Amplitude-Dependent Damping of Gravity--Capillary Waves in Power-Law Fluids: An Energy-Dissipation Framework
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=−CtAn, with Ct=4Kk2(2kΩ)n−1/[(n+1)ρ], and the corresponding spatial law dd A/dd x=−(Ct/vg)An. For finite depth we calculate bulk dissipation using a phase- and depth-resolved quadrature Jn(kh) and formulate a separate oscillatory power--law bottom layer. We solve the latter numerically in periodic state, obtaining dimensionless integrated wall losses I0.6≃0.5432 and I1.4≃0.4708 and recovering the exact Newtonian benchmark I1=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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BibTeX
@misc{hassan2026amplitudedependent,
title = {{Amplitude-Dependent Damping of Gravity--Capillary Waves in Power-Law Fluids: An Energy-Dissipation Framework}},
author = {Sabbir Hassan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34351},
eprint = {2610.34351},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34351},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Amplitude-Dependent Damping of Gravity--Capillary Waves in Power-Law Fluids: An Energy-Dissipation Framework
AU - Sabbir Hassan
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.34351
UR - https://arxiv.org/abs/2610.34351
AB - 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≃0.5432 and I_1.4≃0.4708 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.
ER -