INSTABILITIES AND BIFURCATIONS OF VON KARMAN SIMILARITY SOLUTIONS IN SWIRLING VISCOELASTIC FLOW

We derive an asymptotic model that describes the swirling flow of a vi scoelastic fluid between a relating cone and a stationary plate when t he gap angle, alpha, is small and inertia is neglected. The model, whi ch uses the Phan-Thien Tanner (PTT) constitutive law, is valid in the limit alpha-->0 and for Deborah number, De, order unity. We show that the model admits similarity solutions of von Karman type. A solution c orresponding to a viscometric how is obtained. This base flow, which e xhibits shear thinning if the PTT parameter epsilon not equal 0. is li nearly stable if the Deborah number De is less than a critical value D e(c) and unstable if De > De(c). The critical Deborah number is a decr easing function of the retardation parameter beta, and an increasing f unction of epsilon. The method of Lyapunov-Schmidt is used to determin e the nature of bifurcation when De is close to De(c). Our analysis sh ows that there is a supercritical pitchfork bifurcation at De = De(c).