Spatiotemporal linear stability of viscoelastic free shear flows

Abstract

In this thesis, we have explored the temporal and spatiotemporal stability analyses offree shear, viscoelastic flows in the limit of low to moderate Reynolds number (Re) and Weissenberg number (We). The description of the six chapters are as follows:•The first chapter introduces the fundamental difference in the instability arising from Newtonian versus non Newtonian fluids: Newtonian fluid undergoes instabilities with increasing Re, although a direct transition to instability is also possible via bypass transition. Contrarily, non Newtonian fluids exhibit both inertial and purely elastic instabilities that arise even when the effect of inertia is too small to drive an instability in a Newtonian fluid at the same flow condition. Past research by several groups are highlighted.•The second chapter discusses the Compound Matrix Method (CMM) which is used to numerically integrate the eigenvalue problem using auxiliary variables emerging from the Orr Sommerfeld equation (OSE).•In the third chapter, a description of the different types of instabilities is included. The temporal modes refer to cases where the instability in the complex frequency is determined as a function of real wave number. The convectively unstable modes give rise to wave packets moving away from the source and ultimately leaving the medium in its undisturbed state. Absolutely unstable modes, by contrast, are gradually contaminated everywhere from a point-source disturbance. Evanescent modes (or the direct resonance mode) arise if the two coalescing modes originate from waves propagating in the same direction.• The fourth chapter highlights the stability analyses of antisymmetric, free shear, viscoelastic flows in the dilute regime, obeying the Oldroyd-B constitutive equation. The temporal stability analysis indicates that with increasing We, (a) the entire range of the most unstable mode is shifted toward longer waves, (b) the vorticity structure contours are dilated, and (c) the residual Reynolds stresses are diminished. The spatiotemporal analyses show that the free shear flow of dilute polymeric liquids is either (absolutely/convectively) unstable for all Re or the transition to instability occurs at comparatively low Re. •In the fifth chapter, we provide a detailed comparison of the temporal and the spatiotemporal linearized analyses of free shear, viscoelastic flows in the limit of low to moderate Reynolds number and Elasticity number obeying four different types of stress-strain constitutive equations: Oldroyd-B(ε=0,a=1), Upper Convected Maxwell(ε=0,a=1,ν=0), Johnson-Segalman (ε=0,a=0.5) and Phan-Thien Tanner(ε=0.5,a=0.5). The temporal stability analysis indicates (a) elastic stabilization at higher values of elasticity number and (b) a non-monotonic instability pattern at low to intermediate values of elasticity number for the JS as well as the PTT model. The spatiotemporal phase diagram divulge the familiar regions of inertial and elastic turbulence, a recently verified region of elastoinertial turbulence and the unfamiliar temporally stable region for intermediate values of Reynolds and Elasticity number.• In the concluding chapter, we highlight the challenges that we have faced and intend to face in future numerical simulations as well as our future problems demandinga full spatiotemporal stability analyses: (a) Rayleigh-Plateau, describing the on set of the detachment of a droplet, (b) Saffman-Taylor, or the formation of patterns in a morphologically unstable interface between two fluids in a porous medium, (c)Faraday instability, or an unstable state of a flat hydrostatic surface due to a critical vibration frequency