Asymptotic analysis of a thin fluid layer flow between two moving surfaces
aa r X i v : . [ m a t h . A P ] J a n , Asymptotic analysis of a thin fluid layer flow between twomoving surfaces ⋆ J. M. Rodr´ıguez a, ∗ , R. Taboada-V´azquez b a Department of Mathematics, Higher Technical University College of Architecture, Universidade daCoru˜na, Campus da Zapateira, 15071 - A Coru˜na, Spain b Department of Mathematics, School of Civil Engineering, Universidade da Coru˜na, Campus deElvi˜na, 15071 - A Coru˜na, Spain
Abstract
In this paper we study the behavior of an incompressible viscous fluid moving betweentwo very close surfaces also in motion. Using the asymptotic expansion method weformally justify two models, a lubrication model and a shallow water model, dependingon the boundary conditions imposed. Finally, we discuss under what conditions each ofthe models would be applicable.
Keywords:
Lubrication, Shallow waters, Asymptotic analysis.
1. Introduction
The asymptotic analysis method is a mathematical tool that has been widely used toobtain and justify reduced models, both in solid and fluid mechanics, when one or twoof the dimensions of the domain in which the model is formulated are much smaller thanthe others.After the pioneering works of Friedrichs, Dressler and Goldenveizer (see [28] and [30])the asymptotic development technique has been used successfully to justify beam, plateand shell theories (see, for example, [43], [16], [17], [15], [5], [54], and many others).This same technique has also been used in fluid mechanics to justify various types ofmodels, such as lubrication models, shallow water models, tube flow models, etc. (see,for example, [25], [24], [18], [3], [37], [55], [36], [31], [6], [2], [29], [26], [32], [33], [9], [23],[45]-[50], [21], [22], [34], [35], [40], [41], [42], [10], [11], and many others). ⋆ This work has been partially supported by Ministerio de Econom´ıa y Competitividad of Spain, undergrant MTM2016-78718-P with the participation of FEDER, and the European Union’s Horizon 2020Research and Innovation Programme, under the Marie Sklodowska-Curie Grant Agreement No 823731CONMECH. ∗ Corresponding author
Email addresses: [email protected] (J. M. Rodr´ıguez), [email protected] (R.Taboada-V´azquez)
Preprint submitted to Elsevier January 26, 2021 n this work, we are interested in justifying, again using the asymptotic developmenttechnique, a lubrication model in a thin domain with curved mean surface. Followingthe steps of [3], but with a different starting point, we devote sections 2 and 3 to this jus-tification. During the above process we have observed that, depending on the boundaryconditions, other models can be obtained, which we show in section 4. In this sectionwe derive a shallow water model changing the boundary conditions that we had imposedin section 3: instead of assuming that we know the velocities on the upper and lowerboundaries of the domain, we assume that we know the tractions on these upper andlower boundaries.Thus, two new models are presented in sections 3 and 4 of this article. These modelscan not be found in the literature, as far as we know. In addition, the method used tojustify them allows us to answer the question of when each of them is applicable. Insection 5 we discuss the models yielded, as well as the difference between one modeland another depending on the boundary conditions, reaching the conclusion that themagnitude of the pressure differences at the lateral boundary of the domain is key whendeciding which of the two models best describes the fluid behavior.
2. Derivation of the model
Let us consider a three-dimensional thin domain, Ω εt , filled by a fluid, that varies withtime t ∈ [0 , T ], given byΩ εt = (cid:8) ( x ε , x ε , x ε ) ∈ R : x i ( ξ , ξ , t ) ≤ x εi ≤ x i ( ξ , ξ , t ) + h ε ( ξ , ξ , t ) N i ( ξ , ξ , t ) , ( i = 1 , , , ( ξ , ξ ) ∈ D ⊂ R (cid:9) (1)where ~X t ( ξ , ξ ) = ~X ( ξ , ξ , t ) = ( x ( ξ , ξ , t ) , x ( ξ , ξ , t ) , x ( ξ , ξ , t )) is the lower boundsurface parametrization, h ε ( ξ , ξ , t ) is the gap between the two surfaces in motion, and ~N ( ξ , ξ , t ) is the unit normal vector: ~N ( ξ , ξ , t ) = ∂ ~X∂ξ × ∂ ~X∂ξ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∂ ~X∂ξ × ∂ ~X∂ξ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (2)The lower bound surface is assumed to be regular, ∂ ~X∂ξ × ∂ ~X∂ξ = ~ ∀ ( ξ , ξ ) ∈ D ⊂ R , ∀ t ∈ [0 , T ] , (3)and the gap is assumed to be small with regard to the dimension of the bound surfaces.We take into account that the fluid film between the surfaces is thin by introducing asmall non-dimensional parameter ε , and setting that h ε ( ξ , ξ , t ) = εh ( ξ , ξ , t ) (4)where h ( ξ , ξ , t ) ≥ h > , ∀ ( ξ , ξ ) ∈ D ⊂ R , ∀ t ∈ [0 , T ] . (5)2 .2. Construction of the reference domain Let us consider Ω = D × [0 ,
1] (6)a domain independent of ε and t , which is related to Ω εt by the following change ofvariable: t ε = t (7) x εi = x i ( ξ , ξ , t ) + εξ h ( ξ , ξ , t ) N i ( ξ , ξ , t ) (8)where ( ξ , ξ ) ∈ D and ξ ∈ [0 , { ~a , ~a , ~a } ~a ( ξ , ξ , t ) = ∂ ~X ( ξ , ξ , t ) ∂ξ (9) ~a ( ξ , ξ , t ) = ∂ ~X ( ξ , ξ , t ) ∂ξ (10) ~a ( ξ , ξ , t ) = ~N ( ξ , ξ , t ) (11)In Appendix A we obtain ∂x εi ∂ξ j = a ji + εξ ∂h∂ξ j a i + εξ h ∂a i ∂ξ j , ( i = 1 , , j = 1 ,
2) (12) ∂x εi ∂ξ = εha i , ( i = 1 , ,
3) (13) ∂x εi ∂t = ∂x i ∂t + εξ ∂h∂t a i + εξ h ∂a i ∂t , ( i = 1 , ,
3) (14) ∂t ε ∂ξ = ∂t ε ∂ξ = ∂t ε ∂ξ = 0 , ( i = 1 , ,
3) (15) ∂t ε ∂t = 1 , (16)where a ij = ~a i · ~e j , ( i, j = 1 , , { ~e , ~e , ~e } is the canonical basis of R , and (cid:18) ∂ξ ∂x ε , ∂ξ ∂x ε , ∂ξ ∂x ε (cid:19) = α ~a + β ~a + γ ~a (17) (cid:18) ∂ξ ∂x ε , ∂ξ ∂x ε , ∂ξ ∂x ε (cid:19) = α ~a + β ~a + γ ~a (18) (cid:18) ∂ξ ∂x ε , ∂ξ ∂x ε , ∂ξ ∂x ε (cid:19) = α ~a + β ~a + γ ~a (19)3 ξ ∂t ε = − ( α ~a + β ~a ) · ∂ ~X∂t + εξ h ∂~a ∂t ! (20) ∂ξ ∂t ε = − ( α ~a + β ~a ) · ∂ ~X∂t + εξ h ∂~a ∂t ! (21) ∂ξ ∂t ε = − ( α ~a + β ~a ) · ∂ ~X∂t + εξ h ∂~a ∂t ! − εh~a · ∂ ~X∂t − ξ h ∂h∂t (22) ∂t∂x εi = 0 ( i = 1 , ,
