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

Then considering

; = 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

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

Considering = metric abscissa and = metric ordinate

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

where

(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:

Continuity:

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

the position of the lagrangian particle, in the A, is given by the

= String viscous Tension :

where , , then ,

hence, get an unidimensional N+S equation :

While if the flow density is unsteady, considering

,

where V = average speed , D = system dimension , ,

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

assuming ,

TO BE CONTINUED........