Bifurcation of solutions through a contact manifold in bidisperse models
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 b...
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| Format: | Article |
| Language: | English |
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Frontiers Media S.A.
2023-06-01
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| Series: | Frontiers in Applied Mathematics and Statistics |
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| Online Access: | https://www.frontiersin.org/articles/10.3389/fams.2023.1199011/full |
| _version_ | 1797796683958976512 |
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| author | Stefan Berres Pablo Castañeda |
| author_facet | Stefan Berres Pablo Castañeda |
| author_sort | Stefan Berres |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-03-13T03:37:48Z |
| format | Article |
| id | doaj.art-bc38c66ac0b64348b8531a696ace22b7 |
| institution | Directory Open Access Journal |
| issn | 2297-4687 |
| language | English |
| last_indexed | 2024-03-13T03:37:48Z |
| publishDate | 2023-06-01 |
| publisher | Frontiers Media S.A. |
| record_format | Article |
| series | Frontiers in Applied Mathematics and Statistics |
| spelling | doaj.art-bc38c66ac0b64348b8531a696ace22b72023-06-23T16:07:03ZengFrontiers Media S.A.Frontiers in Applied Mathematics and Statistics2297-46872023-06-01910.3389/fams.2023.11990111199011Bifurcation of solutions through a contact manifold in bidisperse modelsStefan Berres0Pablo Castañeda1Department of Information Systems, Universidad del Bío-Bío, Concepción, ChileAcademic Department of Mathematics, ITAM, Ciudad de México, MexicoThis 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.https://www.frontiersin.org/articles/10.3389/fams.2023.1199011/fullsystem of non-linear conservation lawsquasi-umbilic pointcontact manifoldHugoniot locusRiemann problem |
| spellingShingle | Stefan Berres Pablo Castañeda Bifurcation of solutions through a contact manifold in bidisperse models Frontiers in Applied Mathematics and Statistics system of non-linear conservation laws quasi-umbilic point contact manifold Hugoniot locus Riemann problem |
| title | Bifurcation of solutions through a contact manifold in bidisperse models |
| title_full | Bifurcation of solutions through a contact manifold in bidisperse models |
| title_fullStr | Bifurcation of solutions through a contact manifold in bidisperse models |
| title_full_unstemmed | Bifurcation of solutions through a contact manifold in bidisperse models |
| title_short | Bifurcation of solutions through a contact manifold in bidisperse models |
| title_sort | bifurcation of solutions through a contact manifold in bidisperse models |
| topic | system of non-linear conservation laws quasi-umbilic point contact manifold Hugoniot locus Riemann problem |
| url | https://www.frontiersin.org/articles/10.3389/fams.2023.1199011/full |
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