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A novel definition of the caputo fractional finite difference approach for Maxwell fluid

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Abstract

This paper focuses on modeling and analyzing a nonlinear fractional Maxwell fluid across a vertical plate using a novel definition of the Caputo fractional derivative. Specifically, the newly formulated fractional definition is tailored for the finite difference method. Buoyancy effects are considered to bring the model closer to the natural occurrences of the magnetic field. The novel fractional derivative definition of the designed model has been implemented, and the results have been further analyzed using theoretical and numerical methods. The range of fractional parameters \(\alpha \) and \(\beta \) is \(0<\alpha \le 1\) and \(0<\beta \le 1\), respectively. Significant theorems are also included in the study. A simulation was performed to determine which fractional derivative operator is more accurate than the others. The designed definition demonstrates a high convergence rate and decreased computational cost, both of which are primary objectives. Graphs have also been plotted to illustrate the velocity and temperature distributions as the physical parameters vary.

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No data were created or analysed in the course of this research.

Abbreviations

\(\textbf{M}_{nf}\) :

Maxwell nano-fluid

V :

Velocity

\(\rho _{nf}\) :

Density for nanofluid

B :

Magnetic field

\(\rho \) :

Pressure

\(\textbf{I}\) :

Identity matrix

\(\mu _{nf}\) :

Density viscosity

\(\textbf{E}\) :

Electric field

Grashof number:

Gr \( =\frac{g(\rho \beta )_f({\textbf {T}}-{\textbf {T}}_\infty )}{u_0}\)

Hartman n:

Ha \( =\frac{M\nu _f}{u_0^2}\)

\(\textbf{J}^\alpha \) :

Current density

e :

Internal energy

\(\tau _1\) :

Stress tensor

\(\textbf{A}_1 \) :

Rivline–Ericksen tensor

\(\kappa _{nf}\) :

Thermal conductivity

\(\textbf{S}_t\) :

Extra stress

\(\epsilon ^\alpha _1 \) :

Buoyancy forces

\(\varpi \) :

Angular velocity

Prandt number:

Pr \( =\frac{\rho c_{\rho \mu _f}}{k_f}\)

Fractional:

\(\alpha ,\beta \)

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Authors and Affiliations

  1. School of Electrical and Electronic Engineering, The University of Adelaide, Adelaide, Australia

    Tamour Zubair

  2. Department of Mathematics, Riphah International University, Faisalabad, 44000, Pakistan

    Syeda Alishwa Zanib

  3. Department of Mathematics, University of Management and Technology, Lahore, 54770, Pakistan

    Muhammad Imran Asjad

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  1. Tamour Zubair
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  2. Syeda Alishwa Zanib
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  3. Muhammad Imran Asjad
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Correspondence to Tamour Zubair.

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Zubair, T., Zanib, S.A. & Asjad, M.I. A novel definition of the caputo fractional finite difference approach for Maxwell fluid. Comp. Appl. Math. 43, 238 (2024). https://doi.org/10.1007/s40314-024-02728-0

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