EPJ E. Soft Matter and Biological Physics. EPJ.org. Thin viscoelastic dewetting films of Jeffreys type subjected to gravity and substrate interactions

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1 EPJ E Soft Matter and Biological Pysics EPJ.org your pysics journal Eur. Pys. J. E (9) : DOI./epje/i9-77- Tin viscoelastic dewetting films of Jeffreys type subjected to gravity and substrate interactions Valeria Barra, Sariar Afkami and Lou Kondic

2 Eur. Pys. J. E (9) : DOI./epje/i9-77- Regular Article THE EUROPEAN PHYSICAL JOURNAL E Tin viscoelastic dewetting films of Jeffreys type subjected to gravity and substrate interactions Valeria Barra a, Sariar Afkami b, and Lou Kondic Department of Matematical Sciences, New Jersey Institute of Tecnology, Newark, NJ, 7, USA Received May 8 and Received in final form 7 September 8 Publised online: January 9 c EDP Sciences / Società Italiana di Fisica / Springer-Verlag GmbH Germany, part of Springer Nature, 9 Abstract. Tis work presents a study of te interfacial dynamics of tin viscoelastic films subjected to te gravitational force and substrate interactions induced by te disjoining pressure, in two spatial dimensions. Te governing equation is derived as a long-wave approximation of te Navier-Stokes equations for incompressible viscoelastic liquids under te effect of gravity, wit te Jeffreys model for viscoelastic stresses. For te particular cases of orizontal or inverted planes, te linear stability analysis is performed to investigate te influence of te pysical parameters involved on te growt rate and lengt scales of instabilities. Numerical simulations of te nonlinear regime of te dewetting process are presented for te particular case of an inverted plane. Bot gravity and te disjoining pressure are found to affect not only te lengt scale of instabilities, but also te final configuration of dewetting, by favoring te formation of satellite droplets, tat are suppressed by te slippage wit te solid substrate. Introduction Wetting or dewetting penomena are of great importance for different types of scientific and industrial processes, suc as painting, coating or printing. For tis reason, freeboundary or interfacial flows ave been intensively studied (see, for instance, te reviews [,] and references terein). To capture te interface instabilities, te position of te interface, or boundary between te different pases, needs to be modeled and found as a part of te solution of te equations governing te fluid flow []. Te goal of te present work is to provide te matematical description and a numerical investigation of te interfacial dynamics for free-boundary flows of tin layers of fluids, in te particular case in wic te liquid of interest is a viscoelastic fluid of Jeffreys type [], subjected to bot te disjoining pressure wit te substrate [5] and te gravitational force. Among te broad spectrum of natural or syntetic tin layers of liquids, viscoelastic films are ubiquitous. Polymeric liquids, in particular, are one example of viscoelastic liquids, constituted by a Newtonian (viscous) solvent and a non-newtonian (polymeric) solute, and, possess interesting features, suc as te stress relaxation and fading memory [6]. As oter suspensions or emulsions, poly- Contribution to te Topical Issue Flowing Matter, Problems and Applications, edited by Federico Tosci, Ignacio Pagonabarraga Mora, Nuno Araujo, Marcello Sega. a vb8@njit.edu b sariar.afkami@njit.edu meric liquids are considered one type of complex fluids. Viscoelastic liquids are, in turn, part of te wider class of non-newtonian fluids. Tey are caracterized by te fact tat te stress tensor is dependent on te strain rate via a relationsip tat is generally nonlinear, and tat can be, as in te model used in te present work, differential. Historically, te foundations of te long-wave (or lubrication) teory were laid in a pioneering work by Reynolds [7], tat analyzed te beavior of a viscous liquid confined between a solid substrate and a fluid-lubricated slipper bearing [8]. Subsequently, te understanding of te interfacial instability penomena arising in polymeric films dewetting substrates as motivated many teoretical and experimental studies, see, e.g., [9 ]. For te case in wic te gravitational body force is neglected, te long-wave formulation for tin viscoelastic films of Jeffreys type, subjected to substrate interactions, was developed by Rauscer et al. [], and subsequently treated in regimes of weak and strong slippage wit te substrate by Blossey et al. [], as well as in te book [5]. A numerical investigation of te formulation provided by Rauscer et al. [] was performed by te autors in [6]. Moreover, numerical studies concerning wetting and dewetting tin viscoelastic films, in te case of absence of gravity and slippage wit te substrate, were carried out, for instance, by Tomar et al. [7], and, more recently, by Benzaquen et al. [8], wo ave used te Maxwell model [9] to describe viscoelastic stresses. However, to te best of our knowledge, te derivation of a tin-film equation for viscoelastic films of Jeffreys type under te effects of bot

3 Page of Eur. Pys. J. E (9) : te gravitational force and te disjoining pressure, tat also includes te slippage wit te substrate, is missing in te literature. Hence, suc a derivation is proposed in te present study. Among te different types of interfacial instabilities, te ones due to tin films flowing down inclined planes ave been widely studied for te case of viscous liquids (see, for instance, [ 6]). Most recently, dripping penomena due to te competition between te Rayleig- Taylor [7] (RT) and Kapitza [8] instabilities ave been explored for viscous films flowing down planes, inclined at angles larger tan π/ wit respect to te base, using te Weigted Residual Integral Boundary Layer model [9]. Moreover, in recent times, some industrial applications, suc as te manufacturing of very tin (and possibly flexible) displays, ave motivated similar investigations involving complex fluids, suc as nematic liquid crystals [], variable-viscosity fluids [], or sear-tinning liquids []. Moreover, Kaus and Becker ave carried out a numerical investigation focusing on te RT instabilities of bi-layers of Maxwell fluids deposited on viscous ones [], in wic tey confirmed tat elasticity speeds up te growt of te RT instability. However, to te best of our knowledge, a similar numerical investigation