Eur. Phys. J. B 6, 277–292 (1998) THE EUROPEAN PHYSICAL JOURNAL B EDP Sciences c Springer-Verlag 1998
Modeling film flows down inclined planes
C. Ruyer-Quila and P. Mannevilleb Laboratoire d’Hydrodynamiquec, Ecole´ Polytechnique, 91128 Palaiseau, France
Received: 16 April 1998 / Revised: 29 June 1998 / Accepted: 2 July 1998
Abstract. A new model of film flow down an inclined plane is derived by a method combining results of the classical long wavelength expansion to a weighted-residuals technique. It can be expressed as a set of three coupled evolution equations for three slowly varying fields, the thickness h, the flow-rate q,andanew variable τ that measures the departure of the wall shear from the shear predicted by a parabolic velocity profile. Results of a preliminary study are in good agreement with theoretical asymptotic properties close to the instability threshold, laboratory experiments beyond threshold and numerical simulations of the full Navier–Stokes equations.
PACS. 47.20M Interfacial instability – 47.20K Nonlinearity
1 Introduction present themselves as stream-wise surface undulations free of span-wise modulations (“two-dimensional” waves) In addition to being involved in a wide variety of techni- emerging from a supercritical (i.e., continuous) bifurca- cal applications (chemical reactors, evaporators, etc.), the tion. Farther from threshold, they saturate at finite ampli- dynamics of fluid films is an interesting topic in itself. As tudes and, depending on control parameters, may develop a matter of fact, thin films flowing down inclined surfaces secondary instabilities involving span-wise modulations exhibit a rich phenomenology [1] and offer a good testing (“three-dimensional” instabilities) [3] or first evolve into ground for the study of the transition to turbulence. Insta- localized “solitary” structures that subsequently desta- bilities take place at low flow rates, which gives a unique bilize [4]. For the moment we will focus on the two- opportunity to analyze the development of waves at the dimensional case where the hydrodynamic fields depend surface of the fluid into large-amplitude strongly nonlin- only on the cross-stream and stream-wise coordinates, ear localized structures such as solitary pulses and fur- y and x respectively (see Fig. 1), leaving the three- ther to study their disorganization into developed spatio- dimensional problem for future study. temporal chaos via secondary instabilities. A trivial solution to the flow equations is easily found It turns out that, for the regimes we are interested in the form of a steady uniform parallel flow with parabolic in, the height of the waves remains small when compared velocity profile, often called Nusselt’s solution, where the to their wavelength. This motivates a long standing prac- work done by gravity is exactly consumed by viscous tice [5,6] of studying them by means of asymptotic ex- dissipation. Thin films at low flow rate over sufficiently pansions in powers of a small parameter , usually called steep surfaces turn out to be unstable against long wave- the film parameter. Starting from this long-wave expan- length infinitesimal perturbations, i.e. wavelength large sion, a certain number of models have then been derived when compared to the thickness of the flow. This is con- since the pioneering work of Kapitza [7], e.g. [8–10] for firmed by a general study of the relevant Orr–Sommerfeld the most recent ones, see the review by Demekhin et al. equation which shows that short-wavelength shear insta- [11] for earlier attempts. The simplest useful result ob- bilities of the Tollmien–Schlichting type are only relevant tained in this way is a partial differential equation called for flows over planes at vanishingly small inclination an- Benney’s equation [12] to be written explicitly later (36). gles and very high flow rates [2]. Governing the local thickness of the film h(x, t)interms In the following we will thus be concerned with long of its space-time derivatives, it will be written here simply n wavelength interfacial instability modes, the dynamics as ∂th = G(h ,∂xmh), where G involves various algebraic of which is essentially controlled by viscosity and sur- powers and differentiation orders (n, m)ofh. Within this face tension effects. Close to the threshold these waves approach the film evolution is modeled in terms of lubrica- tion theory, which results in the enslaving of flow variables a e-mail: [email protected] to the local film height, i.e. a reduction to some effective b e-mail: [email protected] dynamics for the interface through the elimination of de- c CNRS UMR n◦ 7646. grees of freedom associated to velocity field. 278 The European Physical Journal B
The simplification brought by this reduction has mainly permitted a first study of the nonlinear develop- ment of waves using the tools of dynamical systems the- ory [13], study that was continued using the celebrated Kuramoto–Sivashinsky (KS) equation [14], obtained in the y present context by taking the limit of small amplitude 0 modulations [15,16]. By contrast with the KS equation, Benney’s equation can lead to a non-physical evolution z with the development of finite-time singularities [13,17]. y x Such an evolution strongly limits its use to a narrow neigh- borhood of the threshold where the KS equation —that β does not behave so wildly— is expected to give already valuable results. u 0 Numerical investigation of the full Navier-Stokes (NS) equations, though conceivable, is cumbersome owing to x the presence of a free boundary. Though it can (should) Fig. 1. Fluid film flowing down an inclined plane: definition serve as a check point for models in the two-dimensional of the geometry. case, the only one to be reliably implemented up to now [18,19], the numerical approach does not give much insight into the mechanisms of chaotic wave motion and for the coefficients appearing in the expansion, the inter- pattern formation. An intermediate level of modeling is pretation of such studies is not straightforward and one obtained in terms of so-called boundary layers equations is often confined to a comparison of the obtained output (BL) [20], i.e. reduced NS equations incorporating the with that of concurrent models and numerical solutions condition that stream-wise gradients are small when com- of BL or NS equations, or with the results of laboratory pared to cross-stream variations. Though one is left with a experiments. problem that has the same dimensionality as the original In this paper, after having recalled the governing equa- one, it is a little simpler and leads to lighter computations giving realistic results [21]. tions (Sect. 2) of the two-dimensional problem to which we will restrict, we briefly repeat the first steps of the Ben- A subsequent level of modeling is achieved by assum- ney’s gradient expansion for the dynamics of film flows ing a specific shape for the velocity profile and averaging (Sect. 3) that will be useful to us afterward. We then de- the stream-wise momentum equation in order to relate the velop our model in two steps, mostly for pedagogical rea- h thickness h of the film to the local flow rate q = 0 u(y) dy. sons (Sects. 4 and 5). We follow the same general strategy The first such integral boundary layer model was derived as Yu et al. [9] and use BL equations as a starting point, by Shkadov [22]. We will reobtain it below as systemR (49, which has the interest of focusing on the appropriate long 50). The assumption about the velocity profile and the wavelength properties of the flow right from the begin- averaging procedure are in fact two specific ingredients of ning. We also use polynomials to expand the velocity field a more general method for solving the BL equations in but, to stay closer to the physics of the problem, instead of terms of weighted residuals [23] instead of standard dis- choosing some general systematic expressions of increas- crete methods (e.g., finite differences). The limitations of ing degree, we prefer to take the specific polynomials that Shkadov’s model come from the lack of freedom in the appear in Benney’s gradient expansion and to introduce description of the hydrodynamic fields and the too rus- combinations of coefficients of the lowest order terms that tic character of the consistency condition expressed via may be given an immediate physical interpretation. The the averaging. In spite of these limitations, problems that first-order problem (Sect. 4) involves two polynomials, the could only be approached through black-box numerical zeroth-order parabolic profile and a correction issued from computations of full or reduced NS equations can now be Benney’s expansion. Relevant coefficients involve the flow dealt with a set of partial differential equations with re- rate q, and a new field called τ measuring the departure of duced