Wave regimes and mass transfer in two-layer falling films

Transcription

1 Wave regimes and mass transer in two-layer alling ilms by GÖKÇEN ÇEKİÇ A thesis submitted to The University o Birmingham or the degree o Doctor o Philosophy School o Mathematics The University o Birmingham September 214

2 University o Birmingham Research Archive e-theses repository This unpublished thesis/dissertation is copyright o the author and/or third parties. The intellectual property rights o the author or third parties in respect o this work are as deined by The Copyright Designs and Patents Act 1988 or as modiied by any successor legislation. Any use made o inormation contained in this thesis/dissertation must be in accordance with that legislation and must be properly acknowledged. Further distribution or reproduction in any ormat is prohibited without the permission o the copyright holder.

3 Abstract In this thesis two-layer ilm lows down a vertical wall are studied. The integral method is used to derive an approximate system o evolution equations modelling long-wave lows o the ilm. Then amilies o nonlinear steady-travelling periodic waves are computed. Computed waves have qualitatively similar behaviour to those ound in one-layer ilms but the quantitative characteristics o the waves strongly depend on additional similarity parameters in the two-layer ilms. In particular the average location o the interace between the layers aects the biurcation scheme o the waves. To select the wave regimes which can be used to compare with experiments systematic transient computations have been carried out to create a map o the attracting wave regimes so-called dominating waves. For dominating wave regimes the mass transer problem or a weakly soluble gas is solved. In particularly the absorption problem is systematically studied.

4 Dedicated to three magniicent women in my lie: my grandmother mother and sister.

5 Acknowledgements Above all I would like to state my special thanks and appreciation to Dr. Grigory Sisoev or his splendid supervision and patience. I will be orever grateul to him or the time which he has spent educating me. A special thanks to my mother and sister or their love and encouragement. Words cannot express how indebted I am to them. The support o my mother and sister has enabled me to ulil my ambitions. I would also like to thank the rest o my amily who supported me to come abroad or doing PhD. I would like to thank my co-supervisor Pro. Yulii Shikhmurzaev or his ruitul discussions. I would also like to thank all o my riends and colleagues at the School o Mathematics at the University o Birmingham who made my time here interesting and enjoyable. I am very lucky to have good riends in Birmingham special and best riends in Turkey who make me eel like at home in Birmingham. I eel gratitude to all o them or cheering me up through all times. Finally I grateully acknowledge the inancial support o the IDB.

6 Contents 1 Introduction One-layer ilm lows Two-layer ilm lows Mass transer Evolution Equations Basic Equations Equations and boundary conditions Dimensionless variables Parameters and basic solution Linear stability o the waveless low Evolution equations New variables Approximate equations Integral method Linear stability Results Asymptotic behaviour o very long waves Equations iv

7 2.4.2 Fixed points Solutions in neighbourhood o a ixed point Periodic Steady-Travelling Waves Eigenvalue problem Results Dominating Waves Space-periodic solutions Results Mass Transer Problem statement Dimensional orm Dimensionless orm Approximate problem Velocity proile Diusion in regime o regular waves Numerical method Results Conclusion 115 A Numerical algorithm to solve the Orr-Sommereld problem 118 B Nonlinear system or Fourier coeicients o steady-travelling space-periodic waves 122 C Velocity o harmonics 125 v

8 D Solving diusion problem 127 List o Reerences 129 vi

9 List o Figures 1.1 Physical picture o two-layer low Wave velocities a) and ampliication actors b) in the case o the one-layer ilm at Re = We =.26 when δ =.2. The results do not depend on the value o H The stretching parameter κ 2 a) and the Reynolds numbers Re 1) and Re 2) b) at H =.6 or water-benzene system Dependencies o the stretching parameter κ 2 a) and the Reynolds number Re b) on the ilm parameter δ in the case water-benzene system at H = Dependencies o the stretching parameter κ 2 a) and the Reynolds number Re b) on the ilm parameter δ in the case water-benzene system at H = Dependencies o the stretching parameter κ 2 a) and the Reynolds number Re b) on the ilm parameter δ in the case benzene-water system at H = Wave velocities a) and ampliication actors b) in the case when the same liquid is in both layers at δ =.2 and dierent values o H vii

