Theoretical and Numerical Stability Analysis of the Liquid Metal Pinch Using the Shallow Water Approximation

The pinch instability for a cylindrical jet of liquid metal passed through by an axial electrical current is investigated. Besides the pinch effect originating from surface tension, the Lorentz force, created by the axial current density and the corresponding azimuthal magnetic field, causes an electromagnetic pinch effect. This effect has drawn attention in electrical engineering, because it can be used in the construction of liquid metal current limit- ers with self-healing properties. In this paper a simple model is derived using the shallow water approximation： the equations describing the full system are reduced to two one-dimensional evolution equations for the axial velocity and the radius of the jet. A stability analysis for this reduced system is carried out yielding critical current density and the growth rate for the instability. To investigate the nonlinear behaviour of the pinch instability for finite perturbations simulations, the shallow water model are performed.