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1. Traveling Wave Solutions to the Free Boundary Incompressible Navier‐Stokes Equations

   https://doi.org/10.1002/cpa.22084  

   Leoni, Giovanni ; Tice, Ian ( October 2023 , Communications on Pure and Applied Mathematics)

   In this paper we study a finite‐depth layer of viscous incompressible fluid in dimension , modeled by the Navier‐Stokes equations. The fluid is assumed to be bounded below by a flat rigid surface and above by a free, moving interface. A uniform gravitational field acts perpendicularly to the flat surface, and we consider the cases with and without surface tension acting on the free interface. In addition to these gravity‐capillary effects, we allow for a second force field in the bulk and an external stress tensor on the free interface, both of which are posited to be in traveling wave form, i.e., time‐independent when viewed in a coordinate system moving at a constant velocity parallel to the rigid lower boundary. We prove that, with surface tension in dimension and without surface tension in dimension , for every nontrivial traveling velocity there exists a nonempty open set of force and stress data that give rise to traveling wave solutions. While the existence of inviscid traveling waves is well‐known, to the best of our knowledge this is the first construction of viscous traveling wave solutions.

   Our proof involves a number of novel analytic ingredients, including: the study of an overdetermined Stokes problem and its underdetermined adjoint problem, a delicate asymptotic development of the symbol for a normal‐stress to normal‐Dirichlet map defined via the Stokes operator, a new scale of specialized anisotropic Sobolev spaces, and the study of a pseudodifferential operator that synthesizes the various operators acting on the free surface functions. © 2022 The Authors.Communications on Pure and Applied Mathematicspublished by Wiley Periodicals LLC.

    

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2. Whitham modulation theory for the Zakharov–Kuznetsov equation and stability analysis of its periodic traveling wave solutions

   https://doi.org/10.1111/sapm.12651  

   Biondini, Gino ; Chernyavsky, Alexander ( February 2024 , Studies in Applied Mathematics)

   We derive the Whitham modulation equations for the Zakharov–Kuznetsov equation via a multiple scales expansion and averaging two conservation laws over one oscillation period of its periodic traveling wave solutions. We then use the Whitham modulation equations to study the transverse stability of the periodic traveling wave solutions. We find that all periodic solutions traveling along the first spatial coordinate are linearly unstable with respect to purely transversal perturbations, and we obtain an explicit expression for the growth rate of perturbations in the long wave limit. We validate these predictions by linearizing the equation around its periodic solutions and solving the resulting eigenvalue problem numerically. We also calculate the growth rate of the solitary waves analytically. The predictions of Whitham modulation theory are in excellent agreement with both of these approaches. Finally, we generalize the stability analysis to periodic waves traveling in arbitrary directions and to perturbations that are not purely transversal, and we determine the resulting domains of stability and instability.

    

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3. Whitham modulation theory for generalized Whitham equations and a general criterion for modulational instability

   https://doi.org/10.1111/sapm.12398  

   Binswanger, Adam_L ; Hoefer, Mark_A ; Ilan, Boaz ; Sprenger, Patrick ( May 2021 , Studies in Applied Mathematics)

   The Whitham equation was proposed as a model for surface water waves that combines the quadratic flux nonlinearityof the Korteweg–de Vries equation and the full linear dispersion relationof unidirectional gravity water waves in suitably scaled variables. This paper proposes and analyzes a generalization of Whitham's model to unidirectional nonlinear wave equations consisting of a general nonlinear flux functionand a general linear dispersion relation. Assuming the existence of periodic traveling wave solutions to this generalized Whitham equation, their slow modulations are studied in the context of Whitham modulation theory. A multiple scales calculation yields the modulation equations, a system of three conservation laws that describe the slow evolution of the periodic traveling wave's wavenumber, amplitude, and mean. In the weakly nonlinear limit, explicit, simple criteria in terms of generalandestablishing the strict hyperbolicity and genuine nonlinearity of the modulation equations are determined. This result is interpreted as a generalized Lighthill–Whitham criterion for modulational instability.

    

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4. Stokes waves with constant vorticity: I. Numerical computation

   https://doi.org/10.1111/sapm.12250  

   Dyachenko, Sergey A. ; Hur, Vera Mikyoung ( February 2019 , Studies in Applied Mathematics)

   Periodic traveling waves are numerically computed in a constant vorticity flow subject to the force of gravity. The Stokes wave problem is formulated via a conformal mapping as a nonlinear pseudodifferential equation, involving a periodic Hilbert transform for a strip, and solved by the Newton‐GMRES method. For strong positive vorticity, in the finite or infinite depth, overhanging profiles are found as the amplitude increases and tend to a touching wave, whose surface contacts itself at the trough line, enclosing an air bubble; numerical solutions become unphysical as the amplitude increases further and make a gap in the wave speed versus amplitude plane; another touching wave takes over and physical solutions follow along the fold in the wave speed versus amplitude plane until they ultimately tend to an extreme wave, which exhibits a corner at the crest. Touching waves connected to zero amplitude are found to approach the limiting Crapper wave as the strength of positive vorticity increases unboundedly, while touching waves connected to the extreme waves approach the rigid body rotation of a fluid disk.

    

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5. On the dynamics of a free surface of an ideal fluid in a bounded domain in the presence of surface tension

   https://doi.org/10.1017/jfm.2018.885  

   Dyachenko, Sergey A. ( February 2019 , Journal of Fluid Mechanics)

   We derive a set of equations in conformal variables that describe a potential flow of an ideal two-dimensional inviscid fluid with free surface in a bounded domain. This formulation is free of numerical instabilities present in the equations for the surface elevation and potential derived in Dyachenko et al.  ( Plasma Phys. Rep. vol. 22 (10), 1996, pp. 829–840) with some restrictions on analyticity relieved, which allows to treat a finite volume of fluid enclosed by a free-moving boundary. We illustrate with a comparison of numerical simulations of the Dirichlet ellipse, an exact solution for a zero surface tension fluid. We demonstrate how the oscillations of the free surface of a unit disc droplet may lead to breaking of one droplet into two when surface tension is present. 

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