Journal of Non-Newtonian Fluid Mechanics, 25 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSR (1987) 239-259 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands zyxwvutsrqponmlkjihgf 239 zyxwvuts STOKES’ FIRST PROBLEM FOR VISCOELASTIC FLUIDS LUIGI PREZIOSI and DANIEL D. JOSEPH Aerospace Engineering and Mechanics, University of Minnesota, 110 Union Street SE, Minneapolis, MN 55455 (U.S.A.) (Received July 17, 1986; in revised form December 11, 1986) The theory given in this paper is based on a generalization of Boltzmann’s equation of linear viscoelasticity in which the presence of a Newtonian viscosity is acknowledged. The solution of Stokes’ first problem for this kind of fluid, with a viscosity and a relaxation kernel, are derived here for the first time. The formulas given in this paper form a basis for the numerical interpretation of the idea of an effective viscosity and relaxation modulus. 1. Introduction Stokes’ first problem is concerned with diffusion of vorticity in a fluid occupying a semi-infinite region above a plate which undergoes a step increase of velocity from rest. This problem is the same as the diffusion of heat when the temperature at the boundary is suddenly increased. In this paper, we study this problem when the fluids are viscoelastic and satisfy a constitutive equation in which the effects of relaxing elasticity and Newtonian viscosity are simultaneously acknowledged. Our solutions are presented for their intrinsic value and as a contribution to the theory of the wave-speed meter (see Joseph et al. [l]). The wave-speed meter is a rheomet- rical instrument for measuring the speed of shear waves in liquids. Wave speeds and elastic moduli for many different liquids were measured by Joseph et al. [2]. The constitutive equation used here is a generalization of the one used by Boltzmann, in which the excess stress 7 is ‘determined by 7=2pO[u(x, t)] +~mG(~)D[u(x, t--s)] ds, 0.1) 0377-0257/87/$03.50 0 1987 Elsevier Science Publishers B.V.