Creeping films with vortices overstrongly undulated bottomsM. Scholle, A. WierschemandN. Aksel, Bayreuth, Germany

Dedicated to Prof. Dr.–Ing. Dr.techn.E.h. Jurgen Zierep,former editor ofActa Mechanica,on the occasion of his 75th birthday.

Summary. We study the influence of an undulated bottom profile on steadytwo–dimensionalgravity driven film flows of a Newtonian fluid. Traditional approaches towards this topic arebased on lubrication approximation, on special perturbation methods or on numerics. However,lubrication approximation and perturbation methods deliver acceptable results only within theirrange of validity. Especially if the bottom is strongly undulated, conventional analytical meth-ods fail. Neither can the classical separation solution of the biharmonic equation in terms of aninfinite series be applied because of massive convergence problems if the waviness exceeds alimit. In this paper we present an analytical method based ona representation of the solutionof Stokes equations in terms of holomorphic functions. Applying the complex function theory,convergence problems are avoided and the problem is reducedto solving ordinary differentialequations and integral equations at the boundaries only.Our calculations show the creation, formation and evolution of vortices if waviness and filmthickness exceed critical values. A detailed parameter study on size and strength of the vorticesis shown. Moreover, we present a quantitative study on the effect of the vortices on the flowrate. Our calculations show very good agreement with experimental results.

1 Introduction

Film flows occur in many technical processes as well as in nature. In the large field of coatingtechniques (see e.g. [1], [2] and [3]), especially where thin liquid films are forced to spread overa solid surface, they are of a great importance. Such flows arefound e.g. in the manufacturing ofelectronic devices. Therefore, the gravity driven film flow is one of the most studied systems inhydrodynamics. Nearly all contributions to this field are related to film flows over a flat bottomand their stability, whereas only a few papers are related tofilm flows over non–flat bottoms.We refer to Pozrikidis & Thoroddsen [4], who discuss the influence of a small particle fixed atthe bottom on the film flow, Decreet al. [5], who consider a topography of steps, and Wilson& Duffy [6], who consider bottom variations transverse to the flow direction.

We consider the steady gravity–driven film flow of an incompressible Newtonian fluid onan inclined sinusoidal bottom of infinite extensions. Experimental results in a flow channel arealready presented by Wierschemet al. [7]. A further experimental study on film flow alongan inclined periodic wall with rectangular corrugations isshown by Vlachogiannis & Bon-tozoglou [8]. In this paper the problem is treated analytically. Several different methodicalapproaches have been developed which may be classified by means of three categories:

The first category of contributions to this topic is based upon the lubrication approxima-tion which was used e.g. by Stillwagon & Larson [9]. Lubrication approximation allows forreduction of the problem to only one equation for the local film thickness whereas the velocityfield in leading order becomes locally parabolic due to the assumption that the flow has a smallaspect ratio ’typical length inz–direction per typical length inx–direction’. For a periodic bot-tom shape the latter assumption comes out to be a long–wave approximation being admissibleto weakly varying bottom shapes, only. A study on the accuracy of the lubrication approxima-tion for a film over a step–down is found in Gaskellet al. [10]. Films over strongly undulatedbottom profiles, like the films studied in [7], cannot be covered by lubrication theory.

The second category consists of various methodical approaches based on more sophisti-cated specialperturbation theories [11, 12, 13]. These different methods are distinguished by

1

means of different coordinate systems, scalings and perturbation parameters. Since terms ofhigher orders are neglected, any perturbation theory is applicable only for small perturbationparameters, e.g. if either the amplitude of the bottom wavesis much smaller than the wavelengthor if the film thickness is much smaller than the minimum curvature radius of the bottom shape.The latter case requires a positive local inclination angleas an additional geometrical restric-tion. A detailed experimental study by Wierschemet al. [14] quantifies the range of validity ofthe above–mentioned theories. For films over strongly undulated bottom profiles, however, weare outside the range of validity for perturbation theory inany case since the three geometricallengths, namely the wavelength, the amplitude of the bottomprofile and the film thickness areall of the same order. This is the situation we consider in this paper.

The third category of theoretical investigations on film flows over undulated bottoms arenumerical methods which deliver reasonable results without the above–mentioned geometricalrestrictions. Starting from full Navier–Stokes equations, e.g. Malamataris & Bontozoglou [15]make FEM computations. For creeping flows Pozrikidis [16] studies the influence of the bottomon the surface by making use of the boundary–integral method. He illustrates surface variationsin terms of flow rate, inclination angle, wave amplitude and surface tension.

In contrast to Pozrikidis we focus our attention on streamline patterns, especially on thegeneration and evolution of vortex structures, and on the influence of the vortices on the flowrate. We start with an exact solution of the biharmonic equation in terms of an infinite series. In-finite series of a similar type are also often used in numerical studies, see e.g. Bontozoglou [17]:He shows free surface profiles and flow structure in terms of Reynolds number and bottom am-plitude.

For strong undulations, however, the infinite series diverges. Therefore, we developed analternative method: The series representation of the solution of the biharmonic equation is re-placed by a more general representation in terms of holomorphic functions. This allows forapplying the whole power of complex function theory to the problem. Thereby, the problem isreduced to a system of ordinary differential equations and integral equations for functions ofa single variable, thus reducing the 2–dimensional problemto a 1–dimensional one. The so-lutions, which are obtained by a Fourier analysis, allow forqualitative and quantitative studiesof various phenomena: We find formation of recirculation vortices in the valleys of the bottomprofile in nice agreement with experimental results. For asymptotically thick films our stud-ies on size, strength and number of the vortices reveal that acritical value of the waviness isrequired for vortex generation. Keeping the waviness fixed we show that for decreasing filmthickness, size and strength of the vortices are reduced andthat vortices can even vanish forsufficiently thin films. Finally, the effect of the vortices on the flow rate is discussed.

This paper is organized as follows: In§2 the classical separation solution in terms of Carte-sian coordinates is constructed. It is shown that its range of validity is restricted to weakundulations. The above mentioned alternative approach based on complex function theory isdeveloped in§3. Special attention is paid in§4 to limit cases. Finally, results are presented anddiscussed in§5.

2 Formulation

2.1 Geometry and scaling

We consider the gravity–driven steady film flow of an incompressible Newtonian fluid on aninclined wavy plane of infinite length. In figure 1 a flow with a mean film thicknessH ona sinusoidal bottom shape is depicted with amplitudeA, wavelengthλ and mean inclinationangleα. Since a description of the flow in figure 1 in terms of local Cartesian coordinates isnot admissible because of a cross–over of the coordinate axes, it is inevitable to use Cartesiancoordinates with thex–axis parallel to the mean inclination and thez–axis perpendicular to it.

2

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α

A JJJJ]

HJJJ

JJ]

��

���+

�����3

λ

?~g

b

��

��

��+x

JJ

JJ

JJJ]z

JJ

JJ]z′

6z′′

Figure 1: Sketch of the flow geometry

We consider a two–dimensional flow geometry:

~v = u(x, z)~ex + w(x, z)~ez (1)

For the length scalings we use the wavelengthλ for both directions:

x =2π

λx (2)

z =2π

λz (3)

Accordingly, the characteristic velocityU is the same for both directions,

u =1

Uu , (4)

w =1

Uw , (5)

in order to leave the continuity equation invariant, i.e.

∂u

∂x+∂w

∂z= 0 . (6)

As characteristic velocity we choose

U :=gλ2 sinα

8π2η. (7)

Furthermore, the pressure is scaled by

p =λ

2πηUp , (8)

with the characteristic shear stress2πηU/λ.

