{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T11:20:19Z","timestamp":1753874419079,"version":"3.41.2"},"reference-count":51,"publisher":"AIP Publishing","issue":"11","funder":[{"name":"COMPETE 2020 and FCT","award":["MIT-EXPL\/TDI\/0038\/2019"],"award-info":[{"award-number":["MIT-EXPL\/TDI\/0038\/2019"]}]},{"name":"FCT","award":["FCT\/2022.00753.CEECIND"],"award-info":[{"award-number":["FCT\/2022.00753.CEECIND"]}]},{"name":"FCT","award":["UIDB\/00013\/2020","UIDP\/00013\/2020"],"award-info":[{"award-number":["UIDB\/00013\/2020","UIDP\/00013\/2020"]}]},{"name":"FCT","award":["2022.06672.PTDC"],"award-info":[{"award-number":["2022.06672.PTDC"]}]},{"name":"ALiCE","award":["LA\/P\/0045\/2020"],"award-info":[{"award-number":["LA\/P\/0045\/2020"]}]},{"name":"FCT","award":["UIDB\/00532\/2020","UIDP\/00532\/2020"],"award-info":[{"award-number":["UIDB\/00532\/2020","UIDP\/00532\/2020"]}]}],"content-domain":{"domain":["pubs.aip.org"],"crossmark-restriction":true},"short-container-title":[],"published-print":{"date-parts":[[2023,11,1]]},"abstract":"<jats:p>In multiphase flows, accurately modeling the interaction between the liquid phase of complex fluids and a porous medium of solid spheres poses a fundamental challenge. The dynamics of moderately dense non-colloidal suspensions constituted by static random arrays of mono-disperse spherical particles in non-linear viscoelastic fluids is studied numerically. This numerical study consists of about 9000 different systems, in which the volume fraction \u03d5\u2009(0.04\u2264\u03d5\u22640.2) of the dispersed solid phase, the Reynolds number Re(5\u2264Re\u226450), the solvent viscosity ratio \u03b2(0.05\u2264\u03b2\u22640.9), the Weissenberg number Wi(0.5\u2264Wi\u22644), and the mobility parameter of the Giesekus model \u03b1\u2009(0.1\u2264\u03b1\u22640.5) were varied to understand the particle's interactions with the viscoelastic suspending fluid. We aim to investigate the relationship between the volume fraction of the dispersed solid phase and the non-linear rheology of shear-thinning viscoelastic fluids with the normalized average drag force \u27e8F\u27e9. In addition, by assessing the flow patterns predicted numerically, we were able to provide a characterization of the velocity and stress fields as a function of the simulation parameters.<\/jats:p>","DOI":"10.1063\/5.0171761","type":"journal-article","created":{"date-parts":[[2023,11,22]],"date-time":"2023-11-22T11:19:02Z","timestamp":1700651942000},"update-policy":"https:\/\/doi.org\/10.1063\/aip-crossmark-policy-page","source":"Crossref","is-referenced-by-count":3,"title":["Numerical simulations of suspensions of rigid spheres in shear-thinning viscoelastic fluids"],"prefix":"10.1063","volume":"35","author":[{"ORCID":"https:\/\/orcid.org\/0009-0004-0072-0865","authenticated-orcid":false,"given":"O.","family":"Ayar","sequence":"first","affiliation":[{"name":"Faculty of Engineering - University of Porto (FEUP) 1 , Rua Dr. Roberto Frias s\/n, 4200-465 Porto, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0340-3556","authenticated-orcid":false,"given":"C.","family":"Fernandes","sequence":"additional","affiliation":[{"name":"Transport Phenomena Research Center (CEFT), Faculty of Engineering - University of Porto (FEUP) 2 , Rua Dr. Roberto Frias s\/n, 4200-465 Porto, Portugal"},{"name":"Centre of Mathematics (CMAT), University of Minho 3 , Campus of Gualtar, 4710-057 Braga, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5477-3226","authenticated-orcid":false,"given":"L. L.","family":"Ferr\u00e1s","sequence":"additional","affiliation":[{"name":"Centre of Mathematics (CMAT), University of Minho 3 , Campus of Gualtar, 4710-057 Braga, Portugal"},{"name":"Department of Mechanical Engineering (Section of Mathematics), Faculty of Engineering - University of Porto (FEUP) 4 , Rua Dr. Roberto Frias s\/n, 4200-465 