A conservative bulk-surface closure for soluble-surfactant falling films: an analytical and numerical mass-balance audit
Abstract
Adsorption and desorption redistribute soluble surfactant between a liquid film and its free surface, but cannot change the total inventory in a closed periodic domain. We audit this requirement in nonlinear weighted-residual models for soluble-surfactant-laden falling films. Although a previous correction restores conservation of the nonlinear surface Marangoni flux, a separate inconsistency remains in the depth-integrated bulk diffusion. We derive the resulting mass imbalance and construct explicit positive periodic concentration fields that exhibit it. The missing contribution arises from the streamwise concentration gradient appearing in the normal diffusive flux at a sloping interface. Including this contribution restores local conservation of the coupled bulk-surface transport system while leaving the hydrodynamic equations, the leading-order adsorption-desorption closure, and the linear stability properties of the uniform equilibrium unchanged. In an independent no-slip calculation using the parameters of an earlier travelling-wave study, the reference model loses 0.0228832% of its initial surfactant mass by t=4500, whereas the corrected run has an absolute relative mass change of 1.2×10−15. The main hump and capillary ripples remain closely comparable at t≈4500. This benchmark therefore shows that agreement with travelling-wave calculations does not, by itself, guarantee that a reduced model respects the conservation laws inherited from the underlying transport problem. Although the drift is small in this case, the corrected equations conserve the reduced total inventory for every smooth periodic solution. We also distinguish this exact conservation property from higher-order compatibility with the full curved-interface boundary conditions.
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BibTeX
@misc{mukhopadhyay2026conservative,
title = {{A conservative bulk-surface closure for soluble-surfactant falling films: an analytical and numerical mass-balance audit}},
author = {Sanghasri Mukhopadhyay and Asim Mukhopadhyay},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.37869},
eprint = {2609.37869},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.37869},
}
RIS
TY - UNPB
TI - A conservative bulk-surface closure for soluble-surfactant falling films: an analytical and numerical mass-balance audit
AU - Sanghasri Mukhopadhyay
AU - Asim Mukhopadhyay
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.37869
UR - https://arxiv.org/abs/2609.37869
AB - Adsorption and desorption redistribute soluble surfactant between a liquid film and its free surface, but cannot change the total inventory in a closed periodic domain. We audit this requirement in nonlinear weighted-residual models for soluble-surfactant-laden falling films. Although a previous correction restores conservation of the nonlinear surface Marangoni flux, a separate inconsistency remains in the depth-integrated bulk diffusion. We derive the resulting mass imbalance and construct explicit positive periodic concentration fields that exhibit it. The missing contribution arises from the streamwise concentration gradient appearing in the normal diffusive flux at a sloping interface. Including this contribution restores local conservation of the coupled bulk-surface transport system while leaving the hydrodynamic equations, the leading-order adsorption-desorption closure, and the linear stability properties of the uniform equilibrium unchanged. In an independent no-slip calculation using the parameters of an earlier travelling-wave study, the reference model loses 0.0228832% of its initial surfactant mass by t=4500, whereas the corrected run has an absolute relative mass change of 1.2×10^−15. The main hump and capillary ripples remain closely comparable at t≈4500. This benchmark therefore shows that agreement with travelling-wave calculations does not, by itself, guarantee that a reduced model respects the conservation laws inherited from the underlying transport problem. Although the drift is small in this case, the corrected equations conserve the reduced total inventory for every smooth periodic solution. We also distinguish this exact conservation property from higher-order compatibility with the full curved-interface boundary conditions.
ER -