(a) Show that the time rate of increase of momentum in an infinitesimal volume element d³r = dx dy dz inside a gas of number density n, as a result of particles of mass m entering d³r with average velocity u, is given by –∇ · (nmuu) d³r.

(b) If the infinitesimal volume element d³r moves with the average particle velocity u, show that, because of the work done by the kinetic pressure dyad P, the particle energy inside d³r increases at a time rate given by –∇ · (u · Ρ)d³r.

(c) Verify, by expansion, that

where n̂ denotes an outward unit vector, normal to the surface bounding the volume element.

Transcribed Image Text:

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