Solving Navier-Stokes with Maclaurin Series

In this paper I propose a method for founding solutions of Navier-Stokes equations. Purpose of the research is to solve equations giving form to relations between pressure, velocity and stream. Starting from the fact we do not know the form of functions we give a general representation in Maclaurin Series and prove that with reasonable values of parameters, representation holds and therefore has meaning in continuum. Then we solve the system of equations with respect to the pressure and match equations relation between parameters: matches of equations are possible because of the physical dimen-sions of equations. Then values of Continuity Equation are verified. The result is a polynomial finite and that coincides with the function in continuum, or is anyway one of its representation. The result under hydrostatic condition returns Stevino formula.


Introduction
The most of knowledge can be found in Reference [1] [2] and [3] with the expression of Navier-Stokes Equations, further are used some well known formulas that can be found almost everywhere but they are contained in the article and you can find links in others: References [4] (see method to solve differential equation with power series [5], for turbulence [6], Navier Stokes State [7], incompressible Navier Stokes [8], Multivariable Taylor [9], Cauchy product [10], Fluid Dynamics Knowledge [11] [12] [13] [14]). At the end of the article the results of the pressure match a well known formula when velocity vanishes. Further an Appendix is provided.
The fluid motion is governed by knowing the velocity vector for most. The velocity vector is one of the parameters of the Navier-Stokes equation. Together with continuity equation they constitute a system of differential equations in the unknowns ( ) , , , p x y z t (pressure) and ( ) , , , v x y z t (velocity) in the variables of space ( ) , , x y z and time t. The system for incompressible fluid can be written like for a detailed explanation of the system (see the Appendix and References).

Method Overview
Consider the Navier-Stokes equation They are three equations. We give the V vector a form in Maclaurin series, then substitute the value of V in the three Navier-Stokes equations and then try to resolve each of this equation with respect to pressure with integration of each equation in the respective variable. We match the form of the pressure in the three equations and try to find a relation of parameters. After a partial comparison those parameters result in a relation. Applying Continuity Equation gives values. If the expansion for V is finite then the polynomial coincides with function of V.
We give an old style to composition but not less correct.

Maclaurin Expansion for Velocity
We don't know the form of the function but we know that the function has four variables x,y,z,t. If we want to find analytic solution we can represent functions u, v, w of V u v w = ⋅ + ⋅ + ⋅ i j k as Taylor Series around point   (  )   1  2  3  4 , , , a a a a = a (see we don't know this assumption is legal now). Any finite polynomial in Taylor series corresponds to the same polynomial: if we find terms in the polynomial expansion and those identify a finite polynomial we have found the function. The form of Taylor function in multidimensional variable is of the kind (we used a non compact form another could be compact with Hessian Matrix, see Wikipedia): x t = , with a compact form for summation, with notation α α α β α γ α δ = = = = to be clearer this does not prevent previous consideration.

1 st Equation of Navier-Stokes
Consider the formula of the Navier-Stokes equations  and his first equation We have the form of the unknown u but we have not yet pressure. To solve we substitute the unknown in the equations of Navier stokes, respectively u, v, w. Then we solve the integral of the pressure in the three variables respectively x, y, z and confront physical equations (all elements multiplied by , µ ), found parameters values. We express each term separately because of the long expression and substitute the values of V(u, v, w) in each term. We put a tag at the end of each expression as: 2 nd addend right member, that refers at right member the second addend in the expression. Substituting the value of V = (u, v, w) in the equation we have terms a at left member and b at right member.
( ) This series must be solved with Cauchy product S α β γ δ = + + + and n n n n n S α β γ δ = + + + then (2) became Note that we set c k n c − = we reported anyway this notation to remember which one refers to the first summation and which one to second. Other terms are then then right member Then solve respect to p (

2 nd Equation of Navier Stokes
Now consider the second equation of Navier-Stokes In the same way values of V are substitute and integration is done with respect dy. Terms are integrated with respect to y gives

3 rd Equation of Navier Stokes
Now consider the third equation of Navier-Stokes analogously gives terms.

Equations Comparison
Now we can make a comparison of three equation's terms to find , , , Consider parameters of a given

Satisfy Continuity Equation
Now consider the velocities with substitution of parameter found above.

Result
Substituting the parameter in the V equations we obtain: