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Approximating pressure-driven Stokes flow using the principle of minimal excess dissipation

Stokes' equations model microscale fluid flows including the flows of nanoliter-sized fluid samples in lab-on-a-chip systems. Helmholtz's dissipation theorem guarantees that the solution of Stokes' equations in a given domain minimizes viscous dissipation among all incompressible vector fields that are compatible with the velocities imposed at the domain boundaries. Helmholtz's dissipation theorem directly guarantees the uniqueness of solutions of Stokes flow, and provides a practical method for calculating approximate solutions for flow around moving bodies. However, although generalization of the principle to domains with mixtures of velocity and stress boundary conditions is relatively straight-forward (Keller et al., 1967), it appears to be little known. Here we show that the principle of minimal excess dissipation can be used to derive accurate analytical approximations for the flows in microchannels with different cross-section shapes including when the channel walls are engineered to have different distributions of slip boundary conditions. In addition to providing a simple, rapid, method for approximating the conductances of micro-channels, analysis of excess dissipation allows for a comparison principle that can be used, for example, to show that the conductance of a channel is always increased by adding additional slip boundary conditions.