Abstract
We introduce a mathematical model with a meshfree numerical method to describe contactline motion in lubrication theory. We show how the model resolves the singularity at the contact line, and generates smooth profiles for an evolving, spreading droplet. The model describes well the physics of droplet spreading–including Tanner’s Law for the evolution of the contact line. The model can be configured to describe complete wetting or partial wetting, and we explore both cases numerically. In the case of partial wetting, the model also admits analytical solutions for the droplet profile, which we present here.
Article highlights

We formulate a mathematical model to regularize the contactline singularity for droplet spreading.

The model can be solved using a fast, accurate meshfree numerical method.

Numerical simulations confirm that the model describes the quantitative aspects of droplet spreading well.
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1 Introduction
When a droplet of fluid (surrounded by a gaseous atmosphere) is deposited on a substrate, it spreads until it reaches an equilibrium configuration. At equilibrium, the angle between the liquidgas interface and the solid surface is measured (conventionally, through the liquid), to yield the equilibrium contact angle \(\theta _{\mathrm {eq}}\). If the angle is less than \(90^\circ\), the substrate is deemed hydrophilic, whereas if the angle exceeds \(90^\circ\), the substrate is deemed hydrophobic [1]. Droplet spreading then describes the dynamic phase before the attainment of this equilibrium.
In droplet spreading, the point of contact between the substrate and the gasliquid interface is in motion. And yet this contradicts the classical noslip assumption in viscous fluid flow, which stipulates that there should be no relative motion between a substrate in contact with a fluid [2, 3]. The resolution of this paradox is that there is missing physics, and that on a sufficiently small scale, there is slip, the dynamics of which are governed by the interactions between the fluid molecules and the substrate molecules [4]. These molecularlevel interactions can be incorporated into a macroscopic fluid model via a socalled regularization technique. Broadly, there are three regularization techniques in the literature – the slip length [5], the precursor film [1], and the diffuse interface [6, 7]. Although methodologically distinct, these yield the same qualitative and quantitative results when used to model droplet dynamics. This consistency between the different approaches gives a solid justification for the general approach of model regularization.
The purpose of this article is to introduce a novel model regularization, albeit one in the spirit of those just described. The focus of the work is on mode regularization for the case of thinfilm flows (also called lubrication flows). For simplicity, we focus on twodimensional configurations (or equivalently, threedimensional axisymmetric configurations), however, the generalization to three dimensions is straightforward.
Thinfilm flows arise in the context of hydrophilic substrates, where the equilibrium shape of the droplet is such that the typical size of the droplet base R greatly exceeds the maximum droplet height \(h_0\), leading to a small parameter \(\epsilon =h_0/R\), with \(\epsilon \ll 1\). In this context, the Navier–Stokes equations reduce down to a single equation for the interface height (the socalled ThinFilm Equation). In this context also, the interface height is conventionally written as \(z=h(x,t)\). Thus, the coordinate x and is in the plane of the substrate, and the zcoordinate is orthogonal to the substrate (e.g. Fig. 1).
1.1 Motivation for this work
In the context of droplet spreading, the ThinFilm Equation inherits the contactline singularity from the full Navier–Stokes equations; the singularity can be regularized by introducing a slip length [5, 8], or a precursor film [1, 9]. Both these regularizations give accurate and consistent descriptions of contactline spreading [10], albeit with some drawbacks—the classical Navier sliplength model has a logarithmic stress singularity at the contact line. We note, however, that the stress singularity can be alleviated using the approach in Reference [11]. While as the precursorfilm model requires a precursor film to be present that extends indefinitely beyond the droplet core. Although physically, such a precursor film does exist, it has a very small thickness (10–100 nm [9]), meaning that such a small scale must be resolved in the model: in particular, the numerical grid size must be at least as small as the precursorfilm thickness [12]. Furthermore, the resulting equations are quite stiff numerically [13]. Although this approach is just about feasible for millimetrescale droplets, it may not be feasible for larger ones. Beyond the millimetre scale, an unphysically large precursorfilm thickness can be used in numerical investigations (and the results checked for robustness to changes in the value of the precursorfilm thickness). The Diffuse Interface Model has been proposed as a more general regularization of the contactline singularity problem, valid for the full Navier–Stokes equations beyond the lubrication limit [14]. The Diffuse Interface Model has been implemented for droplet spreading in the thinfilm lubrication approximation [6].
Our contribution in this article is to introduce a novel regularization technique similar in spirit to the Diffuse Interface Method—we formulate a theory of droplet spreading involving a smooth interface height \({\overline{h}}\), as well as a sharp interface height h, which interact via a convolution operator and an evolution equation. In a previous work [15], the idea behind this regularization was introduced in the context of thinfilm lubrication flows—the socalled Geometric Thin Film Equation. In the present article, we extend this previous work by introducing a particle method as a novel and highly accurate solution method for the Geometric Thin Film Equation. Also, Reference [15] was for complete wetting—in this work we extend the Geometric ThinFilm Equation to describe partial wetting as well.
The Geometric ThinFilm Equation can be viewed as special case of a mechanical model for energydissipation on general configuration spaces—the derivation of the general model involves methods from Geometric Mechanics such as Lie Derivatives and Momentum Maps [16]—hence the name Geometric ThinFilm Equation. The main advantage of this new method so far has been the nonstiff nature of the differential equations in the model, which leads to robust numerical simulation results. A second advantage (the main focus of the present work) is that the Geometric ThinFilm equation admits socalled particle solutions. These give rise to an efficient and accurate numerical method (the particle method) for solving the model equations.
The basis for the particle method lies in the structure of the Geometric ThinFilm Equation: the model includes a ‘smoothened’ freesurface height \({\overline{h}}(x,t)\) and a ‘sharp’ freesurface height h(x, t), related by convolution, \({\overline{h}}(x,t)=\varPhi *h(x,t)\), where \(\varPhi \ge 0\) is a filter function with a characteristic lengthscale \(\alpha\). The model admits a deltafunction solution for the sharp freesurface height \(h(x,t)=\sum _{i=1}^N w_i \delta (xx_i)\), where \(\delta\) is the Dirac delta function, and \(x_i(t)\) is the (timedependent) centre of the Delta function. The deltafunction centres \(\{x_i(t)\}_{i=1}^N\) satisfy a set of ordinary differential equations:
where \(V_i\) is a velocity function which can be derived from the Geometric ThinFilm Equation. Thus, the screened freesurface height admits a regular solution \({\overline{h}}(x,t)=\sum _{i=1}^N w_i \varPhi (xx_i)\). In this context, the weights \(w_i\) and the deltafunction centres \(x_i\) (with \(i\in \{1,2,\ldots ,N\}\)) can be viewed as pseudoparticles, which satisfy the firstorder dynamics (1). We will demonstrate in this article another advantage of this particlemethod: it is a meshfree method that automatically accumulates particles in regions where \(\partial _{xx}{\overline{h}}\) is large, thereby mimicking the effects of adaptive mesh refinement, with none of the computational overheads associated with that method. A final key advantage of the particle method is positivitypreservation: as \({\overline{h}}(x,t)=\sum _{i=1}^N w_i\varPhi (xx_i)\), with \(w_i\ge 0\) and \(\varPhi \ge 0\), the numerical values of \({\overline{h}}(x,t)\) are guaranteed never to be negative, hence the numerical method is manifestly positivitypreserving.
1.2 This work in the context of environmental fluid mechanics
Before introducing the method, we place the method in the context of Environmental Fluid Mechanics, focusing in particular on droplet spreading on plant leaves. In general, surfaces can be further classified as being (i) superhydrophobic (\(\theta _{\mathrm {eq}}>150^\circ\)), hydrophobic (\(90^\circ<\theta _{\mathrm {eq}}<150^\circ\)), hydrophilic (\(10^\circ<\theta _{\mathrm {eq}}<90^\circ\)), or superhydrophilic (\(\theta _{\mathrm {eq}}<10^\circ\)). Plant leaves display this wide range of contact angles. Superhydrophobic surfaces have been frequently found in wetland plants, where the superhydrophobic surface prevents a buildup of water on the leaves, which could otherwise promote the growth of harmful microorganisms and limit the gas exchange necessary for photosynthesis [17].
