2.1.1. Flow model

As described in, for example, Deegan et al. (Reference Deegan, Bakajin, Dupont, Huber, Nagel and Witten2000), Freed-Brown (Reference Freed-Brown2015) and Moore et al. (Reference Moore, Vella and Oliver2021), since the droplet is assumed to be thin, to leading order the fluid flows within the drop according to the lubrication equations given by

(2.4a–c)\begin{equation} \frac{1}{t_{f}}\,\frac{\partial h}{\partial t} + \boldsymbol{\nabla}\boldsymbol{\cdot}\left(h\bar{\boldsymbol{u}}\right) ={-} E, \quad \bar{\boldsymbol{u}} ={-}\frac{h^{2}}{3}\,\boldsymbol{\nabla} p, \quad p ={-}\frac{1}{{{Ca}}}\,\nabla^{2}h, \end{equation}

for $(x,y)\in \varOmega$, $0< t<1$, where $\boldsymbol {\nabla } = (\partial /\partial x,\partial /\partial y)^{\textrm {T}}$. Here, $\bar {\boldsymbol {u}}(x,y,t)$ is the leading-order depth-averaged velocity of the droplet, the leading-order pressure $p(x,y,t)$ is independent of $\hat {z}$, and the capillary number is

(2.5)\begin{equation} {{Ca}} = \frac{U^{*}\mu^{*}}{\sigma^{*}\delta^{3}}. \end{equation}

Equations (2.4) are supplemented by boundary conditions that require the droplet thickness to vanish and there to be no flux of liquid through the pinned contact line, so that

(2.6a,b)\begin{equation} h = h\bar{\boldsymbol{u}}\boldsymbol{\cdot}\boldsymbol{n} = 0 \quad \mathrm{on} \ \partial\varOmega, \end{equation}

where $\boldsymbol {n}$ is the outward pointing unit normal to $\partial \varOmega$, while we must prescribe the initial droplet profile, $h_{0}$ say, so that

(2.7)\begin{equation} h(x,y,0) = h_{0}(x,y) \quad \mathrm{for} \ (x,y) \in \varOmega. \end{equation}

We note that, in the absence of suitable regularization (such as imposing a Navier slip condition, rather than no slip on the substrate), a local analysis (along the lines of that described in, for example, Saxton Reference Saxton2016) reveals that the contact angle is unbounded at the contact line for both the kinetic and diffusive models. Fortunately, these effects are localized, so we neglect them in the present analysis.

2.1.2. Solute model

As we have assumed that the solute is inert and dilute, its transport is governed by the competing effects of advection and diffusion; the strength of this competition is expressed by the solutal Péclet number

(2.8)\begin{equation} {{Pe}} = \frac{U^{*}R^{*}}{D^{*}}, \end{equation}

where $D^{*}$ is the solutal diffusion coefficient. Since the droplet is thin, the diffusive flux across the droplet thickness has typical size $\phi _{{init}}^{*}D^{*}/\delta ^{2}R^{*2}$, while advection and diffusion parallel to the substrate have typical size $\phi _{{init}}^{*}U^{*}/R^{*}$. Provided that $\delta ^{2}\,{{Pe}}\ll 1$, therefore, diffusion across the droplet thickness dominates, and the solute concentration $\phi$ is independent of $\hat {z}$ to leading order. Hence, as described in detail elsewhere (see, for example, Wray et al. Reference Wray, Papageorgiou, Craster, Sefiane and Matar2014; Pham & Kumar Reference Pham and Kumar2017; Moore et al. Reference Moore, Vella and Oliver2021), $\phi (x,y,t)$ satisfies

(2.9)\begin{equation} \frac{1}{t_{f}}\,\frac{\partial}{\partial t}(h\phi) + \boldsymbol{\nabla}\boldsymbol{\cdot}\left(h\phi\bar{\boldsymbol{u}} - \frac{1}{{{Pe}}}\,h\,\boldsymbol{\nabla}\phi\right) = 0, \end{equation}

for $(x,y)\in \varOmega$ and $0< t<1$, which is to be solved subject to the boundary condition of no solute flux through the contact line, given by

(2.10)\begin{equation} \left(h\phi\bar{\boldsymbol{u}} - \frac{1}{{{Pe}}}\,h\,\boldsymbol{\nabla}\phi\right)\boldsymbol{\cdot}\boldsymbol{n} = 0 \quad \mathrm{for} \ (x,y) \in \partial\varOmega, \end{equation}

together with the initial condition

(2.11)\begin{equation} \phi(x,y,0) = 1 \quad \mathrm{for} \ (x,y)\in\varOmega. \end{equation}

We reiterate that the reduced model is valid so long as $\delta ^{2}\,{{Pe}}\ll 1$. For thicker droplets with $\delta ^{2}\,{{Pe}}$ of order unity or larger, other effects such as the capture and transport of solute along the droplet free surface may be relevant (Kang et al. Reference Kang, Vandadi, Felske and Masoud2016).

2.1.3. Evaporation models

The appropriate evaporation model depends on the chemical and thermal properties of the liquid and substrate, as well as properties of the surrounding air, such as the ambient humidity and the ease with which vapour can be transported away from the droplet. It is the purpose of this paper not to determine what is the most appropriate evaporation model, but rather to investigate the formation of the nascent coffee ring in different regimes. For this reason, we will concentrate on two illustrative cases by considering the kinetic and diffusive evaporative models, as in Moore et al. (Reference Moore, Vella and Oliver2021). These models are often considered to be the two limiting cases that bracket most realistic evaporation scenarios (Murisic & Kondic Reference Murisic and Kondic2011).

As described in detail by Murisic & Kondic (Reference Murisic and Kondic2011), in a kinetic evaporation model, evaporation is limited by the liquid phase alone, and the dimensionless evaporative flux in such cases is well approximated by $E = 1/(1 + h/\mathcal {K})$, where $\mathcal {K}$ is a parameter that depends on the thermodynamic properties of the system. For water evaporating on silicon, Murisic & Kondic (Reference Murisic and Kondic2011) find that $\mathcal {K} \approx 10$, and since $h\lesssim 1$, we can therefore reasonably take the evaporative flux to be uniform across the drop, setting

(2.12)\begin{equation} E = 1 \quad \text{for all} \ (x,y)\in\varOmega. \end{equation}

We note that a constant evaporative flux may also occur in other situations where the evaporation model is not diffusion-limited, for example, for water droplets evaporating on a hydrogel bath (Boulogne, Ingremeau & Stone Reference Boulogne, Ingremeau and Stone2016).

