Unifying the roll waves

Area of Science:

• Fluid dynamics
• Rheology
• Non-Newtonian fluid mechanics

Background:

• Free surface flows are crucial in engineering and natural phenomena.
• Surface wave instability occurs when inertial forces dominate, quantified by the Reynolds number.
• Existing models for fluid rheology lack a unified framework for stability analysis.

Purpose of the Study:

• To develop a generalized framework for analyzing the linear stability of free surface flows for any generalized Newtonian fluid.
• To derive a universal Orr-Sommerfeld equation applicable to a wide range of fluid rheologies.
• To establish novel analytical expressions for wave celerity and critical Reynolds number, independent of specific rheological models.

Main Methods:

• Development of new dimensionless quantities to minimize rheology dependence.
• Derivation of the Orr-Sommerfeld stability equation for generalized Newtonian fluids.
• Application of long-wave expansion to obtain analytical expressions for wave celerity and critical Reynolds number.

Main Results:

• A novel, rheology-independent analytical expression for wave celerity and critical Reynolds number was derived.
• The derived expressions were validated against experimental and numerical data for shear-thinning, shear-thickening, and viscoplastic fluids.
• The analytical results demonstrated excellent agreement with existing literature, confirming their accuracy and broad applicability.

Conclusions:

• The study provides a unified and computationally efficient method for assessing free surface flow stability across diverse fluid rheologies.
• The derived analytical expressions offer significant advancements for predicting wave behavior and instability thresholds.
• The framework's resilience extends to non-monotonous rheologies, enhancing its practical utility in various scientific and engineering fields.