A particularly wellstudied plant with superhydrophobic a leaf surface is the Lotus plant (Nelumbo nucifera), with a contact angle of about \(160^\circ\) [17, 18]. The leaves of the Lotus plant also demonstrate a low contactangle hysteresis, such that water droplets roll off the leaf surface when at a low tilt angle (\(4^\circ\)). During rolling, contaminating particles are picked up by the water droplets, and are then removed with the droplets as they roll off [17]. The leaf structure of the Lotus plant has been studied using Scanning Electron Microscopy, and reveals a hierarchical microstructure made up of micronscale pillars (cell papillae) and a randomly covered by a smaller branchlike nanostructure [19] (the wax crystal, with microstructures \(\sim 100\,\mathrm {nm}\) in diameter) [20]. Such biological microsturctures are the inspiration for engineered superhydrophobic surfaces [20].
In contrast, plants with superhydrophilic leaf surfaces are often found in tropical regions [21]. A water droplet that spreads on a superhydrophilic surface will spread and form a wide, flat droplet—effectively a thin film. Evaporation in such films is more efficient than in a spherical droplet, due to the increase of the waterair interface. Thus, water evaporates from a superhydrophilic leaf much faster than that from a hydrophilic or superhydrophobic one, thereby keeping the leaf dry, reducing the accumulation of harmful microorganisms on the leaf surface, and increasing the gas exchange with the environment, for the purpose of photosynthesis [17]. A wellstudied hydrophilic plant is Ruellia devosiana, wherein the superhydrophilic property of the leaf surface is due both to the leaf microstructure, and to a secretion of surfactants by the leaf, which both promote spreading [22].
Other plants which exploit hydrophilicity are the carnivorous plants of the Nepenthes genus (e.g. Fig. 2), the perisotone of which is a fully wettable, waterlubricated anisotropic surface [23]. Insects landing on the peristone or rim of the plant effectively ‘aquaplane’ down the plant rim [23] before being captured by the viscoelstic fluid inside the pitcher [24].
We emphasize here that both the superhydrophobic and superhydrophilic plant surfaces are extreme cases. In a comprehensive investigation of 396 plant species out of 85 families growing in three different continents at various elevations (including 792 leaf surfaces total), only forty leaf surfaces (5.1% out of 792) were found in these extremes, including 24 superhydrophobic surfaces and 16 superhydrophilic surfaces [21]. Thus, most leaf surfaces lie inside these extremes. The dropletmodel introduced in this paper is most relevant to hydrophilic cases where the equilibrium contact angle is small. Hence, droplet spreading over the leaves of Ruellia devosiana should be well described by our model, but not droplet spreading over lotus leaves. However, we anticipate that future applications of the regularization technique introduced herein will be able to handle more such cases.
Finally, we note that several researchers have further investigated droplet spreading using lubrication theory for nonisothermal configurations, which have important applications in cooling—both environmental and industrial. Readers are referred to standard references [25, 26]; furthermore, it can be anticipated that the methods developed herein can be extended to such complex systems as part of future work.
1.3 Plan of the paper
In Sect. 2 we present the classical ThinFilm Equation as a model of droplet spreading. Certainly, this is a very wellestablished topic, however, we briefly summarize this topic here because it enables us to clearly mark out the point of departure of the present work. Thereafter, in Sect. 3 we introduce the Geometric ThinFilm Equation as a regularized model which enables contactline motion. In Sect. 4 we introduce the particle method for generating numerical solutions of the Geometric ThinFilm Equation. In Sect. 5 we present results for complete wetting. In Sect. 6 we demonstrate how the Geometric ThinFilm Equation can be extended to the case of partial wetting, and we present numerical results for that case also. Concluding remarks are given in Sect. 7.
2 Review of classical theory
In this section we review the classical theory of the Thin Film Equation, including the problem of the contactline singularity. The purpose of this review is to put the Geometric ThinFilm Equation into the context of the classical theory; the theoretical formulation of the Geometric ThinFilm Equation is therefore presented subsequently in Sect. 3.
2.1 Classical thinfilm equation
We review the derivation of the classical Thin Film Equation for a flow in two dimensions, with spatial coordinates x and z (we refer the reader Reference [27] for the details). The startingpoint is the kinematic condition valid on the free surface \(z=h(x,t)\):
The fluid flow is assumed to be incompressible, such that \(u_x+v_y=0\). The incompressiblity condition can be integrated once to give
Eqs. (2)–(3) can be combined to give
In the lubrication limit, the velocity u(x, z, t) satisfies the equations of Stokes flow, hence
where P is the fluid pressure and \(\mu\) is the constant dynamic viscosity. We integrate the first equation of the pair in (5) once with respect to z to obtain
The standard interfacial condition is that the viscous stress \(\partial u/\partial z\) should vanish on the free surface \(z=h(x,t)\). Thus, Eq. (6) becomes \(\partial u/\partial z=\mu ^{1}(\partial p/\partial x)(zh)\). Applying the noslip boundary condition
the uvelocity profile becomes
hence
The pressure P is identified with the Laplace pressure, \(P=\gamma \partial _{xx}h\), where \(\gamma\) is the surface tension and \(h_{xx}\) is the interfacial curvature in the longwave limit. Hence, Eq. (9) becomes:
Substituting Eq. (10) into Eq. (4) gives:
2.2 Contactline singularity
In the context of droplet spreading, it is desirable to propose a similarity solution to Eq. (11), corresponding to a selfsimilar droplet that retains some overall structural properties even as the base of the droplet spreads out. Dimensional analysis indicates that the similarity solution should be:
where \(h_0\) and R are as given in Fig. 1 and \(t_0\) is a timescale to be determined. Substitution of Eq. (12) into Eq. (11) yields:
The timescale \(t_0\) is chosen to be the capillary timescale, such that
Thus, Eq. (13) becomes:
The appropriate dropletspreading boundary conditions for Eq. (14) are \(f(0)=1\), \(f'(0)=0\),
where \(\eta _0\) corresponds to the outermost extent of the droplet. Thus, the position a at which the (microscopic) contact line touches down to zero is described by \(a(t)/R=\eta _0 (t/t_0)^{1/7}\). Unfortunately, the similarity solution (12) with the boundary conditions (15) fails to exist; instead, \(f(\eta )\) degenerates into a Dirac delta function centred at \(\eta =0\), and the droplet does not spread [28].
The reason for this failure is that the noslip condition (7) is inconsistent with the phenomenon of droplet spreading. When the model (11) is applied to droplet spreading, the physics which permits slip to occur on sufficiently small scales is missing. The missing physics is then put into the model as part of a regularization. For instance, by allowing for slip on a sufficiently small scale \(\ell _{\mathrm {s}}\), Eq. (11) becomes:
Equation (16) is the ThinFilm Equation with a sliplength model. Through a procedure of matched asymptotic expansion, the leadingorder behaviour of the dynamic contact angle as shown in Reference [29] is given by
where \(\mathrm {Ca} = (\mu /\gamma )(\mathrm {d}x_{cl}/\mathrm {d}t)\) is the capillary number based on the contactline velocity; here \(x_{cl}\) is the position of the contact line, which depends on t. Here also, b is a constant. From this description, it can be inferred that
Hence the leadingorder behaviour of Eq. (17) is given by \(x_{cl}\sim t^{1/7}\), which agrees with the socalled Tanner’s Law. Similar calculations were done in Reference [30] for radially symmetric droplets in three spatial dimensions, from which the spreading rate \(x_{cl}\sim t^{1/10}\) was found.
