Abstract The Herschel‐Bulkley‐Extended fluid model is widely employed in the petrochemical industry to characterize the rheological behaviour of non‐Newtonian fluids, including drilling muds and foams. In this study, flow equations are firstly established for the fully developed laminar pipe flow of Herschel‐Bulkley‐Extended fluid, and analytical solutions are derived via the perturbation analysis and validation with numerical simulations. The effects of key rheological parameters (yield stress, consistency coefficient, and shear‐thinning index) and flow parameters on velocity profile, flow rate, and pressure gradient are quantified by systematic analysis. Results indicate that flow velocity is inversely related to yield stress and the consistency coefficient, while correlating positively with the pressure gradient. Notably, the shear‐thinning index induces a transitional reversal in the velocity profile trend. Flow rate declines with higher yield stress, consistency coefficient, and shear‐thinning index but rises with increasing pressure gradient and pipe radius. The pressure gradient systematically intensifies with all three rheological parameters. These findings provide a quantitative framework for predicting Herschel‐Bulkley‐Extended fluid transport behaviour and contribute to a deeper understanding of non‐Newtonian laminar flow. The proposed approach supports hydraulic optimization in drilling operations and improves the design of multiphase foam transport systems. Furthermore, it lays the groundwork for extending the analysis of Herschel‐Bulkley‐Extended fluid dynamics to hierarchical flow networks, nanocapillary systems, and slip‐governed transport regimes.
Sun et al. (2026) studied this question.