tat uses te Jeffreys viscoelastic model is not available in te literature. Te goal of tis study is not only to investigate te effects of viscoelasticity on te interfacial flows of tin viscoelastic films anging on inverted planes, but also to analyze te competing effects of te different forces at play. For tin-film approximations, te van der Waals attraction/repulsion interaction force is used to model te film breakup and te consequent dewetting process, as well as to impose te contact angle wit te solid substrate. Tis force induces an equilibrium film on te solid substrate, leading to a prewetted (also called precursor) layer in nominally dry regions. We remark tat in te absence of oter forces, suc as gravity, te liquid-solid interaction force is te only possible driving mecanism of dewetting. Moreover, we notice tat te dynamics of dewetting processes can be divided in two regimes: te initial stage of te evolution, caracterized by amplitudes in te interface tickness tat are small relative to te initial eigt of te film and wose growt is analytically predicted by a Linear Stability Analysis (LSA); and te developed pase of te interface evolution, caracterized by amplitudes tat are no longer small and terefore cannot be described in terms of linear asymptotic approximations and require a fully nonlinear dynamic description. In tis work, we outline te teoretical and numerical study concerning te interfacial flow of two-dimensional tin viscoelastic films under te effects of bot te gravitational force and te disjoining pressure. We derive ere a novel governing equation describing te evolution of te interface of tin viscoelastic films lying on planes tat can ave an arbitrary inclination wit respect to te base, obtained as a long-wave approximation of te Navier-Stokes equations, wit te Jeffreys model for viscoelastic stresses. For te particular cases of orizontal or inverted planes, te LSA is performed to assess te effects of te different pysical parameters involved on te dynamics governing te linear regime, and compare teoretical predictions of te early stage of te dynamics, wit te numerical results obtained. Te competing effects of te pysical parameters involved on te lengt and time scales of instabilities are analyzed, in te linear and nonlinear regimes. We find tat, in te linear regime, te critical and most unstable wavenumbers are neiter dependent on te viscoelastic parameters, nor on te slip lengt, but only on te interactions induced by te disjoining pressure and te gravitational force. Moreover, we provide numerical simulations of te evolution of te interface in te nonlinear regime, for te particular case of an inverted plane. In tis regime, we find tat te gravitational force and te disjoining pressure affect te equilibrium configuration attained by te dewetted films, by favoring te formation of satellite droplets, and by forming a ump in te nominally dry central region, destabilizing te precursor film. Moreover, we notice tat te slippage wit te solid substrate suppresses te secondary droplets, observed in regimes of microgravity. We empasize tat te Jeffreys model (togeter wit te Maxwell model [9] and te Newton model for viscous fluids [6]) is appropriate to describe liquids in wic displacement gradients are small [6,]. More broadly, generalizations of te Jeffreys model, tat include nonlinear objective stress rates (in place of te simple time derivative of te stress) to describe te advection and rotation of te stress tensor wit te flow, are considered; for instance, te Oldroyd-B model [6] tat uses an upper-convected time derivative, or te corotational Jeffreys model [5] tat involves te Jaumann derivative. However, in te context of te present work, and similarly in studies concerning tin viscoelastic dewetting films in te absence of gravity (see, e.g., [, 6, 7]), te liquid films are considered to be initially at rest and undergo a spontaneous, relatively slow, dewetting flow. In [6,7], it as been found tat viscoelastic dewetting films exibit a slow, viscous response, especially in te early times and final stages of evolution. As Tomar et al. ave discussed in [7], results obtained from nonlinear simulations tat use linear viscoelastic models witout te upper-convected terms are expected to be qualitatively accurate. In particular, tis applies to te regions of te decay of te capillary ridge (as also discussed by Rauscer et al. in []), and in te central ole were secondary lengt scales of instabilities form and slowly coalesce (tat is te region were most of te attention is focused on in te present work). Altoug limited, bot te Maxwell and te Jeffreys models ave been applied to and proven to be useful for te analysis of a broad range of materials (see, e.g., [6]), and analyses using tese models ave served as baselines for studies tat ave considered teir expansions including nonlinearities. Te remainder of tis paper is organized as follows. In sect., we outline te matematical modeling, wose detailed description and tin-film approximation are given in te appendix. In sect., we outline te numerical metods employed. In sect., we discuss our numerical results for te linear and nonlinear regimes, and finally, in sect. 5, we draw our conclusions.

4 Eur. Pys. J. E (9) : Page of Matematical formulation We present ere te governing equation (wose detailed derivation is sown in te appendix) for a viscoelastic fluid lying on a plane, inclined at an angle α wit respect to te positive x-axis, and subjected to te disjoining pressure and weak slip regime wit te substrate. We consider an incompressible liquid, wit constant density ρ, surrounded by a gas pase assumed to be inviscid, dynamically passive, and of constant pressure. Te equations of conservation of momentum and mass, respectively, for te liquid pase are Liquid Gas Fig.. Scematic of a fluid interface and boundary conditions at te interface of te fluid and te solid substrate, for α = π (inverted plane). ρ( t v + v v)= (p + Π)+ σ + F b, v =, (a) (b) were v =(v (x,y,t),v (x,y,t)) is te velocity field in te Cartesian xy-plane (as per convention, te x-axis is parallel to te plane, and te y-axis is perpendicular to te plane), and =( x, y ); σ is te (symmetric) stress tensor, p is te pressure, Π = Π() is te disjoining pressure due to liquid-solid interaction forces (we note tat Π = except at te liquid-gas interface), and F b =(ρg sin α, ρg cos