Mathematics problems: Navier-Stokes equations

posted by @ulaulaman about #MillenniumProblems #NavierStokes #MukhtarbayOtelbayev #physics #mathematics

In fluid mechanics, the Navier-Stokes equations, developed by Claude-Louis Navier and George Gabriel Stokes, describe the motion of a fluid into the space. Given its velocity $\vec{v}$, the pressure $p$, and the kinematic viscosity $\nu$, in presence of an external force $\vec{f}$, the particles' motion in the fluid could be described by the following vector equation: \[\frac{\partial \vec{v}}{\partial t} + ( \vec{v} \cdot \vec \nabla ) \vec{v} = -\vec \nabla p + \nu \Delta \vec{v} +\vec{f}(\vec{x},t)\] The trouble is that, to obtain solutions of this equation, we must introduce approximations that simplify the search of them: for example, a major difficulty is to determine the solutions in the presence of some turbulence. To this problem, that it has a physical nature, we must add another mathematical question: the difficulty in proving, given the initial conditions, the existence of smooth solutions for the equations. Given these difficulties, the Clay Mathematics Institute included it in the list of the seven Millennium Problems:

In three space dimensions and time, given an initial velocity field, there exists a vector velocity and a scalar pressure field, which are both smooth and globally defined, that solve the Navier–Stokes equations.

The bidimensional motions of a heavy metal moshing

posted by @ulaulaman via @LuciaMarino81 about #heavymetal #collectivemotions #physics #fluiddynamics

During the APS March Meeting 2013, Matthew Bierbaum with Jesse Silverberg, James P. Sethna and Itai Cohen from Cornell University, presented a curious study about the heavy metal mosh pits: studing the people during the moshing, the researchers find two types of collective motions:

mosh pits, in which participants collide with each other randomly in a manner resembling an ideal gas, and circle pits, in which participants run collectively in a circle forming a vortex of people.

The results are published on arXiv (Collective Motion of Moshers at Heavy Metal Concerts):

Human collective behavior can vary from calm to panicked depending on social context. Using videos publicly available online, we study the highly energized collective motion of attendees at heavy metal concerts. We find these extreme social gatherings generate similarly extreme behaviors: a disordered gas-like state called a mosh pit and an ordered vortex-like state called a circle pit. Both phenomena are reproduced in flocking simulations demonstrating that human collective behavior is consistent with the predictions of simplified models.

(via Lucia Marino)