space dimensionality (1 instead of 2). Accordingly, the wall shear from the shear predicted by a parabolic ve- properties of nonlinear waves can again be studied in their locity profile. At this order, τ is slaved to h and q and can rest-frame using the tools of dynamical systems theory [24, be eliminated adiabatically, yielding a set of two partial 25]. Better approximations of the flow have to be devel- differential equations (9, 58) with the same structure as oped in order to get more realistic results. This requisite Shkadov’s model but different coefficients. With respect to has lead to the derivation of improved models by weighted the latter, the advantage of our first-order model is to give residual methods [9,10], by expanding the hydrodynamic a more accurate description of the vicinity of the instabil- fields on a functional basis of the cross-stream variable ity threshold, and in particular to predict the critical flow y, finding relations between the coefficients of the (trun- rate exactly. At second order (Sect. 5), τ becomes a degree cated) expansion from the NS or BL equations by some of freedom for its own and four well-chosen supplementary specific projection rule, and further applying the resulting polynomials are introduced, the coefficients of which can set of equations to concrete problems such as the structure be eliminated to yield a system of three equations (78– of solitary waves. In the absence of clear physical meaning 80) governing h, q,andτ. Sections 6 and 7 are devoted C. Ruyer-Quil and P. Manneville: Modeling film flows down inclined planes 279 to a discussion of our achievements focusing on a quali- continuity condition. Integrating (3) over the interval [0,h] tative comparison of experimental and numerical results we obtain: in the linear and nonlinear regimes, with preliminary nu- h h merical simulations of (78–80) and on the physical under- 0= (∂ u+∂ v)dy = ∂ udy+v −v standing that can be expected from our model. x y x h 0 Z0 Z0 h h
=∂th+ u h∂xh+ ∂xudy =∂th+∂x udy 2 Governing equations " Z0 # Z0
The geometry is defined in Figure 1: the inclined plane using v h given by (4) and v 0 = 0 from (7). Defining the makes an angle β with the horizontal. As usual, xˆ, yˆ, local instantaneous flow rate as and ˆz are unit vectors in the stream-wise, cross-stream, h(x,t ) and span-wise directions respectively. Here we only con- q(x, t)= u(x, y, t)dy, (8) sider the two-dimensional case where the solution is inde- 0 pendent of coordinate z, the extension to the full three- Z dimensional case does not present conceptual difficulties. we arrive at the integral condition The basic (2D) equations read ∂th + ∂xq =0. (9) ρ [∂tu+u∂xu+v∂yu]=−∂xp+ρg sin β+µ (∂xx +∂yy)u, (1) System (1-7) admits a trivial solution corresponding to a steady constant-thickness film, often called the Nusselt so- ρ [∂tv+u∂xv+v∂yv]=−∂yp−ρg cos β+µ (∂xx + ∂yy)v, lution (hence the subscript “N” in the following). Assum- (2) ing ∂t ≡ 0and∂x ≡0, one simply gets v ≡ 0, p h = pa ∂xu + ∂yv =0, (3) and
where u and v denote x and y velocity components, and µ∂yyu+ρg sin β =0,∂yp=−ρg cos β, u 0 =0,∂yuh=0, p the pressure. ρ is the density, µ the viscosity, and g the intensity of the gravitational acceleration. These equations must be completed with boundary which, for a film of thickness hN yields: conditions at y =0ory=h. They will be denoted as w or w where w(x, y, t) is a generic name for the pres- ρg sin β 0 h u(y)= y(2h − y),p(y)=p +ρg cos β(h − y), sure field, the velocity components and their derivatives. 2µ N a N
The first such condition:
∂th + u h ∂xh = v h, (4) where the atmospheric pressure pa is set to zero in the following. The corresponding flow rate is given by simply expresses the fact that the interface h(x, t)isa material line. The continuity of the stress at y = h adds h ρg sin βh3 q = u(y)dy = N , two more equations. The normal component reads N 3µ Z0 γ∂ h 2µ xx + ∂ h ∂ u +∂ v from which an average velocity uN can be defined by qN = 3/2 2 x y h x h h u , i.e. u = ρg sin βh2 /3µ. 1+(∂ h)2 1+(∂xh) N N N N x h At this stage, it is usual to turn to dimensionless
h i 2 equations. Different scalings can be used. The first and −(∂xh) ∂xu −∂yv +p −pa =0 (5) h h h most obvious one takes hN and hN /uN as length and i time units, see note [26]. Here, we will take another scal- where coefficient γ is the surface tension and the term in ing defined without reference to the flow by construct- ∂xxh describes the curvature of the interface (pa is the ing the length and time units from g (LT −2)andthe atmospheric pressure). For the tangential component one kinematic viscosity ν = µ/ρ (L2T −1). Taking for conve- gets nience g sin β instead of g, this yields L = ν2/3(g sin β)−1/3 1/3 −2/3 2 and T = ν (g sin β) . The velocity unit is then 0=2∂ h ∂ v −∂ u + 1−(∂ h) ∂ u +∂ v . x y h x h x y h x h U = LT −1 =(νg sin β)1/3. For the pressure, we get h i (6) ρ(νg sin β)2/3. The surface tension is then measured by 4/3 1/3 Finally, the no-slip condition at the rigid bottom, y =0, the Kapitza number Γ = γ ρν (g sin β) .Infact, reads: Kapitza was concerned with vertical planes for which β = π/2sothatthefactorsin βdid not appear in his u = v =0. (7) definition. It is a matter of convenience to include it or 0 0 not. The two numbers, with and without, are of the same
It will turn interesting to replace the kinematic condition order of magnitude as long as one does not consider nearly (4) at y = h by an equivalent equation derived from the horizontal planes. 280 The European Physical Journal B
Inserting the corresponding variable changes we obtain When developing the calculation systematically, we should notice first that the status of the kinematic in- ∂tu+u∂xu+v∂yu=−∂xp+1+(∂xx +∂yy)u, (10) terface condition (4) or its integral version (9) is different
∂tv+u∂xv+v∂yv=−∂yp−B+(∂xx +∂yy)v, (11) from that of the other equations. As a matter of fact, the sought-after solution has to be seen as a functional of h where B =cotβand, for the normal-stress boundary con- and its successive space-time derivatives, all considered as dition at y = h independent quantities. Once it is found at a given or- der, the result can be inserted in (9) (or (4)), which then
Γ∂xxh 2 presents itself as a constraint relating h and its succes- 3/2 + 2 ∂xh ∂yu h +∂xv h sive partial derivatives, i.e. an evolution equation for h. 1+(∂ h)2 1+(∂xh) Furthermore, it is immediately seen that this equation is x h formally one order higher than the solution found. So, h i 2 −(∂xh) ∂xu h −∂yv h +p h =0, (12) already at zeroth order we get the following nontrivial re- lation i while the continuity condition (3), the kinematic condition (0) 2 (4) at y = h and the remaining boundary conditions (6,7) ∂th + ∂xq ≡ ∂th + h ∂xh =0. (14) are left unchanged. In this unit system where g sin β = ν = However, letting H = h2 leads to ∂ H+H∂ H =0,i.e. the ρ = 1, the Nusselt flow rate given by q = u h = 1 h3 t x N N N 3 N Burgers equation, an equation known to produce shocks is numerically equal to the Reynolds number R as defined and hence steep gradients incompatible with the slow- in note [26]. variation assumption. Continuing the expansion is there- fore necessary for finding gradient-limiting terms playing the role of viscous dissipation in the Burgers case. Though 3 Gradient expansion the result is not guaranteed to be well-behaved, let us re- view the first steps of the expansion. Laboratory experiments show that the Nusselt solution At first order we get: may not be relevant, being possibly unstable against waves (1) (1) (0) (0) (0) (0) (0) at the surface of the film. However, as long as the flow rate ∂yyu −v ∂yu =∂tu +u ∂xu +∂xp , (15) is not too large, the interface remains smooth at the scale ∂ v(1) − ∂ p(1) =0, (16) of the film thickness as measured locally by h(x, t). This yy y (1) (0) feature can be introduced as a supplementary assumption ∂yv = −∂xu , (17) and solutions to the equations can be searched in the form of a systematic expansion in powers of a formal parame- to be solved with the appropriate boundary conditions: ter ε expressing the smallness of the stream-wise space (1) (1) p − 2∂yv =0, (18) derivative ∂x. h h (1) At order zero, the problem simply reads ∂yu =0, (19) h (1) (1) (0) (0) (0) (a) u 0 =0, (b) v 0 =0. (20) ∂yyu = −1,∂yp = −B, ∂yu h =0, (0) (0) (0) u =0,p=0, (Note that ∂yu = 0 has been taken into account to 0 h h simplify the r.h.s. of (18).) Equation (17) yields v(1) = so that the Nusselt solution is recovered locally with − 1 y2∂ h when making use of (20b). The stream-wise h(x, t) as reference height: 2 x correction u(1) is then obtained from (15) together with 1 boundary conditions (19, 20a) and the pressure correction u(0)(y)= y(2h − y),v(0) ≡ 0,p(0) = B(h − y), from (16) subject to (18). We obtain 2 (13) (1) 1 1 3 2 1 1 4 3 u (x, y, t)= y −h y ∂th+ y −h y h (0) 1 3 2 3 6 4 yielding a local flow rate q = 3 h . This is an exact solu- tion to the problem, provided that the thickness gradient 1 +B y2−hy ∂ h, (21) is strictly zero. When this is no longer the case, correc- 2 x tions have to be introduced. Assuming that h ≡ h(x, t) p(1)(x, y, t)= (y+h)∂ h. (22) but remains slowly varying, one now looks for a solution − x “close to” the stationary uniform flow in the form: To complete the calculation at first order it remains to