Porto, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6585-6609","authenticated-orcid":false,"given":"M. A.","family":"Alves","sequence":"additional","affiliation":[{"name":"Transport Phenomena Research Center (CEFT), Faculty of Engineering - University of Porto (FEUP) 2 , Rua Dr. Roberto Frias s\/n, 4200-465 Porto, Portugal"},{"name":"Associate Laboratory in Chemical Engineering (ALiCE), Department of Chemical Engineering, Faculty of Engineering - University of Porto (FEUP) 5 , Rua Dr. Roberto Frias s\/n, 4200-465 Porto, Portugal"}]}],"member":"317","published-online":{"date-parts":[[2023,11,22]]},"reference":[{"key":"2023112211164048900_c1","doi-asserted-by":"publisher","first-page":"80","DOI":"10.1016\/j.jnnfm.2014.09.014","article-title":"Particle dynamics in viscoelastic liquids","volume":"215","year":"2015","journal-title":"J. Non-Newtonian Fluid Mech."},{"key":"2023112211164048900_c2","doi-asserted-by":"publisher","first-page":"5395","DOI":"10.1007\/s00366-022-01626-5","article-title":"Finite volume simulations of particle-laden viscoelastic fluid flows: Application to hydraulic fracture processes","volume":"38","year":"2022","journal-title":"Eng. Computers"},{"key":"2023112211164048900_c3","doi-asserted-by":"publisher","first-page":"415","DOI":"10.1146\/annurev-chembioeng-080615-033630","article-title":"Complex fluids and hydraulic fracturing","volume":"7","year":"2016","journal-title":"Annu. Rev. Chem. Biomol. Eng."},{"key":"2023112211164048900_c4","first-page":"1","article-title":"A review of the rheology of filled viscoelastic systems","volume":"1","year":"2003","journal-title":"Rheol. Rev."},{"key":"2023112211164048900_c5","doi-asserted-by":"publisher","first-page":"271","DOI":"10.1016\/0377-0257(82)80019-0","article-title":"The motion of rigid particles in viscoelastic fluids","volume":"7","year":"1980","journal-title":"J. Non-Newtonian Fluid Mech."},{"key":"2023112211164048900_c6","doi-asserted-by":"publisher","first-page":"1379","DOI":"10.1122\/1.5024698","article-title":"Mechanism of shear thickening in suspensions of rigid spheres in Boger fluids. Part II: Suspensions at finite concentration","volume":"62","year":"2018","journal-title":"J. Rheol."},{"key":"2023112211164048900_c7","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1002\/aic.16575","article-title":"On the rheology of particle suspensions in viscoelastic fluids","volume":"65","year":"2019","journal-title":"AIChE J."},{"key":"2023112211164048900_c8","doi-asserted-by":"publisher","first-page":"113","DOI":"10.1038\/s41378-020-00218-x","article-title":"Viscoelastic microfluidics: Progress and challenges","volume":"6","year":"2020","journal-title":"Microsyst. Nanoeng."},{"key":"2023112211164048900_c9","doi-asserted-by":"publisher","first-page":"169","DOI":"10.1007\/s00397-009-0404-6","article-title":"A differential model for the rheological properties of concentrated suspensions with weakly viscoelastic matrices","volume":"49","year":"2010","journal-title":"Rheol. Acta"},{"key":"2023112211164048900_c10","doi-asserted-by":"publisher","first-page":"013301","DOI":"10.1103\/PhysRevFluids.1.013301","article-title":"Stress in a dilute suspension of spheres in a dilute polymer solution subject to simple shear flow at finite Deborah numbers","volume":"1","year":"2016","journal-title":"Phys. Rev. Fluids"},{"key":"2023112211164048900_c11","doi-asserted-by":"publisher","first-page":"013301","DOI":"10.1103\/PhysRevFluids.3.013301","article-title":"Einstein viscosity with fluid elasticity","volume":"3","year":"2018","journal-title":"Phys. Rev. Fluids"},{"key":"2023112211164048900_c12","doi-asserted-by":"publisher","first-page":"1331","DOI":"10.1122\/1.2998219","article-title":"Rotation of a sphere in a viscoelastic liquid subjected to shear flow. Part I: Simulation results","volume":"52","year":"2008","journal-title":"J. Rheol."},{"key":"2023112211164048900_c13","doi-asserted-by":"publisher","first-page":"459","DOI":"10.1122\/1.3073052","article-title":"Rotation of a sphere in a viscoelastic liquid subjected to shear flow. Part II. Experimental results","volume":"53","year":"2009","journal-title":"J. Rheol."