Micropumps for drug delivery systems: a new semi-linear elliptic boundary-value problem

In this paper, a new differential model for electrostatic membrane micropumps for drug delivery systems is presented. In particular, a new two-dimensional nonlinear second-order differential model with singularity has been determined in steady-state conditions in which the electric field magnitude has been considered proportional to the mean curvature of the membrane. Then, a result of the existence of at least one solution for the model has been obtained, although, concerning the uniqueness of the solution, it is not guaranteed. Moreover, the stability of the solutions has been studied highlighting that when a solution exists, then it is unstable. Moreover, the problem was numerically solved by means of three-stage Lobatto IIIa formula achieving the ranges of the electromechanical parameters of the material constituting the membrane with or without ghost solutions. Finally, a criterion to select the intended use of the micropump starting from the electromechanical properties of the membrane and a criterion to choice the material constituting the membrane starting from the intended use of the micropump are presented.