Exact solutions for Couette and Poiseuille flows for fourth grade fluids T. Hayat, R. Ellahi, Islamabad, Pakistan, and F. M. Mahomed, Johannesburg, South Africa Received December 27, 2005; revised July 25, 2006 Published online: October 25, 2006 Ó Springer-Verlag 2006 Summary. Exact solutions for four types of flows between two parallel plates are presented, viz. Couette flow, plug flow, Poiseuille flow and generalized Couette flow. The nonlinear second-order ordinary dif- ferential equation for the velocity field is solved exactly in each case. These solutions are compared to those found by perturbation and homotopy analysis methods by Siddiqui et al. [1]. 1 Introduction Interest in the study of non-Newtonian fluids has been mainly motivated by their importance in many technological and industrial problems. Numerous authors cite a wide variety of appli- cations involving non-Newtonian fluids that include synthetic fibres, food stuffs, flow of polymer solutions, the extrusion of molten plastics and drilling oil and gas wells. Some inter- esting recent studies involving non-Newtonian fluids have been discussed by Hayat et al. [2]–[4], Fetecau and Fetecau [5]–[7], Tan and Xu [8], Tan and Masuoka [9], [10] and Chen et al. [11]. More recently Siddiqui et al. [1] considered the problem of steady plane Couette flows between two parallel plates sliding with respect to each other. They derived four different problems of a fourth grade fluid depending upon the relative motion of the sliding plates, i.e., (i) one plate moves and the other is at rest (simple Couette flow), (ii) both plates move with the same speed and in the same direction (plug flow), (iii) both plates are stationary and flow is due to the imposition of pressure gradient (Poiseuille flow), and (iv) either of the two plates moves with constant speed with external pressure gradient (generalized plane Couette flow). Siddiqui et al. [1] used the perturbation and homotopy analysis method (HAM) to obtain approximate solutions. In this paper we derive exact solutions of the same problems and compare our results with those of Siddiqui et al. [1]. The outline of the paper is as follows. In the next section we present the basic equations given in Siddiqui et al. [1]. Section 3 gives the exact solutions for four problems of flows mentioned earlier. In Sect. 4 we give a comparison of the exact and approximate solutions. Then in Sect. 5 we provide concluding remarks. 2 Basic equations We present the basic equations derived in Siddiqui et al. [1]. The x; y ð Þ coordinate system is used, where x is the direction of motion of the fluid between two infinite parallel plates Acta Mechanica 188, 69–78 (2007) DOI 10.1007/s00707-006-0400-1 Printed in The Netherlands Acta Mechanica