The sliplength model therefore provides for a resolution of the contactline singularity. However, the stress \(\gamma h_{xx}\) remains singular at the contact line. For these reasons, an alternative regularization of the ThinFilm Equation has been proposed, namely the PrecursorFilm model [1, 9]. Following Reference [15], in this work we present the Geometric ThinFilm Equation as an alternative regularization, the advantage of this approach as we reveal in subsequent sections is the remarkable simplicity of the numerical solutions produced by this model.
3 Geometric thinfilm equation: theoretical formulation
In the framework of the Geometric ThinFilm Equation, the startingpoint is the assumption that there is missing physics on a small scale. Instead of modelling the missing physics, it is parametrized. As such, h(x, t) is used to denote the interface location in a crude model with missing physics – which we call here the ‘noisy’ interface location. The noisy interface location is to be smoothened by a filtering operation, to produce a smoothened, more accurate, estimate of the interface location, which we denote by \({\overline{h}}(x,t)\). The noisy interface location may be different from the true interface location – for instance, the noisy interface location may be zero, whereas the true interface location may be close to, but different from zero – as would be the case if the noisy interface location was obtained through an incomplete model with missing smallscale physics.
3.1 Model A
To take account of the fact that \({\overline{h}}\) represents a smoother description of the interface location than h, we propose that h and \({\overline{h}}\) be connected via the expression
where \(\eta\) is a fluctuating quantity with mean zero and variance \(\sigma ^2\). Then, \({\overline{h}}(x,t)\) can be made into an accurate estimate of the interface location by minimizing the total interfacial energy
subject to a fixedvarianceconstraint:
Here, S is the system size (we take \(S\rightarrow \infty\) in what follows), and \(\gamma\) is a positive constant representing the surface tension. The constraint (21) enforces a fixed level of uncertainty between the model with missing physics and the smoothened model. In practice, we minimize the surface area in the longwave limit: in this limit \(\partial _x{\overline{h}}\) is small, and the square root in Eq. (20) can be well approximated by \(1+(1/2)\partial _x{\overline{h}}^2\). Thus, the energy in Eq. (20) becomes
As only energy differences are important, the constant term here can be ignored. Furthermore, as the droplet profile rapidly decreases far from the droplet core at \(x=0\), we can take \(S\rightarrow \infty\) in the foregoing analysis, this then gives the required functional form for the energy in the longwave limit:
Equation (22) with constraint (21) (with \(S\rightarrow \infty\)) is a constrained minimization problem—to solve it, one would introduce an energy functional with a Lagrange multiplier:
One would then compute
yielding
In practice, solving Eq. (25) yields inconsistent results, as it implies that \(h={\overline{h}}\). But h and \({\overline{h}}\) live in different function spaces (h is noisy, \({\overline{h}}\) is smooth), so Eq. (25) cannot be correct. Instead, we can study the dynamics, whereby \({\mathcal {L}}[h,{\overline{h}}]\) in Eq. (23) reduces over time. The dynamics are highly conditioned:

The evolution of h and \({\overline{h}}\) must be such that \({\mathcal {L}}\) tends to a minimum over time;

The integrals \(\int _{\infty }^\infty h(x,t)\mathrm {d}x\) and \(\int _{\infty }^\infty {\overline{h}}(x,t)\mathrm {d}x\) must be conserved quantities, reflecting underlying principles of conservation of fluid mass.
Under these conditions, the evolution equation for h must be of a generic conservativegradientdescent type, hence:
where \(M\ge 0\) is a mobility function to be determined. The evolution equation for \({\overline{h}}\) may be similar. However, for simplicity, we may assume that \({\overline{h}}\) relaxes instantaneously to a smoothened form of h, hence \(\delta {\mathcal {L}}/\delta {\overline{h}}=0\), hence
or
Equation (28) establishes a natural smoothing operation and hence, smoothing kernel for the formulation, namely, the Helmholtz kernel.
We now use \(\delta {\mathcal {L}}/\delta h=\lambda (h{\overline{h}})\). We compare this to Eq. (27), and get \(\delta {\mathcal {L}}/\delta h=\gamma \partial _{xx}{\overline{h}}\). Substitution of these results into Eq. (26) yields:
The physical model for h is completed by specifying the mobility. This is done by reference to the classical theory in Sect. 2. However, instead of \(M=(1/3\mu )h^2\) we take
the reason for using \({\overline{h}}^2\) in the mobility becomes apparent when we look at particlelike solutions of the regularized model (Sect. 4). Finally, the value of the Lagrange multiplier \(\lambda\) is chosen at each point in time to reflect the model uncertainty:
We refer to this model with a fixed level of uncertainty as Model A.
3.2 Model B
In practice, recomputing the Lagrange multiplier \(\lambda\) at each time t is a difficult task numerically. However, an equivalent model can be formulated by introducing an unconstrained functional,
Here, \(\alpha\) is a fixed parameter of the model. The dynamical equation is the same as before (Eq. (29)), as is the mobility; however, now \({\overline{h}}\) is computed as
We refer to this model with uncertainty on a small scale \(\alpha\) as Model B. Here, we have introduced the standard notation for smoothing kernels [31]:
and we explicitly use K for the Helmholtz kernel, such that
Although Model A and Model B are different, there is a onetoone relationship between them, and they are equivalent – e.g. \(\lambda\) in Eq. (31) is clearly a \(\left[ \text {lengthscale}\right] ^2\) which depends on time. We therefore identify
and the required uncertainty on a small lengthscale \(\alpha\) in the second description of the model is the average value of Eq. (34):
Due to the computational efficiency, Model B is preferred in this work.
The kernel solution (33) can be substituted back into the expression \({\mathcal {L}}[h,{\overline{h}}]\) to give:
This can in turn be written in several further ways:

1.
The innerproduct pairing of \(\partial _x h\) with \(\partial _x {\overline{h}}\):
$$\begin{aligned} \ell [h]&{\mathop {=}\limits ^{\text {Eq.~(36)}}}&\tfrac{1}{2}\gamma \int _{\infty }^\infty \left[ \left( \partial _x{\overline{h}}\right) ^2+\alpha ^2\left( \partial _{xx}{\overline{h}}\right) ^2\right] \mathrm {d}x,\\&{\mathop {=}\limits ^{\text {IBP}}}&\tfrac{1}{2}\gamma \int _{\infty }^\infty \partial _x {\overline{h}}\left( 1\alpha ^2\partial _x^2\right) \partial _x{\overline{h}}\,\mathrm {d}x,\\= & {} \tfrac{1}{2}\gamma \int _{\infty }^\infty \partial _x {\overline{h}}\partial _x\underbrace{\left( 1\alpha ^2\partial _x^2\right) {\overline{h}}}_{=h}\,\mathrm {d}x. \end{aligned}$$Here, ‘IBP’ stands for ‘integration by parts’. Hence,
$$\begin{aligned} \ell [h]=\tfrac{1}{2}\gamma \int _{\infty }^\infty \partial _x h \partial _x{\overline{h}}\,\mathrm {d}x. \end{aligned}$$(37) 
2.
The weighted innerproduct pairing:
$$\begin{aligned} \ell [h]=\tfrac{1}{2}\gamma \langle \partial _x{\overline{h}},\partial _x{\overline{h}}\rangle _K=\tfrac{1}{2}\gamma \int _{\infty }^\infty \partial _x {\overline{h}}\left( 1\alpha ^2\partial _x^2\right) \partial _x{\overline{h}}\,\mathrm {d}x. \end{aligned}$$(38)The pairing \(\langle \cdot ,\cdot \rangle _K\) defines a Reproducing Kernel Hilbert Space [32].
Equation (29) now reads:
3.3 Higherorder smoothing
For the purpose of generating particlelike solutions of the regularized model (e.g. Sect. 4), smoothing with the Helmholtz kernel is not sufficient. Therefore, in this paper, we work with a higherorder smoothing. We take \(\ell\) as before (specifically, Eq. (36)), with evolution Eq. (39) and smoothing kernel \({\overline{h}}=K*K*h\) – this is a straightforward extension of the basic model. We therefore summarize in one place the model studied in this work:
The aim of the remainder of the paper is to explore numerically the solutions of Eq. (40)—we will use \(\varPhi =K*K\) to denote the double Helmholtz kernel; specifically,
Hence, \({\overline{h}}=\varPhi *h\). The need for such higherorder smoothing will become apparent in Sect. 4.
