Three-dimensional nonlinear dynamics of a thin liquid film on a spinning ellipsoid

Covering a surface with a thin coating layer that may serve a protective, a functional, or a decorative purpose is routine in many industrial applications. Many coating processes have been optimized to obtain quality coatings on flat, rigid surfaces. In spin coating, for example, the centrifugal force is used to distribute the dispensed fluid uniformly on a rotating surface. Spin coating on a flat surface which has benefited from decades of research is therefore a well-established procedure.¹ Achieving a quality coating on a complex-shape substrate is, however, more challenging because of the interplay between the fluid film, the substrate shape, and its motion kinematics resulting from translation and/or rotation.

We are interested here in the dynamics of a thin liquid film on an arbitrary-shaped surface subjected to rotation around a vertical axis, and we therefore draw from two distinct but sometimes intersecting areas of the literature on thin liquid films: one related to the effects of substrate shape and the other related to substrate rotation.

The dynamics of thin film flows is most commonly described by using a depth-integrated or depth-averaged formulation such as the lubrication or the shallow-water approximation which simplify the problem of having to solve the Navier–Stokes equations in an evolving domain to a simpler set of partial differential equations (PDEs) on a fixed domain. Perhaps not surprisingly, the vast majority of studies to date focus on substrate shapes with a high degree of symmetry (planar or axisymmetric) or which can be described naturally with the usual coordinate systems (cartesian, cylindrical, spherical) to allow further simplification of the governing partial differential equations.

For example, the rimming flow of a thin film on the inside of a rotating cylinder or the coating flow on the outside has been extensively studied in Refs. 2–4, where the authors explored the structure of stationary-state solutions of the lubrication approximation in the plane flow configuration, and by O'Brien⁵ through a stability analysis of these solutions. The Rayleigh–Taylor instability of thin viscous films coated on the inside of a horizontal non-rotating cylindrical substrate was recently explored.⁶ The plane flow assumption was removed in Leslie, Wilson, and Duffy,⁷ who considered the three-dimensional flow of a thin, slowly varying ring of fluid. The vanishingly small Reynolds number assumption, implicit in the above references, was alleviated by Morad and Zhukov⁸ through the derivation of a shallow-water type approximation. While the flow in the above references was in a plane perpendicular to the axis of rotation, the flow down a vertical cylinder rotating about its axis was considered in Refs. 9–11, where the authors studied the effects of centrifugal or Coriolis forces on the stability of the flow. The fingering instability in the non-rotating case was investigated numerically by Mayo, McCue, and Moroney.¹² 

Other flow configurations where the substrate curvature has been shown to have a strong impact on the fluid film dynamics include the flow in a funnel^13,14 and in corners.^15,16

The flow of thin films around a sphere has also received much attention as an archetypical example of interplay between the flow and substrate curvature. The dynamics of thin films driven by gravity on the outer surface of a sphere was investigated in Refs. 17 and 18 and exploited by Lee et al.¹⁹ for the rapid fabrication of hemispherical elastic shells by coating the outside of a sphere with a polymer solution. The development of the Rayleigh–Taylor instability for a thin film coating the inside of a spherical substrate was treated by the same group.²⁰ The case with fluid injection at the top, relevant to spherical water fountain, was explored by D'Alessio and Pascal.²¹ Kang, Nadim, and Chugunova²² added rotation in the formulation of the thin film equations and showed that the problem admits different types of stationary state. The transient dynamics of the wetting front on a tow was studied by Shepherd, Sellier, and Boujo.²³ 

