Structure of shock waves in magnetohydrodynamics
The mathematical question of the existence of structure for fast, slow, intermediate, switch-on, and switch-off magnetohydrodynamic shock waves is stated in terms of the following system of ordinary differential equations: ..mu.. dx/sub 1/ = x/sub 1/ - delta x/sub 2/, nu dx/sub 2/ = delta x/sub 1/ + Vx/sub 2/ + epsilon, ..mu../sub 1/ dV = 1/2 x/sub 2//sup 2/ + V - J + p(V, T), k dT = -1/2(x/sub 1//sup 2/ 2 delta x/sub 1/x/sub 2/ + Vx/sub 2//sup 2/) - epsilon x/sub 2/ - 1/2 V/sup 2/ + JV - E + e(V,T), where the symbols ..mu.., nu, ..mu../sub 1/ and kappa denote the viscosity parameters (which are always non-negative), while delta J > O, epsilon greater than or equal to 0 and E are constants. The variables V and T, corresponding to volume and temperature are naturally positive, while p(V,T) and e(V,T) correspond to pressure and internal energy, respectively. Let S be the entropy of the system and consider p as a function of V and S. It is known that the above system under the hypotheses: p/sub v/ < 0, p/sub vv/ > 0,p/sub s/ > 0 admits (at most) four rest points, say, u/sub i/ 0 less than or equal to i less than or equal to 3, ordered by increasing density. The problems considered in this dissertation are: (1) to show that, for all values of the viscosities there is an orbit running from u/sub 0/ to u/sub 1/ and likewise an orbit running from u/sub 2/ to u/sub 3/; (2) to show that, for the limiting case when epsilon = 0, and for all values of the viscosities, there is an orbit running from anti u/sub 0/ to anti u/sub 1/ and similarly, an orbit running from anti u/sub 2/ to anti u/sub 3/. Here anti u/sub i/ is the limit of u/sub i/ as epsilon tends 0.
- Research Organization:
- Michigan Univ., Ann Arbor (USA)
- OSTI ID:
- 5666886
- Resource Relation:
- Other Information: Thesis (Ph. D.)
- Country of Publication:
- United States
- Language:
- English
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