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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)].

35B40 Asymptotic behavior of solutions to PDEs
76D99 Incompressible viscous fluids
35Q35 PDEs in connection with fluid mechanics
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