3

2.2 Conventional methods and their limitations

Expressing the velocity field by a stream function

u =∂ψ

∂z, (9)

w = −∂ψ

∂x, (10)

the continuity equation (6) is fulfilled and the Stokes equations in dimensional form

~0 = −∇p+ η∇2~v + ~g , (11)

after applying the curl operator are reduced to the biharmonic equation

{

∂2

∂x2+

∂2

∂z2

}2

ψ = 0 (12)

for the stream function.At the bottom, which in non–dimensional representation is characterized byz = b(x), the

velocity field must fulfill the no–slip condition

~v = ~0 , (13)

where the given functionb(x) determines the bottom shape byz = b(x). Decomposition ofEq. (13) in normal and tangential component delivers

{

∂ψ

∂x+∂ψ

∂zb′(x)

}

z=b(x)

= 0 , (14)

{

∂ψ

∂xb′(x)−

∂ψ

∂z

}

z=b(x)

= 0 . (15)

The prime denotes the derivative. The normal component (14)is integrable and can be writtenalternatively as the algebraic condition

ψ (x, b(x)) = ψb = const , (16)

i.e. the bottom shape is a streamline. It is convenient to setψb = 0.The free surface of the flow, which in non–dimensional representationz = f(x) is charac-

terized by a functionf(x), is not given a priori — it has to be calculated from the kinematicboundary condition which conveniently is written in its integral form

ψ (x, f(x)) = ψs = const , (17)

i.e. the free surface has to be a streamline. The integrationconstantψs is the non–dimensionalflow rate, i.e.:

ψs =

f(x)∫

b(x)

u(x, z) dz . (18)

Note, that for a given film thickness the flow rateψs is not given a priori: Byh0 := (3ψs/2)1/3

the reference film thickness of a plane film with the same flow rate as the film over the undula-tions is defined. This dimensionless reference thickness isdifferent fromh, which is the meangeometrical thickness of the film over undulations.

Finally, at the free surfacez = f(x) the dynamic boundary condition in dimensional form

~n{

[p− ps + σκ] 1− η[

∇⊗ ~v + (∇⊗ ~v)T]}

= ~0 , (19)

4

which is the equilibrium between pressure jump at the surface, surface tension and viscousforces, has to be fulfilled. Byps the pressure of the surrounding is denoted. For the presentflow geometry and scaling the normal and tangent component ofEq. (19) read

p− ps +1

Ca

f ′′(x)√

1 + f ′(x)23 + 2

∂2ψ

∂x∂z+ f ′(x)

(

∂2ψ

∂z2−∂2ψ

∂x2

)

= 0 , (20)

[

1− f ′(x)2]

(

∂2ψ

∂z2−∂2ψ

∂x2

)

− 4f ′(x)∂2ψ

∂x∂z= 0 , (21)

(see e.g. [18]) where thecapillary number Ca is defined as

Ca :=ηU

σ. (22)

The capillary number has to be understood as a measure for theratio ’viscous shear stress atsurface per surface tension’. Substituting the reference velocityU according to definition (7)we obtain the alternative representation

1

Ca=

8π2σ

gλ2 sinα= 4π2

L2

λ2cotα (23)

in terms of the wavelengthλ, the inclination angleα and thegeneralized capillary length

L :=

√

2σ

g cosα, (24)

being the capillary length (see e.g. [19]) with the reduced gravity g cosα.We now make use of the classical analytical method for solving linear PDE’s, namely the

variable separation method. The superposition of all periodic solutions of Eq. (12) obtained byvariable separation results in the infinite series (see e.g.[20])

ψ = Az2 +Bz3 ++∞∑

n=−∞

e−nz[

(Rn + zQn) e−inx + c.c.

]

, (25)

with the real–valued coefficientsA, B and the complex–valued coefficientsQn, Rn, whichwill be determined later. Here, ’c.c.’ denotes the complex conjugate. It should be mentionedthat spectral representations of the stream function according to (25) are also often used innumerical works, see e.g. Bontozoglou [17] who studies filmsat comparatively high Reynoldsnumbers (Re ≈ 102).

After re–inserting (25) into Stokes equations (11) and taking the periodicity of the pressureinto consideration the coefficientB and the pressurep result in

B = −1

3, (26)

p = p1 − 2 cotαz ++∞∑

n=−∞

2in e−nz[

Qne−inx − c.c.

]

, (27)

with p1 being an integration constant. The coefficientsA, p1, ψs, Qn andRn are determinedfrom the above boundary conditions.

2.3 Results and limitations of the method

By inserting the general solution (25), (27) of the field equations into the boundary condi-tions (15), (16), (17), (20) and (21), the problem is reducedto finding a solution of an al-gebraic system of equations for the unknown coefficientsQn andRn and for the integration

5

constantsA, p1 andψs. The resulting algebraic equations, however, are non–linear and rathercomplicated, in general.

In this subsection, we focus our attention on a film flow with a nearly flat surfacef(x) ≈h := 2πH/λ as a special case. This is a reasonable approximation for thick films h ≫ 1,i.e. much thicker as the wavelength of the bottom corrugation, see figure 2. The general case

Figure 2: Thick film with nearly flat surface.

of films with a curved surface and arbitrary thickness is treated in §3. Note that under theconditionh≫ 1 the Reynolds number,

Re :=2gH3 sinα

2η2, (28)

which is independent ofh = 2πH/λ, may remain small and the assumption of a creeping flowis not violated.

According to figure 2 we make the a priori assumption that the bottom undulations influencethe flow inside a region near the bottom whereas outside this region the flow is nearly a planefilm flow. As a consequence each term with negative damping in (25) and (27) has to vanish,i.e.Qn = Rn = 0 for n < 0. Then the stream function and the pressure near the surfacez ≈ hare approximately given by

ψ (x, z) = Az2 −1

3z3 + 2zℜQ0 + 2ℜR0 +O

(

e−h)

, (29)

p (x, z) = p1 − 2 cotαz +O(

e−h)

, (30)

containing only the four unknown constantsA, Q0, R0 andp1. The symbol′ℜ′ denotes thereal part. Eqs. (29) and (30) show that the accuracy of our approximation is given by theexpressionexp(−h). Thus, the criterionh ≫ 1 for a ’thick film’ can be replaced by the muchweaker criterionexp(−h) ≪ 1 for a ’sufficiently thick film’. As a consequence of Eqs. (29)and (30) the normal component (20) of the dynamic boundary condition simplifies to

p1 − ps − 2h cotα = 0 +O(

e−h)

, (31)

and its tangential component (21) delivers

2A− 2h = 0 +O(

e−h)

. (32)

6

Both conditions are approximately fulfilled forA = h andp1 = ps+2h cotα. Since the streamfunction near the surface (29) does not depend onx, the kinematic boundary condition (17) isalso approximately fulfilled for a flat surfacef(x) = h and delivers

ψs =2

3h3 + 2hℜQ0 + 2ℜR0 +O

(

e−h)

(33)

as integration constantψs.In contrast to kinematic and dynamic boundary condition theno–slip condition (13) at the

bottom has to be fulfilled exactly. For a harmonic bottom shape

b(x) = −a cos x (34)

the no–slip condition is evaluated in Appendix A. As result we obtain for the coefficientsQn

andRn the algebraic set of equations

∞∑

n=0

[

M1R0n Rn +M1Q

0n Qn

]

=a2

2, (35)

∞∑

n=0

[

M1RmnRn +M1Q

mnQn

]

= ahδm1 +a2

4δm2 , m ∈ N , (36)

∞∑

n=0

[

M2RmnRn +M2Q

mnQn

]

= 0 , m ∈ N , (37)

R0 + aQ0 +

∞∑

n=1

(−1)n [Rn + aQn] exp (−na) =a3

6−ha2

2, (38)

where the corresponding matrix elementsM1Rmn, M1Q

mn, M2Rmn andM2Q

mn are defined accordingto Appendix A by Eqs. (A.10)–(A.13). Byδmn the Kronecker–delta is denoted.

