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Simultaneous evaporation and imbibition of a droplet on a fully flooded porous substrate

David Craig1,*, Alexander W. Wray1,†, Khellil Sefiane2,‡, and Stephen K. Wilson3,1,§

  • 1Department of Mathematics and Statistics, University of Strathclyde, Livingstone Tower, 26 Richmond Street, Glasgow G1 1XH, United Kingdom
  • 2School of Engineering, Institute for Multiscale Thermofluids, University of Edinburgh, The King's Buildings, Mayfield Road, Edinburgh EH9 3JL, United Kingdom
  • 3Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, United Kingdom

  • *Contact author: david.craig@strath.ac.uk
  • †Contact author: alexander.wray@strath.ac.uk
  • ‡Contact author: k.sefiane@ed.ac.uk
  • §Contact author: sw3197@bath.ac.uk
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Phys. Rev. Fluids 11, 093601 – Published 14 September, 2026

DOI: https://doi.org/10.1103/jxdz-7h5w

Abstract

A mathematical model for the evolution of, and deposition from, a thin particle-laden droplet on an infinitely thick, isotropic, fully flooded, porous substrate with interconnected pores undergoing simultaneous evaporation and imbibition is formulated and analyzed. In particular, analytical expressions for the evolution of the droplet, as well as for the flow within the droplet and the substrate, and for the transport and deposition onto the substrate of the particles are obtained for droplets evolving in four different modes. While the physical mechanisms driving evaporation (namely the diffusion-driven flow of vapor in the atmosphere away from the free surface of the droplet) and imbibition (namely the pressure-driven flow of liquid in the substrate away from the base of the droplet) are rather different, perhaps rather unexpectedly, it is found that there are a number of qualitative and quantitative similarities as well as differences in the resulting behavior of the droplet as it loses mass to its environment. For example, it is shown that a droplet undergoing pure imbibition in the constant contact radius mode never completely imbibes (i.e., it has an infinitely long lifetime), but if it evaporates (either with or without imbibition also occurring), then it has a finite lifetime, and increasing the strength of evaporation and/or imbibition shortens its lifetime. It is also shown that in the regime in which diffusion of particles is faster than axial advection but slower than radial advection of particles, the final deposit of particles left behind on the substrate after the droplet has completely evaporated and/or imbibed is independent of both the nature and the strength of the physical mechanism(s) driving the mass loss from the droplet. Not only are these results of theoretical interest, but they are also relevant to a wide variety of practical applications, including ink-jet printing, DNA chip manufacturing, and disease diagnostics, that would benefit from an improved ability to predict and/or control the final deposit pattern from a droplet undergoing simultaneous evaporation and imbibition.

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References (121)

  1. Z. Chen, K. Peng, and B. Xu, Material assembly by droplet drying: From mechanics theories to applications, Droplet 2, e76 (2023).
  2. R. Zenit, Some fluid mechanical aspects of artistic painting, Phys. Rev. Fluids 4, 110507 (2019).
  3. P. Tseng, C. Murray, D. Kim, D. Di Carlo, Research highlights: Printing the future of microfabrication, Lab Chip 14, 1491 (2014).
  4. D. Brutin and V. M. Starov, Recent advances in droplet wetting and evaporation, Chem. Soc. Rev. 47, 558 (2018).
  5. H. Gelderblom, C. Diddens, and A. Marin, Evaporation-driven liquid flow in sessile droplets, Soft Matter 18, 8535 (2022).
  6. S. K. Wilson and H.-M. D'Ambrosio, Evaporation of sessile droplets, Annu. Rev. Fluid Mech. 55, 481 (2023).
  7. T. Gambaryan-Roisman, Liquids on porous layers: Wetting, imbibition and transport processes, Curr. Opin. Colloid Interface Sci. 19, 320 (2014).
  8. P. Johnson, A. Trybala, and V. M. Starov, Kinetics of spreading over porous substrates, Colloids Interfaces 3, 38 (2019).
  9. A. Oron, S. H. Davis, and S. G. Bankoff, Long-scale evolution of thin liquid films, Rev. Mod. Phys. 69, 931 (1997).
  10. R. V. Craster and O. K. Matar, Dynamics and stability of thin liquid films, Rev. Mod. Phys. 81, 1131 (2009).
  11. S. H. Davis, The importance of being thin, J. Eng. Math. 105, 3 (2017).
  12. S. H. Davis and L. M. Hocking, Spreading and imbibition of viscous liquid on a porous base. I, Phys. Fluids 11, 48 (1999).
