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Effects of Dufour and fractional derivative on unsteady natural convection flow over an infinite vertical plate with constant heat and mass fluxes

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Abstract

In this paper, we analyze the effects of Dufour number and fractional-order derivative on unsteady natural convection flow of a viscous and incompressible fluid over an infinite vertical plate with constant heat and mass fluxes. The fractional constitutive model is obtained using fractional calculus approach. The Caputo fractional derivative operator is used in this problem. The dimensionless system of equations has been solved by employing Laplace transformation technique. Closed form solutions for concentration, temperature and velocity are presented in the form of Wright function and complementary error function. Effects of fractional and physical parameters on temperature and velocity profiles are illustrated graphically.

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Abbreviations

u :

Velocity of the fluid

T :

Temperature of the fluid

C :

Concentration of the fluid

T ∞ :

Temperature of the fluid far away from the plate

C ∞ :

Species concentration of the fluid far away from the plate

g :

Acceleration due to gravity

C p :

Specific heat at a constant pressure

C S :

The concentration susceptibility

Gr :

Thermal Grashof number

Gm :

Mass Grashof number

k :

Thermal conductivity of the fluid

k T :

Thermal diffusion ratio

D :

Mass diffusivity

q w :

Heat flux per unit area at the plate

j :

Mass flux per unit area at the plate

L :

Characteristic length \( \left( { = (\nu^{2} /g)^{{\frac{1}{3}}} } \right) \)

β T :

The volumetric coefficient of thermal expansion

β C :

The volumetric coefficient of concentration expansion

Pr:

Prandtl number, (= μCp/k)

D f :

Dufour number

Sc :

Schmidt number

N :

Buoyancy ratio parameter

μ :

Dynamic viscosity

ρ :

Fluid density

ν :

Kinematic viscosity of the fluid \( \left( { = \frac{\mu }{\rho }} \right) \)

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Acknowledgements

The author Nehad Ali Shah is highly thankful to ASSMS, GC University Lahore, and Higher Education Commission of Pakistan for facilitating this work. The author Thanaa Elnaqeeb is grateful to Zagazig University for facilitating this work. The author Shaowei Wang is grateful to National Natural Science Foundation of China (Grant No. 11672164) for supporting this work.

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Authors and Affiliations

  1. Abdus Salam School of Mathematical Sciences, GC University, Lahore, Pakistan

    Nehad Ali Shah

  2. Department of Mathematics, Faculty of Science, Zagazig University, Zagazig, 44519, Egypt

    Thanaa Elnaqeeb

  3. School of Civil Engineering, Shandong University, Jinan, 250061, People’s Republic of China

    Shaowei Wang

Authors
  1. Nehad Ali Shah
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  2. Thanaa Elnaqeeb
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  3. Shaowei Wang
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Correspondence to Nehad Ali Shah.

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Communicated by Antonio José Silva Neto.

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Shah, N.A., Elnaqeeb, T. & Wang, S. Effects of Dufour and fractional derivative on unsteady natural convection flow over an infinite vertical plate with constant heat and mass fluxes. Comp. Appl. Math. 37, 4931–4943 (2018). https://doi.org/10.1007/s40314-018-0606-6

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  • DOI: https://doi.org/10.1007/s40314-018-0606-6

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