Thus, the whole problem has been reduced to solving the linear set of equations (35)–(38).For the determination of the solution we truncate the above infinite set of equations at a finiteorderN and make use of the computer algebra system MAPLE [21] to calculate the coeffi-cientsQn andRn. Figure 3 shows the resulting streamlines for two differentwave amplitudesat a mean film thickness ofh = 2π. Note, that according toexp(−h) = exp(−2π) ≈ 0.0019the criterion for sufficiently thick films is fulfilled.

For a comparatively weak waviness ofa = 2π/12 we recognize that the streamlines nearthe bottom follow the bottom shape whereas the streamlines become more and more flat whenapproaching the surface as expected from the exponential decay in Eq. (29) with increasingdistance from the bottom structure.

For a higher waviness ofa = 2π/5, however, we encounter a serious problem: The infi-nite series in Eq. (25) does not converge within the whole region of definition: According tofigure 3b the streamlines can properly be calculated within the upperpart of the flow regionwhereas in the lower part extreme oscillations of the streamfunction (25) take place whichmake the calculation of the streamlines within this region impossible. For illustration, the val-ues of the stream function which result from a calculation withN = 64 terms, are plotted alongthe linez = −a/2 between the pointsA andB. The amplitude of the oscillations is of the orderof 1014. We point out that this problem cannot be solved by increasingN : Taking into accountmore terms in Eq. (25) the values ofψ (x, z) remain unchanged in the upper region whereasthe amplitude of the oscillations of the stream function in the lower part of the flow reachesastronomical orders of magnitude.

The explanation for the divergence of the series (25) is obvious: Forz < a the streamfunction is well–defined only within the interval− arccos(−z/a) ≤ x ≤ +arccos(−z/a)whereas its series representation (25) is not defined in the full interval −π ≤ x ≤ +π. Thus,

7

(a) s ss s

A B

ψ(x,−a

2)

(b)

Figure 3: Calculated streamlines for film thicknessh = 2π and two different values of thewaviness: (a) ata = 2π/12 we obtain a proper result, (b) ata = 2π/5 below the linez ≈ 0.118no streamline can be calculated. The order of truncation isN = 16 for (a) andN = 64 for (b).

by the representation (25) we meet a serious domain perturbation which cannot be solved byan analytic continuation or Taylor series expansion in the vicinity of bottom contour. Theclassical method of ’transfer of boundary condition’, van Dyke [22], works properly only forweak waviness [23]. Note, that the above limitations are also relevant for numerical workswhich make use of a spectral representation of the type (25).

3 Solution by means of complex function theory

3.1 Basic formulation

Complex function theory is a powerful tool for potential flows since all solutions of the two–dimensional Laplace equation can be identified with holomorphic functions. However, complexfunction theory can also be applied successfully on Stokes flows: We e.g. refer to the articles ofSiegel [24] and Cummings [25] on bubbles in Stokes flow. Making use of conformal mappingand Laurent expansion, the authors derive asymptotic solutions. In contrast to this our approachis based on determining the unknown holomorphic functions directly by boundary conditionsinstead of using conformal mapping.

We introduce the complex coordinateξ and the two complex functionsQ(ξ) andR(ξ) as:

ξ :=z + ix

2, Q(ξ) :=

+∞∑

n=−∞

Qne−2nξ , R(ξ) :=

+∞∑

n=−∞

Rne−2nξ (39)

Now, the stream function (25) and the pressure (27) are rewritten in terms ofξ,Q(ξ), R(ξ) as

ψ = A(

ξ + ξ)2

−

(

ξ + ξ)3

3+

[

R(ξ) +(

ξ + ξ)

Q(ξ) + c.c.]

, (40)

p = p1 − 2 cotα(

ξ + ξ)

− i[

Q′ (ξ)− c.c.]

, (41)

where the bar denotes the complex conjugate. The main advantage of this alternative repre-sentation is that (40) fulfills the biharmonic equation (12)for arbitrary holomorphic functions

8

Q(ξ) andR(ξ). A representation of the functions by infinite series according to Eqs. (39) is notrequired. Thus, Eq. (25) can be considered a special case of Eq. (40). The two functionsQ(ξ)andR(ξ) are completely determined by their values at the boundaries: By means of the Cauchyintegrals

Q(ξ) =1

2πi

∮

Γ

Q(ζ)

ζ − ξdζ R(ξ) =

1

2πi

∮

Γ

R(ζ)

ζ − ξdζ (42)

along an arbitrarily closed pathΓ the values of the two functions at any pointξ can be calculatedprovided thatξ lies within the inner region of the closed loopΓ. In figure 4 the geometry of theflow in complex representation is shown. The bottom is definedby ξ = β(x) = 1

2 [b(x) + ix]

ξ = β(x)

ξ = ϕ(x)

Γ2

Γ1

�������

AA

AA

AAK

ℑ

ℜ

Figure 4: Geometry and notations of the flow in complex representation.

and the free surface byξ = ϕ(x) = 12 [f(x) + ix]. The pathΓ1, e.g. , is adequate for calcu-

lating any values of the functionsQ andR inside the flow region by the Cauchy integrals (42).However, any pathΓn which includes2n − 1 periods would work equally well. A Cauchyintegral along the pathΓn is conveniently decomposed in four integrals, namely one integralalong the bottom, one along the free surface and two along thestraight lines passing throughthe flow region. The latter ones are for sufficiently large numbersn of the orderO (1/n). Thus,by taking the limitn → ∞ the Cauchy integrals (42) reduce to integrals over the bottom β(x)and the surfaceϕ(x) with dζ = β′(x)dx anddζ = ϕ′(x)dx respectively:

Q(ξ) =1

2πi

−∞∫

+∞

Qβ(x)

β(x)− ξβ′(x) dx+

1

2πi

+∞∫

−∞

Qϕ(x)

ϕ(x) − ξϕ′(x) dx (43)

R(ξ) =1

2πi

−∞∫

+∞

Rβ(x)

β(x)− ξβ′(x) dx+

1

2πi

+∞∫

−∞

Rϕ(x)

ϕ(x) − ξϕ′(x) dx (44)

Here, we have introduced the abbreviations

Qβ(x) := Q (β(x)) , Qϕ(x) := Q (ϕ(x)) ,

Rβ(x) := R (β(x)) , Rϕ(x) := R (ϕ(x)) . (45)

In contrast to the series representation (25), we do not leave the domain of definition in ourcomplex formulation: By the path integrals (43, 44) the region of definition of the two func-tionsQ andR is exactly the domain between bottom and surface. Since the question of the

9

existence of an analytic continuation of the two functions outside the flow domain is not rele-vant, convergence problems do not occur in complex formulation.

Compared to the complex function theory for two–dimensional potential flows, there is anessential difference: The stream functionψ itself is not a holomorphic function — it is rather acombination of two holomorphic functionsQ andR according to Eq. (40).

Considering the representation (40) of the stream function, the associated wall shear stress

τw = ∇2ψ∣

∣

ξ=β(x)= 2 [h− b(x)] + 2ℜ

[

1

β′(x)

d

dxQβ(x)

]

results directly from the boundary values ofQ along the bottom.We remark that the two functionsQ andR are not unique: The stream function (40) and

the pressure (41) are invariant with respect to the substitutions

Q(ξ) −→ Q(ξ) + iCQ , CQ ∈ R ,

R(ξ) −→ R(ξ) + iCR , CQ ∈ R .

These gauge transformations also act on the boundary valuesof the two functions, i.e.

Qβ(x) −→ Qβ(x) + iCQ , Qϕ(x) −→ Qϕ(x) + iCQ , CQ ∈ R , (46)

Rβ(x) −→ Rβ(x) + iCR , Rϕ(x) −→ Rϕ(x) + iCR , CR ∈ R , (47)

without having any effect on the physics of the system.

