The flow of nonlinear viscoelastic fluids between oscillating parallel plates is investigated. The investigation features time-dependent analysis of a complicated viscoelastic material modeled based on the Johnson-Segalman constitutive relation. Given the rheological parameters of certain material known from experiments, the coefficients of Johnson-Segalman constitutive equation model for the material are evaluated by fitting the data. The problem is first formulated by writing the governing equations for the flow between two independently oscillating parallel plates, i.e. oscillating Couette flow. The velocity and stress are represented by symmetric and antisymmetric Chandrasekhar functions in space. Both inertia and normal stress effects are included. A numerical scheme is applied to solve the governing equations in time domain projected by Galerkin method. For given Reynolds number and viscosity ratio, one critical Weissenberg numbers is found at which an exchange of stability occurs between the Couette and other steady flows. The model is capable of predicting the nonlinear amplitude-dependent behavior of viscoelastic flows under single and multiple-frequency excitations.

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