(0) (1) (2) express the kinematic condition in integral form (9). With u = u (h(x, t),y)+u (x, y, t)+u (x, y, t)+..., h u = u(0) +u(1), we obtain for q = u(y) dy an expression (1) (2) 0 v = v (x, y, t)+v (x, y, t)+..., of the form 1 h3 + a(h)∂ h + b(h)∂ h, hence the evolution 3 t Rx p = p(0)(h(x, t),y)+p(1)(x, y, t)+p(2)(x, y, t)+..., equation for h(x, t): 2 where the corrections are formally of order 1, 2, ... ∂th + h ∂xh + ∂x [a(h)∂th + b(h)∂xh]=0, (23) C. Ruyer-Quil and P. Manneville: Modeling film flows down inclined planes 281
5 4 3 6 1 3 where a(h)=−24 h and b(h)=−40 h − 3 Bh .The and expression of u(1) (21) and equation (23) can be put in (2) (2) simpler forms by noting that the bracketed contribution p h − 2∂yv h = −Γ∂xxh, (32) is already of first order, so that ∂th can be replaced by its ∂ u(2) =4∂ h∂ u(0) ∂ v(1) , (33) 2 y h x x h − x h zeroth order estimate −h ∂xh. This leads to: (2) (2) (a) u 0 =0, (b) v 0 =0 . (34) (1) 1 1 4 1 3 3 1 2 u = y − hy +h y h+B y −hy ∂xh, 3 8 2 2 where (19), (17) and the fact that ∂ u(0) = 0 have been y h (24) used to simplify (32) and (33), respectively. As before, v(2) (2) 2 1 2 6 3 is derived from (31) subject to (34b), u from (29) using ∂th + h ∂xh + ∂x h − Bh ∂xh =0. (25) 3 5 v(2) just found with boundary conditions (33, 34a), which is enough to determine the evolution equation for h(x, t) Looking for a nearly uniform solution h(x, t)=hN+η(x, t) at this order (see [28] for a detailed computation). In the we obtain at lowest order: same time, the pressure p(2) is obtained from (30, 32). So, 1 2 surface tension effects supposed to smooth out the steep ∂ η + h2 ∂ η + h3 h3 − B ∂ η =0. (26) t N x 3 N 5 N xx space gradients of interface modulations now enter the solution via the boundary condition (32) on p(2).Wehave This shows that in a reference frame moving at velocity thus to continue the expansion and observe that they will 2 (2) V = hN =3uN, with coordinate ξ = x − Vt, the interface enter the solution at third order through a term −∂xp modulations are controlled by a diffusion equation: in the equation governing u(3) playing the same role as (1) the term −∂xp in (29). ∂tη = D∂ξξη, (27) It is easy to demonstrate recursively that at each order n the velocity field u(n) can be written in the form of where m a polynomial in y, h and its derivatives ∂x h. A careful 1 3 2 3 examination of the derivation process also shows that, for D = h B − h , (28) 3 N 5 N n ≥ 2, the term of highest degree in y appearing in u(n) is of degree 4n and originates from inertial interactions which leads to the known result that fluid films flowing between the Nusselt flow profile u(0) and its correction down an inclined plane are unstable against waves (D<0) (n−1) (0) (n−1) (n) (0) at order n − 1, u , via u ∂xu + v ∂yu = when their thickness becomes larger than some thresh- y u(0)∂ u(n−1) − ∂ u(0) ∂ u(n−1) dy. Moreover, it can be old value h given by: h3 = 5B (= 3R according to x y 0 x Nc Nc 2 c seen that, if c is the coefficient of the term y4n in u(n), the definition of the Reynolds number R in note [26]). In n R terms of the flow rate (usual control parameter) we can then 6 1 3 5 write D = 5 qN (qNc − qN)whereqNc = 3hNc = 6B. 4n − 1 Flows along vertical planes are therefore always unstable cn+1 = − cn for n ≥ 1, 2(4n + 1)(4n + 3)(4n +4) since θ = π/2 implies B = 0 and hence qNc =0,sothat D = −6q2 < 0 for all q . Whereas the stabilizing ef- 5 N N so that c = − 1 , c = 1 etc., showing that fect of gravity is obvious from the expression of D,the 2 4480 3 1520640 the contributions of these highest-degree terms become origin of the destabilizing contribution is more difficult to quickly negligible in the evolution equation for h at order trace back. Instability comes via the term ∂ u(0) in (15) t n, ∂ h + ∂ (q(0) + ...+q(n))=0. [27] converted into a space derivative term using (14) and t x However, the line of thought followed up to now sticks partly compensated by the other source terms in (15). to a strict gradient expansion. Accordingly the different At this stage, gradient-limiting terms playing the role terms retained at a given order in the final evolution equa- of viscosity for the Burgers equation are effective only for tion for h are supposed to scale as the corresponding pow- infinitesimal interface fluctuations of thin films at suffi- ers of the gradients ∂ and ∂ . As soon as the latter are ciently low flow rate (h
Boundary condition (12) then reads p h = −Γ∂xxh so plays the role of an effective degree of freedom governed that the pressure is no longer given as in (13) but rather by a Benney-like evolution equation. Another approach is by then needed to deal with the dynamics of the film in a context where this enslaving is partly relaxed and other (0) p = B(h − y) − Γ∂xxh. (35) effective degrees of freedom are introduced, under the con- straint that these new variables should remain slowly vari- The solution at first order has to be modified accordingly, able in x and t and that exact results of the gradient which yields expansion should be recovered in the appropriate limit. Up to now, the hydrodynamic fields (u, v, p) could be ex- 2 1 2 6 3 3 ∂ h + h ∂ h + ∂ h − Bh ∂ h + Γh ∂ 3h =0. panded on a special set of polynomials in y with slowly t x 3 x 5 x x varying coefficients functions of h(x, t) and its derivatives. (36) If the flow modulations are sufficiently slow, these fields instead of (25). should not be far from their estimates obtained by the gra- Considering infinitesimal modulations, in lieu of (27) dient expansion. In other terms, the residue of a Galerkin we now get expansion —or of an approximation derived from a more general weighted residual method— based on these poly- ∂tη = D∂ξξη − K∂ξξξξη, (37) nomials should be intrinsically small. The coefficients of the expansion would then be considered as the sought- 1 3 with K = 3 hN Γ = qN Γ . As expected from Fourier after effective degrees of freedom, and they would be gov- analysis with perturbations ∝ exp(ikx), the last term in erned by equations generalizing the expressions asymp- (37) damps out short-wavelength fluctuations, at a rate totically valid when modulations are infinitely slow. The −Kk4 that dominates the destabilizing contribution |D|k2 required extension would give some latitude of evolution when D is negative. The two terms counterbalance each to these coefficients around the asymptotic value obtained other exactly for a certain cut-off wave-vector kc given from the gradient expansion. The model developed below 2 6 by kc = |D|/K = 5 (qN − qNc)/Γ . This wave-vector de- is an attempt to implement this general idea in the most fines a scale for which, in order of magnitude, capillary “economical” way. effects enter the problem and work to limit the divergence Let us begin with the set of equations consistent at of space gradients. A low order truncation of the expan- first order except for surface tension effects that, though sion will therefore be acceptable if kc is physically small formally of higher order, are included here owing to their enough, which will always be the case close to the insta- gradient-limiting role, as discussed above. The problem to bility threshold qNc. be solved reads: Equation (36), usually called the Benney equation, is thus expected to govern the interface modulations with ∂tu + u∂xu + v∂yu + ∂xp − ∂yyu − 1=0, (38) space gradients at most of order kc as long as kc 1, i.e. ∂ p + B − ∂ v =0, (39) at flow rates close to the threshold in a range depending on y yy the value of Γ . The linear argument above using (37) gives ∂xu + ∂yv =0, (40) hints on the behavior of solutions to (36) only because the instability turns out to be supercritical, so that a weakly- with boundary conditions nonlinear theory accounts for the continuous growth of p + Γ∂ h 2∂ v =0, (41) the amplitude of modulations in the neighborhood of the h xx − y h threshold. In fact, owing to the strong nonlinearities as- ∂yu h =0, (42) sociated with the high powers of h present, this neigh- u =0,v =0, (43) borhood is quite narrow and typical solutions to (36) dis- 0 0 play finite-time singularities not so far from the threshold. and of course the kinematic condition at the interface From the additive nature of the contribution of the Γ -term which, in integral form (9), accounts for mass conserva- in (12) leading directly to (35), it is clear that, depending tion on average over the thickness. on the order in ∂x one decides