3.2 The complex equations

By the Eqs. (40), (41), (43) and (44) stream function and pressure are represented in terms offive functionsQβ(x),Qϕ(x), Rβ(x), Rϕ(x) andf(x). These functions are derived from the3boundary conditions and from the2 holomorphic conditions forQ andR.

First, we decompose the dynamic boundary condition (19) into z– andx–component. Theresulting two equations can be interpreted as real and imaginary part of the complex equation

i

[

p− ps +1

Ca

f ′′(x)√

1 + f ′(x)23

]

ϕ′(x) +∂2ψ

∂ξ2ϕ′(x) = 0 (48)

which is valid along the free surfaceξ = ϕ(x). We decompose the surface according to

f(x) = h+ g′(x) (49)

additively into a mean thicknessh and a surface oscillationg′(x). The functiong′(x) is takenas a first order derivative of a periodic functiong(x), which is well–defined except for itsmeans valueg0 :=< g >=

∫ +π−π g(x)dx/(2π). After inserting the pressure (41) and the stream

function (40) into Eq. (48), the dynamic boundary becomes a second order ordinary differentialequation

p1 − ps2

− cotαh+ i (A− h) +d

dx

{

− (cotα+ i) g(x) +[A− h− g′(x)]2

2

−i

2tanα

[

p1 − ps2

− cotαh− cotαg′(x)

]2

+1

2Ca

g′′(x)− i√

1 + g′′(x)2

+R′ (ϕ(x)) +[

h+ g′(x)]

Q′ (ϕ(x)) +Q (ϕ(x)) −Q (ϕ(x))

}

= 0 . (50)

10

By taking the mean value of Eq. (50) we derive the identities

A = h (51)

p1 = ps + 2h cotα (52)

for the two integration constantsA andp1 being in accordance with Eqs. (32) and (31). Afterre–inserting these identities, the dynamic boundary condition (50) takes the form of a total dif-ferential which can be integrated with respect tox. By adequate gauging according to Eq. (46)and adequate choosing the mean valueg0 of the functiong the integration constant can becompensated. Thus, the integration of the dynamic boundarycondition (50) delivers after mul-tiplying with ϕ′(x) finally

d

dx

{

Rϕ(x) +[

h+ g′(x)]

Qϕ(x)}

−ℑQϕ(x)− g′′(x)ℜQϕ(x) (53)

−cotα+ i

2

[

g′′(x) + i]

[

g(x) +i

2g′(x)2

]

+

√

1 + g′′(x)2

4Ca= 0 .

This ordinary differential equation is linear and of first order with respect to the functionsQϕ(x) andRϕ(x) but non–linear and of second order with respect to the function g(x).

Next, reconsidering Eqs. (40) and (51) the kinematic boundary condition (17) results in thesimple algebraic equation

h2 g′(x)−1

3g′(x)3 + 2ℜ

{

Rϕ(x) +[

h+ g′(x)]

Qϕ(x)}

= ψs . (54)

containing the functionsQϕ(x), Rϕ(x) andg′(x).By inserting the stream function (40) into the no–slip conditions (14, 15) and combining

both equations as real part and imaginary part of one complexequation we derive the first orderordinary linear differential equation

d

dx

[

h

2b(x)2 −

1

6b(x)3 +Rβ(x) + b(x)Qβ(x)

]

+i

[

h b(x)−1

2b(x)2 + ℜQβ(x)− b′(x)ℑQβ(x)

]

= 0 , (55)

containing the two functionsQβ(x) andRβ(x). Furthermore, withψb = 0 the mean value ofEq. (16) reads

h 〈b(x)2〉 −1

3〈b(x)3〉+ 2〈ℜRβ(x)〉 + 2〈b(x)ℜQβ(x)〉 = 0 . (56)

However, there is still one degree of freedom left for gauging by the transformation (47) whichallows for choosing the mean value of eitherℑRβ(x) orℑRϕ(x) arbitrarily. By the choice

〈ℑRβ(x)〉 = 0 , (57)

e.g. , the two functionsRβ(x) andRϕ(x) are uniquely gauged. Any other gauging, however,is also possible.

The set of equations (53)–(57) is completed by considering the holomorphy of the func-tionsQ(ξ) andR(ξ) which gives rise for Cauchy’s theorem

∮

Γ1

F (ξ)Q(ξ)dξ = 0 ,

∮

Γ1

F (ξ)R(ξ)dξ = 0 , (58)

which have to be fulfilled for any holomorphic test functionF (ξ). Here, the closed pathΓ1

includes one period of the flow according to figure 4. If we restrict F (ξ) to be2π–periodic,

11

then the contributions to the path integrals in Eq. (58) which result along the straight lines ofΓ1

through the flow region (see figure 4) annihilate each other. The remaining contributions arethe path integrals along the bottom and the free surface. Thus, Eq. (58) simplifies to

+π∫

−π

F (ϕ(x))Qϕ(x)ϕ′(x) dx −

+π∫

−π

F (β(x))Qβ(x)β′(x) dx = 0 , (59)

+π∫

−π

F (ϕ(x))Rϕ(x)ϕ′(x) dx−

+π∫

−π

F (β(x))Rβ(x)β′(x) dx = 0 . (60)

Thus, the set of equations for the five unknown functions is set up by: The no–slip condi-tion and the dynamic boundary condition are represented by the two ordinary differential equa-tions (55) and (53), the kinematic boundary condition by thealgebraic equation (54) and theholomorphic coupling between bottom and surface by the two integral equations (59) and (60).Supplemented by the two mean–value conditions (56) and (57)this system allows for deter-mining the boundary valuesQβ(x),Qϕ(x), Rβ(x) andRϕ(x) of the holomorphic functionsQandR and the integral of the surface shapeg(x). Thus, by means of the complex formulationa two–dimensional problem has been reduced to a one–dimensional one.

3.3 Solution by discretization

In order to reduce the problem to a set of algebraic equationswe choose a representation of thefive unknown functions in terms of Fourier series as follows:

Qβ(x) =+∞∑

m=−∞

qβm exp(−imx) Qϕ(x) =+∞∑

m=−∞

qϕm exp(−imx)

Rβ(x) =

+∞∑

m=−∞

rβm exp(−imx) Rϕ(x) =

+∞∑

m=−∞

rϕm exp(−imx)

g(x) =

+∞∑

m=−∞

gm exp(−imx) (61)

Note, that not all coefficients are independent: Since the functionsg(x) is real–valued its coef-ficients have to fulfill the conditiong−m = gm. The bottom shape is assumed to be harmonicaccording to Eq. (34). Now, the series representations (61)have to be inserted in the five com-plex Eqs. (55), (53), (54), (59) and (60) and the two mean–value conditions (56) and (57).

By means of Fourier decomposition the boundary conditions (55), (53) and (54) and thetwo mean–value conditions (56) and (57) reduce to a non–linear algebraic set of equations forthe respective Fourier coefficients in Eq. (61). This set of equations is given in Appendix B.