10 2.7 Wave velocity a) and ampliication actors b) or the surace mode in the case o water-benzene system at δ =.2 and H =.6. The solutions o the Orr-Sommereld problem and the integral method are shown by dashed curves and solid curves respectively Wave velocity a) and ampliication actors b) or the interace mode in the case o water-benzene system at δ =.2 and H =.6. The solutions o the Orr-Sommereld problem and the integral method are shown by dashed curves and solid curves respectively Wave velocity a) and ampliication actors b) or the surace mode in the case o water-benzene system at δ =.1 and H =.3. The solutions o the Orr-Sommereld problem and the integral method are shown by dashed curves and solid curves respectively Wave velocity a) and ampliication actors b) or the interace mode in the case o water-benzene system at δ =.1 and H =.3. The solutions o the Orr-Sommereld problem and the integral method are shown by dashed curves and solid curves respectively Neutral curves o the surace mode curves 1 and 2) and the interace mode 3 and 4) in the case o water-benzene system at H =.6. Curves 1 and 3 corresponds to solutions o the integral method and curves 2 and 4 denote solutions o the Orr-Sommereld problem Neutral curves o the surace mode curves 1 and 2) and the interace mode 3 and 4) in the case o water-benzene system at H =.3. Curves 1 and 3 corresponds to solutions o the integral method and curves 2 and 4 denote solutions o the Orr-Sommereld problem viii

11 2.13 Fixed points o the steady low in the case o a) water-benzene system and b) benzene-water system at H =.6. Curves 1 2 and 3 denote the biurcating ixed points o the irst type biurcating ixed points o the second type and waveless low respectively Fixed points o the steady low in the case o water-benzene system at H =.6. Curves 1 2 and 3 denote the biurcating ixed points o the irst type biurcating ixed points o the second type and waveless low respectively Ampliication actors λ r o small perturbations in the neighbourhood o the ixed point corresponding to the waveless low in the case o water-benzene system at H =.6 and δ =.1. Solid curves correspond to real values o λ and dashed curves correspond to real parts o complex conjugate values o λ Biurcation scheme o the wave amilies at δ =.1 and H = Examples o waves belonging to the amily γ 1 11 at s =.19 a) s =.7 b) and s =.9632 c). Panels a) b) and c) correspond to the points A B and C respectively in Fig Examples o waves belonging to the amily γ 2 21 at s =.18 a) s =.4 b) and s =.4616 c). Panels a) b) and c) correspond to the points D E and F respectively in Fig Examples o waves belonging to the amily γ at s =.74 a) s =.25 b) and s =.3125 c). Panels a) b) and c) correspond to the points G H and I respectively in Fig Projections o 6-dimensional phase trajectory o the amily γ+11 3 in 3- dimensional subspace h 1) h 1)) ) ) h 1) a) and h 2) h 2)) ) ) h 2) b) at s =.74 correspond to the panel a) in Fig ix

12 3.6 Biurcation scheme o the wave amilies at δ =.1 and H = Biurcation scheme o the wave amilies at δ =.1 and H = Projections o 6-dimensional phase trajectory o the amily γ 11 1 in 3- dimensional subspace h 1) h 1)) ) ) h 1) a) and h 2) h 2)) ) ) h 2) b) at s =.19 correspond to Fig. 3.2a Examples o waves belonging to the amilies γ 1 11 a) γ 3 21 b) and γ c) at s =.2 δ =.1 and H = Projections o 6-dimensional phase trajectory o the amily γ+11 2 in 3- dimensional subspace h 1) h 1)) ) ) h 1) a) and h 2) h 2)) ) ) h 2) b) at s =.2 correspond to the panel c) in Fig Maximum ilm thicknesses o biurcating amilies γ 11 1 γ 21 2 γ+11 3 at δ =.1 and H =.3. The ull and open circles correspond to results o transient computations Examples o waves belonging to the amily γ 11 1 at s =.9 a) amily γ 21 2 at s =.462 b) and amily γ+11 3 at s =.323 c) or the case corresponding to Fig Maximum irst layer thicknesses in the case shown in Fig Maximum ilm thicknesses o biurcating amilies γ 11 1 γ 21 2 γ+11 3 at δ =.1 and H =.6. Full and open circles show results o transient computations Maximum irst layer thicknesses in the case shown in Fig Dependency o the maximum ilm layer thickness on time or dierent values o s at δ =.1 and H = Dependency o the maximum irst layer thickness in the cases shown in Fig x