},{"volume-title":"Understanding Viscoelasticity: An Introduction to Rheology, Graduate Texts in Physics","year":"2012","key":"2023112211164048900_c14"},{"key":"2023112211164048900_c15","doi-asserted-by":"publisher","first-page":"351","DOI":"10.1016\/0045-7825(95)00988-4","article-title":"Simulation of multiple spheres falling in a liquid-filled tube","volume":"134","year":"1996","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"2023112211164048900_c16","doi-asserted-by":"publisher","first-page":"82","DOI":"10.1007\/s10404-019-2245-7","article-title":"Numerical simulations on the dynamics of a particle pair in a viscoelastic fluid in a microchannel: Effect of rheology, particle shape, and confinement","volume":"23","year":"2019","journal-title":"Microfluid. Nanofluid."},{"key":"2023112211164048900_c17","doi-asserted-by":"publisher","first-page":"7146","DOI":"10.1103\/PhysRevE.61.7146","article-title":"Navier-Stokes simulation with constraint forces: Finite-difference method for particle-laden flows and complex geometries","volume":"61","year":"2000","journal-title":"Phys. Rev. E"},{"key":"2023112211164048900_c18","doi-asserted-by":"publisher","first-page":"64","DOI":"10.1115\/1.1530636","article-title":"A method for particle simulation","volume":"70","year":"2003","journal-title":"J. Appl. Mech."},{"key":"2023112211164048900_c19","doi-asserted-by":"publisher","first-page":"166","DOI":"10.1016\/j.jcp.2005.06.021","article-title":"An explicit finite-difference scheme for simulation of moving particles","volume":"212","year":"2006","journal-title":"J. Comput. Phys."},{"key":"2023112211164048900_c20","doi-asserted-by":"publisher","first-page":"89","DOI":"10.1002\/cnm.468","article-title":"Interface tracking finite volume method for complex solid-fluid interactions on fixed meshes","volume":"18","year":"2001","journal-title":"Commun. Numer. Methods Eng."},{"key":"2023112211164048900_c21","doi-asserted-by":"publisher","first-page":"239","DOI":"10.1146\/annurev.fluid.37.061903.175743","article-title":"Immersed boundary methods","volume":"37","year":"2005","journal-title":"Annu. Rev. Fluid Mech."},{"key":"2023112211164048900_c22","doi-asserted-by":"publisher","first-page":"104218","DOI":"10.1016\/j.jnnfm.2019.104218","article-title":"A closure model for the drag coefficient of a sphere translating in a viscoelastic fluid","volume":"277","year":"2020","journal-title":"J. Non-Newtonian Fluid Mech."},{"key":"2023112211164048900_c23","doi-asserted-by":"publisher","first-page":"75","DOI":"10.1016\/j.jnnfm.2017.04.007","article-title":"A novel Lattice Boltzmann method for the dynamics of rigid particles suspended in a viscoelastic medium","volume":"244","year":"2017","journal-title":"J. Non-Newtonian Fluid Mech."},{"key":"2023112211164048900_c24","doi-asserted-by":"publisher","first-page":"109487","DOI":"10.1016\/j.jcp.2020.109487","article-title":"An immersed boundary-lattice Boltzmann method for fluid-structure interaction problems involving viscoelastic fluids and complex geometries","volume":"415","year":"2020","journal-title":"J. Comput. Phys."},{"key":"2023112211164048900_c25","doi-asserted-by":"publisher","first-page":"109","DOI":"10.1016\/j.cma.2018.12.001","article-title":"A novel algorithm of immersed moving boundary scheme for fluid\u2013particle interactions in DEM\u2013LBM","volume":"346","year":"2019","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"2023112211164048900_c26","doi-asserted-by":"publisher","first-page":"26","DOI":"10.1016\/j.partic.2022.06.004","article-title":"Fully resolved simulations of viscoelastic suspensions by an efficient immersed boundary-lattice Boltzmann method","volume":"75","year":"2023","journal-title":"Particuology"},{"key":"2023112211164048900_c27","doi-asserted-by":"publisher","first-page":"121609","DOI":"10.1063\/1.4993610","article-title":"SPH modeling and simulation of spherical particles interacting in a viscoelastic matrix","volume":"29","year":"2017","journal-title":"Phys. Fluids"},{"key":"2023112211164048900_c28","doi-asserted-by":"publisher","first-page":"1993","DOI":"10.1122\/1.4896896","article-title":"Scaling laws for the flow of generalized Newtonian suspensions","volume":"58","year":"2014","journal-title":"J. Rheol."