Stepwise mathematical derivation of the Herschel–Bulkley laminar fluid flow equations—in pipes

Research output: Contribution to conferencePaperpeer-review


  • Rahman Ashena
  • Foued Badrouchi
  • Siamak Mishani
  • Fatemeh Sotoudeh
  • Sahand Nekoeian

External Organisational units

  • Asia Pacific University of Technology and Innovation (APU), Kuala Lumpur, Malaysia
  • University of North Dakota (UND), Grand Forks, USA
  • The University of British Columbia
  • Independent Consultant, Ottawa


Stepwise derivation of flow equations of the Herschel–Bulkley (HB) model is not available in the literature. These
equations are crucial for mechanical, chemical and petroleum engineering academia and industries where fundamental
works on non-Newtonian fluids may be done to reach future models and estimation methods. Therefore, this work focuses
on derivation of laminar flow equations and estimation methods of HB fluids through pipes. In this work, first, stepwise
derivation of the HB fluid flow parameters consisting of fluid velocity, flow rate, average velocity and relative velocity
equations is presented, followed by a straightforward mathematical model for use in numerical solution. Next, stepwise
mathematical derivation of the laminar pressure drop equations by Merlo et al. (An innovative model for drilling fluid
hydraulics. Paper presented at the SPE Asia Pacific oil and gas conference, Kuala Lumpur, Malaysia, 1995) and Gjerstad
and Time (SPE J 20:1–18, 2014) is presented, and finally practical and user-friendly calculation procedures for different
estimation methods are presented. The step-by-step derivation procedures presented in this work contribute to effective
learning for engineering students and practitioners in addition to providing a clear example derivation guideline for future
researchers to reach other more accurate non-Newtonian hydraulics models and estimation methods.


Original languageEnglish
Publication statusPublished - 26 Aug 2022