13 4.8 Examples o developed oscillating waves at s =.25 a) s =.5 b) and s =.9 c) at δ =.1 and H = Examples o waves or s =.5 at t = 1.5 a) and t = 6 b) at δ =.1 and H = Examples o waves at interace or s =.47 at t = 5.8 a) t = 51.4 b) and t = 52.4 c) at δ =.1 and H = Projections o the phase trajectories at δ =.1 H =.3 and s =.5 corresponding to Fig. 4.6 and 4.7 at planes o Fourier coeicients or the ilm layer a) and the irst layer b) Dependency o the maximum ilm layer thickness on time or dierent values o s at δ =.1 and H = Dependency o the maximum irst layer thickness in the cases shown in Fig Examples o developed oscillating waves at s =.3 a) s =.5 b) and s =.8 c) at δ =.1 and H = Examples o waves or s =.5 at t = 27.9 a) t = 3 b) and t = 31.8 c) at δ =.1 and H = Projections o the phase trajectories at s =.8 corresponding to Figs and 4.13 at planes o the irst Fourier coeicients or ilm layer a) and irst layer b) at δ =.1 and H = The interace and wall concentrations a) the average concentrations in the layers b) the local luxes at the surace c) and interace d) or the waveless low at δ =.1 and H =.3 in water-benzene system Global a) and local b) instant proiles o the gas concentration or the waveless low at H = xi

14 5.3 Instant proiles o the interace and surace a) the interace and surace concentrations b) the average concentrations in the layers c) the local luxes at the surace d) and interace e) at δ =.1 H =.3 and s =.6 in water-benzene system Instant proiles o the interace and surace a) and the local luxes b) at the inlet area in the case shown in Fig Global a) and local b) instant proiles o concentration at the inlet area in the case shown in Fig Instant proiles o the interace and surace a) and the local luxes b) at the outlet area in the case shown in Fig Instant proiles o concentration at the outlet area in the case shown in Fig Instant proiles o the interace and surace a) the interace and wall concentrations b) the average concentrations in the layers c) the local luxes at the surace d) and interace e) at δ =.1 H =.3 and s =.3 in water-benzene system Instant proiles o the interace and surace a) and the local luxes b) at the inlet area in the case shown in Fig Global a) and local b) instant proiles o concentration at the inlet area in the case shown in Fig Minimum and maximum values o λ 2) a) and λ 1) b) at H = Instant proiles o the interace and surace a) the interace and wall concentrations b) the average concentrations in the layers c) the local luxes at the surace d) and interace e) at H =.6 and s = Minimum and maximum values o λ 2) a) and λ 1) b) at H = xii

15 5.14 Instant proiles o the interace and surace a) the interace and wall concentrations b) the average concentrations in the layers c) the local luxes at the surace d) and interace e) at δ =.1 H =.6 and s =.5 at the initial conditions with C 1) y) = 1 C 2) y) = Instant proiles o the interace and surace a) and the local luxes b) at the inlet area in the case shown in Fig Instant proiles o concentration at the inlet area in the case shown in Fig xiii

16 List o Tables 2.1 Physical properties o water and benzene Parameters o water-benzene and benzene-water systems Complex conjugate and real values o λ at δ =.1 and H =.3 abc) and δ =.1 and H =.7 d) xiv

17 Chapter 1 Introduction Hydrodynamics o one-layer ilms and immiscible two-layer ilms are intensively studied as they hold practical interest or many applications in environmental lows. It is also a key element o technological processes in chemical engineering bioengineering pharmaceutical industry and others. In particular two-layer ilms are used in many technologies which acilitate mass and heat transer between two viscous liquids which have dierent physical properties. In previous theoretical studies o two-layer ilm lows linear and weakly nonlinear models are exploited due to the complexity o the nonlinear problem. In this thesis developed rom [8] a strongly nonlinear model is used to solve the hydrodynamical problem and diusion problem. Hydrodynamics o a two-layer ilm lowing down a vertical wall is investigated under the assumptions that liquids in the layers are viscous immiscible and incompressible. The superposition o the irst layer adjacent to the second layer indicates a liquid-liquid interace. A sketch o the studied low is demonstrated in Fig A review o the theoretical and experimental works relevant to the problem studied in the thesis is discussed in this chapter. Then the thesis is organized as ollows: In Chapter 2 the approximate evolution equations modelling lows at real-lie values o the 1

18 Figure 1.1: Physical picture o two-layer low. similarity parameters are derived. Linear stability o the waveless low is analysed in the ramework o the evolution equations and the generalized Orr-Sommereld problem to veriy the model. Ater this in Chapter 3 the method used to compute steadytravelling waves is given and examples o the waves are shown. Attracting low regimes are computed in numerous transient computations and these regimes are compared with ound steady-travelling waves in Chapter 4. Then in Chapter 5 mass transer in regimes o dominating waves is studied. Finally conclusions are provided in Chapter One-layer ilm lows Modelling two-layer ilm lows is based on theoretical models or one-layer ilms which have been developing or a ew decades starting rom seminal works [34] and [35]. For this reason we begin rom a short review o the theory developed or the one-layer ilms. Film low down a vertical wall at moderate low rates or alling ilms have been considered later in numerous experimental and theoretical investigations [3]. In experiments alling ilms demonstrate a wide variety o low regimes which are very sensitive to low conditions. The systematic experimental investigation irstly carried out in [35] demonstrates that there is some critical low rate at which waves on the liquid surace start to 2

