Fractional calculus modelling for unsteady unidirectional flow of incompressible fluids with time-dependent viscosity

TL;DR

This paper introduces a fractional calculus-based model for unsteady, unidirectional flow of incompressible fluids with time-dependent viscosity, providing analytical solutions that incorporate memory effects and power-law viscosity variability.

Contribution

It develops a novel fractional calculus model for fluids with time-dependent viscosity, extending classical equations with memory formalism and deriving explicit analytical solutions.

Findings

Analytic solutions for unsteady flow with time-dependent viscosity.

Model captures power-law viscosity variability.

Explicit solution in a particular case.

Abstract

In this note we analyze a model for a unidirectional unsteady flow of a viscous incompressible fluid with time dependent viscosity. A possible way to take into account such behaviour is to introduce a memory formalism, including thus the time dependent viscosity by using an integro-differential term and therefore generalizing the classical equation of a Newtonian viscous fluid. A possible useful choice, in this framework, is to use a rheology based on stress/strain relation generalized by fractional calculus modelling. This is a model that can be used in applied problems, taking into account a power law time variability of the viscosity coefficient. We find analytic solutions of initial value problems in an unbounded and bounded domain. Furthermore, we discuss the explicit solution in a meaningful particular case.