{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T15:45:25Z","timestamp":1753890325855,"version":"3.41.2"},"reference-count":41,"publisher":"Frontiers Media SA","license":[{"start":{"date-parts":[[2023,6,23]],"date-time":"2023-06-23T00:00:00Z","timestamp":1687478400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100003141","name":"Consejo Nacional de Ciencia y Tecnolog\u00eda","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100003141","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["frontiersin.org"],"crossmark-restriction":true},"short-container-title":["Front. Appl. Math. Stat."],"abstract":"<jats:p>This research focuses on a hyperbolic system that describes bidisperse suspensions, consisting of two types of small particles dispersed in a viscous fluid. The dependence of solutions on the relative position of contact manifolds in the phase space is examined. The wave curve method serves as the basis for the first and second analyses. The former involves the classification of elementary waves that emerge from the origin of the phase space. Analytical solutions to prototypical Riemann problems connecting the origin with any point in the state space are provided. The latter focuses on semi-analytical solutions for Riemann problems connecting any state in the phase space with the maximum packing concentration line, as observed in standard batch sedimentation tests. When the initial condition crosses the first contact manifold, a bifurcation occurs. As the initial condition approaches the second manifold, another structure appears to undergo bifurcation, although it does not represent an actual bifurcation according to the triple shock rule. The study reveals important insights into the behavior of solutions in relation to these contact manifolds. This research sheds light on the existence of emerging quasi-umbilic points within the system, which can potentially lead to new types of bifurcations as crucial elements of the elliptic\/hyperbolic boundary in the system of partial differential equations. The implications of these findings and their significance are discussed.<\/jats:p>","DOI":"10.3389\/fams.2023.1199011","type":"journal-article","created":{"date-parts":[[2023,6,24]],"date-time":"2023-06-24T20:04:24Z","timestamp":1687637064000},"update-policy":"https:\/\/doi.org\/10.3389\/crossmark-policy","source":"Crossref","is-referenced-by-count":0,"title":["Bifurcation of solutions through a contact manifold in bidisperse models"],"prefix":"10.3389","volume":"9","author":[{"given":"Stefan","family":"Berres","sequence":"first","affiliation":[]},{"given":"Pablo","family":"Casta\u00f1eda","sequence":"additional","affiliation":[]}],"member":"1965","published-online":{"date-parts":[[2023,6,23]]},"reference":[{"key":"B1","doi-asserted-by":"publisher","first-page":"41","DOI":"10.1137\/S0036139902408163","article-title":"Strongly degenerate parabolic-hyperbolic systems modeling polydisperse sedimentation with compression","volume":"64","author":"Berres","year":"2003","journal-title":"SIAM J Appl Math."},{"key":"B2","doi-asserted-by":"publisher","first-page":"699","DOI":"10.1002\/1521-4001(200210)82:10<699::AID-ZAMM699>3.0.CO;2-#","article-title":"Model equations and instability regions for the sedimentation of polydisperse suspensions of spheres","volume":"82","author":"B\u00fcrger","year":"2002","journal-title":"Z Angew Math Mech."},{"key":"B3","doi-asserted-by":"publisher","first-page":"587","DOI":"10.1017\/S0956792503005266","article-title":"An n-populations model for traffic flow","volume":"14","author":"Benzoni-Gavage","year":"2003","journal-title":"Eur J Appl Math."},{"key":"B4","doi-asserted-by":"publisher","first-page":"401","DOI":"10.3934\/nhm.2011.6.401","article-title":"An adaptive finite-volume method for a model of two-phase pedestrian flow","volume":"6","author":"Berres","year":"2011","journal-title":"Netw Het Media."},{"key":"B5","doi-asserted-by":"publisher","first-page":"665","DOI":"10.1002\/zamm.200710343","article-title":"On Riemann problems and front tracking for a model of sedimentation of polydisperse suspensions","volume":"87","author":"Berres","year":"2007","journal-title":"Z Angew Math Mech."},{"key":"B6","doi-asserted-by":"publisher","first-page":"47","DOI":"10.1007\/s00707-008-0072-0","article-title":"On the settling of a bidisperse suspension with particles having different sizes and densities","volume":"201","author":"Berres","year":"2008","journal-title":"Acta Mecanica."},{"key":"B7","doi-asserted-by":"publisher","first-page":"2595","DOI":"10.1016\/S0009-2509(99)00536-9","article-title":"Particle segregation during pressure filtration for cast formation","volume":"55","author":"Biesheuvel","year":"2000","journal-title":"Chem Eng Sci."