A Second-Gradient Theory of Fluid Flow
Recently, Eliot Fried and Mort Gurtin have developed general balance equations and boundary conditions for second-grade materials. Their work is set to appear in the Archive for Rational Mechanics and Analysis and is presently available online (DOI: 10.1007/s00205-006-0015-7). The theory essentially blends classical work by Toupin on elastic materials with couple stresses with a modern, nonstandard principle of virtual power developed by Gurtin. Importantly, the basic formulation is independent of constitutive assumptions, and as such, applicable to both solids and fluids.
Fried and Gurtin consider incompressible fluid flow as one such application. The approach effectively generalizes the Navier-Stokes equations to include higher-order gradients of the velocity field. Through constitutive assumptions, material lengths are naturally introduced in the flow equation and higher-order boundary conditions. Fried and Gurtin refer to the former as the gradient length, L, and the latter as the adherence length, l. This work is of interest because recent simulations suggest that at sufficiently small length scales, the classical Navier-Stokes equations and their boundary conditions fail to accurately describe fluid flow. The new theory provides a mechanism to account for these length scale effects, and being continuum-based, promises to be much more efficient than discrete methods such as molecular dynamics.
In particular, Fried and Gurtin consider the case of plane Poiseuille flow and derive analytical expressions for the velocity profile. If one considers laminar flow through a channel of height h, for example, gradient effects play an increasingly important role on the flow with decreasing ratios h/L of physical to gradient lengths. A plot of the flow profiles predicted by the theory is reproduced here in the Figure to the right. The theory allows for a range of flow profiles from the limiting cases of strong (l approaching infinity) and weak (vanishing l) adherence to the classical results predicted by the Navier Stokes equations.