},{"key":"2023112211164048900_c29","doi-asserted-by":"publisher","first-page":"277","DOI":"10.1007\/s10035-011-0255-6","article-title":"Two-scale modeling of granular materials: A DEM-FEM approach","volume":"13","year":"2011","journal-title":"Granular Matter"},{"key":"2023112211164048900_c30","doi-asserted-by":"publisher","first-page":"78","DOI":"10.1016\/j.powtec.2016.09.052","article-title":"Comparative study on mesh-based and mesh-less coupled CFD-DEM methods to model particle-laden flow","volume":"305","year":"2017","journal-title":"Powder Technol."},{"key":"2023112211164048900_c31","doi-asserted-by":"publisher","first-page":"215","DOI":"10.1016\/j.pecs.2012.10.003","article-title":"Lagrangian-Eulerian methods for multiphase flows","volume":"39","year":"2013","journal-title":"Prog. Energy Combust. Sci."},{"key":"2023112211164048900_c32","doi-asserted-by":"publisher","first-page":"430","DOI":"10.3390\/polym14030430","article-title":"A meta-model to predict the drag coefficient of a particle translating in viscoelastic fluids: A machine learning approach","volume":"14","year":"2022","journal-title":"Polymers"},{"key":"2023112211164048900_c33","doi-asserted-by":"publisher","first-page":"499","DOI":"10.1122\/8.0000540","article-title":"Rheology of non-Brownian particle suspensions in viscoelastic solutions. Part 1: Effect of the polymer concentration","volume":"67","year":"2023","journal-title":"J. Rheol."},{"key":"2023112211164048900_c34","doi-asserted-by":"publisher","first-page":"517","DOI":"10.1122\/8.0000541","article-title":"Rheology of non-Brownian particle suspensions in viscoelastic solutions. Part II: Effect of a shear thinning suspending fluid","volume":"67","year":"2023","journal-title":"J. Rheol."},{"key":"2023112211164048900_c35","doi-asserted-by":"publisher","first-page":"69","DOI":"10.1016\/0377-0257(82)85016-7","article-title":"A simple constitutive equation for polymer fluids based on the concept of deformation-dependent tensorial mobility","volume":"11","year":"1982","journal-title":"J. Non-Newtonian Fluid Mech."},{"key":"2023112211164048900_c36","doi-asserted-by":"publisher","first-page":"147","DOI":"10.1080\/01495728408961817","article-title":"Enhancements of the simple method for predicting incompressible fluid flows","volume":"7","year":"1984","journal-title":"Numer. Heat Transfer"},{"key":"2023112211164048900_c37","doi-asserted-by":"publisher","first-page":"85","DOI":"10.1016\/j.jnnfm.2016.12.002","article-title":"Stabilization of an open-source finite-volume solver for viscoelastic fluid flows","volume":"239","year":"2017","journal-title":"J. Non-Newtonian Fluid Mech."},{"key":"2023112211164048900_c38","doi-asserted-by":"publisher","first-page":"63","DOI":"10.1016\/j.jnnfm.2017.09.008","article-title":"Improved both sides diffusion (iBSD): A new and straightforward stabilization approach for viscoelastic fluid flows","volume":"249","year":"2017","journal-title":"J. Non-Newtonian Fluid Mech."},{"key":"2023112211164048900_c39","doi-asserted-by":"publisher","first-page":"47","DOI":"10.1002\/fld.428","article-title":"A convergent and universally bounded interpolation scheme for the treatment of advection","volume":"41","year":"2003","journal-title":"Int. J. Numer. Methods Fluids"},{"key":"2023112211164048900_c40","doi-asserted-by":"publisher","first-page":"50","DOI":"10.1017\/S0305004100023197","article-title":"A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type","volume":"43","year":"1947","journal-title":"Math. Proc. Cambridge Philos. Soc."