Exact Solution of Navier-Stokes Equations

In Navier-Stokes equations (NASA’s Navier-Stokes Equations, 3-dimensional-unsteady), wed iscover the exact solution by Newton potential function and time-function. We think the solution likely Newton potential function that be able to solve Laplace equation

Post-Newtonian properties of EMRI with power law potential

There are many astrophysical scenarios where extreme mass ratio inspiral (EMRI) binaries can be surrounded by inhomogenous matter distribution. The distribution of mass can affect the dynamical properties (e.g. orbital frequency, average energy radiation rate, etc.) of the EMRI. In this matter distribution, instead of Kepler–Newton potential, one may consider a more general form of potential i.e. power law potential. Moreover, due to the power law potential, at the Newtonian order itself, the velocity profile of test particles does not fall as much as that predicted by Kepler–Newton potential and this feature of the velocity profile may be observationally important. In this study, we have obtained the first post-Newtonian (1PN) expressions for dynamical quantities and the average energy radiation rate from the circular orbit EMRI which is surrounded by a matter distribution. We show that the energy radiation rate and orbital frequency of EMRI can be significantly different in the presence of power law potential as compared to that in the Kepler–Newton potential, signatures of which may be observed in gravitational waves from EMRI.

Analytic Integration of the Newton Potential over Cuboids and an Application to Fast Multipole Methods

We present simplified formulæ for the analytic integration of the Newton potential of polynomials over boxes in two- and three-dimensional space. These are implemented in an easy-to-use C++ library that allows computations in arbitrary precision arithmetic which is also documented here. We describe how these results can be combined with fast multipole methods to evaluate the Newton potential of more general, non-polynomial densities.