Concurrently, several studies have developed general models to predict the dynamics of thin films on arbitrary surfaces. The seminal work of Schwartz and Weidner²⁴ extends the classical lubrication approximation to include the effects of substrate curvature using a curvilinear coordinate system in the plane flow configuration. The effect of moderate inertia was included by Ruschak and Weinstein²⁵ using the method of von Kármán and Pohlhausen. Based on a rigorous analysis using differential geometry and generalized curvilinear coordinates, Roy, Roberts, and Simpson²⁶ and Thiffeault and Kamhawi²⁷ derived a very general equation governing the three-dimensional flow of a viscous Newtonian fluid upon arbitrarily curved substrates. Rare applications of such generalized modeling framework can be found in Ref. 28, where a curvilinear thin film model was used to simulate the motion of droplets on a virtual leaf surface, and in Ref. 29, which studied the drainage and spreading of thin films under the assumption of negligible surface tension. The effect of substrate motion kinematics was added by Howell³⁰ and that of solidification by Myers, Charpin, and Chapman.³¹ 

Very few studies have considered the combined effects of a complex substrate shape and rotation on the dynamics of a thin film. Notable exceptions include Refs. 32 and 33, where the author derived the lubrication form of the equations governing the fluid motion of a thin liquid film on a curved, rotating, axisymmetric substrate, and Ref. 34, where the dynamics of a thin film on an axisymmetric spinning spheroid was considered. The above studies are however restricted to axisymmetric configurations.

The goal of this study is therefore to extend our previous work³⁴ to non-axisymmetric substrates. The governing equation for a thin liquid film deposited on a smooth closed surface of a substrate rotating around its vertical axis at constant angular velocity is determined, and the corresponding lubrication model is derived as an extended version of the equation presented in Ref. 27. We particularly focus on the thin film flow on a spinning ellipsoid as a typical example of a non-axisymmetric and non-constant curvature substrate. A parametric analysis is then performed to shed light on the effect of the centrifugal force on two basic ellipsoidal substrates, namely, a disk-like and a rugby ball-like substrate.

The outline of this work is as follows. In Sec. II, we define the physical setup, determine the relevant mathematical model, and derive the lubrication equation for the film thickness over a solid substrate, rotating around the vertical axis at constant angular velocity. In Sec. III, we utilize this lubrication model for a tri-axial ellipsoid by defining the preferred parameterization and the solution methodology. In Sec. IV, we discuss the film thickness distribution on different substrates by increasing the amplitude of the centrifugal forcing, before drawing concluding remarks in Sec. V.

In this section, we formulate the lubrication model for a thin liquid film deposited on a closed body, rotating around the vertical axis. The liquid is assumed to be an incompressible, Newtonian fluid of characteristic thickness h₀, density ρ, viscosity μ, and surface tension σ. The substrate is assumed to be smooth, and of characteristic length scale R. Its surface $S ( x )$ is described in three-dimensional (3D) space by employing an arbitrary curvilinear parameterization $x = ( x 1 , x 2 )$⁠. The free surface of the liquid is defined as $χ ( x , t ) = S ( x ) + h ( x , t ) n$⁠, where $h ( x , t )$ is the instantaneous local film thickness, and n is the unit vector normal to the surface S. The angular velocity is constant, and the velocity vector in Euclidean space is expressed via $ω = { 0 , 0 , ω }$⁠, where we use curly brackets to distinguish Cartesian coordinates from substrate-oriented coordinates $( x 1 , x 2 , y )$⁠. The system is subjected to gravitational, capillary, and viscous forces along with the centrifugal one. The configuration of the problem for an ellipsoid with the semi-axes length (A, B, C) is represented in Fig. 1.

This part is devoted to recalling the elements of differential geometry including metric coefficients, curvature terms, and the differential operators employed to state the lubrication model on a curved substrate.

A generic equation combining capillary and gravitational effects on stationary substrates of arbitrary shape has already been derived in the framework of the lubrication theory, assuming both the film parameter (⁠ $ε = h 0 / R$⁠) and the Reynolds number (⁠ $Re = ρ u 0 h 0 / μ$⁠) to be small (i.e., thin film and negligible inertia).^26,27,30,31 Here, we extend the derivation to rotating substrates. A proper formulation of the total body force is derived, and a procedure similar to that of Roy, Roberts, and Simpson²⁶ and Thiffeault and Kamhawi²⁷ is followed. The reader is referred to Ref. 27 for a detailed derivation. Here, for the sake of brevity, only the steps that differ are presented. In what follows, we express the total body force acting on the system and insert it in the Navier–Stokes equations. We then derive an evolution equation for the film thickness in accordance with the lubrication approximation.