In a diffusive evaporative model, evaporation is limited by the transport of the liquid vapour away from the air–liquid interface. Following, for example, Deegan et al. (Reference Deegan, Bakajin, Dupont, Huber, Nagel and Witten2000) and Hu & Larson (Reference Hu and Larson2002), we assume that this process is quasi-steady and dominated by diffusion, so that the dimensionless vapour concentration $c$ satisfies the mixed boundary value problem

(2.13) \begin{equation} \left. \begin{aligned} \displaystyle{\nabla^{2}c} & = 0, \quad \text{in } z>0, \\ \displaystyle{c(x,y,0)} & = 1, \quad \text{for } (x,y) \in \varOmega, \\ \displaystyle{\frac{\partial c}{\partial z}(x,y,0)} & = 0, \quad \text{for } (x,y)\notin \varOmega, \\ \displaystyle{c} & \rightarrow 0, \quad \text{as } x^{2}+y^{2}+z^{2}\rightarrow\infty. \end{aligned} \right\} \end{equation}

Here, we have used the fact that the droplet is thin to linearize the boundary conditions onto the substrate, $z = 0$. The dimensionless evaporative flux is then given by

(2.14)\begin{equation} E ={-}\frac{\partial c}{\partial z}(x,y,0), \quad \text{for} \ (x,y)\in\varOmega. \end{equation}

2.3.1. Local behaviour of the flow model

We can use the contact line coordinate system to determine the local behaviour of the droplet free surface profile, evaporative flux and liquid flow. These results will be of vital importance in determining the correct asymptotic structure as $\varepsilon \rightarrow 0$ in § 3. We introduce the contact angle

(2.22)\begin{equation} \theta_{c}(s,t) = \lim_{n\rightarrow0^{+}}\frac{\partial h}{\partial n}(s,n,t), \end{equation}

which we assume exists and is non-zero for $0< t<1$, so that the droplet contact angle decays linearly with time according to (2.16). We therefore write $\theta _{c}(s,t) = (1-t)\,\psi (s)$, where $\psi (s)$ is the initial contact angle profile. We note that the non-axisymmetry of the droplet may mean that $\psi$ varies with $s$ (see, for example, the variations in contact angle for polygonal drops in Sáenz et al. Reference Sáenz, Wray, Che, Matar, Valluri, Kim and Sefiane2017, and the case study of elliptical droplets in § 4). The contact angle acts as a degree of freedom in the sense that it is determined globally rather than locally. We note that we have abused notation in the sense that the contact angle introduced here is for the leading-order droplet profile in the small capillary number limit, as governed by (2.19), rather than that for the original model, governed by (2.4), (2.6) and (2.7) – the latter does not exist in the absence of regularization, as mentioned above.

The local behaviour of the liquid flow depends strongly on the chosen evaporative model. For a kinetic evaporative flux, with $E = 1$, a local analysis of (2.17) and (2.20), and using the fact that $h\sim \theta _{c}(s,t)n$ as $n\rightarrow 0^{+}$, implies that, close to the contact line,

(2.23a,b)\begin{equation} \bar{u}_{s} \sim \frac{-3n}{\theta_{c}(s,t)^{2}}\,\frac{\partial}{\partial s}\theta_{c}(s,t), \quad \bar{u}_{n} \sim \frac{-1}{\theta_{c}(s,t)}\ \text{as} \ n\rightarrow0^{+}. \end{equation}

For a diffusive evaporative flux, the change from a Dirichlet to a Neumann boundary condition in (2.13) dictates that

(2.24)\begin{equation} E(s,n) \sim \frac{\chi(s)}{\sqrt{n}} \quad \text{as} \ n\rightarrow0^{+}, \end{equation}

where $\chi (s)$ is a degree of freedom. Then, by a similar analysis, we can show that the local behaviour of the liquid velocity at the contact line is given by

(2.25a,b)\begin{equation} \bar{u}_{s} \sim \frac{4\theta_{c}(s,t)^{2}}{3}\,\frac{\partial}{\partial s}\left(\frac{\chi(s)}{\theta_{c}(s,t)^{3}}\right)\sqrt{n}, \quad \bar{u}_{n} \sim \frac{-2\chi(s)}{\theta_{c}(s,t)}\,\frac{1}{\sqrt{n}} \text{ as } n\rightarrow0^{+}. \end{equation}

It is clear that the singularity in the diffusive evaporative flux drives a stronger normal velocity close to the contact line than the kinetic evaporative flux. Moreover, while in both cases the local velocity behaviour depends on the shape of the free boundary at the contact line – with the degree of freedom $\theta _{c}(s,t)$ being independent of the evaporation model – the coefficient of the normal velocity singularity in the diffusive regime also depends on the coefficient $\chi (s)$ of the evaporative flux singularity. The latter behaviour was noted by Sáenz et al. (Reference Sáenz, Wray, Che, Matar, Valluri, Kim and Sefiane2017) to have a strong effect on the local coffee-ring profile.

2.3.2. Local formulation of the solute problem and the integrated mass variable

In the region in which the $(s,n)$ coordinate system is well-defined, the solute transport equation (2.9) can be written as

(2.26)\begin{equation} \frac{1}{t_{f}}\,\frac{\partial}{\partial t}(a h\phi) + \frac{\partial}{\partial s}\left(h\phi\bar{u}_{s} - \frac{\varepsilon h}{a}\,\frac{\partial\phi}{\partial s}\right) + \frac{\partial}{\partial n}\left(ah\phi\bar{u}_{n}-\varepsilon ah\,\frac{\partial\phi}{\partial n}\right) = 0, \end{equation}

where the scale factor $a = 1+\kappa (s)\,n$, while the no-flux boundary condition (2.10) becomes

(2.27)\begin{equation} h\phi\bar{u}_{n}-\varepsilon h\,\frac{\partial\phi}{\partial n} = 0 \quad \text{on} \ n = 0. \end{equation}

To facilitate our matched asymptotic analysis, we will find it convenient to introduce the integrated mass variable, defined by

(2.28)\begin{equation} \mathcal{M}(s,n,t) = \int_{0}^{n}a(s,\nu)\,\phi(s,\nu,t)\,h(s,\nu,t)\,{\rm d}\nu. \end{equation}

If we integrate (2.26) in the normal direction from $0$ to $n$ and apply (2.27), then we see that the integrated mass variable satisfies

(2.29)\begin{equation} \frac{1}{t_{f}}\,\frac{\partial\mathcal{M}}{\partial t} + \frac{\partial}{\partial s}\int_{0}^{n} \left( h\phi\bar{u}_{s} - \frac{\varepsilon h}{a}\,\frac{\partial\phi}{\partial s}\right) {\rm d}\nu + ah\phi\bar{u}_{n} - \varepsilon ah\,\frac{\partial\phi}{\partial n} = 0, \end{equation}

while, by the initial condition (2.11), we have

(2.30)\begin{equation} \mathcal{M}(s,n,0) = \alpha \int_{0}^{n}a(s,\nu)\,H(s,\nu)\,{\rm d}\nu, \end{equation}

where $\alpha$ and $H$ are given by (2.18) and (2.19), respectively.

