TY - JOUR
T1 - A model ODE for the exponential asymptotics of nonlinear parasitic capillary ripples
AU - Shelton, Josh
AU - Trinh, Philippe H.
N1 - Funding: Engineering and Physical Sciences Research Council (EPSRC grant no. EP/V012479/1 to J.S. and P.H.T.); Engineering and Physical Sciences Research Council (EPSRC grant no. EP/W522491/1 to J.S.).
PY - 2024/6/13
Y1 - 2024/6/13
N2 - In this work, we develop a linear model ordinary differential equation (ODE) to study the parasitic capillary ripples present on steep Stokes waves when a small amount of surface tension is included in the formulation. Our methodology builds upon the exponential asymptotic theory of Shelton & Trinh (J. Fluid Mech., vol. 939, 2022, A17), who demonstrated that these ripples occur beyond-all-orders of a small-surface-tension expansion. Our model equation, a linear ODE forced by solutions of the Stokes wave equation, forms a convenient tool to calculate numerical and asymptotic solutions. We show analytically that the parasitic capillary ripples that emerge in solutions to this linear model have the same asymptotic scaling and functional behaviour as those in the fully nonlinear problem. It is expected that this work will lead to the study of parasitic capillary ripples that occur in more general formulations involving viscosity or time-dependence.
AB - In this work, we develop a linear model ordinary differential equation (ODE) to study the parasitic capillary ripples present on steep Stokes waves when a small amount of surface tension is included in the formulation. Our methodology builds upon the exponential asymptotic theory of Shelton & Trinh (J. Fluid Mech., vol. 939, 2022, A17), who demonstrated that these ripples occur beyond-all-orders of a small-surface-tension expansion. Our model equation, a linear ODE forced by solutions of the Stokes wave equation, forms a convenient tool to calculate numerical and asymptotic solutions. We show analytically that the parasitic capillary ripples that emerge in solutions to this linear model have the same asymptotic scaling and functional behaviour as those in the fully nonlinear problem. It is expected that this work will lead to the study of parasitic capillary ripples that occur in more general formulations involving viscosity or time-dependence.
KW - Exponential asymptotics
KW - Gravity-capillary waves
KW - Parasitic capillary ripples
KW - Free-surface flows
UR - https://arxiv.org/abs/2309.11779
U2 - 10.1093/imamat/hxae016
DO - 10.1093/imamat/hxae016
M3 - Article
SN - 0272-4960
VL - 89
SP - 318
EP - 342
JO - IMA Journal of Applied Mathematics
JF - IMA Journal of Applied Mathematics
IS - 2
ER -