Comment on “On the inverse problem of capillary imbibition through nonuniform axial geometry” [Phys. Fluids 36 , 011701 (2024)]

Abstract

The inverse problem of capillary filling involves the identification of the radius profile of nonuniform capillaries from measurements of fluid imbibition kinematics, which requires solving a nonlinear integral equation. About ten years ago, we have shown1 that the problem entails multiple solutions, provided that no data on the capillary radius are known beforehand. Furthermore, in order to identify the right solution, we suggested measuring the capillary filling in the opposite direction as well. This idea was later used to obtain an analytical solution to the inverse problem.2 In contrast, a recent paper3 proposed another procedure to solve the inverse problem, claiming that a unique solution can be derived from fluid imbibition data in a single direction. The aim of this Comment is to revise that claim and reinforce the fact that the inverse problem inherently yields multiple solutions, unless additional information is available, such as the capillary radius at some point. In what follows, the mathematical formulation of the problem is presented, accompanied by an explicit illustration of the family of solutions associated with a given imbibition dynamics. Subsequently, the arguments and equations reported in Ref. 3 are critically studied to elucidate the misleading analysis. Finally, some concluding remarks are provided.