• 5

SHOCKS

At a shock front propagating in a magnetized fluid at an angle 0 with respect to the magnetic induction B, the jump conditions are 13,14

(1)   pU = pU = q-

(2)   pU2 + p + B2/2p = ptJ2+p + B2/2p-

(3)   pUV - BllB±/p = pUV - B||B±/M;

(4)   B,| = %

(5)   UB± - VB\\ = UB± - VBy;

(6)   ±(U2 + V2)+w + (UB2 - VB{lB±)/ppU

= ±(U2 + V2) +w + (UB2 — V B\\B_\_) / ppU.

Here U and V are components of the fluid velocity normal and tangential to the front in the shock frame; p = 1/v is the mass density; p is the pressure; Bl = B sin By = B cos 9; p is the magnetic permeability (p = 4tt in cgs units); and the specific enthalpy is w = e + pv, where the specific internal energy e satisfies de = Tds — pdv in terms of the temperature T and the specific entropy s. Quantities in the region behind (downstream from) the front are distinguished by a bar. If B = 0, then15

(7)   U — U = [{p — p)(v — v)}1/2\

(8)   (p - p)(v - v)~1 = q2\

(9)   w — w = ^(p — p)(v + v);

(10)   e — e = \ (p + p)(v — v).

In what follows we assume that the fluid is a perfect gas with adiabatic index 7 = 1 + 2/n, where n is the number of degrees of freedom. Then p = pRT/m, where R is the universal gas constant and m is the molar weight; the sound speed is given by Cs2 = (dp/dp)s = 7pv\ and w — ^e — 'ypv/(7 — 1). For a general oblique shock in a perfect gas the quantity X = r-1 (U/Va)2 satisfies14

(11) (X-(3/a)(X-cos2 6)2 = X sin2 0 j[l + (r - l)/2a] X - cos2 6>|, where r = p/p, a = \ [7 + 1 - (7 - l)r], and (3 = CS2/VA2 = 4tv^P/B2.

The density ratio is bounded by

(12)   1 < r < (7 + l)/(7- I)-

If the shock is normal to B (i.e., if 0 = tt/2), then

(13)   U2 = (r/a) [Cs2 + VA2 [1 + (1 - 7/2)(r - 1)]};

(14)   U/U = B/B = r;

(15) V = V;

(16) p = p + (l- r~1)pU2 + (1 - r2)B2/2p.

If 6 = 0, there are two possibilities: switch-on shocks, which require (3 < 1 and for which

(17)  U2 = rVA2-

(18)  U = Va2/U-

(19) B2 = 2B2(r - l)(a - /3);

(20) V = UB±/Bll-,

(21) p = p + pU2( 1 - a + (3){1 - r-1),

and acoustic (hydrodynamic) shocks, for which

(22)  U2 = (r/a)Cs2;

(23)  U = U/r;

(24) V = B± = 0;

(25) p = p + pU2(l - r-1).

For acoustic shocks the specific volume and pressure are related by

(26) v/v = [(7 + 1)P + (7 - l)p] / [(7 - 1)P + (7 + l)p]- In terms of the upstream Mach number M = U/Cs,

(27) p/p = v/v = U/U = (7 + l)M2/[(7 - 1)M2 + 2];

(28) p/p = (27M2 -7+ l)/(7 + 1);

(29) T/T = [(7 - 1)M2 + 2](27M2 -7 + l)/(7 + 1)2M2;

(30)  M2 = [(7 - 1)M2 + 2]/[27M2 -7 + I]. The entropy change across the shock is

(31) As = s - s = cv ln[(p/p)(p/p)7],

where cv =                 — l)m is the specific heat at constant volume; here R is the

gas constant. In the weak-shock limit (M —► 1),

zQOx A                27(7-1) 2 3 167^                                                          3

(32) As ^ c,                           (M -1) ^                                    (M - 1) .

3(7 + 1)                                                                   3(7 + 1 )m

The radius at time t of a strong spherical blast wave resulting from the explo­sive release of energy E in a medium with uniform density p is

(33) RS = Co(Et2/p)1/f5,

where Co is a constant depending on 7. For 7 = 7/5, Co = 1.033.

Авторы: 1379 А Б В Г Д Е З И Й К Л М Н О П Р С Т У Ф Х Ц Ч Ш Щ Э Ю Я

Книги: 1908 А Б В Г Д Е З И Й К Л М Н О П Р С Т У Ф Х Ц Ч Ш Щ Э Ю Я