  13. S. H. Davis and L. M. Hocking, Spreading and imbibition of viscous liquid on a porous base, Phys. Fluids 12, 1646 (2000).
  14. R. F. Mangel, Jr. and E. Baer, The evaporation of water drops from a “Teflon” surface, Chem. Eng. Sci. 17, 705 (1962).
  15. R. G. Picknett and R. Bexon, The evaporation of sessile or pendant drops in still air, J. Colloid Interface Sci. 61, 336 (1977).
  16. K. S. Birdi, D. T. Vu, and A. Winter, A study of the evaporation rates of small water drops placed on a solid surface, J. Phys. Chem. 93, 3702 (1989).
  17. C. Bourgès-Monnier and M. E. R. Shanahan, Influence of evaporation on contact angle, Langmuir 11, 2820 (1995).
  18. R. G. Larson, Transport and deposition patterns in drying sessile droplets, AIChE J. 60, 1538 (2014).
  19. R. D. Deegan, O. Bakajin, T. F. Dupont, G. Huber, S. R. Nagel, and T. A. Witten, Capillary flow as the cause of ring stains from dried liquid drops, Nature (London) 389, 827 (1997).
  20. R. D. Deegan, O. Bakajin, T. F. Dupont, G. Huber, S. R. Nagel, and T. A. Witten, Contact line deposits in an evaporating drop, Phys. Rev. E 62, 756 (2000).
  21. R. D. Deegan, Pattern formation in drying drops, Phys. Rev. E 61, 475 (2000).
  22. H. Hu and R. G. Larson, Evaporation of a sessile droplet on a substrate, J. Phys. Chem. B 106, 1334 (2002).
  23. H. Hu and R. G. Larson, Analysis of the microfluid flow in an evaporating sessile droplet, Langmuir 21, 3963 (2005).
  24. H. Hu and R. G. Larson, Analysis of the effects of Marangoni stresses on the microflow in an evaporating sessile droplet, Langmuir 21, 3972 (2005).
  25. H. Hu and R. G. Larson, Marangoni effect reverses coffee-ring depositions, J. Phys. Chem. B 110, 7090 (2006).
  26. E. Sultan, A. Boudaoud, and M. Ben Amar, Diffusion-limited evaporation of thin polar liquid films, J. Eng. Math. 50, 209 (2004).
  27. E. Sultan, A. Boudaoud, and M. Ben Amar, Evaporation of a thin film: Diffusion of the vapour and Marangoni instabilities, J. Fluid Mech. 543, 183 (2005).
  28. N. J. Shirtcliffe and M. I. Newton, Analysis of droplet evaporation on a superhydrophobic surface, Langmuir 21, 937 (2005).
  29. Y. O. Popov, Evaporative deposition patterns: Spatial dimensions of the deposit, Phys. Rev. E 71, 036313 (2005).
  30. G. J. Dunn, S. K. Wilson, B. R. Duffy, S. David, and K. Sefiane, A mathematical model for the evaporation of a thin sessile liquid droplet: Comparison between experiment and theory, Colloids Surf. A 323, 50 (2008).
  31. G. J. Dunn, S. K. Wilson, B. R. Duffy, S. David, and K. Sefiane, The strong influence of substrate conductivity on droplet evaporation, J. Fluid Mech. 623, 329 (2009).
  32. R. Zheng, A study of the evaporative deposition process: Pipes and truncated transport dynamics, Eur. Phys. J. E 29, 205 (2009).
  33. J. Freed-Brown, Evaporative deposition in receding drops, Soft Matter 10, 9506 (2014).
  34. J. M. Stauber, S. K. Wilson, B. R. Duffy, and K. Sefiane, On the lifetimes of evaporating droplets, J. Fluid Mech. 744, R2 (2014).
  35. J. M. Stauber, S. K. Wilson, B. R. Duffy, and K. Sefiane, Evaporation of droplets on strongly hydrophobic substrates, Langmuir 31, 3653 (2015).
  36. J. M. Stauber, S. K. Wilson, B. R. Duffy, and K. Sefiane, On the lifetimes of evaporating droplets with related initial and receding contact angles, Phys. Fluids 27, 122101 (2015).
