Linear Stability Theory Applied to Three-Dimensional Viscoelastic Fluid Flows - Furlan, Laison Junio da Silva (2022)
Several flows of practical interest are viscoelastic fluids. Due to the industrial applicability of this type of flow, it is desirable to know how this flow propagates if disturbances appear in the system. Assuming that disturbances are introduced into the system, depending on the characteristics of the flow and the fluid, it may transition to a turbulent state, which can damage structures and even pipelines to rupture. The interaction between inertial, viscous and elastic forces strongly affects the hydrodynamics of viscoelastic fluids. The Linear Stability Theory (LST) technique investigates the propagation of disturbances in the flow. This technique solves an eigenvalue/ eigenvector problem, where the most unstable eigenvalue carries information regarding the stability of the flow studied (the amplification rate of the Tollmien-Schlichting waves). This eigenvalue problem is solved using the EIG function in MATLAB software. By directly solving this eigenvalue problem, the entire eigen spectrum is obtained, among them the eigenvalue that carries the stability information of the flow. The eigenfunctions associated with this eigenvalue are also obtained, allowing the analysis of the energy of the disturbances. There are many ways to analyze the stability of a flow. The most common is through the amplification rate of disturbances and the analysis of the energy of these disturbances. In the present research, the laminar-turbulent transition is studied by investigating the propagation of Tollmien-Schlichting waves. It adopted an incompressible viscoelastic fluid flow in a three-dimensional channel. The constitutive equations adopted were the UCM (Upper-Convected Maxwell), the Oldroyd-B, the Giesekus, and the linear Phan-Thien-Tanner (LPTT) models. The stability analysis is performed by analyzing the amplification rate of the disturbances and building a stability diagram. This diagram is called the neutral stability curves diagram. In the Oldroyd-B model increasing the solvent contribution in the mixture stabilizes the flow. For the UCM model, the increase in elasticity destabilizes the flow both for lower and higher frequencies. For the Giesekus model, the higher amount of polymer in the mixture stabilizes the flow for lower values of this models parameter and as the value of this parameter increases, the flow becomes more unstable.