Eddy genesis and transformation of Stokes ow in a double-lid driven cavity F Gu¨rcan 1 , P H Gaskell 2 * , M D Savage 3 and M C T Wilson 2 1 Department of Mathematics, University of Erciyes, Kayseri, Turkey 2 School of Mechanical Engineering, The University of Leeds, UK 3 Department of Physics and Astronomy, The University of Leeds, UK Abstract: Stokes ow is considered in a rectangular driven cavity of depth 2H and width 2L , with two stationary side walls and two lids moving in opposite directions with speeds U 1 and U 2 . The ow is governed by two control parameters: the cavity aspect ratio, A ˆ H =L , and the speed ratio, S ˆ U 1 =U 2 . The solution for the streamfunction is expressed as an in nite series of Papkovich–Faddle eigenfunctions, which is then expanded about any stagnation point to reveal changes in the local ow structure as A and S are varied. An …S , A † control space diagram is constructed, which exhibits an intricate structure due to the intersection and con uence of several critical curves representing ow bifurcations at degenerate critical points. There are eight points where two critical curves intersect and the ow bifurcations are described and interpreted with reference to the theoretical work of Bakker (Bifurcations in Flow Patterns, Kluwer Academic, 1991) and Br ø ns and Hartnack (Phys. Fluids , 1999, 11, 314). For a speed ratio in the range ¡1 4 S < 0 the various ow transformations are tracked as A increases in the range 0 < A < 3:2, and hence the means is identi ed by which new eddies appear and become fully developed. It is shown that for S =0, the number of eddies increases from 1 to 3 via several key ow transformations, which become more complicated as jS j is reduced. Keywords: uid mechanics, ow structure, bifurcation, stagnation point NOTATION A cavity aspect ratio ˆ H =L A n , B n streamfunction expansion coef cients H half cavity height (m) L half cavity length (m) n summation index s n streamfunction expansion eigenvalues S speed ratio ˆ U 1 =U 2 u dimensionless uid velocity U 1 , U 2 top and bottom lid speeds respectively (m/s) x ˆ…x , y † dimensionless Cartesian coordinates …X , Y † dimensional Cartesian coordinates (m) f n 1 streamfunction expansion eigenfunctions c streamfunction 1 INTRODUCTION Examples of Stokes ows within rectangular cavities driven by either one or a pair of translating lids may be found in various manufacturing processes, including coating systems [ 1, 2], polymer melts [ 3] and ceramic tape casting [ 4]. In addition to their practical relevance, cavity ows have proved to be a fertile topic of study due to their propensity to exhibit complex ow structures [ 5] in simple geometries. Stokes ow in a rectangular cavity driven by the uniform motion (speed U 1 ) of one side wall has been studied by several authors, forming the subject of numerous computational investigations and encompass- ing the application of a wide variety of numerical techniques [ 6–9]. However, it was Joseph and Sturges [ 10] who were the rst to formulate and solve the Stokes ow problem analytically by expressing the streamfunc- tion, c, as an in nite series of Papkovich–Faddle eigenfunctions and determining the associated series coef cients using Smith’s biorthogonality condition [ 11]. They concentrated much of their effort on a particular case for direct comparison with the (now classical) The MS was received on 27 May 2002 and was accepted after revision for publication on 14 November 2002. * Corresponding author: School of M echanical Engineering, The University of Leeds, Leeds LS2 9JT, UK. 353 C06902 # IMechE 2003 Proc. Instn Mech. Engrs Vol. 217 Part C: J. Mechanical Engineering Science