α), were, te gravitational acceleration constant is positive (g >) for te reference system depicted in fig.. To model te stresses, we use a generalization of te Maxwell model [9] for viscoelastic liquids: te Jeffreys model []. Te Jeffreys model, togeter wit te Maxwell model, belongs to a class of linear viscoelastic differential models. Viscoelastic materials can be described as mecanical systems were material points are connected by daspots (representing energy dissipating devices), springs (representing energy storing devices), or any combination of te two devices. For a Maxwell material [9], te system comprises a spring and a daspot in series, and for a Kelvin-Voigt material [7], a spring and a daspot in parallel. For a Jeffreys fluid, te system is composed of anoter daspot connected in parallel to a Maxwell system [7 9] (see fig. ). Te constitutive law for te Jeffreys model is given by σ + λ t σ =η( ǫ + λ t ǫ), () were ǫ is te strain rate tensor, e.g. ǫ ij =( v j / x i + v i / x j )/, wit i,j = {,}, andη is te dynamic viscosity coefficient. In tis model, te response to te deformation of a viscoelastic liquid is caracterized by two time constants, λ and λ,terelaxation time and te retardation time, respectively, related by λ = λ η s /(η s + η p ). Here, η s and η p are te viscosity coefficients of te Newtonian solvent and te polymeric solute, respectively, suc tat η = η s + η p (see fig. ). Noting tat te ratio η s /(η s + η p ), we ave tat λ λ [6]. We observe tat te Maxwell model [9] is recovered from te Jeffreys model wen λ =, and te Newtonian constitutive relation for viscous fluids [6] is obtained wen λ = λ. We empasize tat it is believed to be quite restrictive to expect a polymeric liquid of a broad molecular weigt distribution to be caracterized in terms of a single relaxation time [7]. In fact, te relaxation time occurring in Fig.. Jeffreys model represented as a mecanical system, were G represents te sear modulus, and η s and η p te viscosity coefficients of te Newtonian solvent and te polymeric solute, respectively. te Jeffreys (and Maxwell) constitutive relation is interpreted as te longest relaxation time exibited by a polymeric liquid [7]. In te present work, te Jeffreys model is preferred over te simpler Maxwell model to be able to describe features suc as te retardation due to te Newtonian solvent in te solution. In sect., we will analyze te influence of te relaxation and retardation time constants, as well as te oter pysical parameters, on te dynamics and morpologies of te dewetting films. Te system of eqs. () is subjected to boundary conditions at te free surface, represented parametrically by te function f(x,y,t) =y (x,t) =, and boundary conditions at te solid substrate (y = ). At te latter, we apply te non-penetration and te Navier slip boundary conditions, wit te slip lengt coefficient denoted by b. As also discussed in [,], long-wave models for tin films can be derived in different slip regimes. In tis work, we will focus on te weak slip regime. Te stress balance at te interface is given by (σ (p + Π)I) n = γκn, () were I is te identity matrix. In te absence of motion, tis condition describes te jump in te pressure across te interface wit outward unit normal n (wose definition is given in te appendix), and a local curvature, κ = n, due to te surface tension γ. Te form of te disjoining pressure, Π = Π(), used in tis work is given by te

5 Page of Eur. Pys. J. E (9) : following expression: Π = γ( cos θ e) M [( ) m ( ) m ], () were represents te equilibrium film tickness induced by te van der Waals attraction/repulsion interaction force, θ e is te equilibrium contact angle, formed between te fluid interface y = (x,t) and te solid substrate, and M =(m m )/[(m )(m )], were m and m are constants suc tat m >m >. In tis work, we coose m =andm =, as also widely used in te literature, for instance by te autors in [ ], but oter values can be modeled as verified in [5]. Te interested reader can find te derivation of te governing equation for te evolving interface (x,t)in te appendix, were all quantities and scalings are defined. We report ere its final dimensionless form in wic we use te compact notation ( )/ x =( ) x,and ( )/ t =( ) t, (similarly for iger-order spatial or temporal derivatives later on in te text) [( ( + λ t ) t +(λ λ ) + [( + λ t ) ( p x + S) ] +( + λ t )b ( p x + S) x ] ) Q R t =, (5) in wic we use te dimensionless form of te pressure, p, given in te appendix, and were Q and R satisfy te equations ( + λ t )Q = p x S, (6a) ( + λ t )R = (p x S). (6b) In eqs. (5) and (6), we ave used S = ρgl ε Vη C = ρgl ε Vη sin α, cos α. x (7a) (7b) Te reader can find te meaning of te scaling factors L, Γ, V, togeter wit te definition of te small parameter ε, in te appendix. Let B = ρgl /Γ ρgl ε /V η = O() (unless specified differently) be te Bond number, a dimensionless quantity representing te importance of gravity relative to surface tension. We note tat for te particular cases in wic te plane as a small inclination α wit te base, i.e., forα = εα or α = π + εα,te parameters in eqs. (7) can be expressed as S Bα and C B, respectively. We notice tat for te case in wic λ = λ (tat corresponds to a Newtonian fluid), eqs. (5) and (6) reduce to te governing equation for tin viscous films flowing down inclined planes, as outlined in [5, 6]. Moreover, we remark tat, for te particular cases for wic we present our results in tis work (i.e.,forα =,π), te term S cancels. However, it is retained from now on, to allow for easy extensions of te present study in future researc endeavors, tat may consider arbitrary values of te inclination angle α. Numerical metods To discretize eq. (5), we isolate te time derivatives from te spatial ones, so tat we can apply an iterative sceme to find te approximation to te solution at te new time step. We do so by differentiating te spatial derivatives and, assuming te partial derivatives of (x,t)tobecontinuous, obtaining [( ) ] λ tt + ( p x + S) + b { [( ) + +(λ λ ) Q R ( ) +( x ) t (λ λ ) Q R [( ) ] +λ t ( p x + S) x x x ]} t +λ t [( b ( p x + S) ) x] =. (8) Te spatial domain [,Λ] is discretized by a staggered structured grid, in wic te first- and tird-order derivatives are defined at te cell-centers, and te second- and fourt-order ones at te grid points. Following te natural order from left to rigt, adjacent vertices are associated to te indices