More than 150 years have passed since Navier and Stokes developed a set of partial differential equation called Navier-Stokes equation which describe the motion of viscous fluid. However, the exact or analytical solution of these differential equations is still unsolved. The Clay Mathematics Institute in May 2000 made this problem one of its seven Millennium Prize problems in mathematics. It offered a US$1,000,000 prize to the first person providing a solution for a specific statement of the problem.

Prove or give a counter-example of the following statement:
In three space dimensions and time, given an initial velocity field, there exists a vector velocity and a scalar pressure field, which are both smooth and globally defined, that solve the Navier–Stokes equations.

Click here to check out the details of the problem.

What is Navier-Stokes Equation?

As mentioned earlier, Navier-Stokes equation in fluid dynamics or aerodynamics is a set of partial differential equation that describe the motion of incompressible viscous Newtonian fluid. These are the generalized form of Euler equation. Euler’s equation  devised by Swiss mathematician Leonhard Euler in the 18th century describes the flow of incompressible and frictionless fluids. The Euler’s equation is as follows;

Euler equation
Euler’s Equation

French engineer Claude-Louis Navier in 1821 introduced the element of viscosity (friction) for the more realistic and vastly more difficult problem of viscous fluids and further improvements were done byBritish physicist and mathematician Sir George Gabriel Stokes in this work. The Navier-Stokes Equation is written as;

Navier-Stokes Equation
Navier-Stokes Equation

where where u is the fluid velocity vector, P is the fluid pressure, ρ is the fluid density, υ is the kinematic viscosity, and ∇2 is the Laplacian operator. Both of these equations are actually the application of Newton’s Second law of motion in fluid flow with the forces modelled according to those in a viscous Newtonian fluid—as the sum of contributions by pressure, viscous stress and an external body force.

The Navier Stokes equation in all 3 dimensions combined with the continuity equation can be used to describe the fluid flow. There are 4 equations and 4 unknowns( P, u, v, w), where u, v and w are the component of velocities in x, y and z directions. This shows that these equations must be sufficient to solve any fluid models.

Why is Navier-Stokes unsolvable?

Only one out of the seven Millennium Prize Problems has been solved till date. The problem has two requirements; existence of solution globally defined and smooth solution. Here smooth solutions means that for any little change in the initial value of the problem, the output should also change in a similar way.

Turbulence, vortices and the chaotic nature of the three dimensional flow is the reason why it is hard to obtain smooth solutions to the problem. The convective term in the equation is related to the turbulence in fluid flow. Smooth solutions have been found when fluid is modelled such that convective term is weaker compared to the diffusion term. However, the problem requires smooth solution for every practical cases of fluid flow.

Some efforts have been made throughout many years towards solving this problem.

  • Smooth and globally defined solutions were obtained for two dimensional fluid flow by 1960.
  • For sufficiently small value of initial velocity, smooth and globally defined solutions can be obtained as the diffusion is dominant over convection in this case.
  • Jean Leray in 1934 proved the existence of so-called weak solutions to the Navier–Stokes equations, satisfying the equations in mean value, not pointwise.
  • Terence Tao in 2016 published a finite time blowup result for an averaged version of the 3-dimensional Navier–Stokes equation. He writes that the result formalizes a “supercriticality barrier” for the global regularity problem for the true Navier–Stokes equations, and claims that the method of proof hints at a possible route to establishing blowup for the true equations.

Now one can argue that we are already applying Navier-Stokes equation to model fluid flow problems in CFD which has been successfully applied to different areas like aviation, meteorological research, marine applications and so on. However, CFD uses approximate solution of the Navier-Stokes equation using various techniques like finite difference method, Reynolds averaging or making assumptions and simplifications to the Navier-Stokes equation.

Click here to learn about CFD and the steps involved; http://geniuserc.com/introduction-to-cfd-analysis-and-steps-involved/

What will happen if it is solved?

Now the question may arise if there is actually the need of exact smooth and globally defined solution when the approximate solutions are reliable enough to fly an airplane into the sky. The inability to solve this problem shows us that we are missing something. The Navier-Stokes equation simply apply the conservation of mass and Newton’s second law of motion in a correct way and we have 4 equations and 4 unknowns. Thus, there must be a solution which we are not able to obtain yet. This is why this problem is raised and one million dollar price tag is given to encourage Mathematicians to develop that missing tool or technique to solve this partial differential equation.

Once the Navier-Stokes Equation is solved as specified in the problem, we will have a simple set of expressions for scalar velocities and pressure that can be used to understand any kind of fluid flow problem. Solving the fluid flow problems will be as simple as finding the volume of sphere for a given radius and centre. The only problem is we don’t know the equation to calculate volume of sphere yet.

I would be proud if one of my readers will eventually solve this Million Dollar problem.

By Nishchal Poudel

Nishchal is currently studying Bachelor in Aerospace Engineering at IOE Pulchowk Campus, Nepal.

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