19 appear or at least to be observed. I the low rate is less than the critical one the wave regime does not exist or cannot be measured since its small amplitude. In addition to this it was demonstrated that there is existence o two principal wave types: periodic sinusoidal waves and solitary waves each is travelling with its own constant velocity. These so-called regular waves can take on dierent shapes amplitudes and velocities depending on low conditions. [34] analysed linear and weakly nonlinear lows o a one-layer ilm down an inclined plane in the ramework o the boundary layer approximation unortunately a convective term was mistakenly omitted in the momentum equation). Linear stability analysis was irst ormulated comprehensively by [75] where the Orr-Sommereld problem [14] was solved by Yih with an expansion in powers o αre where Re is the Reynolds number and α is the wavenumber. Yih s numerical results showed that the low down a vertical plane is unstable or Re > 1.5. However it was proved in [5] that showed the neutral stability curve and values o the wave velocity were incorrect. In his revision o Yih s ormulation and method Benjamin [5] calculated the neutral stability curves and he obtained the critical Reynolds number Re cr = 5 cot θ)/6 where θ is the inclination angle. In particular it means that the vertical low is always unstable since Re cr = in this case. Nevertheless when Re is small the unstable waves ampliication actors become very small and that o wavelengths become very big. Benjamin s study provided to estimate the value o Re at which observable waves should irst develop on a vertical water ilm as well as the length and the velocity o the waves. Dependence o the neutral curves on the similarity parameters in the one-layer ilms was studied in [76] using the asymptotic method. In particular there are two important outcomes: a) The α = axis is always a part o the neutral curve b) When Re = 5 cot θ)/6 the biurcation point exists on α = or the neutral stability 3

20 curve. As mentioned above the principal approach o theoretical investigation based on use o a long-wave approximation was suggested in [34]. The majority o theoretical results used to describe experimental data were reached in the ramework o the Kapitza-Shkadov evolution equations derived in [52] by the integral method. This method was used to enable the consideration o the wave low o the liquid as a non-linear problem and it allowed to speciy the parameters o the wave regime amplitude velocity length and requency. In urther theoretical papers it has been ound that the Kapitza-Shkadov model possesses a two-parametric maniold o solutions in the orm o steady-travelling periodic waves and this maniold includes wave regimes observed in experiments [65 53]. The character o branching periodic steady-travelling wave solutions was analysed in the pioneering work [6] and urther publications see [9] where the review can be ound. Stability and biurcations o the wave amilies were studied in [ ]. Studies o evolution o waves on a vertical alling ilm were carried out in many works see [1] and reerences in there. In [ ] the amilies o the steady-travelling space-periodic waves and their biurcations were revealed. In addition to these analysis periodic waves in phase space and their mechanical meaning were discussed in [55 57]. The maniold o steady-travelling space-periodic waves has nonuniqueness o solutions when more than one solutions exist at given values o the similarity parameters and the wavenumber. To use these solutions to describe experimental data it was necessary to identiy theoretical waves observed in experiments. To select such wave regimes theoretically two principal approaches have been used. The irst approach is to study the linear stability analysis o nonlinear waves see [67 69] where stable wave regimes or regimes with minimum ampliication actor are ound or the comparison with experiments. Another approach see [ ] is to ind attractors o the evolution equations. The latter 4

21 method has led to discovering so-called dominating waves such that each is a solution with astest velocity and largest amplitude among all steady-travelling space-periodic waves existing at given values o the similarity parameters and the wave requency. Attracting oscillating regimes were also computed i the wavenumber is in some neighbourhood o old biurcations o a amily o steady-travelling waves. This approach led to a map o dominating waves [64 63] corresponding to the experimental data [63 53]. Since Kapitza & Kapitza s experiments [35] or one-layer ilms it has been known that it is impossible to observe steady-travelling periodic waves whose wave lengths exceed some critical value. In other words existing o maximum inite spacing between principal humps in a sequence o quasi-solitary waves was showed experimentally. It was seen that the space between the humps decreases owing to the hydrodynamic instability or small values o wavenumber and the critical wavelength was computed in [54]. For many years the Kapitza-Shkadov model [52] has been used as the main tool to explain experimental data and provides modelling possible low regimes in one-layer ilm lows despite deriving other models. We just note that there are another approaches to the problem. For example in [48 49] weighted-residual method combined with standard long-wavelength expansion was used or ilm lows down inclined planes. The inite element method was also used to analyse the ull Navier-Stokes equations in [5] and comparisons between the long-wave expansion and inite element method or increasing Reynolds number were given or large and small amplitude waves. Existing knowledge o the wavy low o alling thin liquid ilms was summarized in monographs [3] and [9] where numerous examples o lows o one-layer alling ilms were given. 5