For a discretization of the two integral equations (59) and (60) we have to make an adequatechoice for the test functionsF (ξ) as elements of a discrete set{Fn(ξ) |n ∈ Z} of 2π–periodicholomorphic functions which are independent of each other.It is near at hand to choose expo-nential functions

Fn (ξ) = exp(2nξ) . (62)

Having taken into account Eq. (61) as well as Eq. (62) the two integral equations (59) and (60)result in the algebraic set of equations

+∞∑

m=−∞

[

Eϕn,m q

ϕm − Eβ

n,m qβm

]

= 0 , (63)

+∞∑

m=−∞

[

Eϕn,m r

ϕm − Eβ

n,m rβm

]

= 0 , (64)

12

with the matrix elementsEβn,m andEϕ

n,m given as

Eβn,m :=

δ0m −a

2(δ1m − δ−1m) , n = 0

m

nIn−m (−na) , n 6= 0 ,

(65)

Eϕn,m :=

δ0m + im2g−m , n = 0

m exp (nh)

2πn

+π∫

−π

exp(

ng′(x))

exp (i[n−m]x) dx , n 6= 0 ,(66)

with In being the modifiednth order Bessel functions. Now, Eqs. (63) and (64) form to-gether with Eqs. (B.1)–(B.6) in Appendix B a complete set of nonlinear algebraic equationsfor the determination of the coefficientsqβm, qϕm, rβm, rϕm (m = −∞, · · · ,+∞), gm, Sm(m = 0, 1, · · · ,∞) and the integral flow rateψs. Its solution is calculated by means of com-puter algebra, e.g. [21]. For the software implementation we replace each of the infinite Fourierseries in Eq. (61) by a finite sum up to the orderN in order to reduce the above infinite algebraicsystem to a finite system

Fp

(

qβm, rβm, q

ϕm, r

ϕm, gm, Sm, ψs

)

= 0 , p = 1, · · · , 10N + 7 (67)

of algebraic equations being linear inqβm ,rβm, qϕm, rϕm andψs and non–linear ingm andSm.Multi–dimensional Newton iteration is used to solve the setof equations (67).

4 Limit case studies

4.1 Approximation for films with weakly curved surface

Subsequently we pay special attention to film flows the free surface of which is only weaklycurved, i.e.|f(x) − h| ≪ 1 for −π < x < π with h = 2πH/λ. According to§2.3, theinfluence of the bottom on the surface decreases exponentially with increasing film thickness.Thus, the surface is weakly curved if the film is sufficiently thick, i.e. exp(−h) ≪ 1. Viceversa, the influence of the free surface on the bottom can be neglected in this case, which leadsto a decoupling of the equations and therefore simplifies thepresent problem to a great extend.

For the coefficients at the surface we start with the ad hoc assumption

qϕm = O (exp(−h))

rϕm = O (exp(−h)) (68)

gm = O (exp(−h))

for m 6= 0, which will be verified later in a self–consistent manner. This immediately in-ducesg′(x) = O (exp(−h)) and exp(ng′(x)) = 1 + O (exp(−h)). As a consequence thematrix elements (66) are approximately given by

Eϕn,m = exp(nh)δnm +mO (exp ([n− 1]h)) . (69)

Thus, the holomorphy conditions (63, 64) take the simplifiedform

+∞∑

m=−∞

m

nIn+m (na) qβm = 0 +O (exp (−[n+ 1]h)) , (70)

+∞∑

m=−∞

m

nIn+m (na) rβm = 0 +O (exp (−[n+ 1]h)) , (71)

13

and

qϕ0 = qβ0 −a

2

(

qβ1 − qβ−1

)

+O (exp(−2h)) , (72)

rϕ0 = rβ0 −a

2

(

rβ1 − rβ−1

)

+O (exp(−2h)) , (73)

qϕn = exp(−nh)

+∞∑

m=−∞

m

nIn−m (−na) qβm +O (exp(−2h)) , (74)

rϕn = exp(−nh)

+∞∑

m=−∞

m

nIn−m (−na) rβm +O (exp(−2h)) , (75)

with n ∈ N. Furthermore, from the kinematic boundary condition (B.3)we derive form = 0the identity

ψs =2

3h3 + 2ℜ [rϕ0 + hqϕ0 ] +O (exp(−2h)) , (76)

for the flow rate and form 6= 0 the set of equations

rϕm + rϕ−m + h

[

qϕm + qϕ−m

]

− im[

h2 + 2ℜqϕ0]

gm = 0 +O (exp(−2h)) . (77)

The dynamic boundary condition (B.4) delivers form = 0 the two identities

g0 = 0 +O (exp(−2h)) , (78)

ℑqϕ0 =1

4Ca+O (exp(−2h)) , (79)

and form 6= 0 the set of equations

mrϕm +

(

mh−1

2

)

qϕm +1

2qϕ−m +

1

2

[

cotα+m2

2Ca+ i

]

gm = O (exp(−2h)) . (80)

In a next step we simplify the above equations by neglecting all terms of orderO (exp(−2h))

and taking into account only a finite number of coefficientsqβ−N , · · · , q

βN andrβ

−N , · · · , rβN ,

N ∈ N. The holomorphy conditions (70, 71) thus reduce to

+N∑

m=−N

mIn+m (na) qβm = 0 , n = 1, · · · , N (81)

+N∑

m=−N

mIn+m (na) rβm = 0 , n = 1, · · · , N , (82)

and the no–slip boundary condition (B.1) takes the form

1

2

(

qβm + qβ−m

)

+a

4

[

(2m+ 1) qβm+1 + (2m− 1) qβm−1 − qβ−m−1 + qβ

−m+1

]

−ah

2

[

δm,1 + δm,−1 +a

2(δm,2 − δm,−2)

]

−a2

8(δm,2 + δm,−2 + 2δm,0)

−a3

16(δm,1 + δm,3 − δm,−3 − δm,−1) = mrβm , (83)

for qβm ,m = −N, · · · , N andrβm ,m = −N, · · · ,−1, 1, · · · , N . The set of equations (81)–(83) is completely decoupled from the remaining equations,especially from kinematic anddynamic boundary conditions at the free surface. This meansthat in leading order the velocityfield in the vicinity of the bottom is not influenced by the freesurface. Therefore, Eqs. (81)–(83)allow for determining the coefficientsqβ

−N , · · · , qβ−N and rβ

−N , · · · , rβ−1, r

β1 , · · · , r

βN . Their

14

solution is easily obtained by using computer algebra, e.g.[21]. The numberN is chosen toreach a given accuracy. For determining the coefficientrβ0 we need the supplementary relation

rβ0 =a

2ℜ(

qβ1 + qβ−1

)

−a2h

4(84)

which results from the mean value conditions (B.2) and (B.6).Having once determined the coefficientsqβm andrβm for the bottom values ofQ andR the

remaining coefficientsqϕm, rϕm andgm for the surface can be calculated straight forward: FromEqs. (72)–(75) it follows by neglecting all terms of orderO (exp(−2h))

qϕ0 = qβ0 −a

2

(

qβ1 − qβ−1

)

, (85)

rϕ0 = rβ0 −a

2

(

rβ1 − rβ−1

)

, (86)

qϕ1 = exp(−h)

+N∑

m=−N

mIm−1 (−a) qβm , (87)

rϕ1 = exp(−h)

+N∑

m=−N

mIm−1 (−a) rβm . (88)

The coefficientsgm for the surface shape as well as the remaining coefficientsqϕm, rϕm withnegative indexm are obtained as solutions of Eqs. (77) and (80). They read explicitly

g1 = −2rϕ1 + hqϕ1

cotα+ (2Ca)−1 − i [h2 + 2ℜqϕ0 ], (89)

qϕ−1 = (1− 2h)qϕ1 − 2rϕ1 − (cotα+ (2Ca)−1 + i)g1 , (90)

rϕ−1 = (1 + 2h)rϕ1 + 2h2qϕ1 +

[

(1 + h)(

cotα+ (2Ca)−1)

+ ih]

g1 . (91)

Note, thatqϕm, rϕm, gm = O (exp(−2h)) for |m| ≥ 2. According to Eqs. (49) and (61) the free

surface readsf(x) = h+ 2ℑ [g1 exp(−ix)] +O (exp(−2h)) , (92)

with g1 given by Eq. (89) and is therefore always sinusoidal, independent of the bottom shape,provided that the criterionexp(−h) ≪ 1 is fulfilled.