3.7.1. Kinetic regime

The simplified geometry allows us to evaluate the leading-order flow solution explicitly. In the kinetic evaporative regime, (2.15)–(2.20) give

(3.42a–e)\begin{equation} H = \frac{1-r^{2}}{4}, \quad P = P_{0} - \frac{48}{1-r^{2}}, \quad \bar{u}_{r} = \frac{{\rm \pi} r}{4(1-t)}, \quad t_{f} = \frac{1}{\rm \pi}, \quad \alpha = \frac{8}{\rm \pi}. \end{equation}

Here, $P_{0}$ is an arbitrary constant and $\bar {u}_{r}$ is the radial velocity. (We note that there is a slight difference between (3.42a–e) and the corresponding results in Moore et al. (Reference Moore, Vella and Oliver2021); this is due to the fact that the droplet aspect ratio $\delta$ in that paper is defined so that it contains an additional $2/{\rm \pi}$.)

Now, noting that for an axisymmetric droplet, $s = \theta$, $n = 1-r$, $\kappa = -1$ and $\bar {u}_{n} = -\bar {u}_{r}$, we can use (3.42a–e) to determine the leading-order solute concentration in each region, finding from (3.1a,b), (3.10) and (3.24),

(3.43)$$\begin{gather} \phi_{0} = \frac{1}{\sqrt{1-t}}\left(\frac{1-\sqrt{1-t}\,r^{2}}{1-r^{2}}\right), \end{gather}$$

(3.44)$$\begin{gather}\hat{\phi}_{0} = \frac{{\rm \pi}^{2}}{64(1-t)^{3}}\left[1-\sqrt{1-t}-\frac{t}{2}\right]\exp\left(\frac{-{\rm \pi}(1-r)}{4\varepsilon(1-t)}\right), \end{gather}$$

(3.45)$$\begin{gather}\mathring{\phi}_{0} = \exp\left(\frac{-{\rm \pi}(1-r)}{4\varepsilon(1-t)}\right)\left[\frac{{\rm \pi}^{2}}{64\varepsilon}\,\frac{(1-\sqrt{1-t}-t/2)}{(1-t)^{3}} +\frac{\rm \pi}{8}\,\frac{(1-\sqrt{1-t})}{(1-t)^{3/2}}\,{Ei}\left(\frac{{\rm \pi}(1-r)}{4\varepsilon(1-t)}\right)\right]. \end{gather}$$

Combining these expressions and evaluating the overlap contributions, we find that an additive composite expansion for the solute concentration is given by

(3.46)\begin{align} \phi_{{comp}} & = \phi_{0}(r,t) + \frac{1}{\varepsilon}\,\mathring{\phi}_{0}(r,t) + \frac{1}{\varepsilon^{2}}\hat{\phi}_{0}(r,t) - \frac{(1-\sqrt{1-t})}{2\sqrt{1-t}}\,\frac{1}{1-r} \nonumber\\ & \quad -\frac{1}{\varepsilon^{2}}\,\frac{{\rm \pi}^{2}(1-\sqrt{1-t}-t/2)}{64(1-t)^{3}}\exp\left(-\frac{{\rm \pi}(1-r)}{4\varepsilon(1-t)}\right) \nonumber\\ & \quad -\frac{1}{\varepsilon}\,\frac{\rm \pi}{8}\,\frac{(1-\sqrt{1-t})}{(1-t)^{3/2}}\exp\left(-\frac{{\rm \pi}(1-r)}{4\varepsilon(1-t)}\right)\log\left(\frac{{\rm \pi}(1-r)}{4\varepsilon(1-t)}\right). \end{align}

It is worth noting that the composite solution presented here is valid only to $O(1/\varepsilon ^{2})$ in the inner region, while the composite solution derived by Moore et al. (Reference Moore, Vella and Oliver2021) is valid to $O(1)$.

3.7.2. Diffusive regime

Similarly, in the diffusive regime, (2.15)–(2.20) give

(3.47) \begin{align} \left. \begin{aligned} H & = \frac{1-r^{2}}{4}, \quad \bar{u}_{r} = \frac{1}{r(1-t)}\left(\frac{1}{\sqrt{1-r^{2}}} - (1-r^{2})\right), \quad t_{f} = \frac{1}{4}, \quad \alpha = \frac{8}{\rm \pi}, \\ P & = P_{0} - \frac{384}{\rm \pi}\left[\frac{1}{\sqrt{1-r^{2}}} + \frac{1}{3(1-r^{2})^{3/2}} - \frac{1}{2}\log\left(\frac{1-\sqrt{1-r^{2}}}{1+\sqrt{1-r^{2}}}\right) + \log\left(\frac{r}{\sqrt{1-r^{2}}}\right)\right], \end{aligned} \right\} \end{align}

where again, $P_{0}$ is an arbitrary constant. The leading-order solute concentrations in the outer, inner and intermediate regions are then given by

(3.48)\begin{align} \phi_{0} & = \frac{1}{(1-t)^{1/4}\sqrt{1-r^{2}}}\left[1-(1-t)^{3/4}(1-(1-r^{2})^{3/2})\right]^{1/3}, \end{align}

(3.49)\begin{align} \hat{\phi}_{0} & = \frac{1}{24(1-t)^{5}}\left(1-(1-t)^{3/4}\right)^{4/3}\exp\left(-\frac{\sqrt{2(1-r)}}{\varepsilon(1-t)}\right), \end{align}

(3.50)\begin{align} \mathring{\phi}_{0} & = \exp\left(-\frac{\sqrt{2(1-r)}}{\varepsilon(1-t)}\right)\left[\frac{1}{24\varepsilon^{3}(1-t)^{5}}\left(1-(1-t)^{3/4}\right)^{4/3} \right. \nonumber\\ & \quad\left. + \frac{1}{(1-t)^{5/4}}\left(1-(1-t)^{3/4}\right)^{1/3}{Ei}\left(\frac{\sqrt{2(1-r)}}{\varepsilon(1-t)}\right)\right]. \end{align}

