Falling liquid films on longitudinal grooved geometries

Falling liquid films on longitudinal grooved geometries: Integral boundary layer approach I. Mohammed Rizwan Sadiq, Tatiana Gambaryan-Roisman, and Peter Stephan Citation: Phys. Fluids 24, 014104 (2012); doi: 10.1063/1.3675568 View online: http://dx.doi.org/10.1063/1.3675568 View Table of Contents: http://pof.aip.org/resource/1/PHFLE6/v24/i1 Published by the American Institute of Physics.

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PHYSICS OF FLUIDS 24, 014104 (2012)

Falling liquid films on longitudinal grooved geometries: Integral boundary layer approach I. Mohammed Rizwan Sadiq,1,a) Tatiana Gambaryan-Roisman,1,2 and Peter Stephan1,2 1 2

Center of Smart Interfaces, TU Darmstadt, 64287 Darmstadt, Germany Institute for Technical Thermodynamics, TU Darmstadt, 64287 Darmstadt, Germany

(Received 12 April 2011; accepted 5 October 2011; published online 9 January 2012)

Falling thin liquid film on a substrate with complex topography is modeled using a three equation integral boundary layer system. Linear stability and nonlinear dynamics of the film in the framework of this model are studied on a topography with sinusoidal longitudinal grooves aligned parallel in the direction of the main flow. The linear stability theory reveals the stabilizing nature of the surface tension force and the groove measure on the film, and the pronounced destabilizing effects of inertia. The evolution of the film thickness is tracked numerically for a vertically falling film on a grooved geometry by choosing wavenumbers corresponding to the unstable mode where the growth rate of instability is maximum. The effect of surface geometry on the temporal evolution of the film dynamics is analyzed on a periodic domain. Numerical investigations agree with the linear stability predictions and show that the longitudinal grooves exert a stabilizing effect on the film and the waviness is suppressed when the steepness of the longitudinal groove C 2012 American Institute of Physics. [doi:10.1063/1.3675568] measure increases. V

I. INTRODUCTION

Thin liquid film flow along a surface exhibits a rich variety of spatial and temporal structures and is one of the most vivid and experimentally accessible examples of the intrinsically unstable extended systems. The falling film problem exhibits a wide range of dynamic phenomena such as formation of waves, ruptures, contact lines, corners, and cusps. Ever since the investigation of Benjamin1 and Yih,2 the dynamics and stability of a thin film flow is a subject of hot pursuit which facilitates to understand the role of waves in the transport of heat, mass, and momentum in the liquid and gas phases. A detailed review of thin film flow problems are available in Cross and Hohenberg,3 Chang et al.,4 Alekseenko et al.,5 Colinet et al.,6 Velarde and Zeytounian,7 and Nepomnyashchy et al.8 Due to the presence of the free surface, manipulation of the entire Navier-Stokes system numerically is a complicated task. The intricacy involved in thin film flow dynamics of a free boundary problem led to the derivation of various asymptotic approaches to describe physics involved in the flow in terms of simple reduced partial differential equations. When the inertial effects are negligible in comparison to the viscous effects, an asymptotic series with respect to a small wave number expansion can be sought to reduce the governing equations into a single nonlinear partial differential equation for the evolution of local film thickness layer,9 thereby enslaving the associated flow variables to the local film thickness. Under the auspices of this lubrication approximation and due to the success of such a model based on its great simplicity, several works have been reported investigating Benney’s long-wave model for film flows at small Reynolds numbers on flat substrates involving various physical mechanisms.10–13 Additional assumption of small film deformation led to the derivation of Kuramoto-Sivashinksy equation (which has become a prototype for studying weak turbulence or spatio-temporal chaos), the Kawahara equation (which includes dispersion and synchronizes the irregular patterns of the Kuramoto-Sivashinksy equation), a)

Present address: Department of Mathematics and Statistics, University of Konstanz, Germany. Author to whom correspondence should be addressed. Electronic mail: [email protected].

