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 :
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:
TO BE CONTINUED........