Analytic model of a resistive magnetohydrodynamic shock without Hall effect

Abstract

An analytic model of a stationary hypersonic magnetohydrodynamic (MHD) shock with an externally applied magnetic field is proposed. Basically, original jump conditions at a plane oblique shock, analogous to the Rankine–Hugoniot formulae, with a moderately resistive air plasma downstream are derived. Viscous, thermal and Hall effects are neglected, but the plasma dissociation behind the shock causing a jump of isentropic exponent is also a major input of the model. Then, a shock-fitting procedure with ambient atmospheric conditions is worked out by the coupling of these MHD jumps with thermodynamic correlations and an electric conductivity model. For an application to atmospheric entry problems, the flow behind an axisymmetric blunt-body shock is modelled with a stream function satisfying these MHD jump conditions as boundary conditions. An important feature put into evidence is a similarity rule involving the hypersonic parameter$M_{1}\cos \unicode[STIX]{x1D712}_{1}$, which shows an aerodynamic correspondence between the upstream Mach number$M_{1}$and the velocity angle$\unicode[STIX]{x1D712}_{1}$. It also emerges that curvature effects become important past$30^{\circ }$and the assumption of a spherical shock also becomes untenable past$50^{\circ }$; therefore, we limit the model of shock thickness used in the MHD fitting to$\unicode[STIX]{x1D712}_{1}<50^{\circ }$.

Analytic model of a plane resistive magnetohydrodynamic shock with Hall effect

“…More recently, Hall MHD has become a part of astrophysical magnetohydrodynamics in its own [22], because the Hall effect is likely to develop in most of magnetized celestial bodies, and to impact MHD processes, including turbulence [27], dynamo [29], star formation [7] and magnetic reconnection [15]. The main difference between both situations is that, as pointed out in our previous work by considering the magnetic Reynolds number [3], the magnetic field does not evolve in our problem, while it is strongly flow-dependent in geophysical and astrophysical situations.…”

“…( ) ( ) In addition, following other authors [36][39], in order to be consistent with our previous approach [3], we assume a power-law relationship between the ordinary electrical conductivity σ e and the temperature T:…”

“…In a previous work, we worked out an analytic model of resistive MHD shock without Hall effect, in order to focus on the jumps due to the finite conductivity of the plasma, and to the real gas effects caused by molecular dissociations [3]. Here, we shall add the contribution of the Hall effect for a plane MHD oblique shock, and develop in detail the basic formalism, about which we recently presented results without mathematical proofs [4].…”

“…Then, the plane-shock fitting procedure with Hall effect will be described and illustrated in Sect. 3. In particular, two geometric configurations will be chosen, in order to display the influence of the Hall effect on the shock properties, as the incident Mach number and the magnetic field strength are varied.…”

Numerical solution through mathematical modelling of unsteady MHD flow past a semi-infinite vertical moving plate with chemical reaction and radiation

“…The effects of a magnetic field on the three-dimensional flow of a nanofluid having a suspension of ferrous nanoparticles within the framework of a non-uniformly thickened sheet in a slip flow regime has been analysed [16]. An analytical model of a stationary hypersonic MHD shock with an externally applied magnetic field has also been proposed [17]. Decaying homogeneous and isotropic MHD turbulence has been investigated numerically at large Reynolds numbers [18].…”