In recent years there has been a great deal of interest in mesoscopic systems in physics. The majority of this work has focused on studying the electronic properties of materials. Often these intermediate sized systems display novel physics. Our project explores the non-turbulent flow of water in the smallest uniform channels known to have been fabricated. We have obtained measurements of Poiseuille flow in channels as small as ~ 3 x 3 µm² in cross-section. The technology we have developed for the fabrication of these unusually small flow structures may have applications for research in the areas of biophysics and fluid flow in porous media. These flow structures have allowed us to revisit a surprising problem that exists in the literature concerning Poiseuille flow in channels of order 1 x 100 µm² in cross-section; namely that there appear to be deviations from the theoretically expected flow rates.
Presumably, fluid flow at low Reynolds number is described by the Navier-Stokes equations, and is well understood. Yet, studies have found deviations from this theory in structures as simple as a channel of constant shape and cross-sectional area.¹ Such non-turbulent flow through structures of constant cross-sectional area and shape is called Poiseuille flow and is governed by Poiseuille's law,
ü is the mean velocity of the fluid in the channel, µ is the viscosity of the fluid, and rh is the hydraulic radius of the channel defined by
f is a factor of order unity which depends on the shape of the channel (it is precisely unity for a circular cross-section). Poiseuille's law is the solution to the Navier-Stokes equations in a channel of uniform cross-section given certain assumptions about the nature of the flow in the channel such as a no-slip boundary condition at the walls of the channel and that the flow is laminar in the channel. For structures in which the continuum mechanics of a fluid is a reasonable approximation, we do not expect to find any deviations from the Navier-Stokes relations. However, such deviations have been reported in structures whose dimensions are much larger than any characteristic length scale associated with the fluid. (Such a length scale in a simple fluid would be a few Å.) Because deviations from Poiseuille's law are not to be expected, this apparent deviation should be examined closely.
Using a simple, table-top experiment to study Poiseuille flow of water, we found no deviations from Poiseuille's law in our flow structures. To measure Poiseuille flow in the smallest structures yet studied, we developed a technique for fabricating ultra-small flow structures of order 3 x 3 µm² in cross-section and for measuring volume flow rates as small as ~ 1 x 10-6 cm³/s. Our channels were fabricated using photolithographic techniques on a glass substrate. Channels were made in photoresist sandwiched between two glass cover slides as described in a manuscript² which has been submitted to Physical Review E. For more details on the experimental set-up we refer the reader to this paper.
 |
 |
| Schematic of the lithographic method used to make flow channels. A photoresist coated glass slide is first exposed to light (a). The photoresist in the exposed regions is then removed with a developer solution (b). The result is a flow channel pattern; for the present work we made two square inlet/outlet regions (approximately 3 mm on a side), and connected them with one or two narrow channels (c). | High resolution optical micrograph showing a flow channel which was 125 µm long and 4 µm wide. |
To summarize our work, we have revisited a seemingly well understood problem and developed novel fabrication methods for studying Poiseuille flow in extremely small channels. We have found no deviations from Poiseuille's law. This is to be expected because our channels, although small, are still larger than any characteristic length scale associated with the fluid. Moreover, the fabrication technique we have developed has several important advantages that could be useful for research in other areas. First, our fabrication technique allows one to create channels of arbitrary shape. Thus, one can study the effect of geometry on the movement of fluids. The flow structure geometry can even be extended to approximate that of a porous medium, and such studies are currently underway. Secondly, our fabrication technique should permit us to make extremely small channels that can approach 100 Å in one or both transverse directions. Channels of this size may allow the study of mesoscopic effects in fluid systems.