We can use these to form an additive composite expansion, given by

(3.51)\begin{align} \phi_{{comp}} & = \phi_{0}(r,t) + \frac{1}{\varepsilon}\,\mathring{\phi}_{0}(r,t) + \frac{1}{\varepsilon^{4}}\,\hat{\phi}_{0}(r,t) - \frac{1}{\sqrt{2}(1-t)^{1/4}\sqrt{1-r}}\left(1-(1-t)^{3/4}\right)^{1/3} \nonumber\\ & \quad -\frac{1}{\varepsilon^{4}}\,\frac{1}{24(1-t)^{5}}\left(1-(1-t)^{3/4}\right)^{4/3}\exp\left(-\frac{\sqrt{2(1-r)}}{\varepsilon(1-t)}\right) \nonumber\\ & \quad -\frac{1}{\varepsilon}\,\frac{1}{(1-t)^{5/4}}\left(1-(1-t)^{3/4}\right)^{1/3}\exp\left(-\frac{\sqrt{2(1-r)}}{\varepsilon(1-t)}\right)\log\left(\frac{\sqrt{2(1-r)}}{\varepsilon(1-t)}\right). \end{align}

3.7.3. Comparisons to numerical results

To check the validity of our asymptotic analysis, we compare profiles of the solute mass $m = h\phi _{{comp}}$ against a numerical solution of (2.9)–(2.11). While the axisymmetry greatly facilitates the numerical solution, the thinness of the boundary layer as discussed in § 3.2 means that care has to be taken with resolution close to the contact line: we use the numerical scheme validated in Moore et al. (Reference Moore, Vella and Oliver2021).

The results are shown for ${{Pe}} = 200$ in figure 2. In the figure, one can clearly see the transport of the solute mass from the droplet bulk towards the contact line as the droplet evaporates, leading to ring formation in the boundary layer. Moreover, it is evident that the asymptotics do an excellent job of capturing the dynamics, particularly as $t$ increases. This gives us confidence in using our asymptotic results to consider the nascent coffee-ring characteristics for more complicated geometries, where numerical solutions of (2.9)–(2.11) are much more computationally challenging (see, for example, Sáenz et al. Reference Sáenz, Wray, Che, Matar, Valluri, Kim and Sefiane2017).

Figure 2.

Profiles of the solute mass as an axisymmetric droplet evaporates under (a,b) a kinetic evaporative flux, and (c,d) a diffusive evaporative flux with ${{Pe}} = 200$. In each panel, the bold black curve represents the initial mass profile $m(r) = 2(1-r^{2})/{\rm \pi}$. Also shown are plots at time intervals of 0.1 up to $t = 0.9$ in which solid blue curves represent the results from the numerical solution of (2.9)–(2.11), and dashed red curves show the composite mass profiles $m = h\phi _{{comp}}$. Panels (b,d) display a doubly-logarithmic plot of the mass profile near the contact line.

4.3.1. Kinetic evaporation

For the kinetic evaporative model, we combine (2.23a,b) and (4.9) to show that

(4.14)\begin{equation} \bar{u}_{\mu} \sim \frac{{\rm \pi}(1+a)^{2}}{4(1-t)}[1+(2a+a^{2})\sin^{2}\nu]^{{-}1/2}, \end{equation}

as $\mu ^{*}-\mu \rightarrow 0$. To illustrate the effects of the droplet geometry on the velocity profile, we display the normal velocity close to the contact line (4.14) as a function of the elliptical polar angle $\nu$ in figure 3(a) for different eccentricities. We have scaled $(1-t)\bar {u}_{\mu }$ by $\delta _{0}/\delta _{a}$ so that for each curve, the droplet has the same initial volume and perimeter, with just the eccentricity of the ellipse changing (cf. (4.5)). The axisymmetric case is illustrated by the purple line. Initially, as we increase $a$, it is clear that the velocity is increased in the regions with higher curvature, namely close to the semi-major axis, while being diminished close to the semi-minor axis. However, since $\delta _{a}$ increases with $a$, eventually the velocity is lower than the equivalent axisymmetric problem around the whole contact line. Nevertheless, the velocity is still relatively stronger along the more highly curved parts of the boundary, and this disparity grows as $a$ increases, that is, for more eccentric ellipses, the stronger the flow in the direction of the semi-major axis in comparison to that along the semi-minor axis.

Figure 3.

The angular dependence of the normal velocity close to the contact line for (a) a kinetic evaporative model as given by (4.14), and (b) a diffusive evaporative model as given by (4.15). In each panel, we vary the shape of the elliptical contact set by changing $a$, but scale in such a way that each droplet has the same initial volume and perimeter. For illustration, we show the different droplet footprints in the inset to (b). Displayed are the axisymmetric case, $a = 0$ ($e(0) = 0$, purple), $a = 0.1$ ($e(0.1) \approx 0.417$, blue), $a = 1$ ($e(1) \approx 0.866$, green) and $a = 3$ ($e(3) \approx 0.968$, yellow).

4.3.2. Diffusive evaporation

Similarly, for a diffusive evaporative model, combining (2.25), (4.3), (4.9) and (4.11) gives

(4.15)\begin{equation} \bar{u}_{\mu} \sim \frac{{\rm \pi}(1+a)^{5/2}}{2^{3/2}K(e(a))(1-t)}\,\frac{1}{1+(2a+a^2)\sin^{2}\nu}(\mu^{*}-\mu)^{{-}1/2}, \end{equation}

as $\mu ^{*}-\mu \rightarrow 0$. We plot $(1-t)\sqrt {\mu ^{*}-\mu }\,\delta _{0}\bar {u}_{\mu }/\delta _{a}$ for different ellipse eccentricities in figure 3(b), where we again scale appropriately to fix the initial droplet volume and perimeter. We see very similar behaviour to the kinetic regime: for all ellipse eccentricities, the normal velocity is stronger along the semi-major axis than the semi-minor axis, with this effect being amplified as $a$ increases. For $a$ small, the velocity is also stronger along the semi-major axis compared to the equivalent axisymmetric droplet, but as $a$ gets larger, the velocity is weaker everywhere around the contact line.

4.4.1. Kinetic evaporation

We can use the local expansions for the outer solute concentration, (3.6), the contact angle, (4.8), and the velocity, (4.14), together with (4.3)–(4.13) to express the accumulated mass flux into the contact line for a kinetic evaporative model as

(4.16)\begin{equation} \mathcal{M}_{0}(\nu,0,t) = \frac{1}{{\rm \pi}(1+a)}[1+(2a+a^{2})\sin^{2}\nu]^{1/2}\int_{0}^{t}B(\nu,\tau)\,{\rm d}\tau. \end{equation}

To investigate how $\mathcal {M}_{0}$ varies with $\nu$ and $a$, however, we need to determine $B(\nu,t)$, which must be found numerically. The solution procedure is therefore as follows. First, we solve (2.20) numerically to determine $\bar {\boldsymbol {u}}$ from (2.17). Since (2.20) is independent of $t$, we need do this only once. We can then use the velocity profile to solve the leading-order outer solute problem, (3.1a,b) – again numerically – which then allows us to determine $B(\nu,t)$. Finally, we can use $B(\nu,t)$ to evaluate the integral in (4.16). The details of the numerical methodologies for each step are recorded in Appendix A.

