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This study presents a universal framework for analyzing free surface flow instability in generalized Newtonian fluids. The new analytical expressions accurately predict wave speed and instability thresholds across various rheologies.

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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.