i,i,i+, respectively. Tus, we let x i = x +iδx, i =,,..., N (were N = Λ/Δx,andΔx is te fixed grid size), so tat te endpoints of te pysical domain, and Λ, correspond to te x Δx and x N + Δx cell-centers, respectively. Similarly, we discretize te time domain and denote by n i te approximation to te solution at te point (x i,nδt), were n =,,... indicates te number of time steps, and Δt is te temporal step size, wic can be cosen adaptively to speed up te marcing algoritm wen te solution does not exibit fast temporal variations (see [6,, 7] for a detailed description). In addition to te numerical formulation provided in [6], te discrete versions of eqs. (6a) and (6b) are given by Q n+ i R n+ i Q n i Δt Ri n Δt = Qn i λ λ ( p x + S) n i, = Rn i λ λ n i ( p x + S) n i, (9a) (9b) tat we solve using te forward Euler metod wit initial conditions Q i =andr i =, respectively. Te nonlinear terms and are computed at te cell-centers, as outlined in [6,5,8]. We solve eqs. (8) and (9) for te particular case in wic α = π (according to te setup depicted in fig. ), and at te endpoints of te domain, we impose te x =

6 Eur. Pys. J. E (9) : Page 5 of xxx = boundary conditions, tat reflect te symmetry of te problem. Te condition x = gives te value of at te two gost points x and x N+ outside te pysical domain, i.e. = and N+ = N ; te condition xxx = specifies te two gost points x and x N+, i.e. = and N+ = N (consistent wit [5]). Moreover, similarly to te numerical study in [6], we apply a mixed implicit/explicit finite difference formulation to discretize te nonlinear equation (8). We remark tat, because of te symmetry condition, we can reduce te computational domain to alf te pysical domain. However, in te results tat follow, we sow te numerical solutions on te entire domain for ease of visualization. After Newton s linearization, we obtain a system of equations of te form Aξ = B, tat we numerically solve for te correction term, ξ, using a direct metod [6]. Te initial condition given for (x,) is a known function tat describes te initial perturbation of te fluid interface (see sect. ), and te initial velocity is t (x,) =, describing te fact tat te considered films are initially at rest. Results and discussion. Linear Stability Analysis (LSA) In tis section, we report our results regarding te special case of an inverted plane, for wic α = π, as depicted in fig.. We perturb a flat film of initial tickness by an oscillatory Fourier mode of amplitude δ (suc tat δ ), wit wavenumber k, i.e., welet(x,t) = + δ e ikx+ωt. Performing te LSA on eqs. (5) and (6) leads to te following dispersion relation: λ ω + [ +(K C + isk) ( λ )] ω + λ b ( ) +(K C + isk) + b =, () were we ave defined K C := k k (Π ( ) C). () We consider te real part of te two roots of te dispersion relation (), namely Re{ω } and Re{ω }. One is always negative (indicating stable modes), say Re{ω }, and te oter one as varying sign (describing potentially unstable ones), say Re{ω }. We find ( )] [+K C λ + λ b Re{ω } = λ + Re{ Δ ω } λ, () were we ave defined [ ( )] Δ ω := +(K C + isk) λ + λ b ( ) λ (K C + isk) + b. () Te critical wavenumber, for wic Re{ω } =, satisfies te relationsip k c = Π ( ) C. Hence, troug tis relationsip, we note tat te gravitational term affects te lengt scales of instability. We also remark tat, for te parameters considered in tis work, Π ( ) > and C >. Moreover, we notice tat te wavenumber of maximum growt, as for te case witout gravity [, 6], satisfies te relationsip k m = k c /. In addition, consistent wit te previous studies witout gravity [6,7], we find tat te maximum growt rate, ω m = ω(k m ), is an increasing function of λ and b, wile a decreasing function of λ.. Dewetting of tin viscoelastic films on inverted substrates We outline ere te numerical results for dewetting tin films under te influence of te disjoining pressure and te gravitational force. As anticipated in sect.., we consider te particular case in wic α = π, tat is, for films tat ang on an inverted plane. As described in sect.., we perturb te initially flat fluid interface of tickness, wit a perturbation caracterized by te wavenumber k = k m and δ =., i.e., (x,) = +δ cos(xk m ), and we coose te domain size, Λ, to be equal to te wavelengt of maximum growt, tat is, Λ m =π/k m. Initially, we consider dewetting films in a regime of no-slip wit te solid substrate, and subsequently, we analyze te effects of te substrate slippage on te dewetting dynamics and morpologies. For all te results tat follow, we use a fixed grid size of Δx =.; owever, all results ave been verified to be mes independent. To isolate te effects of te gravitational force, we start by analyzing te beavior of dewetting films for te particular case of te absence of te disjoining pressure wit te substrate. In fig., we present te comparison of te computed growt rates, corresponding to different wavenumbers (red dots), wit te teoretical values predicted by te dispersion relation, given by eq. (), for te following parameters: =, =., θ e = π/, B =, b=. () In fig., in particular, we sow te results for a Newtonian film, i.e., wit λ = λ =,forα = π (blue dotted line) or α = (blue solid line), and a viscoelastic film wit λ =5andλ =., and α = π (blue dased line). We remark tat in tis case, te only driving force for te instability is gravity. As expected, in te absence of te attraction/repulsion force wit te substrate, and for α =, tere is no instability, and te growt rate is negative for all wavenumbers. To compare wit te analytical values, we measure te computed growt rates in te linear regime in wic te amplitude of te interface function grows exponentially. In wat follows, we will sow numerical results for films dewetting an inverted substrate (i.e., for α = π), wit te same fixed initial and equilibrium ticknesses, equilibrium contact angle, and slip coefficient as sown in (), unless specified differently, and will vary te different pysical