We display the resulting accumulated mass flux into the contact line (4.16) in figures 4(a,b). In figure 4(a), we display $\mathcal {M}_{0}$ as a function of the elliptical polar angle $\nu$ at different stages of the evaporation for $a = 1$. Clearly, even at small times, there is a larger mass flux accumulating along the parts of the contact line with higher curvature, consistent with the results of Freed-Brown (Reference Freed-Brown2015) and Sáenz et al. (Reference Sáenz, Wray, Che, Matar, Valluri, Kim and Sefiane2017). This disparity increases as $t$ increases. We display the two extremes by plotting $\mathcal {M}_{0}$ along each semi-major axis for various values of the eccentricity $a$ in figure 4(b). As previously, we have used the scalings (4.5) to compare droplets that have the same initial volume and contact line perimeter. It is apparent from the figure that as $a$ increases, the accumulated mass flux along the semi-major axis increases compared to the axisymmetric case, while the accumulated mass flux along the semi-minor axis decreases compared to the same. These behaviours are accentuated further as the ellipse becomes more eccentric. It is worth stressing that this is in spite of the normal velocity along $\partial \varOmega$ being smaller everywhere than in the equivalent axisymmetric problem for large values of $a$ (as seen in figure 3a).

Figure 4.

The angular dependence of the accumulated mass flux at the contact line under (a,b) a kinetic evaporative flux, as given by (4.16), and (c,d) a diffusive evaporative flux, as given by (4.17). (a,c) Profiles of $\mathcal {M}_{0}(\nu,0,t)$ at $t = 0.1$ (purple), $t = 0.2$ (blue), $t = 0.4$ (green) and $t = 0.8$ (yellow) for an ellipse with $a = 1$ ($e(1) \approx 0.866$). (b,d) The accumulated mass flux along the semi-major (solid) and semi-minor (dashed) axes for different ellipse eccentricities: $a = 0$ (black), $a = 0.1$ ($e(0.1) \approx 0.417$, purple), $a = 1$ ($e(1) \approx 0.866$, blue) and $a = 3$ ($e(3) \approx 0.968$, green). In each of (b,d), we have scaled appropriately so that the droplets have the same perimeter and initial volume.

4.4.2. Diffusive evaporation

The equivalent expression for the accumulated mass flux under a diffusive evaporative model is determined from (4.9), (4.11) and (4.15) to be

(4.17)\begin{equation} \mathcal{M}_{0}(\nu,0,t) = \frac{1}{\rm \pi}\sqrt{\frac{1}{2(1+a)}}[1+(2a+a^{2})\sin^{2}\nu]^{{-}1/4}\int_{0}^{t}B_{d}(\nu,\tau)\,{\rm d}\tau. \end{equation}

Again, we must determine $B_{d}(\nu,\tau )$ numerically by solving the leading-order outer solute advection problem for $\phi _{0}$. The procedure is identical to the that in the kinetic regime, and we display the resulting profiles of $\mathcal {M}_{0}(\nu,0,t)$ in figures 4(c,d). The broad behaviour of accentuated mass accumulation along the semi-major axis is very similar to the kinetic regime, although it is notable that $\mathcal {M}_{0}(\nu,0,t)$ is larger at earlier times in the diffusive regime. While under both evaporative models all of the mass will be driven to the contact line at the dryout time (since diffusion is a lower-order effect in the outer region), the stronger evaporative flux in the diffusive regime means that mass accumulates faster at the contact line in this regime.

4.5.1. Kinetic evaporation

To isolate the role of geometry-induced fluid flow, we first consider the kinetic evaporative model. We display in figure 5 mass profiles along the semi-minor and semi-major axes for an ellipse with $a = 1$ ($e(1) = 0.866$). The droplet has been chosen to have the same initial volume and perimeter as the axisymmetric droplet depicted in figure 2, so that the equivalent Péclet number is ${{Pe}}_{a} \approx 35$. In the figure, the bold black lines represent the initial mass profile, while the blue lines represent the mass profile evolution along the semi-major axis (figure 5a,b) and semi-minor axis (figure 5c,d). As the droplet evaporates, we can see clearly the formation of the nascent coffee ring along both axes, with a characteristic thin, sharp peak growing close to the pinned contact line. It is noticeable that the peak along the semi-major axis is larger than that along the semi-minor axis. At $90\,\%$ of the drying time, the coffee-ring peak is approximately three times higher along the semi-major axis than the semi-minor axis. The coffee-ring effect is also enhanced when compared to the axisymmetric case: the peak along the semi-major axis is approximately $1.4$ times larger than that for the equivalent axisymmetric droplet shown in figure 2, with the peak on the semi-minor axis approximately $2.2$ times smaller.

Figure 5.

Profiles of the solute mass $m = h\phi _{{comp}}$ along the semi-major (a,b) and semi-minor (c,d) axes of an elliptical droplet with $a = 1$ ($e(1) = 0.866$) evaporating under a kinetic flux for ${{Pe}} \approx 35$ (so that the droplet has the same initial volume and perimeter as the axisymmetric droplet shown in figure 2). In each panel, the bold black curve represents the initial mass profile, while plots at time intervals of 0.1 up to $t = 0.9$ are shown as solid blue curves. Panels (b,d) display a doubly-logarithmic plot of the mass profiles near the contact line, where we see the formation of the nascent coffee ring as $t$ increases. It is notable that the coffee-ring effect is much stronger along the semi-major axis due to the strong accumulated mass flux in this region.

It is also worth noting that the peak along the semi-minor axis is slightly further from the contact line compared to the semi-major axis. For example (and accounting for the rescalings in (4.5)), at $90\,\%$ of the drying time, the peak along the semi-major axis is at $2-x \approx 5.8\times 10^{-4}$, while along the semi-minor axis it is at $1-y \approx 1.3\times 10^{-3}$. For reference, the axisymmetric peak location is comparable to the peak on the semi-major axis, with $1-r \approx 6.5\times 10^{-4}$ in figure 2.

We further illustrate the effect of ellipse geometry in figures 6(a,b), where we show the variation of the maximum coffee-ring peak (figure 6a) and its distance from the contact line (figure 6b) along each semi-axis with ellipse eccentricity. Each ellipse has the same initial volume and perimeter, with the Péclet number of the axisymmetric drop taken to be ${{Pe}}_{0} = 200$. The equivalent Péclet numbers for the elliptical drops are then calculated from (4.5). The results are shown at $90\,\%$ of the drying time.

