We report the temporal and spatio-temporal stability analyses of anti-symmetric, free shear, viscoelastic flows obeying the Oldroyd-B constitutive equation in the limit of low to moderate Reynolds number and Weissenberg number. The resulting fourth order Orr-Sommerfeld equation is reduced to a set of six auxiliary equations which are numerically integrated starting from the rescaled far-field conditions, i. e., via the Compound Matrix Method. Numerical results indicate that with increasing Weissenberg number: (a) the peak of the maximal growth rate (i. e., the maximum value of the imaginary component of the growth rate) is reduced, (b) the entire range of unstable spectrum is shifted towards longer waves (i. e., the entire region of temporal instability is gradually concentrated near zero wavenumber), (c) the vorticity structure contours are dilated and (d) the residual Reynolds stresses are diminished. All these observations suggest that viscoelasticity reduces the temporal instability but does not completely suppress it. The Briggs idea of analytic continuation is deployed to classify regions of temporal stability, absolute and convective instabilties as well as evanescent (false) modes, in a finite range of Reynolds number, Weissenberg number and the viscosity coefficient. The main result is that, unlike Newtonian fluids, the free shear flow of dilute polymeric liquids are either (absolutely/convectively) unstable for all Reynolds number or the transition to instability occurs at very low Reynolds number, a finding attributed to the fact that viscoelasticity aggravates free surface flow instabilities. Although this transitional pathway connecting the temporally stable state to the elastoinertially unstable state have been identified by some in-vitro experiments, but until now, has not been quantified theoretically via linear stability analysis.

We provide a detailed comparison of the two-dimensional, temporal, and spatiotemporal linearized analyses of the viscoelastic free shear flows (inhomogeneous flows with mean velocity gradients that develop in the absence of boundaries) in the limit of low to moderate Reynolds number and elasticity number obeying four different types of stress–strain constitutive equations: Oldroyd-B, upper convected Maxwell, Johnson–Segalman (JS), and linear Phan-Thien–Tanner (PTT). The resulting fourth-order Orr–Sommerfeld equation is transformed into a set of six auxiliary equations that are numerically integrated via the compound matrix method. The temporal stability analysis suggests (a) elastic stabilization at higher values of elasticity number {shown previously in the dilute regime [Sircar and Bansal, “Spatiotemporal linear stability of viscoelastic free shear flows: Dilute regime,” Phys. Fluids 31, 084104 (2019)]} and (b) a nonmonotonic instability pattern at low to intermediate values of elasticity number for the JS as well as the PTT model. To comprehend the effect of elasticity, Reynolds number, and viscosity on the temporal stability curves of the PTT model, we consider a fourth parameter, the centerline shear rate, ζc. The “JS behavior” is recovered below a critical value of ζc, and above this critical value, the PTT base stresses (relative to the JS model) are attenuated thereby explaining the stabilizing influence of elasticity. The Briggs idea of analytic continuation is deployed to classify regions of temporal stability and absolute and convective instabilities, as well as evanescent modes, and the results are compared with previously conducted experiments for Newtonian as well as viscoelastic flows past a cylinder. The phase diagrams reveal the two familiar regions of inertial turbulence modified by elasticity and elastic turbulence as well as (a recently substantiated) region of elastoinertial turbulence and the unfamiliar temporally stable region for intermediate values of Reynolds and elasticity number.

We apply visualization and modeling methods for convective and diffusive flows to public school mathematics test scores from Texas. We obtain plots that show the most likely future and past scores of students, the effects of random processes such as guessing, and the rate at which students appear in and disappear from schools. We show that student outcomes depend strongly upon economic class, and identify the grade levels where flows of different groups diverge most strongly. Changing the effectiveness of instruction in one grade naturally leads to strongly nonlinear effects on student outcomes in subsequent grades. Fokker-Planck equation | convection | education