The steady boundary value problem for charged incompressible fluids: PNP/Navier–Stokes systems

The initial-boundary value problem for the Poisson–Nernst–Planck/Navier–Stokes model was investigated in [J.W. Jerome, R. Sacco, Global weak solutions for an incompressible charged fluid with multi-scale couplings: initial-boundary-value problem, Nonlinear Anal. 71 (2009) e2487–e2497], where an existence theory was demonstrated, based upon Rothe’s method of horizontal lines. In this article, the steady case is considered, and the existence of a weak solution is established for the boundary-value problem. This solution satisfies a weak maximum principle for the concentrations relative to the boundary values. As noted in the above-mentioned citation, the model assumes significance because of its connection to the electrophysiology of the cell, including neuronal cell monitoring and microfluidic devices in biochip technology. The model has also been used in other applications, including electro-osmosis. The steady model is especially important in ion channel modeling, because the channel remains open for milliseconds, and the transients appear to decay on the scale of tens of nanoseconds.