Class Of Velocity Fields Research Articles

Finite-time singularity formation for $C^{1,\alpha}$ solutions to the incompressible Euler equations on $\mathbb{R}^3$

It has been known since work of Lichtenstein and Gunther in the 1920s that the 3D incompressible Euler equation is locally well-posed in the class of velocity fields with Hölder continuous gradient and suitable decay at infinity. It is shown here that these local solutions can develop singularities in finite time, even for some of the simplest three-dimensional flows.

An Upper Bound Solution for a Two-Layer Cylinder Subject to Compression and Twist

The choice of a kinematically admissible velocity field has a great effect on the predictive capacity of upper bound solutions. It is always advantageous, in addition to the formal requirements of the upper bound theorem, to select a class of velocity fields satisfying some additional conditions that follow from the exact formulation of the problem. In the case of maximum friction law, such an additional condition is that the real velocity field is singular in the vicinity of the friction surface. In the present paper this additional condition is incorporated in the class of kinematically admissible velocity fields chosen for a theoretical analysis of two - layer cylinders subject to compression and twist. An effect of the angular velocity of the die on process parameters is emphasized and discussed.

Velocity fields that preserve the force-free character of magnetic fields

The flow of a compressible fluid of finite conductivity in the presence of a force-free magnetic field satisfying the relation curl B = α ( r , t) B is analysed. It is shown that only a particular class of velocity fields allows the magnetic field to remain force free in time. In the case of incompressible fluids (or for velocities very small compared to the velocity of sound in the medium) the type of flow is simpler and is the hydromagnetic analog of Beltrami flow, that is represented by a simultaneous set of equations: curl ( B · ▽) ν = α ( B · ▽) ν and curl (ν · ▽) B = α (ν · ▽) B . A new interpretation of the actual physical nature of force-free magnetic fields is offered.