<HTML> <HEAD> <TITLE>index</TITLE> <STYLE>A { COLOR: red } </STYLE> </head> <body bgcolor=white text=green> <br> written and published by Renzo Diomedi // UNDER CONSTRUCTION <br> <br> <br> Assume a Fluid as incompressible, id est if the variations of the density of the fluid do not have appreciable effects, so its density can be considered ,with a good approximation, a constant. Then <br> <br> Continuity + Momentum = <img src="83.png"> + <img src="87.png"> <br> <br> <br> <br> <br> the conservative continuity equation (scalar value) <img src="83.png"> where u, v, w lie on x, y, z and u^x v^y w^z have cosine = 1. <br> <br> the non-conservative Momentum equation (i.e. not exactly measurable as a scalar field) <img src="87.png"> <br> <br> <br> <br> <br> the non-conservative Momentum equation are however decomposable in 3 scalar equations laid along the directions of <i> x, y, z </i> . <br> Then considering <b><i> u, v, w </i></b> as components of the shift vectors <img src="78.png"> along the axes <i> x, y, z, </i> : <br> <br> <img src="104.png"> ; <img src="81.png"> = Density , <img src="80.png"> = Viscosity , <img src="154.png"> = standard gravity (acceleration) <br> <br> The independent variables are 4 : <img src="212.png"> <br> <br> The dependent variables are 10 , of which 3 velocity components <img src="210.png"> + the pressure <img src="211.png"> <br> and 6 dependent variables given by the Stress Tensor: <img src="209.png"> <br> <br> <br> So, as seen above, to know the value of the unknown of the 10 dependent variables we have 4 scalar equations only. <br> <br><br> <br> <br> <br> <br> expansed Continuity eq = <br> <br> <img src="184.png"> <br> <br> momentum eq is also <br> <br> <img src="213.png"> <br> <br> <br> <br> <br> Stress Tensor <br> <br> <img src="79.png"> <br> <img src="77.png"> <br> <br> then, partially derived as in the NS equations: <br> <img src="127.png"> ; <br> <br> <br> Components of the <i>viscous stress state </i> linearly linked to the components of the <i>deformation velocity</i> through Stokes' relations: <br> <br> <img src="82.png"> <br> <br> <img src="84.png"> <br> <br> <br> The Principal stresses and Shear stresses act in a 3d space. They can act in a deformable and curvable sheet, a 2-dimensional space rather than 3-dimensional space. <br> <br> Imagine a lattice composed of equidistant points subjected to stresses and crossed by a flow to analyze. The Turbulence produces lattice distortions, but the points remain at a constant distance among them because only the axial coordinates of reference vary. <br> <img src="174.png"> <br> But how much the lattice is deformed? how much are the axes and coordinates moved? this deviation is the Tension to be calculated. <br> <br> The Metric Tensor <b> g </b> 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. <br> <br> Considering <img src="150.png"> = metric abscissa and <img src="151.png"> = metric ordinate <br> <br> <br> <img src="130.png"> = distance of point from origin, whatever the inclination of the reference axes <br> <img src="131.png"> <br> where <img src="112.png"> <img src="113.png"> <br> <br> <br> (note that : <img src="114.png"> , <img src="115.png"> , <img src="132.png"> , <img src="133.png"> <br> <br> contravariant metric tensor: <img src="134.png"> , covariant metric tensor: <img src="135.png"> ) <br> <br> <br> <br> 2-dimensional viscous stress tensor : <br> <img src="137.png"> <br> <br> <br> hence we can calculate the amount of energy created over a certain period of time by these simplified new version of N-S equations: <br> <br> <br> <br> Continuity: <br> <img src="144.png"> <br> <br> <br> <br> Momentum : <br> <img src="146.png"> <br> <br> <br> 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 <br> <br> <br> <br> <br> <br> <br> <br> <br> We need an unidimensional output to overcome the non-linearity. A lagrangian particle 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 : <br> <img src="173.png"> where <img src="221.png"> <br> the position of the lagrangian particle, in the A, is given by the <b> Space-properTime </b> coordinates <img src="178.png"> <br> <br> <br> <br> <img src="136.png"> = String viscous Tension : <img src="214.png"> <br> <br> <br> <br> <img src="138.png"> where <img src="139.png"> , <img src="140.png"> , then <img src="143.png"> , <br> <br> <br> <br> <br> <br> <br> <br> <br> <br> <br> <br> <br> <br> While if the flow density is unsteady, considering <br> <br> <br> <img src="158.png"> , </i></b> <b><i>p =</i></b> <img src="166.png"> , <br> <br> <br><b><i>Nm (vector value)= J (scalar value) = N </i></b> applied at one rod meter hinged at one end <br> <br> <br> <br> <img src="167.png"> where V = average speed , D = system dimension , <img src="159.png"> , <i>Prandtl's number = </i> <img src="160.png"> <br> <br> <br> <br> <i>Heat Flux = </i> <img src="161.png"> where : <img src="162.png"> warmer face temperature , <img src="163.png"> cooler face temperature , <img src="164.png"> Surface of the matter traversed by the heat , <img src="165.png"> Thickness of the matter traversed by the heat, and applying the metric tensor to the three-dimensional space of <img src="177.png"> we get <img src="180.png"> <br> <br> <br> <br> Linear calculus of the Energy developed (output must be a scalar value) : <br> <br> <img src="171.png"> <br> <br> <img src="218.png"> <img src="219.png"> <img src="220.png"> <br> <br> assuming <img src="216.png"> , <img src="217.png"> <br> <br> <br> <br> TO BE CONTINUED........ <br> <br> <br> It's possible calculate the <a href="turb.htm"><i>turbulence</i></a> where the tensions are known <br> <br> <br> <h4> <p align="center"> <a href="http://www.newmath.org"><i>Home Page</i></a> </p></h4> </body></HTML>