Fractional time derivative on fluid flow through horizontal microchannel filled with porous material

Abstract

Fractional time derivative is considered in the description of the unsteady fluid flow through a horizontal microchannel filled with porous material. The resultant governing equations were solved using the Laplace transform technique and the method of undetermined coefficient in the Laplace domain. The Riemann-sum approximation approach was then utilized to obtain the solution in the time domain. The results were then simulated and presented as line graphs utilizing MATLAB (R2015b) to study the effects of the parameters involved in the fluid flow.

Keywords

• Fractional time derivative
• Pressure driven flow
• Darcy number
• Knudsen number

Gözenekli malzeme ile doldurulmuş yatay mikrokanaldan akışkan akışında kesirli zaman türevi

Abstract

Kesirli zaman türevi, gözenekli malzeme ile doldurulmuş yatay bir mikrokanaldan geçen kararsız akışkan akışının tanımlanmasında dikkate alınmıştır. Elde edilen yönetici denklemler Laplace dönüşümü tekniği ve Laplace alanında belirlenmemiş katsayı yöntemi kullanılarak çözülmüştür. Riemann-toplam yaklaşımı daha sonra zaman alanında çözümü elde etmek için kullanılmıştır. Sonuçlar daha sonra simüle edilmiş ve akışkan akışına dahil olan parametrelerin etkilerini incelemek için MATLAB (R2015b) kullanılarak çizgi grafikler halinde sunulmuştur.

Keywords

• Darcy numarası
• Kesirli zaman türevi
• Knudsen numarası
• Basınç tahrikli akış

References

1. Mathai, A. M. and Haubold, H. J., An introduction to fractional calculus, Nova Science Publishers, Inc., New York, 2017.
2. Guo, B., Pu, X. and Huang, F., Fractional partial differential equations and their numerical solutions, World Scientific Publishing Co. Pte. Ltd., Singapore, 2015.
3. Hamid, M., Zubair, T., Usman, M. and Haq, R. U. 2019. Numerical investigation of fractional-order unsteady natural convective radiating flow of nanofluid in a vertical channel. AIMS Mathematics, 4(5), 1416-1429.
4. Ali, F., Ahmad, Z., Arif, M., Khan, I. and Nisar, K. S. 2020. A time fractional model of generalized Couette flow of couple stress nanofluid with heat and mass transfer: Applications in engine oil. IEEE Access, 8, 146944-146966.
5. Saqib, M., Kasim, A. R. M., Mohammad, N. F., Ching, D. L. C. and Shafie, S. 2020. Application of fractional derivative without singular and local kernel to enhanced heat transfer in CNTs nanofluid over an inclined plate. Symmetry, 12, 768.
6. Saqib, M., Hanif, H., Abdeljawad, T., Khan, I., Shafie, S. and Nisar, K. S. 2020. Heat transfer in MHD flow of maxwell fluid via fractional Cattaneo-Friedrich model: A finite difference approach. Computers, Materials & Continua, 65(3), 1959-1973.
7. Atangana, A. and Bildik, N. 2013. The use of fractional order derivative to predict the ground water flow. Mathematical Problems in Engineering, Volume 2013, 543026.
8. Caputo, M. and Fabrizio, M. 2015. A new definition of fractional derivative without singular kernel. Progress Fractional Differentiation and Applications, 1(2), 73-85.

9. Sene, N. 2022. Analytical investigations of the fractional free convection flow of Brinkman type fluid described by the Caputo fractional derivative. Results in Physics, 37, 105555.
10. Arif, M., Ali, F., Sheikh, N. A., Khan, I. and Nisar, K. S. 2019. Fractional model of couple stress fluid for generalized Couette flow: A comparative analysis of Atangana–Baleanu and Caputo–Fabrizio fractional derivatives. IEEE Access, 7, 88643-88655.
11. Daud, M. B. M., Jiann, L. Y., Shafie, S. and Mahat, R. 2022. Casson fluid convective flow in an accelerated microchannel with thermal radiation using the Caputo fractional derivative. CFD Letters, 14(8), 12-19.
12. Raza, A., Al-Khaled, K., Khan, M. I., Khan, S. U., Farid, S., Haq, A. U. and Muhammad, T. 2021. Natural convection flow of radiative maxwell fluid with Newtonian heating and slip effects: Fractional derivatives simulations. Case Studies in Thermal Engineering, 28, 101501.
13. Daud, M. B. M., Jiann, L. Y., Mahat, R. and Shafie, S. 2022. Application of Caputo fractional derivatives to the convective flow of Casson fluids in a microchannel with thermal radiation. Journal of Advanced Research in Fluid Mechanics and Thermal Sciences, 93(1), 50-63.
14. Kaurangini, M. L. and Jha, B. K. 2011. Unsteady generalized Couette flow in composite microchannel. Applied Mathematics and Mechanics, 32(1), 23-32.
15. Ghadle, K. P., Firdous, K. and Anjum, K. A. 2017. Solution of FPDE in fluid mechanics by ADM, VIM and NIM. American Journal of Mathematical and Computer Modelling, 2(1), 13-23.
16. Ajibade, A. O. 2014. Dual-phase-lag and Dufour effects on unsteady double-diffusive convection flow in a vertical microchannel filled with porous materials. Proceedings of the Institution of Mechanical Engineers, Part E: Journal of Process Mechanical Engineering, 228(4), 272-285.