written and published by Renzo Diomedi // UNDER CONSTRUCTION
NS, composed by a conservative continuity equation or
where u, v, w are aligned on x, y, z and the cosine of their angle has value = 1.
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 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)
TO BE CONTINUED ........
So we have 4 independent variables x,y,z,t and the 4 dependent variables u(x,y,z,t),v(x,y,z,t),w(x,y,z,t),
and p(x,y,z,t) (pressure) and 6 other dependente variables given by the Stress Tensor:
So, to solve the unknown of 10 dependent variables we have 4 scalar equations only.
So we use the metric tensor and the worldsheet to get to reduce the unknown values.
let's start working on stress tensor.
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.
the angles relative to the deformation of the sheet , in x and y direction, are added and form the angle
= distance of point from origin, whatever the inclination of the reference axes
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 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
Nm (vector size)= J (scalar size) = N applied on a one meter-rod hinged at one end
, p = ,
, 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 Energy developed (note that in output a scalar value is requested ) :
we apply the metric tensor to the three-dimensional space of , so
TO BE CONTINUED........