Breaking the Euler Equations

Cup of tea with lemon and spoon.

Mathematicians like to break things. Or, more exactly, they like to know when the equations we use to describe physics break down. One popular target in fluid mechanics are the Euler equations, which describe the motion of frictionless, incompressible flows. Mathematicians have been on the hunt for centuries for situations where these equations predict singularities, points where the velocity or vorticity of a fluid change infinitely quickly. Since that can’t happen in reality (at least as far as we understand it), these singularities indicate weaknesses in our mathematical description and may help uncover fundamental flaws in our understanding.

Despite centuries of effort, the Euler equations withstood mathematical assault… until recently. Since 2013, a series of mathematicians have been successfully chipping away at the Euler equations’ seeming perfection with a series of scenarios that seem to lead to singularities. One is similar to stirring a cup of tea, except that you stir the upper part of the cup in one direction and the bottom half in the opposite. As the flow develops, a singularity occurs where the secondary flows of these two stirring motions collide. For more, check out these two articles over at Quanta. (Image credit: L. Fotios; see also Quanta Magazine 1, 2)

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