In Eqs. (89)–(91) the inclination angleα and the capillary numberCa enter the problemonly in combination

{

cotα+ (2Ca)−1}

. Thus, sufficiently thick films with capillarity looklike a film without capillarity at a smaller inclination angle. This result has also been obtainedby Wang [11]. Furthermore, we note that the above results arein accordance with the basicassumption (68) which therefore is self–consistently fulfilled.

Comparing the thick–film approximation with the mathematical treatment for the generalcase in§3.3 we recognize that for thick films the computational requirements are reduced toa minimum, namely to solving the system (81–83) of purely linear algebraic equations for thetwo sets of coefficientsqβm ,m = −N, · · · , N andrβm ,m = −N, · · · ,−1, 1, · · · , N . Theremaining coefficientsrβ0 , g1, q

ϕ−1, q

ϕ0 , qϕ1 , rϕ

−1, rϕ0 andrϕ1 result successively from Eqs. (84)–(91). Hence, according to definition (61) the boundary values of the functionsQ andR in thestream function (40) and the free surface are determined.

4.2 Asymptotically thick films

We discuss asymptotically thick filmsh ≫ 1 as another relevant case. After rescaling thestream function and the Fourier coefficients according to

ψ = 2hΨ

qβm = 2hQβm qϕm = 2hQϕ

m (93)

rβm = 2hRβm rϕm = 2hRϕ

m

15

the limit h→ ∞ can be applied to the set of algebraic equations (81)–(83) which now read

+N∑

m=−N

mIm+n(na)Qβm = 0 , n = 1, · · · , N (94)

+N∑

m=−N

mIm+n(na)Rβm = 0 , n = 1, · · · , N (95)

mRβm −

1

2

(

Qβm + Qβ

−m

)

−a

4

[

(2m+ 1)Qβm+1 + (2m− 1)Qβ

m−1 − Qβ−m−1 + Qβ

−m+1

]

= −a

4(δm,1 + δm,−1)−

a2

8(δm,2 − δm,−2) , m = −N, · · · , N . (96)

By solving the above set of equation with computer algebra, the coefficientsQβ−N , · · · , Q

βN

andRβ−N , · · · , R

β−1, R

β1 , · · · , R

βN are determined. According to Eq. (84) the zeroth order co-

efficientRβ0 results from

Rβ0 =

a

2ℜ(

Qβ1 + Qβ

−1

)

−a2

8. (97)

After applying the rescaling (93) and the limith → ∞ to the Eqs. (85)–(91) the zeroth ordercoefficients for the surface read

Qϕ0 = Qβ

0 −a

2

(

Qβ1 −Qβ

−1

)

, (98)

Rϕ0 = Rβ

0 −a

2

(

Rβ1 −Rβ

−1

)

, (99)

whereas the respective first order coefficientsQϕ−1, R

ϕ−1,Q

ϕ1 ,Rϕ

1 andg1 vanish.We remark that the limith→ ∞ does not violate the conditionRe ≪ 1 for creeping flows:

Sinceh/(2π) is the aspect ratioH/λ, the limith→ ∞ can be reached byλ→ 0 for a fixed filmthicknessH. From the physical viewpoint the bottom waves become microscopically small andtherefore give rise to a two–dimensional model for roughness. In this context the rescaling (93)makes sure that the values of the stream function in the vicinity of the undulations (see figure 2)are of orderO (1).

The limit h → ∞ reduces the general4–parameter problem with parametersa, h, α, Cato a one–parameter problem: In the linear algebraic equations (94)–(99) for determining thecoefficientsQβ

m, Rβm ,m = −N, · · · , N , Qϕ

0 andRϕ0 only the wavinessa is involved as

parameter. Moreover, the region of definition is now mathematically unbounded in positivez–direction, i.e. a free surface is not present in this limit case. As a consequence, kinematic anddynamic boundary condition simplify to the asymptotic conditions (98, 99) for the functionsQandR.

5 Results and discussion

5.1 Asymptotically thick films

Having solved the set of equations (94)–(99) by computer algebrathe wall shear stress is di-rectly given according to§3.1. By Eqs. (43), (44) and (40) the stream function is determined.

We have calculated streamlines for parameter values ofa from 0 up to 3.8. For smallwaviness we found that the streamlines follow the bottom shape like e.g. the previously cal-culated flow in figure 3a for a = 2π/12 ≈ 0.524. If the waviness exceeds a critical valueof a = a∞1 ≈ 0.769, however, flow separation is observed. For demonstrationtwo flows areshown in figure 5:Here wesee still a flow which follows the bottom fora = 2π/10 ≈ 0.629,whereas fora = 2π/5 ≈ 1.257 we see a symmetric recirculation area in the valley of the

16

�w012

�� ��=2 0 +�=2 +�

(a)�w

012

�� 0 +�

(b)

Figure 5: Streamlines near the bottomand associated wall shear stress of films with (a):a =2π/10 and (b):a = 2π/5. Note, that the free surface is outside this picture.

bottom profile separated from the main flow by a nearly straight separatrix. The observation ofvortex creation is in accordance with numerical results of [16, 15]. Flow reversal has also beenobserved in Stokes flow through wavy tubes, see [26, 27].

The conventional method developed in§2.3 failed for the wavinessa = 2π/5 as was shownin Figure 3b. Moreover, this fundamental phenomenon is outside the scope of all other classicalapproaches, especially lubrication theory which predictsa locally parabolic velocity profile.

Note, that the separatrix in figure 5 is not exactly straight but slightly convex–shaped. Byvariation of the waviness we found that at the beginning of the flow separation the separa-trix becomes more convex, whereas for higher waviness it becomes slightly concave–shaped.Moreover, the straightness of the separatrix seems to be a special feature of sinusoidal bottoms.Recent studies on inharmonic bottom shapes have delivered relevantly curved separatrices.

By increasing the waviness, the vortex becomes larger. If the waviness exceeds a secondcritical valuea = a∞2 ≈ 2.280 a secondary vortex is generated. In figure 6, e.g. , a flowat a = 9π/10 ≈ 2.827 with primary and secondary vortex is shown. The circulationof thesecondary vortex, however, is orders of magnitudes below the circulation of the primary vortex.Furthermore, its separatrix is clearly concave–shaped.

Increasing the waviness again, we determinea = a∞3 = 3.786 as the critical value forcreation of a tertiary vortex. Hence, with increasing waviness an increasing number of vorticesis created similar to Moffatt’s eddies [28] in a sharp corner. However, Moffatt’s solution ofthe Stokes equation is valid for an unbounded flow geometry without any characteristic length,whereas in the present problemλ, A andH are involved as characteristic lengths, which leadsto a different problem. For a quantitative study on the circulation of the vortices we refer to thenext section.

5.2 Films with weakly curved surface

Next we turn to films with finite thicknessh, but with an only weakly curved surface. Accord-ing to§4.1 the criterionexp(−h) ≪ 1 has to be fulfilled. In this case the set (81)–(83) of linearalgebraic equations has to be solved. In figure 7 calculated streamlines of a flow over a sinu-soidal bottom with wavinessa = 2π/5, film thicknessh = 5π and inclination angleα = 10◦

are shown together with path lines detected experimentallyby Wierschemet al. [7]. We seehere a very good agreement between theoretical and experimental results. Again, a symmetric

17

Figure 6: Streamlines near the bottom of an asymptotically thick film with a = 9π/10. Note,that the free surface is outside this picture.

flow separation area in the valley with a nearly straight separatrix is apparent. Note, that thefree surface is outside the picture.

Comparing figure 7 (h = 5π) with the asymptotic case depicted in figure 5 (h → ∞) wefind that both pictures are qualitatively similar. A quantitative difference is the size of the vortexwhich is smaller for finite film thicknessh = 5π.

