Stepwise derivation and comparison of the Herschel–Bulkley Laminar Fluid Flow equations—in Annulus

The Herschel-Bulkley (HB) model is considered one of the most accurate non-Newtonian fluid flow models, useful in various disciplines, including chemical, mechanical and petroleum engineering. Past literature has presented clear derivations and validation of Bingham Plastic and Power Law models, but the HB model’s equations as well as pressure estimation methods were not covered adequately. Such a derivation and validation are important for showing the methodology for future development of more accurate non-Newtonian rheological models by future researchers and students. This work addresses this shortcoming by investigating the HB modelling methodology for the annular flow. Therefore, a methodology flowchart is presented for finding equations for HB fluid’s velocity, flow rate, average velocity and relative velocity equations as wells as their validation and comparison using experimental data, with 32% average absolute percent error between the numerical model and the experimental pressure loss measurements. This indicates that the HB model is not as accurate as previously claimed. This suggests a need for further research to develop new and more precise rheological models that can outperform the HB model in predicting fluid behavior. Next, the methodology of finding equations of two main estimation methods for pressure drop is presented. This research represents a significant foundational contribution, establishing a basis for advancing future studies in non-Newtonian fluids. It offers a methodology, presented in the form of a flowchart, that enhances the clarity of the process for future researchers. This approach paves the way for developing models that could achieve greater accuracy than the HB model.