  37. A. W. Wray, D. T. Papageorgiou, R. V. Craster, K. Sefiane, and O. K. Matar, Electrostatic suppression of the “coffee stain effect”, Langmuir 30, 5849 (2014).
  38. A. W. Wray, B. R. Duffy, and S. K. Wilson, Competitive evaporation of multiple sessile droplets, J. Fluid Mech. 884, A45 (2020).
  39. A. W. Wray, P. S. Wray, B. R. Duffy, and S. K. Wilson, Contact-line deposits from multiple evaporating droplets, Phys. Rev. Fluids 6, 073604 (2021).
  40. X. Du and R. D. Deegan, Ring formation on an inclined surface, J. Fluid Mech. 775, R3 (2015).
  41. S. J. Jafari Kang, V. Vandadi, J. D. Felske, and H. Masoud, Alternative mechanism for coffee-ring deposition based on active role of free surface, Phys. Rev. E 94, 063104 (2016).
  42. F. Boulogne, F. Ingremeau, and H. A. Stone, Coffee-stain growth dynamics on dry and wet surfaces, J. Phys.: Condens. Matter 29, 074001 (2017).
  43. F. Boulogne and B. Dollet, Convective evaporation of vertical films, Soft Matter 14, 1665 (2018).
  44. F. G. H. Schofield, S. K. Wilson, D. Pritchard, and K. Sefiane, The lifetimes of evaporating sessile droplets are significantly extended by strong thermal effects, J. Fluid Mech. 851, 231 (2018).
  45. F. G. H. Schofield, A. W. Wray, D. Pritchard, and S. K. Wilson, The shielding effect extends the lifetimes of two-dimensional sessile droplets, J. Eng. Math. 120, 89 (2020).
  46. H.-M. D'Ambrosio, T. Colosimo, B. R. Duffy, S. K. Wilson, L. Yang, C. D. Bain, and D. E. Walker, Evaporation of a thin droplet in a shallow well: Theory and experiment, J. Fluid Mech. 927, A43 (2021).
  47. H.-M. D'Ambrosio, S. K. Wilson, A. W. Wray, and B. R. Duffy, The effect of the spatial variation of the evaporative flux on the deposition from a thin sessile droplet, J. Fluid Mech. 970, A1 (2023).
  48. H.-M. D'Ambrosio, S. K. Wilson, A. W. Wray, and B. R. Duffy, Effect of gravity-induced shape change on the diffusion-limited evaporation of thin sessile and pendant droplets, Phys. Rev. E 111, 045107 (2025).
  49. H.-M. D'Ambrosio, A. W. Wray, and S. K. Wilson, The effect of contact-line motion on the deposition of particles from an evaporating sessile droplet, J. Fluid Mech. 1034, A22 (2026).
  50. H. Masoud, P. D. Howell, and H. A. Stone, Evaporation of multiple droplets, J. Fluid Mech. 927, R4 (2021).
  51. M. R. Moore, D. Vella, and J. M. Oliver, The nascent coffee ring: How solute diffusion counters advection, J. Fluid Mech. 920, A54 (2021).
  52. M. R. Moore, D. Vella, and J. M. Oliver, The nascent coffee ring with arbitrary droplet contact set: An asymptotic analysis, J. Fluid Mech. 940, A38 (2022).
  53. C. Seyfert, J. Rodríguez-Rodríguez, D. Lohse, and Á. G. Marín, Stability of respiratory-like droplets under evaporation, Phys. Rev. Fluids 7, 023603 (2022).
  54. M. R. Moore and A. W. Wray, Gravitational effects on coffee-ring formation during the evaporation of sessile droplets, J. Fluid Mech. 967, A26 (2023).
  55. A. W. Wray and M. R. Moore, Evaporation of non-circular droplets, J. Fluid Mech. 961, A11 (2023).
  56. A. W. Wray and M. R. Moore, High-order asymptotic methods provide accurate, analytic solutions to intractable potential problems, Sci. Rep. 14, 4225 (2024).
  57. A. W. Wray and M. R. Moore, A novel asymptotically consistent approximation for integral evaporation from a spherical cap droplet, J. Eng. Math. 146, 4 (2024).
  58. E. Yariv, Lifetime of evaporating two-dimensional sessile droplets, Phys. Rev. E 107, L063101 (2023).
  59. A. K. Erdem, F. Denner, and L. Biancofiore, Numerical analysis of the dispersion and deposition of particles in evaporating sessile droplets, Langmuir 40, 13428 (2024).