1070-6631/2012/24(1)/014104/20/$30.00

24, 014104-1

C 2012 American Institute of Physics V

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and their invariants.14 In all these modeling approaches, the Reynolds number was considered to be of O(1) and the equations gave a first insight in the underlying interfacial dynamics. One of the major drawbacks of the above models is that they are valid under the assumption that the flow is weakly unstable. When this restriction is not satisfied, the models produce erroneous results. Although Benney’s model faithfully captures the physics at low Reynolds number, at moderate Reynolds number such a model breaks down and shows unrealistic finite-time blow up behavior for the solitary-like wave solutions in the neighborhood of the parameter space near the onset of the instability threshold.15–18 Also, the Benney, the Kuramoto-Sivashinksy and the Kawahara equations when compared with the experimental investigations of Liu and Gollub19 for the maximal growth rate (predicted for a film flow with Re ¼ 29 and We ¼ 35 on a plane inclined at b ¼ 6.4 to the horizontal plane) were shown to over predict this value by a 45-fold factor.20 The investigations with these models further showed that some of the properties of the weakly unstable films are preserved for the flows with higher Reynolds numbers, nonetheless, a new approach is needed to describe the film dynamics for moderate Reynolds number more realistically. Maintaining the dimensionality of the system similar to the Navier-Stokes equations, a new system of equations to describe the film flow system based on the boundary layer equations21,22 were developed. The integral boundary layer (IBL) model which combines the Ka´rmań-Polhausen averaging technique in the boundary-layer theory with the assumption of a self-similar parabolic velocity profile and considering the spanwise momentum balance to be dominated by hydrostatic forces. This model describes the dynamics of moderate to large amplitude disturbances in thin wavy falling films and has been extensively studied on flat substrates.4,23–31 In contrast to the Benney’s long-wave modelling approach where the relative orders of the film amplitude and the governing dimensionless parameters like the Reynolds number and the Weber number are assigned a priori, the IBL system is derived only with the long-wave expansion and without any overly restrictive stipulations on the order of the amplitude and dimensionless-groups. The topography of the substrate may play an important role in the development of the instability. Film flows on complex patterned substrates with wavy walls have various engineering applications such as two-phase heat exchangers,32–34 absorption columns, distillation trays,35,36 reactor,37 etc. Surface modifications on the substrate topography are found in the heat transfer instruments for the intensification of single-phase heat transfer as well as condensation, evaporation, and boiling phenomena.38–40 Viscous thin film flows on uneven walls arise frequently in a large variety of coating applications including fabrication of microelectronic components, especially in the manufacture of magnetic memory devices, magnetic disks, compact disks, and optical devices. Understanding the effect of wall topography upon the film stability and dynamics is an important step towards the design and optimization of industrial apparatuses, such as heat exchangers, evaporators, and condensers. Experimental studies of the falling film flow over wavy undulated surfaces reveal new phenomena which are not found in the flow of a liquid on a planar surface, such as the formation of surface rollers, standing waves, or hydraulic jumps in the form of steep shocks at the free surface.41 Also, the experimental investigations report about the formation of eddies due to the modification in the flow field.42 Most of the theoretical and numerical studies of falling liquid films on structured surfaces are devoted to surfaces with grooves arranged normal to the flow direction (undulated or wavy surfaces). Using a perturbation analysis, the linear stability of the film flow over a wavy surface was considered by Wierscheim and Aksel43 and Wierscheim et al.44 The analysis was based on long undulations, when the liquid free surface follows the corrugated geometry. Wierscheim et al.44 employed a curvilinear approach to show that the critical Reynolds number for the onset of surface waves is higher in the case of film flows over an undulated topography compared to the case of a planar wall. It was shown that the theoretical results at large inclination angles agreed quantitatively with the experimental results. Trifonov45 performed a numerical investigation on the stability of a viscous liquid film flowing down a periodic surface based on the Navier-Stokes equations in their full statement. A range of corrugation parameters was identified, which stabilizes the flow for moderate Reynolds number. Based on the weighted residual integral boundary-layer model (WRIBL), Oron and Heining46 studied the nonlinear dynamics of the film on an undulated vertical

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wall. The reduced dimensionality of the model showed good efficiency in the analytical and numerical investigations. D’Alessio et al.47 used WRIBL model to investigate the gravity-driven flow over uneven surfaces. Their study showed that the bottom topography has a stabilizing influence on the flow up to a moderate surface tension effect. However, the film flow is destabilized for the strong surface-tension effects provided the wavelength of the bottom undulation is sufficiently short. Ha¨cker and Uecker48 analyzed the weighted residual integral boundary layer equation and its regularized version for film flows based on curvilinear coordinates employed by Wierscheim et al.49 over a wide parametric range. Their study also showed that with increase in the bottom waviness and the inverse Bond number, the critical Reynolds number decreases. It has been shown that the bottom modulation may introduce a short wave instability not present over a flat bottom. Micro and minigrooved wall structures have been recommended with an anticipation to improve thin film evaporation in lean premixed prevaporized combustion technology (LPP).39,50,51 By using LPP, the NOx emission in the modern gas turbine combustors can be considerably reduced. When the wall surfaces of the LPP chamber walls are designed with grooved surfaces, the fuel evaporation rate is found to increase significantly, since the grooves make use of the capillary forces to evenly distribute the liquid thereby preventing the appearance of dry patches. Literature on the stability of the film thickness is still not abundant for the case of micro and ministructured walls with longitudinal topography. Investigations on the effect of surface micro-geometry on the stability of an isothermal film falling in the direction parallel to the grooves were performed by Gambaryan-Roisman and Stephan52,53 in the framework of the longwave theory. Sinusoidal grooved topography was shown to stabilize the film stronger than the triangular grooves.52 However, both the topographies were shown to stabilize the film flow. The investigations also revealed that the micro-geometry stabilized the film when the structured wall is completely covered with fluid. On the other hand, it induces instability when the wall is not completely covered with fluid.53 Experimental investigations for falling films on structured walls54–57 show that the amplitude of the waves in the case of a structured wall is smaller in comparison to the film falling on an unstructured wall. These results agree with the predictions of the linear stability analysis.52,53 However, the above theoretical results obtained using the long-wave theory are applicable in a very limited range of Reynolds number. Recently, the steady deformations of the liquid-gas interface for very small wall corrugations were reported in Gambaryan-Roisman et al.,58 using the Benney type and IBL models for the film flowing over three-dimensional topography. It has been shown that the Benney model adequately describes the free surface deformation of the falling film flowing along the structured plate for Re < 5. Also, it has been observed that opposed to the straight grooves in the longitudinal direction, the grooves with meandering path led to significant deformations of the liquid-gas interface. A comprehensive understanding of the above phenomena can be achieved by numerically investigating the dynamics and stability of falling films on a complex topography. In the present study, the linear stability and the nonlinear dynamics of an isothermal falling film along the sinusoidal longitudinal grooves are investigated for an IBL model. This model reproduces numerous experimental observations and does not exhibit singularities for the solitary wave solutions at moderate Reynolds number,31,59 although it does fail in the vicinity of the linear stability threshold.31 The present analysis forms the first step towards studying the behavior of an isothermal falling film on a micro and minigrooved sinusoidal topography at moderate Reynolds numbers with grooves being arranged parallel to the main flow direction. II. GOVERNING EQUATIONS AND BOUNDARY CONDITIONS

A viscous, incompressible fluid layer falling along a grooved substrate under the influence of the gravitational acceleration g is considered. The grooves are oriented direction in the pyparallel l represents the 1 cos with the gravity induced flow as shown in Fig. 1, where lðx; yÞ ¼ amp 2 k groove shape (periodic in spanwise direction), f(x,y,t) ¼ l(x,y) þ h(x,y,t) is the location of the interface at any time t, and h(x,y,t) describes the local instantaneous film thickness. Also, lamp and k characterize the amplitude and the wavelength of a typical groove of the substrate. Properties of the fluid like the viscosity, density, and the surface tension do not vary, and the surrounding air is considered passive.