We emphasize that the present higherorder smoothing is introduced to give the required degree of regularity to the numerical solutions (e.g. Sect. 4). Therefore, the higherorder smoothing is introduced here for mathematical convenience. However, the higherorder smoothing can be given a physical basis through the construction of appropriate energy functionals, as in Sect. 3.2, although we do not pursue that approach here.
3.4 Discussion
Summarizing our work so far, we have introduced a regularized thinfilm equation where the missing smallscale physics is not modelled, but is instead parametrized. The idea of the model is that h(x, t) provides an incomplete description of the droplet evolution (but which nonetheless contains important physical information, such as the problem dependence on the viscosity \(\mu\) and time t. A refined description of the interface is then obtained via a smoothened interface profile \({\overline{h}}=\varPhi *h\). Overall, the model evolves so as to minimize the interfacial energy (area) while keeping the difference between h and \({\overline{h}}\) as small as possible.
The model as formulated envisages that h(x, t) is an incomplete description of the interface profile (with missing smallscale physics). The missing physics is encoded either as a fixed level of uncertainty between h and \({\overline{h}}\) (model A), or such that the uncertainty in the model description occurs below a lengthscale \(\alpha\) (model B). These two models are equivalent, although model B is preferred for computational simplicity. The model is concerned with physics on the droplet scale, which is typically on the millimetre or micrometre scale. The model does not include explicitly the intermolecular forces between the droplet and the substrate, which occur typically on the nanometre scale [1]. Therefore, the uncertainty in the model can be said to occur on the nanometre scale, with the value of \(\alpha\) chosen accordingly. However, our calculations (e.g. Appendix C) show that a much larger value of \(\alpha\) can be chosen without affecting the largescale droplet spreading.
The Eq. (40) is a variant of the socalled Geometric ThinFilm equation introduced in Reference [15]: by viewing \(a=h\mathrm {d}x\) as a oneform, Eq. (40) can be written as
where \(\pounds _U(a)=\pounds _U(h\mathrm {d}x)=\partial _x(U h)\,\mathrm {d}x\) is the Lie derivative on oneforms; in this instance, \(U=M\partial _x(\delta \ell /\delta h)\) is the pertinent vector field. Equation (42) is then a very simple instance of a general theory of gradientflow dynamics [16] which uses geometric mechanics (Lie Derivatives, Momentum Maps) to formulate an energydissipation mechanical model for general configuration spaces – hence the Geometric ThinFilm equation. Equation (40)–(42) can furthermore be identified as a special case of Darcy’s Law [16], where the generalized force is \(f=\partial _x(\delta \ell /\delta h)\), the Darcy velocity is \(U=Mf\), and the fluxconservative evolution is equation for the conserved scalar quantity h is \(h_t+\partial _x(hU)=0\). These insights will be used in formulating the Geometric ThinFilm equation for the case of partial wetting in Sect. 6, below.
We remark finally here on the intriguing connection between Model A and ‘denoising’ in Image Processing – in Image Processing one is given a noisy image h and it is desired to produce a smoother image \({\overline{h}}\) while keeping the difference between the noisy and the smooth image at a fixed level \(\sigma\). This is achieved by minimizing a functional such as Eq. (23) [33, 34]. Our evolution Eq. (29) for the freesurface height h(x, t) in a thinfilm flow is equivalent to carrying out denoising on a (onedimensional) image in Image Processing.
4 Geometric thinfilm equation: particle solutions
The general theory of geometric dissipative mechanics introduced in Reference [16] includes many examples where discrete, particlelike solutions (such as Eq. (1)) are admitted. Motivated by these examples, we seek simplified solutions of Eq. (40) of the following form:
where N is a positive integer corresponding to a truncation of an infinite sum, \(w_i\ge 0\) are weights to be computed, and \(\delta (\cdot )\) is the Dirac delta function. The motivation for seeking out such highly simplified particle solutions is that they make the task of solving the partial differential Eq. (40) numerically very simple: instead of discretizing a fourthorder parabolictype partial differential equation and solving it numerically, we can solve a set of ordinary differential equations for the deltafunction centres \(x_i(t)\) using standard timemarching algorithms. This simplifies the numerical computations greatly, and is similar to the concept in the Method of Lines. We refer to the weights \(w_i\) together with the deltafunction centres \(x_i(t)\) as the ‘particles’ – thus, we are concerned with a particlesolution of Eq. (40). We describe the construction of such particlesolutions in what follows.
The weights \(w_i\ge 0\) are chosen such that
where \(\phi (x)\) is an arbitrary smooth, integrable test function. This limit can be satisfied by taking
where L is a lengthscale such that the set of all points where \(h_0(x)\) is nonzero is contained in the interval \([L,L]\); we must also take
Then, Eq. (44) is satisfied automatically.
We now multiply both sides of Eq. (40) by the test function \(\phi (x)\), integrate from \(x=\infty\) to \(x=\infty\), and apply vanishing boundary conditions at these limits. We thereby obtain
where \(\langle \cdot ,\cdot \rangle\) denotes the standard pairing of squareintegrable functions:
We substitute Eq. (43) into Eq. (47). Owing to the judicious choice of mobility \({\overline{M}}=(1/3\mu ){\overline{h}}^2\) (cf. Eq. (30)), no instance of the singular solution \(h^N(x,t)\) gets squared (only the smoothened solution \({\overline{h}}^N\) gets squared). After performing some standard manipulations with Dirac delta functions, Eq. (47) becomes:
where now \([({\overline{h}}^N)^2 \partial _{xxx}{\overline{h}}^N]_{x=x_i}\) is taken to mean
Equation (49) is rearranged to give
Equation (51) is true for all test functions \(\phi (x)\) and all initial data \(h_0(x)\) (hence \(w_i\)). Therefore,
Thus, Eq. (43), together with the ordinary differential equations
and initial data
give a socalled singular solution to the Geometric ThinFilm Eq. (40). The centres of the Dirac delta functions \(x_i(t)\) with associated weights \(w_i\) are identified as pseudoparticles, and the velocity \(V_i\) of the \(i^{\text {th}}\) pseudoparticle is identified with the righthand side in Eq. (53),
4.1 Key properties of the particle evolution equations
We notice that in Eq. (55), evaluation of \(\varPhi '''\) is required – this is the rationale for our choice of the double Helmholtz kernel as the smoothing kernel in Eq. (40). Using the single Helmholtz kernel K would not be sufficient, as \(K'''\) is singular at the origin. Furthermore, as the reconstructed interface profile \({\overline{h}}(x,t)=\sum _{i=1}^Nw_i\varPhi (xx_i(t))\) involves the positive weights \(w_i\) and a positive kernel \(\varPhi \ge 0\), the particle method is positivitypreserving: if h and \({\overline{h}}\) are initially positive, then the stay positive for all time. The numerical particle method is manifestly positivitypreserving, this is a key advantage as a numerical method that led to erroneous negative values of h and \({\overline{h}}\) would produce unphysical results.
4.2 Numerical solutions using the particle method
In this paper, we solve the Geometric ThinFilm Eq. (40) numerically using the particle method (55). To demonstrate the accuracy of the novel particle method, we compare the particle method to a standard fullyimplicit finitedifference method. It can be noted from the particle method (specifically Eq. (55)) that N ordinary differential equations are to be solved; each of the N righthandside (RHS) terms requires a summation over all other particles (cf. Eq. (43b) and the second entry in Eq. (55))—this suggests the particle method has computational complexity \(O(N^2)\). However, the number of floatingpoint operations to be performed in evaluating the different RHS terms can be dramatically reduced by symmetry operations, to give an overall computational complexity O(N)—this is the socalled fastparticle method. We give details of the fast particle method and our fullyimplicit finitedifference method in Appendix A.
