Abstract
Pipe flow is one of the most commonly used models to describe fluid dynamics. The concept of fractional derivative has been recently found very useful and much more accurate in predicting dynamics of viscoelastic fluids compared with classic models. In this paper, we capitalize on our previous study and consider space-time dynamics of flow velocity and stress for fractional Maxwell, Zener, and Burgers models. We demonstrate that the behavior of these quantities becomes much more complex (compared to integer-order classical models) when adjusting fractional order and elastic parameters. We investigate mutual influence of fractional orders and consider their limiting value combinations. Finally, we show that the models developed can be reduced to classical ones when appropriate fractional orders are set.
Highlights
Fractional calculus in general and fractional derivative in particular has been gathering an increasing researchers interest recently
One of the fields where fractional calculus has already proven its efficiency is the prediction of dynamics of viscoelastic flows in various constrained geometries, with circular pipe being one of the most popular ones
In the first pair of animations we investigate the influence of fractional order α
Summary
Fractional calculus in general and fractional derivative in particular has been gathering an increasing researchers interest recently. Fractional viscoelastic models, including Maxwell and Zener ones, were considered in close detail in series of works by Schiessel, Friedrich, and others [22,23]. Those authors provided in-depth physical analysis of mathematical concepts introduced via fractional calculus formalism, suggested, developed, and explained various ladder models. We capitalize on the approaches developed in [1] to examine the behavior of fractional viscoelastic Maxwell, Zener, and Burgers models in application to pipe flow.
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