3) (23) ∂t∂t ε = 1 (24)where α i , β i , γ i ( i = 1 , ,
3) are given by (A.21)-(A.28) in Appendix A.Given any function F ε ( x ε , x ε , x ε , t ε ) defined on Ω εt , we can define another function F ( ε )( ξ , ξ , ξ , t ) on Ω using the change of variable F ( ε )( ξ , ξ , ξ , t ) = F ε ( x ε , x ε , x ε , t ε ) (25)and the relation between its partial derivatives is trivially: ∂F ε ∂x εi = ∂F ( ε ) ∂ξ ∂ξ ∂x εi + ∂F ( ε ) ∂ξ ∂ξ ∂x εi + ∂F ( ε ) ∂ξ ∂ξ ∂x εi (26) ∂F ε ∂t ε = ∂F ( ε ) ∂t + ∂F ( ε ) ∂ξ ∂ξ ∂t ε + ∂F ( ε ) ∂ξ ∂ξ ∂t ε + ∂F ( ε ) ∂ξ ∂ξ ∂t ε (27)where ∂ξ j ∂x εi , ∂ξ j ∂t ε are given by (17)-(22). Let us consider an incompressible newtonian fluid, so we can assume that the fluidmotion is governed by Navier-Stokes equations ( i = 1 , , ρ ∂u εi ∂t ε + ∂u εi ∂x εj u εj ! = − ∂p ε ∂x εi + µ (cid:18) ∂ u εi ∂ ( x ε ) + ∂ u εi ∂ ( x ε ) + ∂ u εi ∂ ( x ε ) (cid:19) + ρ f εi (28) ∂u εj ∂x εj = 0 (29)where repeated indices indicate summation ( j takes values from 1 to 3), ρ is the fluiddensity, assumed to be constant, ~u ε = ( u ε , u ε , u ε ) is the fluid velocity, p ε is the pressure, µ is the dynamic viscosity and ~f ε denotes the external density of volume forces.Let us write ~u ε and ~f ε in the new basis (9)-(11) (repeated indices i and k indicatesummation from 1 to 3): ~u ε = u εi ~e i = u k ( ε ) ~a k (30) ~f ε = f εi ~e i = f k ( ε ) ~a k (31)4o we have u εi = ( u k ( ε ) ~a k ) · ~e i = u k ( ε ) a ki (32) f εi = ( f k ( ε ) ~a k ) · ~e i = f k ( ε ) a ki (33)Taking into account (32)-(33), equations (28)-(29) yield ( i = 1 , , ρ ∂ ( u k ( ε ) a ki ) ∂t ε + ∂ ( u k ( ε ) a ki ) ∂x εj ( u k ( ε ) a kj ) ! = − ∂p ( ε ) ∂x εi + µ (cid:18) ∂ ( u k ( ε ) a ki ) ∂ ( x ε ) + ∂ ( u k ( ε ) a ki ) ∂ ( x ε ) + ∂ ( u k ( ε ) a ki ) ∂ ( x ε ) (cid:19) + ρ f k ( ε ) a ki (34) ∂ ( u k ( ε ) a kj ) ∂x εj = 0 (35)Equations (34)-(35) can be written in the reference domain Ω, using (26)-(27) and(17)-(22), as follows (repeated indices indicates summation from 1 to 3; i = 1 , , ∂u k ( ε ) ∂t a ki + u k ( ε ) ∂a ki ∂t + (cid:18) a ki ∂u k ( ε ) ∂ξ l + u k ( ε ) ∂a ki ∂ξ l (cid:19) " − ( α l ~a + β l ~a ) · ∂ ~X∂t + εξ h ∂~a ∂t ! + (cid:18) a ki ∂u k ( ε ) ∂ξ + u k ( ε ) ∂a ki ∂ξ (cid:19) − εh~a · ∂ ~X∂t − ξ h ∂h∂t ! + u k ( ε ) a kj (cid:18) a ki ∂u k ( ε ) ∂ξ l + u k ( ε ) ∂a ki ∂ξ l (cid:19) ( α l a j + β l a j + γ l a j )= − ρ ∂p ( ε ) ∂ξ l ( α l a i + β l a i + γ l a i )+ ν (cid:26)(cid:20) ∂ ( u k ( ε ) a ki ) ∂ξ l ∂ξ m ( α l a j + β l a j + γ l a j )+ ∂ ( u k ( ε ) a ki ) ∂ξ l ∂∂ξ m ( α l a j + β l a j + γ l a j ) (cid:21) ( α m a j + β m a j + γ m a j ) (cid:27) + f k ( ε ) a ki (36) (cid:18) a kj ∂u k ( ε ) ∂ξ l + u k ( ε ) ∂a kj ∂ξ l (cid:19) ( α l a j + β l a j + γ l a j ) = 0 (37) Let us assume that u i ( ε ), f i ( ε ) ( i = 1 , ,
3) and p ( ε ) can be developed in powers of ε ,that is: u i ( ε ) = u i + εu i + ε u i + · · · ( i = 1 , ,
3) (38) p ( ε ) = ε − p − + ε − p − + p + εp + ε p + · · · (39) f i ( ε ) = f i + εf i + ε f i + · · · ( i = 1 , ,
3) (40)5n making this choice, we follow [3], [1], and [20].Before substituting α i , β i , γ i ( i = 1 , ,
3) in (36)-(37), we must develop (A.21)-(A.27)in powers of ε . It is easy to check that α i = α i + εξ hα i + ε ξ h α i + · · · , ( i = 1 ,
2) (41) α = ξ h ( α + εξ hα + ε ξ h α + · · · ) , (42) β i = β i + εξ hβ i + ε ξ h β i + · · · , ( i = 1 ,
2) (43) β = ξ h ( β + εξ hβ + ε ξ h β + · · · ) , (44) γ = 1 εh , γ = γ = 0 , (45)where α = k ~a k A = GEG − F (46) α = ~a · ∂~a ∂ξ − α A A = − g + α A A (47) α n = − α n − A + α n − A A , n ≥ α = β = − ~a · ~a A = − FA (49) α = β = − ~a · ∂~a ∂ξ + α A A = f − α A A (50) α n = β n = − α n − A + α n − A A , n ≥ α = ∂h∂ξ ~a · ~a − ∂h∂ξ k ~a k A = ∂h∂ξ F − ∂h∂ξ GA (52) α = ~a · (cid:20) ∂h∂ξ ∂~a ∂ξ − ∂h∂ξ ∂~a ∂ξ (cid:21) − α A A = − ∂h∂ξ f + ∂h∂ξ g − α A A (53) α n = − α n − A + α n − A A , n ≥ = k ~a k A = EA (55) β = ~a · ∂~a ∂ξ − β A A = − e + β A A (56) β n = − β n − A + β n − A A , n ≥ β = ∂h∂ξ ~a · ~a − ∂h∂ξ k ~a k A = ∂h∂ξ F − ∂h∂ξ EA (58) β = ∂h∂ξ (cid:18) ~a · ∂~a ∂ξ (cid:19) − ∂h∂ξ (cid:18) ~a · ∂~a ∂ξ (cid:19) − β A A = − ∂h∂ξ f + ∂h∂ξ e − β A A (59) β n = − β n − A + β n − A A , n ≥ ε , lead to a series of equationsthat will allow us to determine ~ u , p − , etc.In what follows, we will use the standard summation convention that repeated indicesindicate summation from 1 to 3, unless we indicate otherwise.In this way, we first identify the terms multiplied by ε − : − ρ ∂p − ∂ξ h a i = 0 ( i = 1 , ,
3) (61)so we have ∂p − ∂ξ = 0 (62)As a second step, we identify the terms multiplied by ε − . Multiplying by ~a i , ( i =1 , , µh (cid:18) E ∂ u ∂ξ + F ∂ u ∂ξ (cid:19) = ∂p − ∂ξ (63) µh (cid:18) F ∂ u ∂ξ + G ∂ u ∂ξ (cid:19) = ∂p − ∂ξ (64) ∂p − ∂ξ h = µh ∂ u ∂ξ (65)The terms multiplied by ε − in (37) are: ∂u k ∂ξ a kj h a j = ∂u ∂ξ h = 0 (66)and using this equality in (65), we deduce: ∂p − ∂ξ = 0 (67)7rom the terms multiplied by ε − in (36) we obtain: ρ A h (cid:18) E ∂u ∂ξ + F ∂u ∂ξ (cid:19) u − ~a · ∂ ~X∂t ! + ξ h (cid:20) ∂p − ∂ξ ( Ge − f F ) + ∂p − ∂ξ ( f E − eF ) (cid:21) + ∂p − ∂ξ A = µ A h (cid:18) ∂ u ∂ξ E + ∂ u ∂ξ F (cid:19) + µA h (cid:18) ∂u ∂ξ E + ∂u ∂ξ F (cid:19) (68) ρ A h (cid:18) F ∂u ∂ξ + G ∂u ∂ξ (cid:19) u − ~a · ∂ ~X∂t ! + ξ h (cid:20) ∂p − ∂ξ ( f G − gF ) + ∂p − ∂ξ ( Eg − f F ) (cid:21) + ∂p − ∂ξ A = µ A h (cid:18) ∂ u ∂ξ F + ∂ u ∂ξ G (cid:19) + µA h (cid:18) ∂u ∂ξ F + ∂u ∂ξ G (cid:19) (69) ∂p ∂ξ h = µ ∂ u ∂ξ (70)Finally, the term of order 0 in (37) is:1 h ∂u ∂ξ = − ˆ A A u − ˆ A A u − A A u − ∂u ∂ξ − ∂u ∂ξ + ξ h (cid:18) ∂u ∂ξ ∂h∂ξ + ∂u ∂ξ ∂h∂ξ (cid:19) (71)whereˆ A i = k ~a k (cid:18) ~a · ∂~a i ∂ξ (cid:19) − ( ~a · ~a ) (cid:18) ~a · ∂~a i ∂ξ + ~a · ∂~a i ∂ξ (cid:19) + k ~a k (cid:18) ~a · ∂~a i ∂ξ (cid:19) = 12 G ∂E∂ξ i − ∂F ∂ξ i + 12 E ∂G∂ξ i = 12 ∂ ( EG − F ) ∂ξ i = 12 ∂A ∂ξ i , ( i = 1 , . (72)
3. A new generalized lubrication model