The transformations described in Appendix A rely on the properties of the onedimensional kernel function \(\varPhi (x)\). The analogous kernel function in higher dimensions does not possess the same properties, rendering the onedimensional fastparticle method a somewhat special case. This issue has already been encountered for fastparticle methods for the Camassa–Holm equation [35]—there, the development of a fastparticle method in higher dimensions can be achieved via a fast multipole expansion. The same approach may apply to the Geometric ThinFilm Equation.
Throughout the paper, we use the value \(\alpha =0.05\) in the numerical computations. This is justified in Appendix C, where show that the results of the simulations are insensitive to the choice of the parameter \(\alpha\) – provided \(\alpha\) is sufficiently small. This is advantageous, since the numerical solver must resolve the finest scale in the problem – which is precisely \(\alpha\). Finally, both the particle method and the fullyimplicit finitedifference method are solved in MATLAB. The particle method makes use of MATLAB’s builtin timeevolution algorithms for ordinary differential equations, notably, ODE45 and ODE15s.
5 Complete wetting
In this section we present numerical results for complete wetting for the Geometric ThinFilm Equation. Equation (40) clearly corresponds to the case of complete wetting: the energy functional \({\mathcal {E}}=(1/2)\gamma \int _{\infty }^\infty \partial _x{\overline{h}}^2\mathrm {d}x\) is penalized, meaning that the system evolves to minimize the curved part of the droplet interface, that is, the part of the droplet interface in contact with the surrounding atmosphere.
5.1 Nondimensionalization and initial conditions
To characterize this spreading phenomenon, we solve the Geometric ThinFilm Eq. (40) in dimensionless variables. The interfacial height is made dimensionless on a lengthscale \(h_0\), which is proportional to the initial maximum droplet height \(h_\mathrm{{max}}\). In this section, we take the rather nonstandard value \(h_0=(8/3)h_\mathrm{{max}}\) for \(h_0\), this is done here to compare with Reference [15].^{Footnote 1} The xcoordinate is then made dimensionless on the initial droplet base L. Finally, time is made dimensionless on the capillary timescale \(\tau =(3\mu L/\gamma )(L/h_0)^3\). The ratio \(\epsilon =h_0/L\) must be small, for inertial effects to be negligible, and hence, for the lubrication theory underlying Eq. (40) to be valid. Henceforth, we assume that all of the relevant variables have been made dimensionless in this way. The initial condition for the droplet height therefore reads:
Thus, the droplet area (which is the analogue of droplet volume in two dimensions) is therefore fixed as \(\int h(x,t=0)\,\mathrm {d}x=1/4\). Both the droplet area \(\int h(x,t)\,\mathrm {d}x\) and \(\int {\overline{h}}(x,t)\,\mathrm {d}x\) are conserved under the evolution Eq. (40). The simulations are carried out in a finite spatial domain \(x\in [2,2]\) with periodic boundary conditions – this condition also establishes the limits of integration on the foregoing integrals.
5.2 Results
We solve Eq. (40) with the initial condition (56). The numerical calculations indicate that the particle method and the finitedifference method produce results that are indistinguishable by eye: we therefore show only results for the particle method. A yet more rigorous comparison between the two methods is also presented herein—this analysis also justifies the number of particles used in the calculations as \(N=800\), this can be deemed equivalent to a grid spacing of \(\varDelta x=2L/N=0.005\) in the finitedifference method.
A first set of results is shown in Fig. 3. In Fig. 3a we show a spacetime plot of the smoothened freesurface height \({\bar{h}}\) where the spatial grid is evaluated at the particle positions \(x_i(t)\)—effectively, a discretization of \({\overline{h}}\) on a nonuniform grid. From this plot, the region in space where where \({\overline{h}}\) is significantly different from zero increases over time, demonstrating that the droplet is indeed spreading. Fig. 3b shows a snapshot the filtered surface height and its slope at \(t=50\).
The particle locations are shown explicitly in Fig. 3b—there is a high concentration of particles in the regions of high curvature—this is discussed in more detail in what follows.
In Fig. 4, we plot the particle trajectories \(x_i(t)\) for the solution of the Geometric ThinFilm Equation via the particle method.
Intriguingly, the particles are seen to accumulate at regions where \(\partial _{xx}{\overline{h}}\) is large, giving a higher spatial resolution in regions of high interfacial curvature. When a finitedifference or finitevolume solver is able to execute local grid refinement in regions of where the spatial derivatives are large in magnitude, this is because of adaptive meshrefinement. Here, the particle method demonstrates a tendency to mimic the effect of adaptive meshrefinement, without having to implement adaptivemesh procedures explicitly. Furthermore, in Fig. 4 we have used the builtin MATLAB ODE solvers to generate the spacetime plot, meaning that the adaptive mesh refinement is performed in the temporal as well as the spatial domain.
A further advantage of the particle method is that it provides a numerical description of the contact line by simply following the trajectory of a particle starting near the contact line, say \(x_k(t)\) such that \(x_k(0)=0.5\). This is also shown in Fig. 4, where the contact line is found to satisfy Tanner’s Law, with \(x_k(t)\sim t^{1/7}\). The scaling persists even at late time, indicating that finitesize effects are not important here. Indeed, the particle method is intrinsically free from finitesize effects: since the kernel function used extends to infinity in both directions, the particle method and the accompanying ODE solver seamlessly track the particles as they move away from the origin and hence, as the droplet spreads. This is a key benefit of using the particle method.
In contrast, in the finitedifference method, the position of the contact line is obtained by extrapolation: the value \(x_*(t)=\mathrm {argmax}_{x}[{\overline{h}}(x,t)]\) is computed, the tangent line at the point \((x_*(t),{\overline{h}}(x_*,t))\) is computed, and the nominal position \(x_{\mathrm {ref}}\) of the contact line is taken to be the intersection of the tangent line with the xaxis. This procedure was used in Reference [15], and produces a clear demarcation between the droplet core region, and farfield region, where the droplet profile decays to zero exponentially (we take this opportunity here to clarify the procedure used to compute \(x_{\mathrm {ref}}\), which was not clear in our previous work). Using either the particle method, or the finitedifference method with \(x_{\mathrm {ref}}\) chosen is equivalent: both descriptions reproduce the \(t^{1/7}\) behaviour for the contact line.
The finding that the Geometric ThinFilm Equation satisfies Tanner’s Law of droplet spreading indicates that the regularized model is capturing the largescale physics in the dropletspreading problem. In order to demonstrate this even further, we introduce the function \(f_\alpha (\eta ,t)=t^{1/7}{\overline{h}}(\eta t^{1/7},t)\), with \(\eta t^{1/7}=x\). In Fig. 5 we produce a spacetime plot of \(f_\alpha (\eta ,t)\) – this is seen to relax to a constant profile at late times as \(t\rightarrow \infty\).
Furthermore, the profile of \(f_\alpha (\eta ,t)\) at fixed t (t large) can be compared with a similarity solution of the unregularized problem, \(f^3f'''=\eta f/7\) (cf. Eq. (14)). This ordinary differential equation is then solved with the shooting method together with appropriate initial conditions [15]. The results are shown in Fig. 6.
This figure therefore shows that the Geometric ThinFilm Equation describes the expected largescale dropletspreading physics in the droplet core. Where the classical ThinFilm Equation breaks down at the contact line, the height profile of the Geometric version decays smoothly to zero.
5.3 Rigorous error and performance analysis
We analyse the truncation error associated with the standard fullyimplicit finitedifference method and the particle method. As such, let \({\overline{h}}\) denote the exact solution of Eq. (40), and let \({\overline{h}}_{\varDelta x}\) denote the numerical solution with step size \(\varDelta x\) (finitedifference method), or number of particles \(N=2L/\varDelta x\) (particle method). We assume that the error \(\Vert {\overline{h}}{\overline{h}}_{\varDelta x}\Vert\) depends smoothly on \(\varDelta x\), then
for some constant C and p. Since \({\overline{h}}\) is unknown, we instead compute
Using the triangle inequality, it can be shown that
We take the natural logarithm on both sides of Eq. (59); this gives:
Thus, the rate of convergence (or the order of accuracy) of the numerical method is p; p can be computed from the numerical simulations as the slope of the loglog plot between the error \(\varepsilon\) and the grid spacing \(\varDelta x\).
