Can the complex motions in fluid, such as Brownian motion and diffusion, be described with the exact solution of the motion equations of fluid?

This problem is closely related to the famous "Millennium Prize Problems" established by the Clay Mathematics Institute of Cambridge, Massachusetts，for celebrating mathematics of new millennium. One of them is about the Navier-Stokes equation. This problem was introduced shortly and vividly in the website of the Clay Institute as follows:

Waves follow our boat as we meander across the lake, and turbulent air currents follow our flight in a modern jet. Mathematicians and physicists believe that an explanation for and the prediction of both the breeze and the turbulence can be found through an understanding of solutions to the Navier-Stokes equations. Although these equations were written down in 19th Century, our understanding of them remains minimal. The challenge is to make substantial progress toward a mathematical theory which will unlock the secrets hidden in the Navier-Stokes equations.

Obviously, one of the possible explorations to the this problem is to try to give an exact solution to the Euler equation (which is the simplest case of Navier-Stokes equations) for describing some complex fluid motions.

In the past thirty years, several exact solutions of such kind were given, such as in [1], [2], and [3]. But these solutions usually need some complex and unnatural external force in the Euler equation, i.e., the corresponding complex motions were driven by the complex and unnatural external force (here, the "unnatural external force" means a non-potential force). So, these solutions are somehow not quite satisfactory.

From 2006 to 2008, this problem was also studied by me and my graduate students Weiwei Yu and Minghui Liu. Based on the "pseudo-potential" conception proposed by Weiwei Yu, a kind of exact solution of the Euler equation was found out. This kind of exact solution contains two arbitrary given functions and three arbitrary given parameters, and the external force of the corresponding Euler equation could be zero or any given potential force. Based on the choice of the two functions and three parameters contained in the solution, and based on the KAM theory and Melnikov Method, it is proven that the Brownian motion and diffusion of the fluid can described by the chosen exact solution. The concrete exact solutions and the sketch of the related proofs are introduced in my blog paper <A Series along the Nature and Beauty> in Chinese.  The exact solutions and the obtained second order Melnikov function are also listed on the attached pdf file <Main Mathematical Formulae> in English.

Main Mathematical Formulae.pdf

To show the complex motion (diffusion), an animation (click on the animation to watch it) was made with the software <Mathematica>. In this animation, 40000 fluid particles are initially distributed to four small circles, and the four groups of particles are each dyed with a different color, so that each circle has their own unified color. The animation shows how the 40000 particles move according to the chosen exact solution, and how the four colored circles develop into four different closed curves following the fluid particles on it.

It is a well known fact that if infinitely many particles are continuously distributed on the four circles, following the motions of the fluid particles, the shapes of the four circles will develop into four closed curves (homotopic to the original four circles), while the areas surrounded by them are maintained respectively, and the four closed curves will never intersect each other. This means the true diffusion (or osmosis) can not really happen if the continuity of the curves is not destroyed. However, for a practical fluid, the fluid particles are always with finite number, no matter how large the number is. So, each circles are formed with only a finite number fluid particles. When the "pseudo-continuous" curves are stretched and deformed drastically, the "continuity" of these curves will be destroyed, obviously, and the diffusion (or osmosis) will really happen between the particles distributed on the four closed curves, shown this way by the animation. 

The velocity field described by the chosen exact solution used for the animation is periodic both in time and in the coordinates of the two dimensional plane. It is proven by calculation that the mean value of the velocity over time and over space is zero, while the mean value of the square of the velocity is a positive number if the motion exists. Clearly, the larger the mean value of the square of the velocity is, the stronger the complex motion of the fluid is. Therefore, if the period of time and period of space are small from the view point of macro-scope, then the exact solution obtained can be treated as a module of static water with temperature which is proportional the mean value of the square of the velocity.

References:
[1] T.H.Solomon and J.P. Gollub, Chaotic particle transport in time-dependent Rayleigh-Benard convection , Physical Review A. Vol.38 No. 12, (1988) 6280-6286
[2] S. Wiggins, The dynamical systems approach to Lagrangian transport in oceanic flows, Annu. Rev. Fluid Mech. 37, (2005) 295–328.
[3] N. Malhotra and S. Wiggins, Geometric Structures, Lobe Dynamics, and Lagrangian Transport in Flows with Aperiodic Time-Dependence, with Applications to Rossby Wave Flow,J. Nonlinear Sci. Vol. 8: pp. 401–456 (1998)



Author: Keying Guan
(Science College, Beijing Jiaotong University)
email: keying.guan@gmail.com


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