The influence of magnetic field on the fluid flow in the entrance region of channels: analytical/numerical solution

Abstract

The present paper investigates the steady laminar development flow of a Newtonian incompressible electrically conducting fluid in a plane channel subjected to a uniform transverse external magnetic field. In order to solve the problem, the analytical integral momentum method and numerical finite volume method (FVM) are conducted. In integral momentum solution, the Pohlhausen's fourth-degree velocity profile is assumed. Applying the boundary conditions leads to the appearance of a new dimensionless parameter named auxiliary parameter (AP). It is shown that the value of AP depends on the magnitude of magnetic intensity and the pressure gradient. For each specified value of AP, distinct velocity distribution and as a result, a correlation for computing the magnetic development length is obtained. Besides, the FVM is applied to solve the same problem and the correlation for estimating the magnetic development length is obtained. It can be concluded that for a particular engineering application of magnetohydrodynamics channel flows, the associated value of AP can be determined by the numerical investigation. The results of numerical solution are compared with the analytical results and it can be concluded that the results for AP = − 6 are in agreement with the numerical results. Further, the effect of Hartmann and Reynolds number on the magnetic development length and Lorentz force are illustrated. It was shown that as the Hartmann number increases, the value of development length and Lorentz force are reduced. KeywordsChannel · Integral momentum method · Magnetic development length · Magnetohydrodynamics · Numerical method List of symbols a Half width of channel (m) AP Auxiliary parameter B 0 External magnetic field (T) Ha Hartmann number P Pressure (Pa) Re Reynolds number u x-Component of the velocity (m/s) U Inlet velocity (m/s) u c Potential core velocity (m/s) v y-Component of the velocity (m/s) x Coordinate in the direction of flow (m) x em /D Dimensionless MHD development length y Coordinate in the direction normal to flow (m) ρ Density (kg/m 3 ) σ Electrical conductivity (Ω m) −1 δ Boundary layer thickness (m) υ Kinematic viscosity (m/s 2 ) η Arbitrary variable

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