Abstract
In this paper the coupled fluid-structure interaction problem for incompressible non-Newtonian shear-dependent fluid flow in two-dimensional time-dependent domain is studied. One part of the domain boundary consists of an elastic wall. Its temporal evolution is governed by the generalized string equation with action of the fluid forces by means of the Neumann type boundary condition. The aim of this work is to present the limiting process for the auxiliary \((\kappa,\varepsilon,k)\)-problem. The weak solution of this auxiliary problem has been studied in our recent work (Hundertmark-Zaušková, Lukáčová-Medvid’ová, Nečasová, On the existence of weak solution to the coupled fluid-structure interaction problem for non-Newtonian shear-dependent fluid, J. Math. Soc. Japan (in press)).
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Notes
- 1.
We use here notations \(\|\cdot \|_{p}:=\| \cdot \|_{L^{p}(D)},\|\cdot \|_{1,p}:=\| \cdot \|_{W^{1,p}(D)}\).
- 2.
Since \(\mbox{ $\varphi $}(x_{t+\tau }) =\bar{ \mbox{ $v$}}_{\gamma }(x_{t+2\tau },t + 2\tau ) -\bar{\mbox{ $v$}}_{\gamma }(x_{t+\tau },t+\tau )\), we have to integrate over \(\int _{0}^{T-2\tau }dt\) in the estimate of the term (II), or we define \(\mbox{ $\varphi $}(x_{t+\tau }) = 0\) if t +τ > T.
- 3.
For p = 2 this estimate is valid for \(\mbox{ $\psi $} \in L^{p}(0,T; \mbox{ $V $}) \cap L^{4}((0,T) \times D)\), cf. [6].
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Acknowledgements
The present research has been financed by the DFG project ZA 613/1-1, the Nečas Centrum for Mathematical Modelling LC06052 (financed by MSMT) and the Grant of the Czech Republic, No. P201/11/1304. It has also been partially supported by the 6th EU-Framework Programme under the Contract No. DEASE: MEST-CT-2005-021122 and the DST-DAAD project based personnel exchange program with Indian Institute of Science, Bangalore. We would like to thank Ján Filo (Comenius University, Bratislava) for fruitful discussions on the topic.
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Hundertmark, A., Lukáčová-Medviďová, M., Nečasová, Š. (2016). On the Weak Solution of the Fluid-Structure Interaction Problem for Shear-Dependent Fluids. In: Amann, H., Giga, Y., Kozono, H., Okamoto, H., Yamazaki, M. (eds) Recent Developments of Mathematical Fluid Mechanics. Advances in Mathematical Fluid Mechanics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0939-9_16
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