Pressure–flow rate relationship and its polynomial expansion for laminar flow in a circular pipe based on exponential viscosity-pressure characteristics: An extension of classical Poiseuille's law

Abstract

Laminar flow in circular pipes is widespread in various fields. Poiseuille's law is the classical equation describing the pressure–flow rate relationship for laminar flow in circular pipes. However, the fluid viscosity is treated as a constant in Poiseuille's law. Therefore, Poiseuille's law cannot be used to accurately analyze fluids that have viscosities that vary exponentially with pressure, such as hydraulic oils and lubricating oils. In this study, with the exponential viscosity-pressure characteristics, a total of four simple and explicit equations are given for calculating the flow rate or pressure difference of the pipe, and corresponding polynomial expansions are derived based on the Taylor series. Experimental tests and computational fluid dynamics simulations are carried out to verify the correctness of the theoretical equations, with error of less than 6% and 2%, respectively. An error analysis of the theoretical equations for different numbers of polynomial terms is also performed. The results show that the proposed theoretical equations all degenerate to the classical Poiseuille's law when the number of polynomial terms is taken to be 1, and the relative errors are less than ±5% for viscosity changes less than 10%. When the number of terms is 2, the relative error is less than ±5% for viscosity changes less than 40%. In the calculation of connection pipelines of a deep-sea hydraulic actuator, the difference in pressure loss calculated with or without viscosity change is 31.47% and reaches up to 5.7202 MPa, which shows the practical value of this research in piping systems.