},{"key":"2023112211164048900_c41","doi-asserted-by":"publisher","first-page":"949","DOI":"10.1002\/nme.1620200511","article-title":"A robust incomplete Choleski-conjugate gradient algorithm","volume":"20","year":"1984","journal-title":"Int. J. Numer. Methods Eng."},{"key":"2023112211164048900_c42","doi-asserted-by":"publisher","first-page":"158","DOI":"10.1016\/S0021-9991(02)00052-9","article-title":"Incomplete LU preconditioning for large scale dense complex linear systems from electromagnetic wave scattering problems","volume":"185","year":"2003","journal-title":"J. Comput. Phys."},{"volume-title":"Preconditioned Conjugate Gradient Methods for Solving Systems of Algebraic Equations, Laboratory Note: Central Electricity Research Laboratories","year":"1981","key":"2023112211164048900_c43"},{"article-title":"The Open Source CFD Toolbox","year":"2004","author":"OpenFOAM","key":"2023112211164048900_c44"},{"key":"2023112211164048900_c45","doi-asserted-by":"publisher","first-page":"47","DOI":"10.1016\/j.jnnfm.2014.08.005","article-title":"Numerical simulation of the viscoelastic flow in a three-dimensional lid-driven cavity using the log-conformation reformulation in OpenFOAM","volume":"212","year":"2014","journal-title":"J. Non-Newtonian Fluid Mech."},{"key":"2023112211164048900_c46","doi-asserted-by":"publisher","first-page":"281","DOI":"10.1016\/j.jnnfm.2004.08.008","article-title":"Constitutive laws for the matrix-logarithm of the conformation tensor","volume":"123","year":"2004","journal-title":"J. Non-Newtonian Fluid Mech."},{"key":"2023112211164048900_c47","doi-asserted-by":"publisher","first-page":"23","DOI":"10.1016\/j.jnnfm.2004.12.003","article-title":"Time-dependent simulation of viscoelastic flows at high Weissenberg number using the log-conformation representation","volume":"126","year":"2005","journal-title":"J. Non-Newtonian Fluid Mech."},{"key":"2023112211164048900_c48","doi-asserted-by":"publisher","first-page":"831","DOI":"10.1007\/s00397-014-0800-4","article-title":"A model for the shear viscosity of non-colloidal suspensions with Newtonian matrix fluids","volume":"53","year":"2014","journal-title":"Rheol. Acta"},{"key":"2023112211164048900_c49","doi-asserted-by":"publisher","first-page":"213","DOI":"10.1017\/S0022112001005948","article-title":"The first effects of fluid inertia on flows in ordered and random arrays of spheres","volume":"448","year":"2001","journal-title":"J. Fluid Mech."},{"key":"2023112211164048900_c50","first-page":"1","article-title":"On the effect of the internal friction of fluids on the motion of pendulums","volume-title":"Mathematical and Physical Papers","year":"2009"},{"key":"2023112211164048900_c51","doi-asserted-by":"publisher","first-page":"415","DOI":"10.1063\/1.4823194","article-title":"Data reduction and error analysis for the physical sciences","volume":"7","year":"1993","journal-title":"Comput. Phys."}],"container-title":["Physics of Fluids"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/pubs.aip.org\/aip\/pof\/article-pdf\/doi\/10.1063\/5.0171761\/18224419\/113327_1_5.0171761.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/pubs.aip.org\/aip\/pof\/article-pdf\/doi\/10.1063\/5.0171761\/18224419\/113327_1_5.0171761.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,11,22]],"date-time":"2023-11-22T11:19:18Z","timestamp":1700651958000},"score":1,"resource":{"primary":{"URL":"https:\/\/pubs.aip.org\/pof\/article\/35\/11\/113327\/2923253\/Numerical-simulations-of-suspensions-of-rigid"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,11,1]]},"references-count":51,"journal-issue":{"issue":"11","published-print":{"date-parts":[[2023,11,1]]}},"URL":"https:\/\/doi.org\/10.1063\/5.0171761","relation":{},"ISSN":["1070-6631","1089-7666"],"issn-type":[{"type":"print","value":"1070-6631"},{"type":"electronic","value":"1089-7666"}],"subject":[],"published-other":{"date-parts":[[2023,11]]},"published":{"date-parts":[[2023,11,1]]},"article-number":"113327"}}