Equation (16) is an extended version of the model of Roy, Roberts, and Simpson²⁶ and Thiffeault and Kamhawi²⁷ because it includes rotation and the order-ε term f. The lubrication equation derived without this term has been shown to be valid not only for gravity-driven flows on various stationary substrates,²⁹ but also on rotating spheroids.³⁴ Despite the fact that a number of studies on rotating spheres and spheroids neglect f and yet exhibit good agreement with the solution of the full Navier–Stokes equations,^22,23,34 for consistency, this correction of order ε should not be discarded.

The main outcome of our study is the recent lubrication model Eq. (16) that describes the complex dynamics of a thin liquid layer on a rotating substrate of arbitrary shape which can be simplified to various forms. For instance, on a constant curvature body (such as a sphere or a cylinder), the capillary term reduces to $Q 1 = ( h 3 ∇ 3 h )$ and admits a constant solution as h = 1 when the total body force vanish. If the total body force is constant, the system again grants a uniform solution on a constant curvature substrate. However, on a non-constant curvature substrate, even in a micro-gravity field, h = 1 is not a solution. Equation (16) for a non-constant curvature, rotating body reveals highly nonlinear terms and investigating the system behavior via a uniform initial condition highlights critical differences with the constant curvature case.

After formulating the lubrication equation that models the evolution of the film thickness on a rotating body, we focus on investigating the combined effects of curvature, rotation, and gravity on ellipsoidal substrates. The ellipsoid is chosen as a relevant illustrative example as it highlights key features of interest in our study: the combined effects of rotation and spatially distributed curvature on the dynamics of a fluid film on a non-axisymmetric substrate. In Secs. III A, III B, and IV, we discuss the parameterization of the ellipsoid, introduce two generic surface profiles, explain the solution methodology, and perform a parametric analysis.

The whole range of geometry-related components, including the differential operators, should be defined to construct and solve Eqs. (16). This can be done by setting either an orthogonal basis or a non-orthogonal one. For a tri-axial ellipsoid, one can use ellipsoidal or spherical polar coordinates, respectively.

Although the ellipsoidal coordinates appear more advantageous and convenient, as it is the natural way to describe the substrate via the geodesics and derive a simpler equation via the orthogonal parameterization, it has very basic disadvantages. One of them is the singularities at four umbilical points, and another one is the difficulty of relating critical locations on the ellipsoidal plane. While it is straightforward to locate the poles and the equator on a plane spanned by the spherical polar coordinates, it requires some extra effort to locate them on the ellipsoidal one. Hence, in the present work, we use the spherical polar coordinates to construct the film equation and to discuss the results on a basis spanned by $( θ , ϕ )$⁠. Figure 3 displays the mapping of the polar plane on an ellipsoidal surface for $( θ , ϕ ) = ( x 1 , x 2 )$⁠. The north and the south poles correspond to θ = 0 and $θ = π$⁠, respectively, whereas $θ = π / 2$ represents the equator (solid red line) and acts as a separator for the lower and upper hemispheres.

At this stage, to conduct a systematic investigation, we consider two non-axisymmetric ellipsoidal substrates, i.e., the disk-like and rugby ball-like objects illustrated in Fig. 3, labeled S₁ and S₂, respectively. The semi-axes of these ellipsoids are determined by $( C , C / 2 , C )$ and $( C , 2 C , C )$⁠, respectively, which corresponds to an oblate (S₁) and a prolate (S₂) spheroids with two equal semi-axes.

