Meccanica (2006) 41:501–508 DOI 10.1007/s11012-006-0007-6 Pulsatile pipe flow of pseudoplastic fluids Irene Daprà · Giambattista Scarpi Received: 2 July 2005 / Accepted: 23 January 2006 / Published online: 6 October 2006 © Springer Science+Business Media B.V. 2006 Abstract This note is concerned with a laminar pipe flow of a non-Newtonian fluid under the action of a small pulsating pressure gradient superposed to a steady one. The constitutive law describing the rheological behaviour of the fluid is the so-called power law (Ost- wald–de Waele). An approximated analytical solution is found for the velocity, as power series of the ampli- tude of the periodic disturbance. The analytic solution is compared with a direct numerical solution and the perfect accord of the values obtained is underscored. Keywords Ostwald–de Waele fluid · Pulsatile pipe flow · Power law · Fluid mechanics Introduction This paper deals with the unsteady flow of a non- Newtonian fluid in a pipe: the problem has been faced by many authors even recently. The motion generated superposing a small periodical variation to a constant pressure gradient has acquired also technical and prac- tical uses, because it allows the growth of the rate of flow with a limited increase of power. The non-linear- ity of the constitutive equation of pseudoplastic fluids I. Daprà (B ) · G. Scarpi DISTART Idraulica, University of Bologna, via Risorgimento 2, 40136 Bologna, Italy e-mail: irene.dapra@mail.ing.unibo.it makes the variation of the discharge during the phase of increase of the pressure gradient exceed the decrease occurring when it falls. The unsteady pipe flow of a non Newtonian fluid has been studied in the past either for numerical calculation purposes or analytically. Proposed numerical solutions include those of Balmer and Fiorina [1] and of Warsi [2] which study the start-up and the pulsatile flow of a power-law fluid in a circular tube using an implicit finite difference method: our analytical solution is coherent with the numerical results of Warsi. Adusumilli and Hill [3] use explicit finite difference technique to investigate some unsteady flows of a truncated power law fluid in a pipe; Pontrelli [4] studies the pulsatile flow in a tube using three different models to simulate the rheologi- cal behaviour of the blood, applying an implicit finite difference method. Edwards et al. [5] use an explicit finite difference scheme to derive velocity profiles for a power law fluid in some unsteady laminar flows, Mai and Davis [6] study numerically a pulsatile flow of a two-phase fluid in a pipe. Ramkissoon [7] uses an Old- royd model to analyse analytically some unsteady axial flows. Fewer analytical papers include an early one of Rajagopal [8] who gives an exact solution for pulsatile flow of a second grade fluids; Pascal [9] obtains the solution to some unsteady flows (Stokes’s first prob- lem, Couette flow) for a power law fluid; Fetecau [10] gives an exact analytical solution for transient flows of an Oldroyd-B fluid in pipe-like domains; Fetecau and Fetecau [11] give an exact solution for unsteady flow of a second grade fluid which can be regarded as an