A consistent description of electro-magneto-hydrodynamic flows in narrow slits

A consistent framework to study the flow of a Newtonian fluid in a narrow charged slit, in the presence of imposed external electric and magnetic fields is presented here. We separately consider cases of externally imposed fields and fields induced as a result of purely pressure driven flows. It is established that the 3-dimensional nature of the Lorentz forces acting on the fluid inevitably leads to a multi-dimensional flow field, wherein pressure gradient and electric fields are induced along the width of the confinement to conserve mass and current flux. These induced fields and gradients are strongly coupled to each other and also depend on the surface charge density as well as the applied magnetic field strength. We argue that such complex multi-dimensionality, intrinsic to the problem, has not been taken into account by most of the previous studies. Our analysis depicts that consideration of the 3-dimensional Lorentz forces reveals a number of non-trivial features, particularly in relation to the induced electric fields, while at the same time it leads to a general conservative and consistent framework for electro-magneto-hydrodynamic flows. One of the central outcomes of our formulation is the prediction of a magnetic field driven anisotropy in the induced streaming potential despite the absence of any non-uniformity in the surface properties, hitherto not reported in the existing studies.