# Numerical Methods for Second-Gradient Fluids

Figure 1: Flow profiles ranging from classical Navier-Stokes to those predicted by the second-gradient theory with weak and strong adherence boundary conditions.

The rational design of micro- and nano-fluidic devices requires a methodology for efficiently and accurately simulating fluid flow at small length scales. Atomistic simulations indicate that, for devices with characteristic dimensions below 10~nm, the Navier--Stokes equations and its classical boundary conditions fail to adequately predict even time-averaged flow rates. While new devices and experimental observations continue to emerge, techniques for modeling this class of phenomena are severely under-developed. Current approaches rely on molecular dynamics or hybrid continuum-MD techniques, neither of which are capable of providing results for sufficiently long time scales.

Recently, we have developed a finite-element method for discretizing the governing equations associated with a second-gradient theory of fluid flow (Fried and Gurtin, 2005). This extended continuum description allows for predicting fluid flow at length scales between the molecular length scale and the critical length scale at which the classical theory of incompressible viscous fluids breaks down The theory introduces additional terms to the Navier--Stokes equations involving the second-gradient of the velocity field. Coupled with boundary conditions that incorporate a material length scale, this theory gives rise to predictions of velocity profiles for channel flow that compare favorably with recent predictions based on hybrid continuum-MD methods (Figure 1). Further, it allows for the classical velocity profile to be recovered as a particular limiting case.

Our numerical method is novel in that we discretize the fourth-order equations using classical finite elements, albeit with appropriate interior penalizations. This allows us to ensure continuity of higher order terms in a weak sense. This approach is much more efficient than those based on smooth-splines or Hermite polynomials. Further, it has the advantage of being available over arbitrary, unstructured domains.

Figure 2: Flow through a step predicted by our numerical method. As the gradient length L is increased relative to the physical length h, a marked decrease in flow rates is predicted (click for larger image).

This work has already provided some interesting predictions.
Figure 2a provides velocity contours for flow through a step, obtained using our
finite element code. Interestingly, our theory predicts a
*decrease* in flow rates as the change in step height is introduced
at small length scales where the magnitude of the gradient
length L approaches the physical length (Figure 2b). This result is
counter-intuitive, and could have significant ramifications for the
optimization of micro- and nano-fluidic devices.

Much of this work has stemmed from a collaboration with Eliot Fried at Washington University in St. Louis. The work has been supported through generous grants from NSF and the Department of Energy.

## References

- Kim TY, Dolbow J, Fried E A Numerical Method for a Second-Gradient Theory of Incompressible Fluid Flow JOURNAL OF COMPUTATIONAL PHYSICS 223(2): 551-570 2007