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.

Appendices

Appendix A: Simulations

We perform numerical simulations in COMSOL Multiphysics, rewriting (2.1) as a system of second-order equations as

$$\begin{aligned} d_a \varvec{v}_t + \nabla \cdot \varvec{\Gamma } = \mathbf {f}, \end{aligned}$$

(A.1)

where

$$\begin{aligned} d_a = \begin{pmatrix} 1&{}0&{}0 \\ 0&{}0&{}0 \\ 0&{}0&{}1 \end{pmatrix}, \quad \varvec{v} = \begin{pmatrix} H \\ P \\ Q \end{pmatrix}, \quad \varvec{\Gamma } = \begin{pmatrix} ({H^4}/{3Q}) \nabla P \\ \nabla H \\ \frac{1}{3}H^3 \nabla P - {Pe}^{-1}H \nabla (Q/H) \end{pmatrix}, \quad \mathbf {f} = \begin{pmatrix} 0 \\ P \\ 0 \end{pmatrix}, \end{aligned}$$

(A.2)

where $Q\equiv HM$. The computational domain is a square $[-L, L] \times [-L, L]$ around which a no-flux boundary condition $\varvec{n}\cdot {\varvec{\Gamma }} =0$ is applied, where $\varvec{n}$ is the outward-facing normal. Simulations are terminated before the boundary has any direct influence. We used COMSOL’s built-in mesh generator to create an unstructured triangular mesh on the computational domain and its default second-order elements to discretise (A.1, A.2). Convergence tests were undertaken to ensure that reported results were grid independent. The 2D scheme was also tested against a 1D finite-difference solver that used the method of lines and the Matlab time-stepping routine ode23t in an axisymmetric problem (see the solid and dotted lines in Fig. 5; both methods quickly regularise the initial pressure jump in (2.2).). We used the routines $\mathtt {g05zr}$ and $\mathtt {g05zs}$ in the NAG toolbox for MATLAB to generate realisations of the stationary Gaussian random field $\mathcal {G}$ with covariance (2.3) by circulant embedding [24]. Integrals in (2.4) were evaluated by defining $\mathcal {A} = \{(x,y):H(t,x,y)>1.01\eta \}$. We collected statistics from simulations initiated from multiple realisations of the initial viscosity field in order to characterise the distribution of outcomes.

Appendix B: Emulation

A GP emulator of the simulation outputs (implemented in the Python package GP_emu_UQSA²) was constructed following [19]. The simulator is modelled as a noisy regression problem with inputs $\mathbf {x}_i$ (representing model parameters in realisation i) leading to a noisy output $y_i$, of the form

$$\begin{aligned} y_i = f(\mathbf {x}_i) + \delta _i,\quad \delta _i\sim \mathcal {N}(0,r(\mathbf {x}_i)),\quad i=1,\ldots ,N. \end{aligned}$$

(B.1)

Here f represents the nonlinear model and r the heteroscedastic noise in the model (i.e. the parameter-dependent variability that is propagated forwards from the initial conditions to the final state). For datasets of each quantity of interest $\{y_i\}$ (i.e. multiple predictions of $A(t_\mathrm{f})$ or $D_y(t_\mathrm{f})$), the functions $f(\mathbf {x})$ and $r(\mathbf {x})$ must be learned from training data. GP priors are placed on f and r, using exponential covariance functions with hyperparameters (variances and correlation lengths) that are also learned from training data. Once determined, the model provides a posterior distribution for outputs $y^*_j$ at parameters $\mathbf {x}_j^*$ as a multivariate normal distribution. The latent variables ($r(\mathbf {x}_i)$) are determined using a most-likely approach [19]. The hyperparameters are determined using an expectation-maximisation algorithm, in which a GP is first trained on the simulation data $\{\mathbf {x}_i, y_i\}$ using a prediction for the mean of the variance at each point $\{ \mathbb {E}[r_i] \}$ (initially set to zero as unknown). This ‘mean GP’ is used to estimate the actual pointwise variance $\{ r_i \}$ at each $\mathbf {x}_i$, and a different ‘noise GP’ is then trained on the $\{\mathbf {x}_i, z_i\}$, where $z_i = \log (r_i)$, in order to obtain a smooth estimate of the mean of the noise $\{ \mathbb {E}[r_i] \}$, which allows the ‘mean GP’ to be retrained. This iterative procedure is repeated until convergence. In this problem, because the variation of the mean function is much smaller than the variance arising from the random initial conditions (unlike most machine-learning problems, where the variance of the signal typically exceeds that of the noise), learning the exact form of the mean function was not possible. To ensure that the ‘mean GP’ fitted the data well, a random sample from a noiseless GP was added to the data, preserving the noise levels; adding different samples preserved the results of fitting the noise.