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FIG. 1. (Color online) Schematic representation of the fluid flow on a grooved topography.

The system of equations governing the fluid flow problem is made dimensionless by choosing the length scale h0 as the transverse length scale for a flow whose mass flux is q0 ¼ qgh30 sin b=3l and the velocity is u0 ¼ gh20 sin b=3. The longitudinal and the lateral length scales are chosen as k. Here, ¼ l/q is the kinematic viscosity where l and q represent the dynamic viscosity and the density of the liquid, respectively, and g is the gravitational acceleration. The scales are defined as30,31 lÞ ¼ ðx=k; y=k; z=h0 ; h=h0 ; l=h0 Þ; ð x; y; z; h; ¼ ðu=u0 ; v=u0 ; wk=ðu0 h0 ÞÞ; ð u; v; wÞ ðp; tÞ ¼ p=ðqu20 Þ; tu0 =k : Conservation of mass for the fluid and the components of momentum balance are z ¼ 0; ux þ vy þ w uz ¼ eRe px þ 3 þ e2 uxx þ uyy þ uzz ; eRe ut þ uux þ vuy þ w vz ¼ eRe py þ e2 vxx þ vyy þ vzz ; eRe vt þ uvx þ vvy þ w xx þ e2 w yy þ w t þ uw x þ vw y þ w w z ¼ Re zz ; pz 3 cot b þ e e2 w e2 Re w

(1) (2) (3) (4)

where the subscripts denote partial differentiation. The equations described above require appro x; yÞ to priate boundary conditions on the free interface z ¼ fð x; y; tÞ and the rigid surface z ¼ lð complete the definition of the problem. No slip and no penetration conditions are imposed on the solid-liquid interface, which states u ¼ v ¼ 0

and

¼ 0 on w

z ¼ l:

(5)

The normal stress condition on the deformable interface is p ¼

i 2e 2 h 2 y vz þ ew 2 þ uy þ vx fxfy fxðuz þ ew e þ v Þ f þ w f f u y y x x z x y ReN 2 2 3e S on z ¼ f; 2 3 fxx 1 þ e2 fy2 þ fyy 1 þ e2 fx2 2e2 fxfyfxy Re N gh3

(6)

0 where Re ¼ 302 sin b and S ¼ qrh 3l2 are the Reynolds and the inverse Crispation numbers with r being the surface tension. The second term on the right hand side of the equality within brackets in Eq. (6) represents the mean film curvature and N ¼ ð1 þ fx2 þ fy2 Þ1=2 measures the metric. The balance of tangential stresses on the deformable free surface z ¼ f are given by z uxÞ þ 1 e2 fx2 uz þ e2 w x e2 fy uy þ vx e2 fxfy vz þ ew y ¼ 0; 2e2 fxðw (7)

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z vy þ 1 e2 fy2 vz þ w y e2 fx uy þ vx e2 fxfyðuz þ ew xÞ ¼ 0; 2e2 fy w

(8)

and the kinematic boundary condition at the free surface is defined as ¼ ht þ ufx þ vfy on w

z ¼ f:

(9)

x; yÞ ¼ L0 ð1 cosðp yÞÞ, where L0 ¼ lamp/h0. Here, The scaled bottom profile is of the form lð 2 h0 the ratio e ¼ k 1 and represents a small parameter. Also, in order to focus attention on the l lamp 2 weak bottom waviness, L0 ¼ amp h0 ¼ ek is of OðeÞ by considering lamp =k ¼ Oðe Þ, where L0 represents the groove measure. The bar notation is dropped from this point onwards for convenience. III. INTEGRAL BOUNDARY LAYER MODEL

Mathematical analysis of the free surface flow problem is complicated due to the fact that the location of the interface is not known a priori and must be determined as part of the solution. This has led to the development of variety of approximate mathematical models and numerical techniques based on the non-dimensional parameters governing the film flow system. However, the limits of validity of such models and the approximate equations are still a matter of active research. In falling film problems, the strong nonlinear interaction of long waves gives rise to short wave components which eventually trigger an additional wave suppression mechanism due to the force of inertia.14,60 The long-wave evolution equation captures only the partial effects of inertia. The effects of inertia are maintained in the boundary layer equations by eliminating pressure due to thin-film assumption and then leaving the inertial terms in the streamwise and spanwise components of the momentum balance unaffected. Although boundary layer equations significantly simplify the complete Navier-Stokes equations, they represent a coupled system for the velocity field along the streamwise and spanwise directions and does not represent a simple equation for the deformable interface. This difficulty is overcome by reducing the dimensionality of the boundary layer equations while keeping the model accurate at Oð1Þ by assuming self similar velocity profiles61,62 " " # # 3q1 z l 1 z l 2 3q2 z l 1 z l 2 and v ¼ ; (10) u¼ h h h 2 h h 2 h beneath the film layer, which Ð f persists evenÐ ffor large Reynolds numbers when the free surface is no longer flat.5 Here, q1 ¼ l udz and q2 ¼ l vdz represent flow rates in the x and y directions. These velocity profiles satisfy the no-slip condition at the solid-liquid interface and the zero tangential stress condition at the liquid-gas interface. Depth averaging momentum equations across the film layer22–24,30,31,61,62 leads to the following system of three equations: ht þ q1x þ q2y ¼ 0; 2 6 q1 6 q1 q2 3 q1 3 cot b h 2 hfx þ Weh r2 f x ; þ ¼ q1t þ 5 h x 5 h y Re Re h 6 q1 q2 6 q22 3 q2 3 cot b q2t þ hfy þ Weh r2 f y ; þ ¼ 5 h x 5 h y Re h2 Re 3S where We ¼ Re 2 ¼

ð3FiÞ1=3 1=3 ðRe5 sin bÞ

(11) (12) (13)

