An integral method for the boundary layer of MHD non-Newtonian power-law fluid in the entrance region of channels

Abstract

This paper presents the analytical solutions to laminar flow of MHD Newtonian and non-Newtonian power-law fluids in the entrance region of channels. The boundary layer growth and velocity profile of developing flow in a two-dimensional channel, under the influence of a uniform magnetic field, are investigated. The direction of the magnetic field is assumed perpendicular to the flow. For each case, a novel and useful non-dimensional correlation for computing the magnetic entrance length is proposed, using the integral equations method. In addition, the effect of different parameters on the magnetic entrance length, boundary layer thickness and thus core velocity and pressure loss are studied. It was found that with the increase of the Hartmann number, the entrance length declined. Furthermore, the entrance length decreases while the powerlaw index and magnetic interaction parameter increase. As well as, the results have shown that the augmentation of the magnetic interaction parameter leads to greater pressure drop in comparison with the hydrodynamic flow. Keywords Entrance length Á Integral method Á MHD channel Á Power-law fluid 1 Introduction Magnetohydrodynamics (MHD) apparatus such as channels in reactors, MHD flow meters, MHD generators, pumps, etc., have a significant role in engineering applications. The distance along the channel length, where the centerline velocity reaches 99.9% of its final magnetic fully Technical Editor: Cezar Negrao.

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

“…It is observed that there is no variation in the results for 100 × 80 grid. 0.0054 0.0045 Present study (second order) 0.026 0.0156 Present study (third order) 0.042 0.021 Taheri et al [44] (sinusoidal) 0.04 0.0208 Schlichting [47] 0.04 -Hwang et al [35] 0.0422 0.0188 Roidt and Cess [37] 0.0454 0.0167 Bodoia and Osterle [48] 0.044 -Eventually, applying a surface fitted on the datasets, the correlation of the magnetic development length as a function of the Reynolds and Hartmann number is obtained as:…”

“…For analytical solution, according to the boundary layer approximations, the governing equations including continuity and momentum are defined as [44][45][46]:…”

“…As in the potential core, the viscous term vanishes, also v equals to zero, according to Eq. (1b), the pressure gradient can be related to the core velocity as [44]:…”

“…the integral form of the momentum equation including the magnetic term (Lorentz body force) can be written as [44]:…”

“…Indeed, in the previously published work of the authors' [44], the Newtonian and power-law non-Newtonian MHD channel flow in the entrance region was investigated. An integral momentum solution was conducted to solve the governing equations.…”