22 1.2 Two-layer ilm lows Studies on the one-layer ilm low discussed in Section 1.1 was extended in [32] to the case o low o a heterogeneous system which consists o two layers o viscous incompressible and immiscible liquids. Long wave instability o the two-layer ilm low was studied in this paper in the case o equal dynamic viscosities and dierent densities o the liquids. It was indicated that there are eects o the existence o the ilm surace and the interace on the hydrodynamic stability o the system. Kao analysed these eects with respect to surace disturbances and shear waves. Furthermore a relation between the wavenumber α and the Reynolds number Re or given ratios o the densities o the liquids depth rate and inclined angle was obtained. The axis α = in the α Re)-plane is part o the neutral stability curve and it shows that the neutral oscillation can exist when Re = similar to the case o the one-layer ilm. The biurcation point o this curve on α = was ound. The critical Reynolds number or the two-layer low where the density o the upper liquid is more than that o the lower liquid was calculated. It was also obtained that the stratiication may lead to stability or instability depending on whether the density o the second layer is less or higher than that o the irst layer. Finally surace tension has a stabilising inluence or long waves and small Reynolds numbers. The stability o low down an inclined plane under the role o the interace mode was also analysed in [31]. This mode is signiicant or the governing o the low. When the ratio ρ 2) /ρ 1) where ρ 1) and ρ 2) are densities o the irst liquid adjoin to the wall and the second liquid with the ree surace respectively is small especially 1/1 interace and surace modes o instability were showed to compete each other to control the low. In particular i ρ 2) /ρ 1) =.1 at h 2) h 1) )/h 1) < 1.6 where h 1) and h 2) are thicknesses o the irst layer and the ilm respectively the stability is governed by the surace mode but or h 2) h 1) )/h 1) > 1.6 the stability is governed by the interace mode and the 6

23 similar results at ρ 2) /ρ 1) =.3 were demonstrated. The general case o two modes in low with viscous stratiication was presented in [33] using the same method. It was ound that the surace mode has a aster velocity than the interace mode in the long wave estimation. The interace instability happens under a critical Reynolds number or the surace mode. For instance when Re the interace mode exists. The interace mode was named as inertialess instability because o the inertialess character o the destabilisation. Linear stability o two-layer low down an inclined plane was numerically investigated in the ramework o the Orr-Sommereld problem in [62]. Two unstable modes associated with the ree surace and the interace were computed at moderate values o the Reynolds number. It was shown that the modes have dierent physical meaning and compete each other depending on values o the similarity parameters. In examples it was shown that the interace mode demonstrates the property o the Rayleigh-Taylor instability [14] depending on the ratio o the liquids densities in the case o small wall inclination and the eect o the density ratio depends on the value o the inclination. On the other hand the interace mode is unstable i the less viscous liquid is in the region between the wall and interace and it is stable in the opposite case. [38] analysed the limit case o zero value o the Reynolds number and absence o the surace and interace tensions by using asymptotic method and it has been demonstrated that it is not suicient to investigate the behaviour o the system at all small wavenumbers i a consideration o the wavelength o the unstable mode is necessary. A signiicant conclusion o this paper is that when the less viscous layer is contiguous to the inclined wall the low is unstable at zero value o the Reynolds number. The most reinorced wave was demonstrated to have a wavelength comparable to the second layer thickness. In addition to this the interace mode is unstable and this instability is owing to a stratiication in the liquids viscosity. The low was shown to be stable to all disturbance 7

