written and published by Renzo Diomedi // UNDER CONSTRUCTION

NS, composed by a conservative continuity equation that since the flow is incompressible (if the variations in density of the fluid do not have appreciable effects and therefore the density can be considered with a good approximation, a constant) is

and a non-conservative Momentum equation not exactly measurable as a scalar field, but divisible by 3 scalar equations laid and projected along the directions x, y, z which returned values not coinciding.

So we have 4 independent variables x,y,x,t and the 4 dependent variables u,v,w, and p (pressure) and the 6 given by the Stress Tensor considering u, v, w components of the shift vectors along axes x, y, z, , we have: ; = Density , = Viscosity

The stress tensor works as:

then, if the components of the viscous stress state are linearly linked to the components of the deformation velocity through Stokes' relations,

= = = = whereas

so finally the viscous stress tensor is :

so we saw this tensor as a generally irregular pyramid with triangular base , but if instead we tried to consider it as a deformable and curvable sheet, we may have a new vision about these equations
Let examine a lagrangian particle that in its path creates a string instead the eulerian cube. Then we have

T = 2-dimensional stress tensor


c = velocity speed

= distance of point from origin, whatever the inclination of the reference axes = (using the Einstein notation)

The Metric Tensor g expresses the property of a structure geometrically curvable with the points of its lattice at a distance always equal in relation to the structural components themselves

then contravariant metric tensor
then covariant metric tensor


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