A number of analytical models exist for both electro-hydrodynamics (EHD), the study of fluid flows containing electric charges under the influence of an electric field and negligible magnetic field, and magneto-hydrodynamics (MHD), the study of fluid flows containing no free electric charges that are under the influence of a magnetic field and no electric field. At the present, there are no practical yet consistent models for the combined electro-magneto-hydrodynamic (EMHD) effects which most often occur in actual situations. This work represents an attempt to develop such a fully consistent analytical model for multi-dimensional, steady and unsteady, compressible and incompressible flows of electrically conducting fluids under the simultaneous or separate influence of externally applied and internally generated steady or unsteady electric and magnetic fields. The approach is based on the fundamental laws of continuum mechanics and thermodynamics with all assumptions clearly stated and consistently applied. The resulting second order EMHD model allows for non-linear polarization and magnetization of the medium. The new model is therefore superior to the existing EMHD models and represents a tractable set of equations suitable for detailed numerical discretization and integration.

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