{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:26:45Z","timestamp":1760059605515,"version":"build-2065373602"},"reference-count":40,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2025,6,24]],"date-time":"2025-06-24T00:00:00Z","timestamp":1750723200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Northern Border University, Arar, KSA","award":["NBU-FFR-2025-2768-01"],"award-info":[{"award-number":["NBU-FFR-2025-2768-01"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This study presents an innovative approach to quantifying uncertainty in Herschel\u2013Bulkley (H-B) fluid flow through rectangular ducts, analyzing four scenarios: uncertain apparent viscosity (Case I), uncertain pressure gradient (Case II), uncertain boundary conditions (Case III) and uncertain apparent viscosity and pressure gradient (Case IV). Using the stochastic finite difference with homogeneous chaos (SFDHC) method, we produce probability density functions (PDFs) of fluid velocity with exceptional computational efficiency (243 times faster), matching the accuracy of Monte Carlo simulation (MCS). Key statistics and maximum velocity PDFs are tabulated and visualized for each case. Mean velocity shows minimal variation in Cases I, III, and IV, but maximum velocity fluctuates significantly in Case I (63.95\u2013187.45% of mean), Case II (50.15\u2013156.68%), and Case IV (63.70\u2013185.53% of mean), vital for duct design and analysis. Examining the effects of different parameters, the SFDHC method\u2019s rapid convergence reveals the fluid behavior index as the primary driver of maximum stochastic velocity, followed by aspect ratio and yield stress. These findings enhance applications in drilling fluid management, biomedical modeling (e.g., blood flow in vascular networks), and industrial processes involving non-Newtonian fluids, such as paints and slurries, providing a robust tool for advancing understanding and managing uncertainty in complex fluid dynamics.<\/jats:p>","DOI":"10.3390\/axioms14070492","type":"journal-article","created":{"date-parts":[[2025,6,24]],"date-time":"2025-06-24T08:50:57Z","timestamp":1750755057000},"page":"492","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Uncertainty Quantification of Herschel\u2013Bulkley Fluids in Rectangular Ducts Due to Stochastic Parameters and Boundary Conditions"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5064-1119","authenticated-orcid":false,"given":"Osama Hussein","family":"Galal","sequence":"first","affiliation":[{"name":"Engineering Mathematics and Physics Department, Faculty of Engineering, Fayoum University, Fayoum 63514, Egypt"}]},{"given":"Eman","family":"Alruwaili","sequence":"additional","affiliation":[{"name":"Department of mathematics, Faculty of Science, Northern Border University, Arar 73222, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2025,6,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"291","DOI":"10.1007\/BF01432034","article-title":"Konsistenzmessungen von Gummi-Benzoll\u00f6sungen","volume":"39","author":"Herschel","year":"1926","journal-title":"Kolloid-Zeitschrift"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"128","DOI":"10.1016\/j.jnnfm.2006.07.009","article-title":"Flow of Herschel-Bulkley fluids through the Marsh cone","volume":"139","author":"Nguyen","year":"2006","journal-title":"J. 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