J. Fluid Mech. (1999), vol. 397, pp. 203–229. Printed in the United Kingdom c  1999 Cambridge University Press 203 The effect of boundaries on the plane Couette flow of granular materials: a bifurcation analysis By P. R. NOTT 1 , M. ALAM 1 , K. AGRAWAL 2 , R. JACKSON 2 AND S. SUNDARESAN 2 1 Department of Chemical Engineering, Indian Institute of Science, Bangalore 560012, India 2 Department of Chemical Engineering, Princeton University, Princeton, NJ 08544, USA (Received 28 October 1998 and in revised form 20 May 1999) The tendency of granular materials in rapid shear flow to form non-uniform structures is well documented in the literature. Through a linear stability analysis of the solution of continuum equations for rapid shear flow of a uniform granular material, performed by Savage (1992) and others subsequently, it has been shown that an infinite plane shearing motion may be unstable in the Lyapunov sense, provided the mean volume fraction of particles is above a critical value. This instability leads to the formation of alternating layers of high and low particle concentrations oriented parallel to the plane of shear. Computer simulations, on the other hand, reveal that non-uniform structures are possible even when the mean volume fraction of particles is small. In the present study, we have examined the structure of fully developed layered solutions, by making use of numerical continuation techniques and bifurcation theory. It is shown that the continuum equations do predict the existence of layered solutions of high amplitude even when the uniform state is linearly stable. An analysis of the effect of bounding walls on the bifurcation structure reveals that the nature of the wall boundary conditions plays a pivotal role in selecting that branch of non-uniform solutions which emerges as the primary branch. This demonstrates unequivocally that the results on the stability of bounded shear flow of granular materials presented previously by Wang et al. (1996) are, in general, based on erroneous base states. 1. Introduction The formation of non-uniform structures in rapid granular flows has been observed in many recent experiments and computer simulations (Hopkins & Louge 1991; Savage 1992a; Goldhirsch, Tan & Zanetti 1993; Miller, O’Hern & Behringer 1996). As continuum equations of motion, such as those derived from the kinetic theory of granular materials (for example, see Lun et al. 1984, and Jenkins & Richman 1985a), are commonly used to model such flows, it is of interest to determine whether these structures can be captured by such equations in a qualitatively correct manner. Mello, Diamond & Levine (1991) examined plane shearing motion using continuum equations derived from granular kinetic theory and investigated the propagation of disturbances with wave vectors in the direction of the vorticity. Instabilities which arise during the cooling of an initially uniform granular material as a result of inelastic collisions have been examined by Goldhirsch et al. (1993) and McNamara (1993). The stability of the solution of the continuum equations representing an infinite plane