A harmonically-driven, incompressible, electrically conducting, and viscous liquid metal magnetohydrodynamic flow through a thin walled duct of rectangular cross section interacting with a uniform magnetic field traverse to its motion direction is numerically investigated. Chebyshev spectral collocation method is used to solve the Navier-Stokes equation under the Inductionless approximation for the magnetic field in the gradient formulation for the electric field. Flow is considered fully developed in the direction perpendicular to the applied magnetic field and laminar in regime. Validation of numerical calculations respect to analytical calculations is established. Flow structure and key magnetohydrodynamic features regarding eventual alternating power generation application in a rectangular channel liquid metal magnetohydrodynamic generator setup are numerically inquired. Influence of pertinent parameters such as Hartmann number, oscillatory interaction parameter and wall conductance ratio on magnetohydrodynamic flow characteristics is illustrated. Particularly, it is found that in the side layer and its vicinity the emerging flow structures/patterns depend mainly on the Hartmann number and oscillatory interaction parameter ratio, while the situation for the Hartmann layer and its vicinity is less eventful. A similar feature has been discussed in the literature for the steady liquid metal flow case and served as rationale for developing the composite core-side-layer approximation to study the magnetohydrodynamics of liquid metal flows usable in direct power generation. In this study that approximation is not considered and the analysis is performed on liquid metal oscillatory (i., e., unsteady) flows usable in alternating power generation. Conversely, in terms of prospective practical applicability the formulation developed and tested with these calculations admits the implementation of a load resistance and walls conductivity optimization. That means that besides representing a numerical study on the magnetohydrodynamics of the oscillatory flow under consideration, absent in the literature for the parametric ranges reported, the formulation presently implemented can also be applicable to study the performance of an alternating liquid metal magnetohydrodynamic generator in the rectangular channel configuration.

The performance characterization of a liquid metal magnetohydrodynamic alternate generator is numerically investigated via its electric isotropic efficiency. The model consists of a harmonically driven liquid metal oscillatory flow confined to a thin-walled closed rectangular duct interacting with a uniform magnetic field transverse to its motion and attached to a load resistance. Spectral collocation method is used to solve the properly boundary-conditioned Navier-Stokes equation under Inductionless approximation for the magnetic field with implementation of gradient formulation for the electric field. Flow is considered fully developed in the direction perpendicular to the applied uniform magnetic field (i. e., motion direction), incompressible, viscous, and laminar in regime. Currently, there are neither purely analytical or experimental results on this problem, but ours were cross-referenced with those from a one-dimensional analytical model as close as possible to it, finding reasonable correspondence. Dimensional estimates on the power production of prospective mesoscale devices having in mind household application are provided for different liquid metals as well.

‎‎‎Given a fuzzy normed space, we will introduce the notion of fuzzy near best approximation as a generalization of the notion of fuzzy best approximation. Some basic properties are characterized and also many examples for illustration are presented. Also, the hereditary properties of the fuzzy near best approximation on direct sum and tensor product of linear spaces are discussed.

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