  60. D. Jeon, G. Oh, S. Mun, J. S. Park, and B. M. Weon, Acceleration of sessile droplet evaporation by wind, Phys. Fluids 37, 072130 (2025).
  61. Y. Mei and X. Zhou, Evaporation dynamics of wavy sessile droplets, J. Fluid Mech. 1010, A38 (2025).
  62. J. E. O'Brien, K. L. Maki, X. Li, and M. J. Schertzer, Quantifying evaporative-driven flow inside drying droplets with pinned contact lines, J. Eng. Math. 150, 9 (2025).
  63. V. M. Starov, S. R. Kostvintsev, V. D. Sobolev, M. G. Velarde, and S. A. Zhdanov, Spreading of liquid drops over dry porous layers: Complete wetting case, J. Colloid Interface Sci. 252, 397 (2002).
  64. T. Pham and S. M. Kumar, Imbibition and evaporation of droplets of colloidal suspensions on permeable substrates, Phys. Rev. Fluids 4, 034004 (2019).
  65. L. Espín and S. M. Kumar, Droplet spreading and absorption on rough, permeable substrates, J. Fluid Mech. 784, 465 (2015).
  66. J. P. Burelbach, S. G. Bankoff, and S. H. Davis, Nonlinear stability of evaporating/condensing liquid films, J. Fluid Mech. 195, 463 (1988).
  67. N. Murisic and L. Kondic, On evaporation of sessile drops with moving contact lines, J. Fluid Mech. 679, 219 (2011).
  68. C. Larsson and S. Kumar, Comparison of one-sided and diffusion-limited evaporation models for thin liquid droplets, J. Fluid Mech. 976, A25 (2023).
  69. P. He and B. Derby, Controlling coffee ring formation during drying of inkjet printed 2D inks, Adv. Mater. Interfaces 4, 1700944 (2017).
  70. V. Dugas, J. Broutin, and E. Souteyrand, Droplet evaporation study applied to DNA chip manufacturing, Langmuir 21, 9130 (2005).
  71. Y. J. P. Carreón, M. Ríos-Ramírez, R. E. Moctezuma, and J. González-Gutiérrez, Texture analysis of protein deposits produced by droplet evaporation, Sci. Rep. 8, 9580 (2018).
  72. M. Layani, M. Gruchko, O. Milo, I. Balberg, D. Azulay, and S. Magdassi, Transparent conductive coatings by printing coffee ring arrays obtained at room temperature, ACS Nano 3, 3537 (2009).
  73. K. Sefiane, Patterns from drying drops, Adv. Colloid Interface Sci. 206, 372 (2014).
  74. D. Mampallil and H. B. Eral, A review on suppression and utilization of the coffee-ring effect, Adv. Colloid Interface Sci. 252, 38 (2018).
  75. M. Parsa, S. Harmand, and K. Sefiane, Mechanisms of pattern formation from dried sessile drops, Adv. Colloid Interface Sci. 254, 22 (2018).
  76. D. Lohse, Fundamental fluid dynamics challenges in inkjet printing, Annu. Rev. Fluid Mech. 54, 349 (2022).
  77. S. P. Thampi and M. G. Basavaraj, Drying drops of colloidal dispersions, Annu. Rev. Chem. Biomol. Eng. 14, 53 (2023).
  78. Á. G. Marín, H. Gelderblom, D. Lohse, and J. H. Snoeijer, Order-to-disorder transition in ring-shaped colloidal stains, Phys. Rev. Lett. 107, 085502 (2011).
  79. Á. G. Marín, H. Gelderblom, D. Lohse, and J. H. Snoeijer, Rush-hour in evaporating coffee drops, Phys. Fluids 23, 091111 (2011).
  80. T. A. H. Nguyen, A. V. Nguyen, M. A. Hampton, Z. P. Xu, L. Huang, and V. Rudolph, Theoretical and experimental analysis of droplet evaporation on solid surfaces, Chem. Eng. Sci. 69, 522 (2012).
  81. D. P. Siregar, J. G. M. Kuerten, and C. W. M. Van Der Geld, Numerical simulation of the drying of inkjet-printed droplets, J. Colloid Interface Sci. 392, 388 (2013).
