Normally most of what I read in Phys. Rev. Lett. (PRL) is somewhat beyond the scope of vdov.net. However, two days ago there was a neat advance (arXiv here) on a very simple application of molecular dynamics to the microscale swimming problem. Microscale swimming is quite a bit different than what we associate with the swimming of people, animals and large machines. The Reynold’s number associated these types of processes are generally large (Re >> 1), whereas in a microscale swimming process the Reynold’s number of usually orders of magnitude below 1. Thus the importance of inertia in a microscale swimming process is effectively zero, the consequence of which is that a viscous-only flow is fully reversible.

In a viscous only flow, the Navier-Stokes equations are simplified as the inertial components (and therefore any time derivatives) must be zero. The standard Navier-Stokes equation for a Newtonian fluid reads as,

This is a lot simpler than you think: the left hand side of the equation is characteristic of inertial forces, and the right hand size consists of a pressure gradient term, viscosity term and a body forces term (gravity, etc.). In the limit of very low Reynold’s number, these equations simplify to the Stokes equations, given as (neglecting any body forces),

With continuity equation for an incompressible fluid as,

we have a relatively nice description of the problem. So what does this all mean for microscale swimming? These Stokes equation have no time dependence and the solutions to the equations are time-reversible. So, in a viscous enough fluid at relatively low velocity, swimming in the traditional sense won’t work. Any motion in one direction will be completely countered by the equal and opposite motion. So a flapping wing or paddle is pretty useless in these types of conditions. Certainly you or I couldn’t swim at low Reynold’s number. So microscale species develop inventive ways of getting around this problem. A lot of these solutions are detailed in the classic Life at Low Reynold’s Number (I found a free link to the paper … AIP would like to charge you for it). There are many methods of solution to the Stokes equations, the details of which I will certainly not discuss here. If you want to know more about some of the more interesting ones (Boundary Element, Streamfunction or Green’s Function methods), take a graduate class in numerical methods or fluid dynamics.

This is all great and good. What the paper in PRL has done is simply apply a reasonable molecular dynamics (MD) approach to the power and efficiency of these problems. It’s an extremely simple application of MD to a pretty easily understood phenomena, and therefore perhaps appropriate here on vdov.net. In the paper, they investigate various biological and microbiotic designs (such as biflaps motion, flagellum, legs and snaking motion), and extract infomation about the power vs. efficiency of each design. This information has some pretty important ramifications, some of which might include some interesting work on the efficiency of microscale therapeutics. The paper is extremely easy to read and the majority of the conclusions drawn should be understandable to the reader without any simulation or fluids experience.

Why then, you might ask, would I care about any of this? Well it turns out this type of fluid to molecular reduction is precisely what I am trying to do to study the collisions of droplets in a spray or on a surface. The problem specification in my situation is perhaps more involved and I certainly won’t detail it here, but this is a wonderful example on how MD and computational fluids can talk to each other. Secondly, I’m looking to put more science on the vdov.net front page, and this paper seemed like a nice place to start. All of you current or future PhDs out there are welcome/encouraged to post anything interesting at any time.

Cheers.