Reynolds wrote, in 1886, a seminal work on lubrication theory (see [44]), where heintroduced heuristically the Reynolds equation. This two-dimensional equation describ-ing the stationary flow of a thin layer of fluid is considered to be the key element formodelling lubrication phenomena. Since then, we can find numerous works in whichmore general physical models have been considered.Most models dedicated to the study of thin film flow, specially in lubrication, arederived from the Stokes equation. These first works were focused on stationary modelsin which the gap and the boundary conditions were fixed with respect to time (see [3], [18],[24]). These assumptions were considered no longer valid in some devices, so variationwith respect to time of the domain was introduced (see [4]). In the same way, in somecases, the inertial effects can not be ignored (see [12]), so the studies using Navier-Stokes8quation, as ours, turned out to be relevant (see [1], for example). It was in 1959, in[25], when full Navier-Stokes equations were used firstly. Various boundary conditionsfor the velocity of the surfaces (see [27]), and other types of generalizations have alsobeen studied (see [13], [14], [19] or [38]).In this work, as we have stated in the previous section, we will use Navier-Stokesequations to derive a new generalized lubrication model. We are considering a three-dimensional thin domain, that varies with time, whose mean surface can be chosenwithout any restriction (in particular, neither the lower boundary surface, nor the upperboundary surface, need to be flat). With respect to boundary conditions, we assumethat the fluid slips at the lower surface ( ξ = 0), and at the upper surface ( ξ = 1), butthere is continuity in the normal direction, so the tangential velocities at the lower andupper surfaces are known, and the normal velocity of each of them must match the fluidvelocity. u εk ~e k = u k ( ε ) ~a k = V ( ε ) ~a + V ( ε ) ~a + ∂ ~X∂t · ~a ! ~a on ξ = 0 (73) u εk ~e k = u k ( ε ) ~a k = W ( ε ) ~a + W ( ε ) ~a + ∂ ( ~X + εh~a ) ∂t · ~a ! ~a on ξ = 1 (74)where V ~a + V ~a is the tangential velocity at the lower surface and W ~a + W ~a is thetangential velocity at the upper surface. So we have, u k ( ε ) = V k ( ε ) ( k = 1 ,
2) on ξ = 0 (75) u ( ε ) = ∂ ~X∂t · ~a on ξ = 0 (76) u k ( ε ) = W k ( ε ) ( k = 1 ,
2) on ξ = 1 (77) u ( ε ) = ∂ ( ~X + εh~a ) ∂t · ~a on ξ = 1 (78)If we assume, in the same way as in (38)-(40), that V i ( ε ) = V i + εV i + ε V i + · · · ( i = 1 ,
2) (79) W i ( ε ) = W i + εW i + ε W i + · · · ( i = 1 ,
2) (80)we yield from (73)-(74): u lk = V lk ( k = 1 , , l = 0 , , , . . . ) on ξ = 0 (81) u lk = W lk ( k = 1 , , l = 0 , , , . . . ) on ξ = 1 (82) u = ∂ ~X∂t · ~a on ξ = 0 (83) u l = 0 ( l = 1 , , . . . ) on ξ = 0 (84) u = ∂ ~X∂t · ~a on ξ = 1 (85) u = ∂ ( h~a ) ∂t · ~a = ∂h∂t on ξ = 1 (86) u l = 0 ( l = 2 , , . . . ) on ξ = 1 (87)9rom (63)-(64) we can deduce: ∂ u ∂ ( ξ ) = h µA (cid:18) E ∂p − ∂ξ − F ∂p − ∂ξ (cid:19) (88) ∂ u ∂ ( ξ ) = h µA (cid:18) G ∂p − ∂ξ − F ∂p − ∂ξ (cid:19) (89)As p − does not depend on ξ (see (62)), we can integrate the previous equations in ξ and impose (81)-(82) u = h ( ξ − ξ )2 µA (cid:18) G ∂p − ∂ξ − F ∂p − ∂ξ (cid:19) + ξ ( W − V ) + V (90) u = h ( ξ − ξ )2 µA (cid:18) E ∂p − ∂ξ − F ∂p − ∂ξ (cid:19) + ξ ( W − V ) + V (91)From (66), (83) and (85) we know: u = ∂ ~X∂t · ~a (92)Now, we yield the following equation by substituting u i ( i = 1 , ,
3) into equation(71) by their expressions (90)-(92), integrating over ξ from 0 to 1, and evaluating byusing (84) and (86): ∂∂ξ (cid:20) h A (cid:18) G ∂p − ∂ξ − F ∂p − ∂ξ (cid:19)(cid:21) + ∂∂ξ (cid:20) h A (cid:18) E ∂p − ∂ξ − F ∂p − ∂ξ (cid:19)(cid:21) = 12 µ ∂h∂t + 12 µ hA A ∂ ~X∂t · ~a ! − h ˆ A ( A ) (cid:18) G ∂p − ∂ξ − F ∂p − ∂ξ (cid:19) − h ˆ A ( A ) (cid:18) E ∂p − ∂ξ − F ∂p − ∂ξ (cid:19) + 6 µ h ˆ A A ( W + V ) − µ ∂h∂ξ ( W − V ) + 6 µh ∂∂ξ ( W + V )+ 6 µ h ˆ A A ( W + V ) − µ ∂h∂ξ ( W − V ) + 6 µh ∂∂ξ ( W + V ) (93)If we denote bydiv( f , f ) = ∂f ∂ξ + ∂f ∂ξ (94) ∇ f = (cid:18) ∂f∂ξ , ∂f∂ξ (cid:19) (95) ~V = ( V , V ) , ~W = ( W , W ) , ( ˆ A , ˆ A ) = 12 ∇ A (96) M = (cid:18) G − F − F E (cid:19) (97)10nd we take into account thatdiv( ~ω ) + 12 A ∇ A · ~ω = 1 √ A div( √ A ~ω ) (98)we arrive at the equation:1 √ A div (cid:18) h √ A M ∇ p − (cid:19) = 12 µ ∂h∂t + 12 µ hA A ∂ ~X∂t · ~a ! − µ ∇ h · ( ~W − ~V ) + 6 µh √ A div( √ A ( ~W + ~V )) (99)that can be considered a generalization of Reynolds equation. Remark 1.
We claim that (99) is a new generalized Reynolds equation because, if weconsider the classic assumptions to derive Reynolds equations, we re-obtain the classicReynolds equation from (99). For example, in [18], [3], [20] and [1] the domain consideredis independent of time, x = 0 in (1), the upper surface is fixed ( ~W = ~
0) and the lowersurface is moving in the x -direction with constant velocity ( ~V = ( s, ~X , such that E = G = 1 and F = 0, and then equation (99) writes as the classical Reynolds equation:div (cid:0) h ∇ p − (cid:1) = 6 µs ∂h∂ξ (100)In [4] time is taken into account, allowing h to depend on time, and then the term ∂h∂t appears: div (cid:0) h ∇ p − (cid:1) = 12 µ ∂h∂t + 6 µs ∂h∂ξ (101) Remark 2.
The matrix M and the coefficients A , A , that appear in (99), dependonly on the geometry of the surface parametrized by ~X . In fact, the matrix 1 A M is theinverse of the matrix of the first fundamental form of ~X , and the term A A = − K m (see(A.41)). Remark 3.
Equation (99) must be completed with boundary conditions at ∂D , usuallythe value of p − at ∂D . Remark 4.
Equation (99) can be re-scaled, and then p ε is approximated by p − ,ε = ε − p − , solution of1 √ A div (cid:18) ( h ε ) √ A M ∇ p − ,ε (cid:19) = 12 µ ∂h ε ∂t ε + 12 µ h ε A A ∂ ~X∂t · ~a ! − µ ∇ h ε · ( ~W − ~V ) + 6 µh ε √ A div( √ A ( ~W + ~V )) (102)11 emark 5. We must point out that the expression1 √ A div( √ A ~ω ) = ω , + ω , (103)is exactly the covariant divergence of ~ω , where ω α,β stands for the covariant devirative of ω α with respect to ξ β .
