Z. angew. Math. Phys. 59 (2008) 834–847 0044-2275/08/050834-14 DOI 10.1007/s00033-007-6133-8 c  2007 Birkh¨auser Verlag, Basel Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP Exact solutions for the flow of an Oldroyd-B fluid due to an infinite flat plate D. Vieru, Corina Fetec˘au and C. Fetec˘au ∗ Abstract. The unsteady flow of an Oldroyd-B fluid due to an infinite flat plate, subject to a translation motion of linear time-dependent velocity in its plane, is studied by means of the Laplace transform. The velocity field and the associated tangential stress corresponding to the flow induced by the constantly accelerating plate as well as those produced by the impulsive motion of the plate are obtained as special cases. The solutions that have been determined, in all accordance with the solutions established using the Fourier transform, reduce to those for a Newtonian fluid as a limiting case. The similar solutions for a Maxwell fluid are also obtained. Keywords. Infinite plate, velocity field, tangential stress, exact solutions, Oldroyd-B fluid. 1. Introduction The investigation of the exact solutions for the equations of motion of non-New- tonian fluids, as well as for Navier–Stokes equations, is very important for many reasons. They provide a standard for checking the accuracies of many approximate methods which can be numerical or empirical. Although the computer techniques make the complete integration of the equations of motion of these fluids feasible, the accuracy of the results can be established only by comparison with an exact solution. The exact solutions can be also used as tests to verify numerical schemes that are developed to study more complex unsteady flow problems. Because of a great diversity in the physical structure of non-Newtonian fluids, many models have been proposed to describe the response characteristics of fluids that cannot be described by the classical Navier—Stokes fluid model. One of them corresponds to the Oldroyd-B model that is characterized by three material constants, a viscosity μ, a relaxation time λ and a retardation time λ r (<λ). The Cauchy stress T in such a model is given by [1-3] T = -pl + S, S + λ( ˙ S - LS - SL T )= μ  A + λ r ( ˙ A - LA - AL T )  , (1) where -pl denotes the indeterminate spherical stress, S is the extra-stress tensor, * Author for correspondence