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arXiv · 2610.34351

Amplitude-Dependent Damping of Gravity--Capillary Waves in Power-Law Fluids: An Energy-Dissipation Framework

Sabbir Hassan
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Abstract

Classical gravity--capillary theory locates the restoring-force crossover, but the wave amplitudes and depths for which energy losses remain measurable depend on the constitutive response and on where viscous dissipation occurs. We develop a weak-loss mechanical-energy model for small-steepness progressive surface waves in memoryless power--law liquids. The full invariant of the leading inviscid strain field yields the deep-water bulk law $\dd A/\dd t=-C_tA^n$, with $C_t=4Kk^2(2kΩ)^{n-1}/[(n+1)ρ]$, and the corresponding spatial law $\dd A/\dd x=-(C_t/v_g)A^n$. For finite depth we calculate bulk dissipation using a phase- and depth-resolved quadrature $J_n(kh)$ and formulate a separate oscillatory power--law bottom layer. We solve the latter numerically in periodic state, obtaining dimensionless integrated wall losses $I_{0.6}\simeq0.5432$ and $I_{1.4}\simeq0.4708$ and recovering the exact Newtonian benchmark $I_1=1/2$. These values make it possible to evaluate conditional finite-depth bulk--wall dominance boundaries rather than only their amplitude exponents. The theory is organized into restoring-force, attenuation, local propagation-length, and dissipation-mechanism regime maps; at fixed wave steepness it additionally predicts a capillary-side maximum of normalized temporal damping for $0<n<2/3$. Grid, time-step, domain, and energy-work checks support the reduced wall calculation, while the coupled bulk--wall approximation still requires independent free-surface numerical or experimental validation. The results provide a reproducible single-fluid reference for subsequent two-liquid interfacial-wave modeling.

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Sabbir Hassan. 2026-10-25. Amplitude-Dependent Damping of Gravity--Capillary Waves in Power-Law Fluids: An Energy-Dissipation Framework. https://arxiv.org/abs/2610.34351

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