The finite element method for the space fractional magnetohydrodynamic flow and heat transfer on an irregular domain

Abstract

We consider the magnetohydrodynamic flow and heat transfer of a classical Newtonian fluid in a straight channel with fixed irregular cross section. A spatial fractional operator is introduced to modify the classical Fourier's law of thermal conduction, and we obtain the space fractional coupled model. With the help of the finite element method, the coupled model is solved numerically. Finally, a special numerical example is proposed to verify the stability and efficiency of the presented method.

References

• S. Aman, Q. Al-Mdallal, and I. Khan. Heat transfer and second order slip effect on MHD flow of fractional Maxwell fluid in a porous medium. J. King Saud Uni. Sci. 32.1 (2020), pp. 450–458. doi: 10.1016/j.jksus.2018.07.007
• W. Bu, Y. Tang, Y. Wu, and J. Yang. Finite difference/finite element method for two-dimensional space and time fractional Bloch–Torrey equations. J. Comput. Phys. 293 (2015), pp. 264–279. doi: 10.1016/j.jcp.2014.06.031
• X. Chi and H. Zhang. Numerical study for the unsteady space fractional magnetohydrodynamic free convective flow and heat transfer with Hall effects. App. Math. Lett. 120, 107312 (2021). doi: 10.1016/j.aml.2021.107312
• T. G. Cowling. Magnetohydrodynamics. New York: Interscience, 1957
• W. Fan, F. Liu, X. Jiang, and I. Turner. A novel unstructured mesh finite element method for solving the time-space fractional wave equation on a two-dimensional irregular convex domain. Frac. Calc. Appl. Anal. 20.2 (2017), pp. 352–383. doi: 10.1515/fca-2017-0019
• L. Feng, F Liu, I. Turner, Q. Yang, and P. Zhuang. Unstructured mesh finite difference/finite element method for the 2D time-space Riesz fractional diffusion equation on irregular convex domains. Appl. Math. Model. 59 (2018), pp. 441–463. doi: 10.1016/j.apm.2018.01.044
• L. Feng, F. Liu, I. Turner, and L. Zheng. Novel numerical analysis of multi-term time fractional viscoelastic non-Newtonian fluid models for simulating unsteady MHD Couette flow of a generalized Oldroyd-B fluid. Frac. Calc. Appl. Anal. 21.4 (2018), pp. 1073–1103. doi: 10.1515/fca-2018-0058
• C. Li and A. Chen. Numerical methods for fractional partial differential equations. Int. J. Comp. Math. 95.6–7 (2018), pp. 1048–1099. doi: 10.1080/00207160.2017.1343941
• C. Li and F. Zeng. Finite difference methods for fractional differential equations. Int. J. Bifur. Chaos 22.4, 1230014 (2012). doi: 10.1142/S0218127412300145
• Y. Liu, X. Chi, H. Xu, and X. Jiang. Fast method and convergence analysis for the magnetohydrodynamic flow and heat transfer of fractional Maxwell fluid. App. Math. Comput. 430, 127255 (2022). doi: 10.1016/j.amc.2022.127255
• H. Zhang, F. Liu, and V. Anh. Galerkin finite element approximation of symmetric space-fractional partial differential equations. App. Math. Comput. 217.6 (2010), pp. 2534–2545. doi: 10.1016/j.amc.2010.07.066
• H. Zhang, F. Zeng, X. Jiang, and G. E. Karniadakis. Convergence analysis of the time-stepping numerical methods for time-fractional nonlinear subdiffusion equations. Frac. Calc. Appl. Anal. 25.2 (2022), pp. 453–487. doi: 10.1007/s13540-022-00022-6
• M. Zhang, M. Shen, F. Liu, and H. Zhang. A new time and spatial fractional heat conduction model for Maxwell nanofluid in porous medium. Comput. Math. Appl. 78.5 (2019), pp. 1621–1636. doi: 10.1016/j.camwa.2019.01.006
• L. Zheng, Y. Liu, and X. Zhang. Slip effects on MHD flow of a generalized Oldroyd-B fluid with fractional derivative. Nonlin. Anal.: Real World Appl. 13.2 (2012), pp. 513–523. doi: 10.1016/j.nonrwa.2011.02.016