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A physical model of a vacuum arc is developed, which is used to construct a 1-D model and a 2-D two-fluid mathematical model. The 1-D model is based on the principle of compensation of radial magnetic forces. This model is used to derive analytical expressions for the critical values of current Icr and of the external magnetic field Bcr, which define the range of steady passage of current. The model describes both supersonic and subsonic modes of plasma flow. The possibilities of coexistence of these modes are discussed. The 2-D magnetohydrodynamic model is based on the method of trajectories, where a set of partial equations is reduced to a set of ordinary differential equations that are written for derivatives along the lines of the total current. The 2-D model is used to calculate supersonic flows of current for different geometries of the discharge gap and different magnitudes of the external magnetic field B0. Two-dimensional distributions of discharge parameters are obtained, including those of magnetic field components in view of the magnetic field of the Hall current. The distributions of current density are calculated, as well as the dependences Icr(B0) which tend to saturate in high values of B0.