4. A new thin fluid layer model
Thin fluid layer models are widely used for the analysis and numerical simulationof a large number of geophysical phenomena, such as rivers or coastal flows and otherhydraulic applications. Saint-Venant firstly derived in his paper [51] a shallow watermodel, since then numerous authors have studied this type of models (see, for example,[39], [53], [7]-[9], [26], [29], [33]), on many occasions using asymptotic analysis techniquesto justify them (see [2],[45]-[50]).With this aim, in this section we will study what happens when, instead of consideringthat the tangential and normal velocities are known on the upper and lower surfaces, aswe have done in (73)-(74), we assume that the normal component of the traction on ξ = 0 and on ξ = 1 are known pressures, and that the tangential component of thetraction on these surfaces are friction forces depending on the value of the velocities on ∂D . Therefore, we assume that ~T ε · ~n ε = ( σ ε ~n ε ) · ~n ε = − π ε on ξ = 0 , (104) ~T ε · ~n ε = ( σ ε ~n ε ) · ~n ε = − π ε on ξ = 1 , (105) ~T ε · ~a i = ( σ ε ~n ε ) · ~a i = − ~f εR · ~a i on ξ = 0 , ( i = 1 ,
2) (106) ~T ε · ~v εi = ( σ ε ~n ε ) · ~v εi = − ~f εR · ~v εi on ξ = 1 , ( i = 1 ,
2) (107)where ~T ε is the traction vector and σ ε is the stress tensor given by σ εij = − p ε δ ij + µ ∂u εi ∂x εj + ∂u εj ∂x εi ! , ( i, j = 1 , ,
3) (108)vectors ~n ε , ~n ε are, respectively, the outward unit normal vectors to the lower and theupper surfaces, that is ~n ε = s ~a (109) ~n ε = − s ~v ε k ~v ε k (110)where s = − s = 1 (111)12s fixed ( ~n ε = ~a or ~n ε = − ~a , depending on the orientation of the parametrization ~X ),and ~v ε = ~a + ε (cid:18) ∂h∂ξ ~a + h ∂~a ∂ξ (cid:19) (112) ~v ε = ~a + ε (cid:18) ∂h∂ξ ~a + h ∂~a ∂ξ (cid:19) (113) ~v ε = ~v ε × ~v ε (114)From the identities (112)-(114), we also have the equalities: ~v ε = ~a × ~a + ε (cid:20) ∂h∂ξ ( ~a × ~a ) + h (cid:18) ~a × ∂~a ∂ξ (cid:19) + ∂h∂ξ ( ~a × ~a ) + h (cid:18) ∂~a ∂ξ × ~a (cid:19)(cid:21) + ε (cid:20)(cid:18) ∂h∂ξ ~a + h ∂~a ∂ξ (cid:19) × (cid:18) ∂h∂ξ ~a + h ∂~a ∂ξ (cid:19)(cid:21) (115) k ~v ε k = k ~a × ~a k + εh (cid:20) ~a · (cid:18) ~a × ∂~a ∂ξ (cid:19) + ~a · (cid:18) ∂~a ∂ξ × ~a (cid:19)(cid:21) + O ( ε ) (116)Typically, the friction force is of the form ~f εRα = ρ C εR k ~u ε k ~u ε on ξ = α, ( α = 0 ,
1) (117)where C εR is a small constant. Let us assume that it is of order ε , that is, C εR = εC R (118)Now, taking into account (108), (17)-(19), we have the following development inpowers of ε : σ ij ( ε ) = − ∞ X r = − ε r p r δ ij + µ ∞ X r =0 ε r "(cid:18) ∂u rk ∂ξ l a ki + u rk ∂a ki ∂ξ l (cid:19) ∂ξ l ∂x εj + (cid:18) ∂u rk ∂ξ l a kj + u rk ∂a kj ∂ξ l (cid:19) ∂ξ l ∂x εi = − ε − p − δ ij + ε − (cid:26) p − δ ij + µ (cid:20) a j h ∂u k ∂ξ a ki + a i h ∂u k ∂ξ a kj (cid:21)(cid:27) − p δ ij + µ (cid:20) a j h ∂u k ∂ξ a ki + a i h ∂u k ∂ξ a kj + X l =1 (cid:18) ∂u k ∂ξ l a ki + u k ∂a ki ∂ξ l (cid:19) ( α l a j + β l a j ) + ξ h ∂u k ∂ξ a ki ( α a j + β a j )+ X l =1 (cid:18) ∂u k ∂ξ l a kj + u k ∂a kj ∂ξ l (cid:19) ( α l a i + β l a i ) + ξ h ∂u k ∂ξ a kj ( α a i + β a i ) ε (cid:26) − p δ ij + µ (cid:20) a j h ∂u k ∂ξ a ki + a i h ∂u k ∂ξ a kj + X l =1 (cid:18) ∂u k ∂ξ l a ki + u k ∂a ki ∂ξ l (cid:19) ( α l a j + β l a j ) + ξ h ∂u k ∂ξ a ki ( α a j + β a j )+ X l =1 (cid:18) ∂u k ∂ξ l a kj + u k ∂a kj ∂ξ l (cid:19) ( α l a i + β l a i ) + ξ h ∂u k ∂ξ a kj ( α a i + β a i )+ ξ h X l =1 (cid:18) ∂u k ∂ξ l a ki + u k ∂a ki ∂ξ l (cid:19) ( α l a j + β l a j ) + ξ ∂u k ∂ξ a ki ( α a j + β a j )+ ξ h X l =1 (cid:18) ∂u k ∂ξ l a kj + u k ∂a kj ∂ξ l (cid:19) ( α l a i + β l a i ) + ξ ∂u k ∂ξ a kj ( α a i + β a i ) + · · · (119)If we assume, now, that π ( ε ) = ∞ X r =0 ε r π r (120) π ( ε ) = ∞ X r =0 ε r π r (121) ~f R ( ε ) = ∞ X r =1 ε r ~f rR (122) ~f R ( ε ) = ∞ X r =1 ε r ~f rR (123)condition (104) can be written (using (120), (119), (109)) as:( σ ij ( ε ) a j ) a i = − ε − p − − ε − p − − p + µ h ∂u ∂ξ + ε (cid:18) − p + µ h ∂u ∂ξ (cid:19) + · · · = − ( π + επ + · · · ) on ξ = 0 (124)and we can deduce: p − = 0 on ξ = 0 (125) p − = 0 on ξ = 0 (126) − p + µ h ∂u ∂ξ = − π on ξ = 0 (127) − p + µ h ∂u ∂ξ = − π on ξ = 0 (128)From (62) and (125) we obtain p − = 0 (129)14nd, analogously, from (67) and (126), we have p − = 0 (130)Substituting p − into equations (63)-(64) we yield ∂ u ∂ξ = ∂ u ∂ξ = 0 (131)Let us denote, as in section 3, by V ~a + V ~a the tangential velocity to the lowersurface, and by W ~a + W ~a the tangential velocity to the upper surface. Thus we haveagain (73)-(78), but now V ( ε ), V ( ε ), W ( ε ), W ( ε ), ∂h∂t are unknown, they are not dataas in section 3.Let us assume the equalities (79)-(80) once more. Then we re-obtain (81)-(87). Now,from (131) and (66), we deduce u i = ( W i − V i ) ξ + V i ( i = 1 ,
2) (132) u = ∂ ~X∂t · ~a (133)Now, if we substitute u i by their expressions (132)-(133) into (71), we integrate over ξ from 0 to 1 and we evaluate using (84) and (86), we obtain2 ∂h∂t − ( ~W − ~V ) · ∇ h + h √ A div (cid:16) √ A ( ~W + ~V ) (cid:17) + 2 hA A ∂ ~X∂t · ~a ! = 0 (134)From (68)-(69), (129)-(130) and (132)-(133), we have ∂ u i ∂ξ = − hA A ( W i − V i ) ( i = 1 ,
2) (135)and integrating twice we yield u i = − hA A ( W i − V i )( ξ − ξ ) + ( W i − V i ) ξ + V i ( i = 1 ,
