Start-up flow of a viscoelastic fluid in a pipe with fractional Maxwell's model

TL;DR

This paper derives an exact solution for the start-up flow of a fractional Maxwell's viscoelastic fluid in a pipe, revealing conditions under which the fluid behaves solid-like or fluid-like over time.

Contribution

It provides a novel exact analytical solution for fractional Maxwell's fluid flow in a pipe and offers insights into its long-term behavior based on fractional parameters.

Findings

Flow tends to rest as time approaches infinity for most parameters.

Fractional Maxwell's fluid exhibits solid-like or fluid-like behavior depending on parameter β.

Oscillations can occur before reaching asymptotic behavior.

Abstract

Unidirectional start-up flow of a viscoelastic fluid in a pipe with fractional Maxwell's model is studied. The flow starting from rest is driven by a constant pressure gradient in an infinite long straight pipe. By employing the method of variable separations and Heaviside operational calculus, we obtain the exact solution, from which the flow characteristics are investigated. It is found that the start-up motion of fractional Maxwell's fluid with parameters $\alpha$ and $\beta$, tends to be at rest as time goes to infinity, except the case of $\beta=1$. This observation, which also can be predicted from the mechanics analogue of fractional Maxwell's model, agrees with the classical work of Friedrich and it indicates fractional Maxwell's fluid presents solid-like behavior if $\be\neq 1$ and fluid-like behavior if $\be=1$. For an arbitrary viscoelastic model, a conjecture is proposed to give an intuitive way judging whether it presents fluid-like or solid-like behavior. Also oscillations may occur before the fluid tends to the asymptotic behavior stated above, which is a common phenomenon for viscoelastic fluids.