Density-driven instabilities of variable-viscosity miscible fluids in a capillary tube

Area of Science:

• Fluid Dynamics
• Rheology
• Instability Phenomena

Background:

• Investigates the stability of miscible fluids in an unstable configuration (heavier fluid above lighter fluid) within a vertical capillary tube.
• Considers the initial interface thickness as a key parameter influencing fluid mixing and stability.
• Examines scenarios with both constant and variable fluid viscosities.

Purpose of the Study:

• To perform a linear stability analysis on variable-viscosity miscible fluids in an unstable configuration.
• To understand the influence of initial interface thickness and mobility ratio on instability dynamics.
• To compare stability characteristics between constant and variable viscosity fluids.

Main Methods:

• Employs three-dimensional Stokes equations coupled with a convection-diffusion equation for the concentration field.
• Utilizes cylindrical coordinates for the analysis.
• Analyzes dispersion relations and eigenfunctions to determine instability modes and growth rates.

Main Results:

• For constant viscosity, the 3D mode with azimuthal wave number one is most unstable, with a critical Rayleigh number of approximately 920.
• Variable viscosity: instability growth rate is independent of which fluid is more viscous; eigenfunctions shift towards the less viscous fluid.
• High Rayleigh numbers and mobility ratios show a crossover to axisymmetric perturbations, with thicker interfaces becoming more unstable in variable viscosity cases.

Conclusions:

• Variable viscosity significantly alters instability dynamics compared to constant viscosity, particularly regarding interface thickness effects.
• Thicker interfaces can promote instability in variable-viscosity systems, a phenomenon not observed in constant-viscosity systems.
• The study highlights the complex interplay between viscosity contrast, interface thickness, and perturbation modes in fluid mixing.