skip to main content


This content will become publicly available on October 1, 2024

Title: Traveling Wave Solutions to the Free Boundary Incompressible Navier‐Stokes Equations
Abstract

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.

 
more » « less
Award ID(s):
1653161
NSF-PAR ID:
10470423
Author(s) / Creator(s):
;
Publisher / Repository:
Wiley
Date Published:
Journal Name:
Communications on Pure and Applied Mathematics
Volume:
76
Issue:
10
ISSN:
0010-3640
Page Range / eLocation ID:
2474 to 2576
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. Inspired by the recent realization of a two-dimensional (2-D) chiral fluid as an active monolayer droplet moving atop a 3-D Stokesian fluid, we formulate mathematically its free-boundary dynamics. The surface droplet is described as a general 2-D linear, incompressible and isotropic fluid, having a viscous shear stress, an active chiral driving stress and a Hall stress allowed by the lack of time-reversal symmetry. The droplet interacts with itself through its driven internal mechanics and by driving flows in the underlying 3-D Stokes phase. We pose the dynamics as the solution to a singular integral–differential equation, over the droplet surface, using the mapping from surface stress to surface velocity for the 3-D Stokes equations. Specializing to the case of axisymmetric droplets, exact representations for the chiral surface flow are given in terms of solutions to a singular integral equation, solved using both analytical and numerical techniques. For a disc-shaped monolayer, we additionally employ a semi-analytical solution that hinges on an orthogonal basis of Bessel functions and allows for efficient computation of the monolayer velocity field, which ranges from a nearly solid-body rotation to a unidirectional edge current, depending on the subphase depth and the Saffman–Delbrück length. Except in the near-wall limit, these solutions have divergent surface shear stresses at droplet boundaries, a signature of systems with codimension-one domains embedded in a 3-D medium. We further investigate the effect of a Hall viscosity, which couples radial and transverse surface velocity components, on the dynamics of a closing cavity. Hall stresses are seen to drive inward radial motion, even in the absence of edge tension. 
    more » « less
  2. null (Ed.)
    We introduce a mathematical modeling framework for the conformational dynamics of charged molecules (i.e., solutes) in an aqueous solvent (i.e., water or salted water). The solvent is treated as an incompressible fluid, and its fluctuating motion is described by the Stokes equation with the Landau–Lifschitz stochastic stress. The motion of the solute-solvent interface (i.e., the dielectric boundary) is determined by the fluid velocity together with the balance of the viscous force,hydrostatic pressure, surface tension, solute-solvent van der Waals interaction force, and electrostatic force. The electrostatic interactions are described by the dielectric Poisson–Boltzmann theory.Within such a framework, we derive a generalized Rayleigh–Plesset equation, a nonlinear stochastic ordinary differential equation (SODE), for the radius of a spherical charged molecule, such as anion. The spherical average of the stochastic stress leads to a multiplicative noise. We design and test numerical methods for solving the SODE and use the equation, together with explicit solvent molecular dynamics simulations, to study the effective radius of a single ion. Potentially, our general modeling framework can be used to efficiently determine the solute-solvent interfacial structures and predict the free energies of more complex molecular systems. 
    more » « less
  3. Abstract

    The equation for a traveling wave on the boundary of a two‐dimensional droplet of an ideal fluid is derived by using the conformal variables technique for free surface waves. The free surface is subject only to the force of surface tension and the fluid flow is assumed to be potential. We use the canonical Hamiltonian variables discovered and map the lower complex plane to the interior of a fluid droplet conformally. The equations in this form have been originally discovered for infinitely deep water and later adapted to a bounded fluid domain.The new class of solutions satisfies a pseudodifferential equation similar to the Babenko equation for the Stokes wave. We illustrate with numerical solutions of the time‐dependent equations and observe the linear limit of traveling waves in this geometry.

     
    more » « less
  4. Abstract

    We formulate the two-dimensional gravity-capillary water wave equations in a spatially quasi-periodic setting and present a numerical study of solutions of the initial value problem. We propose a Fourier pseudo-spectral discretization of the equations of motion in which one-dimensional quasi-periodic functions are represented by two-dimensional periodic functions on a torus. We adopt a conformal mapping formulation and employ a quasi-periodic version of the Hilbert transform to determine the normal velocity of the free surface. Two methods of time-stepping the initial value problem are proposed, an explicit Runge–Kutta (ERK) method and an exponential time-differencing (ETD) scheme. The ETD approach makes use of the small-scale decomposition to eliminate stiffness due to surface tension. We perform a convergence study to compare the accuracy and efficiency of the methods on a traveling wave test problem. We also present an example of a periodic wave profile containing vertical tangent lines that is set in motion with a quasi-periodic velocity potential. As time evolves, each wave peak evolves differently, and only some of them overturn. Beyond water waves, we argue that spatial quasi-periodicity is a natural setting to study the dynamics of linear and nonlinear waves, offering a third option to the usual modeling assumption that solutions either evolve on a periodic domain or decay at infinity.

     
    more » « less
  5. Abstract

    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.

     
    more » « less