The Navier-Stokes Equations represent two fundamental concepts encapsulated in equations that have left physicists scratching their heads around the world in search of a million-dollar prize.
Sally has prepped the house to her guests’ liking for the upcoming party. The moment someone rings the bell, she realizes that a musty smell is lingering in the air. Her hand immediately reaches out for the can of air freshener and spritzes a bit in all the rooms before walking confidently to the door to receive her guests. Later, a guest asks for a cup of tea. She gets the kettle boiling and dips the tea bag into the hot cup of water. While stirring in a cube of sugar, Sally begins to wonder how exactly the fragrance from earlier spread across the room and how the sugar is dissolving in the cup of tea…
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What Sally is wondering about is the flow of fluids. Fluids are broadly defined as anything that can flow. They can take the shape of the container in which they are placed. Gases and liquids are both considered fluids, though gas also has flexibility in terms of volume. When considering a solid, say a rock flowing down a stream, you can see how the rock is moving, because the molecules that make up the rock are closely packed together and don’t deform easily in water. However, if you take the fragrant molecules from a can and disperse them to the air, they flow in a random motion. Every time you dissolve a cube of sugar into a cup of tea, the motion of the sugar particles is different every single time, i.e., the movement is chaotic in nature.
Newton’s Second Law of Motion
The laws of physics that govern all things solid are also applicable to fluids. The famous laws of motion developed by Newton are often quoted to explain how things behave in a Newtonian frame of reference. The second law is used when the summation of forces acting on an object are unbalanced, which gives rise to acceleration. Mathematically, this is denoted as force being equal to mass multiplied by acceleration.
Conservation of mass
Consider some amount of fluid, let’s say water, represented by of density flowing through a pipe of area with velocity . Therefore, the flow of water flowing through a pipe of area and where , keeping the mass of water constant, would result in a change of the velocity of the fluid flowing through the pipe. Since density is inherent in the fluid, the only variable that can be altered is velocity, given a pipe with two different cross-sectional area sizes. This is the conservation of mass of a fluid in fluid dynamics.
So, why bother learning about the above-mentioned concepts of physics? Because they are fundamental when it comes to fluid dynamics. What if I told you that you could earn a million dollars simply by solving the equations outlining these primary concepts? In the year 2000, the Clay Mathematics Institute declared 7 open problems called the Millennium Prize Problems. Anyone who could solve these problems would receive 1 million USD. As of now, only one problem has been solved—The Poincaré Conjecture.
So, what is that literal million-dollar question still leaving physicists and mathematicians scratching their heads, even though they involve concepts typically studied in high school? The answer is the Navier-Stokes equations.
The Navier-Stokes equations
Claude-Louis Navier and George Gabriel Stokes provided partial differential equations for depicting the motion of fluids in the 19th century. The equations can be written in a compact form as:
where, is the effect of mass in each direction of the 3D space, is the pressure acting on the fluid, is the density of the fluid, and is the combined external forces acting on the fluid. We’re currently in the 21st century, yet we have not been able to fully understand the Navier-Stokes equations. The reason for this uncertainty is turbulence.
We’ve often heard of turbulence when flying on an airplane, and it is not a pleasant experience. Turbulence is the irregular motion caused by the eddies in the air and chaotic changes in both pressure and velocity. For chaotic systems, a minute change in an initial condition of the system can result in a significantly magnified change in terms of output. The systems exhibit nonlinear behavior.
Usually, for practical purposes, the equations are approximated and averaged for a certain area to derive solutions for a particular system. The Navier-Stokes equations do work in the real world but we have yet to crack the solution for the equations presented in their true form. Mathematics often helps us find alternate ways to approach a problem. Hence, these equations yet remain to be solved by someone or some people who can help further the understanding for such an innocuous open problem.
The Millennium Prize problems have captivated the “reel” world too, as the Navier-Stokes equations make an appearance in the movie ‘Gifted’ (2017). While there are countless brilliant minds turned to these problems, with the hope of securing the major cash prize and not to forget the high of solving the apparent unsolvable bringing along the fame and glory of the academic community, six millenium problems are up for grabs. So the next time you’re bored on a Sunday and are looking for some side work, put your mind to these incredible puzzles and see if you’re actually a mathematical savant!