to introduce it, other gra- dient terms in h will appear in (36), which will play an Integrating (39) with the help of boundary conditions (41–42) we get p = B(h − y)+∂yv+∂yv −Γ∂xxh and effective role if kc is too large, i.e. in general K too small, h further eliminate ∂ p from (38). Because ∂ v = −∂ u is a hence Γ too small. Taking these terms into account does x y x not solve the problem of finite-time singularities, whose first order term, its derivative is of second order and can origin may be attributed to the strongly nonlinear charac- be dropped of. Therefore, our set of equations read ter of the evolution equation for h, which involves rapidly increasing powers of h as the gradient expansion proceeds. ∂tu + u∂xu + v∂yu − ∂yyu =1−B∂xh + Γ∂x3h, (44) ∂xu + ∂yv =0, (45) 4 First-order model with boundary conditions (42–43). (44–45) is sometimes called boundary-layer equations (BL). In the gradient expansion, the flow variables are sup- Let us now consider the averaging of equation (44) that posed to be strictly enslaved to the local thickness h which gives the balance of x-momentum (von K´arm´an’s equation C. Ruyer-Quil and P. Manneville: Modeling film flows down inclined planes 283 in the context of boundary layers). We obtain: which differs from equation (36) by the coefficient in front of h6. This discrepancy leads to an overestimation of h the critical Reynolds number Rc,IBL = B [29]. Proko- [∂tu+u∂xu+v∂yu−∂yyu]dy =h+Γh∂x3h−Bh∂xh, piou et al. [8] and Lee and Mei [10] tried to get rid of 0 Z (46) this discrepancy by starting from systems of equations more complete than (44–45), while keeping the zeroth- which can be transformed into order parabolic velocity profile. These approaches failed to recover the correct critical Reynolds number prediction h h 5 2 Rc = 6 B and led back to Rc,IBL. The cause of this failure ∂ udy+∂ u dy =h ∂ u +Γh∂ 3h Bh∂ h. t x − y 0 x − x thus seems to be the lack of flexibility of the assump- Z0 Z0 (47) tion about the velocity profile rather than the omission of terms in the set of primitive equations. Transformation of the l.h.s. is similar to that leading to In the spirit of Shkadov’s assumption, a whole family of models analogous to (49–50), with just different numer- (9). The term ∂yu 0, representing the shear at the wall, will be denoted τw in the following. On the l.h.s. we rec- ical constants, would be obtained by using velocity profiles h ognize q = u(y) dy and we can define a new averaged expressed in terms of a single, possibly better fitted, func- 0 tion of the reduced variable ζ = y/h, u = a(x, t) g(ζ). field r = h u2(y) dy. With these notations (47) reads: 0 R Such velocity profiles are sometimes called “similar” solu- R tions in boundary-layer theory [30]. The defect of such an ∂tq + ∂xr = h (1 + Γ∂x3h−B∂xh) − τw. (48) approach comes from a somewhat arbitrary freezing of the effective degrees of freedom involved in the velocity field. Assuming a given velocity profile, one arrives at a set of Here we want to relax part of this constraint by assuming two equations (9) and (48) for two unknowns h and q,since that u is a superposition of functions with slow variable r can then be computed from q. Simply taking Kapitza’s coefficients so that the quantities r and τ are less rigidly parabolic profile u(y) 1 ζ(2 ζ)whereζ=y/h,wehave w ∝ 2 − related to q and h. 6 2 2 r = 5(q /h), and τw =3q/h . Inserting these estimates So, keeping in mind the results of the long-wavelength in (48) we obtain Shkadov’s model [22]: expansion, we may reasonably admit that Shkadov’s sim- ple assumption, valid at order zero, can be improved by ∂th = −∂xq, (49) correcting the parabolic profile with the polynomials that q 12 q 6 q2 appear in the gradient expansion at higher orders, or more ∂ q = h−3 − ∂ q+ −Bh ∂ h+Γh∂ 3h. t h2 5 h x 5 h2 x x precisely, linearly independent combinations of these poly- (50) nomials adequately chosen for better computational ease. So, let us expand u as
Now, taking (49, 50) as a set of primitive equations, let (0) (1) us consider slow modulations to the uniform solution that u(x, y, t)=b0(x, t)f (y/h)+b1(x, t)f (y/h), (52) verifies q = 1 h3 = q(0). We are now in position to perform 3 where f (0)(ζ) 1ζ2+ζand f (1)(ζ) a gradient expansion parallel to the previous one by as- ≡−2 ≡ (0) (1) (2) (1) (2) 1 1 4 3 2 (0) suming q = q +q +q +...,whereq , q , etc. are 6 4 ζ − ζ + ζ . Whereas u in (13) is obviously formally of order 1, 2, etc. At lowest order we obtain for proportional to f (0), a little algebra is necessary to check (0) 1 3 2 2 (0) (1) the l.h.s. of (50): ∂tq = ∂t( 3 h )=h ∂th=−h ∂xq = that u in (24) can indeed be written as some specific 2 2 4 (0) (1) −h (h ∂xh)=−h ∂xh, and for the r.h.s.: combination of f and f (cf. note [31]). Apart from the fact that the so-far unknown fields b0 2 (1) (0) (0) and b1 are supposed to be slow functions of x and t,by q 6 q 12 q (0) 3 −3 2 + 2 −Bh ∂xh− ∂xq +Γh∂x h, definition of q they must fulfill h "5 h # 5 h h 1 1 q = u(y) dy = h b + b . (53) 3 0 15 1 where, as before, the surface-tension term has been intro- Z0 duced earlier than dictated by its formal order. This leads At this stage we have three unknowns, h, b0,andb1 but to the estimate from (53) we see that we can pass to a set formed by h, q and b , with b given by b =(3q/h) 1 b . Inserting this 1 1 1 1 0 0 − 15 1 (1) 6 3 3 2 q = h ∂xh − Bh ∂xh + Γh ∂x3h, in τ ≡ ∂ u = b /h we get τ =(3q/h ) − (b /15h). In 9 3 3 w y 0 0 w 1 that way, b1 appears as a correction to the shear at the (0) (1) plate that would be created by a parabolic velocity profile which, when replaced in ∂th + ∂x q + q =0,gives the following evolution equation corresponding to a film with thickness h and flow rate q. To make this explicit, let us re-define b1 as b1 = −15hτ so 2 1 1 6 3 3 that ∂th + h ∂xh + ∂x h − Bh ∂xh + Γh ∂x3h =0, 3 3 3q τ = + τ. (54) (51) w h2 284 The European Physical Journal B
What we have just done is to pass from the original alge- for the two unknowns h and q. Having the same structure braic variables b0 and b1 to the more physically minded as Shkadov’s model but with slightly different coefficients, variables q and τ. Accordingly (52) can be rewritten as it will be called “modified Shkadov model”. The correc- tions arise from a better account of the fluctuations of 3q (0) (1) τw introduced via the third derivative term in (56) by u = + hτ f (y/h) − 15hτf (y/h). (55) 3 (1) h the ζ -term in f , which can be traced back to the ∂th contribution of u(1) in (21) already at the origin of the But, h, q,andτstill make a set of three variables for instability of the plane interface. which we have only two equations (9, 48). Several strate- Now, let us consider a gradient expansion of (9, 58) gies can be followed to find equations for the unknowns by similar to the previous ones by assuming q = q(0) + q(1) + the method of weighted residual. One can for example av- q(2) + ···. We are this time led to erage the governing equations with a series of weights. (In the classical Galerkin method the set of weight functions (1) 2 6 1 3 1 3 q = h ∂ h − Bh ∂ h + Γh ∂ 3h, is simply the set of the basis functions, further assumed 15 x 3 x 3 x to fulfill the boundary conditions.) Here equation (48) is (0) (1) just the averaged version of (44) with the trivial uniform which, when replaced in ∂th+∂x q + q =0,givesus weight and, as shown previously, (9) is nothing but the back the Benney equation (36). So, by construction, the averaged continuity equation (3) simplified by using the near-critical behavior is correctly predicted by our mod- kinematic interface evolution equation (4). An alternate ified Shkadov model, which no longer overestimates the method consists in imposing conditions at special points value of the stability threshold. of the domain (collocation). Here, to obtain complemen- Though predictions in the asymptotic domain far be- tary equations, we will follow this last method and im- yond the near-critical region are not improved by the cor- pose the fulfillment of additional conditions at y =0and rection, a step has clearly been made in the right direction. y = h. These relations will be obtained by differentiat- In particular, the derivation implies that the shear at the ing the equations of motion with respect to y and further wall is a slowly varying quantity. At order zero, q is slaved evaluating them at the boundaries (see [32]). Conditions to h (or the reverse) and τw does not fluctuate (inject- at y = 0 can be interpreted as relations giving the coeffi- ing a parabolic profile for u in (56) gives ∂t(∂yu 0)=0). At order one, q and h are two independent slowly vary- cients of a Taylor expansion of the solution as a function of y. ing effective degrees of freedom while τw remains rigidly A condition useful at first order is obtained from (44) linked to q and h through (57). Let us now proceed to differentiated with respect to y: the next step. Anticipating the result, we will introduce four additional fields b2, ... , b5 associated with the ve- ∂tyu + u∂xyu + v∂yyu − ∂y3 u =0, locity corrections introduced by the gradient expansion at second order. Supplementary conditions at boundaries further evaluated at y = 0, which gives: analogous to (56) will be derived which, when imposed to the solution will allow us to eliminate these supplemen- 3 tary fields so that we will be left with three equations for ∂t(∂y u 0) − ∂y u 0 =0. (56) three unknowns, h, q and τ, that will play the role of an
