Model of continuumVortex lines in the continuum - line direction instantaneous axis of rotation of the particles are on them.
Vortex filaments in a continuum - Dedicated mass, each element surface boundary set according vortex line [103].
Suppose that for all the forces acting on the environment there is a potential strength.
We define potential 0-forms:
Kinematic viscosity
or Magnetic weight
is 0-form of primal nodes.
Thin magnetic solenoid - thread magnetized liquid crystal, which can be represented as a line current I is 0-form of dual nodes.
Faraday's law of induction makes use of the magnetic flux
through a hypothetical surface
whose boundary is a wire loop. Since the wire loop may be moving, we write
)
for the surface. The magnetic flux is defined by a surface integral:
} \mathbf{B}(\mathbf{r}, t) \cdot d \mathbf{A})
is primal node (0-form),
where dA is an element of surface area of the moving surface
)
, B is the magnetic field (also called "magnetic flux density"), and B·dA is a vector dot product (the infinitesimal amount of magnetic flux through the infinitesimal area element dA). In more visual terms, the magnetic flux through the wire loop is proportional to the number of magnetic flux lines that pass through the loop.
The product of the cross section of the speed of rotation for an infinitely thin filament (2-form) to all over her constantly and stores the value in the movement of the thread [103]. In other words, the magnetic mass of filaments remain unchanged during the motion.
Or for NS:
is primal edges (1-form),
is primal 2-cell,
Vortex filaments it is dual node (not real vector):
is dual 0-form.
Hill's spherical vortexH is 1-form may be referred to as the normal velocity form of the Vortex filaments in a continuum or the flux, since it represents the velocity normal to the triangles’ faces (i.e. dual edges).
We denote the velocity 1-form
defined on the primal edges by v; i.e.
)
. The velocity 1-form v represents the velocity tangential to the triangles edges.
[103] H.Helmholtz, Zwei hydrodynamische Abhandlungen. Monatsberichte d. konigl. Acad. d. Wiss. zu Berlin, 1868, S. 215-228