7 Page 6 of Eur. Pys. J. E (9) : Fig.. Comparison of te computed growt rates, corresponding to different wavenumbers (red dots), wit te prediction of te LSA, for te parameters given in (), and, in particular, for a Newtonian film, i.e., witλ = λ =,andα = π (blue dotted line) or α = (blue solid line); and for a viscoelastic film wit λ = 5 and λ =., and α = π (blue dased line). 8 6 (a).5 parameters to investigate teir isolated effects. In fig., we investigate te dynamics of tin viscoelastic films, in te absence of te disjoining pressure. We compare te evolution of a Newtonian film, wit λ = λ = (blue solid curve wit circles), wit te one of viscoelastic films, wit λ =. and λ = (cyan dotted curve wit triangles), λ = 5 (green dased curve wit squares), and λ = (red solid curve wit crosses). Figure (a) sows te results at time t =, and fig. (b) at time t =. Consistent wit te results obtained in [] (and references terein), we observe tat te viscoelastic film wit te igest relaxation time exibits te fastest dynamics in te development of te RT instability. In fact, in fig. (a) te interface of te film wit λ = as significantly developed, wile te oter films are still in te initial pase of small interfacial amplitude. Eventually, all viscoelastic films reac te near-equilibrium configuration, depicted in fig. (b). A similar beavior was observed by te autors in [6] for te case of regular dewetting processes in te absence of gravity, for wic te instability was solely driven by te disjoining attraction/repulsion force wit te solid substrate, and for wic elasticity was found to facilitate te dynamics of te dewetting process. Analogously to te results obtained in [6], we ave confirmed ere tat λ does not significantly influence te dynamics or te attained morpologies. To investigate tis, in fig. 5, we plot te evolution of different viscoelastic films wit te same parameters as in (), in te absence of te disjoining pressure, wit λ =andλ =. (blue solid curve wit circles), λ =. (cyan dotted curve wit triangles), λ =.5 (green dased curve wit squares), and λ = (red solid curve wit crosses), respectively. We can see tat bot in fig. 5(a), at time t =.7,and in fig. 5(b), at time t =, te profiles of te different film interfaces completely overlap. Terefore, from now on, to investigate te material beavior of te dewetting (b) Fig.. Evolution of different viscoelastic films wit te same parameters as in (), in te absence of te disjoining pressure, and: λ = λ = (blue solid curve wit circles), λ =. and λ = (cyan dotted curve wit triangles), λ = 5 (green dased curve wit squares), and λ = (red solid curve wit crosses), respectively; in (a), at time t =, and in (b), at time t =. viscoelastic films, we will focus our attention on varying te relaxation time, rater tan te retardation time. As described earlier, te disjoining pressure induces an equilibrium (precursor) layer on te substrate. Hence, in te absence of tis interaction wit te substrate, we expect te tickness of te dewetting films to reac zero as t. In fig. 6, we plot te temporal evolution (in logaritmic scales) of te minimum tickness of viscoelastic films in te absence of te disjoining pressure, wit te same parameters as in () and λ =., λ = (blue solid curve), λ = (cyan das-dotted curve), λ =5 (green dased curve), and λ = (red dotted curve), respectively. We empasize tat, in fig. 6, we sow te late times of te evolution wit t [, 7 ], for wic exponential growt of te interface amplitude is not expected. Te sligt variations in te beavior of te different viscoelastic films are noticeable only in te (relatively) early times of te evolution sown in te inset of fig. 6, for wic exponential growt of te interface amplitude is observed..5.5

8 Eur. Pys. J. E (9) : Page 7 of (a) Fig. 6. Temporal evolution, in logaritmic scales, of te minimum tickness of viscoelastic films wit te same parameters as in () and λ =., λ = (blue solid curve), λ = (cyan das-dotted curve), λ = 5 (green dased curve), and λ = (red dotted curve), respectively. Te inset sows a magnification for te early times of te evolution, for wic min exibits an exponential decay. Wen te disjoining pressure is zero, te tickness reaces zero for times tat are long compared to te ones considered in tis work. (b) Fig. 5. Evolution of different viscoelastic films wit te same parameters as in (), in te absence of te disjoining pressure, wit λ = and λ =. (blue solid curve wit circles), λ =. (cyan dotted curve wit triangles), λ =.5 (green dased curve wit squares), and λ = (red solid curve wit crosses), respectively; in (a), at time t =.7, and in (b), at time t =. Moreover, our results suggest a power law scaling, tat is, min t s, were s is found to be /, for te data sown in fig. 6. Commonly, only te separate effects of te gravitational force or te disjoining pressure are analyzed. However, for liquid films of micro scale, te crossover region, were te effects of te two forces are comparable, can be considered. Hence, in fig. 7, we investigate te competing effects of te disjoining pressure and te gravitational force on te dynamics of dewetting films on an inverted plane. We compare te evolutions of two dewetting films: a Newtonian liquid, wit λ = λ = (blue dased curve) and a viscoelastic one, wit λ =, λ =. (red solid curve), in fig. 7(a) at time t =5.5, and in fig. 7(b) at time t =. Bot films form a large droplet in te center of te dewetting region, tat is not present wen disjoining pressure is not considered (cf. fig. (b) in wic only a small ump appears in te ole region). Moreover, in fig. 7(a), we notice tat te viscoelastic film exibits.5 two small secondary umps in te interface on te sides of te central droplet. Eventually, in fig. 7(b), te small umps disappear, and te viscoelastic film attains a nearequilibrium configuration wit a central droplet tat is sligtly larger tan one of te Newtonian film. We empasize tat figs. and 7 include films wit te same viscoelastic parameters, i.e., λ = λ = (blue curves) and λ =, λ =. (red curves), and only differ in te absence or presence of te disjoining pressure, respectively. As discussed in sect.., bot te disjoining pressure and te gravitational term affect te wavenumbers (and terefore te wavelengts) of maximum instability. However, for te cases presented in figs. and 7, te wavelengt of maximum growt is not significantly different. For te former, Λ m =8.89, and, for te latter, Λ m =8.8. To demonstrate tat te morpologies observed so far are independent of te particular initial perturbation and domain size cosen (i.e., for te fastest growing wavelengt, Λ m =π/k m ), we analyze a dewetting viscoelastic film, wit te same parameters as in (), and, in particular, λ =5,λ =., on a computational domain Λ =5 Λ m. Te film is initially perturbed by different wavelengts, Λ i =Λ m /i, wit i =,,...,5, suc tat 5 (x,) = + δ A i cos(πx/λ i ), (5) i= were te amplitudes A i are randomly cosen in te range [,]. In fig. 8, we plot te film evolution, at tree different times: t = (black solid curve), t = (blue dased curve), and t = (red dotted curve). We can see tat, at t =, a pattern is formed by satellite droplets tat are separated by an average distance d 8.8, wic is close

