Stochastic Velocity Assessment of Robertson-Stiff Fluids in Rectangular Ducts under Uncertain Parameters and Boundary Conditions

Document Type : Research Paper

Author
Osama H. Galal
Engineering Mathematics and Physics Department, Faculty of Engineering, Fayoum University, Fayoum 63514, Egypt
10.22055/jacm.2025.48324.5150
Abstract
This study presents a pioneering stochastic modelling and analysis of Robertson-Stiff (R-S) fluid flow in a rectangular duct, addressing a critical gap in non-Newtonian fluid mechanics by quantifying velocity uncertainty under three distinct scenarios: uncertain dimensionless viscosity (Case I), uncertain pressure gradient (Case II), and uncertain boundary conditions (Case III). The Stochastic Finite Difference with Homogeneous Chaos (SFDHC) method is developed and applied, demonstrating superior computational efficiency compared to the Monte Carlo Simulation (MCS) approach, which is used for validation. Both methods generate probability density functions (PDFs) of fluid velocity, revealing key statistical metrics for maximum velocity across all cases. The results indicate significant velocity variability in Cases I and II, with standard deviations reaching approximately 80% of the coefficient of variation (COV). Specifically, at a COV of 20%, Case I exhibits minimum and maximum velocities of 71.66% and 191.06% of the mean, respectively, while Case II shows 41.40% and 152.70% of the mean. In contrast, Case III displays near-deterministic behavior at the duct’s center, attributed to boundary effects. A detailed parametric analysis under uncertain viscosity conditions identifies the aspect ratio as the dominant factor influencing maximum stochastic velocity statistics, followed by the flow behavior index and dimensionless shear stress. This research introduces a robust framework for uncertainty quantification in R-S fluid dynamics, offering critical insights for engineering design by highlighting how uncertainty in fluid properties impacts flow behavior. These findings enhance the reliability of ducted flow systems in practical applications—such as drilling fluid management, polymer processing, food and pharmaceutical production, cement slurries, and the paints and coatings industry—where non-Newtonian fluids are prevalent, thereby promoting safer and more efficient engineering solutions.
Keywords
Stochastic fluids
Non-Newtonian fluids
Robertson-Stiff
Stochastic finite difference method
Polynomial chaos expansion
Stochastic boundary conditions
Subjects

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Journal of Applied and Computational Mechanics
Volume 12, Issue 1
Winter 2026
Pages 189-201