International Journal of Non-Linear Mechanics 38 (2003) 501–511 MHD ow of a third-grade uid due to eccentric rotations of a porous disk and a uid at innity T.Hayat a ,TahiraHaroon a ,S.Asghar a ,A.M.Siddiqui b; * a Department of Mathematics, Quaid-I-Azam University, Islamabad, Pakistan b Department of Mathematics, Pennsylvania State University, York Campus, York, PA 17403, USA Received 20 November 2000; received in revised form 5 August 2001 Abstract The problem of magnetohydrodynamics (MHD) ow of a conducting, incompressible third-grade uid due to non-coaxial rotations of a porous disk and a uid at innity in the presence of a uniform transverse magnetic eld is considered.Anexactanalysisiscarriedouttomodelthegoverningnon-linearpartialdierentialequation.Anumerical solution of the third-order non-linear partial dierential equation has been obtained. Several graphs and tables have been drawn to show the inuence of porosity , magnetic parameter N , material parameters  and  on the velocity distribution. ? 2002 Elsevier Science Ltd. All rights reserved. 1. Introduction The ow between eccentric rotating disks has been examined by a number of investigators. Berker [1] rst found an exact solution for a New- tonianuidinthistypeofowwhichhedenedas “pseudo plane motions” when the disks rotate with the same angular velocity. However, this study by Berker was omitted in later papers and this mis- take was corrected in [2]. Berker [3,4] showed that there is the existence of an innity of non-trivial solutions to the Navier–Stokes equations between twoinniteparallelplatesrotatingaboutacommon axis or about dierent axes. He assumed the ow is symmetric with respect to the origin for a sin- gle solution. The studies on non-Newtonian uids * Corresponding author. Fax: +1-717-771-8404. E-mail address: ams5@psu.edu (A.M. Siddiqui). for this type of ow have also been investigated. MaxwellandCharto[5]pointedoutthatitispos- sible to determine the complex dynamic viscosity of an elastico-viscous liquid by using the orthogo- nal rheometer which is an instrument consisting of two parallel plates which rotate with the same an- gular velocity about two axes normal to the plates but not coincident. Blyler and Kurtz [6] and Bird and Harris [7] established various studies to obtain the complex viscosity of the uid. The inertia of the uid was neglected in Refs. [5–7]. Abbott and Walters [8] both introduced an exact solution for a Newtonianuidincludinginertiaeectsandexam- ined the ow of a viscoelastic uid assuming that the distance between the axes is small. The same problem was investigated for a second-grade uid byRajagopal[9,10],RajagopalandGupta[11],for aBKZuidbyRajagopalandWineman[12],fora K-BKZ uid by Dai et al. [13]. Separately, G o g us 0020-7462/03/$-see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII:S0020-7462(01)00075-0