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), (velocity components) 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

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:






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........


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