The threshold of instability of Taylor–Couette flow of a Newtonian fluid in quasi-periodic modulation with two immense frequencies using spectral Chebychev collocation methods

Abstract

This study investigates the dynamic flow of a Newtonian fluid through two coaxial cylinders, each rotating at speeds Ω1(t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\Omega }_{1}(t)$$\\end{document} (inner cylinder) and Ω2(t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\Omega }_{2}(t)$$\\end{document} (outer cylinder). We derive equations of motion for disturbances in balance, yielding a controlled system characterized by parameters such as Taylor number, wave number, frequency ratio, and interior cylinder frequency. We introduce numerical techniques for solving this system, employing spectral Chebychev collocation for spatial resolution and a combined approach of Floquet theory and Runge–Kutta method for temporal resolution. Our refined approach enables comprehensive analysis of fluid dynamics within the rotating coaxial cylinders, showcasing the interplay of various control parameters.