written and published by Renzo Diomedi // UNDER CONSTRUCTION


NS, composed by a conservative continuity equation that since the flow is incompressible (if the variations in density of the fluid do not have appreciable effects and therefore the density can be considered with a good approximation, a constant), and a non-conservative Momentum equation not exactly measurable as a scalar field, but divisible by 3 scalar equations laid and projected along the directions x, y, z which returned values not coinciding.

So we have 4 independent variables x,y,x,t and the 4 dependent variables u,v,w, (velocity components) and p (pressure) and the 6 given by the Stress Tensor:



then, considering u, v, w as components of the shift vectors along the axes x, y, z, :
; = Density , = Viscosity , = standard gravity (acceleration)
Stress tensor :
;



if the components of the viscous stress state are linearly linked to the components of the deformation velocity through Stokes' relations :

whereas




so we saw this tensor as a generally irregular pyramid with triangular base , but if instead we tried to consider it as a deformable and curvable sheet, we may have a new vision about these equations



2-dimensional viscous stress tensor :



The Metric Tensor g expresses the property of a geometrically curvable structure with the points of its lattice at a distance always equal in relation to the structural components themselves:

= distance of point from origin, whatever the inclination of the reference axes = (using the Einstein notation)
where

then then then

contravariant metric tensor: , covariant metric tensor:

so we can maintain the position of the points created in the cube, transferring them to a deformable sheet


hence, the 10 dependent variables can be reduced to 6 , while the equations used to give them a solution are 3 , so the difference is reduced from 6 to 3 :

Continuity:

Momentum:

where


= metric abscissa
= metric ordinate


hence we can calculate the amount of energy created over a certain period of time by these simplified new version of N-S equations:



Then we examine a lagrangian particle that in its path creates a String. Nambu-Goto equation analyzes the behavior of the string and the energy produced by it, proportional to the minimum area of the worldsheet area. So we apply the N-G's Action : where


= 2-dimensional viscous stress tensor :






where , and the cosine of is , then

hence, considering x = metric abscissae and y = metric ordinates we have the unidimensional N+S equation :

TO BE CONTINUED........


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