3

is the Weber number and Fi ¼ gqr3 4 denotes the film number.5,63 In

deriving the above system, long-waves are assumed in the x and y directions. In the remainder of the study, “e” is dropped. Although it might seem that the slope of the waves appear to be of Oð1Þ by dropping e, it is already shown that the solitary waves do obey the long-wave assumption by doing so.30,31 The terms to the left of the equality in Eqs. (12) and (13) describe the influence of inertia on the physical system. The second and the third terms on the right hand side in Eqs. (12) and (13) represent the effects of hydrostatic pressure and the force of surface tension on a grooved geometry. The gravitational effects and the shear stress exhibited by the topography come from the first

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term in Eq. (12) on the right hand side of the equality. And, the first term in Eq. (13) on the right side of the equality is due to the contribution from the wall shear stress. IV. LINEAR STABILITY ANALYSIS: TEMPORAL FORMULATION

Knowledge of instabilities arising in a film flow system related to various applications is prerequisite to understand the flow behavior. The stability analysis of the IBL system (11)–(13) when the undisturbed state is perturbed provides the first insight into the underlying interfacial dynamics of the flow on a longitudinal topography. To this cause, the undisturbed state of the falling film is considered to be homogeneous in x and t, and, therefore, the system described by Eqs. (11)–(13) admits the trivial solution, h ¼ h0 ðyÞ 6¼ 0;

q1 ¼ h30 ðyÞ;

and

q2 ¼ 0;

(14)

where the function h0(y) satisfies the following homogeneous third order ordinary differential equation: d3 3 cot b d ðh0 ðyÞ þ lðyÞÞ ¼ 0: ðh0 ðyÞ þ lðyÞÞ 3 ReWe dy dy

(15)

The bottom topography depends only on the y-coordinate and is considered to be l(y) ¼ L0‘(y), where ‘ðyÞ ¼ 12 ð1 cosðpyÞÞ. Equation (15) is a linear third order ordinary differential equation for the film thickness h0(y) and is influenced by the capillary pressure and the hydrostatic effect. The solution of Eq. (15) is of the form rffiffiffiffiffiffiffiffiffiffiffiffiffi ! rffiffiffiffiffiffiffiffiffiffiffiffiffi ! 3 cot b 3 cot b y þ c3 sinh y lðyÞ; h0 ðyÞ ¼ c1 þ c2 cosh ReWe ReWe where c1, c2, and c3 are constants to be determined. Imposing periodic boundary conditions (h0( 1) ¼ h0( þ1); h0y( 1) ¼ h0y( þ1); h0yy( 1) ¼ h0yy( þ1)) yields c2Ð¼ c3 ¼ 0 with c1 yet 1 remaining an unknown. By considering the volume per unit width to be 1 h0 ðyÞ ¼ 2 L0 , the unknown c1 is determined as 1. This would mean that the maximum height of h0(y) varies up to 1 in the vertical direction even while considering flows on different groove measures. Also, it should be remarked that, since L0 ’ OðeÞ, h0(y) can be expanded asymptotically in the form h0 ðyÞ ¼ 1 þ L0 h1 ðyÞ þ OðL20 Þ, where h1(y) is the correction at the first order to the base state of the flow over a planar geometry satisfying46,64 d 3 h1 ðyÞ 3 cot b dh1 ðyÞ d3 ‘ðyÞ 3 cot b d‘ðyÞ ¼ : þ 3 dy ReWe dy dy3 ReWe dy

(16)

Assuming the undulated bottom ‘(y) to be influencing periodic inhomogeneities in Eq. (16), a periodic solution of the form h1 ðyÞ ¼ A 12 ð1 cosðpyÞÞ þ B 12 ð1 sinðpyÞÞ is sought, where A and B are unknowns.46,64 By equating the like terms on both the sides, it is found that A ¼ 1 and B ¼ 0. The profile l(y) considered above steepens with increase in L0 and widen with decrease in the groove measure. However, the function h0(y) traces a steepening crest for 0 < L0 < 1. Focusing on the temporal stability analysis, the disturbed state corresponding to the base flow given by Eq. (14) is considered as h ¼ h0 ðyÞ þ d/1 ðyÞeiðkxþmyaCL tÞþXt ;

(17)

q1 ¼ h30 ðyÞ þ d/2 ðyÞeiðkxþmyaCL tÞþXt ;

(18)

q2 ¼ 0 þ d/3 ðyÞeiðkxþmyaCL tÞþXt ;

(19)

where k and m are real wavenumbers of the disturbance in the x and y directions, respectively, with a2 ¼ k2 þ m2 and x ¼ iaCL þ X being the complex wave celerity. Here, CL and X represent the phase speed and the growth rate of instability. The absolute value of d/i(i ¼ 1,2,3) is

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considered small, so that d 1 and max (j/i(y)j) ¼ O(1). Using Eqs. (17)–(19) in Eqs. (11)–(13) and then linearizing for jd/i j 1 yields a linear system of differential algebraic equations for the disturbance components with variable coefficients given by ik/2 þ

d/3 þ im/3 ¼ x/1 ; dy

d 2 /1 9 2 3ik cot b 3 6ik 6 3 d/1 3 3 2 3 þ ik h h h þ 2ikmWeh Weh þ þ ikm Weh 0 0 0 0 dy2 dy Re 0 Re 5 0 12ik 4 3 6 d/ 12 3 6im 4 h0 þ /2 þ h40 3 þ h0 h0y þ h / ¼ xh20 /2 ; /1 þ dy 5 Re 5 5 5 0 3 d3 / d2 / 3 cot b 2 d/ þ a þ 2m2 We h30 1 Weh30 31 3imWeh30 21 þ dy dy dy Re 3 cot b 6ik 3 þ imh30 þ a2 We /1 þ h4 þ / ¼ xh20 /3 : Re 5 0 Re 3

(20)

ikWeh30

(21)

(22)

System (20)–(22) constitutes an eigenvalue problem for the linear stability with x as an eigenvalue. When h0 ¼ 1 and /1, /2, and /3 are constants, system (20)–(22) becomes the eigenvalue problem studied by Kalliadasis et al.31 for a film flow along a planar substrate under isothermal conditions. A simple algebraic manipulation yields the neutral stability condition (X ¼ 0) a4

