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Here, Q and L are the source and loss densities, respectively (number of particles created or lost in a unit volume during unit time). The physical content of this equation is very simple: a change of the number of particles in a unit volume (∂n/∂t) is due to flow of particles (nv), creation of particles (Q), and destruction of particles (L). Considering some volume V bounded by a surface S and containing a number of particles N, the net number of particles flowing into V per unit time is

dN dt

flow

=

S

n v · dS,

(27)

using the usual convention that the normal to the surface S is directed outward from V . Using Gauss’ theorem, this may be written as

dN dt

flow

=

V

∇ · (n v) dV.

(28)

The change in N due to creation and annihilation of particles must be

dN Q dV dt = V source

(29)

and

dN dt

loss

=

V

L dV.

(30)

The total change in N may clearly be written in two ways:

dN dt

=

dN dt

flow

+

dN dt

dN dt

=

d dt

V

n dV =

From a comparison of these two expressions, equation (26) follows directly.

loss

dV.

dN dt

V

∂n ∂t

source

+

(31)

(32)

3.5.5 Equation of motion

In the fluid picture, the equations of motion for the ion gas and the electron gas are “Newton’s second law per unit volume”. The forces acting on a volume of gas of charged particles are the

electromagnetic forces, the pressure, and possible other forces like gravitation. Thus,

n α m α d v α d t = n α q α ( E + v α × B ) p α + o t h e r f o r c e d e n s i t i e s .

(33)

There is one such equation of motion for each particle species, e.g. for protons and electrons in the two-component plasma. These two equations are known as the two-fluid equations of motion.

The fluid equations of motion (33) looks very similar to the equation of motion (2) for a single particle. Basically, the differences are:

  • Equation (2) considers forces [SI unit: N], equation (33) considers forces per unit volume [SI

unit: N/m3].

  • Equation (2) considers the exact velocity vk of some particular particle k, while equation (33) considers the average velocity vα of many particles of the species α.

  • The first-order statistical effect of the deviation of the individual particle velocities from the average velocity is included in (33) as the pressure term.

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