Equation (56) tells us that the fluctuations of the shear additional independent effective degree of freedom. The at the wall are directly linked to the presence of correc- derivationdetailedinthenextsectioncanbeskippedby tions to the velocity profile beyond the parabolic shape those only interested in the result, equations (78–80). for which they cancel identically. From (55) and (54), fur- ther observing that τ is a first-order correction so that the term ∂tτ is of higher order and can be dropped from (56), 5 Second-order model we get the required supplementary equation Let us first write down the set of equations consistent at 3q τ second order. Clearly all terms in (10), rewritten here for ∂ − 15 =0, (57) t h2 h2 convenience as expressing the correction τ in terms of the time derivative ∂tu + u∂xu + v∂yu + ∂xp − (∂xx + ∂yy)u − 1=0, (59) of the other fields. Inserting the resulting evaluation of τw are relevant. Such is not the case for (11) which can be in (48), using (9) to eliminate ∂th and the (here sufficient) 6 2 simplified as zeroth-order estimate r = 5 (q /h), we get ∂ p − ∂ v + B =0. (60) 5 5 q 7 q q2 5 5 y yy ∂ q = h− − ∂ q+ − Bh ∂ h+ Γh∂ 3h. t 6 2 h2 3 h x h2 6 x 6 x It can indeed be seen that the contribution of the terms (58) ∂tv + u∂xv + v∂yv is in fact of higher order, owing to the differentiation of p with respect to x in (59) as seen from When added to (9), equation (58) completes the model at the calculation leading to (63) below. Of course u and v first order as a system of two partial differential equations remain linked by the continuity equation (3). C. Ruyer-Quil and P. Manneville: Modeling film flows down inclined planes 285
For the boundary conditions we have so that, by definition of q,wehave
p + Γ∂ h 2∂ v =0, (61) 1 1 1 7 1 1 h xx − y h q=h b + b + b + b + b + b . 3 0 45 1 6 2 360 3 840 4 7560 5 −4∂xh∂x u + ∂yu + ∂xv =0, (62) h h h (65) to which we add (43) and of course (9). These new polynomials do not contribute to the evaluation Let us transform (59). Integrating (60) with respect to of τw which is therefore still given by τw = b0/h. y we get p(y)=−By +∂yv+K. The integration constant Sticking to the approach followed for the derivation K, in fact a function of x and t, is obtained from (61) of the first order model, we keep q and τ as primitive which gives K = Bh + ∂ v Γ∂ h, hence p = B(h y h − xx − variables in addition to h. In particular τ remains defined y)+∂ v+∂ v −Γ∂ h. Inserting this expression into 2 y y h xx from τw =(3q/h )+τ, which implies b0 =(3q/h)+hτ. (59) we get: Inserting this in (65) we obtain b = −(15hτ + 15 b − 1 2 2 7 3 1 8 b3 + 56 b4 + 168 b5), so that the expression of u extending (55) to second order reads: ∂tu + u∂xu + v∂yu =1+∂yyu − 2∂xyv + ∂x ∂xu h
−B∂xh + Γ∂x3h, h (63)i 3q 15 7 3 1 u = +hτ f (0) − 15hτ + b + b + b + b h 2 2 8 3 56 4 168 5 where the continuity equation has been used to transform 5 ∂xxu into −∂xyv and −∂yv h into ∂xu h (notice that this (1) (k) last term is a function of x and t through h and is differ- × f + bkf , (66) k=2 entiated with respect to x as such). Averaging (63) over X the thickness h as before we obtain y from which v = − 0 ∂xudy can be derived. To deal with the seven unknowns h, q, τ, bk, k =2,...,5, assumed to ∂ q + ∂ r = h 1+Γ∂ 3h−B∂ h + ∂ ∂ u t x x x x x h be slow functions ofR x and t, we need seven independent equations. In addition to the kinematic condition at the h −τw + ∂yu − 2∂xv , i h h interface (9) and the average x-momentum equation (64) derived earlier, we have the boundary condition (62) on where q, r,andτw are defined as before. (The second the tangential stress at y = h, here used for the first time. boundary term ∂xv resulting from the explicit integra- 0 To get the remaining four equations we follow the same tion of ∂xyv has been dropped since it cancels automat- scheme as for the first-order model and look for collocation ically by virtue of v ≡ 0.) Making use of (62) we can 0 at the boundaries. simplify this equation: explicitly, we have h∂ ∂ u + x x h The equation extending (56) at second order is ob- ∂ u − 2∂ v = h∂ ∂ u + 4∂ h∂ u − ∂ v − y h x h x x h x x h x h tained by differentiating (59) with respect to y,which 2∂ v = h∂ ∂ u + ∂ h∂ u +3∂h∂ u − yields: x h x x h x x h x x h
3∂xv h = ∂x h∂xu h − 3∂xh∂yv h − 3∂xv h = ∂tyu + u∂xyu + v∂ 2 u − ∂ 3 u − 2∂ 2 u =0, (67) ∂ h∂ u − 3∂ v = ∂ h∂ u − 3v . Finally we y y x y x x h x h x x h h arrive at where we have used (60) differentiated with respect to x and the continuity equation (3) to eliminate ∂xyp.Further 3 ∂tq+∂xr =h 1+Γ∂x h−B∂xh −τw+∂x h∂xu h −3v h , evaluating (67) at y =0givesus h i h (64)i 3 2 ∂t(∂yu0)−∂y u0−2∂x yu0 =0. (68) to be compared with its first order counter-part (48). The evaluation of (67) at y = h leads to an independent The two first polynomials to be used in addition to (0) (1) (2) 1 2 (3) 1 1 4 2 equation. The raw condition: f and f are f = 2 ζ and f = 12 − 2 ζ + ζ . (0) (1) (2) (3) The set f ,f ,f ,f is easily seen to form a basis ∂ u + u ∂ u + v ∂ 2 u − ∂ 3 u − 2∂ 2 u =0 ty h h xy h h y h y h x y h of the space of polynomials of degree ≤ 4 that cancel at (4) 1 1 6 5 y = 0. The two last polynomials f = 120 − 6 ζ + ζ (5) 1 1 8 1 7 1 6 and f = ζ − ζ + ζ have been chosen so can be simplified by noting that ∂tyu = ∂t ∂yu − 240 56 7 3 h h as to permit the reconstruction of all the polynomials ∂ u ∂ h and ∂ u = ∂ ∂ u − ∂ u ∂ h.Itis yy h t xy h x y h yy h x entering the gradient expansion at second order. Their easily checked that the kinematic condition v = ∂ h + seemingly exotic coefficients have been fixed by the ad- h t 5 (4) 5 6 (5) 6 u ∂x h at the interface leads to a canceling of terms in ditional constraints: d f /dζ |0 =1,df /dζ |0 =1, h ∂ u . Furthermore (62) shows that ∂ u is already of that slightly simplify the forthcoming computations. The yy h y h velocity component u is expanded as second-order so that its x and t derivatives can be ne-
glected. Accordingly, equation (67) evaluated at y = h 5 simply reads (k) u(x, y, t)= bk(x, t)f (y/h), k=0 ∂y3 u +2∂x2yu =0. (69) X h h
286 The European Physical Journal B
To obtain simple conditions involving more specifically the b3 from (72) and (73), respectively. In fact, the somehow variables associated to the high-degree polynomials f (k), weird definitions of polynomials f (2) to f (5) were intended k =4,5, we will add conditions obtained in the same way to make this final evaluation simpler: instead of having the as (68) but with higher derivatives. The second derivative bk given by the most general (4 × 4) linear system, we get of (59) brings no new condition on the coefficients of these them by elementary substitution owing to the fact that polynomials since d4f (4)/dζ4 and d4f (5)/dζ4 cancel when specific coefficients in this linear system cancel automati- evaluated at ζ = 0. Thus taking the third derivative of cally when the corresponding combinations of derivatives (59), we obtain are evaluated at y =0ory=h. To terminate the derivation of the second-order model, 3 2 5 2 3 ∂t(∂y u 0)+2∂yu 0∂xy u 0 − ∂y u 0 − 2∂x y u 0 =0, it remains to substitute the expressions of the bk in the (70) dynamical equations for h, q and τ,thatistosaythe
kinematic condition at the interface (9), the average x- and next for the fourth derivative: momentum balance equation (64) that now reads
∂t(∂ 4 u )+3∂yu ∂ 3 u +2∂ 2u ∂ 2 u 2 y 0 0 xy 0 y 0 xy 0 6q 2 3q 9∂xq 6q∂xh ∂tq+∂x − hqτ =− 2 −τ +h+∂x − − 2∂xyu ∂y3 u − ∂y6 u − 2∂x2y4 u =0. (71) 5h 35 h 2 h 0 0 0 0 − Bh∂xh + Γh∂x3h, (76) The model at second order is obtained by inserting the ansatz (66) into equations (9,64,62,68–71). In the deriva- and the stress balance at y =0: tion of the first-order model, the time derivative of τ could be neglected since it was of higher order. This allowed us to eliminate it from the equations to get a system of 15τ 3q 6q 1 15 7 − + ∂t + ∂tτ − ∂xx − b2 + b3 two equations for the two unknowns h and q. Here this h2 h2 h2 h3 2 8 derivative is no longer negligible and τ is a genuine dy- 3 1 + b + b =0. (77) namical variable. By contrast, the additional fields are 56 4 168 5 second-order corrections so that their t-orx-derivatives, of still higher order, can be dropped without damage. The At last we obtain derivation, straightforward but (really) tedious, has been performed using mathematica. Let us consider first the ∂th = −∂xq, (78) equations accounting for the enslaving of b2, ..., b5 to h, q, 2 3q 2 6q 6q∂xh 9∂xq and τ, namely (62) and (69–71) taken in that order. We ∂ q =h− −τ +∂ hτq− − + t h2 x 35 5h h 2 obtain −Bh∂xh + Γh∂x3h, (79) 2 b2 b4 b5 3q(∂xh) 3∂xh∂xq 2 + + + − 7 21q 42τ 18q ∂xh 6q∂xq 2qτ∂xh 2 ∂ τ = − − − + + h 30h 210h h h t h h4 h2 5h4 5h3 5h2 3q∂xxh 2 + −∂ q =0, (72) τ∂xq 3q∂xτ 84q(∂xh) 63∂xq∂xh 2h xx + − + − 15h 5h h4 h3 7B∂xh 7Γ∂x3h 2 − + · (80) b4 b5 36q(∂xh) h h −b + + − +12∂ h∂ q+6q∂ h=0, 3 3 15 h x x xx (73) A gradient expansion up to second order assuming q = q(0) + q(1) + q(2) and τ = τ (1) + τ (2) leads to