9 Page 8 of Eur. Pys. J. E (9) : (a) (b) Fig. 7. Evolution in te presence of te disjoining pressure, for films wit te same parameters as in (), and λ = λ = (blue dased curve) compared to a viscoelastic film λ =, λ =. (red solid curve), in (a), at time t =5.5,and in (b), at time t =. to Λ m =8.8, for tis set of parameters. Moreover, we notice tat, as in te case in wic a single wavelengt was considered, te droplets appear to reac a steady configuration, during te late times of observation. We remark tat, for te pysical parameters cosen, te growt rate of te fastest growing wavelengt, ω m.. Terefore, te largest time of evolution plotted in fig. 8 corresponds to t = ωm, tat is muc longer tan te breakup time, comparable to ωm [9]. A furter analysis of te evolution of te satellite droplets is provided in sect... Next, we investigate te effect of reducing te Bond number, B, on te morpologies of te dewetted films. Wile B could be modified in a number of different ways, for te present purposes we may consider tat variations of B are due to cange of g, e.g., we could consider tin films under microgravity conditions. In fig. 9, we plot te evolution of dewetting for a viscoelastic film wit λ = and λ =., in te presence of te disjoining pressure, Fig. 8. Evolution of a viscoelastic film, wit te same parameters as in (), and, in particular, λ =5,λ =., initially perturbed by different wavelengts wit random amplitude. for B =. (top row), and B =. (bottom row). In te results sown so far, we ave considered te no-slip boundary condition wit te solid substrate (given by b = ). We now add to our analysis te effect of slippage, by varying, in fig. 9, te slip coefficient from b =(first column) to b =. (second column). In fig. 9(a), for B =. andb =, at time t =, we notice tree small secondary droplets in te interface on eac side of te main central drop, tat eventually, at time t = 6, disappear. In fig. 9(b), for B =. andb =., we can see tat, at time t = only one central drop is present. In fact, te small droplets observed in fig. 9(a) are flattened by slippage. However, in te steady configuration attained at time t = 6, te slip wit te substrate causes also te central drop to be completely suppressed. We continue our investigation of te microgravity conditions, by furter reducing te Bond number. In fig. 9(c), for B =. and b =, at time t =.7 5, we can see te formation of multiple secondary droplets, tat mostly coalesce in te near-equilibrium configuration, attained at time t = 6. In fact, we can see tat only two satellite droplets remain in te final configuration visualized. Finally, in fig. 9(d), we plot te dewetting film for B =., b =.. By comparing te beavior of tis viscoelastic film wit te one sown in fig. 9(c), tat considers te same times of evolution (t =.7 5 and t = 6 ), we can see ow te slippage wit te substrate suppresses te satellite droplets. Moreover, we notice ow te near-equilibrium configuration displayed in fig. 9(d), at time t = 6,iste same as te one attained at time t =. We remark tat, for te simulations sown in fig. 9, B and b are te only two parameters varied, and all oter parameters in () are fixed.. Droplets analysis Te emergence and evolution of secondary droplets in tin dewetting films as been of interest in te literature (see, e.g., [6,, 9]). Te secondary lengt scales

10 Eur. Pys. J. E (9) : Page 9 of (a) (b) (c) (d) Fig. 9. Evolution in te presence of te disjoining pressure, for a viscoelastic film wit λ =andλ =., for B =. (top row), B =. (bottom row), b = (first column), and b =. (second column); in (a) and (b) for t = (blue solid curve) and t = 6 (red dotted curve), and in (c) and (d) for t =.7 5 (blue solid curve) and t = 6 (red dotted curve). For tese simulations, except for B and b tat ave varied, we ave kept te fixed parameters presented in (). of instabilities, tat are observed in te nominally dry region between te two separating rims, are induced by te interaction force wit te solid substrate (as deduced by comparing te viscoelastic films wit same material parameters in figs. and 7). Moreover, at parity of parameters related to bot viscoelasticity and disjoining pressure, tese secondary droplets are found to be favored by regimes of microgravity and suppressed by iger slippage wit te substrate (as sown in fig. 9). We present ere an analysis of te dependence of te satellite droplets on te different pysical parameters involved. We empasize tat, for te parameter studies tat follow, we vary one of te pysical quantities at a time and keep all oters at teir default value, given by (). In fig., we plot te evolution of te number of satellite droplets, for films wit λ = λ = (blue circles), λ =. and λ = (yellow triangles), λ = 5 (green crosses), and λ = (red diamonds), respectively, in a semilogaritmic scale (linear scale for te y-axis and logaritmic scale for te x-axis). We can see tat viscoelastic films exibit a iger number of droplets compared to te Newtonian one, and tat tese secondary instabilities remain longer for iger values of te relaxation time. Eventually tese droplets coalesce and reac a steady state (for te time of observation of te current numerical experiments, t ωm ). In fig., we plot te number of droplets in time, for te same regimes considered in fig. 9, in a semilogaritmic scale. We can see ow in microgravity conditions, a iger number of droplets is formed, and ow, for iger values of slippage wit te substrate, tese secondary drops are suppressed. We proceed by analyzing te effects of te equilibrium (precursor) tickness,, induced by te disjoining pressure. In fig., we plot te evolution of te number of droplets for a Newtonian film wit different equilibrium ticknesses: =.5 (blue circles),. (yellow triangles),.5 (green crosses),. (red diamonds), in a semilogaritmic scale, for a Newtonian film in fig. (a), and a viscoelastic one wit λ = 5 and λ =.,

