written and published by Renzo Diomedi // UNDER CONSTRUCTION

NS, composed by a conservative continuity equation . If the flow is incompressible, id est if the variations of the density of the fluid do not have appreciable effects, 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 non-linear, divisible by 3 scalar equations laid and projected along the directions x, y, z .

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

So, to solve the unknown of 10 variables we have 4 equations only.

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

Then , considering the components of the viscous stress state are linearly linked to the components of the deformation velocity through Stokes' relations , we have :


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

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)

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

2-dimensional viscous stress tensor :

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




= 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:

but this is not enough to overcome the non-linearity. We need an output unidimensional parameter. 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

= String viscous Tension :

where , and the cosine of is , then

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

While if the flow density is unsteady, considering

, ,


Prandtl's number = ,

Heat Flux = where :

warmer face temperature , cooler face temperature , Surface of the matter traversed by the heat , Thickness of the matter traversed by the heat

considering the linear calculus of the total energy developed:


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