Figure 6.

Variation with eccentricity of the dimensionless peak coffee-ring height (a,c), and its dimensionless distance from the contact line (b,d), at $90\,\%$ of the drying time. Each ellipse has the same initial perimeter and volume. We display results for both the kinetic (a,b) and diffusive (c,d) regimes. In each panel, the red squares indicate results for the semi-minor axis, while the blue circles indicate results for the semi-major axis. In the insets to (a,c), we display the ratio of the current peak height at $e(a)$ to the equivalent axisymmetric droplet to highlight how increasing the eccentricity of the droplet leads to a local maximum in the peak height.

Initially, as the ellipse eccentricity is increased, the peak height along the semi-major axis increases, until reaching a maximum at $a \approx 1.5$ ($e(a) \approx 0.917$), where it is approximately $1.46$ times larger than the equivalent axisymmetric droplet (cf. the inset to figure 6a). For larger eccentricities, the peak height then decreases again, although remaining higher than the axisymmetric case for the eccentricities displayed. It is worth noting that as $e(a)$ approaches unity, the assumptions made in deriving the model begin to break down, with the aspect ratio of the droplet contact set becoming larger than $O(1)$. Moreover, the Péclet number as given by (4.5) decreases with $a$, so the existence of a maximum is not unexpected.

On the other hand, the peak height along the semi-minor axis decreases monotonically as the eccentricity of the ellipse increases. Indeed, as $e(a)$ gets closer to unity, the rate of decrease of the height gets faster. For $a = 3$ ($e(a) \approx 0.968$), the peak height along the semi-minor axis has decreased by almost a factor of $7$.

Along the semi-minor axis, the location of the coffee-ring peak moves radially inwards away from the pinned contact line of the droplet. Correspondingly, according to the analysis of § 3.5, as the ellipse eccentricity increases, the thickness of the nascent coffee ring as measured by the full-width at half-maximum increases. Along the semi-major axis, the distance of the peak from the contact line location decreases slightly at very small eccentricities, before again becoming larger than the equivalent axisymmetric case as $e(a)$ approaches unity. The effect is noticeably weaker than that along the semi-minor axis.

Thus, in summary, on the semi-minor axis, as the eccentricity of the ellipse increases, the coffee ring gets progressively shallower and wider compared to an axisymmetric droplet of the same initial volume and perimeter. On the semi-major axis, the coffee ring initially becomes narrower and higher, before transitioning to a ring that is wider and higher than the equivalent axisymmetric droplet.

These behaviours can be probed further by considering the similarity analysis of § 3.5. The local similarity profile is given by (3.28), which becomes

(4.18)\begin{align} \frac{\hat{m}_{0}}{{{Pe}}_{t}\,\mathcal{M}_{0}(\nu,0,t)} = \frac{{\rm \pi}^{2}(1+a)^{4}}{16}\,\frac{N}{[1+(2a+a^{2})\sin^{2}\nu]}\exp\left(\frac{-{\rm \pi}(1+a)^{2}N}{4[1+(2a+a^{2})\sin^{2}\nu]^{1/2}}\right), \end{align}

where $N = {{Pe}}_{t}\,n$, and $n$ is given by (4.3). The leading-order coffee-ring peak and its location are found from (3.29) to be

(4.19a,b)\begin{align} \frac{m_{{max}}}{{{Pe}}_{t}\,\mathcal{M}_{0}(\nu,0,t)} = \frac{{\rm \pi}(1+a)^{2}}{4\text{e}\,[1+(2a+a^{2})\sin^{2}\nu]^{1/2}}, \quad {{Pe}}_{t}\,n_{{max}} = \frac{4[1+(2a+a^{2})\sin^{2}\nu]^{1/2}}{{\rm \pi}(1+a)^{2}}. \end{align}

It is clear that the right-hand side of (4.19a) is monotonically decreasing in $\nu$. In particular, its value along $\nu = 0$ is $(1+a)^{2}$ times larger than that along $\nu = {\rm \pi}/2$. Thus the geometry-induced flow alone drives an enhanced coffee ring along the semi-major axis. Moreover, we note that since, for a fixed $a$, $\mathcal {M}_{0}(\nu,0,t)$ is a decreasing function of $\nu$ (cf. figure 3c), including the accumulated mass flux accentuates this effect further.

The peak location and hence the full-width at half-maximum (cf. (3.30)) are independent of the accumulated mass flux, so that their behaviour as functions of $\nu$ is governed purely by the local free surface profile. As is seen clearly from (4.19b), as $\nu$ increases, the ring peak is further from the contact line, leading to a thicker coffee ring. The scale factor in terms of distance from the contact line – and hence thickness of the ring – between the semi-major and semi-minor axes is given by $1/(1+a)^{2}$.

All of these features can be seen by plotting the similarity profile (4.18) for a droplet with $a = 1$ for different values of $\nu$, which is shown in figure 7(a): as $\nu$ increases, the mass profiles get progressively shallow and broader, with the location of the peak moving away progressively from the contact line at $n = 0$. The accumulated mass flux $\mathcal {M}_{0}(\nu,0,t)$ simply acts to accentuate the coffee-ring height, with the peak location and the full-width at half-maximum remaining unchanged.

Figure 7.

Similarity profiles of the nascent coffee ring for an elliptical drop with $a = 1$ ($e(1) = 0.866$) for (a) the kinetic evaporative regime as given by (4.18), and (b) the diffusive evaporative regime as given by (4.20).

A previous study by Freed-Brown (Reference Freed-Brown2015) demonstrated numerically that the mass flux of solute is stronger along the semi-major axis of a uniformly evaporating elliptical droplet. Here we have expanded upon this study, illustrating clearly that this increased mass flux combines with the effect of the local droplet profile, leading to an enhanced coffee ring along the more highly curved parts of the contact line. Moreover, this is for a constant evaporative flux, showing that this asymmetry in the solute distribution can be driven by geometry-induced flow alone, which may have useful applications in evaporative-driven patterning processes (Harris et al. Reference Harris, Hu, Conrad and Lewis2007).

4.5.2. Diffusive evaporation

If one also allows the evaporative flux to vary as a result of the droplet geometry, then the asymmetries in the nascent coffee ring become more exaggerated. To show this, we plot in figure 8 mass profiles along the semi-minor and semi-major axes for an ellipse with $a = 1$ ($e(1) = 0.866$) evaporating under a diffusive flux (the droplet again has the same initial volume and perimeter as that in figures 2 and 5). The coffee-ring effect is enhanced significantly by the contact line geometry: the peak of the ring at $90\,\%$ of the drying time is $\approx 13$ times larger along the semi-major axis than the semi-minor axis. Moreover, the peak is also approximately $1.1$ times larger than that in the equivalent axisymmetric diffusive problem.