54q2 ∂ h 18q∂ q b τ 54qτ∂ h (2) 7 3 8 6 127 9 2 x x 4 x q = h − Bh + h (∂xh) 6 − 5 − 5 +15∂t 2 + 4 3 15 315 h h h h h 6hτ∂xiq 36q∂xτ 4 10 7 4 10 − − =0, (74) + h − Bh + h ∂ 2 h, h3 h3 63 63 x (2) 4 4 13 7 2 τ = 2h − Bh + h (∂xh) 2 15 63 q b4 − b5 τ 234qτ∂xh 9∂x + − 15∂t − h6 h6 h3 h5 1 2 8 5 2 8 + h − Bh + h ∂x2 h, h 19iq∂ τ −6τ∂ q 2 105 63 +9 x x =0. (75) h4 yielding the exact second-order evolution equation [6] as As already mentioned, partial derivatives of the bk do not expected. enter these equations so that they can be obtained imme- At this stage, one can notice that the second-order diately as functions of h, q,andτ. Indeed b4 is directly formulation includes the effects of the stream-wise dissi- extracted from (74), next b5 from (75), and finally b2 and pation, while they are absent from the first-order model, C. Ruyer-Quil and P. Manneville: Modeling film flows down inclined planes 287 and more generally from any expansion based on the first- implies that the phase speed c = ω/k is constant and equal order BL equations (44, 45). As a matter of fact, perform- to unity at marginality. As expected, equations (85–86) ing the same derivation at second order with this system, characterize the marginal waves of Benney’s equation (36) one obtains [13] (For comparison, linear stability analysis of Shkadov’s model (49–50) would lead to R − B = Wk2.). ∂th = −∂xq, (81) Following Yih [34], let us consider the temporal stabil- 3q 2 6q2 ity problem in terms of an asymptotic expansion of fre- ∂ q = h − − τ + ∂ hτq − t h2 x 35 5h quency ω in powers of a real wave-number k in the limit k 1. We get −Bh∂xh + Γh∂x3h, (82) 2 7 21q 42τ 18q ∂xh 6q∂xq 2qτ∂xh 2 1 22 44 ∂tτ = − − − + + ω = k + ik2 R− B +k3 BR − R2 h h4 h2 5h4 5h3 5h2 5 3 45 75 τ∂xq 3q∂xτ 7B∂xh 7Γ∂ 3h + − − + x , (83) 2 28 1184 1 15h 5h h h + ik4 − B2R + BR2 − R3 − W + O(k5), 15 27 1125 3 where it can be seen that terms of the form ∂xx() or (87) ∂x()∂x(), are absent. By difference with (78–80), such terms have thus to be attributed to the effects of stream- to be compared with the exact asymptotic expansion of wise viscous dissipation. Orr–Sommerfeld equation, 2 1 10 4 ω = k + ik2 R− B +k3 −1+ BR − R2 5 3 21 7 6 Linear stability analysis 3 471 2 17363 +ik4 B− R − B2R + BR2 Before presenting our preliminary numerical results with 5 224 15 17325 the model equations developed in Sections 4 and 5, let 75872 1 us compare their linear stability properties with the data − R3 − W + O(k5). (88) 75075 3 collected by Liu et al. [33]. Setting w = wN +¯wexp[i(kx− ωt)] where w and wN refer to h and q and their values for the reference Nusselt flow solution, linearizing (9, 58), we Although equation (87) is close to (88), three terms are obtain missing in (87). A careful examination of the calculus shows that these terms mostly originate from second-order ω h¯ = k q,¯ viscous terms omitted in the derivation of (9–58). A parallel stability analysis of the second-order model (78–80) leads to the dispersion relation 5 1 5 5 + i h4 − Bh k −i Γh k3 h¯ = 2 3 2 9 N 6 N 6 N A(k)+B(k)ω+C(k)ω +D(k)ω =0, (89) 5 7 1 with +i h4 k − i h2 ω q.¯ 6 27 N 3 N 1 1 4 2 1 A(k)=k+i R− B k2+ − R2 + BR k3 Compatibility of this system yields the dispersion rela- 5 3 5 1225 105 tion. For easier comparison with experimental results, it 2 1 1 is preferable to turn to a scaling where h is the length + i R − W k4 + Wk5, N 175 3 315 scale. This leads to change k into k/hN, ω into ωhN,and 1 3 to introduce the Reynolds and Weber numbers, R = h 38 9 6 2 1 2 3 N B(k)=−1−i Rk + − + R − BR k and W = Γ/h2 respectively. 35 5 245 35 N 29 3 − i Rk3 − WRk4, 2 1 2 1 4 14 350 105 k + i R − B k − i Wk + −1−i Rk ω 15 3 3 15 9 3 9 3 C(k)=i R− R2k + i Rk2, D(k)= R2. 6 7 35 70 35 +i Rω2 =0. (84) 5 The asymptotic expansion of (89) in the temporal domain Separating real and imaginary parts for real k and ω we (k ∈ R)gives get 2 1 10 4 ω=k+ik2 R− B +k3 −1+ BR − R2 ω = k, (85) 5 3 21 7 6 2 3 376 2 3683 R − B = Wk . (86) +ik4 B− R − B2R + BR2 5 5 175 15 3675 Equation (86) defines the neutral stability curve of the 1238 1 − R3 − W + O(k5), (90) modified Shkadov model in the plane (R, k) while (85) 1225 3 288 The European Physical Journal B
a linear stability analysis of BL equations at second-order 12 including both a heuristic term accounting for pressure variation across the film and viscous stream-wise dissipa- 10 tion, even at large Reynolds number. The slight discrep- ancy between experiments and our theory remains unex- plained but we can remark that it is closely comparable 8 to that already reported in [33]. As a matter of fact, im- proved agreement (light solid line in Fig. 2) can easily be 6 obtained by adjusting the value of the theoretical thresh- old to that extrapolated from experimental data, which
frequency (Hz) ◦ ◦ 4 can be done by changing the slope from β =5.6 to 5.85 , or by adopting a slightly different set of physical constants for the fluid mixture. 2 Spatial linear stability properties (ω ∈ R, k ∈ C)for equations (84) and (89) with or without second-order vis- 0 cous terms, are compared with experimental results in 8 9 10 11 12 13 14 15 Reynolds number Figure 3. For the phase speed (Fig. 3b), predictions of the various approximations are very close to each other Fig. 2. Neutral stability curve for glycerin-water films at ◦ and discrepancies with experimental data (squares) are β =5.6 and Γ = 700.15. The cut-off frequency is shown as not significant. By contrast, for the growth-rate (Fig. 3a) a function of the Reynolds number. The thick solid line is the the agreement is again excellent for the full second-order prediction of the second-order model, the thin one is a fit with model (solid line). With respect to predictions of the first- experimental data of Liu and Gollub [33] displayed as squares. See text for dashed and dot-dashed lines that correspond to order —dashed— and simplified second-order model (81– the first-order and Shkadov’s model respectively. 83) —dot-dashed— it can be seen that the curves remain close to each other. As a consequence of the degeneracy of the marginal problem mentioned above, they intersect pre- cisely at zero growth-rate and phase-speed equal to unity. which is now very close to (88). All terms are present On the contrary, the full second-order model predicts a to the considered order in k and coefficients differ by no significantly lower growth-rate so that the marginal condi- more than 3% from the exact result. The remaining dis- tion is obtained for a smaller phase-speed, which explains crepancy between (90) and the exact asymptotic result the discrepancy illustrated in Figure 2. can be attributed to the omission of the inertial terms ∂tv + u∂xv + v∂yv in the y-momentum equation (11) ow- ing to the fact mentioned earlier that monomials of higher 7 Spatial evolution of nonlinear periodic degree involved in the velocity profile contribute very lit- tle to the coefficients of the gradient expansion at third waves order. Comparison of fully nonlinear solutions to our second- The neutral stability curve of (89), the spatial growth model (78–80) with experiments performed by Liu and rate −Im(k) and phase speed Re(ω/k) have been deter- Gollub [4] has been attempted by numerical simulations mined for a set of parameters corresponding to the ex- corresponding to a periodic upstream forcing. A second- periments of Liu et al. [33] using a standard predictor- order finite-difference quasi-linearized Crank-Nicholson corrector Euler–Newton continuation scheme [35]. Fig- scheme [37] has been implemented to deal with the ure 2 shows excellent agreement between our second- high-order nonlinearities involved. Owing to the convec- order model (thick solid line) and the experimental tive character of the instability that sweeps disturbances data (squares) for the cut-off frequency. Predictions from continuously away from the computational domain, the Shkadov’s model and from our modified first-order model boundary condition at the downstream end could be (9–58) are also displayed as dot-dashed and dashed lines, treated in a non-physical way by truncating the discretized respectively. It is clear from the figure that the latter pre- equations at the relevant nodes. The scheme was seen to dicts the threshold correctly, while the former overesti- be satisfactory in that inaccuracies remained confined to mates it somehow. The increasing discrepancy between a very small downstream numerical boundary layer and the first-order and the second-order predictions must be never invaded the upstream region. The upstream bound- attributed to the neglect of stream-wise viscous dissipa- ary condition was chosen as a periodically forced flow rate tion. As a matter of fact, starting with the BL equations
