Viscous damping of gravity-capillary waves: Dispersion relations and nonlinear corrections

Abstract

We discuss the impact of viscosity on nonlinear propagation of surface waves at the interface of air and a fluid of large depth. After a survey of the available approximations of the dispersion relation, we propose to modify the hydrodynamic boundary conditions to model both short and long waves. From them we derive a nonlinear Schrödinger equation where both linear and nonlinear parts are modified by dissipation and show that the former plays the main role in both gravity and capillarygravity waves while, in most situations, the latter represents only small corrections. This provides a justification of the conventional approaches to damped propagation found in the literature.

Bifurcation Analysis and Propagation Conditions of Free-Surface Waves in Incompressible Viscous Fluids of Finite Depth

“…A bifurcation that is closely related to the one we find here at short wavelengths has already been found by Refs. [21,37]. In those works an infinitely deep fluid was studied.…”

“…Surfactant, pollutant, and fluid film impacts at interfaces are now hot subjects [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35]. Current research on surface waves remains ongoing, with a predominant emphasis on multi-layer models [18], damping rate [2,5,7,[17][18][19][20]36,37], and experimental verifications [17,20]. The majority of research endeavours in this particular domain are founded upon the utilization of viscoelastic fluids [17,21].…”

Reconciling different formulations of viscous water waves and their mass conservation

“…Additionally, various other models hint at the existence and relevance of nonlinear viscosity terms in the propagation equation [22,23]. The order analysis in Cartesian coordinates performed in Appendix C is limited to very low steepness, therefore, an analysis of the magnitude of the nonlinear vortical terms in curved coordinates could provide insight to their relevance in different regimes of viscosity and wave steepness.…”

Comprehensive Perturbation Approach to Nonlinear Viscous Gravity–Capillary Surface Waves at Arbitrary Wavelengths in Finite Depth

“…The impact of viscosity on the nonlinear propagation of surface waves at the interface of air and a fluid of large depth is discussed in [35]. They proposed modified hydrodynamic boundary conditions to model both short and long waves.…”

“…Some longer expressions are relegated to the Appendix A. Le Meur [34] Boussinesq approximation Armaroli [35] X -Ghahraman and…”