2) (136)From (71), using (132), (133) and (98), we can derive an expression for u : u = ξ (cid:20) ( ~W − ~V ) · ∇ h − h √ A div (cid:16) √ A ( ~W − ~V ) (cid:17)(cid:21) − hξ " √ A div( √ A ~V ) + A A ∂ ~X∂t · ~a ! (137)and we can also yield the following expression for p from (70), (137) and (127) p = µξ h (cid:20) − h √ A div (cid:16) √ A ( ~W − ~V ) (cid:17) + ( ~W − ~V ) · ∇ h (cid:21) − µ √ A div( √ A ~V ) − µA A ∂ ~X∂t · ~a ! + π (138)15oundary condition (105) can be written (using (110)) as follows: (cid:0) σ εij v ε j (cid:1) · v ε i = − π ε k ~v ε k on ξ = 1 (139)We use expressions (119) and (121) to substitute σ εij and π ε into the above conditionand we take into account (115), (116), (129), (130), (132), (137) and (138) to simplify.Identifying the terms multiplied by ε we obtain: k ~a × ~a k (cid:26) − µh (cid:20) h √ A div (cid:16) √ A ( ~W − ~V ) (cid:17) − ( ~W − ~V ) · ∇ h (cid:21) − π (cid:27) + 2 µh (cid:20) ( W − V ) (cid:18) ∂h∂ξ ( ~a × ~a ) · ~a + h (cid:18) ~a × ∂~a ∂ξ (cid:19) · ~a (cid:19) +( W − V ) (cid:18) ∂h∂ξ ( ~a × ~a ) · ~a + h (cid:18) ∂~a ∂ξ × ~a (cid:19) · ~a (cid:19)(cid:21) = − π k ~a × ~a k on ξ = 1 (140)Noticing that( ~a × ~a ) · ~a = ( ~a × ~a ) · ~a = (cid:18) ~a × ~a × ~a k ~a × ~a k (cid:19) · ~a = 1 k ~a × ~a k (cid:0) ~a ( ~a · ~a ) − ~a k ~a k (cid:1) · ~a = − A k ~a × ~a k = −k ~a × ~a k (141) (cid:18) ~a × ∂~a ∂ξ (cid:19) · ~a = ( ~a × ~a ) · ∂~a ∂ξ = −k ~a × ~a k ~a · ∂~a ∂ξ = 0 (142) (cid:18) ∂~a ∂ξ × ~a (cid:19) · ~a = ∂~a ∂ξ · ( ~a × ~a ) = − ∂~a ∂ξ · k ~a × ~a k ~a = 0 (143)we finally derive µ √ A div (cid:16) √ A ( ~W − ~V ) (cid:17) + µh ( ~W − ~V ) · ∇ h + π = π (144)Boundary conditions (106) on ξ = 0 can be written using (46), (49), (55), (52), (58),(119) and (122) in this way: ε − µh (cid:18) E ∂u ∂ξ + F ∂u ∂ξ (cid:19) + µ (cid:20) Eh ∂u ∂ξ + Fh ∂u ∂ξ + ∂u ∂ξ + eu + f u (cid:21) + εµ (cid:20) Eh ∂u ∂ξ + Fh ∂u ∂ξ + ∂u ∂ξ + eu + f u − ξ h ∂u ∂ξ ∂h∂ξ + ξ h X l =1 (cid:18) ∂u ∂ξ l + u k ∂~a k ∂ξ l · ~a (cid:19) ( α l E + β l F ) + · · · = s (cid:16) ε ~f R + ε ~f R + · · · (cid:17) · ~a on ξ = 0 (145)16 − µh (cid:18) F ∂u ∂ξ + G ∂u ∂ξ (cid:19) + µ (cid:20) Fh ∂u ∂ξ + Gh ∂u ∂ξ + ∂u ∂ξ + f u + gu (cid:21) + εµ (cid:20) Fh ∂u ∂ξ + Gh ∂u ∂ξ + ∂u ∂ξ + f u + gu − ξ h ∂u ∂ξ ∂h∂ξ + ξ h X l =1 (cid:18) ∂u ∂ξ l + u k ∂~a k ∂ξ l · ~a (cid:19) ( α l F + β l G ) + · · · = s (cid:16) ε ~f R + ε ~f R + · · · (cid:17) · ~a on ξ = 0 (146)We yield from the terms multiplied by ε − in the equations above and the equality(132) that u i = W i = V i ( i = 1 ,
2) (147)Identifying the terms multiplied by ε in (145)-(146), and taking into account (133),(136) and (147), we have W − V = − h " X l =1 α l ∂∂ξ l ∂ ~X∂t · ~a ! + B A V + B A V (148) W − V = − h " X l =1 β l ∂∂ξ l ∂ ~X∂t · ~a ! + B A V + B A V (149)where B ij ( i, j = 1 , ∂h∂t + h √ A div (cid:16) √ A ~V (cid:17) + hA A ∂ ~X∂t · ~a ! = 0 (150) u = − h " B A V + B A V + X l =1 α l ∂∂ξ l ∂ ~X∂t · ~a ! ξ + V (151) u = − h " B A V + B A V + X l =1 β l ∂∂ξ l ∂ ~X∂t · ~a ! ξ + V (152) u = − hξ " √ A div( √ A ~V ) + A A ∂ ~X∂t · ~a ! (153)= ξ ∂h∂t (154) p = − µ √ A div( √ A ~V ) − µA A ∂ ~X∂t · ~a ! + π (155)= 2 µh ∂h∂t + π (156) π = π (157)17ow, we identify the terms multiplied by ε in (145)-(146) and, considering (151)-(153), we obtain: µ (cid:20) A h ∂u ∂ξ + B V + B V (cid:21) = − s h G ( ~f R · ~a ) − F ( ~f R · ~a ) i on ξ = 0 (158) µ (cid:20) A h ∂u ∂ξ + B V + B V (cid:21) = − s [ E ( ~f R · ~a ) − F ( ~f R · ~a )] on ξ = 0 (159)Going back to (139), the terms multiplied by ε yield π = p − µh ∂u ∂ξ + 2 µh (cid:20) ∂h∂ξ ( W − V ) + ∂h∂ξ ( W − V ) (cid:21) + 2 µ " X l =1 ∂∂ξ l ∂ ~X∂t · ~a ! (cid:18) α l ∂h∂ξ + β l ∂h∂ξ (cid:19) + V (cid:18) ∂h∂ξ B A + ∂h∂ξ B A (cid:19) + V (cid:18) ∂h∂ξ B A + ∂h∂ξ B A (cid:19)(cid:21) on ξ = 1 (160)and using (148)-(149), we can simplify (160), and write p = π + 2 µh ∂u ∂ξ on ξ = 1 (161)Taking into account (46), (49), (55), (112)-(116), (119), (123), (133), (141)-(143),(147), (148)-(149) and (151)-(154), we can rewrite conditions (107), identify the termsof order zero, Eh ∂u ∂ξ + Fh ∂u ∂ξ + ∂u ∂ξ + eV + f V = 0 on ξ = 1 (162) Fh ∂u ∂ξ + Gh ∂u ∂ξ + ∂u ∂ξ + f V + gV = 0 on ξ = 1 (163)and the first order terms (here, repeated index k indicates sum from 1 to 3), µ (cid:20) A h ∂u ∂ξ + G ∂ h∂t∂ξ − F ∂ h∂t∂ξ + B V + B V − h ∂h∂t (cid:18) ∂h∂ξ F − ∂h∂ξ G (cid:19) − A X l =1 " ∂V ∂ξ l + X m =1 α m u k (cid:18) ∂~a k ∂ξ l · ~a m (cid:19) α l ∂h∂ξ + β l ∂h∂ξ (cid:19) − A X l =1 (cid:18) ∂h∂ξ ∂V ∂ξ l + ∂h∂ξ ∂V ∂ξ l (cid:19) α l + µ √ A " − hI (cid:0) B V + B V (cid:1) + A u k X l =1 α l (cid:18) ∂~a k ∂ξ l · ~η ( h ) (cid:19) = s (cid:16) G ~f R · ~a − F ~f R · ~a (cid:17) on ξ = 1 (164)18 (cid:20) A h ∂u ∂ξ − F ∂ h∂t∂ξ + E ∂ h∂t∂ξ + B V + B V − h ∂h∂t (cid:18) ∂h∂ξ F − ∂h∂ξ E (cid:19) − A X l =1 " ∂V ∂ξ l + X m =1 β m u k (cid:18) ∂~a k ∂ξ l · ~a m (cid:19) α l ∂h∂ξ + β l ∂h∂ξ (cid:19) − A X l =1 (cid:18) ∂h∂ξ ∂V ∂ξ l + ∂h∂ξ ∂V ∂ξ l (cid:19) β l + µ √ A " − hI (cid:0) B V + B V (cid:1) + A u k X l =1 β l (cid:18) ∂~a k ∂ξ l · ~η ( h ) (cid:19) = s (cid:16) − F ~f R · ~a + E ~f R · ~a (cid:17) on ξ = 1 (165)where I and ~η ( h ) are given by (B.7) and (B.15).From equalities (158)-(159) and (164)-(165) we have (again repeated index k indicatessum from 1 to 3): µ " A h ∂u ∂ξ (cid:12)(cid:12)(cid:12)(cid:12) ξ =1 − ∂u ∂ξ (cid:12)(cid:12)(cid:12)(cid:12) ξ =0 ! + G ∂ h∂t∂ξ − F ∂ h∂t∂ξ l − h ∂h∂t (cid:18) ∂h∂ξ F − ∂h∂ξ G (cid:19) − A X l =1 " ∂V ∂ξ l + X m =1 α m u k (cid:18) ∂~a k ∂ξ l · ~a m (cid:19) α l ∂h∂ξ + β l ∂h∂ξ (cid:19) − A X l =1 (cid:18) ∂h∂ξ ∂V ∂ξ l + ∂h∂ξ ∂V ∂ξ l (cid:19) α l + µ √ A " − hI (cid:0) B V + B V (cid:1) + A u k X l =1 α l (cid:18) ∂~a k ∂ξ l · ~η ( h ) (cid:19) = s h G (cid:16) ~f R + ~f R (cid:17) · ~a − F (cid:16) ~f R + ~f R (cid:17) · ~a i (166) µ " A h (cid:18) ∂u ∂ξ (cid:12)(cid:12)(cid:12)(cid:12) ξ =1 − ∂u ∂ξ (cid:12)(cid:12)(cid:12)(cid:12) ξ =0 ! − F ∂ h∂t∂ξ + E ∂ h∂t∂ξ − h ∂h∂t (cid:18) ∂h∂ξ F − ∂h∂ξ E (cid:19) − A X l =1 " ∂V ∂ξ l + X m =1 β m u k (cid:18) ∂~a k ∂ξ l · ~a m (cid:19) α l ∂h∂ξ + β l ∂h∂ξ (cid:19) − A X l =1 (cid:18) ∂h∂ξ ∂V ∂ξ l + ∂h∂ξ ∂V ∂ξ l (cid:19) β l + µ √ A " − hI (cid:0) B V + B V (cid:1) + A u k X l =1 β l (cid:18) ∂~a k ∂ξ l · ~η ( h ) (cid:19) = s h − F (cid:16) ~f R + ~f R (cid:17) · ~a + E (cid:16) ~f R + ~f R (cid:17) · ~a i (167)19ow, from the terms of order ε in the equation (36), and following the steps outlinedin Appendix C, we obtain the equations below, ∂V i ∂t + X l =1 (cid:0) V l − C l (cid:1) ∂V i ∂ξ l + X k =1 R ik + X l =1 H ilk V l ! V k = − ρ (cid:18) α i ∂π ∂ξ + β i ∂π ∂ξ (cid:19) + ν ( X m =1 2 X l =1 ∂ V i ∂ξ m ∂ξ l J lm + X k =1 2 X l =1 ∂V k ∂ξ l ( L kli + ψ ( h ) ikl )+ X k =1 V k ( S ik + χ ( h ) ik ) + ˆ κ ( h ) i ) + F i ( h ) − Q i ∂ ~X∂t · ~a ! ( i = 1 ,
2) (168)where the different coefficients are defined in Appendix B.
Remark 6.
Equations (168) and (150) allow us to determine h , V and V , once theinitial and boundary conditions have been set. These equations provide a shallow watermodel (see [39], [53], [9], [33], [45]-[50]). Equation (150) represents the conservation ofmass of the fluid. If h is known, then (150) means an additional condition on the velocity ~V and, in that case, the pressure π must be an unknown in (168). Remark 7.
As in (102) (see remark 4) equations (168) and (150) can be re-scaled, towork with h ε instead of h .
5. Conclusions
In this paper, starting from the same initial problem, an incompressible viscous fluidmoving between two surfaces parametrized by ~X and ~X + h ε ~N (see section 2), we obtain,using the asymptotic expansion technique, two different models. The first one is yieldedin section 3, assuming that the fluid velocity is known on the surfaces ~X and ~X + h ε ~N .The second one is derived in section 4, assuming that we know the tractions appliedon the surfaces ~X and ~X + h ε ~N , rather than the fluid velocity, as in section 3. Thissimple change gives rise to two different models: a lubrication model in section 3 and ashallow water model in section 4. This fact exemplifies the importance of the boundaryconditions in partial differential equations, and it tells us which of the two models shouldbe used when simulating flow of a thin fluid layer between two surfaces: if the fluidpressure is dominant (that is, it is of order O ( ε − )), and the fluid velocity is known onthe upper and lower surfaces, we must use the lubrication model obtained in section 3;if the fluid pressure is not dominant (that is, it is of order O (1)), and the tractions areknown on the upper and lower surfaces, we must use the shallow water model obtainedin section 4. In the first case we will say that the fluid is “driven by the pressure” andin the second that it is “driven by the velocity”.In the lubrication model derived in section 3, the pressure is determined by theequation (99), and it depends on the fluid velocity on the upper and lower surfaces ofthe domain, and on the speed at which these surfaces move, as well as on the geometry20f the surface ~X , and on the pressure at ∂D (see remark 3). The fluid velocities insidethe domain are subsequently obtained from the pressure using the equations (90)-(91).In the shallow water model of section 4, the fluid velocities are calculated from equations(168) and (150), and they are determined by the geometry of the surface ~X , as well asby the applied tractions (that is, the pressures π = π and the friction forces), whilethe fluid pressure is obtained now from the expression (156).But, when do we know “a priori” if the fluid is “driven by the pressure” or “driven bythe velocity”, that is, if we should use the lubrication model or the shallow water model?If we look closely at sections 3 and 4, we can say that the lubrication model describesthe fluid behavior when the pressure differences at ∂D are large enough, forcing thefluid movement described in (90)-(91), and that the shallow water model describes thefluid behavior when the pressure differences are small at ∂D , so that the pressure isdetermined by the pressure applied to the upper and lower surfaces of the domain andby its separation velocity (see (156)). 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Appendix A. Change of variable
Let us consider the change of variable (7)-(8) between the original domain (1) andthe reference domain (6). 23ts jacobian matrix is J ε = ∂x ε ∂ξ ∂x ε ∂ξ ∂x ε ∂ξ ∂x ε ∂t∂x ε ∂ξ ∂x ε ∂ξ ∂x ε ∂ξ ∂x ε ∂t∂x ε ∂ξ ∂x ε ∂ξ ∂x ε ∂ξ ∂x ε ∂t∂t ε ∂ξ ∂t ε ∂ξ ∂t ε ∂ξ ∂t ε ∂t and it is clear from (7)-(11) that ∂x εi ∂ξ j = a ji + εξ ∂h∂ξ j a i + εξ h ∂a i ∂ξ j , ( i = 1 , , j = 1 ,
2) (A.1) ∂x εi ∂ξ = εha i , ( i = 1 , ,
3) (A.2) ∂x εi ∂t = ∂x i ∂t + εξ ∂h∂t a i + εξ h ∂a i ∂t , ( i = 1 , ,
