Interdisciplinary Applied Mathematics

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with x G 0 and v G R3. Here F is an external body force, and the term on


the right-hand side represents molecule collisions; it is given by


Q(f,f*)=/    /    |V • n[f(x, v*)f(x, v) — f(x, v* )f(x, v)]dn dv*.


■Jr3 JS+


(15.6)


This represents collisions of two molecules with postcollision velocities v and v*, and corresponding precollision velocities denoted in addition by primes    (see Figure    15.13).    Here,    the    integration    is taken    over    the    three


dimensional velocity space R3 and the hemisphere S + , which includes the particles moving away from each other after the collision. Also, we have the definitions


V = v — v*; v’ = v — n(n • V);    v* = v* + n(n • V),


where n is the unit vector along (v — v’).


The definition of the rest of the terms in equation (15.5) is as follows: The first term is the rate of change of the number of class v molecules in the phase space. The second term shows convection of molecules across a fluid    volume    by    molecular    velocity    v.    The    third term is    convection    of


molecules across the velocity space as a result of the external force F.


Let us now define the quadratic function


ф^) = a + b • v + c|v|2;


then the collision term satisfies the following relations:


(15.7a)


(15.7b)


/ Ф(v)Q(f,f*)dv = 0, Jr3


/(ln f) Q(f,f*) < 0.


The first one represents conservation of mass, momentum, and energy for the a, b, and c terms, respectively. In the second one, known as the Boltzmann inequality, the equal sign applies if (ln f) is collision invariant. This leads to the solution

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