11 Page of Eur. Pys. J. E (9) : Fig.. Evolution of te number of droplets, in a semilogaritmic scale, for different films in te presence of te disjoining pressure, wit te same parameters as in () and λ = λ = (blue circles), λ =. and λ = (yellow triangles), λ =5 (green crosses), and λ = (red diamonds). 5 (a) 5 5 (b) 5 6 Fig.. Evolution of te number of droplets, in a semilogaritmic scale, for different films wit te same parameters as in fig. 9. in fig. (b). We can see ow, in bot cases, a iger equilibrium tickness suppresses te formation of satellite droplets and favors teir coalescence. Te results sown so far, in sect.., concerned dewetting films wit a computational domain equal to te wavelengt of maximum growt, i.e., Λ = Λ m. Finally, we present a quantitative analysis of te evolution of droplets, for te cases in wic te films are initially perturbed by a number of wavelengts wit random amplitude, as described in eq. (5), for different computational domains. In fig., we consider a dewetting viscoelastic film, wit λ =5,λ =., for Λ =5 Λ m (blue circles), Λ = Λ m (yellow triangles), and Λ = Λ m (green crosses). In fig. (a), we plot te evolution of te number of droplets, and, in fig. (b), teir mean distance, bot in a semilogaritmic scale. We can see tat te maximum number of secondary droplets, N max, varies approximately Fig.. Evolution of te number of droplets, in a semilogaritmic scale, for different equilibrium ticknesses: =.5 (blue circles),. (yellow triangles),.5 (green crosses),. (red diamonds); for a Newtonian film, in (a), and a viscoelastic one, wit λ = 5 and λ =., in (b). linearly wit te computational domain. In fact, we can compare te results sown in fig. (a), wit te ones for te same viscoelastic film for wic te computational domain is equal to te single wavelengt of maximum growt rate, Λ = Λ m =8.8 (for tis set of parameters), depicted in fig.. We notice tat, for Λ = Λ m, te maximum number of droplets observed is N max =;forλ =5 Λ m, N max =, for Λ = Λ m, N max = 6, and, for Λ = Λ m, N max = 98. As discussed in sect.., we notice tat, for all te different computational domains, te number of droplets plateaus to a constant value for large times, i.e., t ωm, indicating tat te coalescence of droplets reaces a near-equilibrium state. We empasize tat for times longer tan te ones considered in tis work, i.e., fort, te number of droplets slowly decays to zero (see Glasner and Witelski [5]). In our results, te reaced minimum number of droplets, N min, also appears to depend linearly on te domain size. In fact, for Λ = Λ m,

12 Eur. Pys. J. E (9) : Page of (a) (b) Fig.. Droplets evolution, in a semilogaritmic scale, for a viscoelastic film wit te same parameters as in () and λ = 5 and λ =., on different computational domains: Λ =5 Λ m (blue circles), Λ = Λ m (yellow triangles), and Λ = Λ m (green crosses); in (a), te number of droplets in time; in (b), teir mean distance. For tese simulations, an initial perturbation wit different wavelengts wit random amplitude, as in (5), is considered. N min =,forλ =5 Λ m, N min =5,forΛ = Λ m, N min =, and for Λ = Λ m, N min = 6. Moreover, te mean distance between te droplets is comparable to Λ m =8.8 (as displayed in fig. (b)), except for te Λ case, in wic te mean distance, d.. 5 Conclusions We ave considered a novel long-wave governing equation for te interfacial flow of two-dimensional tin viscoelastic films dewetting substrates (wit a zero or weak slippage) tat can be inclined wit respect to te base, under te effects of te gravitational force and te disjoining pressure, in wic te stresses are described by te Jeffreys model. We ave carried out te linear stability analysis, tat sows tat te viscoelastic parameters and te slippage coefficient do not influence eiter te wavenumber corresponding to te maximum growt rate or te critical one. However, te lengt scales of instabilities are found to be affected by te gravitational contribution. Our numerical results of te computed growt rates in te linear regime are sown to be in agreement wit te teoretical prediction given by te linear stability analysis. We ave provided numerical simulations of tin viscoelastic dewetting films in te particular case in wic tey ang on inverted planes (i.e., for α = π). To isolate te effects of te gravitational force, we ave begun our investigation by analyzing dewetting films in te absence of te disjoining pressure. Te results for te nonlinear regime sow tat, consistent wit previous results tat did not consider gravity effects [6,7], a iger relaxation time speeds up te dewetting dynamics. Furtermore, we ave investigated te competing effects of te gravitational and te attraction/repulsion forces, by considering te evolution of viscoelastic films, dewetting an inverted substrate, in te presence of te disjoining pressure. We ave found tat, at parity of gravitational force (i.e., for te same Bond number), te disjoining pressure induces te formation of satellite droplets. Tese secondary instabilities, favored by small values of te Bond number, are suppressed wen a iger slippage wit te substrate is considered. Moreover, we ave analyzed te influence of te different pysical parameters on te formation and coalescence of te satellite droplets. We ave found tat a iger value of te equilibrium film tickness suppresses te formation of te secondary instabilities. In addition, we ave verified tat our results are independent of te particular initial perturbation and domain size cosen. In fact, by considering dewetting films on a computational domain muc longer tan te fastest growing wavelengt, and by perturbing tem wit different wavelengts possessing random amplitudes, we ave demonstrated tat te mean distance between te droplets is accurately described by te wavelengt of fastest growt. Finally, we ave observed tat te number of satellite droplets and teir distance scale approximately linearly wit te domain size. Future work sall consider te extension of tis numerical investigation in wic inclined planes of arbitrary angle α are considered. In tat case, boundary conditions for wic a constant influx is maintained at te inlet, as in [6], would ave to be included. Moreover, for arbitrary values of α, te form of te perturbation employed for te linear stability analysis sould present a wave-like mode [5,6], rater tan an oscillatory one. Furtermore, extensions of tis investigation to tree spatial dimensions would allow one to describe and capture fingering instabilities, known to arise in te direction transversal to te flow [5,5]. Te autors would like to tank te referees for te very toroug and detailed revision provided.