For a quantitative study of the vortices, we calculate the circulation as measure for thevortex strength which is given as

Γ :=

∮

∂A~v · d~x =

Uλh

π

x2∫

x1

[

∂Ψ

∂z− s′(x)

∂Ψ

∂x

]

z=s(x)

dx , (100)

where the separatrix line is given byz = s(x). Here,A denotes the recirculation area,x1 andx2 the x–positions of the respective triple points. The corresponding dimensionless circula-tion Γ := πΓ/(Uλh) of the primary and the secondary vortex is plotted versus thewavinessain figure 8. Here,Γ∞

1 andΓ∞

2 denote the circulation of primary and secondary vortex ofasymptotically thick films (h → ∞), whereasΓ2π

1 andΓ2π2 denote the circulation of primary

and secondary vortex of films with a fixed flow rateψs = 2(2π)3/3, i.e. of films with the sameflow rate as a plane film flow with finite thicknessh0 = 2π.

Obviously, no vortex is generated below a critical wavinessof a∞1 ≈ 0.769. At this valuethe primary vortex is created for the asymptotic caseh→ ∞, whereas a retarded vortex creationstarts in the film withh0 = 2π at a2π1 ≈ 0.893. In the asymptotic caseh → ∞ the secondary

18

Figure 7: Calculated and measured streamlines fora = 2π/5, h = 5π, α = 10◦ andCa =0.0024.

-a0.5 1.0 1.5 2.0

2.5 3.0 3.5

6Γ

0.2

0.4

0.6

−0.2

−0.4

0

Γ∞

1

500 · Γ∞

2

Γ2π1

500 · Γ2π2

Figure 8: Circulation of primary and secondary vortex vs.a for h→ ∞ and forh = 2π.

vortex is generated if the waviness exceeds a second critical valuea∞2 ≈ 2.280, for the finitecaseh0 = 2π this second critical value isa2π2 ≈ 2.430. The diagram ends ata = a∞3 = 3.786,which is the critical value for creation of a tertiary vortexin the asymptotic case.

Furthermore, the circulation of the finite caseh0 = 2π is always smaller than the circulationof the asymptotic caseh → ∞. Moreover, the circulation of the secondary vortex remainsorders of magnitude below the circulation of the primary vortex in both cases. These resultsare in accordance with the observations of Wierschemet al. [7].

19

5.3 Thin films

The simplified method described in§4.1 can successfully be applied to film flow down to themoderate film thickness of abouth ≈ 2π. For thin filmsh < 2π, however, the validity criterion,exp(−h) ≪ 1, is no longer fulfilled. Thus, according to§3.3 the self–consistent procedure forthe general case has to be used in order to take the nontrivialsurface shape into account. Thisgeneral procedure allows for studying the influence of the film thickness on the vortex forarbitrary cases, especially for thin films.

In the previous sections we have shown that size and strengthof recirculation vorticesare reduced with decreasing film thickness. The question arises if the vortices vanish for asufficiently thin film. For this purpose we have calculated the streamlines for a film flow overa sinusoidal bottom with fixed wavinessa = 2π/5, inclination angleα = 45◦ and capillarynumberCa = 0.2 and with varying film thickness. Some representative results are shown infigure 9: Obviously, the size of the vortex is again reduced bydecreasing the film thickness

h = 3.589

(a)

h = 2.691

(b)

h = 1.704

(c) 0.0 2.5 5.0 7.5 10.0 12.50.00

0.25

0.50

0.75

1.00

Experiment Theory

Dis

tanc

e fro

m V

alle

y [m

m]

Film Thickness [mm](d)

Figure 9: Reduction and suppression of the vortex due to decreasing film thickness.

from h = 3.589 to h = 2.691. In the latter case only a tiny recirculation area remains.Decreasing again the thickness toh = 1.704, the flow separation vanishes completely.

A quantitative comparison of the distance of the separatrixfrom the minimum depth of thebottom contour vs. film thickness with experimental values [7] is shown in figure 9d. Thecritical film thickness for vortex creation here ish ≈ 2.114. For thick films the size of therecirculation vortex tends to an asymptotic value according §5.2.

Furthermore, the sequence in figure 9 shows that the amplitude and the curvature of the freesurface are increasing with decreasing film thickness. Thisis in accordance with the numericalstudies of Pozrikidis [16] and the experimental studies of Wierschemet al. [7].

The reduction of vortex size and circulation is also observed for secondary vortices: In

20

figure 10 an example for a film flow with primary and secondary vortex is shown in comparisonwith experimental results [7]. Again, the theoretically calculated streamlines are in very goodagreement with the experimentally detected path lines. Comparing these results with those for

Figure 10: Primary and secondary vortex of a film flow with wavinessa = 9π/10 ≈ 2.83,thicknessh = 0.89, inclination angleα = 45◦ and capillary numberCa = 1.6.

the same waviness andh → ∞ (see figure 6), we find that the size of the primary vortex aswell as the size of the secondary vortex have become smaller due to the finite film thickness.For more details on the dependence of vortex size on film thickness we refer to [7]. Finally,extremely thin films in which intermolecular forces become important are excluded from ouranalysis.

5.4 Consequences for the flow rate

The question arises, how the flow rate is influenced by the vortices. Reconsidering that accord-ing to Eq. (18) the non–dimensional flow rate equals the valueψs of the stream function at thesurface, we define

h0 :=

[

3

2ψs

]1

3

(101)

asreference thickness of a plane film flow with the same flow rate as the flow over the topogra-phy. We further define themean transport thickness

h′ := h−1

2π

x2∫

x1

[s(x)− b(x)] dx , (102)

as mean thickness of the part of the flow which contributes to the material transport, i.e. themean thickness of the film above the separation areas. In above definition the separatrix of the

21

primary vortex is denoted byz = s(x), x1 ≤ x ≤ x2 with x1 andx2 being thex–positions ofthe corresponding triple points. For a better illustrationthe geometrical meaning of the threequantitiesh0, h andh′ is shown in figure 11.

K

U

h0

K

U

h

U

K

h′

Figure 11: Illustration of the quantitiesh0, h andh′.

By comparing the film over topography with the plane film flow, the difference∆h :=h′ − h0, which is subsequently called the film elevation, is an adequate measure for retardationor acceleration of the material transport: For positive∆h the material transport is retarded, fornegative∆h it is accelerated.

By qualitative considerations we find two relevant aspects which lead to opposite effects:On the one hand the friction is increased in the film by the undulations due to the elongation ofthe solid–fluid interface, on the other hand the friction is reduced on th separatrix.

Quantitative results are shown by a parameter study in figure12: For three different flowrates, characterized by the associated reference thickness, the resulting film elevation is plottedversus the waviness. Note, that the influence of inclinationangle and capillary number isnegligible since the respective films are comparatively thick.

For small waviness the film elevation is monotonously increasing and follows a mastercurve, i.e. it does not depend on the flow rate in this regime. If the waviness exceeds the criticalvalue for the creation of the primary vortex, however, we finda flow–rate–depending surfaceelevation. By slightly increasing the waviness, each of thethree curves reaches a maximum.Beyond the maximum we observe a monotonous decrease due to the reduction of friction byflow separation. The film elevation remains always positive and tends to zero for high waviness.Thus, for creeping flows an optimal material transport is reached for either a perfectly smoothor an extremely undulated bottom.

