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Analysis of Generalized Newtonian Fluids

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Topics in Mathematical Fluid Mechanics

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 2073))

Abstract

In this paper we want to present an optimal existence result for the steady motion of generalized Newtonian fluids. Moreover, we present an optimal error estimate for a FEM approximation of the corresponding steady p-Stokes system. The presented results are based on long lasting cooperations with L. Berselli, L. Diening, J. Málek, A. Prohl and J. Wolf.

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Notes

  1. 1.

    N stands for “nice”.

  2. 2.

    In the following we will not distinguish between a sequence a suitably chosen subsequences, i.e. a subsequence of (an) will be again denoted by (an).

  3. 3.

    Note that it is possible to modify the proof such that the limiting case \(p = \frac{3d} {d+2}\) is included. For that one has to choose a more regular basis of V p (Ω), which is always possible (cf. [28]). We do not go into the details because later (cf. Theorem 25) we will improve the lower bound substantially.

  4. 4.

    Note, that one can also adapt the approach in  [7, Sec. VI.4] to obtain the same results in two dimensions (cf. [3]).

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Acknowledgements

The author have been partially supported by the SFB/TR 71 “Geometric Partial Differential Equations”.

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  1. Institut of Applied Mathematics, Freiburg University, Freiburg, Germany

    Michael Růžička

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Růžička, M. (2013). Analysis of Generalized Newtonian Fluids. In: Topics in Mathematical Fluid Mechanics. Lecture Notes in Mathematics(), vol 2073. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36297-2_4

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