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A semismooth Newton solution of the steady-state non-isothermal Bingham flow with temperature dependent nonlocal parameters. (English) Zbl 1471.76045

Vermolen, Fred J. (ed.) et al., Numerical mathematics and advanced applications. ENUMATH 2019. Proceedings of the European conference, Egmond aan Zee, The Netherlands, September 30 – October 4, 2019. Cham: Springer. Lect. Notes Comput. Sci. Eng. 139, 479-487 (2021).
Summary: In this paper, we discuss the numerical solution of the non-isothermal steady-state Bingham flow considering that the viscosity and the yield limit variate with temperature. In the present contribution, we focus on the asymptotic limit case of high thermal conductivity. In this case, the energy equation collapses into an implicit energy equation which involves the viscosity and the yield stress functions, while the temperature becomes a constant solution for this equation. Once we obtain the coupled limit system of this energy equation and the classical Bingham variational inequality of the second kind, we propose a mixed formulation for the resulting limit variational inequality and a finite element discretization for the resulting system of PDEs. Next, we develop a semismooth Newton algorithm for the coupled flow model. Finally, we carry on several numerical experiments for validating our method.
For the entire collection see [Zbl 1471.65009].

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
76A10 Viscoelastic fluids
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