Analysis of an electroless plating problem

Electroless plating in microfluidic channels is a novel technology at the micrometer scale. As the microchannel depth varies with the flow of the chemicals, care must be taken for the channel not to run dry. Owing to the deposited chemical species the physical domain of the flow changes with time, leading to a free boundary problem. As the motion of the free boundary is small it is modelled by a transpiration approximation. With this simplification, the mathematical model consists of a Navier-Stokes flow and an equation for the concentration of the plating chemical coupled by nonstandard and nonlinear boundary conditions. Existence and uniqueness are proved for the concentration equation, assuming that the flow is given. Numerical analysis is carried out and justifies the proposed numerical schemes and the nonlinear algorithms. The numerical study is performed, in the two-dimensional case, with the finite element method and an implicit Euler time scheme for the coupled problem with the Navier-Stokes flow and the nonlinear concentration equation.