 2 3uRe . (35) This distribution is called the Maxwellian. 4. Slight Disequilibrium We seek a solution to Boltzmann’s equation that is close to the Maxwellian distribution.