Výpočet proudění nestlačitelných tekutin mocninného typu s nízkými hodnotami mocninného indexu
| Thesis title in Czech: | Výpočet proudění nestlačitelných tekutin mocninného typu s nízkými hodnotami mocninného indexu |
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| Thesis title in English: | Computation of flows of incompressible power-law fluids for low values of power-law index |
| Key words: | nestlačitelná tekutina|tekutina mocninného typu|metoda konečných prvků|metoda deflace |
| English key words: | incompressible fluid|power-law fluid|finite-element method|deflation method |
| Academic year of topic announcement: | 2023/2024 |
| Thesis type: | diploma thesis |
| Thesis language: | |
| Department: | Mathematical Institute of Charles University (32-MUUK) |
| Supervisor: | Patrick Farrell |
| Author: | hidden - assigned and confirmed by the Study Dept. |
| Date of registration: | 07.12.2023 |
| Date of assignment: | 08.12.2023 |
| Confirmed by Study dept. on: | 08.12.2023 |
| Advisors: | prof. RNDr. Josef Málek, CSc., DSc. |
| Guidelines |
| The classical power-law models with the power-law exponent approaching its lower bound exhibit very sharp boundary layers near the walls provided that the flow is subject to the no-slip boundary condition. Such flows are computationally challenging and from the analytical viewpoint, at least for three-dimensional flows, it is known that multiple weak solutions exist.
The objective of the thesis is two-fold: to develop efficient computational code capable of computing two-dimensional flows in channels for low values of the power-law exponent, and to survey the results available for the problem, its analytical solutions, and associated PDE theory. For the computational part, a code implemented in the finite element system Firedrake will be developed. 1. Survey of the results and the methods from literature concerning PDE analysis, analytical solutions for unidirectional flows and for finite-element based coputations relevant to flows of incompressible power=law fluids. 2. Implement different variants of nonlinear solver with the objective to compute solution with the low power-law index. 3. Use the deflation method to see the possible existence of the other non-unidirectional solutions for flows in channels. |
| References |
| J. Blechta, J. Málek, K.R. Rajagopal: On the classification of incompressible fluids and a mathematical analysis of the equations that govern their motion, SIAM J. Math. Anal. 52 (2020) 1232–1289.
M. Bulíček, J. Málek, E. Maringová: On unsteady internal flows of incompressible fluids characterized by implicit constitutive equations in the bulk and on the boundary, J. Math. Fluid Mech. 25 (2023) Paper No. 72, 29 pp. J. Burczak, S. Modena, L. Székelyhidi: Non uniqueness of power-law flows. Comm. Math. Phys. 388 (2021) 199–243. P.E. Farrell, A. Birkisson, S.W. Funke: Deflation techniques for finding distinct solutions of nonlinear partial differential equations. SIAM J. Sci. Comput. 37 (2015) A2026–A2045. J. Hron, C. Le Roux, J. Málek, K.R. Rajagopal: Flows of Incom, pressible Fluids subject to Navier’s slip on the boundary, Computers and Mathematics with Applications 56 (2008) 2128–2143. and other literature given by the advisor. |
| Preliminary scope of work in English |
| The classical power-law models with the power-law exponent approaching its lower bound exhibit very sharp boundary layers near the walls provided that the flow is subject to the no-slip boundary condition. Such flows are computationally challenging and from the analytical viewpoint, at least for three-dimensional flows, it is known that multiple weak solutions exist.
The objective of the thesis is two-fold: to develop efficient computational code capable of computing two-dimensional flows in channels for low values of the power-law exponent, and to survey the results available for the problem, its analytical solutions, and associated PDE theory. For the computational part, a starting code implemented in the finite element system Firedrake is available. |