James Constant
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 -px+fx=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 y1,y2 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) ........ 
................. 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 y1,y2
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 ............................. ................... equation (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 -px+fx=0
which means -p+f= constant and thus 
constant. However,
while setting -px+fx=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 y1,y2
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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