Figure 7 shows the rate of convergence of the finitedifference method and the particle method – here we use the \(L^1\) norm applied to Eqs. (59)–(60) (our choice of norm is obtained because the particle solutions are expected to converge weakly in an \(L^1\) function space, see Reference [36]). The finitedifference method is implemented with a step size of \(\varDelta t=0.01\), while the particle method uses the ODE45 solver in MATLAB, hence an adaptive time step. Both methods use the same numerical parameter of, \(\alpha =0.05\), with final time \(T=1\), and periodic boundary condition on the spatial domain \(x\in [1,1]\). From this figure, both the particle method and the standard finitedifference method are estimated to be secondorder accurate in the spatial domain, albeit that the absolute error is smaller for the finitedifference method.
Furthermore, in Fig. 8 we evaluate the execution time of the different numerical methods to see if any one method outperforms the rest. A comparison of the average execution time over 10 runs between the finitedifference method, the direct implementation of the particle method (computational complexity \(O(N^2)\)), and the fast implementation particle method (computational complexity O(N)). The numerical parameters used are the same as the one used in the convergence analysis and the calculations are performed on an Intel i79750H with 6 hyperthreaded cores.
Thus, although the fastparticle method is less accurate for fixed N (Fig. 7), the fastparticle method is more efficient (Fig. 8). Therefore, the a fixed execution time (but varying N), the fastparticlemethod and the finitedifference method achieve a similar level of accuracy, albeit that the finitedifference method still has the edge. However, beyond this standard performance analysis, the particle method still retains a key benefit, namely that it is free from finitesize effects.
6 Partial wetting
In this section, we extend the Geometric ThinFilm Equation to the case of partial wetting, where a droplet on a substrate spreads initially before assuming an equilibrium shape. This requires the inclusion of an extra, stabilizing term, to Eq. (40). We derive this additional term. Then, we construct an analytical solution for the equilibrium droplet shape. Finally, we use both the finitedifference method and the particle method to simulate transient droplet spreading, up to the point where the droplet assumes its equilibrium shape.
6.1 Theoretical formulation
The startingpoint for the theoretical formulation is to consider an unregularized description of the droplet, with h(x) as the droplet profile. Then, the unregularized energy associated with a droplet of radius r is:
Here, \(\gamma _{la}\) is the surface tension between the air and the liquid droplet (previously referred to as \(\gamma\)), \(\gamma _{ls}\) is the surface tension between the liquid and the substrate, and \(\gamma _{as}\) is the surface tension between the air and the substrate; S is an arbitrary lengthscale denoting the extent of the system in the lateral direction. In the longwave limit, \(\sqrt{1+h_x^2}\) is expanded as \(1+(1/2)h_x^2\), and Eq. (61) becomes:
The three surfacetension coefficients are related via the LaplaceYoung condition,
where \(\theta _{\mathrm {eq}}\) is the equilibrium contact angle. Thus, Eq. (62) can be rewritten as
Now, inspired by the replacements \(h\rightarrow {\overline{h}}\) in Sect. 3, we propose herein a regularized energy,
where w is an estimate of the size of the droplet footprint, based on the interfacial profile h, and on the smoothened interfacial profile, \({\overline{h}}=\varPhi *h\).
We estimate the size of the droplet footprint as
where \(A_0\) is the constant droplet volume, \(A_0=\int _{\infty }^\infty h(x,t)\mathrm {d}x=\int _{\infty }^\infty {\overline{h}}(x,t)\mathrm {d}x\), and c is an O(1) parameter to be determined. The estimate in Eq. (66) is dimensionally correct, but also yields good agreements with some model droplet profiles: for instance, if h were a parabolic cap, \(h(x)=\max \{0,(3A_0/4r)[1(x/r)^2]\}\), then we would have (by direct calculation) \(\langle h,{\overline{h}}\rangle =3A_0^2/5r + O(\alpha ^2)\), and
i.e. a width proportional to the droplet footprint 2r, with a constant of proportionality \(6/5 + O(\alpha ^2)\) close to one. Thus, the regularized energy becomes:
where \(\chi =c(1\cos \theta _{\mathrm {eq}})\) is an O(1) constant, which will be selected a priori in what follows; we also use \(\gamma\) instead of \(\gamma _{la}\), for consistency with the previous sections. Finally, the constant term in the energy has been dropped in Eq. (67), because only energy differences are important for the purpose of deriving evolution equations.
To derive the evolution equation associated with Eq. (67), we use the framework of Darcy’s Law introduced previously in Sect. 3. Thus, the generalized force associated with Eq. (67) is:
and the Darcy velocity is therefore \(U=Mf\), where M is the mobility; we again take \(M=(1/3\mu ){\overline{h}}^2\). Thus,
The evolution equation for h which conserves \(\int _{\infty }^\infty h\mathrm {d}x\) is thus:
hence
6.2 Nondimensionalization
We nondimensionalize Eq. (68) using the lengthscale \(h_0=\sqrt{A_0 \tan \theta _{\mathrm {eq}}}\) in the vertical direction and \(L=\sqrt{A_0/\tan \theta _{\mathrm {eq}}}\) in the lateral direction. Thus, h is made dimensionless on \(h_0\), x is made dimensionless on L, and time is made dimensionless on the capillary timescale \((3\mu L/\gamma )(\tan ^3\theta _{\mathrm {eq}})\). In dimensionless variables, Eq. (68) now reads:
Also, in dimensionless variables, \(\int _{\infty }^\infty h(x,t)\mathrm {d}x=1\). Also in this context, the ratio \(\epsilon =h_0/L\) is precisely \(\tan \theta _{\mathrm {eq}}\); strictly speaking therefore, \(\theta _{\mathrm {eq}}\) should be small, for inertial effects to be negligible, and hence, for the lubrication theory underlying Eq. (69) to be valid.
6.3 Equilibrium solution
Equation (69) has an equilibrium solution with \(\partial h/\partial t=0\). In this limiting case, Eq. (69) reduces to
where \(\xi ^2\) is a positive constant,
Equation (70) has an analytical solution, parametrized by \(\xi\), and by a radius r:
Correspondingly,
Here, \(B_1,B_2,C_1\), and \(C_2\) are constants of integration. These constants are fixed by imposing continuity of \({\overline{h}},{\overline{h}}_x,{\overline{h}}_{xx}\) at \(x=r\), and also by imposing \(\int _{\infty }^\infty {\overline{h}}\, \mathrm {d}x=1\). These give four conditions in four unknowns. Hence, \(\{B_1,B_2,C_1,C_2\}\) are fixed in terms of r. The value of r is in turn fixed by imposing continuity of \({\overline{h}}_{xxx}\) at \(x=r\), this gives a simple rootfinding condition:
The details of this calculation are provided in Appendix B. Equation (74) gives r as a function of \(\xi\). However, \(\xi\) is not arbitrary, but is instead fixed by its own rootfinding condition (Eq. (71)). Although this procedure is somewhat involved, the point remains: the Geometric ThinFilm Equation with partial wetting admits an analytical equilibrium solution in terms of elementary functions (via Eqs. (72)–(73)). Furthermore, the elementary solution (72) for \({\overline{h}}(x)\) coincides with the expression for a paraboliccap droplet in the core region \(x\rightarrow 0\),
It is noted however, that this expression is valid only in the core region \(x\rightarrow 0\). In particular, the analytical solution \({\overline{h}}(x)\) does not converge to a paraboliccap profile in the limit as \(\alpha \rightarrow 0\); rather, \({\overline{h}}(x)\) maintains its cosine shape in this limit. Therefore, in order to reproduce the paraboliccap profile in the limit as \(\alpha \rightarrow 0\), it would be necessary to make a more judicious choice for estimate of the width of the droplet footprint in Eq. (67). Finding the optimal choice for the functional form of the width of the droplet footprint will be the subject of future work.
