Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 1708.04307

On tidal energy in Newtonian two-body motion

Shuang Miao·Sohrab Shahshahani
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this work, which is based on an essential linear analysis carried out by Christodoulou, we study the evolution of tidal energy for the motion of two gravitating incompressible fluid balls with free boundaries obeying the Euler-Poisson equations. The orbital energy is defined as the mechanical energy of the two bodies' center of mass. According to the classical analysis of Kepler and Newton, when the fluids are replaced by point masses, the conic curve describing the trajectories of the masses is a hyperbola when the orbital energy is positive and an ellipse when the orbital energy is negative. The orbital energy is conserved in the case of point masses. If the point masses are initially very far, then the orbital energy is positive, corresponding to hyperbolic motion. However, in the motion of fluid bodies the orbital energy is no longer conserved because part of the conserved energy is used in deforming the boundaries of the bodies. In this case the total energy $\tilde{\mathcal{E}}$ can be decomposed into a sum $\tilde{\mathcal{E}}:=\widetilde{\mathcal{E}_{{\mathrm{orbital}}}}+\widetilde{\mathcal{E}_{{\mathrm{tidal}}}}$, with $\widetilde{\mathcal{E}_{{\mathrm{tidal}}}}$ measuring the energy used in deforming the boundaries, such that if $\widetilde{\mathcal{E}_{{\mathrm{orbital}}}}<-c<0$ for some absolute constant $c>0$, then the orbit of the bodies must be bounded. In this work we prove that under appropriate conditions on the initial configuration of the system, the fluid boundaries and velocity remain regular up to the point of the first closest approach in the orbit, and that the tidal energy $\widetilde{\mathcal{E}_{{\mathrm{tidal}}}}$ can be made arbitrarily large relative to the total energy $\tilde{\mathcal{E}}$. In particular under these conditions $\widetilde{\mathcal{E}_{{\mathrm{orbital}}}}$, which is initially positive, becomes negative before the point of the first closest approach.

Explore related subjects

math.APmath-phmath.MPphysics.flu-dyn

Keep this discovery

BibTeXRIS

Shuang Miao, Sohrab Shahshahani. 2017-08-14. On tidal energy in Newtonian two-body motion. https://arxiv.org/abs/1708.04307

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Well-posedness of the two-dimensional unsteady Prandtl system in Sobolev space with degenerate critical points

This paper is devoted to the well-posedness of classical Prandtl equations in a finite order Sobolev space. For a initial data with degenerate critical points and general outflow, we obtain the local-in-time existence and uniqueness of the solution to the Prandtl equations in a Sobolev space, by introducing a new iteration scheme and linear cancelation. This result shows that Oleinik's monotonicity condition is not a necessary condition for the Prandtl equations to be well-posed in Sobolev spaces and provides evidence to demonstrate that zero shear stress does not necessarily lead to boundary layer separation in two-dimensional unsteady boundary layers.

math.AP↗
arXiv2026

Global existence and time decay for a bipolar Euler-Poisson system with one pressureless and undamped fluid

We study the Cauchy problem for a three-dimensional bipolar Euler--Poisson system in which one fluid is pressureless and undamped, while the other is subject to momentum relaxation. For sufficiently small smooth perturbations of a constant equilibrium, we prove the global existence and uniqueness of smooth solutions under an irrotationality assumption on the initial velocity of the pressureless fluid, together with algebraic time-decay estimates. The main difficulty is that the velocity of the pressureless fluid is dissipated only indirectly through the Poisson coupling, and this mechanism degenerates strongly at high frequencies, leading to a regularity-loss structure. We overcome this difficulty by combining refined Green-function estimates, a low--middle--high frequency decomposition, and high-order nonlinear energy estimates adapted to the asymmetric regularity hierarchy. The result establishes a global small-data theory for this asymmetric regime, in which pressure and damping are simultaneously absent from the same fluid.

math.AP↗
arXiv2026

Boundary layer of 2D Chemotaxis Navier-Stokes equations with logarithmic Sensitivity. II. viscous vanishing limit

This is the second part of a two-part work concerning boundary layer solutions to the coupled Chemotaxis-Navier-Stokes system in the two-dimensional half-space. In the present work, we address the convergence of boundary layer solutions to singular chemotaxis-fluid equations under slip boundary conditions with respect to the chemical diffusion-viscosity parameter $\varepsilon$ in the two-dimensional half-plane. More precisely, we show that the boundary layer for $\varepsilon>0$ (viscous convection coefficient) converges to the superposition of the outer layer (solution with $\varepsilon=0$) and the inner layer as $\varepsilon\rightarrow0$. The outer and inner profiles are explicitly derived as in the first part\cite{WWZ}. Furthermore, the well-posedness results of the coupled Chemotaxis-Navier-Stokes system in conormal Sobolev spaces will be presented in Appendix. They answer the question mentioned in the first part of the two-part work. This study could help the understanding of the chemotactic movement of aerobic bacteria to the water-air surface observed experimentally in fluids, and enrich the theoretical results of boundary layer in chemotactic fluid models.

math.AP↗