Effect of non-uniform heating and viscous dissipation on natural convective flow of Casson fluid over a vertical cone

Abstract

The non-Newtonian fluid flows find its applications in the production of paints, pharmaceutical products, synthetic lubricant and biological fluid. Because of the significance of this aspect, the present paper describes the consequence of non-uniform surface heating on the natural convection of the Casson fluid past a vertical cone with a viscous dissipation effect. Firstly, the governing partial differential equations are derived in the dimensionless form. After that, the numerical solution is obtained using the Crank–Nicolson technique of the finite difference method. The obtained numerical solutions for the temperature profiles, velocity profiles, local Nusselt number and local skin friction are displayed by employing the graphs in order to judge the impact of the governing physical parameters of the model such as the Casson fluid parameter, Grashof number, Eckert number and semi-vertical angle of the cone. The major result is found that the velocity and temperature profiles are more at the center of the lateral side of the cone.

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Abbreviations

\(C_{p}\) :

Specific heat at constant pressure \(\left[ {{\text{JM}}^{ - 1} {\text{K}}^{ - 1} } \right]\)

\(Ec\) :

Eckert number

\(Gr\) :

Grashof number

\(Nu\) :

Local Nusselt number at surface of the cone

\(Pr\) :

Prandtl number

\(t^{\prime}\) :

Time \(\left[ {\text{T}} \right]\)

\(t\) :

Dimensionless time

\(T^{\prime}\) :

Temperature \(\left[ {\text{K}} \right]\)

\(u^{\prime}, v^{\prime}\) :

Velocity components in \(x^{\prime} {\text{and }}y^{\prime}\) directions, respectively \(\left[ {{\text{LT}}^{ - 1} } \right]\)

\(u,v\) :

Dimensionless velocity components in \(x {\text{and }}y\) directions, respectively

\(x^{\prime}, y^{\prime}\) :

Co-ordinate axes \(\left[ {\text{L}} \right]\)

\(x, y\) :

Dimensionless co-ordinate axes

\(\beta\) :

Casson parameter

\(\rho\) :

Density of fluid \(\left[ {{\text{ML}}^{ - 3} } \right]\)

\(\theta\) :

Dimensionless temperature

\(\mu\) :

Dynamic viscosity \(\left[ {{\text{NTL}}^{ - 2} } \right]\)

\(g\) :

Gravitational acceleration \(\left[ {{\text{LT}}^{ - 2} } \right]\)

\(\nu\) :

Kinematic viscosity of fluid \(\left[ {{\text{L}}^{2} {\text{T}}^{ - 1} } \right]\)

\(\tau\) :

Local skin-friction

\(\beta_{T}\) :

Thermal expansion coefficient \(\left[ {{\text{K}}^{ - 1} } \right]\)

\(\kappa\) :

Thermal conductivity of fluid \(\left[ {{\text{WL}}^{ - 1} {\text{K}}^{ - 1} } \right]\)

References

  1. [1]

    N Casson Rheology of Disperse Systems (eds) A Mill (Pergamon Press, Oxford) p 84 (1959)

  2. [2]

    W P Walwander, T Y Chen and D F Cala Biorheology 12 111 (1975)

    Article  Google Scholar 

  3. [3]

    G K Ramesh, K G Kumar, S A Shehzad and B J Gireesh Canadian Journal of Physics 96 18 (2018)

    ADS  Article  Google Scholar 

  4. [4]

    S U Khan, N Ali, T Mushtaq, A Rauf and S A Shehzad Thermal Science 23 3365 (2019)

    Article  Google Scholar 

  5. [5]

    V Vatsa and A Kumar Heat Transfer-Asian Research 48 1204 (2019)

    Article  Google Scholar 

  6. [6]

    T Y Na and J P Chiou Applied Scientific Research 35 409 (1979)

    MathSciNet  Article  Google Scholar 

  7. [7]

    T Watanabe Acta Mechanica 87 1 (1991)

  8. [8]

    K A Yih International Journal of Heat and Mass Transfer 42 4299 (1999)

  9. [9]

    A J Chamkha Numerical Heat Transfer, Part A 39 511 (2001)

  10. [10]

    P Bapuji, K Ekambavanan and I Pop Heat and Mass Transfer 44 517 (2008)

