On the inverse problem of capillary imbibition through nonuniform axial geometry

We investigate the inverse problem of capillary imbibition, which consists in determining capillary radius from measurements of imbibition kinematics. The solution of this inverse problem is helpful in determining the internal geometry of micro- and nano-porous materials and the design of autonomous capillary pumps for microfluidic applications. Previous studies stated that the inverse problem is ill-posed in the sense that it has multiple solutions. Therefore, an approach was proposed to solve this problem, which requires measuring the imbibition kinematics in “both” capillary directions. In this Letter, we revisit the inverse problem of capillary imbibition, and two main results are achieved. The first is related to the ill-posedness of the inverse problem. We demonstrate that, contrary to what it was thought up until now, the inverse problem is well-posed and has a unique solution. The second main result relates to the solution of the inverse problem. Based on purely mathematical arguments, we propose an analytical solution of the inverse problem, which requires measuring the imbibition kinematics in only “one” tube direction. The analytical solution is validated using imbibition kinematics data obtained from two different sources: (a) from numerical simulations and (b) from published experimental work. The results show excellent agreement between the capillary radius obtained analytically and the true capillary radius profiles.