An important landmark in the modern study of jets is Lord Raleigh's (1878) explanation of the break up of a liquid jet into drops, a classic example of an "instability" in fluid flow. In 1944, the Russian physicist Lev Landau (who later went on to receive the Nobel Prize for his work on superfluidity) presented an exact solution of the equations of fluid mechanics for a "submerged jet from a point source." A submerged jet is what you have when you relax in your hot tub or when you blow out your birthday candles - it is a jet within a bath of the same fluid. The equations of fluid mechanics (the "Navier-Stokes equations") are nonlinear and it is extraordinarily difficult to derive exact solutions, though approximate ones can be found; using computers, for example. Landau's solution is one of a handful of exact solutions that are known. In 1951, an Englishman, H.B. Squire published the same solution, quite unaware of Landau's work. This is perhaps excusable as communication between Russia (the USSR then) and the west wasn't exactly smooth. In the fluid mechanics community this solution has become known as "The Landau-Squire solution". Scientists are not good in history though, they almost always get it wrong. The so called "Landau solution" was actually discovered by Slezkin (Slezkin, N.A. 1934 "On an exact solution of the equation of viscous flow." Moscow State Univ. Uchenie Zapiski, vol. 2) right on the other side of town and then "re-discovered" twice; approximately once every decade!
We have created in the laboratory what is perhaps the world's tiniest jet. The diameter of our jets range from 40-150 nano meters, that is, a few hundred water molecules lined up in a row. Remarkably, the Landau-Squire solution works almost perfectly for our tiny jet and the theory allows us to calculate the volume flux from it. The Navier-Stokes equations and everything derived from it is supposed to go awry as you approach molecular scales, but no one knows how far down one can push before it breaks. It is a bit like the the next mega earthquake in LA, it is a certainty but nobody knows when it will call. We found that it all works very nicely at the 100 nanometer scale, which is nice. The flow rates turn out to be in the range of tens of pico liters per second. At this rate if you started to fill a 2 liter soda bottle about the time when the first Pyramid was being built in Egypt, your bottle will be about half full now!
The set up for our experiment is shown in the figure below (Panel A). At the heart of the apparatus is a glass "nano capillary" that is fabricated by heating an ordinary glass capillary with a laser and gently pulling it in a machine until it breaks making a fine tip. The SEM (Scanning Electron Microscope) image on the left shows the tip of one such capillary, the white bar is 100 nano meter long or about a fifth of the wavelength of light. The flow is generated by applying an electric voltage across the capillary which generates an electro-osmotic jet. To "see" the jet we build a tiny "anemometer" the spinning wheels that meteorologists mount on tall buildings to measure wind speed. Our anemometer is a polystyrene (a kind of plastic) bead one fiftieth the width of a human hair. The tiny anemometer is mounted on a spindle made of light - a fine laser beam that constitutes an "optical trap". When the bead is positioned in front of the jet, it spins. Panel B is a light microscope image of an experiment in progress (the white bar is 5 microns - a micron is a thousandth of a millimeter).
In order to measure the rotation rate, the bead is manufactured so as to have a little dimple on one side - kind of like a ping pong ball that has been stepped on! An SEM image of the bead is shown in Panel D (the white scale bar is 1 micron). As it spins, a video camera picks up the tiny fluctuations in light (Panel D & E) coming from the dimpled ball. Click the arrow below to see a video of this. You will see three passes of the capillary tip with the voltage set successively to +1 V, -1 V and 0 V. Notice the direction of spin of the bead. The poor quality of the image is not due to bad equipment but because we are at the limit of resolution of the optical microscope! Our technique of measuring the rotation rate from fluctuations of light intensity is in fact quite similar to the technique astronomers use to determine the rotation period of binary stars that are too far away to resolve in telescopes. The measured rotation rate can be compared to the one predicted on the basis of the Landau-Squire solution and the flow rate determined.
The measurements also reveal something that we had not expected. If you reverse the voltage, the flow direction reverses, that is, the capillary now sucks in the fluid. This is of course expected. However, the flow rate at these negative voltages is much lower. In other words, the capillary behaves like a flow rectifier similar to a semiconductor diode, except, it is fluid rather than electrons that are flowing. If you are perhaps reading this blog on your computer, then billions of semiconductor diodes and transistors are working silently making your activity possible. What applications can a "nano fluidic diode" have? I am afraid, I do not have a clue but I am reminded of the inventor who gave us (almost) all things electrical, Michael Faraday. Faraday was once visited by a delegation of government dignitaries who were shown his electric motors and other demos. One of them said "This is all very interesting, but of what possible use are these toys?" Faraday responded: "I cannot say what use they may be, but I can confidently predict that one day you will be able to tax them."
Acknowledgement: The experimental part of the work was carried out in the laboratory of my colleague Ulrich Keyser at the Cavendish Laboratories, Cambridge University in the UK where I was a Leverhulme Visiting Professor. I am grateful to the Leverhulme Trust for making this possible. I would also like to thank the NIH in the US for support. The SEM image of the nanocapillary was taken by Lorenz Steinbock at the Keyser lab.
Our paper was published in Nano Letters
 A Landau-Squire nanojet authors: Nadanai Laohakunakorn, Benjamin Gollnick, Fernando Moreno-Herrero, Dirk Aarts, Roel PA Dullens, Sandip Ghosal, Ulrich F Keyser. Web publication date: 2013/10/14 Journal: Nano Letters
Press Reports: News from McCormick