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Mathematical analysis of second law on Casson fluid through a vertical plate with arbitrary shear stress and exponential heating

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Abstract

In the present era of research, entropy generation is one of the most important topics, which is used to control the irreversibility phenomena during heat transfer. Due to the important application in engineering, atomic reactors and cooling process in different fields, this work aims to study the second law analysis of Casson fluid. Vertical plate geometry was considered, where the plate at the boundary exhibits arbitrary wall shear stress and the fluid lies above the plate. Exponential type heating was produced at the bounding plate whereas natural convection is caused because of buoyancy force. Magnetohydrodynamic (MHD) analysis was also considered perpendicular to the plate. The usual Darcy’s law of Newtonian fluid was modified to Darcy’s law for Casson fluid. The exact analysis was performed using the Laplace transform technique to establish exact solutions for the velocity field and temperature distribution. Results are interpreted physically using various plots and discussed in detail.

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References

  1. M Saqib, I Khan and S Shafie, J. Magn. Magn. 484, 490 (2019)

    Article  ADS  Google Scholar 

  2. K Bashirnezhad et al, Int. Commun. Heat Mass Transf. 73, 114 (2016)

    Article  Google Scholar 

  3. K A Abro, A D Chandio, I A Abro and I Khan, J. Therm. Anal. Calorim. 135(4), 2197 (2019)

    Article  Google Scholar 

  4. V Rajesh, Ann. Fac. Eng. Hunedoara – Int. J. Eng. 8, 426 (2010)

    Google Scholar 

  5. W R Schowalter, AIChE J. 6(1), 24 (1960)

    Article  Google Scholar 

  6. M A Imran, S Sarwar and M Imran, Bound. Value Probl. 2016(1), 1 (2016)

    Article  Google Scholar 

  7. Y Li, S Tung, E Schneider and S Xi, Powder Technol. 196(2), 89 (2009)

    Article  Google Scholar 

  8. J H Lee, S H Lee, C Choi, S Jang and S Choi, Int. J. Micro-nano Scale Tran. 1, (2011)

  9. A Ghadimi, R Saidur and H S C Metselaar, Int. J. Heat Mass Transf. 54, 4051 (2011)

    Article  Google Scholar 

  10. G Ramesh and N K Prabhu, Nanoscale Res. Lett. 6(1), 334 (2011)

    Article  ADS  Google Scholar 

  11. K Khanafer and K Vafai, Int. J. Heat Mass Transf. 54(19), 4410 (2011)

    Article  Google Scholar 

  12. J Fan and L Wang, J. Heat Transf. 133(4), (2011)

  13. R S Vajjha and D K Das, Int. J. Heat Mass Transf. 55(15), 4063 (2012)

    Article  Google Scholar 

  14. M Sheikholeslami and R Ellahi, Int. J. Heat Mass Transf. 89, 799 (2015)

    Article  Google Scholar 

  15. W A Khan and I Pop, Int. J. Heat Mass Transf. 53(11–12), 2477 (2010)

    Article  Google Scholar 

  16. F Mebarek-Oudina, Heat Transf. Asian Res. 48(1), 135 (2019)

    Article  Google Scholar 

  17. J Raza, F Mebarek-Oudina and A J Chamkha, Multidiscip. Model. Mater. Struct. 15, 737 (2019)

    Article  Google Scholar 

  18. S Pramanik, Ain Shams Eng. J. 5(1), 205 (2014)

    Article  Google Scholar 

  19. N A Sheikh, D L C Ching, I Khan, D Kumar and K S Nisar, AEJ 59(5), 2865 (2020)

    Google Scholar 

  20. S Aman, I Khan, Z Ismail, M Z Salleh and I Tlili, Results Phys. 9, 1352 (2018)

    Article  ADS  Google Scholar 

  21. A Bejan, Entropy generation minimization: the method of thermodynamic optimization of finite-size systems and finite-time processes (CRC Press, 2013)