6 Conclusions

By a detailed analysis we have demonstrated that for film flowsover wavy bottoms the con-ventional representation of the stream function as a separation solution in terms of an infiniteseries in Cartesian coordinates does exist only for weak undulations. On the other hand, the lu-brication approximation covers in local coordinates also situations outside its range of validity.For the treatment of film flows over strongly undulated bottomprofiles we have presented analternative method which avoids convergence problems by applying complex function theory.Since the equations derived in this way are ordinary differential equations and integral equa-tions for functions of one variable, the original 2–dimensional problem has been reduced to a

22

h0 = 20π

h0 = 3π

h0 = 2π

1 2 3 a 4 5 6

0.1

0.2

∆h

0.3

0.4

0.5

e

JJ]

vortexcreation

Figure 12: Resulting film elevation∆h vs. wavinessa for three different flow rates given bythe associated reference film thickness.

1–dimensional one. The solution of the equations, which is determined by means of Fourieranalysis and an iterative procedure using computer algebra, reveals vortex structures whichare far beyond the capability of lubrication approximation, other perturbation theories in localcoordinates and the separation solution in Cartesian coordinates.

Our method reveals the generation and evolution of the vortices depending on four param-eters, i.e. waviness, dimensionless film thickness, inclination angle and capillary number. Wehave shown that number and size of vortices increase with increasing waviness and film thick-ness. By an analysis of asymptotically thick films we determined a minimum waviness, i.e. aminimum radius of curvature, for vortex generation. By decreasing the film thickness one canforce the vortices to vanish even at higher undulations of the bottom. Our calculations are inbest agreement to experimental results.

By parameter study on the flow rate we have investigated in theinfluence of the bottom un-dulations and the vortices on the material transport. The main result is that the flow retardationreaches a maximum at a moderate waviness, whereas an optimalmaterial transport is reachedfor either a perfectly smooth or a highly undulated bottom.

Appendix A Evaluation of the no–slip condition in §2.3

Subsequently the no–slip condition is evaluated for a representation of the stream function interms of the infinite series (25). With Eq. (32) andQn = Rn = 0 for n < 0 the x– andz–component of the no–slip condition (13) read

2hb(x) − b(x)2 + 2ℜ

∞∑

n=0

[Qn − nRn − nb(x)Qn] exp (−nb(x)) exp(−inx) = 0 , (A.1)

−2ℑ∞∑

n=0

n [Rn + b(x)Qn] exp (−nb(x)) exp(−inx) = 0 . (A.2)

23

The above equations have to be fulfilled by an adequate choiceof the coefficientsQn, Rn

with n = 0, 1, · · · ,∞ for a given bottom shapeb(x). In the following we choose a sinusoidalbottom shape

b(x) = −a cos x . (A.3)

Due to the symmetryb(−x) = b(x) of the bottom shape the coefficientsQn andRn are real–valued. This can be proven by applying the parity operationx → −x on Eqs. (A.1) and (A.2)which result in

2hb(x) − b(x)2 + 2ℜ

∞∑

n=0

[Qn − nRn − nb(x)Qn] exp(−nb(x)) exp(−inx) = 0 , (A.4)

−2ℑ

∞∑

n=0

n[Rn + b(x)Qn] exp(−nb(x)) exp(−inx) = 0 . (A.5)

Obviously, the quantitiesQn, Rn fulfill the same equations asQn, Rn. Since the solutionof the linear equations (A.1) and (A.2) is unique, the coefficients must equal their complexconjugates, i.e.Qn = Qn andRn = Rn.

After inserting the harmonic bottom shape (A.3) into Eqs. (A.1) and (A.2) and making useof the formula (see e.g. [29])

exp (na cos x) =

+∞∑

m=−∞

Im(na) cos (mx) (A.6)

with Im being the modified Bessel functions of orderm the no–slip conditions can be writtenin terms of Fourier series. By Fourier analysis of the no–slip conditions (A.1, A.2) we obtainthe algebraic set of equations

∞∑

n=0

[

M1R0n Rn +M1Q

0n Qn

]

=a2

2, (A.7)

∞∑

n=0

[

M1RmnRn +M1Q

mnQn

]

= ahδm1 +a2

4δm2 , m ∈ N , (A.8)

∞∑

n=0

[

M2RmnRn +M2Q

mnQn

]

= 0 , m ∈ N , (A.9)

for the coefficientsQn andRn with the corresponding matrix elements being defined as

M1Rmn := −n [Im−n(na) + Im+n(na)] , (A.10)

M1Qmn := Im−n(na) + Im+n(na)

+na

2[Im−n−1(na) + Im−n+1(na) + Im+n−1(na) + Im+n+1(na)] , (A.11)

M2Rmn := n [Im−n(na)− Im+n(na)] , (A.12)

M2Qmn := −

na

2[Im−n−1(na) + Im−n+1(na)− Im+n−1(na)− Im+n+1(na)] . (A.13)

In the above set of equations the coefficientR0 is missing. For the determination ofR0 wecalculate the mean value of Eq. (16) which results in

R0 =a3

6−ha2

2− aQ0 −

∞∑

n=1

(−1)n [Rn + aQn] exp (−na) . (A.14)

24

Appendix B Fourier decomposition of complex boundary condi-tions

By means of the Fourier series representations (61) the boundary conditions in§3.2 are dis-cretizised.

First, after having inserted the series representations inthe no–slip condition (55), a Fourierdecomposition delivers the system

mrβm =1

2

(

qβm + qβ−m

)

+a

4

[

qβm+1 − qβm−1 − qβ−m−1 + qβ

−m+1

]

(B.1)

+ma

2

(

qβm+1 + qm−1

)

−ah

2

[

δm,1 + δm,−1 +a

2(δm,2 − δm,−2)

]

−a2

8

[

δm,2 + δm,−2 + 2δm,0 +a

2(δm,1 + δm,3 − δm,−3 − δm,−1)

]

,

which is valid for arbitrary integer numbersm = −∞, · · · ,+∞. This system of equations issupplemented by the mean–value condition (56) which reads in terms of the series representa-tions (61)

ℜ[

rβ0 −a

2

(

qβ1 + qβ−1

)]

+a2h

4= 0 . (B.2)

In analogy we derive from the kinematic boundary condition (54) the set

rϕm + rϕ−m + h

[

qϕm + qϕ−m

]

− i

+∞∑

l=−∞

(m− l)gm−l

[

qϕl + qϕ−l

]

(B.3)

−imh2gm −i

3

+∞∑

l,n=−∞

l(n− l)(m− n)glgn−lgm−n =

[

ψs −2

3h3

]

δm0 ,

wherem is again an arbitrary integer number. However, only for non–negative numbersm =0, 1, · · · ,∞ the set (B.3) consists of independent equations.

The third set of equations comes from the first integral (53) of the dynamic boundary con-dition. It takes the form

mrϕm +

(

mh−1

2

)

qϕm +1

2qϕ−m +

i

2

+∞∑

l=−∞

(l −m)gm−l

[

(l +m)qϕl + (l −m)qϕ−l

]

+cotα+ i

4

2gm +

+∞∑

l,n=−∞

l(n−l)(m−n)2glgn−lgm−n − i

+∞∑

l=−∞

(l−m)(3l−2m)glgm−l

+i

4CaSm = 0 (B.4)

wherem = −∞, · · · ,+∞ is an arbitrary integer number. The quantitiesSm are the Fouriercoefficients of the term

√

1 + g′′(x)2 in Eq. (53). In order to determine the coefficientsSm thequadratic system of equations

+∞∑

m=−∞

SmSl−m = δ0l +

+∞∑

m=−∞

m2(l −m)2gmgl−m (B.5)

has to be solved. The set of equations (B.4) is supplemented by the second mean–value condi-tion (57) which reads in terms of the series representations(61)

ℑrβ0 = 0 . (B.6)

25

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Authors’ address: M. Scholle, A. Wierschem and N. Aksel, Department of AppliedMechanicsand Fluid Dynamics, University of Bayreuth, D–95440 Bayreuth, Germany (E–mail: tms@uni-bayreuth.de)

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