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Fractional Calculus and Applied Analysis

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Novel numerical analysis of multi-term time fractional viscoelastic non-newtonian fluid models for simulating unsteady MHD Couette flow of a generalized Oldroyd-B fluid

Libo Feng / Fawang Liu / Ian Turner
  • School of Mathematical Sciences, Queensland University of Technology GPO Box 2434, Brisbane, Qld 4001, Australia
  • Australian Research Council Centre of Excellence for Mathematical and Statistical Frontiers (ACEMS) Queensland University of Technology (QUT) Brisbane, Brisbane, Australia
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/ Liancun Zheng
Published Online: 2018-10-29 | DOI: https://doi.org/10.1515/fca-2018-0058


In this paper, we consider the application of the finite difference method for a class of novel multi-term time fractional viscoelastic non-Newtonian fluid models. An important contribution of the work is that the new model not only has a multi-term time derivative, of which the fractional order indices range from 0 to 2, but also possesses a special time fractional operator on the spatial derivative that is challenging to approximate. There appears to be no literature reported on the numerical solution of this type of equation. We derive two new different finite difference schemes to approximate the model. Then we establish the stability and convergence analysis of these schemes based on the discrete H1 norm and prove that their accuracy is of O(τ + h2) and O(τmin{3–γs,2–αq,2–β}+h2), respectively. Finally, we verify our methods using two numerical examples and apply the schemes to simulate an unsteady magnetohydrodynamic (MHD) Couette flow of a generalized Oldroyd-B fluid model. Our methods are effective and can be extended to solve other non-Newtonian fluid models such as the generalized Maxwell fluid model, the generalized second grade fluid model and the generalized Burgers fluid model.

MSC 2010: Primary 26A33; Secondary 35Q35; 65M06; 65M12; 76W05

Key Words and Phrases: multi-term time derivative; finite difference method; fractional non-Newtonian fluids; generalized Oldroyd-B fluid; Couette flow; stability and convergence analysis


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About the article

Received: 2018-01-01

Published Online: 2018-10-29

Published in Print: 2018-08-28

Citation Information: Fractional Calculus and Applied Analysis, Volume 21, Issue 4, Pages 1073–1103, ISSN (Online) 1314-2224, ISSN (Print) 1311-0454, DOI: https://doi.org/10.1515/fca-2018-0058.

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