John W. Dooley, Physics Department, Millersville University

Go directly to derivation of Euler applications.

Euler's equation is difficult to solve in general, but a number of interesting solutions exist. (For every solution except the sound wave, we will assume that the density of the fluid is constant.)

If the velocity field is zero everywhere, and constant in time, we have the hydrostatic condition, and Euler's equation leads to the familiar laws of hydrostatics, with pressure in a fluid increasing in proportion to depth.

If the velocity field is constant in time, and the pressure is constant in space, we can calculate the velocity field of a waterfall.

If the rate of rotation is constant in time, we can calculate the velocity and pressure fields in a bucket of fluid rotating on its axis. We can also calculate the parabolic shape of the surface of the fluid in the bucket and the density distribution in tornadoes. This analysis has been applied to the manufacture of parabolic mirrors of glass and of mercury, by spinning the mirror blank until its surface takes the desired shape.

Remarkably, we can correctly calculate the tides, which occur on a water covered planet, by requiring the water to accelerate towards the source of an external gravitational field. The results match fairly well with the tides observed on earth.

Taking the viscous force into account, we can calculate the velocity field for a fluid flowing between flat plates, and in a circular pipe. We can integrate this flow to find the rate at which a pipe delivers fluid at its output end. (The equation representing this result is called Poiseuille's equation.)

Finally there is a famous case in which Euler's equation can be integrated. We suppose that the density is constant, the velocity field is independent of time, and the only force terms are the pressure term and the gravitational term. We are left with a differential equation to integrate.

This is still difficult to integrate, until we make one final requirement. We use a definite integral, with respect to an increment , between two points, A and B. We insist upon a strategic restriction: The path of integration must be such that (at every point on the path) is always parallel to the velocity field vector .

A path like this is called a "streamline." A drop of dye placed in the fluid will trace out a streamline as it is swept along by the velocity field. This is because during a time interval , the dye moves a distance .

In this case, Euler's equation integrates to "Bernoulli's equation:"

where is the height of the location above some reference level.

One confusing detail about Bernoulli's equation needs discussion: In elementary treatments it is often stated that, because of Bernoulli's equation, high velocity causes low pressure. This is like saying that the high velocity of bullet leaving a gun caused the low pressure of the gas outside the barrel of the gun. The field treatment embodied in Euler's and Bernoulli's equations does not discuss cause and effect. The field view tells us what parameters "go together" without implying that one causes the other. If we trace back to the roots of these equations in Newton's laws, we can extract a cause and effect statement: Forces are said to cause accelerations. In the same sense, pressure gradients cause changes in velocity; not the other way around.

The requirement that the two points in Bernoulli's equations, A and B, must lie on the same streamline can be removed: In regions where the curl of the velocity is everywhere zero, Bernoulli's equation applies for any pair of points.

Euler's equation can also be solved for fluids in which the curl of the velocity field is not zero, so that Bernoulli's equation has limited application. Knowing the velocity field in a rotating paint bucket lets us calculate the pressure field within the bucket. Extension of this result describes the creation of parabolic telescope mirrors from molten glass, or from mercury.

In addition, phenomena such as dust devils and supercoducting vortices (for which the curl of the relevent fields are not zero) may be approached with Euler's equation.

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