We specify the union of particular arcs as prime meridians of S₁ and S₂, to aid the visualization and the discussions. Formally, the prime meridian is determined by the arc that corresponds to $ϕ = 0$⁠. Here, the prime meridian is determined by $P M 1 = s ( θ , 0 ) ∪ s ( θ , π )$ on S₁, and it is set to $P M 2 = s ( θ , π / 2 ) ∪ s ( θ , 3 π / 2 )$ on S₂, represented by the solid and dashed blue lines in Fig. 3. The parametric analysis will be performed for the above defined structures S₁ and S₂, and the results will be reported with the help of these specific curves (equator and prime meridians) through the mapping of the polar plane on ellipsoids, as shown in Fig. 3.

We use the fixed value $ε = 0.0025$⁠, which corresponds for instance to a film thickness $h 0 = 10 − 4$ m and a characteristic length scale R = 0.04 m. A typical simulation takes about 30 s for 1602 degrees of freedom.

In this section, we perform a parametric study to illustrate the effects of gravity, capillarity, and rotation on the film dynamics. With respect to our choice of characteristic length R = C, the dimensionless semi-axes are defined by $( a , b , c ) = ( A / C , B / C , 1 )$⁠. For our ellipsoids S₁ and S₂, $( a , b , c ) = ( 1 , γ , 1 )$ with $γ = 0.5$ and 2, respectively.

In what follows, we study flows driven purely by capillarity (without gravity nor rotation) in order to highlight the correlation between the substrate curvature and the film thickness. In this case, the orientation of the body is not important. Later, we will specify the orientation of the substrate, as it is crucial when studying the effect of the gravitational and centrifugal forces.

We begin our investigation by considering the non-rotating case without gravity, such that the flow is purely driven by capillary forces. By setting $( B o , Ω ) = ( 0 , 0 )$⁠, we ensure that the body forces vanish (⁠ $Q 2 = 0$⁠) and implement a constant viscosity $μ h ( t ) = 1$ to investigate the thickness profiles on S₁ and S₂.

Figure 4 represents the projection of the mean curvature of the surface (κ) and of the film thickness (h) at $t = 10 7$ on the polar plane, and their distribution on the 3D representation of the substrate. The ellipsoid S₁ has its largest curvature along its prime meridian, whereas S₂ has its highest curvature localized at two points where the equator intersects the prime meridian. In both cases, the liquid, which is uniformly distributed at t = 0, migrates from regions of high curvature toward that of low curvature, as one would expect. The film thickness has its maximum on localized points of S₁ (intersection of PM₂ and the equator) and on a ring on S₂ (along PM₁), which corresponds to regions of low curvature. One would expect the film to thin indefinitely and dewetting to eventually occur when the film thickness becomes very small and inter-molecular forces not included in this model become dominant. Capturing this dewetting process is however beyond the scope of the present work. We conclude that the evolution characteristics observed for the capillary driven flow coincide with the physical insight.

Prior to presenting results for non-axisymmetric substrates, we briefly discuss the thin film dynamics on axisymmetric ellipsoids (spheroids). The flow on rotating spheroids can be simulated in 2D if assumed independent of $ϕ$⁠.^22,23,34 Here, we solve the same problem in 3D and provide examples of thickness distributions on the two substrates used in Ref. 34.

To obtain an axisymmetric flow in the presence of gravity and rotation, we rotate S₁ and S₂ by an angle π around the X-axis. The semi-axes are set to $( a , b , c ) = ( 2 , 2 , 1 )$ for S₁ and $( a , b , c ) = ( 0.5 , 0.5 , 1 )$ for S₂, corresponding to oblate and prolate spheroids, respectively. Figure 5 demonstrates the effect of rotation on these axisymmetric substrates. We employ the same parameter set as in our former study.³⁴ 

Both panels of Fig. 5 show the stationary thickness profiles and the thickness distribution on the substrate for different values of Ω, together with the cross section of the film thickness along $ϕ = 0$ in the lower section, at $t = 10 7$⁠. This time is much larger than the curing time, such that viscosity can be considered as large enough for the solution not to change significantly in practice. In the following, we denote large-time solutions as “stationary solutions.” The left panel represents the results for an oblate spheroid, in which the dimensionless angular velocity varies as $Ω = 0 , 20 , 30$⁠. On the steady substrate (Ω = 0), the film forms a relatively thicker distribution on the lower hemisphere and a thinner coating on the upper part since the fluid drains by gravity. As the ellipsoid rotates, the thickest region moves slightly below the equator while thinning elsewhere as a consequence of the centrifugal force that tends to draw the fluid away from the axis of rotation.