(44–45) that also neglects this contribution, one is led to q(0,t)=qN(1 + A cos ωt) (81–83) at second-order. A dispersion relation with the same structure as (89) but with missing terms correspond- to fit with Liu and Gollub’s flow-rate control via the en- ing to it can be derived (see note [36]), and it can be trance pressure manifold. The length of the wave inception checked that marginal properties are still given by (85–86), region depends on the forcing amplitude. The conversion which translates into the dashed line in Figure 2. A similar of the relevant physical quantity into a numerical param- improvement was obtained by Yu et al. [9] who developed eter being out of reach, we have chosen to compare the C. Ruyer-Quil and P. Manneville: Modeling film flows down inclined planes 289
further downstream is also well captured by the simula- x 10−3 tion. Figure 5 compares laboratory and computer results 5 for a low-frequency forcing ending in near-solitary sta- tionary wave-trains. The comparison is again very good, 4 though our numerics predict amplitudes somewhat larger than in the experiment. The capillary ripples downstream 3 the main hump are also slightly more pronounced and the wavelength of the stationary train a little smaller. The 2 use of phase-sensitive averaging technique that might have smoothed the steepest parts of the experimental wave pro- 1 files could partly explain the larger discrepancies observed growth rate in Figure 5 when compared to Figure 4. Our results are 0 also in good agreement with direct numerical simulations performed by Ramaswamy et al. (see Figs. 15 and 17 of −1 [19]) who also report larger amplitudes of the fronts. We have also compared our numerical results to ex- −2 perimental ones of Alekseenko et al. [38]. Though their 0 0.05 0.1 0.15 0.2 0.25 0.3 geometry was cylindrical, the thickness-to-diameter ratio wavenumber was sufficiently large for curvature effects to be negligible. (a) Figure 6 displays the profile and streamlines of a station- ary large-amplitude near-solitary wave for parameters in 1.01 Figures 5, 6, and Table 1 of [38]. Streamlines show evi- dence of a large recirculation region inside the hump. Such 1 recirculations have been inferred from experimental wave 0.99 profiles by numerical reconstruction using the NS equa- tions [41] and were already present in solutions to the BL 0.98 equations [21]. 0.97 Alekseenko et al. measured the mean film thickness hhi, the peak height hmax, the wavelength λ, the speed c 0.96 of the waves, and the mean surface velocity hUi. Compar- phase velocity ison with our data is given in Table 1. Though the speed 0.95 and wavelength were provided with the experimental data, 0.94 the forcing frequency (the genuine control parameter) was not given. The uncertainty about its value may partly ex- 0.93 plain the differences between experimental and numerical 0.92 values. At any rate, it is remarkable that the mean film 0 0.05 0.1 0.15 0.2 0.25 0.3 wavenumber thickness hhi is much smaller than the Nusselt film thick- ness hN =0.582 mm for both the experiment and the sim- (b) ulation. The wave profile in Figure 6 is strikingly similar Fig. 3. Dispersion of linear waves for glycerin-water films to the sketches in [38]. By contrast, solutions of Shkadov’s with β =4.6◦,R=15.33 and Γ = 797.2 corresponding to model by Tsvelodub and Trifonov [24] (cf. Fig. 15 of [24]a) the experiments of Liu and Gollub (squares) and predictions of and of BL equations by Chang et al. [39,40] exhibit a large various models. (a) Spatial growth rate. (b) Phase velocity. The number of capillary ripples in front of the main hump. This solid lines derive from the dispersion relation of the second- feature can probably be attributed to the lack of disper- order model. Dashed lines and dot-dashed lines corresponds to sion associated to the second-order stream-wise viscous the first-order model and the second-order model with stream- dissipation terms omitted in BL equations. wise second-order dissipation terms omitted.
8 Conclusion waves at corresponding amplitude levels without trying to adjust the distance to the origin of the flow, in other terms From a theoretical point of view, the dynamics of film the abscissa frame is given up to some unknown transla- flows down inclined planes can be studied at different tion. We first present results corresponding to a multi- approximation levels, ranging from direct simulations of peaked wave regime in Figure 4. The form and amplitude the full NS equations to the simple KS equation. While of the numerical wave profiles are in striking agreement the latter gives an over-simplified picture of the problem, with the experiment. In particular, the small depression appropriate for near-infinitesimal thickness modulations appearing on the primary peak, passing the primary front of the flat film basic flow at the limit of strong inter- and then locking its phase to form a subsidiary peak of the facial effects, the former turn out to be untractable in wave-front is evident both in the simulation and in the ex- any realistic 3-dimensional situation, hence the need of periment. The modulation of the experimental wave-train modeling at some intermediate level. Fortunately enough, 290 The European Physical Journal B
1.2
1.0
0.8
0.6 15 25 35 45 55 65
1.2
1.0
0.8
0.6 55 65 75 85 95 105
1.2
1.0
0.8
0.6 90 100 110 120 130 140
Fig. 4. Comparison between experiments (left) and simulation (right). The plane is inclined to 6.4◦ from horizontal, liquid is glycerin-water and Reynolds number R =19.33 [4]. The figure shows three snapshots of the film thickness at three different locations on the plane from upstream (top) to downstream (bottom) at forcing frequency 4.5 Hz. Units are identical in the experiment and the simulation (here h0 ≡ hN ).
1.6
1.3
1.0
0.7 35 45 55 65 75 85
1.6
1.3
1.0
0.7 75 85 95 105 115 125
1.6
1.3
1.0
0.7 110 120 130 140 150 160
Fig. 5. Comparison between experiment (left) and simulation (right) for low-frequency forcing at 1.5 Hz. Parameters are identical to those in Figure 4. C. Ruyer-Quil and P. Manneville: Modeling film flows down inclined planes 291
less, which makes them much easier to analyze or simu- 2 late. The simplest such model due to Shkadov [22] just h involves h and the local flow rate q = 0 u(y) dy and is obtained by assuming a parabolic flow profile and av- eraging the BL equations at lowest orderR over the film 1.5 thickness. Though it yields qualitatively good results in the nonlinear regime, it fails to predict the critical be- havior quantitatively. In order to overcome its limitations N we have developed a more sophisticated approach that 1
h / also rests on averaged equations but deals with a more flexible velocity profile based on the polynomials appear- ing in Benney’s expansion. Among various weighted resid- 0.5 ual strategies, we have chosen a method of collocation at boundaries. The resulting additional conditions are identi- ties that have to be fulfilled by the solutions, which deter- mine relations between unknown coefficients in the expan- 0 sion of the velocity field, further understood as successive 0 5 10 15 20 25 30 35 terms (obtained by identification) of a Taylor series ap- proximation to this field. We have developed several other Fig. 6. Simulation of a periodically forced vertical film at variants of the method of weighted residuals, obtaining f =2.034 Hz, ν =7.210−6 m2/sandγ/ρ =57.610−6 m3/s2 models with similar structure. The interest of the present (i.e. R =12.4andΓ = 193.6) corresponding to the data strategy lies in the fact that the gradient expansion of the of Alekseenko, et al. [38] and leading to the formation of a model matches exact results asymptotically. Linear sta- near-solitary stationary wave-train. Streamlines are plotted in bility analysis has shown that second-order modeling cor- a reference frame moving at the speed of the wave. Notice the rectly predicts the threshold, the spatial phase-speed and large recirculation region in the hump. Wave characteristics are growth-rate beyond criticality in the range of parameters listed in Table 1. corresponding to laboratory experiments. Subsequent nu- merical simulation of the spatial evolution of forced films did not show finite-time blow-up often encountered with at sufficiently low Reynolds numbers instabilities develop Benney-like equations