3) (A.3) ∂t ε ∂ξ = ∂t ε ∂ξ = ∂t ε ∂ξ = 0 , (A.4) ∂t ε ∂t = 1 (A.5)We can compute ( J ε ) − = ∂ξ ∂x ε ∂ξ ∂x ε ∂ξ ∂x ε ∂ξ ∂t ε ∂ξ ∂x ε ∂ξ ∂x ε ∂ξ ∂x ε ∂ξ ∂t ε ∂ξ ∂x ε ∂ξ ∂x ε ∂ξ ∂x ε ∂ξ ∂t ε ∂t∂x ε ∂t∂x ε ∂t∂x ε ∂t∂t ε (A.6)writing its components in the basis { ~a , ~a , ~a } : (cid:18) ∂ξ ∂x ε , ∂ξ ∂x ε , ∂ξ ∂x ε (cid:19) = α ~a + β ~a + γ ~a (A.7) (cid:18) ∂ξ ∂x ε , ∂ξ ∂x ε , ∂ξ ∂x ε (cid:19) = α ~a + β ~a + γ ~a (A.8) (cid:18) ∂ξ ∂x ε , ∂ξ ∂x ε , ∂ξ ∂x ε (cid:19) = α ~a + β ~a + γ ~a (A.9) (cid:18) ∂t∂x ε , ∂t∂x ε , ∂t∂x ε (cid:19) = α ~a + β ~a + γ ~a (A.10)24nd using that ( J ε ) − J ε = I (A.11)Taking into account that ~a i · ~a = 0 , ( i = 1 , , (A.12) k ~a k = 1 (A.13) ~a · ∂~a ∂ξ i = 0 ( i = 1 ,
2) (A.14)and introducing the following notation for the coefficients of the first and second funda-mental forms of the surface parametrized by ~X (here t acts only as a parameter): E = ~a · ~a (A.15) F = ~a · ~a (A.16) G = ~a · ~a (A.17) e = − ~a · ∂~a ∂ξ = ~a · ∂~a ∂ξ (A.18) f = − ~a · ∂~a ∂ξ = − ~a · ∂~a ∂ξ = ~a · ∂~a ∂ξ = ~a · ∂~a ∂ξ (A.19) g = − ~a · ∂~a ∂ξ = ~a · ∂~a ∂ξ (A.20)we deduce from (A.11): α = k ~a k + εξ h (cid:18) ~a · ∂~a ∂ξ (cid:19) A ( ε ) = G − εξ hgA ( ε ) (A.21) β = − ~a · ~a + εξ h (cid:18) ~a · ∂~a ∂ξ (cid:19) A ( ε ) = − F − εξ hfA ( ε ) (A.22) α = − ~a · ~a + εξ h (cid:18) ~a · ∂~a ∂ξ (cid:19) A ( ε ) = β (A.23) β = k ~a k + εξ h (cid:18) ~a · ∂~a ∂ξ (cid:19) A ( ε ) = E − εξ heA ( ε ) (A.24) γ i = 0 ( i = 1 ,
2) (A.25) α = − ξ h (cid:18) α ∂h∂ξ + α ∂h∂ξ (cid:19) (A.26) β = − ξ h (cid:18) β ∂h∂ξ + β ∂h∂ξ (cid:19) (A.27) γ = 1 εh (A.28)25 ξ ∂t ε = − ( α ~a + β ~a ) · ∂ ~X∂t + εξ h ∂~a ∂t ! (A.29) ∂ξ ∂t ε = − ( α ~a + β ~a ) · ∂ ~X∂t + εξ h ∂~a ∂t ! (A.30) ∂ξ ∂t ε = − ( α ~a + β ~a ) · ∂ ~X∂t + εξ h ∂~a ∂t ! − εh~a · ∂ ~X∂t − ξ h ∂h∂t (A.31) α = β = γ = 0 (A.32) ∂t∂x εi = 0 , ( i = 1 , ,
3) (A.33) ∂t∂t ε = 1 (A.34)where A ( ε ) = k ~a k k ~a k − ( ~a · ~a ) + εξ h (cid:20) k ~a k (cid:18) ~a · ∂~a ∂ξ (cid:19) + k ~a k (cid:18) ~a · ∂~a ∂ξ (cid:19) − ( ~a · ~a ) (cid:18) ~a · ∂~a ∂ξ + ~a · ∂~a ∂ξ (cid:19)(cid:21) + ε ξ h (cid:20)(cid:18) ~a · ∂~a ∂ξ (cid:19) (cid:18) ~a · ∂~a ∂ξ (cid:19) − (cid:18) ~a · ∂~a ∂ξ (cid:19) (cid:18) ~a · ∂~a ∂ξ (cid:19)(cid:21) = EG − F + εξ h ( − Ge − Eg + 2 f F ) + ε ξ h (cid:0) eg − f (cid:1) (A.35)If we denote by A = k ~a k k ~a k − ( ~a · ~a ) = EG − F = k ~a × ~a k (A.36) A = k ~a k (cid:18) ~a · ∂~a ∂ξ (cid:19) + k ~a k (cid:18) ~a · ∂~a ∂ξ (cid:19) − ( ~a · ~a ) (cid:18) ~a · ∂~a ∂ξ + ~a · ∂~a ∂ξ (cid:19) = − eG − gE + 2 f F (A.37) A = (cid:18) ~a · ∂~a ∂ξ (cid:19) (cid:18) ~a · ∂~a ∂ξ (cid:19) − (cid:18) ~a · ∂~a ∂ξ (cid:19) (cid:18) ~a · ∂~a ∂ξ (cid:19) = eg − f (A.38)then we obtain that A ( ε ) = A + εξ hA + ε ξ h A (A.39)We remark (see [52]) that A , A and A are related to the Gaussian curvature ( K G )of the surface parametrized by ~X and its mean curvature ( K m ), since K G = eg − f EG − F = A A (A.40) K m = eG + gE − f F EG − F ) = − A A (A.41)Furthermore, the principal curvatures of ~X are the solutions of the equation A K n + A K n + A = 0 (A.42)26 ppendix B. Coefficients definition In this appendix, we introduce some coefficients that depend only on the lower boundsurface parametrization, ~X and other coefficients that depend both on the parametriza-tion and on the gap h . We will use these coefficients throughout this article.In addition to the coefficients that will be defined below, others have been introducedin the body of the paper and in Appendix A: the coefficients of the first and secondfundamental forms of the surface parametrized by ~X (denoted by E, F, G and e, f, g ,respectively), defined in (A.15)-(A.17) and (A.18)-(A.20) from the basis { ~a , ~a , ~a } (see(9)-(11)), the coefficients α i , β i and γ i ( i = 1 , ,
3) in (A.21)-(A.28), and their develop-ment in powers of ε in (41)-(60), A ( ε ) and its development in powers of ε in (A.36)-(A.39),along with its relation with the Gaussian curvature and the mean curvature of the surfaceparametrized by ~X in (A.40)-(A.41), and, finally, the definition of ˆ A i ( i = 1 ,
2) in (72).The following coefficients depend only on the parametrization ~X : B = Ge − F f (B.1) B = Gf − F g (B.2) B = Ef − F e (B.3) B = Eg − F f (B.4) C l = α l ~a · ∂ ~X∂t ! + β l ~a · ∂ ~X∂t ! ( l = 1 ,
2) (B.5) H ilk = α i (cid:18) ~a · ∂~a k ∂ξ l (cid:19) + β i (cid:18) ~a · ∂~a k ∂ξ l (cid:19) ( i, l = 1 , k = 1 , ,
3) (B.6) I = (cid:18) ~a × ∂~a ∂ξ (cid:19) · ~a + (cid:18) ∂~a ∂ξ × ~a (cid:19) · ~a (B.7) J lm = α l α m E + ( β l α m + α l β m ) F + β l β m G = α l δ m + β l δ m ( A ) ( l, m = 1 ,
2) (B.8) L kli = X m =1 (cid:20)(cid:18) ∂α l ∂ξ m δ m + ∂β l ∂ξ m δ m + α l H mm + β l H mm (cid:19) δ ki + 2 H imk J lm (cid:3) ( i, l = 1 , k = 1 , ,
3) (B.9) Q ik = α i (cid:18) ~a · ∂~a k ∂t (cid:19) + β i (cid:18) ~a · ∂~a k ∂t (cid:19) − X l =1 H ilk C l ( i = 1 , k = 1 , ,
3) (B.10) R ik = Q ik + H ik ∂ ~X∂t · ~a ! ( i = 1 , k = 1 ,
2) (B.11)27 ik = I √ A − A A (cid:20) α i (cid:18) ∂~a k ∂ξ · ~a (cid:19) + β i (cid:18) ∂~a k ∂ξ · ~a (cid:19)(cid:21) + X m =1 2 X l =1 (cid:20)(cid:18) α i (cid:18) ~a · ∂ ~a k ∂ξ l ∂ξ m (cid:19) + β i (cid:18) ~a · ∂ ~a k ∂ξ l ∂ξ m (cid:19)(cid:19) J lm + (cid:18) ∂α l ∂ξ m δ m + ∂β l ∂ξ m δ m + α l H mm + β l H mm (cid:19) H ilk (cid:21) − √ A (cid:20)(cid:18) ~a × ∂~a ∂ξ (cid:19) + (cid:18) ∂~a ∂ξ × ~a (cid:19)(cid:21) · (cid:18) α i ∂~a k ∂ξ + β i ∂~a k ∂ξ (cid:19) ( i, k = 1 ,
2) (B.12)
Remark 8.