An increase in the angular velocity for the prolate spheroid also leads to a migration of the maximum film thickness drawn toward the equator, but in contrast to the oblate spheroid case, this peak has a smaller magnitude and is distributed over a wider range forming a smooth hump. The minimum film thickness at the poles, on the other hand, is found to be smaller for the prolate spheroid than for the oblate one, indicating a stronger likelihood of dewetting. These observations coincide with those obtained with the axisymmetric model.³⁴ Next, we continue our investigation on non-axisymmetric substrates, coming back to the substrates presented in Sec. IV A.

The film thickness on the non-axisymmetric substrates S₁ and S₂ is represented in Figs. 6 and 7, respectively, for increasing values of Ω = 0, 10, 20, 30, 40.

Figure 6 confirms that on S₁, and without rotation, the liquid thins on the upper part of the prime meridian and thickens around the south pole. The response is minor for Ω = 10, except for a slight shift of the thicker regions drawn toward the equator on the prime meridian. The film is seen to accumulate in the southern hemisphere along the prime meridian with the maximum migrating from the south pole drawn toward the equator along this prime meridian. Two localized humps are clearly visible for the case with the highest angular velocity—the red region on the bottom left panel of Fig. 6. It can be expected that further increase in the angular velocity would lead to the generation of ligament that would break up into satellite droplet, a topological change that the lubrication approximation is unable to capture.

In Fig. 7, a more complex pattern of the film thickness is formed on S₂ for the non-rotating case. The film reaches both its maximum and minimum on the prime meridian of S₂, just above and below the equator for Ω = 0, as substrate curvature and gravity compete to redistribute the fluid. Introducing rotation accumulates the liquid toward the localized regions of the maximum thickness of the static case on the prime meridian for increasing values of $Ω ≥ 20$⁠. These regions of maximum film thickness move closer to the equator, while still remaining slightly below the equatorial curve for increasing values of the angular velocity. For both S₁ and S₂, as the fluid is increasingly drawn toward two local maxima, the fluid thins out more and more away from these regions.

For a more quantitative representation, we represent thickness profiles on the equator and on the prime meridians in Figs. 8 and 9, respectively. Figure 8 shows that the minimum film thickness on the equator, located at the intersection with prime meridians for S₁ and S₂ in the absence of rotation, is altered to develop into a maximum by imposing rotation. In addition, deviations from the uniform thickness increase as the angular velocity increases, i.e., the maximum value increases, and the minimum value decreases. For Ω = 10 (red line), the film is almost uniform on the equator. Figure 9 displays the thickness along the prime meridians, namely, $ϕ = 0$ for S₁ and $ϕ = π / 2$ for S₂. On both substrates, the region of maximum thickness is transferred toward the equator (⁠ $θ = π / 2$⁠) for increasing Ω while the film thins at both poles.

In Fig. 10, we describe the evolution of the maximum thickness with a different perspective. On both substrates, the maximum thickness $max ( h )$ increases (blue lines) with Ω, and the location θ of maximum thickness moves from the south pole toward the equator (red lines). However, there is a threshold angular velocity, $Ω ≃ 30$⁠, below which the maximum thickness does not vary significantly on S₁, i.e., the maximum remains at the south pole. We also observe a sharp variation of the location of maximum thickness, for $Ω ≃ 30$ on S₁ and as soon as $Ω > 0$ on S₂.

Finally, and to illustrate the time dynamics, the temporal evolution of the film thickness on the prime meridians is presented in Fig. 11 for Ω = 30. The final time is set to $t = 10 7$⁠, in which the film is already cured on both substrates. It can be observed that on both S₁ and S₂, the location of the maximum on the southern hemisphere does not vary much but the magnitude of the maximum is much greater on S₂ compared to S₁. On the other hand, whilst the film is seen to thin at the southern pole for S₂, it thickens with time on S₁.