24 wavelengths at zero-reynolds when the more viscous luid is next to the inclined plane. The calculations o this paper was extended in [11] to the case o thicker second layers or zero-reynolds number lows. The Orr-Sommereld problem was used to investigate the interace and surace modes in the cases when both interace and surace tensions were zero or the only surace tension was zero. The roles o the inertia and viscous stratiication were illustrated with plots o the ampliication actors and neutral curves. When the lower layer is less viscous the stability o both the surace and interace modes can not occur simultaneously and wavy motion resulting rom the instability can happen at any Reynolds number. For the contrary coniguration with the more viscous luid contiguous to the wall under the certain conditions the low can become linearly stable. [26] studied on the inertialess linear interace instability o inclined two-layer ilm lows at small inclination angle. The inertialess interace instability under the same zero- Reynolds number approximation as [38] was analysed under neglecting the surace tension eects and the density stratiication eects through a linear temporal approach were studied. It was demonstrated that increasing the density rate ρ 2) /ρ 1) > 1 ) has a destabilising inluence but decreasing the density rate rom 1 ρ 2) /ρ 1) < 1 ) has unsubstantial inluence. Furthermore the spatio-temporal nature absolute or convective) o the instability o the two-layer ilm low and where the stability is convective were investigated. The study o the properties o the spatially ampliying waves in the region o parameters was ocused in [26 27]. The eect o inertia on the temporal and spatiotemporal instabilities o the two-layer ilm low was also studied in [27] and the eects o inertia and density stratiication on both interace and surace modes were characterized or dierent depth and viscous ratios. A transition rom convective to absolute instability [51] was appointed or the wall s small inclinations and or increasing inertia various depth density and viscosity ratios and the results compared to experimental data. In particular it was detected that the absolute 8

25 instability in the two-layer low is associated with he Rayleigh-Taylor instability [14] o the second mode as previously shown in [62]. The absolute instability appears in ρ 2) > ρ 1) in the case o relatively small inclination angles. The eect o viscosity stratiication in two-layer lows was studied in [29] to designate the physical mechanism o suppression o the inertialess instability by using a kinetic energy balance approach. The mechanism o the long waves on the surace and interace at zero and very low values o the Reynolds number was discussed in this study. They also ound that the shear stress at the unperturbed ree surace is necessary to cause the inertialess interacial instability. In [18] an explanation o the mechanism or the long wave inertialess instability o a two-layer low which was extended the study o [66] or a one-layer ilm was clariied. The mechanism was represented by examining the longitudinal perturbation velocity correlated with the surace and interace variations. It was shown that the velocity is stated with the composition o three parts; related to the shear stress at the ree surace the continuity case at the interace and the pressure induced by gravity. Another relevant area o research is the interace instability between two viscous lows. This type o instability was irst studied in [77] by the asymptotic method [76] in the case o the plane Couette-Poiseulle low. In a speciic case o liquids with equal densities it was shown that the low is unstable or any small value o the Reynolds number and the instability is supplied by either the moving boundary or the pressure gradient. In [24] a parallel low o two viscous liquids o equal density in ininite domains divided by a lat interace was studied and it was shown that there exists a short wave instability in the absence o the surace tension. This mechanism o instability is comparatively small and it can be stabilised by the surace tension. As illustrated in [22] the surace tension at the interace should be unrealistically small to observe the interace instability in the unbounded stratiied Couette low. In the case when one o liquids is bounded by a wall 9

26 and another liquid is unbounded there were ound a long wave interace instability in [23]. In [25] a weakly nonlinear equation modelling the plane Couette-Poiseulle low o two liquids was derived and some examples o wave evolution were computed. A special case o the two-layer Couette low with high dynamic viscosities ratios was modelled in [45] using an evolution equation derived by the integral method. The integral method was also used to model non-linear solitary waves in two-layer plane lows driven by the gravity [58] and the pressure [59]. Some attention was also paid to stability o interace waves in [44] where a gas-liquid waves were studied. Solitary and periodic waves in an interace between two liquids were observed in a cylindrical Couette low at high ratio o the dynamic viscosities [17] and in microchannel [79]. In [3] enhancement and suppression o instability in a two-layer ilm low on a vertical wall was studied theoretically. The results rom this paper show that the instability in a two-layer ilm could be suppressed by oscillation o suitable requency and amplitudes to the wall although the stability could be occurred only in particular low parameters. Except rom these parameters the instability is enhanced. In this area the wave number and the ampliication actors o wave raised considerably. These results can be carried with to improve the mass and heat transer rom the ilm. At small Reynolds numbers spatio-temporal instability o two-layer ilm was analysed in [7]. In this study the low has neutral stability in the limit o zero Reynolds number despite the past studies have showed the low is unstable in the same limit. New mechanisms o instability related to the convective nature o the disturbance were studied. [4] analysed the linear stability o two-layer shear-thinning ilm liquids and studied on the our parameter Carreau inelastic model [7]. It was ound that when the viscosity is stronger in the second layer the changes on the shear-thinning properties in this layer does not strongly aect the stability. On the contrary the shear-thinning properties aect the long-wave surace instability and the short and long-wave interace instabilities in the 1

