An analytical solution of the inverse problem of capillary imbibition

The inverse problem of capillary imbibition involves determination of the capillary geometry from the measurements of the time-varying meniscus position. This inverse problem is known to have multiple solutions, and to ensure a unique solution, measurements of imbibition kinematics in both directions of the capillary are required. We here present a closed-form analytical solution of the inverse problem of determining the axially varying radius of a capillary from experimental data of the meniscus position as a function of time. We demonstrate the applicability of the method for solving the inverse capillary imbibition problem for two cases, wherein the data for imbibition kinematics are obtained (i) using numerical simulations and (ii) from published experimental work. In both cases, the axially varying capillary radius predicted by the analytical solution agrees with the true capillary radius. In contrast to the previously proposed iterative methods for solving the inverse capillary imbibition problem, the analytical method presented here yields a direct solution. This analytical solution of the inverse capillary imbibition problem can be helpful in determining the internal geometry of micro- and nano-porous structures in a non-destructive manner and design of autonomous capillary pumps for microfluidic applications.