Numerical analysis for viscoelastic fluid flow with distributed/variable order time fractional Maxwell constitutive models

doi:10.1007/s10483-021-2796-8

摘要/Abstract

摘要： Fractional calculus has been widely used to study the flow of viscoelastic fluids recently, and fractional differential equations have attracted a lot of attention. However, the research has shown that the fractional equation with constant order operators has certain limitations in characterizing some physical phenomena. In this paper, the viscoelastic fluid flow of generalized Maxwell fluids in an infinite straight pipe driven by a periodic pressure gradient is investigated systematically. Consider the complexity of the material structure and multi-scale effects in the viscoelastic fluid flow. The modified time fractional Maxwell models and the corresponding governing equations with distributed/variable order time fractional derivatives are proposed. Based on the L₁-approximation formula of Caputo fractional derivatives, the implicit finite difference schemes for the distributed/variable order time fractional governing equations are presented, and the numerical solutions are derived. In order to test the correctness and availability of numerical schemes, two numerical examples are established to give the exact solutions. The comparisons between the numerical solutions and the exact solutions have been made, and their high consistency indicates that the present numerical methods are effective. Then, this paper analyzes the velocity distributions of the distributed/variable order fractional Maxwell governing equations under specific conditions, and discusses the effects of the weight coefficient ϖ(α) in distributed order time fractional derivatives, the order α(r, t) in variable fractional order derivatives, the relaxation time λ, and the frequency ω of the periodic pressure gradient on the fluid flow velocity. Finally, the flow rates of the distributed/variable order fractional Maxwell governing equations are also studied.

关键词: distributed order time fractional derivative, variable order time fractional derivative, finite difference scheme, viscoelastic fluid

Abstract: Fractional calculus has been widely used to study the flow of viscoelastic fluids recently, and fractional differential equations have attracted a lot of attention. However, the research has shown that the fractional equation with constant order operators has certain limitations in characterizing some physical phenomena. In this paper, the viscoelastic fluid flow of generalized Maxwell fluids in an infinite straight pipe driven by a periodic pressure gradient is investigated systematically. Consider the complexity of the material structure and multi-scale effects in the viscoelastic fluid flow. The modified time fractional Maxwell models and the corresponding governing equations with distributed/variable order time fractional derivatives are proposed. Based on the L₁-approximation formula of Caputo fractional derivatives, the implicit finite difference schemes for the distributed/variable order time fractional governing equations are presented, and the numerical solutions are derived. In order to test the correctness and availability of numerical schemes, two numerical examples are established to give the exact solutions. The comparisons between the numerical solutions and the exact solutions have been made, and their high consistency indicates that the present numerical methods are effective. Then, this paper analyzes the velocity distributions of the distributed/variable order fractional Maxwell governing equations under specific conditions, and discusses the effects of the weight coefficient ϖ(α) in distributed order time fractional derivatives, the order α(r, t) in variable fractional order derivatives, the relaxation time λ, and the frequency ω of the periodic pressure gradient on the fluid flow velocity. Finally, the flow rates of the distributed/variable order fractional Maxwell governing equations are also studied.

Key words: distributed order time fractional derivative, variable order time fractional derivative, finite difference scheme, viscoelastic fluid

中图分类号:

Yanli QIAO, Xiaoping WANG, Huanying XU, Haitao QI. Numerical analysis for viscoelastic fluid flow with distributed/variable order time fractional Maxwell constitutive models[J]. Applied Mathematics and Mechanics (English Edition), 2021, 42(12): 1771-1786.

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