Blood flow simulation. Image: Argonne National Laboratory

Inside the Inscrutable Physics of a Heart Murmur

The hushed rumble of disrupted blood flow hides a mathematical monster.

The sound is not unlike the distorted low-end bass of an overpowered speaker or jagged whoosh of air against a microphone diaphragm. Next to the even, angular tocks of heart valves opening and closing, the rumble of a heart murmur is difficult to mistake.

But the causes are varied and not always pathological, ranging from everyday exercise to congenital defects to rheumatic fever. In every case, it has to do with unwelcome knots of blood flow ruled by an aspect of fluid dynamics that has befuddled mathematicians and physicists like few other problems: turbulence.

Coming up with a complete mathematical description of turbulence, in which a fluid can be described at every point using what are known as the Navier-Stokes equations, is one of seven Millennium Prize problems put forth by the Clay Mathematics Institute. Showing that solutions to these equations always exist (or don't) then comes with a million dollar bounty.

Money aside, the problem of turbulence is an old thorn: a great unsolved problem pertaining to physics that are both omnipresent and enormously powerful. Until not too long ago, turbulence was thought to maybe even be unsolvable.

I first got a look into the world of abnormal blood flow—and turbulence, generally—via an (unfinished) piece I was working on covering the mathematician Terence Tao's work on the Navier–Stokes equations. Despite well over a century of effort, the greatest brains in math have been unable to prove or disprove whether these equations can fully account for the whole of motion in a turbulent fluid. They usually work very well for describing fluids—blood included—but whether they can do this completely remains to be seen.

The problem is that turbulence would seem to demand "singularities," or points where the fluid becomes undefined. Under certain very common conditions a fluid should twist around on itself forever, it seems, or until it's packed in enough velocity and energy that it just explodes. Which should be happening all of the time: everyday fluids just exploding.

As blood moves around the body it does so in a reasonably orderly fashion (and without exploding). This is called laminar flow. It's what you probably think of when you think of flow in the first place: a fluid moving in concert, generally in the same direction. For blood, plumbing, and most any other sort of engineering, laminar flow is desirable. It's efficient and unchaotic.

You can imagine an idealized flow geometrically as the graph of some equation (function), where, rather than a line or a curve passing through a limited set of points in two-dimensions (or three or more), the flow exists as a field. This means that it exists at every point on the graph. With fluids, by "exists" we really mean that it has velocity at every point, which is a magnitude (or strength) and a direction. So, we can look at a fluid and we should be able to find out what it's doing at discrete points of our choosing: where it's going and how fast.

But fluids can be disrupted, and beyond a certain threshold is when things get weird. For blood, that might be an unusual constriction in a blood vessel. Things are cruising along as normal, but all of a sudden there's a lot less volume to occupy. So, the blood as it passes through the region increases in velocity—it speeds up. Then, after the blood vessel opens back up, it slows down again. This slowing is not instantaneous; all of the blood doesn't return to a uniform velocity at once. This is simply inertia at work. Old physics.

Where the narrower blood vessel meets the wider one, eddies form. The blood has inertia as it blasts out of the constriction, and so it leaves little whorls of stalled blood at the boundary. The faster blood shears against this stalled blood and vortices form: turbulence. This boundary is called an eddy line and it's a function of both the inertia of a fluid and its viscosity (ooziness, basically).

An eddy line is the birth of chaos. Ultimately, it's this chaotic motion that's rumbling against the inner walls of a blood vessel, like the pounding of tiny fists. The same physics are mirrored at all scales, from upper-atmospheric currents to deep-ocean convection patterns to the gnarliest whitewater rapids. Speaking from experience, an eddy line can really fuck a kayak up.

In fluid dynamics, the transition to turbulence is pretty specific. If the ratio between the inertia of a fluid given some specified conditions (velocity, density, volume) and the viscosity of that fluid is above a certain point (the Reynolds number), the flow of the fluid in question will make a transition from an orderly, unified laminar flow to chaotic turbulent flow. A smooth, swift current becomes a line of swirling vortices.

A quote somewhat dubiously attributed to the physicist Werner Heisenberg (on his death bed, no less) goes like this: "When I meet God, I am going to ask him two questions: Why relativity? And why turbulence? I really believe he will have an answer for the first."

The quote is also attributed to the fluid dynamics pioneer Horace Lamb in this form: "I am an old man now, and when I die and go to Heaven there are two matters on which I hope for enlightenment. One is quantum electrodynamics and the other is the turbulent motion of fluids. And about the former I am really rather optimistic."

Whoever actually said it, you get the idea.

Part of the problem is in the very nature of the math involved, which is based on the most difficult variant of the most difficult variety of an already excruciating branch of mathematics known as differential equations: nonlinear partial differential equations. Take an equation with multiple variables (say velocity, viscosity, temperature, and area) and then add in a bunch of derivatives of those variables. (A derivative, which is the fundamental concept of calculus, takes a variable and then restates that variable as its rate of change. So: the derivative of my location in three-dimensional space corresponds to how that location is changing through time.)

Differential equations involving only a single variable are holy terrors, while multivariable differential equations, or partial differential equations, are a near-psychedelic experience. Here, as employed by the Navier-Stokes equations, they're even worse because they're non-linear, which means that they output values that are not directly proportional to their inputs.

In the most general sense, if I were to take a variable x and for every input value, I doubled it, that would be linear. If I were to take each of those input values and, say, multiplied them by themselves three times, that would be non-linear. An exponential increase is non-linear. So, it's easy to see how non-linearity can make for some very extreme situations with even the most gradual increases of the input value.

Solving an equation like this becomes a really unfriendly thing really fast, especially to computers. Simulating the most basic real-world turbulence is computationally taxing, which makes simulating a great deal of real-world phenomena computationally prohibitive or at least very intense—examples range from the field equations behind Einstein's general theory of relativity to models of biological neurons to waves of water.

Physically, what happens within an eddy line is that as vortices form, they interact with other vortices, with the result being shear. Shear produces more eddies and more vortices, albeit at smaller and smaller scales. Eventually, these vortices shrink to molecular scales and diffuse, thanks to viscosity. The seemingly endless self-similar vortex replication is broken. Fluids don't just explode.

But we don't really have a good way of explaining that mathematically or, in a sense, physically. While we can come up with a good statistical model for the dissipation of turbulence, there is no theoretical formulation to match. The solutions determine the problem. Eventually, the tiny fists of a heart murmur lose steam, which is a good thing, but deep inside the rumble a mystery persists.