    ADS  Article  Google Scholar 

  11. [11]

    S E Ahmed and A Mahdy World Journal of Mechanics 2 272 (2012)

    ADS  Article  Google Scholar 

  12. [12]

    M A Husain and S C. Paul Heat and Mass transfer 37 167 (2001)

    ADS  Article  Google Scholar 

  13. [13]

    D A Kumar and S Roy Applied Mathematics and Computation 155 545 (2004)

    MathSciNet  Article  Google Scholar 

  14. [14]

    S M A Harbi Applied Mathematics and Computation 170 64 (2005)

  15. [15]

    C Y Cheng International Communications in Heat and Mass Transfer 35 39 (2008)

  16. [16]

    A M Rashad, M A E Hakiem and M M M Abdou Computers and Mathematics with Applications 62 3140 (2011)

    MathSciNet  Article  Google Scholar 

  17. [17]

    A Kumar and A K Singh International Journal of Energy & Technology 3 1 (2011)

    Article  Google Scholar 

  18. [18]

    Vanita and A Kumar Alexandria Engineering Journal 55 1211 (2016)

  19. [19]

    S Ostrach Boundary Layer Research (ed) H Gortler (Symposium, Freiburg) (1957)

  20. [20]

    B Gebhart Journal of Fluid Mechanics 14 225 (1962)

  21. [21]

    J Koo and C Kleinstreuer International Journal of Heat and Mass Transfer 47 3159 (2004)

    Article  Google Scholar 

  22. [22]

    M M Rahman, M J Uddin and A Aziz International Journal of Thermal Sciences 48 2331 (2009)

    Article  Google Scholar 

  23. [23]

    A Sadeghi and M H Saidi International Journal of Heat and Mass Transfer 53 3782 (2010)

    Article  Google Scholar 

  24. [24]

    O D Makinde International Journal of Numerical Methods for Heat and Fluid Flow 8 1291 (2013)

  25. [25]

    S A Shehzad, T Hussain, T Hayat, M Ramzan and A Alsaedi Journal of Central South University 22 360 (2015)

    Article  Google Scholar 

  26. [26]

    T M Ajayi, A J Omowaye and I L Animasaun Journal of Applied Mathematics 2017 1697135 (2017)

    Article  Google Scholar 

  27. [27]

    R Kumar, R Kumar, S A Shehzad and M Sheikholeslami International Journal of Heat and Mass Transfer 120 540 (2018)

    Article  Google Scholar 

  28. [28]

    R Kumar, R Kumar, R Koundal, S A Shehzad and M Sheikholeslami Communications in Theoretical Physics 71 779 (2019)

    ADS  MathSciNet  Article  Google Scholar 

  29. [29]

    B Mahanthesh, S A Shehzad, T Ambreen and S U Khan Journal of Thermal Analysis and Calorimetry (2020)

  30. [30]

    S Roy and T Basak International Journal of Engineering Science 43 668 (2005)

    Article  Google Scholar 

  31. [31]

    D Mythili, R Sivaraj, M M Rashidi and Z Yang Journal of Naval Architecture and Marine Engineering 12 125 (2015)

    Article  Google Scholar 

  32. [32]

    J Bai, J Pan, G Wu and L Tang International Journal of Heat and Mass Transfer 138 1320 (2019)

    Article  Google Scholar 

  33. [33]

    A H Pordanjani, S Aghakhani, A A Alnaqi and M Afrand International Journal of Mechanical Sciences 152 99 (2019)

    Article  Google Scholar 

  34. [34]

    R Rajora, A Kumar Heat Transfer (2020) (under printing). DOI: https://doi.org/10.1002/htj.21979

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Acknowledgements

Author Arun Kumar Singh thankfully acknowledges the financial relief from UGC, New Delhi, as a Research Fellowship during this work.

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Singh, A.K., Kumar, A. & Singh, A.K. Effect of non-uniform heating and viscous dissipation on natural convective flow of Casson fluid over a vertical cone. Indian J Phys (2021). https://doi.org/10.1007/s12648-020-01984-0

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Keywords

  • Natural convection
  • Vertical cone
  • Casson fluid
  • Non-uniform heating
  • Viscous dissipation