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Solutions with Wright Function for Time Fractional Free Convection Flow of Casson Fluid

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Abstract

A fractional model of Casson fluid coupled with energy equation is developed. Casson fluid in the presence of heat transfer is considered over an oscillating vertical plate with constant wall temperature. Definition of fractional Caputo derivative is used in the mathematical formulation of the problem. Exact solutions via Laplace transform are obtained and presented in terms of Wright function. Parametric studies were undertaken, and the obtained solutions are illustrated through plots for various physical parameters.

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Abbreviations

u :

Dimensional velocity

t :

Dimensional time

x :

Coordinate axis parallel to the plate

y :

Coordinate axis normal to the plate

g :

Acceleration due to gravity

k :

Thermal conductivity of the fluid

\(\tau \) :

Dimensionless time

\(\alpha \) :

Fractional parameter

f :

Amplitude of the plate oscillations

\(c_\mathrm{p} \) :

Specific heat at constant pressure

T :

Temperature of the fluid

\(T_\infty \) :

Ambient temperature

\(T_\mathrm{w} \) :

Wall temperature

Pr :

Prandtl number

v :

Dimensionless velocity

\(\xi \) :

Dimensionless coordinate axis normal to the plate

s :

Laplace transform parameter

H(t):

Unit step function

\(\omega \) :

Frequency of the plate oscillation

\(\theta \) :

Dimensionless temperature

\(\rho \) :

Density of the fluid

\(\nu \) :

Kinematic viscosity

\(\mu \) :

Viscosity

\(\beta \) :

Casson fluid parameter

\(\beta _\mathrm{T}\) :

Coefficient of thermal expansion

\(C_\mathrm{f}\) :

Skin friction

Nu :

Nusselt number

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Authors and Affiliations

  1. Department of Mathematics, City University of Science and Information Technology, Peshawar, Pakistan

    Farhad Ali, Nadeem Ahmad Sheikh & Muhammad Saqib

  2. Basic Engineering Sciences Department, College of Engineering Majmaah University, Majmaah, Saudi Arabia

    Ilyas Khan

Authors
  1. Farhad Ali
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  2. Nadeem Ahmad Sheikh
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  3. Ilyas Khan
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  4. Muhammad Saqib
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Correspondence to Farhad Ali.

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Ali, F., Sheikh, N.A., Khan, I. et al. Solutions with Wright Function for Time Fractional Free Convection Flow of Casson Fluid. Arab J Sci Eng 42, 2565–2572 (2017). https://doi.org/10.1007/s13369-017-2521-3

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  • DOI: https://doi.org/10.1007/s13369-017-2521-3

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