  22. A Bejan, Int. J. Energy Res. 26(7), 545 (2002)

    Google Scholar 

  23. A S Butt and A Ali, Eur. Phys. J. Plus 128(5), 51 (2013)

    Article  Google Scholar 

  24. A Khan, F UlKarim, I Khan, F Ali and D Khan, Results Phys. 8, 1283 (2018)

    Article  ADS  Google Scholar 

  25. N S Gibanov, M A Sheremet, H F Oztop and K Al-Salem, J. Magn. Magn. Mater. 452, 193 (2018)

    Article  ADS  Google Scholar 

  26. M Saqib, F Ali, I Khan, N A Sheikh and A Khan, Arab. J. Sci. Eng. 44(1), 531 (2019)

    Article  Google Scholar 

  27. R Ellahi, S Z Alamri, A Basit and A Majeed, J. Taibah Univ. Sci. 12(4), 476 (2018)

    Article  Google Scholar 

  28. M Maskaniyan, M Nazari, S Rashidi and O Mahian, Therm. Sci. Eng. Prog. 6, 186 (2018)

    Article  Google Scholar 

  29. M I Afridi, M Qasim and O D Makinde, J. Heat Transf. 141(2), 1 (2019)

    Article  Google Scholar 

  30. W A Azhar, D Vieru and C Fetecau, Heat Transf. Res. 49(15), 1507 (2018)

    Article  Google Scholar 

  31. A Z Şahin, Int. Commun. Heat Mass Transf.  19(3), 349 (1992)

    Article  Google Scholar 

  32. A L Lare, Int. J. Heat Mass Transf. 2(2), 63 (2015)

    Google Scholar 

  33. Y T Zuo, Therm. Sci. 25, 2405 (2021)

    Article  Google Scholar 

  34. Y-T Zuo and H-J Liu, Facta. Univ. Ser. Mech. Eng. 19, 271 (2021)

    Google Scholar 

  35. Y Mei, Y Q Liu and J H He, Therm. Sci. 25, 4817 (2021)

    Article  Google Scholar 

  36. J H He, G M Moatimid and D R Mostapha, J. Electroanal. Chem. 895, 115 (2021)

    Article  Google Scholar 

  37. J H Tian and K Jiang, Numer. Heat Transf. A 72(2), 141 (2017)

    Article  ADS  Google Scholar 

  38. A Kumar, J V Ramana Reddy, V Sugunamma and N Sandeep, Multidiscip. Model. Mater. Struct. 14, 999 (2018)

  39. B Mahanthesh, N S Shashikumar, B J Gireesha and I L Animasaun, J. Comput. Des. Eng. 6(4), 551 (2019)

    Google Scholar 

  40. M Saqib, I Khan, Y M Chu, A Qushairi, S Shafie and K S Nisar, Appl. Sci. 10(11), 3886 (2020)

    Article  Google Scholar 

  41. M Saqib, A R M Kasim, N F Mohammad, D L C Ching and S Shafie, Symmetry 12(5), 768 (2020)

    Article  Google Scholar 

  42. M Saqib, H Hanif, T Abdeljawad, I Khan, S Shafie and K S Nisar, Comput. Mater. Contin, 65(3), 1959 (2020)

    Google Scholar 

  43. M Saqib, I Khan and S Shafie, J. Magn. Magn. Mater. 484, 490 (2019)

    Article  ADS  Google Scholar 

  44. M Saqib, S Shafie, I Khan, Y M Chu and K S Nisar, Symmetry 12(4), 663 (2020)

    Article  Google Scholar 

  45. A M Megahed, Eur. Phys. J. Plus 130(4), 1 (2015)

    Article  Google Scholar 

  46. M Atlas, S Hussain and M Sagheer, Eur. Phys. J. Plus 134(1), 33 (2019)

    Article  Google Scholar 

  47. E K Ghiasi and R Saleh, Pramana – J. Phys. 92, 12 (2019)

    Article  ADS  Google Scholar 

  48. A Hiremath, H Basha, B Kethireddy, G J Reddy and N V Narayanan, Pramana – J. Phys. 93, 20 (2019)

    Article  ADS  Google Scholar 

  49. M Qasim and S Noreen, Eur. Phys. J. Plus 129(1), 1 (2014)

    Article  Google Scholar 

  50. M Saleem, M N Tufail and Q A Chaudhry, Pramana – J. Phys. 95, 28 (2021)

    Article  ADS  Google Scholar 

  51. A Hussanan, M Z Salleh, I Khan and R M Tahar, J. Nanofluids 6(4), 784 (2017)

    Article  Google Scholar 

  52. J H Merkin, J. Eng. Math. 19(3), 189 (1985)

    Article  Google Scholar 

Download references

Acknowledgements

The authors acknowledge the financial support provided by the Center of Excellence in Theoretical and Computational Science (TaCS-CoE), KMUTT. This research project is supported by Thailand Science Research and Innovation (TSRI) Basic Research Fund: Fiscal year 2022 under project number FRB650048/0164.

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Authors and Affiliations

  1. Fixed Point Research Laboratory, Fixed Point Theory and Applications Research Group, Center of Excellence in Theoretical and Computational Science (TaCS-CoE), Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha Uthit Rd., Bang Mod, Thung Khru, Bangkok,  10140, Thailand

    Dolat Khan, Poom Kumam, Wiboonsak Watthayu & Muhammad Arif

  2. Center of Excellence in Theoretical and Computational Science (TaCS-CoE), Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha Uthit Rd., Bang Mod, Thung Khru, Bangkok,  10140, Thailand

    Poom Kumam

  3. Department of Medical Research, China Medical University Hospital, China Medical University, Taichung,  40402, Taiwan

    Poom Kumam

  4. Institute of Computer Sciences and Information Technology, The University of Agriculture, Peshawar, Pakistan

    Arshad Khan

  5. Department of Mathematics, College of Science Al-Zulfi, Majmaah University, Al-Majmaah,  11952, Saudi Arabia

    Ilyas Khan

Authors
  1. Dolat Khan
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  2. Poom Kumam
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  3. Wiboonsak Watthayu
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  4. Arshad Khan
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  5. Ilyas Khan
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  6. Muhammad Arif
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Corresponding author

Correspondence to Poom Kumam.

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Khan, D., Kumam, P., Watthayu, W. et al. Mathematical analysis of second law on Casson fluid through a vertical plate with arbitrary shear stress and exponential heating. Pramana - J Phys 96, 106 (2022). https://doi.org/10.1007/s12043-022-02343-w

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  • DOI: https://doi.org/10.1007/s12043-022-02343-w

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