The equilibrium contact angle is now computed as:
where \({\overline{h}}_x\) is expressed in dimensionless variables and \(x_{\mathrm {ref}}>0\) is a reference point. Since \(\tan \theta _{\mathrm {eq}}=\epsilon\) in the chosen dimensionless variables, this requires:
Following the procedure described in Sect. 5, we choose the reference point \(x_{\mathrm {ref}}\) to be the positive value of x which maximizes \({\overline{h}}_x\), thus \(x_{\mathrm {ref}}=\pi /(2\xi )\). Thus, we require \(B_1\xi =1\). But \(B_1\) and \(\xi\) depend parametrically on \(\chi\), hence we require \(B_1(\chi )\xi (\chi )=1\). This therefore fixes the model parameter \(\chi\) as a global constant, \(\chi \approx 1.1602\) (for filter width \(\alpha =0.05\)). The resulting equilibrium droplet profile is shown in Fig. 9. Values of \(\chi\) for different values of \(\alpha\) are given in Table 1.
6.4 Results
For the simulations of partial wetting, we use the initial condition
with \(r_0=0.5\), and \(\int _{\infty }^\infty h(x,t=0)\,\mathrm {d}x=1\). We also take \(\alpha =0.05\). We use both the particle method and the finitedifference method: the results are the same in each case. In Fig. 10a, we show a spacetime plot of the solution for the partial wetting case up to \(t=10\); panel (b) shows a snapshot of the droplet profile at \(t=10\). Figure 11 is based exclusively on the particle method: here we show a loglog plot of the particle trajectories. At intermediate times, the particle trajectories are parallel to the path \(x=t^{1/7}\) before attaining a steady state at late times. Thus, in the case of partial wetting, the system obeys Tanner’s law at intermediate times, until at late times, the partial wetting stabilizes the droplet and it assumes its equilibrium shape.
The foregoing statement that the droplet spreading obeys Tanner’s Law at intermediate times until the onset of equilibrium also applies to the Cox–Voinov Law [9] of droplet spreading, which in equation form is \([\theta (t)]^3=[\theta _{\mathrm {eq}}]^3+c{\dot{x}}_{\mathrm {cl}}\log (x_{\mathrm {cl}}/d)\). Here, \(\theta (t)\) is the dynamic contact angle, which is obtained from the slope of the interface profile h(x, t) at some appropriate location x, and c and d are constants. In order to validate the applicability of the Cox–Voinov law to the droplet spreading within the framework of the Geometric ThinFilm Equation, we operationally define the contact angle as \(\theta (t)=\max _x [\partial _x{\overline{h}}(x,t)])\). The tangent line to \({\overline{h}}(x,t)\) at \(\mathrm {argmax}_x[\partial _x{\overline{h}}(x,t)]\) is constructed, and the contact line is then taken to be the intersection of this tangent line with the xaxis. This is the same procedure as in Sect. 5. A plot of \([\theta (t)]^3\) constructed in this way is shown in Fig. 12. Shown also is a plot \(1+c{\dot{x}}_{\mathrm {cl}}\log (x_{\mathrm {cl}}/d)\) – here, \(\theta _{\mathrm {eq}}=1\), and c and d are bestfit constants. Overall, there is good agreement between the two curves at intermediate times – consistently with the behaviour of the trajectories Fig. 10. There is some disagreement at late times, however, this may be expected, in view of the somewhat imprecise operational definition of \(\theta (t)\) and \(x_{\mathrm {cl}}\).
6.5 Rigorous error and performance analysis
We carry out a rigorous error analysis of both the fullyimplicit finitedifference method, and the particle method, for the case of partial wetting. Because there is an analytical, equilibrium solution valid at late times, we analyze the results of the numerical simulations at such late time (specifically, \(t=100\)), when the numerical solutions are close to the equilibrium state. In this case, the equation for the rate of convergence of the numerical method is simply
where \({\overline{h}}\) denotes the analytical equilibrium solution, and \({\overline{h}}_{\varDelta x}\) denotes the numerical equilibrium solution (or what amounts to the same, the numerical solution at \(t=100\)). Figure 13 shows the rate of convergence for the finitedifference method and the particle method. Both methods have \(\alpha =0.05\) and were performed on the spatial domain \(x\in [2,2]\) for various spatial resolutions \(\varDelta x\). We have used the MATLAB solver ODE15s for the particle method. Again, we observed both the finitedifference method and the particle method to be secondorder accurate in the spatial domain.
Finally, we have looked at the performance of the different numerical methods (fullyimplicit finitedifference method, particle method, and ‘fast’ particle method): the results are similar to what was observed in the case of complete wetting (Fig. 14).
7 Conclusions
Summarizing, we have introduced a new mathematical model to describe contactline motion which has the same regime of validity as conventional lubrication theory. The model involves using a ‘smooth’ interface profile \({\overline{h}}\) and a sharp interface profile h. The smooth interface profile \({\overline{h}}\) is connected to the sharp interface profile h via convolution, while h is defined through an evolution equation which couples both interface profiles, and which drives the droplet spreading. It is possible to assign a physical interpretation to the model: the equation for h is a model with missing smallscale physics (below a lengthscale \(\alpha\)), while \({\overline{h}}\) is a complete model of the physics at lengthscales greater than \(\alpha\). By minimizing the difference between the two descriptions, a convolution relation between \({\overline{h}}\) and h emerges. Furthermore, by formulating the evolution equation so that it involves both descriptions of the interface profile, the contactline singularity is resolved. There is no need to model the missing smallscale physics explicitly: instead, it is parametrized through the lengthscale \(\alpha\).
A further advantage of the new formulation is that it naturally suggests a meshfree numerical method for the purpose of simulating the model numerically. Furthermore, the mathematical model involves equations which are nonstiff, and are straightforward to solve numerically (we include a repository of the code as part of this work; see Reference [37]). We have conducted numerical simulations for both complete wetting and partial wetting using this new meshfree method (as well as a conventional finitedifference method), and have found that the model describes well the physics of droplet spreading – including Tanner’s Law for the evolution of the contact line. Remarkably, in the case of partial wetting, the model also admits a simple analytical solution for the equilibrium profile.
Beyond droplet spreading, the model may find further applications in describing families of droplets (the analytical equilibrium solution already possesses such ‘multipledroplet’ solutions), multicomponent systems, and problems in droplet evaporation. The model’s intrinsically nonstiff equations (as well as the absence of a precursor film extending to infinity) may in the future simplify the description of such complex physical systems.
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Acknowledgements
This publication has emanated from research supported in part by a Grant from Science Foundation Ireland under Grant Number 18/CRT/6049. LON has also been supported by the ThermaSMART network. The ThermaSMART network has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska–Curie grant agreement No. 778104.
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Appendix
Appendix
1.1 A Detailed description of numerical methods
In this Appendix, we describe in detail the numerical methods used in the paper. For definiteness, we focus on the case of complete wetting, although the description carries over to the case of partial wetting as well. We start by describing the fullyimplicit finitedifference method, and then describe the fastparticle method.
