Navier-Stokes equation

The Navier-Stokes equations are complex mathematical calculations that attempt to describe fluid dynamics—the study of the movement of fluids such as liquids and gases. The equations were developed in the nineteenth century through the work of French engineer Claude-Louis Navier and British physicist George Gabriel Stokes. The Navier-Stokes equations are an attempt to apply Isaac Newton's second law of motion to fluids; however, the interactions and resistance in fluid particle movement make for a complex and incomplete solution. The equations are useful to predict simpler aspects of fluid dynamics such as modeling of weather systems, ocean currents, the movement of water through pipes, and the movement of air over airplane wings. In more complicated instances, however, Navier-Stokes equations are simply too difficult to solve. The answers to the equations have proven so elusive that in 2000 a mathematics institute in the United States offered a million-dollar prize for a working solution.

Background

In 1687, English physicist Sir Isaac Newton published his revolutionary work, Mathematical Principles of Natural Philosophy, in which he outlined his three laws of motion. Newton's laws were instrumental in creating modern physics and became the basis for several significant fields of scientific study. His first law stated that a body at rest will remain at rest, and a body in motion will remain in motion unless it is acted upon by an external force. This means that objects will not move, change direction, or stop moving of their own accord; they must first be influenced by another force. His third law stated that for every action, there is an equal and opposite reaction. In other words, one force is always accompanied by another force that acts in the opposite direction.

It was Newton's second law that is directly related to the Navier-Stokes equation. This law stated that the force acting on an object is equal to the mass of that object multiplied by its acceleration. The equation can be illustrated with the formula F=ma, with F representing force, m mass, and a acceleration. Mass is the amount of matter in an object, while acceleration is the rate at which an object changes its velocity—the distance an object travels over time in a given direction.

Overview

In the eighteenth century, Swiss mathematician Leonhard Euler attempted to use Newton's second law to describe how velocity, pressure, and density relate to an "ideal" fluid. In physics, a fluid is a substance that flows under pressure and changes its shape to adapt to a container. An ideal fluid is one that cannot be compressed, flows without turbulence, and flows without viscosity—resistance caused by friction between fluid particles. Thinner fluids such as water have low viscosity while syrup and other thicker fluids have a high viscosity.

In 1821, Claude-Louis Navier, a French engineer known for his work on designing bridges, began work adapting Euler's equation to account for viscosity. Despite a limited knowledge of the physics involved, Navier managed to arrive at a solution that worked to describe motion in simple fluids. Contemporaries of Navier, French mathematicians Siméon-Denis Poisson and Jean Claude Saint-Venant, also contributed solutions to the problem. In the years after Navier's death in 1836, British physicist Sir George Gabriel Stokes arrived at his own mathematical equation describing the motion of fluids. Despite his results being similar to earlier work by Navier, Poisson, and Saint-Venant, Stokes felt his approach was sufficiently different that he published his own paper in 1845. The calculations later became known as the Navier-Stokes equations.

The equations consist of several independent calculations using four independent variables and six dependent variables. An independent variable is a controlled variable that is not dependant on another variable. In the Navier-Stokes equations, the independent variables are the three spatial coordinates—length, height, and depth represented by x, y, and z—and time. A dependant variable is one whose value is affected by other variables. These include pressure, density, and temperature and three components of directional velocity—measured along the x, y, and z coordinates.

The equations proved relatively reliable in predicting the movement of liquid or gases in two dimensions or at slower velocities. Within certain ranges, they can forecast the fluid velocity and pressure in a given circumstance, such as involving water flow through pipes, in tidal inlets, or air movement over specific environments. In reality, fluid movement through three dimensions is often subject to the complex effects of vortices, turbulence, and high speeds. As a result, the Navier-Stokes equations do not always work for all fluids and all conditions. Mathematicians are unsure if solutions are possible for some conditions without the use of numerical shortcuts to match the numbers with reality. In some cases, the equations sometimes result in unrealistic outcomes such as predictions a fluid will accelerate at infinite speed—a solution mathematicians refer to as "blowing up."

To coincide with the new millennium in the year 2000, the Clay Mathematics Institute, a New Hampshire–based math foundation, offered a $1 million prize to anyone who could solve any of seven historically difficult math problems. Among the institute's Millennium Prize Problems was the Navier-Stokes equations. To claim the prize, a solution must be offered that accounts for fluid movement in all cases, factoring in three dimensions and higher speeds. If the equations cannot account for a particular scenario, the solutions must prove that an answer is not possible and does not provide any unrealistic results. As of June of 2017, the Navier-Stokes equations remained unsolved. Of the original seven problems eligible for the Millennium Prize, only one had been solved.

Bibliography

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