Lukassen, Laura Johanna (2015)
A colored-noise Fokker-Planck equation for non-Brownian particles in shear-induced diffusion.
Technische Universität Darmstadt
Dissertation, Erstveröffentlichung
Kurzbeschreibung (Abstract)
The topic of the present work are non-Brownian particles in shear flow. As reported in literature, the occurring phenomenon in this context is shear-induced diffusion which takes place in the absence of the well-known Brownian diffusion. Diffusive processes can be described stochastically in terms of a stochastic differential equation (Langevin equation or Langevin-like equation) or a differential equation for the probability density, in second order referred to as Fokker-Planck equation. It is known that in contrast to Brownian diffusion, the shear-induced diffusion is a long-time diffusion which poses a challenge to the stochastic description of this phenomenon. The present work analyzes the problem of non-Brownian particles in shear-induced diffusion with regard to the Markov property of the treated variables. This concludes that the Fokker-Planck equation so far derived in pure position space may not be sufficient. In order to ensure the Markov process property, a Fokker-Planck equation extended to coupled position colored-noise velocity space is derived. Throughout the extension, the colored-noise velocity is modeled as an Ornstein-Uhlenbeck process. These first two steps were also treated in the author’s Master thesis (Lukassen 2012). A detailed substantiation of this approach is published in (Lukassen & Oberlack 2014b) including a new multiple time scale analysis and a Gaussian solution. The multiple time scale analysis results in the dimensionless form of the equation of motion which serves as a starting point for the derivation of the new colored-noise Fokker-Planck equation. As a next step, this coupled Fokker-Planck equation is integrated over velocity space and approximated to yield a reduced position-space Fokker-Planck equation. It is shown that such a reduction as in the present work is only possible under certain conditions concerning the correlation time. The resulting position-space equation is analyzed and compared to the traditional position-space models. The reduced form exhibits additional correction terms. In an outlook, possible extensions of the presented model are discussed with exemplary simulation results. Chapter 5 and 6 as a whole are based on the author’s publication (Lukassen & Oberlack 2014b).
| Typ des Eintrags: | Dissertation | ||||
|---|---|---|---|---|---|
| Erschienen: | 2015 | ||||
| Autor(en): | Lukassen, Laura Johanna | ||||
| Art des Eintrags: | Erstveröffentlichung | ||||
| Titel: | A colored-noise Fokker-Planck equation for non-Brownian particles in shear-induced diffusion | ||||
| Sprache: | Englisch | ||||
| Referenten: | Oberlack, Prof. Martin ; Bothe, Prof. Dieter | ||||
| Publikationsjahr: | 2015 | ||||
| Datum der mündlichen Prüfung: | 15 April 2015 | ||||
| URL / URN: | http://tuprints.ulb.tu-darmstadt.de/4587 | ||||
| Kurzbeschreibung (Abstract): | The topic of the present work are non-Brownian particles in shear flow. As reported in literature, the occurring phenomenon in this context is shear-induced diffusion which takes place in the absence of the well-known Brownian diffusion. Diffusive processes can be described stochastically in terms of a stochastic differential equation (Langevin equation or Langevin-like equation) or a differential equation for the probability density, in second order referred to as Fokker-Planck equation. It is known that in contrast to Brownian diffusion, the shear-induced diffusion is a long-time diffusion which poses a challenge to the stochastic description of this phenomenon. The present work analyzes the problem of non-Brownian particles in shear-induced diffusion with regard to the Markov property of the treated variables. This concludes that the Fokker-Planck equation so far derived in pure position space may not be sufficient. In order to ensure the Markov process property, a Fokker-Planck equation extended to coupled position colored-noise velocity space is derived. Throughout the extension, the colored-noise velocity is modeled as an Ornstein-Uhlenbeck process. These first two steps were also treated in the author’s Master thesis (Lukassen 2012). A detailed substantiation of this approach is published in (Lukassen & Oberlack 2014b) including a new multiple time scale analysis and a Gaussian solution. The multiple time scale analysis results in the dimensionless form of the equation of motion which serves as a starting point for the derivation of the new colored-noise Fokker-Planck equation. As a next step, this coupled Fokker-Planck equation is integrated over velocity space and approximated to yield a reduced position-space Fokker-Planck equation. It is shown that such a reduction as in the present work is only possible under certain conditions concerning the correlation time. The resulting position-space equation is analyzed and compared to the traditional position-space models. The reduced form exhibits additional correction terms. In an outlook, possible extensions of the presented model are discussed with exemplary simulation results. Chapter 5 and 6 as a whole are based on the author’s publication (Lukassen & Oberlack 2014b). |
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| Alternatives oder übersetztes Abstract: |
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| Freie Schlagworte: | Shear-induced diffusion, non-Brownian particles, colored noise, Fokker-Planck equation | ||||
| URN: | urn:nbn:de:tuda-tuprints-45873 | ||||
| Sachgruppe der Dewey Dezimalklassifikatin (DDC): | 500 Naturwissenschaften und Mathematik > 510 Mathematik 500 Naturwissenschaften und Mathematik > 530 Physik 600 Technik, Medizin, angewandte Wissenschaften > 620 Ingenieurwissenschaften und Maschinenbau |
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| Fachbereich(e)/-gebiet(e): | 16 Fachbereich Maschinenbau 16 Fachbereich Maschinenbau > Fachgebiet für Strömungsdynamik (fdy) Exzellenzinitiative Exzellenzinitiative > Graduiertenschulen Exzellenzinitiative > Graduiertenschulen > Graduate School of Computational Engineering (CE) Zentrale Einrichtungen |
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| Hinterlegungsdatum: | 05 Jul 2015 19:55 | ||||
| Letzte Änderung: | 20 Feb 2019 08:20 | ||||
| PPN: | |||||
| Referenten: | Oberlack, Prof. Martin ; Bothe, Prof. Dieter | ||||
| Datum der mündlichen Prüfung / Verteidigung / mdl. Prüfung: | 15 April 2015 | ||||
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| Suche nach Titel in: | TUfind oder in Google | ||||
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