},{"key":"B8","doi-asserted-by":"crossref","first-page":"1166","DOI":"10.1016\/0009-2509(79)85026-5","article-title":"Hindered settling in a multiple-species particle system","volume":"34","author":"Masliyah","year":"1979","journal-title":"Chem Eng Sci."},{"key":"B9","doi-asserted-by":"publisher","first-page":"1815","DOI":"10.1016\/j.apm.2008.03.021","article-title":"On models of polydisperse sedimentation with particle-size-specific hindered-settling factors","volume":"33","author":"Basson","year":"2009","journal-title":"Appl Math Modell."},{"key":"B10","first-page":"35","article-title":"Sedimentation and fluidization: part I","volume":"32","author":"Richardson","year":"1954","journal-title":"Trans Inst Chem Engrs"},{"key":"B11","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-662-22019-1","volume-title":"Hyperbolic Conservation Laws in Continuum Physics.","author":"Dafermos","year":"2000"},{"key":"B12","doi-asserted-by":"publisher","first-page":"159","DOI":"10.1002\/num.20423","article-title":"A secular equation for the Jacobian matrix of certain multi-species kinematic flow models","volume":"26","author":"Donat","year":"2010","journal-title":"Num Methods Partial Diff Equat."},{"key":"B13","doi-asserted-by":"crossref","first-page":"409","DOI":"10.1090\/psapm\/067.2\/2605236","article-title":"On the spectrum of a rank two modification of a diagonal matrix for linearized fluxes modelling polydisperse sedimentation","volume-title":"Hyperbolic Problems. Theory, Numerics and Applications","author":"Berres","year":"2009"},{"key":"B14","doi-asserted-by":"publisher","first-page":"2186","DOI":"10.1137\/09076163X","article-title":"Hyperbolicity analysis of polydisperse sedimentation models via a secular equation for the flux Jacobian","volume":"70","author":"B\u00fcrger","year":"2010","journal-title":"SIAM J Appl Math."},{"key":"B15","first-page":"415","article-title":"On singular points for convex permeability models","volume-title":"Hyperbolic Problems: Theory, Numerics, Applications","author":"Casta\u00f1eda","year":"2014"},{"key":"B16","doi-asserted-by":"crossref","first-page":"1009","DOI":"10.1137\/0148059","article-title":"The Riemann problem near a hyperbolic singularity: the classification of solutions of quadratic Riemann problems","volume":"48","author":"Isaacson","year":"1988","journal-title":"SIAM J Appl Math."},{"key":"B17","first-page":"147","article-title":"Multiphase flow models with singular Riemann problems","volume":"11","author":"Isaacson","year":"1992","journal-title":"Mat Apl Comput."},{"key":"B18","first-page":"791","article-title":"Classification of the umbilic point in immiscible three-phase flow in porous media","volume-title":"Hyperbolic Problems: Theory, Numerics, Applications","author":"Matos","year":"2014"},{"key":"B19","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1002\/cpa.3160400202","article-title":"The classification of 2 x 2 systems of non-strictly hyperbolic conservation laws with application to oil recovery","volume":"40","author":"Schaeffer","year":"1987","journal-title":"Comm Pure and Appl Math."},{"key":"B20","doi-asserted-by":"crossref","first-page":"437","DOI":"10.2118\/16965-PA","article-title":"Extension of Stone's method 1 and conditions for real characteristic three-phase flow","volume":"4","author":"Fayers","year":"1989","journal-title":"SPE Reservoir Eng."},{"key":"B21","doi-asserted-by":"crossref","first-page":"250","DOI":"10.1016\/0022-0396(86)90037-9","article-title":"Analysis of a singular hyperbolic system of conservation laws","volume":"65","author":"Isaacson","year":"1986","journal-title":"J Differ Equ."},{"key":"B22","doi-asserted-by":"publisher","first-page":"14","DOI":"10.1007\/s10915-020-01392-w","article-title":"A class of Lagrangian-Eulerian shock-capturing schemes for first-order hyperbolic problems with forcing terms","volume":"86","author":"Abreu","year":"2021","journal-title":"J Sci Comput."},{"key":"B23","doi-asserted-by":"publisher","first-page":"171","DOI":"10.1007\/s10596-016-9556-5","article-title":"On a universal structure for immiscible three-phase flow in virgin reservoirs","volume":"20","author":"Casta\u00f1eda","year":"2016","journal-title":"Comput Geosci."},{"key":"B24","doi-asserted-by":"publisher","first-page":"99","DOI":"10.1007\/s11242-009-9508-9","article-title":"The solution by the wave curve method of three-phase flow in virgin reservoirs","volume":"83","author":"Azevedo","year":"2010","journal-title":"Transp Porous Media."},{"key":"B25","doi-asserted-by":"publisher","first-page":"10319","DOI":"10.1029\/2019WR025264","article-title":"Three-phase fractional-flow theory of foam-oil displacement in porous media with multiple steady states","volume":"55","author":"Tang","year":"2019","journal-title":"Water Resour Res."