We present a robust calculation leading to experimentally convenient and accurate ways of detecting, measuring, and characterizing velocity-slippage in viscoelastic-fluids, unlike previous inaccurate Mooney-type techniques. Herein, the unsteady Navier-Stokes equation for viscoelastic-fluid is solved while highlighting the rheological ramification. Our results emphasize phase-lags amongst shear stresses and strains, key output used in differing slip-types. The viscoelastic fluid is non-aging and isothermal, and we obtain an exact solution of the non-trivial flow profile, without assuming a linear Couette profile as customarily used. Moreover, the Navier-type slip boundary condition is considered. This approach is similar to that done in our earlier paper [Azese, "Measurement and characterization of slippage and slip-law using a rigorous analysis in dynamics of oscillating rheometer: Newtonian fluid," Phys. Fluids 30, 023103 (2018)] where instead it was a Newtonian fluid. Accordingly, the sample fluid is trapped in the Couette-gap, where one of them is stationary and the other is steadily oscillating with an amplitude Re (Reynolds-number) and angular speed Omega = R-o (Roshko number), thus Couette-rheometry. We showcase an alternative way to obtain a steady-periodic solution, matching the long-time solution obtained in our earlier paper. We obtain the unsteady solution for this viscoelastic case and also use the alternative method to obtain the steady-periodic version, later used in obtaining the velocity and stress at the walls. Interestingly, we note the influence of R-e, R-o, and W-i (Weissenberg Number) on this analysis. The equations and plots presented evidently show the influence of the slip. We conclude with reverse algorithms, Fourier-transform, Lissajous-figures, and Mooney-like procedures, capable of reproducing the slip-parameters, leading to a systematic measurement-and-characterization of the slip, useful in the calibration of rheological devices. Published under license by AIP Publishing.
