Exact Solution for an Unsteady Isothermal Flow Behind a Cylindrical Shock Wave in a Rotating Perfect Gas with an Axial M
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Journal of Engineering Physics and Thermophysics, Vol. 93, No. 6, November, 2020
EXACT SOLUTION FOR AN UNSTEADY ISOTHERMAL FLOW BEHIND A CYLINDRICAL SHOCK WAVE IN A ROTATING PERFECT GAS WITH AN AXIAL MAGNETIC FIELD AND VARIABLE DENSITY G. Nath
UDC 541.12
An exact solution of the problem on the propagation of a cylindrical shock wave in a rotating perfect gas with an axial magnetic field in the case of an isothermal flow is obtained. The initial density, magnetic field strength, and the initial angular velocity in the ambient medium are assumed to vary according to the power law. An exact similarity solution obtained by the McVittie method for an isothermal flow in a rotating gas is first reported. Similarity transformations are used to transform a system of partial differential equations into a system of ordinary differential equations, and then the product solution of McVittie is used to obtain the exact solution. The effects of the values of the gas specific heat ratio, rotational parameter, and of the strength of the initial magnetic field are discussed. It is shown that the shock velocity increases and the shock strength decreases with increase in the values of these parameters. The effect of variation in the value of the initial density index is also studied. The obtained solutions show that the radial fluid velocity, density, pressure, and the magnetic field strength tend to zero as the axis of symmetry is approached. Keywords: shock waves, exact solution, rotating medium, perfect gas, axial magnetic field. Introduction. The self-similar formulations of the problem on adiabatic motion in the context of the nonrotating gas model of stars were studied in [1–4]. In the study of astrophysical phenomena, the problem of magnetogasdynamic shock waves in a rotating interplanetary atmosphere is of special significance. The experimental studies and astrophysical observations show that the outer atmosphere of planets and stars rotates due to the rotation of the planets and stars themselves. Macroscopic motion with a supersonic speed occurs in the interplanetary atmosphere, so that shock waves are generated there. Thus, the rotation of planets or stars considerably affects the process in their outer layers; therefore the questions related to explosions in a rotating gas atmosphere are of specific astrophysical importance. The problem of the shock wave propagation in a rotating atmosphere was considered, using similarity or nonsimilarity methods, by many authors, e.g., in [5–9]. Magnetic fields permeate the universe and play a crucial role in a number of astrophysical situations, probably affecting astrophysical plasma (see [10, 11]). An important role is played by magnetic fields and radiation flux in momentum and energy transport which can rapidly release energy in flares. A detailed review in the field of magnetogasdynamic flows can be seen in paper of Shang [12]. Lock and Mestel [13] analyzed the annular self-similar solutions in ideal magnetogasdynamics by casting the ideal magnetogasdynamic equations to a three-d
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