We cot b 2 þ a k2 ¼ 0 3 Re

(23)

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi with phase speed CL ¼ 3. For a vertical planar substrate, k2 þ m2 ¼ k ð3=WeÞ gives the neutral stability curves.65 System (20)–(22) represents equations for the perturbation amplitudes, which assesses the stability of the flow in both the longitudinal and transverse directions. Since the stability of the film flow along the longitudinal direction is the subject of interest, m is set to zero in the remainder of the investigation. For a two-dimensional film flowing along the x-direction on a planar surface with constant thickness (h0 ¼ 1), system (20)–(22) can be reduced to an algebraic eigenvalue problem and its principal determinant set equal to zero gives the non-trivial solution. The necessary and sufficient condition for the non-trivial solution of the system yields the dispersion relation, and a small k expansion of this dispersion relation up to the third order gives x, the complex wave celerity. The onset of instability is obtained by equating the first order term in k to zero which gives the critical condition Re cot b ¼ 0. This condition underestimates the Hopf bifurcation threshold instead of the correct critical condition.65 However, Re ¼ 0 when the plane is vertical. This implies that for all Reynolds numbers, the flow is unstable and, therefore, IBL does not introduce an error for a vertically falling film31 in the absence of surface tension effect. The Weber number appears only at the order three in the small k expansion. The surface tension is the only physical effect that limits the growth rate of short waves (k ! 0) and then prevents them from breaking, which implies that k2We ¼ Oð1Þ, where k corresponds to waves observed experimentally. Retaining the capillary force at the free surface, the critical condition for the onset of instability reads (see Eq. (8) in Kalliadasis et al.31) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 cot b k¼ 1 (24) We Re with the phase velocity CL ¼ 3 at the leading order (Eq. (24) is a direct consequence of Eq. (23) with m ¼ 0). Equation (24) introduces an error of about 20% in magnitude compared to the exact linear stability results obtained from the long-wave Benney model valid for small Reynolds numbers, which reads66

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FIG. 2. (Color online) Neutral stability curves of the IBL and long-wave model (LW) considered in Sadiq and Usha66 (see Fig. 6 in their paper). The solid curves correspond to the flow on a vertical plane and the broken lines represent the flow on a plane inclined at b ¼ 45 . The weber number is taken as We ¼ 44.96 to reproduce the result in Sadiq and Usha.66

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 6 cot b : k¼ We 5 Re

(25)

When We ¼ 0 in Eq. (25), the flow is unstable for all Reynolds numbers on a vertical plane as shown by Eq. (24). Figure 2 shows the slight discrepancy of the IBL model in comparison to the stability results of the long-wave model for a low-Reynolds number flow, where the unstable region is below each curve. It is seen that when the surface tension effects are included, the IBL linear stability results deviate even when the plane is vertical. Nevertheless, it is worth studying the nonlinear dynamics of this model since the solitary wave solutions of the IBL model on a planar substrate have shown similarity with that of the long-wave model up to an Oð1Þ, above which the long-wave model breaks down, but the IBL does hold.31 A. Results and discussion

The linear stability of the system (20)–(22) is solved numerically using the Chebyshev spectral collocation technique in a periodic domain adopting QZ algorithm67–69 to capture the wave characteristics in the region of the wave formation.5 A liquid with properties q ¼ 998.21 kg/m3, ¼ 1.002 10 6 m2/s, and r ¼ 7.275 10 2 kg/s2 with Fi ¼ 3.9186 1010 is chosen,70 which corresponds to water at 20 C for analyzing the stability properties. The combined effects of inertia, hydrostatic force, and the film numbers are analyzed and presented in Figs. 3 and 4 in terms of the growth rate curves. The calculations in Fig. 4 are performed by considering two more fluids, the magnitude of whose film numbers vary between 106 1011, namely, the water-glycerin mixture with Fi ¼ 7.249 106 and the liquid nitrogen on the saturation line at the atmospheric pressure71 with Fi ¼ 1.231 1011 along with the water at 20 C. With increase in the value of Reynolds number and with the angle of inclination, it is observed that the growth rate of the disturbance amplitude increases and the long-wave range of unstable wavenumbers increases. However, with increase in Fi, the unstable band of wavenumbers decreases and the film tends to become more stable (see Fig. 4). For a fixed wavenumber, the growth rate of instability decreases with increase in the measure of groove depth. Further, there exists a long-wave range

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FIG. 3. (Color online) Growth rate curves on a structured and unstructured topography for water at 20 C when b ¼ 90 (———–) and when b ¼ 45 (- - - - - -) for different Reynolds numbers; (a) Re ¼ 2, (b) Re ¼ 5, (c) Re ¼ 10, and (d) Re ¼ 15.

of wavenumbers for a particular groove measure below which the flow destabilizes and this range is smaller for a grooved substrate compared to a planar wall. Since falling films down vertical walls are more unstable compared to the film flow on an inclined substrate, the remainder of the discussion is restricted to the film flow analysis on a

FIG. 4. (Color online) Growth rate curves on the structured and unstructured topographies inclined at (a) b ¼ 45 and (b) b ¼ 60 for the water-glycerin mixture (—–), water at 20 C (- - -), and nitrogen on the saturation line at the atmospheric pressure (-.-.-.-) when Re ¼ 20.

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FIG. 5. (Color online) (a) Phase velocity and (b) frequency as a function of wavenumber for water at 20 C when b ¼ 90 : (—–) Re ¼ 5 and (- - -) Re ¼ 15.

vertical wall. The phase velocity plots at which a wave phase propagates at a certain frequency and the associated frequency plots are shown in Fig. 5. In Fig. 5(a), the phase velocity decreases up to a certain wavenumber and increases beyond it. The phase velocity always remains smaller for the range of unstable wavenumbers corresponding to the low Reynolds number limit even with increase in the inertial effect magnitude. Beyond this limit, the phase velocity increases. Also, the linear phase velocity for the case of longitudinal grooves is observed to be smaller than that corresponding to a planar wall. This is true for all wavenumbers corresponding to the unstable mode (see Fig. 6 for the unstable range of wavenumbers). The trend displayed by the phase velocity in

FIG. 6. (Color online) Neutral stability curves for water at 20 C when b ¼ p/2 as a function of groove measure and Reynolds numbers and (b) linear stability curves corresponding to maximal growth rate.

