Stability study of a constant-volume thin film flow.

Area of Science:

• Fluid dynamics
• Non-Newtonian fluid mechanics
• Surface phenomena

Background:

• Investigating fluid spreading on inclines is crucial for understanding various natural and industrial processes.
• Common models often assume constant fluid flux, which doesn't apply to time-dependent base flows of constant fluid volumes.
• Linear stability analysis is a standard tool, but applying it to time-dependent flows requires advanced methods.

Purpose of the Study:

• To develop and apply a consistent linear stability analysis method for time-dependent fluid flow down an incline.
• To investigate the evolution of perturbations and their wavelengths in such flows.
• To determine the influence of fluid volume and substrate inclination on flow stability.

Main Methods:

• Developed a novel method for linear stability analysis by simultaneously solving the base flow and perturbation time evolution.
• Employed a finite-difference numerical method combined with a newly developed integral method.
• Performed spectral analysis to study the time-dependent growth rates of flow modes.

Main Results:

• Perturbations were found to propagate at the same velocity as the leading contact line after an initial transient phase.
• The wavelength of maximum amplitude (lambda_max) asymptotically approached a stable value, matching experimental results.
• lambda_max showed good agreement with experimental data for small cross-sectional fluid areas (small Bond numbers), differing from larger fluid volumes.

Conclusions:

• The developed method provides a robust framework for analyzing the stability of time-dependent fluid flows.
• The study elucidates the asymptotic behavior of perturbation wavelengths and their dependence on fluid volume and inclination.
• Findings offer valuable insights into the physics of thin film spreading and provide a basis for further experimental and theoretical investigations.