Explicit Solutions provided for Navier-Stokes Type Equations and their Relation to the Heat Equation, Burger's Equation, and Euler's Equation
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
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.
The Navier-Stokes Equation
Consider differential equations with even and odd derivatives, for example the Navier-Stokes equations are
(3) .......... u=u(x,y,z,t)
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
(4) ... u=u(x,y,z,t)
and similarly for the y and z components of velocity v and w. Equation (4) assumes that force f has a gradient.
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).
If the last two terms on the right hand side of the first of equations (5) are dropped, the resulting equation is Burger's equation
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
In summary, nonlinear transformation (7) eliminates the nonlinear term in Burger's equation (6) and reduces it to the linear heat equation (8) which has an explicit solution (9) thus providing an explicit solution (10) for Burger's equation.
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
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
and using the last two of equations (5) to obtain solutions for v and w. Solution (17) is identical to solution (10) for Burger's equation.
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
provides two roots D1,D2 of equation (20). The solution of the linear homogenous equation is, therefore
(22) ....... .......... ...........
in which y1,y2 are known functions of x,t and c1,c2 are arbitrary constants. By the method of variation of constants, we write the particular solution of equation (19)
(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
which can be solved for Ax,Bx
(25) ...... ........ ........
and integrating to obtain
(26) ..... ....... ........
Solution (23) then takes the form
(27) ...... ..........
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.
In the present theory, solutions u,v,w are obtained using each component of vector velocity equation (3), i.e., we have three sets of solutions u,v,w at each point x,y,z,t. The solutions, therefore, must be identical.
................ x component of .................. y component of .............. z component of ............................. ................... equation (5) .......................... equation (5) ..................... equation (5) ........................
in which the descending diagonals are solutions u,v,w in terms of x,y,z,t and the remaining second order non-diagonal terms are solutions of u,v,w in terms of second to first derivative velocity ratios.
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.
6 Earl Coddington, Ordinary Differential Equations, Dover Publications, Inc. 1961. pages 66-68.
7 Kay above Chapter 14.
Copyright © 2003 by James Constant
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