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Fig. 5(a) agrees with the behavior found in the experimental observations of Helbig et al.,57 (Fig. 6 in their report). Fig. 4(b) shows the wave frequency computed as a function of wavenumber in the corresponding unstable domain for a fluid on a planar wall and on a grooved wall. The predictions of the linear stability analysis reveal that the frequency grows and behaves almost as a linear function of the wavenumber and remains smaller for the case of a film flow on a grooved wall compared to a planar wall. The locus of (kc, L0), where kc represents the critical wavenumber at which X vanishes on the k > 0 axis, traces the stability curves (Fig. 6(a)). In the region below each curve, the flow is unstable (the growth rate of instability is positive) and in the region above each curve, the flow is stable. At each point on the locus of (kc, L0), the flow is neutrally stable. It is seen that the band of unstable wavenumbers increases with increase in Re and decreases with increase in the magnitude of groove measure. Figures 7(a)–7(c) display the maximal growth rate and the phase velocity of the most unstable wave for different groove measures. The growth rate corresponding to the most unstable mode increases and the associated phase velocity decreases with increase in the value of Re. The result in Fig. 7(a) corresponding to the flow on a planar substrate shows that the maximal growth rate increases as a function of Reynolds number as observed in Fig. 3 considered in Kliakhandler,20 who has shown that the maximal growth rate increases for Re varying up to exp(2.3) approximately. The data given in Table I in Pierson and Whitaker72 for film flows down smooth walls up to Re ¼ 10 in the moderate Reynolds number regime shows that the phase velocity value decreases with increase in the value of Reynolds number, and the temporal growth rate value increases as a function of Re. This behavior is shown in the figures presented in Figs. 7(a) and 7(c). Also, the phase velocity curves of the most unstable wave, in agreement with the results shown in Fig. 5(a), decreases with increase in the groove measure. It is inferred from Fig. 7(a) that with the decrease in the value of the groove measure, the growth rate diminishes slowly and this behavior is supported by Fig. 7(b). The curves in Fig. 7(d) present the frequency (fw ¼ kmCL/(2p)) plot corresponding to Xmax. It should be remarked that for a flow on a planar substrate, the frequency plot as a function of Reynolds number shows similar tendency as observed in the experiments of Liu and Gollub73 and traces a similar graph as seen in the numerical investigation of the spatial stability analysis carried by Ruyer-Quil and Manneville.74 Pierson and Whitaker72 pointed out that the

FIG. 7. (Color online) Growth rate corresponding to the mode where the maximum instability is attained, and the frequency and phase velocity curves at maximum linear growth rate for water at 20 C when b ¼ p/2.

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temporal stability results do not differ in a large magnitude in comparison to the spatial stability predictions. In view of this, the present stability results show that the number of waves falling down a grooved substrate is less in comparison to the case corresponding to a smooth wall and the dominant wave frequency increases with increase in Re. These results are in accordance with Figs. 9 and 10 with the experimental investigations of Helbig et al.,57 which have been performed for grooves having triangular gutters aligned longitudinal in the flow direction. If Re > 15, the effect of Re on the frequency becomes rather weak (see Fig. 7(e)). The effect of L0 on the frequency is substantial. From the above observations through Figs. 3–7, it can be concluded that the film flow is stabilized when the groove measure increases even at moderate Reynolds number for angles of inclination for which cot b 0 and for Film numbers corresponding to real fluids considered in the range whose magnitude vary between 106–1011. V. NONLINEAR EVOLUTION: NUMERICAL INVESTIGATION

Numerical investigation of the full Navier-Stokes system, though conceivable, remains a difficult task owing to the presence of a free boundary in addition to the velocity and pressure terms remaining as unknowns. The IBL model is a simplification of the Navier-Stokes system and heuristically captures the relevant features of the original model qualitatively. Despite the complexity of IBL model, the number of unknown variables is reduced and represent either as an averaged variable (flow rate) or interfacial quantity (free-surface). Hence, they are easily amenable to numerical experiments. When the amplitude of the disturbance is small, but finite, the nonlinear effects become important. The knowledge of the mechanism responsible for the transfer of energy from a basic state to the disturbance state can be understood by solving the nonlinear system governing the evolution of the film flow. The development of the flow behavior from an initial monochromatic perturbation can be traced down numerically through efficient numerical algorithms until the growth of the disturbance amplitude reaches a saturation stage, beyond which the wave pattern remains unchanged.11–13 For two-dimensional falling films on planar substrates based on the low Reynolds number modeling strategy, a weakly nonlinear stability analysis of the model by the method of multiple scales through a rigorous theoretical approach66,75–79 predicts the occurrence of a wavenumber ks in the vicinity of criticality which separates the supercritical stable and explosive regimes. In the supercritical stable region (ks < k < kc, where ks ¼ kc/2 (Refs. 79–81)), the threshold amplitude of the flow is finite and a nonlinear analysis of the base flow in this region leads to surface structures that are either time independent waves of permanent form that propagate11,13,66,82 or timedependent modes whose amplitude slightly oscillates.66,82,83 In short, the analysis predicts that the evolution of waves strongly depend on the initial choice of the wavenumber considered.11,13,78,81,83 In the absence of any concrete mathematical theory to predict the flow behavior based on a weak nonlinear stability theory, system (11)–(13) is solved numerically using MATLAB by means of the Fourier spectral method and choosing the wavenumber corresponding to the maximal growth rate. Spatial derivatives are approximated spectrally in which the nonlinear terms and the products are evaluated in the real space, and the derivatives are computed in the Fourier space.84 This is done by adopting a discrete Fourier transform via a fast Fourier transform85 thereby reducing the number of computations needed for N points from 2N2 to 2N log2(N).86 The computational domain for ðx; yÞ 2 ½0; 2p=k ½1; 1 is first mapped onto [0, 2p] [0, 2p] by means of simple linear transformations and the computations are performed in 32 32 mesh points. The temporal march was proceeded by incorporating a fourth order Runge-Kutta scheme with a small step length, 10 5 in the temporal direction to cater for stability requirements. The simulations were checked by doubling the mesh points and an error tolerance of 10 8 is set in order to avoid the growth of relative error and to ensure that the scheme is consistent to achieve convergence. The initial disturbance is considered as f(x, y, t ¼ 0) ¼ 1 þ d cos kx with d ¼ 0.1 so that this monochromatic wave initially represents the perturbation to the undisturbed state corresponding to the film thickness for all groove measures. The initial conditions for the flow rates are chosen as q1 ðx; y; t ¼ 0Þ ¼ h30 ð yÞ and q2(x, y, t ¼ 0) ¼ 0.

