written and published by Renzo Diomedi // UNDER CONSTRUCTION
Continuity + Momentum =
considering the flow as incompressible, id est if the variations of the density of the fluid do not have appreciable effects,
its density can be considered ,with a good approximation, a constant.
the conservative continuity equation (scalar value) where u, v, w lie on x, y, z and the cosine of their angle has value = 1.
the non-conservative Momentum equation (i.e. not exactly measurable as a scalar field) are however divisible by 3 scalar equations
laid along the directions of x, y, z . considering u, v, w as components
of the shift vectors along the axes x, y, z, :
; = Density , = Viscosity , = standard gravity (acceleration)
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:
Considering the components of the viscous stress state linearly linked
to the components of the deformation velocity through Stokes' relations:
So, to solve the unknown of 10 dependent variables we have 4 scalar equations only.
expansed Continuity eq + Momentum eq =
TO BE CONTINUED ........
the stress tensor is 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 keeping equal in relation to the structural components themselves.
so we can maintain the position of the points created in the cube, transferring them to a deformable sheet
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:
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 :
Considering the worldSheet as an Eulerian continuous, and saw the
Transformation equations for a Tension status ,
TO BE CONTINUED............................................................
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........