We investigate the problem of classification of solutions for the steady Navier-Stokes equations in any cone-like domains. In the form of separated variables, $$u(x,y)=left( begin{array}{c} varphi_1(r)v_1(theta) varphi_2(r)v_2(theta) end{array} right) ,$$ where $x=rcostheta$ and $y=rsintheta$ in polar coordinates, we obtain the expressions of all smooth solutions with $C^0$ Dirichlet boundary condition. In particular, it shows that (i) some solutions are found, which are H{o}lder continuous on the boundary, but their gradients blow up at the corner; (ii) all solutions in the entire plane of $mathbb{R}^2$ like harmonic functions or Stokes equations, are polynomial expressions.
Collective systems across length scales display order in their spatiotemporal patterns. These patterns contain information that correlates with their orders and reflects the system dynamics. Here we show the collective patterns and behaviors of up to 250 micro-rafts spinning at the air-water interface and demonstrate the link between order and information in the collective motion. These micro-rafts display a rich variety of collective behaviors that resemble thermodynamic equilibrium phases such as gases, hexatics, and crystals. Moreover, owing to the unique coupling of magnetic and fluidic forces, a number of collective properties and functions emerge as the micro-rafts interact with magnetic potential and nonmagnetic floating objects. Our findings are relevant for analyzing collective systems in nature and for designing collective robotic systems.
Control on microscopic scales depends critically on our ability to manipulate interactions with different physical fields. The creation of micro-machines therefore requires us to understand how multiple fields, such as surface capillary or electro-magnetic, can be used to produce predictable behaviour. Recently, a spinning micro-raft system was developed that exhibited both static and dynamic self-assembly [Wang et al. (2017) Sci. Adv. 3, e1602522]. These rafts employed both capillary and magnetic interactions and, at a critical driving frequency, would suddenly change from stable orbital patterns to static assembled structures. In this paper, we explain the dynamics of two interacting micro-rafts through a combination of theoretical models and experiments. This is first achieved by identifying the governing physics of the orbital patterns, the assembled structures, and the collapse separately. We find that the orbital patterns are determined by the short range capillary interactions between the disks, while the explanations of the other two behaviours only require the capillary far field. Finally we combine the three models to explain the dynamics of a new micro-raft experiment.
This note is devoted to investigating Liouville type properties of the two dimensional stationary incompressible Magnetohydrodynamics equations. More precisely, under smallness conditions only on the magnetic field, we show that there are no non-trivial solutions to MHD equations either the Dirichlet integral or some $L^p$ norm of the velocity-magnetic fields are finite. In particular, these results generalize the corresponding Liouville type properties for the 2D Navier-Stokes equations, such as Gilbarg-Weinberger cite{GW1978} and Koch-Nadirashvili-Seregin-Sverak cite{KNSS}, to the MHD setting.
