On Measuring the Weissenberg Effect in Complex Fluids

Within the framework of the Cauchy law of motion, we explore an approach to measuring the Weissenberg effect in complex fluids by using the general Weissenberg number GN_We put forth by Huang et al. (2019). First, we analyze and compare the applications of the primary Weissenberg number N_We and the general Weissenberg number GN_We in two typical viscometric flows and in two non-viscometric flows given by Huilgol (1971) and by Huilgol and Triver (1996), respectively, using an incompressible fluid of grade 2 and the incompressible Reiner-Rivlin fluid. Second, we use both N_We and GN_We to carry out detailed analyses of the three normal stress differences N_1, N_2, and N_3 = N_1 +N_2 by employing the experimental results of Gamonpilas et al. (2016), Singh and Nott (2003), Zarraga et al. (2000), Couturier et al. (2011), and Dai et al. (2013). These results indicate that GN_We outdoes N_We in comprehensively characterizing the Weissenberg effect, a.k.a. the normal stress effect or the elastic effect, in complex fluids in both the viscometric and the non-viscometric flows. Third, we show that the kinematical vorticity number V_K(x,t), namely the Truesdell number, plays a vital role in setting up a necessary condition for the measurement of the Weissenberg effect. From a general, theoretical standpoint, we introduce an intrinsic orthonormal basis (e_1,e_2,e_3) in the same sense of Serrin (1959), which coincides with the conventionally used orthonormal basis if the flow is viscometric, to calculate GN_We so as to measure the Weissenberg effect in a laminar flow of complex fluids, provided that in the flow field there exists at least one spatial point x with some neighborhood in which the Truesdell number V_K(x,t) > 0.