Force-free vector fields

As argued by Reed [4], the Beltrami vector field originated in hydrodynamics and is force-free. It is one of the three basic types of field solenoidal, complex lamellar, and Beltrami. These vector fields originated in hydrodynamics and describe the properties of the velocity field, flux or streamline, v, and the vorticity V x v. The Beltrami field is also a Magnus force free fluid flow and is expressed in hydrodynamics as... 

This vector field condition is sometimes referred to as Beltrami fluid flow, and was previously treated in a similar exposition by the author in 1995 [1], There it was indicated that Beltrami vector field flow is representative of a certain class of vector fields that are termed force-free. This type of field topology was first brought to prominence by Eugenio Beltrami in his 1889 paper Considerations on Hydrodynamics. [2], This type of morphology describes a regime of fluid... 

However, in a Beltrami field, the vorticity and velocity vectors are parallel or antiparallel, resulting in a zero Magnus force. The Beltrami condition (1) is therefore an equivalent way of characterizing a force-free flow situation, and vice versa. 

Here E, D, and P represent, respectively, the electric field, electric induction (or displacement), and electric polarization vectors P (D - )/4x. The integration must be carried out over all space penetrated by the electrostatic field. Equation (5.6.1), while correct, is awkward in several respects. First, there is the need to integrate over all space, including the region outside the system of interest. In the presence of a medium, the electric lines of force not only are present within the specimen, but also bulge out in all directions away from the system these effects must be included in (5.6.1). Second, there is a tendency in the literature to associate the first term in (5.6.1b) with the establishment of the electric field in free space, and the second term with the reaction of the medium to the electric field. This is wrong The quantity D is subject to direct experimental control because it is linked by Maxwell s equation to the presence of free charges by contrast, E is in part a reaction field that also includes the... 

Graphs of the expressions (6.76), (6.77) and (6.78) for force on a dislocation in a traction free layer for the three components of the Burgers vector are shown in Figure 6.36. A predictable feature of this graph is that the force is zero at the midplane position, as it must be due to symmetry. Therefore, this is an equilibrium position for all three cases. While this equilibrium position is unstable for / 0 and bz / 0, it is surprising to find that this position is a stable equilibrium position for 6 /0. Apparently, this is due to the bending stress field which arises in the layer when the dislocation is formed. 

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