  82. X. Man and M. Doi, Ring to mountain transition in deposition pattern of drying droplets, Phys. Rev. Lett. 116, 066101 (2016).
  83. A. Zigelman and O. Manor, A model for pattern deposition from an evaporating solution subject to contact angle hysteresis and finite solubility, Soft Matter 12, 5693 (2016).
  84. A. Zigelman and O. Manor, The deposition of colloidal particles from a sessile drop of a volatile suspension subject to particle adsorption and coagulation, J. Colloid Interface Sci. 509, 195 (2018).
  85. T. Pham and S. Kumar, Drying of droplets of colloidal suspensions on rough substrates, Langmuir 33, 10061 (2017).
  86. M. R. Moore, H.-M. D'Ambrosio, and A. W. Wray, Making a deposit: The role of substrate adsorption in coffee-ring formation in sessile evaporating droplets, J. Fluid Mech. 1025, A29 (2025).
  87. R. T. van Gaalen, C. Diddens, H. M. A. Wijshoff, and J. G. M. Kuerten, Marangoni circulation in evaporating droplets in the presence of soluble surfactants, J. Colloid Interface Sci. 584, 622 (2021).
  88. R. T. van Gaalen, C. Diddens, D. P. Siregar, H. M. A. Wijshoff, and J. G. M. Kuerten, Absorption of surfactant-laden droplets into porous media: A numerical study, J. Colloid Interface Sci. 597, 149 (2021).
  89. R. T. van Gaalen, H. M. A. Wijshoff, J. G. M. Kuerten, and C. Diddens, Competition between thermal and surfactant-induced Marangoni flow in evaporating sessile droplets, J. Colloid Interface Sci. 622, 892 (2022).
  90. X. Yang, Z. Jiang, P. Lyu, Z. Ding, and X. Man, Deposition pattern of drying droplets, Commun. Theor. Phys. 73, 047601 (2021).
  91. A. Matavž, U. Uršič, J. Močivnik, D. Richter, M. Humar, S. Čopar, B. Malič, and V. Bobnar, From coffee stains to uniform deposits: Significance of the contact-line mobility, J. Colloid Interface Sci. 608, 1718 (2022).
  92. O. Schumacher, C. Ates, M. Börnhorst, R. Koch, and P. Stephan, Deposit formation from evaporating urea-water droplets on substrates of different wettability, J. Colloid Interface Sci. 634, 1 (2023).
  93. N. C. J. Coombs, J. E. Sprittles, and M. V. Chubynsky, Colloidal deposits from evaporating sessile droplets: Coffee ring versus surface capture, Phys. Rev. Fluids 9, 064304 (2024).
  94. N. C. J. Coombs, M. V. Chubynsky, and J. E. Sprittles, Colloidal deposits from evaporating sessile droplets: A computationally efficient framework for predicting the final deposit shape, Phys. Rev. E 110, 064607 (2024).
  95. H.-M. D'Ambrosio, A. W. Wray, and S. K. Wilson, The effect of particle-substrate adsorption on the deposition of particles from a thin evaporating sessile droplet, J. Eng. Math. 151, 1 (2025).
  96. R. Dou, T. Wang, Y. Guo, and B. Derby, Ink-jet printing of zirconia: Coffee staining and line stability, J. Am. Ceram. Soc. 94, 3787 (2011).
  97. R. Dou and B. Derby, Formation of coffee stains on porous surfaces, Langmuir 28, 5331 (2012).
  98. F. Boulogne, F. Ingremeau, J. Dervaux, L. Limat, and H. A. Stone, Homogeneous deposition of particles by absorption on hydrogels, Europhys. Lett. 112, 48004 (2015).
  99. F. Boulogne, F. Ingremeau, L. Limat, and H. A. Stone, Tuning the receding contact angle on hydrogels by addition of particles, Langmuir 32, 5573 (2016).
  100. A. Nilghaz, L. Zhang, and W. Shen, Coffee stains on paper, Chem. Eng. Sci. 129, 34 (2015).
  101. K. N. Al-Milaji, R. L. Hadimani, S. Gupta, V. K. Pecharsky, and H. Zhao, Inkjet printing of magnetic particles toward anisotropic magnetic properties, Sci. Rep. 9, 16261 (2019).
