Stability Analysis of Nonlinear Fluid Flows through Mathematical and Computational Approaches

In a mathematical and computational approach to this research, the instability of nonlinear fluid flows is investigated. Four advanced numerical algorithms named Finite Volume Method (FVM), Lattice Boltzmann Method (LBM), Spectral Element Method (SEM), and Weighted Essentially Non-Oscillatory (WENO) scheme are utilized to analyze the dynamics of the complex fluid systems. Fluid flow under different such conditions is then simulated using these algorithms as varying Reynolds numbers, thermal gradients, and non-Newtonian characteristics. The results show that changes in viscosity, Reynolds number, and thermal conditions lead to highly sensitive stability of the flow. As an example, the flow is unstable over Reynolds numbers larger than 1500, and the flow oscillations increase in intensity with increased thermal gradients. The relative error of the FVM was 2.5% high accuracy, while the WENO scheme had a better performance on the sharp gradient when compared with other methods (1.8%). Algorithmic performance comparison of the SEM and LBM revealed that due to superior computational efficiency for large scale simulations (20-30%), the SEM and LBM achieved processing time savings of a factor of 20 to 30. Indeed, the study helps provide robust framework for predicting and controlling fluid dynamics in systems of complexity. Such findings are critically important for aerospace, energy systems and environmental engineering applications.