Použití deflační metody ke konstrukci bifurkačních diagramů pro proudění viskoleastických tekutin

• Thesis title in Czech: Použití deflační metody ke konstrukci bifurkačních diagramů pro proudění viskoleastických tekutin
• Thesis title in English: Bifurcation analysis of viscoelastic flows using deflation method
• Academic year of topic announcement: 2022/2023
• Thesis type: diploma thesis
• Department: Mathematical Institute of Charles University (32-MUUK)
• Supervisor: doc. Mgr. Vít Průša, Ph.D.
• Author: hidden - assigned and confirmed by the Study Dept.
• Date of registration: 08.03.2023
• Date of assignment: 07.06.2023
• Confirmed by Study dept. on: 28.06.2023
• Advisors: RNDr. Karel Tůma, Ph.D.

Guidelines

The preliminary work plan is the following:

(1) Get familiar with methods for construction of Lyapunov functionals guaranteeing sufficient conditions for flow stability, see Dostalík et al. (2019) and Dostalík and Průša (2022). Compare the results with linearised stability analysis.
(2) Get familiar with the deflation method for construction of bifurcation diagrams, see Farrell et al. (2015) and Boullé et al. (2022), proceed from simple cases to more complex scenarios.
(3) Implement a numerical scheme for solution of the governing equations for the given viscoelastic fluid.
(4) Use the deflation method and the numerical scheme from the previous step and construct the bifurcation diagram for the flow of a viscoelastic fluid in between two rotating cylinders.

The thesis objective is quite ambitious and it might be adjusted as the thesis progresses.

References

Boullé, N., V. Dallas, and P. E. Farrell (2022). Bifurcation analysis of two-dimensional Rayleigh–Bénard convection using deflation. Phys. Rev. E 105, 055106.

Datta, S. S., A. M. Ardekani, P. E. Arratia, A. N. Beris, I. Bischofberger, G. H. McKinley, J. G. Eggers, J. E. López-Aguilar, S. M. Fielding, A. Frishman, M. D. Graham, J. S. Guasto, S. J. Haward, A. Q. Shen, S. Hormozi, A. Morozov, R. J.

Poole, V. Shankar, E. S. G. Shaqfeh, H. Stark, V. Steinberg, G. Subramanian, and H. A. Stone (2022). Perspectives on viscoelastic flow instabilities and elastic turbulence. Phys. Rev. Fluids 7, 080701.

Dostalík, M. and V. Průša (2022). Non-linear stability and non-equilibrium thermodynamics–there and back again. J. Non-Equilib. Thermodyn. 47 (2), 205–215.

Dostalík, M., V. Průša, and K. Tůma (2019). Finite amplitude stability of internal steady flows of the Giesekus viscoelastic rate-type fluid. Entropy 21 (12).

Farrell, P. E., A. Birkisson, and S. W. Funke (2015). Deflation techniques for finding distinct solutions of nonlinear partial differential equations. SIAM J. Sci. Comp. 37 (4), A2026–A2045.

Preliminary scope of work in English

Viscoelastic fluids are encountered in many applications ranging from biology and geophysics to polymer processing. The governing equations for these fluids are more complex than those for the standard Navier--Stokes fluids, in particular, the governing equations for viscoelastic fluids contain additional variables and nonlinear terms. Due to the presence of additional nonlinear terms, the stability analysis of flows of these fluids is more challenging than for the Navier--Stokes fluid. For example, the flows of viscoelastic fluids can become unstable even in the small Reynolds number regime, wherein the instability is not triggered by the standard convective term, but by another nonlinear term in the equations. This leads to a phenomenon called ``elastic turbulence''.

Thesis objective is to analyse stability/instability of flows of viscoelastic fluids. In the ideal case the thesis will lead to the construction of bifurcation diagram for the flow of a viscoelastic fluid (Giesekus fluid) in between two rotating cylinders (Taylor--Couette like geometry).