  102. Y. Hwang, S. Kim, C. Lee, S. Kwak, G. Lee, and J. Kim, Monte Carlo simulation of the coffee-ring effect on porous papers, Theor. Comput. Fluid Dyn. 37, 627 (2023).
  103. M. Pack, H. Hu, D.-O. Kim, X. Yang, and Y. Sun, Colloidal drop deposition on porous substrates: Competition among particle motion, evaporation, and infiltration, Langmuir 31, 7953 (2015).
  104. W. Li, W. Ji, D. Lan, K. Wu, and Y. Wang, Absorption induced ordered ring and inner network structures on a nanoporous substrate, RSC Adv. 10, 22595 (2020).
  105. A. Kumar, D. Sen, A. Das, and J. Bahadur, Pattern of an evaporated colloidal droplet on a porous membrane dictated by competitive processes of flow and absorption, Langmuir 38, 7121 (2022).
  106. M. G. Wijburg, S. Wang, and A. A. Darhuber, Transport and evaporation of aqueous co-solvent solutions in thin porous media, Colloids Surf. A 656, 130268 (2023).
  107. S. Wang and A. A. Darhuber, A numerical model for the transport and drying of solutions in thin porous media–coffee-stain effect and solute ring formation, Colloids Surf. A 682, 132839 (2024).
  108. J. Niu, L. Qi, H. Lian, J. Luo, R. Zhang, and X. Chao, Revisiting the inhomogeneity in drop-on-demand printing of graphene: An effective route for overcoming the coffee-ring effect, Surf. Interfaces 46, 104036 (2024).
  109. H. Gelderblom, H. A. Stone, and J. H. Snoeijer, Stokes flow in a drop evaporating from a liquid subphase, Phys. Fluids 25, 102102 (2013).
  110. T. Nguyen, M. Hampton, and A. Nguyen, Evaporation of nanoparticle droplets on smooth hydrophobic surfaces: The inner coffee ring deposits, J. Phys. Chem. C 117, 4707 (2013).
  111. A. Trybala, A. Okoye, S. Semenov, H. Agogo, R. G. Rubio, F. Ortega, and V. M. Starov, Evaporation kinetics of sessile droplets of aqueous suspensions of inorganic nanoparticles, J. Colloid Interface Sci. 403, 49 (2013).
  112. J. R. Moffat, K. Sefiane, and M. E. R. Shanahan, Effect of TiO2 nanoparticles on contact line stick-slip behavior of volatile drops, J. Phys. Chem. B 113, 8860 (2009).
  113. M. E. R. Shanahan and K. Sefiane, Kinetics of triple line motion during evaporation, in Contact Angle, Wettability and Adhesion, edited by L. K. Mittal (CRC Press, London, UK, 2009), Vol. 6, pp. 19–31.
  114. D. Orejon, K. Sefiane, and M. E. R. Shanahan, Stick–slip of evaporating droplets: Substrate hydrophobicity and nanoparticle concentration, Langmuir 27, 12834 (2011).
  115. J. Eggers and L. M. Pismen, Nonlocal description of evaporating drops, Phys. Fluids 22, 112101 (2010).
  116. G. G. Wells, É. Ruiz-Gutiérrez, Y. Le Lirzin, A. Nourry, B. V. Orme, M. Pradas, and R. Ledesma-Aguilar, Snap evaporation of droplets on smooth topographies, Nat. Commun. 9, 1380 (2018).
  117. S. Whitaker, Flow in porous media I: A theoretical derivation of Darcy's law, Transp. Porous Media 1, 3 (1986).
  118. O. E. Jensen and J. B. Grotberg, The spreading of heat or soluble surfactant along a thin liquid film, Phys. Fluids 5, 58 (1993).
  119. S. K. Wilson and B. R. Duffy, Mathematical models for the evaporation of sessile droplets, in Drying of Complex Fluid Drops: Fundamentals and Applications, edited by D. Brutin and K. Sefiane (The Royal Society of Chemistry, London, UK, 2022), pp. 47–67.
  120. MATHEMATICA, Version 13.0.0 (Wolfram Research, Inc., Champaign, IL 2025), www.wolfram.com/mathematica.
  121. D. Craig, The simultaneous evaporation and imbibition of a droplet on a porous substrate, Ph.D. thesis, University of Strathclyde, 2026.

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Phys. Rev. Fluids 11, 093601– Published 14 September, 2026
  • Received 7 November 2025
  • Accepted 22 July 2026
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