13 Page of Eur. Pys. J. E (9) : Autor contribution statement All te autors were involved in te preparation of te manuscript. All te autors ave read and approved te final manuscript. Appendix A. Derivation We define te outward unit normal, n,as n = (( x ) +) / ( x,). (A.) Te kinematic boundary condition is given by Df/Dt = f t +v f = (were we ave used te material derivative D( )/Dt), and in wic substituting f(x,y,t) = y (x,t) gives t (x,t)= (x,t) v (x,y)dy. (A.) x As anticipated, te boundary conditions at te solid substrate are described by te non-penetration condition for te normal component of te velocity and te Navier slip boundary condition for te tangential one, respectively v =, v = b η σ, (A.) were for te Navier slip condition we use te Newtonian sear stress, tat is dominant in regions away from te contact line [5]. We observe tat b = implies a no-slip boundary condition wit te substrate. Te influence of tis parameter on te morpology of te dewetting front in lubrication models as been investigated bot teoretically and experimentally, for instance see [,]. We nondimensionalize te system of governing equations using common scalings for long-wave formulations x = Lx, (y,,,b)=h(y,,,b ), (A.) v = Vv, v = εv v, (p,π)=p(p,π ), (A.5) (t,λ,λ )=T(t,λ,λ ), (A.6) ( ) σ σ = η σ σ T σ σ ε σ ε σ, (A.7) were H/L = ε is te small parameter. To balance pressure, viscous and capillary forces, te pressure is scaled wit P = η/(tε ), te surface tension wit γ = Γγ, were Γ = Vη/ε, and te time wit T = L/V. Following te formulation in [], we can consider te scaled surface tension, γ, to be equal to one, by coosing te velocity so tat te inverse capillary number is Ca = ε γ/(vη) = [5], and, subsequently, L = γε T/η.Moreover, we notice tat, by coosing te vertical lengt scale, H, to be equal to te initially flat fluid interface eigt, we can simplify te dispersion relations in sect.., wit te value =. However, we refrain from applying tis simplification to allow for generalization to arbitrary coices of te reference film tickness. We note tat one could also consider oter scaling factors. For instance, if gravity is considered to be te sole driving force for instabilities, one could set te Bond number B =. Tis leads to te capillary lengt scale, i.e., L =(γ/(ρg)) /. Subsequently, te pressure would be scaled by P = γh/l = ρgh, te time by T = ηγ/h (ρg), te velocity determined by V = L/T, and te normalized surface tension, similarly to te former case, would be γ =. However, in te current work, we are interested in te competing mecanisms of te disjoining pressure and te gravitational force tat togeter drive te instabilities and affect teir lengt scales; moreover, as discussed in sect.., we want to be able to vary te Bond number to be able to analyze te beavior of films under te microgravity conditions. For tis reason, we prefer te former coice of scalings, given by eqs. (A.) (A.7). To avoid a cumbersome notation, we drop te superscript and consider for te rest of tis document all quantities to be dimensionless. Te incompressibility condition (b) is invariant under rescalings, wile te dimensionless forms of eq. (a) for te x and y components, respectively, are ε Re Dv Dt = σ ε x + σ ε Re Dv Dt = ε y p x + S, ( σ x + σ y ) p y C, (A.8a) (A.8b) were Re = ρv L/η is te Reynolds number, assumed to be of order /ε or smaller. In eqs. (A.8) we ave used S and C, wose definitions are given in eqs. (7), and te extended notation for te derivatives ( )/ x and ( )/ y (tis version will be used from now on, wenever needed to avoid double subscripts). Te dimensionless distinct components of te stress tensor given by te Jeffreys model, eq. (), satisfy ( + λ t ) σ =(+λ t ) v x, ( + λ t ) σ =(+λ t ) v y, ( + λ t ) σ =(+λ t ) v y +ε ( + λ t ) v x. (A.9a) (A.9b) (A.9c) Te kinematic boundary condition, eq. (A.), is invariant under rescaling, wile te non-penetration condition and te Navier slip boundary condition for te velocity components parallel to te substrate, given in eq. (A.), in dimensionless form are v =, v = bσ, (A.) were in te weak slip regime b = O() []. Te leadingorder terms in te governing equations (A.8a) and (A.8b), respectively, are σ y = p x S, p y = C. (A.a) (A.b)

14 Eur. Pys. J. E (9) : Page of Te leading-order terms of te normal and tangential components of te stress balance at te free surface, y = (x,t), eq. (), respectively are and p = xx Π, on y = (x,t), (A.) σ =, on y = (x,t). (A.) From eq. (A.b) we know tat te pressure is a linear function of y (see also [, 5]). Hence, by integrating eq. (A.b) and by using te boundary condition for te pressure at te interface, given by te leading-order term of te normal component of te pressure balance, eq. (A.), we obtain p = xx Π C(y ). (A.) Te x component of te pressure gradient is given by differentiating eq. (A.) in te x direction, obtaining p x = xxx Π x + C x. (A.5) Te nondimensional form of Π in eqs. (A.) and (A.) is given by Π = γ( cos θ e) ε M [( ) m ( ) m ], (A.6) were, we remark tat all te quantities are considered to be normalized. Moreover, in order for te expression in eq. (A.6) to be O(), we expand cos θ e, by considering θ e = εθ e,sotat cos θ e ε θ e /. Tus, we can recast Π γθ e M [( ) m ( ) m ]. (A.7) For te particular set of parameters considered in te present study (see sect. and ()), we notice tat γθ e /M γ( cos θ e )/M = O() (were te latter expression as been used in tis work). Integrating eq. (A.a) from y to (x,t), we obtain σ =(y )p x (y )S. (A.8) Substituting tis form of σ, eq. (A.8), into eq. (A.9c), we obtain (up to te leading order) ( + λ t )[p x (y ) S(y )] = ( + λ t ) v y. (A.9) Integrating eq. (A.9) from to y and using te corresponding boundary conditions at te substrate, given in eq. (A.), we obtain ( + λ t )(v + bp x bs)= [( ) ] y ( + λ t ) y (p x S). (A.) Integrating eq. (A.) from y =toy = (x,t) gives [ ] (x,t) ( + λ t ) v dy + b p x b S λ t (v (y = (x,t)) + bp x bs)= [ ] ( + λ t ) (p x S) + λ t (p x S). (A.) Taking te spatial derivative of te latter equation and substituting it into te kinematic boundary condition (A.), we obtain a long-wave approximation in terms of v and (x,t) t + λ [ tt + x (v (y = (x,t)) t )] = ( ) x [( + λ t ) p x S +( + λ t ) ( b p x b S ) ] {[ ] } x λ (p x S)+λ b (p x S) t. (A.) To write tis in a closed form relation for (x,t), we note tat eq. (A.) can be written in a more compact form as a linear ordinary differential equation for v (assuming all oter quantities known at a given time), as v v + λ t = ( + λ t )(bp x bs) [( ) ] y +( + λ t ) y (p x S). (A.) One can simply solve eq. (A.) by integrating in time, obtaining v = λ t e t t λ f(x,y,t )dt, (A.) wit f equal to te rigt-and side of eq. (A.). Integration by parts can be performed to recast eq. (A.) at y = (x,t), and finally one finds te dimensionless form of te governing equation (5). References. P.-G. de Gennes, Rev. Mod. Pys. 57, 87 (985).. R. Scardovelli, S. Zaleski, Annu. Rev. Fluid Mec., 567 (999).. G. Tryggvason, R. Scardovelli, S. Zaleski, Direct Numerical Simulations of Gas-Liquid Multipase Flows (Cambridge University Press, Cambridge, ).. H. Jeffreys, Te Eart: Its Origin, History, and Pysical Constitution (Cambridge University Press, Cambridge, 95). 5. J. Israelacvili, Intermolecular & Surface Forces (Academic Press, London, 985).