1549301742-The_Theory_of_Difference_Schemes__Samarskii

526 Difference Methods for Solving Nonlinear Equations

involves the system of gas dynamics equations in Lagrangian variables ( s, t):

(1)

av op

• (the law of the impuls conservation),
  at OS

(2)

dx

• dt == v'

1 ox

(3) (the law of thr mass conservation),

p as

8 v^2 8 ow

(4) at (E + 2) = -as (pv) - as (the law of the energy conservation),

(5) p = p(p, T), E = E(p, T) (the state equation),

where w is the heat flow.

A combination of the second and third equations we have tnentioned

above gives

(6)

by observing that equation (2) can be excluded from the governing system

through the possible separate integration.

(7)

The expression for the heat flow

oT

w = -x(p, T) p as ,

where x = x(p, T) is a coefficient of heat conductivity, is intended to com-

plete the above system of equations. It should be noted here that x can

usua.lly be expresse,cl through the power function of T and p.

The functions p(p, T), E(p, T), x(p, T) for this system of equations

must be given.

For example, the state equations of the ideal gas are of the form p =

RpT and E = E(T). The readers can encounter E = c 0 T, where Rand c 0

are constants, R/c 0 = / - 1, /is constant, so that

(8) E = pj((! - l)p).

In this context, two limiting cases of interest are as follows: