The Underlying Unity between Magnetostatics, Fluid Flow and Deformation of Solids
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The study of deformation in solids, fluid flow and magnetostatics are all shown to possess a common genesis. In particular, this seemingly disparate group of disciplines can be reduced to a study of line defects. For solids, these defects are termed dislocations, whereas in fluid flow and magnetostatics they take the form of fluid and magnetic vortices, respectively. The mathematical formulations of all three are basically identical. For example, the Burgers vector, dislocation density, stress, and elastic constants associated with solids are replaced by circulation, vorticity, density and momentum, respectively in fluids, which in turn are replaced by current, current density, permeability and induction, respectively in magnetostatics. It is then possible to use various surface distributions of such defects to represent any type of stress, velocity or magnetic field. In this way, an externally stressed solid is shown to be equivalent to flow through a pipe, which in turn is analogous to a solenoid. Still further, a stressed hole, flow about a solid body and the Meissner effect are all demonstrated to be mathematically equivalent. This equivalence is underscored in a more basic manner by expressing the elastic distortion, fluid velocity and magnetic field in terms of similar gauge transformations.
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