Analysis of a Two-Fluid Taylor–Couette Flow with One Non-Newtonian Fluid

• Published in: Journal of nonlinear science Vol. 32; no. 2
• Main Authors: Lienstromberg, Christina, Pernas-Castaño, Tania, Velázquez, Juan J. L.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.04.2022; Springer Nature B.V
• Subjects: Analysis; Circles (geometry); Classical Mechanics; Concentric cylinders; Couette flow; Economic Theory/Quantitative Economics/Mathematical Methods; Fluid films; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Newtonian fluids; Non Newtonian fluids; Rheological properties; Rotating cylinders; Rotation; Shear stress; Shear thickening (liquids); Shear thinning (liquids); Stresses; Surface tension; Theoretical; Thickening; Thin films; Viscous fluids; 76A20; Degenerate parabolic equation; Taylor–Couette flow; 76A05; Long-time asymptotics; 35K35; Non-Newtonian fluid; 35K65; Weak solution; 35Q35; Thin-film equation; Power-law fluid; 35B40
• ISSN: 0938-8974; 1432-1467
• DOI: 10.1007/s00332-021-09750-0

We study the dynamic behaviour of two viscous fluid films confined between two concentric cylinders rotating at a small relative velocity. It is assumed that the fluids are immiscible and that the volume of the outer fluid film is large compared to the volume of the inner one. Moreover, while the outer fluid is considered to have constant viscosity, the rheological behaviour of the inner thin film is determined by a strain-dependent power-law. Starting from a Navier–Stokes system, we formally derive evolution equations for the interface separating the two fluids. Two competing effects drive the dynamics of the interface, namely the surface tension and the shear stresses induced by the rotation of the cylinders. When the two effects are comparable, the solutions behave, for large times, as in the Newtonian regime. We also study the regime in which the surface tension effects dominate the stresses induced by the rotation of the cylinders. In this case, we prove local existence of positive weak solutions both for shear-thinning and shear-thickening fluids. In the latter case, we show that interfaces which are initially close to a circle converge to a circle in finite time and keep that shape for later times.