## Monday, February 13, 2012

### Stokes Flow

Anyone working with polymers is intimately aware of laminar flow. The viscosity of polymer is so high that achieving turbulent flow is pretty much impossible, at least for molten polymers. Solutions, and all the different ilks of dispersions are low enough in viscosity that turbulence can be reached, but for a typical molten polymer, turbulence isn't happening. Looking at the Reynolds number
Re = ρ v D / μ
(ρ is the density, v is the velocity, D is the diameter and μ is the viscosity) for different conditions can help show this. For circular pipes, turbulence can occur if Re gets above about 2100, so for a typical molten polymer with viscosity of say, 107 Pa s viscosity and a density of 1 kg/m3 in a 1 m pipe, the velocity would need to be over 21 x 106 m/s for turbulence to occur (that is 7% the speed of light)! Up the pipe diameter to 1000 m (the size of a nice river), and the velocity would still need to exceed a supersonic speed of 21,000 m/s.

Molten polymers actually under flow are at the opposite extreme: the Reynolds number is much less than one, a condition known as Stokes flow or creeping flow. When this happens, the inertial forces become negligible and the viscous forces dominate. The Navier-Stokes equations become very simple and in fact any time dependency in the equations disappear. This then means mathematically that the reverse flow is identical to the forward flow. While people can see this from the equations and are with it in an intellectual sense, seeing a reversible flow in real world situations is often quite shocking. Look at this video for a clear example:
That the initial conditions are not perfectly matched at the end of the video is due to a small amount of diffusion (Brownian motion) that is always present - just as if the initial colors would eventually spread out and mix over time in the absence of any flow field. Nonetheless, this would be a cool demo to put together. The colors are eye catching and while people may question the validity of it on video ("oh, they just ran the tape backwards"), this is one case where seeing would be believing.