Spatio-temporal linear stability of viscoelastic saffman-taylor flows

Abstract

This thesis is about an analytical study of viscoelastic fingering that solves the problem of predicting the finger width when a Newtonian fluid (with negligible viscosity) drives a non-Newtonian (power-law) fluid. The finger thinning and widening phenomena in non-Newtonian fluids have been empirically explained in a large number of prior studies, but an analytical expression derived via a single, unified theory explaining both of these features has remained elusive until now. This work aims at deriving an expression and contrast the findings with the in vitro and in silico data that are already accessible. The dispersion relation is also derived in a rectilinear channel for the power-law fluids, which is utilized in the linearized model. The description of the five chapters is as follows: • The classical Saffman-Taylor instability in a Hele-Shaw cell is introduced in the first chapter, along with a summary of earlier findings from direct numerical simulations as well as experimental and analytical findings for both Newtonian and non-Newtonian fluids. It gives an overview of both a qualitative and a quantitative investigation of viscous fingering in the linear and non-linear regimes. Additionally, its significant interest in a wide range of fields, including physics, biology, applied mathematics, and industrial research, is highlighted. • In the second chapter, the basics of the various categories of instability modes are discussed briefly. The phrase "temporal modes" refers to situations where the real wave number determines the instability in the complex frequency. The convectively unstable modes result in wave packets that eventually leave the medium in its undisturbed state after traveling away from the source. In contrast, a point-source disturbance gradually contaminates all of the absolutely unstable modes. If both of the merging modes develop from waves traveling in the same direction, evanescent modes (or the direct resonance mode) will appear. To discern between these instabilities, elementary knowledge of the branch or pinch point and the Cusp-Map diagram is required. An example of the Briggs’ contour integral method is used to discover the flow-material properties contributing to stability-instability regions. • The third chapter highlights an analytical approach to the problem of predicting the finger width of a simple fluid driving a non-Newtonian (power-law) fluid. The analysis is based on the Wentzel-Kramers-Brillouin (WKB) approximation by representing the deviation from the Newtonian viscosity as a singular perturbation in a parameter, leading to a solvability condition at the fingertip, which selects a unique finger width from the family of solutions. This solvability theory provides a reasonable mechanism for the selection of the pattern, and it is done by constructing a function called the cusp function, whose zeros will determine the possible solutions for the pattern. It is found that the relation between the dimensionless finger width, Λ and the dimensionless group of parameters containing the viscosity and surface tension, ν, has the form: Λ ∼ 1 2 −O(ν −1/2 ) for shear thinning case, and Λ ∼ 1 2 +O(ν 2/(4−n) ) for shear thickening case, in the limit of small ν. This theoretical estimate is compared with the existing experimental finger width data as well as the one computed with the linearized model, and a good agreement is found near the power-law exponent, n = 1. The issues that have been encountered during this study, as well as some recent developments and potential future concerns requiring thorough spatiotemporal stability assessments on the dynamics of fingering patterns, are discussed in the final chapter. The potential future work includes but is not limited to the impact of changing the viscosity contrast parameter on the dynamics of fingered structures, the competition between the typical Saffman-Taylor single-finger stationary solution, and other attractor systems characterized by closed bubbles dominates the long-time asymptotics, the existence of topological singularities in the form of interface pinch-off, wetting effects, and applications to other issues like interface roughening in the fluid invasion of porous media, recent results on rotating Hele-Shaw flows, the impact of boundaries like elastic boundaries and occlusions, the evolution of the moving interface, etc.