Linear stability of free shear flow of viscoelastic liquids

Abstract

The effects of viscoelasticity on the hydrodynamic stability of plane free shear flow are investigated through a linear stability analysis. Three different rheological models have been examined : the Oldroyd-B, corotational Jeffreys, and Giesekus models. We are especially interested in possible effects of viscoelasticity on the inviscid modes associated with inflexional velocity profiles. In the inviscid limit, it is found that for viscoelasticity to affect the instability of a flow described by the Oldroyd-B model, the Weissenberg number, We, has to go to infinity in such a way that its ratio to the Reynolds number, G K We/Re, is finite. In this special limit we derive a modified Rayleigh equation, the solution of which shows that viscoelasticity reduces the instability of the flow but does not suppress it. The classical Orr-Sommerfeld analysis has been extended to both the Giesekus and corotational Jeffreys models. The latter model showed little variation from the Newtonian case over a wide range of Re, while the former one may have a stabilizing effect depending on the product sWe where s is the mobility factor appearing in the Giesekus model. We discuss the mechanisms responsible for reducing the instability of the flow and present some qualitative comparisons with experimental results reported by Hibberd et al. (1982), Scharf (1985a, b) and Riediger (1989). Problem definitionThe mixing-layer flow configuration is standard, as shown in figure 1. U, (respectively U,) is the free-stream velocity of the upper (lower) flow. We denote by uo = i(U, -U,)the free-stream velocity in a reference frame moving with the average velocity of the flow +(q + &), and by S the momentum thickness of the mixing layer. In all the

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