27 dierent cases. [71] examined the van der Waals interacial instability o two superposed thin layers o luids on a vertical substrate. The author arrived the evolution equations by using lubrication theory and classiied the state o the system into our dierent cases: irst layer rupture second layer rupture double layer rupture and mixed layer rupture. The mixed layer rupture is qualitatively dierent rom the double layer rupture. The double layer rupture in which the two luids rupture together and orm compound droplets on the surace o the substrate. The mixed layer rupture where the second layer ills in the gap created when the irst layer goes to rupture. 1.3 Mass transer In this section some selected works on mass transer in lowing liquid ilms are reviewed. Mass transer in one-layer alling ilms have been intensively studied or decades see monograph [36] and review [37]. The researches have been stimulated by numerous applications in particular there is a principal problem o gas absorption by lowing liquid ilm. As has been revealed in experiments [ ] wavy ilms demonstrate larger capacity to absorb a gas in comparison with theoretical predictions or the waveless low and this result has been measured or ilms lowing in dierent regimes. In particular the wave requency aects the developing wave regime and thus the rate o gas absorption. The ampliication o the absorption was ound in many other types o lows or example a cocurrent ilm-gas low [39 4] and in horizontal wavy ilms [41 19]. The gas absorption and its ampliication in wavy ilms have been also studied. The penetration theory [21] and the theory based on surace renewal [12] could not explain this phenomenon. The absorption problem or the waveless low was solved in [46]. A ew approaches used simpliied models to solve the mass transer problem in wavy 11

28 regimes. Kapitza s solution [34] was used in [47] and an experimentally observed waves were used in [72] to model the gas absorption. Systematic computations o the absorption on wavy regimes in alling ilms were carried out in [6 43] or a alling ilm and in [61] or close problem o low over a spinning disk. The main result is that there is a strong dependence o the absorption rate on the wave requency wavenumber) and this is due to nonuniorm distribution o a local gas lux along a wave with maximum at troughs o the wave. A numerical simulations were carried out or a vertical alling ilm in [74] and the numerical experiments were compared with the previous ones. The review o the relevant literature can be also ound in [2] where the absorption problem was solved by Volume-o-Fluid Method where results o [6 43] were conirmed. 12

29 Chapter 2 Evolution Equations 2.1 Basic Equations In this section the Navier-Stokes equations [1 16] and relevant boundary conditions describing a two-layer ilm lowing down a vertical wall are given. Then the dimensionless variables are deined and the system are made dimensionless. The basic solution describing the waveless steady low is ound and properties o this solution in the phase space are studied Equations and boundary conditions We consider a liquid ilm lowing down a vertical wall. The ilm consists o two layers o liquids and the layer attached to the wall will be reerred as the irst layer and the layer with the ree surace will be reerred as the second layer. We assume that both liquids are incompressible viscous and immiscible. To describe the low the Cartesian coordinate system x ỹ) is introduced such that the x-axis coincides the wall and points downstream and the ỹ-axis points to the liquid bulk. Then the ull Navier-Stokes system is written in the orm 13

30 Liquid Density Viscosity Surace tension g/cm 3 cm 2 /s g/s 2 Water Benzene Table 2.1: Physical properties o water and benzene. ũ x + ṽ ỹ = ũ t + ũ ũ x + ṽ ũ ỹ = 1 ) p 2ũ ρ j) x + νj) x + 2 ũ + g 2 ỹ 2 ṽ t + ũ ṽ x + ṽ ṽ ỹ = 1 ) p 2ṽ ρ j) ỹ + νj) x + 2 ṽ 2 ỹ 2 ỹ = : ũ = ṽ = ỹ = h 1) x t ) : ỹ = h 2) x t ) : h 1) t 1) h + ũ x = ṽ [ p nn] σ1) ς 1) = [ p nτ ] 2 1 = [ũ] 2 1 = [ṽ]2 1 = h 2) t 2) h + ũ x = ṽ p nn σ 2) ς 2) = p nτ = 2.1) where ũ x ỹ t ) and ṽ x ỹ t ) are the velocity components in x and ỹ directions respectively p x ỹ t ) is the pressure h x 1) t ) and h x 2) t ) are the thicknesses o the irst layer and the ilm respectively. The notation [] 2 1 2) 1) denotes the jump in quantity rom the value in the irst liquid 1) to the value in the second liquid 2). The problem includes the ollowing dimensional parameters: the densities ρ 1) and ρ 2) viscosities ν 1) and ν 2) o the irst and second liquids and the surace tension σ 1) at the interace and the surace tension σ 2) at the ilm surace. Also the gravity g is contained in the system. As an example o the similarity parameters o the two-layer ilm we consider water and benzene whose physical properties are summarized in Table 2.1. The surace tension 14