Appendix C: Axisymmetric spreading in the uniform viscosity limit

Fig. 5

Axisymmetric drop spreading with uniform viscosity ($M=1$), with $V=\pi /2$ and $\eta =0.01$. The solid and dotted lines (which are almost indistinguishable) compare respectively the 2D COMSOL solver and a 1D finite difference code. The dashed and dot-dash lines compare respectively the leading-order approximation (4.5) and the refined approximation (C.4)

We revisit here the classical problem (4.1) with uniform viscosity ($M=1$), for which an initially axisymmetric drop remains axisymmetric, to illustrate how pressure gradients across the bulk of the drop influence spreading rates. Setting $H = H(r, T)$, where r is the radial coordinate, we match solutions in the inner and outer regions as before in the limit $\epsilon \rightarrow 0$. In the outer region we take $H\approx H_0+\epsilon H_{(1)}+\cdots$, $\mathbf {U}\approx \mathbf {U}_0+\epsilon \mathbf {U}_{(1)}+\cdots$, where $H_0$ and $\mathbf {U}_0$ are given by (4.2). The correction to the film thickness satisfies

$$\begin{aligned} \frac{a_T}{a} r = \frac{H_0^2}{3} \left( \frac{1}{r}\left( rH_{(1)r}\right) _r \right) _r, \end{aligned}$$

(C.1)

where we apply $H_{(1)}\rightarrow 0$ as $r\rightarrow a$, $H_{(1)r}\rightarrow 0$ as $r\rightarrow 0$ and $\int _0^a rH_{(1)}\,\mathrm {d}r=0$. Integrating, one finds that

$$\begin{aligned} H_{(1)r}=\frac{3\pi ^2 a^7 a_T}{16 V^2} \left[ \zeta r - \frac{\log \left( 1-(r^2/a^2)\right) }{r} \right] \end{aligned}$$

(C.2)

for some constant $\zeta$. Integrating the volume constraint by parts reveals that $\zeta =-2/a^2$. Near the contact line, it follows that

$$\begin{aligned} H_r^3\approx -\left( \frac{4V}{\pi a^3} \right) ^3 -9 \epsilon a_T \left( 2+\log \left( 1-(r/a)\right) + \log 2\right) + \cdots \end{aligned}$$

(C.3)

Matching with (4.4) and writing $a_t=\epsilon a_T$ leads to

$$\begin{aligned} \left( \frac{4V}{a^3 \pi }\right) ^3 = 9 a_t \left( \ln (1/\eta ) + \ln (a/2) + \tfrac{1}{3} \ln (3a_t) - 1 + 3^{2/3} \alpha \right) . \end{aligned}$$

(C.4)

Retaining the leading-order term as $\eta \rightarrow 0$ on the right-hand side of (C.4) recovers (4.5). The higher-order corrections are analogous to those derived (for slip) by for example [14] and other authors.

The corrections in (C.4) provide an important boost to the accuracy of this approximation. Figure 5 illustrates the point, showing an axisymmetric drop-spreading simulation for $\eta =0.01$ using (A.1) and comparing it to (4.5) and the refined prediction (C.4). The planar analogue of (C.4) is given in [10]; Fig. 2 of that paper illustrates much closer agreement between asymptotics and numerics once $\eta$ is reduced by a factor of 10 to 0.001. Our numerical scheme (for non-axisymmetric spreading) could not resolve such a thin precursor film with the resources available to us. Significantly, the higher-order corrections in (C.4) are not essential in order to capture the dominant effects of nonuniform viscosity such as the direction of drift (Fig. 2b) and the dependence of variance in drop area and drift on correlation length (Fig. 3).