**James Constant **

**math@coolissues.com**

**Explicit
Solutions provided for Navier-Stokes Type Equations and their
Relation to the Heat Equation, Burger's Equation, and Euler's
Equation**

**Introduction**

**It is well known that linear differential equations with
real coefficients lead to real dispersion relations only if they
consist entirely of even derivatives or they consist entirely of
odd derivatives or they consist of imaginary odd derivatives and
even derivatives. This occurs because we have a direct
correspondence between the equation and the dispersion relation
through the correspondence**

**(1) **

**Examples are the Klein-Gordon equation (even derivatives),
the equation for flexural vibrations of a beam (odd and even
derivatives), the linearized Kortweg- deVries equation (odd
derivatives), and Schrodinger's equation (imaginary odd
derivative and even derivatives).**^{1}** Such equations involve elementary solutions**

**where **** is the wavenumber, **** is the frequency, and
A is the amplitude, a constant in simple** **cases.**

**in which u, p are unknown velocity and
pressure vectors, letter subscripts denote partial differentials,
f is externally applied force (e.g. gravity), **** is a positive
coefficient (the viscosity), **** is the gradient
operator, and ****is
the Laplacian operator. The first of equations (3) is Newton's
law f=ma for a fluid element subject to the external force f and
to forces of pressure and friction, and **** is the condition for
an incompressible fluid. **

**The x-component of equations (3)
reduce to**

**Equations (3) and (4) apply in
principle both to laminar and turbulent flows although, because
of the impossibility of following all the minor fluctuations in
velocity associated with turbulence and because of the difficulty
and lack of solutions in turbulence problems, they cannot be used
directly to solve problems in turbulent flow.**^{2}** ****There is no satisfactory theory for turbulent flow.
For these reasons, more efficient solutions to the Navier-Stokes
equations are needed. Indeed, ****a prize is available
to provide a mathematical theory which provides a better
understanding of ****the Navier-Stokes
equations.**^{3}** ****And, since we have
two unknowns u and p and one equation, solutions of equations (3)
and (4) simply means finding u as a function of p as well as
x,y,z,t.**

**Equation (4) can be replaced by three
equations and three unknowns**

**so that if a solution u is found,
solutions for v and w are available. The justification for
equations (5) assumes that u, the x component of velocity, is
mainly a function of x so that its y and z derivatives tend to
zero. This assumption says that u=u(x,y,z,t) does not vary
greatly when y and z change. Certainly, the last two equations
(5) approach zero as ****. They also approach zero if the vector
velocity is mainly in the x direction since then ****. With these
assumptions, adding the last two second order equations to first
equation (5) results in equation (4).**

**which, by using the nonlinear
transformation**

**may be reduced to the linear heat
equation **

**The general solution of the linear
heat equation (8) is well known**

**in which **** is the initial
temperature, a known function from initial condition t=0. The
solution for u is obtained from equation (7)**

**in which **** is a known function
from initial condition t=0.**^{4}

**Explicit Solutions for
the Navier-Stokes Equation**

**I now propose to obtain an explicit
solution for the Navier-Stokes equation, first of equations (5),
following Whitham's method of obtaining an explicit solution for
Burger's equation (6).**^{5}** Whitham's method consists of setting **** so that first equation
(5) may be integrated to **

**in which **** is evaluated between 0
and x, and then introducing**

**to obtain a linear heat equation **

**Solution With Constant
Coefficient**** **

**A constant coefficient **** means that -p**_{x}**+f**_{x}**=0
so that the first equation (5) reduces to Burger's equation with
solution provided by equation (10). Independently, substitution
of**

**in equation (13) reduces it to the
heat equation**

**(15) **.....

**whose solution is provided in equation
(9). The solution for **** is equation (14) and the solution for **** is equation (12) which
can now be written as **

**and since **** the explicit solution
for the Napier-Stokes equation, first of equations (5), when **** is constant is**

**In summary, the explicit solution of
the Navier-Stokes first equation (5), equation (17) with **** constant, requires
that the velocity u is proportional to a function **** in which **** is provided in
equation (9) and leads to solution (10) of Burger's equation. **

**Solution With Variable
Coefficient**** **** **

**in equation (13) produces the linear
nonhomogeneous equation**

**Note that when ****=constant or
when t=0, equation (19) reduces to the heat equation (15) whose
solution is provided in equation (9). The corresponding linear
homogeneous equation ****can be represented by
an operator**

**(22) **....... .......... ** **...........

**(23)** ....... ** **.......... ..........**
**

**in which y**_{1}**,y**_{2}**
are known functions of x,t and A,B are undetermined functions of
x,t. The method of finding A,B is straightforward.**^{6}**
Set equation (23) in equation (19) and obtain two equations**