Coefficients B il and H il are related in the following way: H il = B il A ( i, l = 1 ,
2) (B.13)The following coefficients depend on the parametrization ~X and on function h : F i ( h ) = f i + s ρh ( ~f R + ~f R ) · (cid:0) α i ~a + β i ~a (cid:1) ( i = 1 ,
2) (B.14) ~η ( h ) = ∂h∂ξ ( ~a × ~a ) + h (cid:18) ~a × ∂~a ∂ξ (cid:19) + ∂h∂ξ ( ~a × ~a ) + h (cid:18) ∂~a ∂ξ × ~a (cid:19) (B.15) ψ ( h ) ijl = 1 h (cid:20)(cid:18) α l ∂h∂ξ + β l ∂h∂ξ (cid:19) δ ij + ∂h∂ξ j (cid:0) α l δ i + β l δ i (cid:1)(cid:21) ( i, j, l = 1 ,
2) (B.16) χ ( h ) ik = 1 h ( ∂h∂ξ " X l =1 H ilk α l − √ A ( ~a × ~a ) · (cid:18) α i ∂~a k ∂ξ + β i ∂~a k ∂ξ (cid:19) + ∂h∂ξ " X l =1 H ilk β l − √ A ( ~a × ~a ) · (cid:18) α i ∂~a k ∂ξ + β i ∂~a k ∂ξ (cid:19) ( i = 1 , , k = 1 , ,
3) (B.17) κ ( h ) i = − h (cid:20) α i ∂∂t (cid:18) h ∂h∂ξ (cid:19) + β i ∂∂t (cid:18) h ∂h∂ξ (cid:19)(cid:21) + " ∂∂ξ ∂ ~X∂t · ~a ! (cid:18) L i − A A α i (cid:19) + ∂∂ξ ∂ ~X∂t · ~a ! (cid:18) L i − A A β i (cid:19) + ∂ ~X∂t · ~a ! ( χ ( h ) i + X m =1 2 X l =1 (cid:20)(cid:18) α i (cid:18) ~a · ∂ ~a ∂ξ l ∂ξ m (cid:19) + β i (cid:18) ~a · ∂ ~a ∂ξ l ∂ξ m (cid:19)(cid:19) J lm + (cid:18) ∂α l ∂ξ m δ m + ∂β l ∂ξ m δ m + α l H mm + β l H mm (cid:19) H il (cid:21) − √ A (cid:20)(cid:18) ~a × ∂~a ∂ξ (cid:19) + (cid:18) ∂~a ∂ξ × ~a (cid:19)(cid:21) · (cid:18) α i ∂~a ∂ξ + β i ∂~a ∂ξ (cid:19)(cid:27) ( i = 1 ,
2) (B.18)ˆ κ ( h ) i = κ ( h ) i − (cid:18) α i ∂∂ξ (cid:18) h ∂h∂t (cid:19) + β i ∂∂ξ (cid:18) h ∂h∂t (cid:19)(cid:19) ( i = 1 ,
2) (B.19)28here δ ij is the Kronecker Delta. Appendix C. Derivation of equations to calculate ~V Let us identify the terms of order ε in the equation (36). We simplify that equa-tion, taking into account (20)-(22), (133), (147) and (151)-(156). Then we multiply theequation obtained by a i and we yield: X k =1 (cid:18) ∂V k ∂t ( ~a · ~a k ) + V k (cid:18) ~a · ∂~a k ∂t (cid:19)(cid:19) + X l =1 3 X k =1 (cid:18) ∂V k ∂ξ l ( ~a · ~a k ) + V k (cid:18) ~a · ∂~a k ∂ξ l (cid:19)(cid:19) (cid:0) V l − C l (cid:1) = − ρ ∂p ∂ξ + ν ( X m =1 2 X l =1 3 X k =1 ∂∂ξ m (cid:20) ∂ ( V k ~a k ) ∂ξ l ( α l a j + β l a j ) (cid:21) · ~a ( α m a j + β m a j )+ 1 h A A X k =1 ∂u k ∂ξ ( ~a · ~a k ) + 1 h X k =1 ∂ u k ∂ξ ( ~a · ~a k ) ) + X k =1 f k ( ~a · ~a k ) (C.1)where we have denoted by V = u (to achieve a more compact expression), and coeffi-cients C l , ( l = 1 , a i and a i , we obtain, respectively: X k =1 (cid:18) ∂V k ∂t ( ~a · ~a k ) + V k (cid:18) ~a · ∂~a k ∂t (cid:19)(cid:19) + X l =1 3 X k =1 (cid:18) ∂V k ∂ξ l ( ~a · ~a k ) + V k (cid:18) ~a · ∂~a k ∂ξ l (cid:19)(cid:19) (cid:0) V l − C l (cid:1) = − ρ ∂p ∂ξ + ν ( X m =1 2 X l =1 3 X k =1 ∂∂ξ m (cid:20) ∂ ( V k ~a k ) ∂ξ l ( α l a j + β l a j ) (cid:21) · ~a ( α m a j + β m a j )+ 1 h A A X k =1 ∂u k ∂ξ ( ~a · ~a k ) + 1 h X k =1 ∂ u k ∂ξ ( ~a · ~a k ) ) + X k =1 f k ( ~a · ~a k ) (C.2) ∂V ∂t + X k =1 V k (cid:18) ~a · ∂~a k ∂t (cid:19) + X l =1 ∂V ∂ξ l + X k =1 V k (cid:18) ~a · ∂~a k ∂ξ l (cid:19)! (cid:0) V l − C l (cid:1) = − ρ h ∂p ∂ξ + ν ( X m =1 2 X l =1 3 X k =1 ∂∂ξ m (cid:20) ∂ ( V k ~a k ) ∂ξ l ( α l a j + β l a j ) (cid:21) · ~a ( α m a j + β m a j )+ 1 h A A ∂h∂t + 1 h ∂ u ∂ξ (cid:27) + f (C.3)29ext we multiply equation (C.1) by α and we add equation (C.2) multiplied by α to get: ∂V ∂t + X l =1 α l X k =1 (cid:18) V k (cid:18) ~a l · ∂~a k ∂t (cid:19)(cid:19) + X l =1 ∂V ∂ξ l + X m =1 α m X k =1 V k (cid:18) ~a m · ∂~a k ∂ξ l (cid:19)! (cid:0) V l − C l (cid:1) = − ρ X l =1 α l ∂p ∂ξ l + ν ( X m =1 2 X p =1 α p X l =1 3 X k =1 ∂∂ξ m (cid:20) ∂ ( V k ~a k ) ∂ξ l ( α l a j + β l a j ) (cid:21) · ~a p ( α m a j + β m a j )+ 1 h A A ∂u ∂ξ + 1 h ∂ u ∂ξ (cid:27) + f (C.4)In the same way, we multiply equation (C.2) by β and we add equation (C.1) mul-tiplied by β to obtain: ∂V ∂t + X l =1 β l X k =1 (cid:18) V k (cid:18) ~a l · ∂~a k ∂t (cid:19)(cid:19) + X l =1 ∂V ∂ξ l + X m =1 β m X k =1 V k (cid:18) ~a m · ∂~a k ∂ξ l (cid:19)! (cid:0) V l − C l (cid:1) = − ρ X l =1 β l ∂p ∂ξ l + ν ( X m =1 2 X p =1 β p X l =1 3 X k =1 ∂∂ξ m (cid:20) ∂ ( V k ~a k ) ∂ξ l ( α l a j + β l a j ) (cid:21) · ~a p ( α m a j + β m a j )+ 1 h A A ∂u ∂ξ + 1 h ∂ u ∂ξ (cid:27) + f (C.5)We yield the following equations by integrating (C.4)-(C.5) over ξ from 0 to 1, andusing expressions (151)-(152), (148)-(149) and (166)-(167): ∂V ∂t + X l =1 (cid:0) V l − C l (cid:1) ∂V ∂ξ l + X k =1 Q k + X l =1 H lk V l ! V k = − ρ X l =1 α l ∂p ∂ξ l + ν ( X m =1 2 X l =1 ∂ V ∂ξ m ∂ξ l J lm + X k =1 2 X l =1 ∂V k ∂ξ l ( L kl + ψ ( h ) kl )+ X k =1 V k ( S k + χ ( h ) k ) + κ ( h ) ) + F ( h ) (C.6)30 V ∂t + X l =1 (cid:0) V l − C l (cid:1) ∂V ∂ξ l + X k =1 Q k + X l =1 H lk V l ! V k = − ρ X l =1 β l ∂p ∂ξ l + ν ( X m =1 2 X l =1 ∂ V ∂ξ m ∂ξ l J lm + X k =1 2 X l =1 ∂V k ∂ξ l ( L kl + ψ ( h ) kl )+ X k =1 V k ( S k + χ ( h ) k ) + κ ( h ) ) + F ( h ) (C.7)where H ilk , J lm , L kli , Q ik , S ik , F i ( h ), ψ ( h ) ikl , χ ( h ) ik and κ ( h ) i are given by (B.6),(B.8)-(B.10), (B.12)-(B.14) and (B.16)-(B.18).Finally, from last equations, taking into account that V = u , (133), (156) andrearranging terms, we obtain ∂V i ∂t + X l =1 (cid:0) V l − C l (cid:1) ∂V i ∂ξ l + X k =1 R ik + X l =1 H ilk V l ! V k = − ρ (cid:18) α i ∂π ∂ξ + β i ∂π ∂ξ (cid:19) + ν ( X m =1 2 X l =1 ∂ V i ∂ξ m ∂ξ l J lm + X k =1 2 X l =1 ∂V k ∂ξ l ( L kli + ψ ( h ) ikl )+ X k =1 V k ( S ik + χ ( h ) ik ) + ˆ κ ( h ) i ) + F i ( h ) − Q i ∂ ~X∂t · ~a ! ( i = 1 ,
2) (C.8)where R ik and ˆ κ ( h ) ii