Evaporation of non-circular droplets

Wray, Alexander; Moore, M. R.

doi:10.1017/jfm.2023.229

Cited by 7 publications

(8 citation statements)

References 65 publications

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“…It is well known that droplet geometry plays a strong role in the behaviour of the evaporative flux (Sáenz et al. 2017; Wray & Moore 2023) and the transient and final deposit profiles (Freed-Brown 2015; Sáenz et al. 2017; Moore et al.…”

Section: Summary and Discussionmentioning

confidence: 99%

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Gravitational effects on coffee-ring formation during the evaporation of sessile droplets

Moore

Wray

2023

J. Fluid Mech.

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We consider the role of gravity in solute transport when a thin droplet evaporates. Under the physically relevant assumptions that the contact line is pinned and the solutal Péclet number, ${Pe}$ , is large, we identify two asymptotic regimes that depend on the size of the Bond number, ${Bo}$ . When ${Bo} = O(1)$ as ${Pe}\rightarrow \infty$ , the asymptotic structure of solute transport follows directly from the surface-tension-dominated regime, whereby advection drives solute towards the contact line, only to be countered by local diffusive effects, leading to the formation of the famous ‘coffee ring.’ In the distinguished limit in which ${Bo} = O({Pe}^{4/3})$ as ${Pe}\rightarrow \infty$ , this interplay between advection and diffusion takes place alongside that between surface tension and gravity. In each regime, we perform a systematic asymptotic analysis of the solute transport and compare our predictions to numerical simulations. We identify the effect of gravity on the nascent coffee ring, providing quantitative predictions of the size, location and shape of the solute mass profile. In particular, for a fixed Péclet number, as the effect of gravity increases, the coffee ring is diminished in height and situated further from the contact line. Furthermore, for certain values of ${Bo}$ , ${Pe}$ and the evaporation time, a secondary peak may exist inside the classical coffee ring. The onset of this secondary peak is linked to the change in type of the critical point in the solute mass profile at the droplet centre. Both the onset and the peak characteristics are shown to be independent of ${Pe}$ .

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Section: Summary and Discussionmentioning

confidence: 99%

“…Such a model has been shown to accurately predict both the total evaporation rate (Hu & Larson 2002) and the pointwise evaporative flux (see, for example, Sáenz et al. 2017; Wray & Moore 2023) for a range of different droplet geometries.…”

Section: Problem Configurationmentioning

confidence: 99%

Gravitational effects on coffee-ring formation during the evaporation of sessile droplets

Moore

Wray

2023

J. Fluid Mech.

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“…In particular, as well as including additional physical effects, such as when the presence of the particles affects the flow within the droplet (see for example Kaplan & Mahadevan 2015) and/or particle adsorption and coagulation (see for example Zigelman & Manor 2018 a ), it would be interesting to investigate the effect of the spatial variation of the local evaporative flux in more complicated geometries such as, for example, a non-axisymmetric droplet (see for example Sáenz et al. 2017; Wray & Moore 2023), a droplet in a well (see for example Vlasko-Vlasov et al. 2020; D'Ambrosio et al.…”

Section: Discussionmentioning

confidence: 99%

“…For example, Popov (2005) and Zheng (2009) modelled the shape of a ring deposit, Tarasevich, Vodolazskaya & Isakova (2011), Vodolazskaya & Tarasevich (2011) and Kaplan & Mahadevan (2015) considered various situations in which the presence of the particles affects the flow within the droplet, Crivoi & Duan (2013) and Crivoi, Zhong & Duan (2015) investigated the effect of particle aggregation on the shape of the final deposit, Sáenz et al. (2017) and Wray & Moore (2023) studied non-axisymmetric contact-line deposits arising from non-axisymmetric droplets, and Wray, Duffy & Wilson (2020) and Wray et al. (2021) analysed non-axisymmetric contact-line deposits arising from multiple interacting droplets.…”

Section: Introductionmentioning

confidence: 99%

The effect of the spatial variation of the evaporative flux on the deposition from a thin sessile droplet

D'Ambrosio,

Wilson,

Wray

et al. 2023

J. Fluid Mech.

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A mathematical model for the effect of the spatial variation of the local evaporative flux on the evaporation of and deposition from a thin pinned particle-laden sessile droplet is formulated and solved. We then analyse the behaviour for a one-parameter family of local evaporative fluxes with the free parameter $n \, (>-1)$ that exhibits qualitatively different behaviours mimicking those that can be obtained by, for example, surrounding the droplet with a bath of fluid or using a mask with one or more holes in it to achieve a desired pattern of evaporation enhancement and/or suppression. We show that when $-1< n<1$ (including the special cases $n=-1/2$ of diffusion-limited evaporation into an unbounded atmosphere and $n=0$ of spatially uniform evaporation), all of the particles are eventually advected to the contact line, and so the final deposit is a ring deposit at the contact line, whereas when $n>1$ all of the particles are eventually advected to the centre of the droplet, and so the final deposit is at the centre of the droplet. In particular, the present work demonstrates that a singular (or even a non-zero) evaporative flux at the contact line is not an essential requirement for the formation of a ring deposit. In addition, we calculate the paths of the particles when diffusion is slower than both axial and radial advection, and show that in this regime all of the particles are captured by the descending free surface before eventually being deposited onto the substrate.

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“…However, despite over 150 years of attention, few exact closed-form solutions are known: only those for a circular disk 12 , 15 and an ellipse 16 (and solutions for non-flat sources 17 – 19 ). There have been attempts to provide solutions for more general planar shapes, including approximate formulations 20 , and asymptotic solutions for monochromatic source boundaries 21 . In addition, solutions have been determined for both the density and capacity for arrays of multiple circular sources 22 – 24 .…”

Section: Introductionmentioning

confidence: 99%

High-order asymptotic methods provide accurate, analytic solutions to intractable potential problems

Wray,

Moore

2024

Sci Rep

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The classical problem of determining the density and capacity of arrays of potential sources is studied. This corresponds to a wide variety of physical problems such as electrostatic capacitance, stress in elastostatics and the evaporation of fluid droplets. An asymptotic solution is derived that is shown to give excellent accuracy for arbitrary arrays of sources with non-circular footprints, including polygonal footprints. The solution is extensively validated against both experimental and numerical results. We illustrate the power of the solution by showcasing a variety of newly accessible classical problems that may be solved in a rapid, accurate manner.

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Evaporation of non-circular droplets

Cited by 7 publications

References 65 publications

Gravitational effects on coffee-ring formation during the evaporation of sessile droplets

Gravitational effects on coffee-ring formation during the evaporation of sessile droplets

The effect of the spatial variation of the evaporative flux on the deposition from a thin sessile droplet

High-order asymptotic methods provide accurate, analytic solutions to intractable potential problems

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