for which all variables except the at long wavelengths, which legitimates approaches resting film-thickness h are adiabatically eliminated. Good quan- on truncated gradient expansions. The resulting simpli- titative agreement with both numerical and experimental fication of NS equations yields so-called boundary layer results was obtained even for large amplitude waves with- equations that are slightly simpler to solve but remain out adjustable parameters. Proper account of the stream- partial differential equations of maximal space dimension wise viscous dissipation was seen to be crucial for the im- (3 or 2 according to whether one considers transverse per- provement of the theoretical predictions. turbations or not). They remain valid up to a point where the assumption about the thickness modulations become In the spirit of dynamical systems theory, a study of untenable, which depends on the strength of interfacial ef- the families of stationary wave solutions to the first and fects but is usually believed to happen above R ∼ 300 [1]. second-order models has been undertaken, also showing The next reduction step is taken by eliminating irrelevant good agreement with the direct numerical simulations of velocity degrees of freedom. Complete elimination of the Salamon et al. [18]. In particular, we were able to recover velocity field ends in Benney-like equations only involv- their bifurcation diagram showing a cusp catastrophe at ing the local film thickness h and its gradients. They are a quantitative level (see [42] for a preliminary report). asymptotically valid but, unfortunately, the lowest signif- All these results thus strongly encourage us not only to icant truncation, namely the Benney equation itself, al- perform a detailed investigation in the parameters space ready displays finite time singularities beyond some crit- R, B and Γ in the case of two-dimensional film-flow but to ical (and rather low) Reynolds number where one-hump turn bravely to the three-dimensional case. The strategy solitary solutions disappear [13]. Less drastic approxima- has been to describe the structure of the flow in terms tions have to keep some trace of flow variables in addi- of a small number of fields with clear physical signifi- tion to h. Within the truncated gradient expansion, the cance, the film thickness h, the instantaneous flow rate velocity field is then projected on a (small) set of func- q and the variable τ measuring the departure of the wall tions. Equations governing the corresponding amplitudes shear stress from the parabolic profile prediction. These are searched for by weighted residual methods, with the quantities serve to reconstruct the full flow field. This ap- goal of obtaining realistic evolutions free of non-physical proach, which comes to replace the dependence on the singularities. Such models should have thus essentially the normal coordinate y by a small set of variables, can be same range of validity as BL equations but now no longer extended straightforwardly to three dimensions. The cor- involve fields depending on the cross-flow coordinate y responding reduction will lead to few partial differential but amplitudes that depend only on the stream-wise x evolution equations in x and z. Besides being much less and transverse z coordinates, i.e., one space-dimension demanding of numerical resources for their simulation, 292 The European Physical Journal B
Table 1. Wave characteristics corresponding to Figure 6. 17. S.W. Joo, S.H. Davis, S.G. Bankoff, Phys. Fluids. A 3, Top line: Experimental observations. Bottom line: Numerical 231–232 (1991). results. 18. T.R. Salamon, R.C. Armstrong, R.A. Brown, Phys. Fluids 6, 2202–2220 (1994). hhi hmax λc c hUi 19. B. Ramaswamy, S. Chippada, S.W. Joo, J. Fluid Mech. mm mm mm mm/s qN /hhi qN /hhi 325, 163–194 (1996). 0.545 1.12 36 460 2.83 1.27 20. E.A. Demekhin, I.A. Demekhin, V.Y. Shkadov, Izv. Ak. 0.514 1.13 34 434 3.20 1.30 Nauk SSSR, Mekh. Zhi. Gaza 4, 9–16 (1983). 21. H.C. Chang, E.A. Demekhin, D.I. Kopelevitch, J. Fluid Mech. 250, 433–480 (1993). 22. V.Y. Shkadov, Izv. Ak. Nauk SSSR, Mekh. Zhi. Gaza 2, 43–51 (1967). the study of these equations should give clearer insight in 23. B.A. Finlayson, The method of weighted residuals and vari- the physical mechanisms at the origin of secondary three- ational principles, with application in fluid mechanics, heat dimensional instabilities observed in experiments [3]. and mass transfer (Academic Press, 1972). The modeling of film flows over inclined planes thus of- 24. (a) Y.Y. Trifonov, O.Y. Tsvelodub, J. Fluid Mech. 229, fers to us a unique opportunity of understanding the tran- 531–554 (1991); (b) O.Y. Tsvelodub, Y.Y. Trifonov, J. sition to turbulence via spatio-temporal chaos in open- Fluid Mech. 244, 149–169 (1992). flow configurations, hinting at other hydrodynamic prob- 25. H.C. Chang, E.A. Demekhin, D.I. Kopelevitch, Physica D lems such as boundary-layer stability. 63, 299–320 (1993). 26. The relative importance of inertia and viscous effects is then characterized by the Reynolds number R = This work was supported by a grant from the D´el´egation G´e- ρuN hN /µ, whereas surface tension γ is measured by the n´erale `a l’Armement (DGA) of the French Ministry of Defense. 2 Weber number W = γ/(ρg sin βhN ) via the stream-wise The authors would like to thank J.M. Chomaz, B.S. Tilley and component of the gravitational acceleration g sin β. I. Delbende for stimulating discussions. Thanks are warmly 27. M.K. Smith, J. Fluid Mech. 217, 469–485 (1990). extended to T. Lescuyer, J. Webert and C. Cossu for their 28. C. Nakaya, Phys. Fluids 18, 1407–1412 (1975). kind and helpful assistance. 29. M. Cheng, H.C. Chang, Phys. Fluids 7, 34–54 (1995). 30. H. Schlichting, Boundary-layer theory (McGraw-Hill, 1955). (1) (1) 31. Polynomial f is chosen such that df /dζ|0 =0, 3 (1) 3 References df /dζ |0 = −1. This choice is in line with those made for the higher degree polynomials used later, which are in 1. H.C. Chang, Annu. Rev. Fluid Mech. 26, 103–136 (1994). turn justified by the fact that we use collocation residuals 2. J.M. Floryan, S.H. Davis, R.E. Kelly, Phys. Fluids 30, 983– at the boundary points ζ =0andζ=1. 989 (1987). 32. J. Villadsen, M.L. Michelsen, Solution of differential equa- 3. J. Liu, B. Schneider, J.P. Gollub, Phys. Fluids 7, 55–67 tion models by polynomial approximation (Prentice-Hall, (1995). 1978). 33. J. Liu, J.D. Paul, J.P. Gollub, J. Fluid Mech. 250, 69–101 4. J. Liu, J.P. Gollub, Phys. Fluids 6, 1702–1712 (1994). (1993). 5. J. Benney, J. Math. Phys. 45, 150–155 (1966). 34. C.S Yih, Phys. Fluids 6, 321–334 (1963). 6. S.P. Lin, J. Fluid Mech. 63, 417–429 (1974). 35. E.L. Allgower, K. Georg, Numerical continuation methods 7. P.L. Kapitza, S.P. Kapitza, Zh. Ekper. Teor. Fiz. 19, 105 (Springer-Verlag, 1990). (1949); Also in Collected papers of P.L. Kapitza, edited by 4 3 2 4 9 2 29 3 36. Namely, 5 k + i 175 Rk in A(k), − 5 k − i 350 Rk in B(k), D. Ter Haar, pp. 690–709. 9 2 and i 70 Rk in C(k). 8. T. Prokopiou, M. Cheng, H.C. Chang, J. Fluid Mech. 222, 37. R.D. Richtmeyer, K.W. Morton, Difference methods for 665–691 (1991). initial value problems (Interscience, 1967). 9. L.-Q. Yu, F.K. Wasden, A.E. Dukler, V. Balakotaiah, 38. S.V Alekseenko, V.Y. Nakoryakov, B.G. Pokusaev, AIChE Phys. Fluids 7, 1886–1902 (1995). J. 31, 1446–1460 (1985). 10. J.-J. Lee, C.C. Mei, J. Fluid Mech. 307, 191–229 (1996). 39. H.C. Chang, E.A. Demekhin, E. Kalaidin, J. Fluid Mech. 11. E.A. Demekhin, M.A. Kaplan, V.Y. Shkadov, Izv. Ak. 294, 123–154 (1995); AIChE J. 42, 1553–1568 (1996). Nauk SSSR, Mekh. Zhi. Gaza 6, 73–81 (1987). 40. Demekhin et al. have shown that the set of control param- 12. Actually, considering only small surface tension effects, eters can be reduced to a single reduced Reynolds number Benney never expressed any truncation of his expansion δ for the BL equations and a vertical plane [20]. The soli- in the form (36). To our knowledge, the first occurrence of tary wave in Figure 6 corresponds to δ =0.056 and speed (36) is in: B. Gjevik, Phys. Fluids 13, 1918–25 (1970). c =7.33 uN , to be compared with results in [39]. 13. A. Pumir, P. Manneville, Y. Pomeau, J. Fluid Mech. 135, 41. F.K. Wasden, A.E. Duckler, AIChE J. 35, 187–195 (1989). 27–50 (1983). 42. C. Ruyer-Quil, P. Manneville, Transition to turbulence of 14. Y. Kuramoto, T. Tsuzuki, Prog. Theor. Phys. 55, 536 fluid flowing down an inclined plane: Modeling and sim- (1976); G.I. Sivashinsky, Acta Astronautica 4, 356 (1977); ulation, Proceedings of the 7th European Turbulence Con- Y. Kuramoto, Prog. Theor. Phys. Suppl. 64, 1177 (1978). ference, Saint Jean Cap Ferrat, June 30-July 3, 1998, in 15. O.Y. Tsvelodub, Izv. Ak. Nauk SSR, Mekh. Zh. Gaza 4, Advances in Turbulence VII , edited by U. Frisch, (Kluwer, 142–146 (1980). 1998), pp. 93-96. 16. H.C. Chang, Phys. Fluids 29, 3142–3147 (1986).