Abstract
The proposal of this paper is to study the effect of magnetic field in stabilizing the hydrodynamic flow and preventing the occurrence of back flow in the Prandtl–Hartmann boundary layer. When both of the initial tangential velocity and upstream velocity are strictly increasing in the normal variable of the boundary, and the normal component of the magnetic field at the boundary is fairly strong, we obtain the global existence of a weak solution to the mixed Prandtl–Hartmann boundary layer problem, even for certain adverse pressure gradient, that may lead to the occurrence of a back-flow point for the classical Prandtl equation. This indicates that the magnetic field has certain stabilization effect on the hydrodynamic flow. On the other hand, under the monotonicity condition, we obtain that a first back-flow point, when the boundary layer evolves in time, should appear on the boundary if it occurs, notably the pressure gradient of the outer flow is not necessarily adverse. Moreover, when the adverse pressure gradient of the outer flow dominates the orthogonal magnetic field, and the initial velocity satisfies certain growth condition, we obtain the existence of a back-flow point of the Prandtl–Hartmann boundary layer.
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Acknowledgements
The authors would like to express their gratitude to Professor Ya-Guang Wang for the valuable discussion on this topic and instructive suggestions on the presentation, and to the referee for the valuable suggestions on improving this manuscript. This research was partially supported by National Natural Science Foundation of China (NNSFC) under Grant No. 11631008.
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Gong, S., Wang, X. On a Global Weak Solution and Back Flow of the Mixed Prandtl–Hartmann Boundary Layer Problem. J. Math. Fluid Mech. 23, 11 (2021). https://doi.org/10.1007/s00021-020-00530-6
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DOI: https://doi.org/10.1007/s00021-020-00530-6