Viscous perturbations of marginally stable Euler flow and finite-time Melnikov theory
نویسندگان
چکیده
The effect of small viscous dissipation on Lagrangian transport in two-dimensional vorticityconserving fluid flows motivates this work. If the inviscid equation admits a base flow in which different fluid regions are divided by separatrices, then transport between these regions is afforded by the splitting of separatrices caused by viscous dissipation. Finite-time Melnikov theory allows us to measure the splitting distance of separatrices provided the perturbed velocity field of the viscous fluid flow stays sufficiently close to vorticity-conserving base flow over sufficiently long time intervals. In this paper, we derive the necessary long-term estimates of solutions to Euler’s equation and to the barotropic vorticity equation upon adding viscous perturbations and forcing. We discover that a certain stability condition on the unperturbed flow is sufficient to guarantee these long time estimates.
منابع مشابه
Melnikov theory for finite-time vector fields
Melnikov theory provides a powerful tool for analysing time-dependent perturbations of autonomous vector fields that exhibit heteroclinic orbits. The standard theory requires that the perturbed vector field be defined, and bounded, for all times. In this paper, Melnikov theory is adapted so that it is applicable to vector fields that are defined over sufficiently large, but finite, time interva...
متن کاملViscous perturbations of vorticity-conserving flows and separatrix splitting
We examine the effect of the breaking of vorticity conservation by viscous dissipation on transport in the underlying fluid flow. The transport of interest is between regimes of different characteristic motion and is afforded by the splitting of separatrices. A base flow that is vorticity conserving is therefore assumed to have a separatrix that is either a homoclinic or heteroclinic orbit. The...
متن کاملA Composite Finite Difference Scheme for Subsonic Transonic Flows (RESEARCH NOTE).
This paper presents a simple and computationally-efficient algorithm for solving steady two-dimensional subsonic and transonic compressible flow over an airfoil. This work uses an interactive viscous-inviscid solution by incorporating the viscous effects in a thin shear-layer. Boundary-layer approximation reduces the Navier-Stokes equations to a parabolic set of coupled, non-linear partial diff...
متن کاملOn the planar extensional motion of an inertially driven liquid sheet
We derive a time-dependent exact solution of the free surface problem for the Navier–Stokes equations that describes the planar extensional motion of a viscous sheet driven by inertia. The linear stability of the exact solution to oneand two-dimensional symmetric perturbations is examined in the inviscid and viscous limits within the framework of the long-wave or slender body approximation. Bot...
متن کاملStability of Two-Dimensional Viscous Incompressible Flows under Three-Dimensional Perturbations and Inviscid Symmetry Breaking
In this article we consider weak solutions of the three-dimensional incompressible fluid flow equations with initial data admitting a one-dimensional symmetry group. We examine both the viscous and inviscid cases. For the case of viscous flows, we prove that Leray-Hopf weak solutions of the threedimensional Navier-Stokes equations preserve initially imposed symmetry and that such symmetric flow...
متن کامل