Exact solutions for oscillatory shear sweep behaviors of complex fluids from the Oldroyd 8-constant framework

Abstract

Large-amplitude oscillatory shear flow (LAOS) is a popular for studying the nonlinear physics of complex fluids. Specifically, the strain rate sweep (also called the strain sweep) is used routinely to identify the onset of nonlinearity. In this paper, we give exact expressions for the nonlinear complex viscosity and the corresponding nonlinear complex normal stress coefficients for the Oldroyd 8-constant framework for oscillatory shear sweeps. We choose the Oldroyd 8-constant framework for its rich diversity of popular special cases (we list 18 of these). We evaluate the Fourier integrals of our previous exact solution to get exact expressions for the real and imaginary parts of the complex viscosity, and for the complex normal stress coefficients, as functions of both test frequency and shear rate amplitude. For our comparisons with data, we use the Spriggs relations to generalize the Oldroyd 8-constant framework to multimode. We explore the role of infinite shear rate viscosity on strain rate sweep responses for the special case of the corotational Jeffreys fluid. We find that raising η ∞ , raises the real part of the complex viscosity, and lowers the imaginary. In our worked examples, we thus first use the corotational Jeffreys fluid, and then, for greater accuracy, we use the Johnson-Segalman fluid, to describe the strain rate sweep response of molten atactic polystyrene. Our generalization yields unequivocally, a longest fluid relaxation time, used to assign Weissenberg and Deborah numbers to each oscillatory shear flow experiment. We then locate each experiment in Pipkin space. () are the real and minus [10] the imaginary parts of the nonlinear complex viscosities, η n * , and first and second normal stress coefficients, Ψ 1,n * and Ψ 2,n * . Some prefer to write Eq. (6) as (see Eq.(3) of [11]):

Series expansion for shear stress in large‐amplitude oscillatory shear flow from oldroyd 8‐constant framework

“…[32,33] Tee and Dealy report one frequency sweep, comprising ten SAOS measurements over the ranges 0:38 < v < 24 rad=s and 0:076 < _ g 0 < 4:8 s À1 , performed using oscillatory cone-plate flow (the Wi De ¼ 0:2 set of points in Figure 12 of Saengow and Giacomin [13] ). [31,32,33] Tee and Dealy also report a set of thirty LAOS measurements, comprising 5 frequency sweeps over the ranges 0:96 < v < 25 rad=s and 2:6 < _ g 0 < 250 s À1 , performed on a concentric cylinder rheometer specially designed for LAOS (the 2:67 Wi De 10 sets of points in Figure 12 of Saengow and Giacomin [13] ). [31,32,33] Specifically, they report frequency sweeps of t 0 v; _ g 0 À Á = _ g 0 , where t 0 is the peak shear stress in LAOS.…”

“…For this application, for the LAOS measurements, we will thus compare t 0 v; _ g 0 À Á = _ g 0 measurements with the h à 1 v; _ g 0 À Á predictions from our results, for the special case of the Johnson-Segalman fluid by following subsection VI.B of Saengow and Giacomin. [13] Here, we apply our results, Equations (34) and (35) (with Tables 3 and 4), for the special case of the Johnson-Segalman fluid, for which (see Table 4 of Saengow et al [6] ):…”

“…For the Johnson-Segalman fluid, we then get the SAOS measurements (Equation (65) of Saengow and Giacomin [13] ):…”

“…This is sometimes further complicated to (see Equation (11) and Figure 14 of Saengow and Giacomin [13] ):…”

2017 winner of The Canadian Journal of Chemical Engineering award for best graduate student paper

“…Chai focused on oscillatory shear performed at large amplitudes (LA), where the molecules are grossly distorted from their shapes at rest. Chai arrived at the first exact mathematical equations for interpreting the stresses associated with these gross distortions in LAOS …”