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FIG. 8. (Color online) Surface wave instability and amplitude profiles reproduced from Sadiq and Usha78 and compared with the IBL model corresponding to the two-dimensional flow for Re ¼ 5, b ¼ p/4, We ¼ 4496, and k ¼ 0.14.

In order to demonstrate the strength of the present numerical scheme, certain results already existing in the literature are reproduced again with the present code and compared with the IBL model. In Fig. 8, figures marked “a,” which were generated using Lee’s three time level scheme in Sadiq and Usha78 (see Figs. 12 and 20), are reproduced with the present numerical scheme for two-dimensional flows on a rigid plane and compared with the figures generated using the IBL model (figures tagged “b”). The effect of the wall inclination angle on the film stability is described by the hydrostatic pressure term. It is seen from Fig. 9 that the instability increases when the magnitude of the wall inclination angle increases, demonstrating that on a vertical wall, the flow is more unstable. The surface waves shown in the inset in Fig. 9 clearly shows the increase in the amplitude of the waves generated on the free surface and agree with the predictions shown in Figs. 3 and 4 corresponding to the linear theory. For the choice of the parameters chosen, it is observed that a periodic solitary wave followed by a one hump capillary ripple emerging at the time of saturation. Figure 10 traces the development of the surface wave instability by tracking the maximal and minimal amplitude of the film flowing on a planar vertical substrate corresponding to the wavenumber responsible for the maximal growth rate under different inertial effects. Because the plane is held vertical, the stabilizing hydrostatic effects are nullified in the system (11)–(13) and the inertial terms contribute to the instability mechanism competing with the force of surface tension. The simple sinusoidal disturbance to the basic state triggers the motion in the fluid. The disturbance amplitude reduces initially (fmax and fmin decrease and increase, respectively, for a short time in all the cases displayed), and for a short period of time, the flow at low Reynolds number is characterized by larger amplitude compared to a moderate Reynolds number flow. With subsequent progress in time, the disturbance amplitude corresponding to the moderate Reynolds number increases in accordance to the predictions shown by the linear theory and over takes the amplitude corresponding to the low-Reynolds number with a significant change in the trend. This shows that the viscous force dominates the scenario for small time and then this trend is taken over by the inertial effect with the subsequent progress in time. It is seen that the wave structure emerges as a time-dependent mode with small oscillations which are damped considerably with decrease in the value of Reynolds number along a vertical plane.

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FIG. 9. (Color online) Two-dimensional view of the surface wave instability and the amplitude profiles for the flow at Re ¼ 5, We ¼ 4496, and k ¼ 0.14: (1) b ¼ p/2, (2) b ¼ p/3, and (3) b ¼ p/4.

The instability mechanism of the system (11)–(13) in the case of a flow on a vertical plane can be understood by observing the shape developed on the interfacial surface (Fig. 11). The gravitational effect first triggers the initial disturbance by bringing the fluid towards the front face of the crest thereby deflecting it more upward and drains the fluid from the rear end, deflecting it downward, which gives rise to the increase in the surface height due to the accumulation of the fluid at the front face of the crest (see Fig. 11 at t ¼ 10). At this juncture, on a streamwise location in the bulk region at a particular instant in time, the forward movement of the disturbance tends to

FIG. 10. (Color online) Maximum and minimum amplitude profiles for a film flow on a vertical planar surface for water at 20 C: (1) Re ¼ 5, (2) Re ¼ 10, and (3) Re ¼ 15.

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FIG. 11. (Color online) Free surface profiles for a film flow on a vertical planar surface for water at 20 C at Re ¼ 5.

increase the surface height. In this location, the flow in the bulk accelerates and attempts to follow the velocity profile of the increased thickness of the layer.87 However, the effects of inertia counteracts the wave motion from accelerating fast enough to follow the fully developed velocity.87 As an implication, the mass flux does not quantify to be the one when the flow is really fully

FIG. 12. (Color online) Maximum amplitude profiles for the flow on a complex topography: (a) Re ¼ 5, (b) Re ¼ 10, and (c) Re ¼ 15. In (b), numbers “1” and “2” correspond to water at 20 C and the liquid nitrogen on the saturation line at the atmospheric pressure. In (a) and (c), the curves correspond to water at 20 C.