We investigated the 3D nonlinear dynamics of a Newtonian liquid deposited as a thin layer on a rotating ellipsoid. The Navier–Stokes equations for a solid body of arbitrary shape, rotating around the vertical axis, are simplified to an evolution equation of the film thickness by applying the lubrication approximation. This lubrication model is employed to examine the deviations on an initially uniform thickness, distributed on two basic structures, namely, S₁ (an oblate shape) and S₂ (a prolate shape), exposed to gravitational, capillary, and centrifugal forces.

First, the correlation between the substrate curvature and the capillary forces is demonstrated. In a zero gravity field, the liquid moved from the high-curvature regions toward the low-curvature parts. Next, the axisymmetric thickness profiles are reviewed on static and rotating spheroids. On a rotating oblate spheroid, the liquid accumulated toward the equator forming a thick ring slightly below the equatorial circle, where the thickness profile exhibit relatively steep humps due to stronger centrifugal forcing. On a prolate spheroid, although the liquid similarly moved to the equator, for a specific value of the angular velocity, the peak is spread over a wider range resulting in smaller deviations of the film thickness from uniformity, whereas a smooth hump formed for a larger velocity. The 3D implementation of the governing equation is validated through comparison of the results with the corresponding axisymmetric problem.³⁴ 

After preliminary discussions, we carried out a parametric analysis for non-axisymmetric ellipsoids as the main focus of this study. We examined thickness variations on two basic ellipsoidal structures by applying rotation and increasing the magnitude of the angular velocity. The investigation of non-axisymmetric ellipsoids points out that the response to the centrifugal force occurs mainly in the neighborhoods of the specific curves: the equator and the prime meridians. The thickness distribution for static substrates displayed distinct pattern formations. On a disk-like structure, the film got thinner on the upper crescent of the prime meridian while thickening on the lower half. On a rugby ball-like ellipsoid, for the non-rotating case, localized maximum and minimum points emerged on the prime meridian, just below and above the equator, respectively, as the draining toward the south pole was not complete due to curing effect. For both objects, the response of the system below a threshold value of the angular velocity was small-scale and acted similar to the static case. Beyond the corresponding threshold values, the location of the maximum points elevated toward the equator and got closer to it for greater angular velocities. The thickness profile exhibits qualitatively identical characteristics on both ellipsoids for a strong-enough forcing, in which steep hump formations develop marginally below the intersection points of the prime meridian and the equator where a thin layer coats elsewhere.

We presented possible alterations of the thickness profile on a curved surface by inducing rotation. Along the investigation of the relative effects of the acting forces, we pointed out that the emplacement of the complex structures (orientation of the object) is crucial when the body forces are active. By orienting an ellipsoid in such a way to align its semi-minor axis either in the horizontal plane or in the vertical direction, we modified the force distribution and generated different profiles on the same substrate.

This work is part of the project “Development of a multi-axis spin-coating system to coat curved surfaces” funded by the New Zealand Ministry of Business, Innovation and Employment Endeavour fund (Grant No. UOCX1904). The funding is gratefully acknowledged.

The authors have no conflicts to disclose.

Selin Duruk: Conceptualization (lead); Formal analysis (lead); Funding acquisition (lead); Investigation (lead); Methodology (equal); Project administration (equal); Software (lead); Supervision (equal); Validation (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (lead). Ross Geoffrey Shepherd: Conceptualization (supporting); Formal analysis (equal); Investigation (supporting); Methodology (supporting); Software (lead); Validation (lead); Writing – review & editing (supporting). Edouard Boujo: Conceptualization (equal); Methodology (equal); Supervision (equal); Writing – review & editing (equal). Mathieu Sellier: Conceptualization (lead); Formal analysis (equal); Funding acquisition (lead); Investigation (supporting); Methodology (equal); Project administration (equal); Supervision (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.