# Drop spreading and drifting on a spatially heterogeneous film: capturing variability with asymptotics and emulation

## Abstract

A liquid drop spreading over a thin heterogeneous precursor film (such as an inhaled droplet on the mucus-lined wall of a lung airway) will experience perturbations in shape and location as its advancing contact line encounters regions of low or high film viscosity. Prior work on spatially one-dimensional spreading over a precursor film having a random viscosity field (Xu and Jensen, Proc R Soc A 472:20160270, 2016) has demonstrated how viscosity fluctuations are swept into a narrow region behind the drop’s effective contact line, where they can impact drop dynamics. In this paper, we investigate two-dimensional drops, seeking to understand the relationship between the statistical properties of the precursor film and those of the spreading drop. Assuming the precursor film is much thinner than the drop and viscosity fluctuations are weak, we use asymptotic methods to derive explicit predictions for the mean and variance of drop area and the drop’s lateral drift. For larger film variability, we use Gaussian process emulation to estimate the variance of outcomes from a restricted set of simulations. Stochastic drift of the droplet is predicted to be the greatest when the initial drop diameter is comparable to the correlation length of viscosity fluctuations.

## Keywords

Drop spreading Gaussian process emulation Surface tension Thin-film flow Uncertainty quantification## Notes

### Acknowledgements

This study was supported by EPSRC Grant No. EP/K037145/1. The GP_emu_UQSA package may be downloaded from https://github.com/samcoveney/maGPy.

## References

- 1.Thornton DJ, Rousseau K, McGuckin MA (2008) Structure and function of the polymeric mucins in airways mucus. Annu Rev Physiol 70:459–486CrossRefGoogle Scholar
- 2.Lai SK, Wang Y-Y, Wirtz D, Hanes J (2009) Micro-and macrorheology of mucus. Adv Drug Deliv Rev 61(2):86–100CrossRefGoogle Scholar
- 3.Jonathan H, Widdicombe JH, Wine JJ (2015) Airway gland structure and function. Physiol Rev 95(4):1241–1319CrossRefGoogle Scholar
- 4.Sims DE, Horne MM (1997) Heterogeneity of the composition and thickness of tracheal mucus in rats. Am J Physiol 273(5):L1036–L1041CrossRefGoogle Scholar
- 5.Kirkham S, Sheehan JK, Knight D, Richardson PS, Thornton DJ (2002) Heterogeneity of airways mucus: variations in the amounts and glycoforms of the major oligomeric mucins muc5ac and muc5b. Biochem J 361(3):537–546CrossRefGoogle Scholar
- 6.Levy R, Hill DB, Forest MG, Grotberg JB (2014) Pulmonary fluid flow challenges for experimental and mathematical modeling. Integr Comp Biol 54(6):985–1000CrossRefGoogle Scholar
- 7.Ostedgaard LS, Moninger TO, McMenimen JD, Sawin NM, Parker CP, Thornell IM, Powers LS, Gansemer ND, Bouzek DC, Cook DP, Meyerholz DK, Abou Alaiwa MH, Stoltz DA, Welsh MJ (2017) Gel-forming mucins form distinct morphologic structures in airways. Proc Natl Acad Sci 114:6842–6847Google Scholar
- 8.Kesimer M, Ehre C, Burns KA, William Davis C, Sheehan JK, Pickles RJ (2013) Molecular organization of the mucins and glycocalyx underlying mucus transport over mucosal surfaces of the airways. Mucosal Immunol 6(2):379–392CrossRefGoogle Scholar
- 9.Chatelin R, Poncet P (2016) A parametric study of mucociliary transport by numerical simulations of 3d non-homogeneous mucus. J Biomech 49(9):1772–1780CrossRefGoogle Scholar
- 10.Xu F, Jensen OE (2016) Drop spreading with random viscosity. Proc R Soc A 472:20160270MathSciNetCrossRefzbMATHGoogle Scholar
- 11.Cox RG (1983) The spreading of a liquid on a rough solid surface. J Fluid Mech 131:1–26MathSciNetCrossRefzbMATHGoogle Scholar
- 12.Miksis MJ, Davis SH (1994) Slip over rough and coated surfaces. J Fluid Mech 273:125–139CrossRefzbMATHGoogle Scholar
- 13.Savva N, Kalliadasis S, Pavliotis GA (2010) Two-dimensional droplet spreading over random topographical substrates. Phys Rev Lett 104(8):084501CrossRefGoogle Scholar
- 14.Hocking LM (1983) The spreading of a thin drop by gravity and capillarity. Q J Mech Appl Math 36(1):55–69MathSciNetCrossRefzbMATHGoogle Scholar
- 15.King JR, Bowen M (2001) Moving boundary problems and non-uniqueness for the thin film equation. Eur J Appl Math 12(3):321–356MathSciNetCrossRefzbMATHGoogle Scholar
- 16.Hocking LM, Rivers AD (1982) The spreading of a drop by capillary action. J Fluid Mech 121:425–442MathSciNetCrossRefzbMATHGoogle Scholar
- 17.Savva N, Kalliadasis S (2009) Two-dimensional droplet spreading over topographical substrates. Phys Fluids 21(9):092102CrossRefzbMATHGoogle Scholar
- 18.Sibley DN, Nold A, Savva N, Kalliadasis S (2015) A comparison of slip, disjoining pressure, and interface formation models for contact line motion through asymptotic analysis of thin two-dimensional droplet spreading. J Eng Math 94(1):19–41MathSciNetCrossRefzbMATHGoogle Scholar
- 19.Kersting K, Plagemann C, Pfaff P, Wolfram B (2007) Most likely heteroscedastic Gaussian process regression. In: Proceedings of the 24th international conference machine learning, pp 393–400Google Scholar
- 20.Filoche M, Tai C-F, Grotberg JB (2015) Three-dimensional model of surfactant replacement therapy. Proc Natl Acad Sci 112(30):9287–9292CrossRefGoogle Scholar
- 21.Kim J, O’Neill JD, Dorrello NV, Bacchetta M, Vunjak-Novakovic G (2015) Targeted delivery of liquid microvolumes into the lung. Proc Natl Acad Sci 112(37):11530–11535CrossRefGoogle Scholar
- 22.Khanal A, Sharma R, Corcoran TE, Garoff S, Przybycien TM, Tilton RD (2015) Surfactant driven post-deposition spreading of aerosols on complex aqueous subphases. 1: high deposition flux representative of aerosol delivery to large airways. J Aerosol Med Pulmon Drug Deliv 28(5):382–393CrossRefGoogle Scholar
- 23.Sharma R, Khanal A, Corcoran TE, Garoff S, Przybycien TM, Tilton RD (2015) Surfactant driven post-deposition spreading of aerosols on complex aqueous subphases. 2: low deposition flux representative of aerosol delivery to small airways. J Aerosol Med Pulmon Drug Deliv 28(5):394–405CrossRefGoogle Scholar
- 24.Lord GJ, Powell CE, Shardlow T (2014) An introduction to computational stochastic PDEs. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar