J. math. fluid mech. 7 (2005) 451–483 1422-6928/05/030451-33 c  2005 Birkh¨auser Verlag, Basel DOI 10.1007/s00021-004-0133-7 Journal of Mathematical Fluid Mechanics Steady Flows of Shear-Dependent Oldroyd-B Fluids around an Obstacle Nadir Arada and Ad´ elia Sequeira Communicated by G. P. Galdi Abstract. This work is concerned with the study of steady flows of an incompressible vis- coelastic fluid of Oldroyd type, with viscosity depending on the second invariant of the rate of deformation tensor in an exterior domain. We establish a result of existence and uniqueness of strong solutions for sufficiently small data and give estimates relating these solutions to those of the corresponding generalized Newtonian fluid. Mathematics Subject Classification (2000). 35Q35, 35M10, 76A10, 76D03. Keywords. Viscoelastic fluids, shear-dependent viscosity, generalized Newtonian fluid, exterior domain. 1. Introduction The mathematical analysis of the equations of motion of non-Newtonian viscoelas- tic fluids is very challenging. The constitutive equations may lead to highly nonlin- ear systems of partial differential equations of a combined elliptic-hyperbolic type (or parabolic-hyperbolic, for unsteady flows) and the complexity of these math- ematical models requires specific techniques of nonlinear analysis to investigate the behaviour of their solutions. Usually the original nonlinear system is written in a decoupled form, composed of a Stokes-like system and of a scalar transport equation, that are studied as two separate linear systems. The solvability for the original problem is obtained using a suitable fixed point argument. This technique has been successfully used in the last years for different viscoelastic fluids of differ- ential and rate type, in several geometries and in the particular case of an exterior domain (see e.g. [6], [8], [15], [16], [17] and the literature cited therein). As far as we know, generalizations of these models incorporating a non-New- tonian viscosity function have been rarely studied from the mathematical point of view. The well-posedness of the equations of motion of a generalized Oldroyd-B fluid with shear-dependent viscosity, recently obtained by N. Arada and A. Se-