# zbMATH — the first resource for mathematics

Steady plane flow of viscoelastic fluid past an obstacle. (English) Zbl 1090.35038
Steady plane flow of certain classes of viscoelastic fluids in exterior domains is studied. The authors consider models for which the viscous part of the stress tensor satisfies the relation $$T = \lambda [(v\cdot \nabla ) T + TW -WT] + B(D,T) = 2 \eta D$$, where $$\lambda$$ and $$\eta$$ are positive constants, $$D$$ is the symmetric and $$W$$ the skew symmetric part of the gradient of velocity $$v$$ and $$B(\cdot ,\cdot )$$ is a bilinear tensor function.
The main result of the paper is the asymptotic structure of the solution near infinity. Denoting by $$v_\infty$$ the nonzero velocity prescribed at infinity (without loss of generality, the only nonzero component is the component in the $$x_1$$ direction) the authors show that $$| v_1- | v_\infty | | \leq C | x| ^{-\frac 12} (1 + s(x))^{-\frac 12}$$ and $$| v_2| \leq C | x| ^{-1} \ln (2+ | x| )^{-1}$$ as $$| x| \to \infty$$ (here $$s(x) = | x| -x_1$$), i.e. the velocity decays as fast as the fundamental Oseen tensor (up to a logarithmic term) provided the force decays sufficiently fast at infinity and the force, the velocity at infinity and the Reynolds and Weisenberg numbers are sufficiently small.
The proof is based on the integral representation of the solution, on estimates of convolutions with kernels corresponding to the fundamental Oseen tensor shown in S. Kračmar, A. Novotný and M. Pokorný [J. Math. Soc. Japan 53, 59–111 (2001; Zbl 0988.76021)] and the properties of the modified Oseen problem studied in R. Farwig, A. Novotný and M. Pokorný [Z. Anal. Anwend. 19, 713–728 (2000; Zbl 0969.35005)].

##### MSC:
 35B40 Asymptotic behavior of solutions to PDEs 76D99 Incompressible viscous fluids 35Q35 PDEs in connection with fluid mechanics
Full Text:
##### References:
 [1] J. Baranger, C. Guillopé and J. C. Saut: Mathematical analysis of differential models for viscoelastic fluids. Rheology for Polymer Melts Processing, Chapt. II, Elsevier Science, Amsterdam, 1996. [2] P. Dutto: Solutions physiquement raisonnables des équations de Navier-Stokes compressibles stationnaires dans un domaine extérieur du plan. Ph.D. thesis, University of Toulon, 1998. [3] R. Farwig, A. Novotný and M. Pokorný: The fundamental solution of a modified Oseen problem. Z. Anal. Anwendungen 19 (2000), 713-728. · Zbl 0969.35005 [4] G. P. Galdi: An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. I: Linearized Steady Problems. Springer Tracts in Natural Philosophy, Vol. 38. Springer-Verlag, New York, 1994. · Zbl 0949.35004 [5] S. Kračmar, M. Novotný and M. Pokorný: Estimates of Oseen kernels in weighted $$L^p$$ spaces. Journal of Mathematical Society of Japan 53 (2001), 59-111. · Zbl 0988.76021 [6] A. Novotný: About the steady transport equation. Proceedings of Fifth Winter School at Paseky, Pitman Research Notes in Mathematics, 1998. [7] A. Novotný: Some Topics in the Mathematical Theory of Compressible Navier-Stokes Equations. Lecture Notes, Lipschitz Vorlesung. Univ. Bonn, to appear. [8] A. Novotný, M. Padula: Physically reasonable solutions to steady Navier-Stokes equations in 3-D exterior domains II ($$v_\infty \neq 0$$). Math. Ann. 308 (1997), 439-489. · Zbl 0883.76072 [9] A. Novotný, M. Pokorný: Three-dimensional steady flow of viscoelastic fluid past an obstacle. J. Math. Fluid Mech. 2 (2000), 294-314. · Zbl 0974.35094 [10] M. Pokorný: Asymptotic behaviour of solutions to certain PDE’s describing the flow of fluids in unbounded domains. Ph.D. thesis, Charles University, Prague & University of Toulon and Var, Toulon-La Garde, 1999. [11] M. Renardy: Existence of slow steady flows of viscoelastic fluid with differential constitutive equations. Z. Angew. Math. Mech. 65 (1985), 449-451. · Zbl 0577.76014 [12] D. R. Smith: Estimates at infinity for stationary solutions of the Navier-Stokes equations in two dimensions. Arch. Rational Mech. Anal. 20 (1965), 341-372. · Zbl 0149.44701 [13] B. O. Turesson: Nonlinear Potential Theory and Weighted Sobolev Spaces. Lecture Notes in Mathematics Vol. 1736. Springer-Verlag, Berlin-Heidelberg, 2000.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.