Figure 8.

Profiles of the solute mass $m = h\phi _{{comp}}$ along the semi-major (a,b) and semi-minor (c,d) axes of an elliptical droplet with $a = 1$ ($e(1) = 0.866$) evaporating under a diffusive flux for ${{Pe}} \approx 35$. In each panel, the bold black curve represents the initial mass profile, while plots at time intervals of 0.1 up to $t = 0.9$ are shown as solid blue curves.

We illustrate how the properties of the nascent coffee ring vary with ellipse eccentricity in figures 6(c,d), where we show how the ring peak and its distance from the contact line change with $e(a)$ for a fixed initial droplet volume and perimeter. The results are displayed at $90\,\%$ of the drying time, with the Péclet number of the equivalent axisymmetric drop taken to be ${{Pe}}_{0} = 200$, and the corresponding parameters for the droplets with elliptical footprints evaluated from (4.5).

The results behave in a qualitatively similar manner to the kinetic regime. In particular, the peak height along the semi-major axis initially increases as we increase the eccentricity, before reaching a maximum. For the diffusive case, the maximum is at $a \approx 0.5$ ($e(a) \approx 0.745)$, with the peak approximately $20\,\%$ larger than the equivalent axisymmetric droplet (cf. the inset to figure 6c). As the eccentricity increases further, the peak height begins to decrease again, eventually decreasing below the axisymmetric peak. It is worth noting, however, that the width of the coffee ring is larger than in the equivalent axisymmetric droplet. Hence for larger eccentricities, we have a shallower but wider coffee ring, as necessitated by the overall increased accumulated mass flux into the contact line along the semi-major axis (cf. figure 4).

For a diffusively evaporating droplet, the similarity profile can be found from (3.39) to be given by

(4.20)$$\begin{gather} \frac{\hat{m}_{0}}{{{Pe}}_{t}^{2}\,\mathcal{M}_{0}(\nu,0,t)} = \frac{{\rm \pi}^{4}(1+a)^{10}}{48K[e(a)]^{4}}\,\frac{N_{d}}{(1+(2a+a^{2})\sin^{2}\nu)^{3}} \nonumber\\ \times \exp\left(\frac{-{\rm \pi}(1+a)^{5/2}\sqrt{N_{d}}}{K(e(a))\,[1+(2a+a^{2})\sin^{2}\nu]^{3/4}}\right), \end{gather}$$

where $N_{d} = {{Pe}}_{t}^{2}\,n$. The coffee-ring peak and its location are given by

(4.21)$$\begin{gather} \frac{m_{{max}}}{{{Pe}}_{t}^{2}\,\mathcal{M}_{0}(\nu,0,t)} = \frac{{\rm \pi}^{2}(1+a)^{5}}{6 \text{e}^{2}\,K[e(a)]^{2}}\,\frac{1}{[1+(2a+a^{2})\sin^{2}\nu]^{3/2}}, \end{gather}$$

(4.22)$$\begin{gather}{{Pe}}_{t}^{2}\,n_{{max}} = \frac{8K[e(a)]^{2}}{{\rm \pi}^{2}(1+a)^{5}}[1+(2a+a^{2})\sin^{2}\nu]^{3/2}. \end{gather}$$

In the diffusive regime, even when we discount the accumulated mass flux into the contact line, there is a more significant strengthening of the coffee-ring effect compared to the kinetic evaporative model. Along the semi-major axis of the ellipse, the peak height is $(1+a)^{3}$ larger, $1/(1+a)^{3}$ closer to the contact line, and $1/(1+a)^{3}$ thinner than along the semi-minor axis. This can be seen clearly in figure 7(b), where we plot the similarity profile (4.20) for different values of $\nu$. Generally speaking, even though there are variations with the angle-like variable $\nu$, the coffee ring in the kinetic regime is much more uniform than that in the diffusive regime.

4.7.1. Kinetic evaporation

Let us suppose that the critical solute concentration at which finite concentration effects are important is given by $\phi _{c}^{*}$. This value will depend upon the solute under consideration. For the purposes of illustration, we will assume that $\phi _{c}^{*} = 0.1$, while we will take indicative values of the initial concentration from Deegan et al. (Reference Deegan, Bakajin, Dupont, Huber, Nagel and Witten2000), with $\phi _{{init}}^{*}$ ranging from $10^{-6}$ to $10^{-2}$.

Now, according to our asymptotic analysis, the maximum value of the solute concentration, $\phi _{{max}}(t)$, occurs at the contact line. Thus we can evaluate the composite profile (3.25) on $\partial \varOmega$ to find that

(4.23)\begin{equation} \phi_{{max}}(\nu,t) = {{Pe}}^{2}\,\frac{\mathcal{M}_{0}(\nu,0,t)}{\theta_{c}(\nu,t)^{3}} + {{Pe}}\,\frac{\gamma\,B(\nu,t)}{\theta_{c}(\nu,t)}+ \lim_{\substack{(x,y)\rightarrow\partial\varOmega \\ n\rightarrow 0}} \left(\phi_{0}(x,y,t) - \frac{B(\nu,t)}{n}\right), \end{equation}

where $\gamma$ is the Euler–Mascheroni constant.

Note that $\phi _{{max}}$ depends on the elliptical polar angle; our previous analysis has indicated that the maximum is higher along the semi-major axis of the ellipse than along the semi-minor axis, and we expect this to translate directly into a reduced range of validity of the dilute model. To investigate this, in figure 9 we plot $\phi _{{max}}(\nu,t)$ as a function of time for an elliptical droplet with $a = 1$ and for different Péclet numbers. In each panel, the value of $\phi _{{max}}(\nu,t)$ along the semi-major axis is given by the dashed curve, while its value along the semi-minor axis is given by the dotted curve. For reference, the equivalent maximum concentration for an axisymmetric droplet of the same volume and perimeter is shown in each panel by the solid curve. Figures 9(a–c) show the results for ${{Pe}} = 200, 400, 800$, respectively, which are defined with respect to the axisymmetric droplet, and the corresponding Péclet number for the elliptical case can be found using (4.5). To help interpret the results, we have also included reference values of $\phi _{c} = \phi _{c}^{*}/\phi _{{init}}^{*}$ as the dashed red lines.

Figure 9.

