Particle Transport in Micro and Nanostructured Arrays: Asymmetric Low Reynolds Number Flow - Nanofluidics: Nanoscience and Nanotechnology - page 51

deeply etched, in fact they are more typically extremely shallow ( h << x,y for all x , y in the
structure), and the approximations used to obtain the biharmonic equation (Equation 3.13)
are not valid. In the case of a very shallow etch a different equation can be derived for the
stream function. First, for the very shallow etch we know that the z dependence of the
velocity function must be a parabolic solution of the 1-D low R e N-S equation:

(3.14)

In this case we can write Equation 3.8 as,

xy z xy Pxy (3.15)

We then use some vector calculus identities to get the Laplacian partial differential
equation for the stream function in the case of very shallow etches, i.e.

xy (3.16)

x ψ for a shallow etched nanofluidic system satisfies
Laplace's equation, and hence solving the stream function for a thin flow, low R e N-S
equation is effectively an electrostatic calculation driven by the boundary values, a huge
simplification! If a solution for the stream function

( , y )

Thus, the stream function

x ψ can be found, the velocity in
the x - y plane can be found from Equation 3.11. The pressure field can be found by solving,
unfortunately, the following equation:

( , y )

(3.17)

If we had not insisted on explicitly making the velocity in the thin plane a function of both
x and y , this equation may have been written much more simply if the flow is only in the x
direction, that is, effectively a 1-D flow where the z -axis etch depth is very shallow. In this
case one obtains the simple equation of lubrication theory, 5

2

∂ P ∂ υ

=

(3.18)

2

∂ x ∂ z

Before nanofabrication came around, this equation was adequate to explain simple thin
laminar flows. However, as we will show by using nanofabrication techniques it is quite
possible to make thin flows with complex (and useful) changes in the x and y directions
that cannot be handled using Equation 3.18. We are therefore forced to develop a more
encompassing model for our problem of an array of obstacles in the x-y plane.

In fact, using the techniques of nanofabrication it is easily possible to create thin
structures which, while they remain in the thin flow limit used to derive Equation 3.17,
have varying etch depths. Such techniques have been exploited by the Craighead group at
Cornell University in variable depth arrays. 6 When the etch depth becomes a function h(z)
Equation 3.17 must be changed to the rather formidable form: