Fluid dynamics and the Euler equations

 When you think about it, the mobility of water, air, or really any fluid, seems strange. Despite being composed of tiny, independently moving molecules, it can move with immense force and moves as a continuous mass. We are aware from firsthand experience that external factors, such as gravity or the push of an oar, as well as a change in pressure from opening a faucet, can cause water to move. Additionally, a body of water experiences movement, such as the ripples that travel across a pond.

The mathematician Leonard Euler wrote a paper titled General principles of the Motions of Fluids, which was published in 1757. The text is quite readable (you can see both the original paper in French and an English translation). Since this article established the foundation for contemporary fluid dynamics—the study of how fluids (liquids and gases) move—the concepts and notation are still familiar to us today. The set of equations presented in the paper—now popularly referred to as the Euler equations—crystallized the mathematical connection between a fluid's velocity, internal pressure, and any external forces—like gravity—that may be exerted on it.

Any fluid that isn't sticky or viscous—to use the technical term—is subject to the Euler equations. To explain the dynamics of a viscous fluid, you require the more challenging Navier-Stokes equations. By eliminating viscosity, you may simplify these equations and return to the Euler equations. Let's use this example to illustrate how the Euler equations work since viscosity has no bearing on how ocean surge, or large waves rolling across the open ocean, moves.

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The Euler equations

The fluid's continuous nature allows the Euler equations to describe the fluid's motion at each instant and place as well as how that motion relates to the internal pressure of the fluid at that location. For our example, we may concentrate on how the water is moving in the wave's direction and examine the 2D vertical slice that results from cutting the wave vertically along its path of travel.

vertical slice of wave

The coordinates (x,y), where x is the horizontal direction and y is the vertical direction, can be used to represent points within the water on this slice. The motion of the water at a place (x,y) and at a time (t) can then be expressed as (u(x,y,t),v(x,y,t)). An arrow, also known as a vector, can be used to characterise the water's velocity at the spot; it has both a direction (the flow's direction) and a magnitude (its speed). For the pressure at location (x,y) and time t, we write P(x,y,t). Pressure is just a number that varies over time and space; it is not a vector.

For this illustration of fluid motion in this 2D slice, the Euler equations are as follows:

The first equation explains how the fluid motion in the horizontal direction interacts with the change in pressure and the change in velocity in this direction. The gravitational constant of acceleration (g) is included in the second equation, which determines vertical directions, to take into consideration the external force of gravity on the movement.

The Euler equations are partial differential equations to take into consideration the rate of change with regard to the various quantities since the fluid's velocity and pressure vary over both time and space. A partial derivative of u(x,y,t) with respect to time, for instance, is used to represent changes in velocity in the horizontal direction.

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Finding the functions u, v, and P whose partial derivatives meet the equations is required to solve the Euler equations. Various boundary conditions, such as whether the water's surface is flat or wavy, will also affect the solution.

It is feasible to create equations that describe how water particles in our example of the ocean swell move using the Euler equations. (Remember to consider these fluid particles at the macroscopic scale; they are so small that you can still think of them as points within the fluid despite containing numerous water molecules.) Ocean waves are not a moving wall of water, contrary to what you might believe, according to the Euler equations. Instead, the water molecules travel in tiny loops that follow the wave but do so at a considerably slower rate.

The particles in an ocean swell describe open loops, thus they move with the wave, albeit more slowly, in accordance with the Euler equations.

Fluids are hard (to solve)

One of the challenges in understanding fluid dynamics is non-linearity, which is revealed by the Euler equations. The fluid's velocity interacts with itself; water moving in one place is influenced by water moving nearby. This interaction may be linear, where the consequences follow the causes in a linear fashion, or it may be non-linear, or more complex. Terms like capture this nonlinear interaction in the Euler equations.

in which the two components u and v of the velocity interact with themselves and each other.

Nonlinearity initially appears in this kind of mathematical equation with the Euler equations (it was the first nonlinear field theory). Additionally, the nonlinearity of the equations makes it nearly hard to solve them in general, together with the difficulties of analytically defining the boundary conditions. There are solutions, but they are simply approximations, that are applicable to real waves, like our example above. Computational fluid dynamics and computer simulations are the mainstays of real-world applications of the Euler equations. And for this reason, methods for solving the associated Navier Stokes equations as well as the Euler equations remain a hotbed of mathematical research.

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