The stability of two-dimensional linear flows of an Oldroyd-type fluid (Q580067)

The linear stability of a class of two-dimensional, homogeneous extensional flows in an unbounded domain is considered. It is an extension of earlier work of the authors on the stability of an ordinary viscous fluid in extensional flow [Phys. Fluids 27, 1094-1101 (1984; Zbl 0585.76045)]. The complete class of two-dimensional linear flows is characterized by two parameters viz. E (shear rate) and \(\lambda\) \((=-1\) for a purely rotational flow to \(\lambda =1\) for pure extensional flow and in between \(=0\) for simple shear flow) for rheological behaviour of an Oldroyd-B model. For certain values of \(\beta\), critical Weissenberg numbers are found above which flows are unstable for all values of the wave number \(\alpha_ 3.\) It is observed that in the central region, the real flow is neither exactly two-dimensional nor unbounded, but stability of an unbounded, linear, two-dimensional flow provide an initial step towards understanding in two and four roll mills for both Newtonian and non- Newtonian fluids. The motivation of present work is in the effects of fluid elasticity on the stability of extensional flows, behaviour of the Oldroyd-Maxwell model in flows with an extensional character to obtain numerical solution of the equation of motion, and comparision between prediction for the Oldroyd model and behaviour of real viscoelastic fluids in so called two and four roll milling flows.