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Applications of non-integer Caputo time fractional derivatives to natural convection flow subject to arbitrary velocity and Newtonian heating

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Abstract

Unsteady time fractional natural convection flow of incompressible viscous fluid due to arbitrary velocity with radiation and Newtonian heating is investigated. By introducing dimensionless variables and functions, the resulting equations are solved by the Laplace transform technique. Exact solutions for temperature and velocity fields are obtained by Caputo time fractional derivatives in dimensionless form, and they are expressed in terms of Robotnov–Hartley function and Wright’s function. The rate of heat transfer on plate is obtained in terms of Nusselt number. Some known solutions from the existing literature are obtained as limiting case when \(\alpha = 1\). At the end, we have seen the significant influence of fractional parameter \(\alpha\) on the temperature and velocity graphically and observed that it has shown an opposite behavior on temperature and velocity for small and large values of time t, respectively. This shows that how fractional parameter \(\alpha\) affects the fluid flow.

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Abbreviations

\(c_{p}\) :

Specific heat at constant pressure

\(g\) :

Gravitational acceleration

\(Gr\) :

Grashof number

\(k\) :

Thermal conductivity

\(Pr\) :

Prandtl number

\(s\) :

Laplace transform parameter

\(T\) :

Fluid temperature

\(T_{\text{w}}\) :

Wall temperature

\(T_{\infty }\) :

Temperature far away from the plate

\(\mu\) :

Dynamic viscosity

\(\nu\) :

Kinematic viscosity

\(\theta\) :

Non-dimensional temperature

\(N_{r}\) :

Radiation parameter

\(Pr_{\text{eff}}\) :

Effective Prandtl number

\(\alpha\) :

Fractional parameter

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Acknowledgements

The authors would like to thank reviewers for careful assessment and pertinent observations. The authors would also acknowledge the University of Management and Technology, Lahore, Pakistan, for the financial support for this research.

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Authors and Affiliations

  1. Department of Mathematics, University of Management and Technology Lahore, Lahore, Pakistan

    M. A. Imran & Maryam Aleem

  2. Abdus Salam School of Mathematical Science, GC, University Lahore, Lahore, Pakistan

    Nehad Ali Shah

  3. Basic Engineering Sciences Department, College of Engineering Majmaah University, P.O. Box 66, Majmaah, 11952, Saudi Arabia

    Ilyas Khan

Authors
  1. M. A. Imran
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  2. Nehad Ali Shah
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  3. Ilyas Khan
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  4. Maryam Aleem
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Corresponding author

Correspondence to Ilyas Khan.

Appendix

Appendix

$$L^{ - 1} \left\{ {\frac{{e^{{ - as^{b} }} }}{{s^{c} }}} \right\} = t^{c - 1} \varPhi (c, - b;\; - at^{ - b} );\quad 0 < b < 1\quad {\text{where}}\quad \varPhi (x,y;z) = \sum\limits_{n = 0}^{\infty } {\frac{{z^{n} }}{\varGamma (n + 1)\varGamma (x - ny)}}$$
(28)
$$L^{ - 1} \left\{ {\frac{1}{{s^{\mu } }}} \right\} = F_{\mu } (t) = \sum\limits_{n = 0}^{\infty } {\frac{{a^{n} t^{(n + 1)\mu - 1} }}{\varGamma [(n + 1)\mu ]};\quad \mu > 0}$$
(29)
$$L^{ - 1} \left\{ {\frac{{s^{a - b} }}{{s^{a} + c}}} \right\} = t^{b - 1} E_{a,\,b} (ct^{a} );\quad E_{a,\,b} (ct^{a} ) = \sum\limits_{k = 0}^{\infty } {\frac{{z^{k} }}{\varGamma (ak + b)};\quad a,b > 0}$$
(30)
$$\begin{aligned} \psi (a,b,t) = L^{ - 1} \left\{ {\frac{{e^{ - a\sqrt s } }}{{s^{2} \left( {\sqrt s + b} \right)}}} \right\} = \frac{1}{b}\left( {t + \frac{{a^{2} }}{2} + \frac{a}{b} + \frac{1}{{b^{2} }}} \right)\;{\text{erfc}}\left( {\frac{a}{2\sqrt t }} \right) - \frac{(2 + ab)\sqrt t }{{b^{2} \sqrt \pi }}e^{{ - \frac{{a^{2} }}{4t}}} - \hfill \\ \quad - \frac{1}{{b^{3} }}e^{{(ab + b^{2} t)}} \;{\text{erfc}}\left( {\frac{a}{2\sqrt t } + b\sqrt t } \right) \hfill \\ \end{aligned}$$
(31)

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Imran, M.A., Shah, N.A., Khan, I. et al. Applications of non-integer Caputo time fractional derivatives to natural convection flow subject to arbitrary velocity and Newtonian heating. Neural Comput & Applic 30, 1589–1599 (2018). https://doi.org/10.1007/s00521-016-2741-6

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  • DOI: https://doi.org/10.1007/s00521-016-2741-6

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