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developed but remains small. The velocity decreases in the rear end of the deflecting crest, however, inertial effects prevent the flow from decelerating fast.87 In this case, the mass flux remains higher than it is expected when the flow is fully developed.87 The fluid accumulates underneath the disturbance crest due to the difference arising in the mass fluxes (see Fig. 11 at t ¼ 20). Thus, the inertial force increases the amplitude and waviness of the film, and as a consequence, the interfacial displacement increases87 (Figs. 10 and 11). The accumulation of the fluid underneath the crest is generally overcome by the hydrostatic pressure term when the wall is held inclined. The hydrostatic pressure inside the bulk region underneath the crest increases in proportion to the local depth of the film and those on the neighboring trough decreases. Because of this difference in the pressure, the fluid flows as directed by the hydrostatic pressure to trough regions where the pressure is low and tend to stabilize the film (see Fig. 9). Now, when the topography is comprised of grooves, the groove measure contributes to the instability mechanism. Maximum amplitude of the surface wave generated while a film flows on a grooved vertical substrate is depicted in Fig. 12. The oscillations emerging in the wave structure sustain to grow and increase with time for a film flowing on a planar substrate when the value of the Reynolds number increases (see Figs. 10 and 12(c)), whereas they decrease and/or subside considerably for a long-time integration when the magnitude of the groove measure increases depending upon the non-dimensional parameters considered. For small time, it is observed that the wave amplitude of a film flowing on a longitudinal groove topography remains higher than the wave amplitude of the film flowing on a planar substrate. This tendency reverses with progress in time. It is seen that the longitudinal grooves stabilize the film flow when the groove measure increases. This numerical observation is in accordance with the theoretical investigations sought in the linear stability theory and agrees with the experimental observations reported in Helbig et al.57 Again, it is revealed that when the magnitude of the film number increases, the flow is stabilized. However, the flow undergoes a series of significant distortions on a grooved topography when the magnitude of the groove measure is large (see Fig. 12(b) for L0 ¼ 0.4). The amplitude

FIG. 13. (Color online) Snap shots of the evolution of waves on a vertically falling film at Re ¼ 10 corresponding to water at 20 C: (a) L0 ¼ 0, (b) L0 ¼ 0.2, and (c) L0 ¼ 0.4; Figures (i) and (ii) represent the wave evolution at time t ¼ 0 and t ¼ 100, respectively, whereas (iii) corresponds to the time at which saturation is attained and the simulations are stopped.

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for the flow of a film with Fi ¼ 1.231 1011 decreases for a short time and then shoots up and remains larger than the case corresponding to the film number associated with water at 20 . This behavior subsequently reverses and comes below the amplitude corresponding to film numbers whose magnitudes are smaller than it, demonstrating thereby the stabilizing mechanism of the force of surface tension on a grooved topography. Thus, the grooves prevent the formation of steep crests and increased waviness by regulating the waviness evenly thereby counteracting the force of inertia jointly with the surface-tension force, and as a consequence, the flow stabilizes. The sequence of snap shots captured during the wave development process at different time instants for a fluid flowing on a grooved substrate are shown in Fig. 13 at Re ¼ 10. It should be remarked that it was observed during the numerical simulation that the surface waves deformed mainly along the streamwise spatial direction (see Figs. 13 and 14), which could be attributed due to the reason that there is no flow across the longitudinal grooves. These snap shots help in understanding the stabilizing mechanism of the substrate with longitudinal grooves. The wave distortions on a film at Re ¼ 15 and L0 ¼ 0.2 are shown in Fig. 14. It is interesting to see the changes in the shape of the surface wave instability generated on the surface of a falling film on a grooved geometry. It is inferred from Fig. 14 that initially the shape of the wave remains almost the same for small units in time, which changes with the progress in time forming into a deflected crest at the front, having a trough at the rear at t ¼ 35. With advance in time, the shape of the wave deforms with a steep upward deflection with an asymmetric back, which is then followed by a stretching at the rear end with a steep deflection at the front. In long-time, a sinusoidal wave with an amplitude larger than its undisturbed state evolves. The physical mechanism responsible for the long-wave instability on the surface of a film can be attributed due to the following factors. The gravitational acceleration triggers the wave motion, and from then onwards the force of inertia destabilizes the fluid taking advantage in the absence from the contribution of the stabilizing hydrostatic effect on a vertical wall. But, when the substrate topography is comprised of longitudinal grooves, the grooves try to counteract the effort of inertia by minimizing the imbalance occurring in the mass flux at the front and the rear end of the wave, and the force of surface-tension acts in its favor by regulating the growth rate of the amplitude of the surface waves thereby stabilizing the fluid and decreasing the waviness.

FIG. 14. (Color online) Wave evolution of a film on a vertical wall at Re ¼ 15 and L0 ¼ 0.2.

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VI. CONCLUSIONS AND PERSPECTIVES

The stability and dynamics of an IBL model on a geometry with longitudinal grooves have been studied. The stability results solved numerically give first hand information about the behavior of the flow and comply with the theoretical findings of Gambaryan-Roisman et al.51 The frequency plots obtained through the linear stability theory show similar analogy revealed in the experimental study of Helbig et al.57 The predictions of the linear stability results are further confirmed for different physical effects by solving the IBL model numerically. It is seen that the force of inertia in principle destabilizes the flow by increasing the amplitude of the wavy disturbance. In certain cases corresponding to the non-dimensional parameters, oscillatory nature of the long-time wave forms was found. For a vertical topography, the film flow is observed to be more unstable since the stabilizing contribution from the hydrostatic effect is absent. The force of surface tension counteracts the destabilizing process and tries to stabilize the film flow. When the magnitude of the surface tension force is increased, the film flow becomes more stable. The stabilizing phenomena can be further improved by having longitudinal grooves on the topography. These longitudinal grooves tend to decrease the waviness of the film. The deeper the groove pockets, more stable is the flow. The experimental investigations of the effect of ministructured grooves on the outer side of an evaporator tube57 and on structured plates54,55 showed that the wall topography affects the development of wavy patterns on the liquid-gas interface. In accordance to this, the knowledge of the present study gains importance in devising surfaces with smart interfaces in falling film apparatuses in advanced engineering applications. The drawback of the IBL model raised by the inaccurate linear stability theory can be overcome by employing more accurate models88–91 to predict the linear stability threshold. Future investigation will be aimed to study the film flow dynamics under non-isothermal conditions and in addressing traveling wave solutions and large domain simulations with natural noise. ACKNOWLEDGMENTS

Financial support for this research was funded by the Center of Smart Interfaces, TU Darmstadt, Germany. I.M.R.S is grateful to Dr. Benoit Scheid for his constructive advice while developing the numerical algorithm. 1

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