written and published by Renzo Diomedi // UNDER CONSTRUCTION

Assume a Fluid as incompressible, id est if the variations of the density of the fluid do not have appreciable effects, so its density can be considered ,with a good approximation, a constant. Then

Continuity + Momentum = +

the conservative continuity equation (scalar value) where u, v, w lie on x, y, z and u^x v^y w^z have cosine = 1.

the non-conservative Momentum equation (i.e. not exactly measurable as a scalar field)

the non-conservative Momentum equation are however decomposable in 3 scalar equations laid along the directions of x, y, z .
Then considering u, v, w as components of the shift vectors along the axes x, y, z, :

; = Density , = Viscosity , = standard gravity (acceleration)

The independent variables are 4 :

The dependent variables are 10 , of which 3 velocity components + the pressure
and 6 dependent variables given by the Stress Tensor:

So, as seen above, to know the value of the unknown of the 10 dependent variables we have 4 scalar equations only.

expansed Continuity eq =

momentum eq is also

Stress Tensor

then, partially derived as in the NS equations:

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

The Principal stresses and Shear stresses act in a 3d space. They can act in a deformable and curvable sheet, a 2-dimensional space rather than 3-dimensional space.

Imagine a lattice composed of equidistant points subjected to stresses and crossed by a flow to analyze. The Turbulence produces lattice distortions, but the points remain at a constant distance among them because only the axial coordinates of reference vary.

But how much the lattice is deformed? how much are the axes and coordinates moved? this deviation is the Tension to be calculated.

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

Considering = metric abscissa and = metric ordinate

= distance of point from origin, whatever the inclination of the reference axes


(note that : , , ,

contravariant metric tensor: , covariant metric tensor: )

2-dimensional viscous stress tensor :

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


Momentum :

Now, the independent variables are 3, the dependent variables are 6 and the output equations are 3. So the imbalance has been reduced from 6 to 3

We need an unidimensional output to overcome the non-linearity. A lagrangian particle 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 A = worldsheet Area.
the position of the lagrangian particle, in the A, is given by the Space-properTime coordinates

= String viscous Tension :

where , , then ,

hence, get an unidimensional N+S equation :

While if the flow density is unsteady, considering

, p = ,

Nm (vector value)= J (scalar value) = N applied at one rod meter hinged at one end

where V = average speed , D = system dimension , , 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, and applying the metric tensor to the three-dimensional space of we get

Linear calculus of the Energy developed (output must be a scalar value) :

assuming ,


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