},{"key":"B26","doi-asserted-by":"publisher","first-page":"335","DOI":"10.1142\/S0219891613500124","article-title":"Riemann solutions for vertical flow of three phases in porous media: simple cases","volume":"10","author":"Rodr\u00edguez-Berm\u00fadez","year":"2013","journal-title":"J Hyperbolic Differ Equ."},{"key":"B27","first-page":"73","article-title":"Stability of singular fundamental solutions under perturbations for flow in porous media","volume":"11","author":"de","year":"1992","journal-title":"Mat Aplic Comp."},{"key":"B28","doi-asserted-by":"crossref","first-page":"303","DOI":"10.1006\/jdeq.1996.0053","article-title":"Structurally stable Riemann solutions","volume":"126","author":"Schecter","year":"1996","journal-title":"J Differ Equ."},{"key":"B29","doi-asserted-by":"publisher","first-page":"523","DOI":"10.1023\/A:1016634307145","article-title":"Classification of codimension-one Riemann solution","volume":"13","author":"Schecter","year":"2001","journal-title":"J Dyn Diff Equ."},{"key":"B30","doi-asserted-by":"publisher","first-page":"713","DOI":"10.1007\/s00033-002-8180-5","article-title":"Capillary instability in models for three-phase flow","volume":"53","author":"Azevedo","year":"2002","journal-title":"Zeitschrift f\u00fcr Angewandte Mathematik und Physik"},{"key":"B31","article-title":"Diffusive Riemann solutions for 3-phase flow in porous media","author":"Lozano Guerrero","year":"2020","journal-title":"CNMAC 2019 - XXXIX Congresso Nacional de Matematica Aplicada e Computacional"},{"key":"B32","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1016\/0022-0396(85)90107-X","article-title":"Stable viscosity matrices for systems of conservation laws","volume":"56","author":"Majda","year":"1985","journal-title":"J Diff Equ."},{"key":"B33","first-page":"881","article-title":"Loss of strict hyperbolicity for vertical three-phase flow in porous media","volume-title":"Hyperbolic Problems: Theory, Numerics, Applications.","author":"Rodr\u00edguez-Berm\u00fadez","year":"2014"},{"key":"B34","unstructured":"Structural stability of nonlinear waves for conservation laws\n            FurtadoF\n          New York University, New York, NY, United States1989"},{"key":"B35","doi-asserted-by":"crossref","first-page":"537","DOI":"10.1002\/cpa.3160100406","article-title":"Hyperbolic systems of conservation laws II","volume":"10","author":"Lax","year":"1957","journal-title":"Commun Pure Appl Math."},{"key":"B36","doi-asserted-by":"publisher","first-page":"1413","DOI":"10.1007\/s00033-014-0469-7","article-title":"Bifurcation under parameter change of Riemann solutions for nonstrictly hyperbolic systems","volume":"66","author":"Matos","year":"2015","journal-title":"Z Angew Math Phys."},{"key":"B37","doi-asserted-by":"crossref","first-page":"781","DOI":"10.1090\/S0002-9947-1983-0716850-2","article-title":"Systems of conservation laws with invariant submanifolds","volume":"280","author":"Temple","year":"1983","journal-title":"Trans Am Math Soc."},{"key":"B38","doi-asserted-by":"publisher","first-page":"105","DOI":"10.1007\/s00574-016-0125-2","article-title":"Identification of shock profile solutions for bidisperse suspensions","volume":"47","author":"Berres","year":"2016","journal-title":"Bull Braz Math Soc."},{"volume-title":"Generic 3-Parameter Families of Vector Fields: Unfoldings of Saddle, Focus, and Elliptic Singularities with Nilpotent Linear Parts","year":"1991","author":"Dumortier","key":"B39"},{"key":"B40","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1007\/s002200050582","article-title":"Bifurcation of nonclassical viscous shock profiles from the constant state","volume":"202","author":"Azevedo","year":"1999","journal-title":"Commun Math Phys."},{"key":"B41","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1023\/A:1022692413112","article-title":"On the influence of viscosity on Riemann solutions","volume":"10","author":"\u010cani\u0107","year":"1998","journal-title":"J Dyn Differ Equ."}],"container-title":["Frontiers in Applied Mathematics and Statistics"],"original-title":[],"link":[{"URL":"https:\/\/www.frontiersin.org\/articles\/10.3389\/fams.2023.1199011\/full","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,6,24]],"date-time":"2023-06-24T20:04:52Z","timestamp":1687637092000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.frontiersin.org\/articles\/10.3389\/fams.2023.1199011\/full"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,6,23]]},"references-count":41,"alternative-id":["10.3389\/fams.2023.1199011"],"URL":"https:\/\/doi.org\/10.3389\/fams.2023.1199011","relation":{},"ISSN":["2297-4687"],"issn-type":[{"type":"electronic","value":"2297-4687"}],"subject":[],"published":{"date-parts":[[2023,6,23]]},"article-number":"1199011"}}