Fluid Flow Development in a Pipe as a Demonstration of a Sequential Change in Its Rheological Properties

Abstract

The sequence of the process of changing the velocity profiles and the laws of resistance during the flow of a fluid in a pipe is considered. With the increasing of the Reynolds number, we obtain the transition of the flow regime from laminar to turbulent. In the presence of small additives of polymers, when the Toms effect is observed in the fluid flow, the turbulent regime changes with a further increase in the Reynolds number to another regime, the rheology of which leads to laminar velocity profiles and corresponding resistance laws. Then, with an increase in the Reynolds number for polymer solutions, the limiting Virk flow regime with its own rheology is reached. All the mentioned flow regimes and all types of rheology can be described using one rheological relation, which is a power-law generalization of Newton’s formula, by changing the values of the power value in this ratio upon reaching the corresponding critical Reynolds numbers. This generalization can be extended to the spatial case of flow and the rheological relation can be represented in tensor form with a further system of differential equations for a fluid flow with an arbitrary rheology. After that, boundary value problems in fluid mechanics can be solved for any fluid flow regime.