partial differential equations / Navier-Stokes equation
Author: Renzo Diomedi

First Hypothesis:

the Fluid is incompressible, the variations of its density are insignificant and the value of its density is a quasi-constant. [1]

The Principal stresses and Shear stresses act in a 3d space. We study them in a space defined by a deformable and curvable sheet.
Suppose a lattice composed of equidistant points subjected to stresses and crossed by a flow to analyze, and that this lattice is deformed by the shift of its axes and its coordinates.
The Turbulence produces lattice distortions, but the points remain at a constant distance beetween them because only the axial coordinates of reference vary.

The 2-dimensional viscous stress metric tensor is :

In this geometric object we consider , , where z is the equivalent of the 3rd axis in a 3d space. So the position of the point in the 3d space is defined by the shift of obtained by the variation of the angle beetween x axis and y axis on the 2d-plane

In this geometric object we consider , , where z is the equivalent of the 3rd axis in a 3d space. So the position of the point in the 3d space is defined by the shift of obtained by the variation of the angle beetween x axis and y axis on the 2d-plane

Imagine a Point that lies in an internal Plane of a Volume used to calculate the turbulence. Then the Deformation of this plane determines the Turbulence in this point (Reynolds number > 4000 ).
We must Calculate the variation of the axial angle in the internal plane chosen into the volume, which has value = 90 in the state of quietness, then we can extract a value of the deformation and relative turbulence
We need to know the physical properties of the micro volume considered , eg the developed heat, and we must use the co-variant coordinates. We can get the values of the co-variant coordinates to know the inclination of the
We can use the Mohr formulas to calculate the angles of the inclination of the chosen plane into the defined volume:

Sigma = Normal tension
Tau = Tangential tension

that derived is

that derived is

= angle beetween normal axis of the square and the oblique plane + the same angle in the opposite side
(note that in Mohr circle , the angle beetween x and y has 180 degrees instead of 90)


then applying this value of this double angle to and we get



where the Tension values we need, they are given by Stoke's relations

on 2-dimesional plane of the Volume.
So, first we must find the value of the tensions and then we apply these values on mohr cicle with metric axes
The half-values of the angles got by is the key to calculate the covariant coordinates
we can choose an appropriate series of planes inside the volume to calculate the turbulence in the different points inside the volume
hence we can calculate the amount of energy created over a certain period of time by 2-dimensional version of N-S equations:


Momentum :

Now, the independent variables are 3, the dependent variables are 6 and the output equations are 3. So the imbalance has been reduced from 6 to 3

Second Hypothesis:

The flow density is unsteady. [2]

The linear calculation of the total developed energy in a defined volume over a defined period of time is:

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.

assuming and using the lattice deformable by metric coordinates variation and its 2-dimensional stress tensor

we get:

we can apply the Nambu-Goto action to this equation if we see the volume traversed by the flow as a series of continuum, instead a single continuum, id est as a 1-dimensional String made by microvolumes rather than particles that run more efficient path. So we have a Lagrangian system that use a series of Eulerian spaces. The energy calculated is proportional to the minimum area of the worldsheet . This Action is a scalar which is the energy developed in a Time, this action is local and it must be defined by an Integral as where = String Tension, = light speed, but conventionally used as an unit, where = 2D-space defined by covariant coordinates and is the proper-time, are functions that determine the worldsheet shape, and is a metric tensor
where , ,
= elements of a space-time vector, in this case

this metric tensor expands the concept of space by inserting the contravariant coordinates.

we do not interested to know the string tension. The string we consider is composed by elements existing at different times. So the action is


So we get



N+S eq = Continuity + Momentum = +
The Continuity equation is conservative , its output is a scalar value. u, v, w lie on x, y, z.
The Momentum equation is non-conservative, not exactly measurable as a scalar field, decomposable in 3 scalar equations, where u, v, w are components of the shift vectors along the axes x, y, z, :
where = Density , = Viscosity , = standard gravity (acceleration)
the Independent variables are 4 =
the Dependent variables are 10 , id est, 3 velocity components + the pressure + 6 dependent variables given by the Stress Tensor:
To get the value of the unknowns, i.e. the 10 dependent variables, we have 4 scalar equations only. Momentum equations is also
where the Stress Tensor is
......................... working as

Then the components of the viscous stress state are linearly linked to the components of the deformation velocity through Stokes' relations:

A remarkable mathematical tool is the Metric Tensor g . It expresses the property of a geometrically curvable structure having the points of its lattice maintaining the same unchanged distance in relation to the structural components themselves. = metric abscissa , = metric ordinate = distance of point from origin, whatever the inclination of the reference axes
, where , , , , , contravariant metric tensor: , covariant metric tensor:


Reynolds nr = where V = average speed , D = system dimension

Prandtl's nr =

Nm (vector value)= J (scalar value) = N applied at one rod meter hinged at one end ,

thermal diffusivity =

pressure =