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

  • 1 School of Mathematical Sciences, Queensland University of Technology, GPO Box 2434, Brisbane, Australia
  • 2 School of Mathematical Sciences, Queensland University of Technology GPO Box 2434, Qld 4001, Brisbane, Australia
  • 3 School of Mathematical Sciences, Queensland University of Technology GPO Box 2434, Qld 4001, Brisbane, Australia
  • 4 Australian Research Council Centre of Excellence for Mathematical and Statistical Frontiers (ACEMS) Queensland University of Technology (QUT) Brisbane, Brisbane, Australia
  • 5 School of Mathematics and Physics University of Science and Technology Beijing, 100083, Beijing, PR China
Libo Feng
  • Corresponding author
  • School of Mathematical Sciences, Queensland University of Technology, GPO Box 2434, Brisbane, Qld 4001, Australia
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, Fawang Liu
  • School of Mathematical Sciences, Queensland University of Technology GPO Box 2434, Brisbane, Qld 4001, Australia
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, 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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and Liancun Zheng
  • School of Mathematics and Physics University of Science and Technology Beijing, Beijing, 100083, PR China
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Abstract

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.

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Fractional Calculus and Applied Analysis (FCAA) is a specialized international journal for theory and applications of an important branch of Mathematical Analysis (Calculus) where differentiations and integrations can be of arbitrary non-integer order.

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