**(24)** ........

**which can be solved for A**_{x}**,B**_{x}

**(26) **.....**
**.......** **........

**Solution (23) then takes the form**

**The solution of equation (19) can be
written as**

**The solution for **** is equation (18) and
the solution for **** is equation (12) which can now be written
as **

**and, since ****, the explicit solution
for the Napier-Stokes first of equations (5) is**

**and using the last two of equations
(5) to obtain solutions for v and w.**

**In summary, the explicit solution of
the Navier-Stokes first of equations (5), equation (30) with **** variable, requires
that the velocity u is proportional to a function **** in which **** ****is provided in
equation (28). However, the Navier-Stokes first equation (5) is
an exception to the general rule that solutions can be defined by
initial conditions. Since ****, equations (25)-(30) do not exist when ****constant or when t=0 or
when x=0 since then W=0, solutions y**_{1}**,y**_{2}**
are no longer independent, and integrals do not converge. For
this reason, solution (30) does not reduce to solution (17) when ****constant, and we cannot
give the velocity u(x,t) an arbitrary initial distribution
u(x,0)=f(x) or u(0,t)=f(t) and expect to find u(x,t) as a
function of f(x) or f(t). Nevertheless, solutions to many
physical problems can still be defined by boundary conditions. **

................ **x component of **..................**
y component of** ..............** z
component of **.............................
...................** e****quation
(5) **..........................**
equation (5) **..................... **equation
(5)** ........................

**There are a number of equations
obtained by omitting, or adding, a term(s) on the right hand side
of the Napier-Stokes first equation (5). With omissions, it would
be hoped-for that the Napier-Stokes solution (30) would lead to
solutions for some related equations. Unfortunately, since ****, this does not occur.**

**The Burger equation is the first of
equations (5) with -p**_{x}**+f**_{x}**=0
which means -p+f= constant and thus **** constant. However,
while setting -p**_{x}**+f**_{x}**=0
in the Napier-Stokes equation produces Burger's equation, this
same setting in the Napier-Stokes solution (30) does not produce
Burger's solution (10). The reason for this is because equations
(26)-(30) do not exist when **** and thus W=0 and solutions y**_{1}**,y**_{2}**
are no longer independent and, therefore, cannot predict the
solution to Burger's equation.**

**The Euler equation is the first of
equations (5) with **** set equal to zero. However, while setting **** in the Napier-Stokes
equation produces Euler's equation, this same setting in the
Napier-Stokes solution (30) does not produce an Euler solution.
The reason for this is because equations (12)-(30) do not exist
when ****
and, therefore, cannot predict the solution to Euler's equation. **

**If the right hand side of equation (5)
is set to zero, the result is**

**which, since ****, gives u=constant.
However, while setting the right hand side of the Napier-Stokes
equation to zero produces equation (34), this same setting in the
Napier-Stokes solution (30) does not produce solution u=constant.
The reason for this is because equations (12)-(30) do not exist,
since ****
and ****
constant and thus W=0, and, therefore, cannot predict the
solution u=constant to equation (34).**

**The Navier-Stokes equations (5) cover
the case of turbulent flow regarded as unstable motion.
Unfortunately, rapidly fluctuating instantaneous velocity u is
impractical to compute in turbulence problems but, for many
engineering purposes, we can replace u by ****where **** is the mean and ****is the turbulent
rapidly fluctuating part of velocity. This requires introducing
the Reynolds stress into equations leading to the derivation of
equations (5). However, while a relationship exists between the
viscous stress p and the derivatives of u leading to the
derivation of equations (5), no relationship is known to exist
between the Reynolds stress r and the derivatives of **** with result that
adding a Reynolds term to equation (5) does not produce a more
tractable equation.**^{7}** The modified equation is **

**in which **** is the divergence of
r. Equation (35) requires replacing second equation (11) by **

**in which p,f and **** are known in many
practical situations. Note that since the y and z components of
stress r are second order effects, the integral in (36) reduces
to r, the x component of stress. The solution to equation (35) is
provided by equation (30) provided equation (36) replaces
equation (11). **

^{1}** G. Whitham, ***Linear
and Nonlinear Waves***, John Wiley & Sons, 1974 pages
366-367.**

^{2} **J. Kay, ***Fluid
Mechanics & Heat Transfer***, 2d ed. Cambridge
University Press, 2d ed. 1962 pages 150-151.**

^{3} **Clay Mathematics
Institute at ****http://www.claymath.org/****
**

^{4} **Whitham above pages
97-98; Lester Ford, ***Differential Equations***,
McGraw-Hill Book Co., 1955 pages 272-274.**

^{5} *Id. *

^{6} **Earl Coddington, ***Ordinary
Differential Equations***, Dover Publications, Inc. 1961.
pages 66-68.**

^{7} **Kay above Chapter
14.**

**Copyright ****©****
2003 by James Constant**

**By the same author:** http://www.coolissues.com/mathematics/sameauthor.htm

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