1.1.1 A.1 Fullyimplicit finitedifference method
Instead of solving Eq. (40), we take the convolution of both sides of the same equation with the filter function \(\varPhi\) and solve the evolution for the smoothened freesurface height \({\overline{h}}(x,t)\):
Here, we have written \(K*K=\varPhi\) (K is the Helmholtz kernel) such that \({\overline{h}}=\varPhi *h\), reflecting the choice of the inversedoubleHelmholtz operator as the smoothing kernel in Eq. (40). We also work in dimensionless variables, such that the prefacor \(\gamma /(3\mu )\) which was present in Eq. (40) is rescaled to one. Eq. (76) is discretized in space using a standard finitedifference method. The derivatives are approximated using standard finitedifference stencils (central differencing, first and secondorder derivatives secondorder accurate in space). The value of \({\overline{h}}\) is therefore known only at the discrete grid points; we collect these values into a vector \(\bar{{\varvec{h}}}\). Similarly, Eq. (76) is discretized in time such that the values of \({\overline{h}}\) are known only at discrete points in time; the fixed interval between each such discrete point in time is the timesetep \(\varDelta t\). The values of the vector \(\bar{{\varvec{h}}}\) at a particular discrete point in time are therefore labelled as \(\bar{{\varvec{h}}}^n\). Thus, in discrete form, Eq. (76) becomes:
Here, \({\mathcal {D}}_1\) and \({\mathcal {D}}_3\) are sparse matrices corresponding to the centred finitedifference approximation of the first and third spatial derivatives, respectively. A similar notation would hold for \({\mathcal {D}}_2\), which corresponds to the centred finitedifference approximation of the second spatial derivative. In this way, \({\mathcal {K}}\) is a convolution matrix, which is constructed as \({\mathcal {K}}=(1\alpha ^2{\mathcal {D}}_2)^{2}\). Furthermore, we use the \(\odot\) notation in Eq. (77) to denote the pointwise multiplication of vectors, e.g. if \({\varvec{a}}\) and \({\varvec{b}}\) are Ndimensional column vectors with entries \({\varvec{a}}=(a_1,\ldots ,a_N)^T\) and \({\varvec{b}}=(b_1,\ldots ,b_N)^T\), then \({\varvec{a}}\odot {\varvec{b}}\) is also an Ndimensional row vector with entries \({\varvec{a}}\odot {\varvec{b}}=(a_1b_1,\ldots ,a_Nb_N)^T\). This notation is useful for the exposition of the numerical method, as the numerical code for solving Eq. (76) is vectorized, and makes use of precisely this pointwise vector multiplication.
Crucially, in the discretized Eq. (77) we have treated all terms on the righthandside fully implicitly (\(\bar{{\varvec{h}}}^{n+1}\) instead of \(\bar{{\varvec{h}}}^{n}\)). This treatment amounts to a backwardsEuler temporal discretization. A fully explicit treatment (Forward Euler, with \(\bar{{\varvec{h}}}^{n}\) instead of \(\bar{{\varvec{h}}}^{n+1}\)) would suffer from a severe hyperdiffusionlimited CFL constraint \(\varDelta t< C \varDelta x^4\), where \(\varDelta x\) is the grid spacing, and C is an O(1) constant. A suitable combination of \(\bar{{\varvec{h}}}^{n}\) and \(\bar{{\varvec{h}}}^{n+1}\) on the righthand side of Eq. (77) (amounting to a Crank–Nicolson scheme) is also a possibility, although due to its simplicity, the backwardsEuler scheme is preferred here. Although the choice of BackwardEuler method makes the numerical code unconditionally stable, it means that at each timestep, a set of nonlinear equations for \(\bar{{\varvec{h}}}^{n+1}\) must be solved. We therefore describe how these nonlinear equations are solved.
We introduce
where \(\bar{{\mathcal {D}}}_1={\mathcal {K}}{\mathcal {D}}_1, {\mathcal {D}}_3={\mathcal {D}}_3{\mathcal {K}}\), and \({\varvec{v}} = {\mathcal {K}}^{1}\bar{{\varvec{v}}}\). Then, solving Eq. (77) for \(\bar{{\varvec{h}}}^{n+1}\) given \(\bar{{\varvec{h}}}^n\) becomes a minimization problem on
We employ a NewtonLinesearch method to solve for \(\bar{{\varvec{h}}}^{n+1}\). This requires us to compute the Jacobian of \({\varvec{F}}(\bar{{\varvec{v}}})\), which is given by
We initialize the guess with \(\bar{{\varvec{v}}}^0=\bar{{\varvec{h}}}^n\). Then, for each iteration I, the descent direction is given by
and the improved guess is updated using
where \(\alpha ^I\) is the optimum step size in the direction \(\delta \bar{{\varvec{v}}}^I\)
The iterative process is continued until the residual \(f(\bar{{\varvec{v}}}^I)\) is sufficiently small, and we set the solution for the next time step by letting \(\bar{{\varvec{h}}}^{n+1} = \bar{{\varvec{v}}}^I\).
1.1.2 A.2 Fast summation algorithm for particle method
Here, we are interested in solving the Geometric ThinFilm Eq. (40) using the particle method. The particle method requires us to compute the evolution equations for the particle trajectories, these are given in Eq. (52). To solve Eq. (52), one is required to evaluate
for \(i=1,\dots ,N\), at each time step. A direct evaluation would require an operational cost of \(O(N^2)\). Now, suppose that \(x_i<x_j\) for all \(i<j\), then the contribution from particles of either side of \(x_i\) becomes a degenerate case of the fast multipole method. We start by defining new variables
Since \(x_ix_j>0\) for \(j=1,\dots ,i1\), we can drop the absolute value in the function \(\varPhi\),
Similarly, we obtain for \(S_2\),
So Eq. (84) can be expressed as
Note that \(a_i, b_i, c_i, d_i\) can be computed at the start of each time step using a recursion formula
We follow the same procedure for Eq. (85) to obtain,
In the case of partial wetting, the term \(\partial _x{\overline{h}}^N\) in Eq. (68) would be decomposed using the same procedure and \(\langle h^N,{\overline{h}}^N\rangle =\sum _{i=1}^N w_i {\bar{h}}^N(x_i)\). This decomposition reduces the complexity of evaluating the Eq. (52) from \(O(N^2)\) to O(N).
1.2 B Derivation of the equilibrium solution for partial wetting
In this Appendix, we look at the equilibrium solution for the droplet in the case of partial wetting. The equilibrium solution was outlined in Sect. 6, and involved expressions for h(x) and \({\overline{h}}(x)\) in terms of elementary functions. We begin by recalling these expressions:
Here, \(B_1\), \(B_2\), \(C_1\), and \(C_2\) (together with r) are constants of integration, which can be obtained via the methods outlined in this Appendix, although we focus herein on deriving the rootfinding condition (74) previously introduced in the main part of the paper.
We start by noticing that the set of boundary conditions

1.
\(A_0=\int h\,\mathrm {d}x\),

2.
\(A_0=\int {\overline{h}}\,\mathrm {d}x\),

3.
\({\overline{h}}'\) continuous at r, and

4.
\({\overline{h}}'''\) continuous at r,
are linearly dependent. From the boundary conditions (2), (3) and (4), we get the following equations:
Taking (96 \(\) 2(97) + (98)) yields boundary condition (1). Therefore choosing any three out of the four boundary conditions yields the same system of equations.
Next, consider the following set of boundary conditions for expression (95):
From these boundary conditions, one can obtain the following two linearly independent equations
In matrix form, this can be expressed as
Requiring the determinant of the matrix to be zero yields Eq. (74).
From the rootfinding condition (74), a value for the parameter \(\xi\) is found in the main paper, via Eq. (71), recalled here as \(\xi ^2=2\chi /\langle h,{\overline{h}}\rangle ^2\). For this purpose, we lastly provide here an analytical expression for \(\langle h,{\bar{h}} \rangle\):
where \({\tilde{B}}_1 = B_1(1+\alpha ^2\xi ^2)^2\).
1.3 C Sensitivity analysis with respect to the parameter \(\alpha\)
In this appendix, we look at the sensitivity of the numerical results for the case considered in Sect. 5. In Fig. 15, we show the effect on the macroscopic contact line of varying \(\alpha\) – these effects are small, and diminish as \(\alpha\) is decreased.
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Pang, K.E., Ó Náraigh, L. A mathematical model and meshfree numerical method for contactline motion in lubrication theory. Environ Fluid Mech 22, 301–336 (2022). https://doi.org/10.1007/s10652021098270
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DOI: https://doi.org/10.1007/s10652021098270