The maximum solute concentration, $\phi _{{max}}(\nu,t)$, for an elliptical droplet with $a = 1$ ($e(a) = 0.866$) evaporating under a constant evaporative flux as given by (4.23). In each plot, the dashed curve displays the maximum concentration on the semi-major axis, $\phi _{{max}}(0,t)$, the dotted curve displays the concentration on the semi-minor axis, $\phi _{{max}}({\rm \pi} /2,t)$, and the solid curve shows the equivalent results for an axisymmetric droplet of the same volume and perimeter. The results are shown for (a) ${{Pe}} = 200$, (b) ${{Pe}} = 400$, (c) ${{Pe}} = 800$, where the Péclet number is defined with respect to the axisymmetric droplet. In each panel, the horizontal dashed red lines denote particular values of the critical concentration $\phi _{c}$. For each curve, the dilute regime lies to the left of the intersection with the red lines.

As is seen clearly in figure 9, our intuition is correct: for a given value of $\phi _{c}$, the solute concentration along the semi-major axis reaches the critical value much earlier than along the semi-minor axis. Moreover, compared to an axisymmetric droplet of the same volume and contact line length, the critical solute concentration is reached sooner along the semi-major axis and later on the semi-minor axis. For a fixed ${{Pe}}$, the time window over which the dilute regime remains valid increases as $\phi _{c}$ increases, while for a fixed $\phi _{c}$, the time window decreases as ${{Pe}}$ increases.

To take a concrete example, let us consider $\phi _{c} = 10^{5}$, which corresponds to a solute that is initially extremely dilute compared to the critical concentration. For ${{Pe}} = 200$, the dilute regime is valid for $\approx 80\,\%$ of the drying time for an axisymmetric drop, while for an elliptical drop, the dilute regime breaks down after $\approx 76\,\%$ of the drying time along the semi-major axis, and after $\approx 87\,\%$ of the drying time along the semi-minor axis. On the other hand, for ${{Pe}} = 800$, the dilute regime is valid for $\approx 60\,\%$ of the drying time for an axisymmetric drop, while it breaks down after $\approx 55\,\%$ of the drying time along the semi-major axis, and after $\approx 75\,\%$ of the drying time along the semi-minor axis, for the equivalent elliptical drop.

It is of note that in each of these cases, the dilute regime takes up a large percentage of the total drying time, indicating that the asymptotic analysis that we have presented here gives a very good account of the nascent coffee-ring formation and, in particular, allows us to predict the solute mass distribution within the droplet when finite concentration effects start to become relevant. Moreover, the solute mass profiles presented in, for example, § 4.5, would be apt for comparison to experimental measurements of transient coffee-ring profiles for an elliptical droplet evaporating under a kinetic evaporative flux.

However, it is worth noting that the time windows over which the dilute regime is valid do significantly reduce as $\phi _{c}$ decreases. For ${{Pe}} = 200$ and $\phi _{c} = 10$, breakdown occurs after $\approx 8\,\%$ of the drying time for the axisymmetric droplet, $\approx 5\,\%$ of the drying time along the semi-major axis of the equivalent elliptical droplet, and $\approx 18\,\%$ of the drying time along the semi-minor axis. Nevertheless, our analysis is appropriate for the time window before breakdown, and provides the initial conditions for the regime in which finite concentration effects are relevant, so it is likely to be an important consideration in understanding the characteristics of the final coffee ring.

We should also note that it may be that finite concentration effects are extremely localized in the model, so that, for example, even in problems where they are relevant close to the highly curved parts of the contact line, the dilute model may still give an excellent description of the coffee-ring dynamics for other parts of the boundary. This seems particularly reasonable given that the solute simply follows the streamlines in the droplet bulk (where the dilute regime is still valid), and the streamlines are independent of $t$. However, these comments do depend on the type of model chosen to incorporate finite concentration effects, and whether such a model causes non-local changes to the liquid flow. We do not seek to address these questions any further here.

4.7.2. Diffusive evaporation

By the composite solution (3.38), the maximum concentration in the diffusive evaporative regime is given by

(4.24) $$\begin{gather} \phi_{{max}}(\nu,t) = {{Pe}}^{4}\,\frac{64\chi(\nu)^{4}\,\mathcal{M}_{0}(\nu,0,t)}{3\theta_{c}(\nu,t)^{5}}+ {{Pe}}\,\frac{4\gamma\,\chi(\nu)\,B_{d}(\nu,t)}{\theta_{c}(\nu,t)}\nonumber\\ + \lim_{\substack{(x,y)\rightarrow\partial\varOmega\\ n\rightarrow 0}} \left(\phi_{0}(x,y,t) - \frac{B_{d}(\nu,t)}{\sqrt{n}}\right). \end{gather}$$

It is apparent immediately that this is two orders of magnitude larger than (4.23) and, accordingly, the time window over which the dilute regime is valid for this mode of evaporation is much smaller, as can be seen in figure 10. For ${{Pe}} = 200$ and $\phi _{c} = 10^{5}$, we see that breakdown occurs for an axisymmetric droplet at just $\approx 1\,\%$ of the drying time, with this value decreasing rapidly as $\phi _{c}$ decreases or ${{Pe}}$ increases. Breakdown along the semi-major axis of the equivalent ellipse occurs at a similar time, while there is an appreciable increase in the validity of the dilute model along the semi-minor axis to $\approx 23\,\%$ of the drying time. Again, this tails off sharply as $\phi _{c}$ decreases. Hence if finite concentration effects can be treated locally, then we may see a sizeable time frame over which the dilute model presented here gives an accurate description of the nascent coffee ring along parts of the contact line with smaller curvature. Overall, however, it is clear that we need to consider these effects much sooner in the diffusive evaporative flux regime.

Figure 10.

The maximum solute concentration, $\phi _{{max}}(\nu,t)$, for an elliptical droplet with $a = 1$ ($e(a) = 0.866$) evaporating under a diffusive flux as given by (4.24). In each plot, the dashed curve displays the maximum concentration on the semi-major axis, $\phi _{{max}}(0,t)$, the dotted curve displays the concentration on the semi-minor axis, $\phi _{{max}}({\rm \pi} /2,t)$, and the solid curve shows the equivalent results for an axisymmetric droplet of the same volume and perimeter. The results are shown for (a) ${{Pe}} = 200$, (b) ${{Pe}} = 400$, (c) ${{Pe}} = 800$, where the Péclet number is defined with respect to the axisymmetric droplet. In each panel, the horizontal dashed red lines denote particular values of the critical concentration $\phi _{c}$.

It is noticeable that there is marginal difference between the breakdown time along the semi-major axis and the equivalent axisymmetric droplet, and indeed some cases where it appears to be reduced along the semi-major axis compared to the axisymmetric case. Given the extremely small time frames under consideration, this is likely a combined effect of the numerical sensitivities in evaluating $\mathcal {M}_{0}(\nu,0,t)$ and $B_{d}(\nu,t)$, and the fact that the equivalent Péclet numbers are significantly smaller for an elliptical droplet (cf. (4.5)).