31 between water and benzene is σ 1) = 33.6 g/s 2. The boundary conditions at the interace and the surace in 2.1) contain the normal stresses p nn = p + 2ρν 1 + ) 2 h x 1 ) 2 h ũ x x + ṽ ỹ h ũ x ỹ + ṽ ) x the tangential stresses p nτ = ρν 1 + ) 2 h x 1 1 h x ) 2 ũ ỹ + ṽ ) + 2 h ṽ x x ỹ ũ ) x and the curvatures ς = 1 + ) 2 h x h x 2 where h = h 1) or h = h 2) and the values o the corresponding liquids are used. The problem 2.1) has a solution describing the steady waveless low denoted by capital letters: y [ H ] 1) : Ũ 1) ỹ) = g {[ ρ 2) ν 1) ρ H 2) + 1) Ṽ 1) ỹ) = ) 1 ρ2) ρ 1) H 1) ] ỹ 1 2ỹ2 } 2.2) P 1) ỹ) = Q 1) H 1) Ũ 1) dỹ = g ) [ ) 2 2 H1) 2ν 1) 3 ρ2) ρ 1) ] H 1) + ρ2) 2) H ρ 1) 15

32 [ y H1) H ] 2) { [1 : Ũ 2) ỹ) = g ) ] 1 + ν2) ρ2) ν 2) ) 2 H1) ν 2) 2 ν 1) ρ 1) ν 1) ) 1 ρ2) ν 2) H 1) ρ 1) ν 1) H2) + H 2) ỹ } 2ỹ2 1 Ṽ 2) ỹ) = P 2) ỹ) = Q H 2) H 1) Ũ 2) dỹ { = g H2) ν H ) 1 1) 2) ) 2 3 ρ2) ν 2) ρ 1) ν 1) ν 2) ν ρ2) ν 2) 1) ρ 1) ν 1) } H 1) H2) + 1 ) 2 H2) 3 ) ) 2 H1) where Q 1) and Q are the low rates o irst and second layers respectively. The irst layer thickness H 1) and the second layer thickness H 2) H 1) are constants. The total low rate o the ilm is Q 2) Q 1) + Q = [ 1 3 H 2) { g ν 2) 2 ) 1 ρ2) ν 2) ρ 1) ν 1) ν 2) ν 3 ) ρ 2) ν 2) 2 H1)) H 2) 1 ) 1) 2 ρ 1) ν 1) 3 H 1) + ] } ) 2 H 1) H2) Dimensionless variables I the length scale is chosen as H c = H 2) and the velocity scale is denoted as U c and deined below then new dimensionless variables can be introduced: t = H c U c t x = H c x ỹ = H c y 2.3) ũ = U c u ṽ = U c v p = ρ 2) U 2 c p h 1) = H c h 1) h 2) = H c h 2). 16

33 Using the dimensionless variables the equations and boundary conditions 2.1) can be rewritten in the ollowing orm u x + v y = u t + u u x + v u y = 1 ) p ρ j) x + νj) 2 u Re x + 2 u 2 y 2 v t + u v x + v v y = 1 ) p y + νj) 2 v Re x + 2 v 2 y 2 ρ j) y = : u = v = y = h 1) x t) : y = h 2) x t) : h 1) t + 1 Fr 2 + u h1) x = v [p nn] σ We ς1) = [p nτ ] 2 1 = [u] 2 1 = [v]2 1 = h 2) t + u h2) x = v p nn 1 We ς2) = p nτ = 2.4) where [ p nn = 2ρj) ν j) 1 + Re [ p nτ = ρj) ν j) 1 + Re ς = [ 1 + ) ] h x ) ] 2 1 [ h 1 x ) ] 2 1 [ h 1 x 2 h x 2 ) ) 2 h v x y h u x y + v ) ] p x ] ) ) 2 u h x y + v ) x + 4 h v x y or h = h 1) or h = h 2). The system contains 6 similarity parameters: Re = U ch c ν 2) ρ = ρ1) ρ 2) We = ρ2) U 2 c H c σ 2) Fr 2 = U 2 c gh c ν = ν1) ν 2) σ = σ1) σ 2) and additional notations: ρ 2) = 1 ρ 1) = ρ ν 2) = 1 and ν 1) = ν. For example Table 2.2 shows values o the similarity parameters ρ ν and σ which 17

34 System ρ ν σ Ka Water 1) Benzene 2) Benzene 1) Water 2) Table 2.2: Parameters o water-benzene and benzene-water systems. only depend on the properties o liquids in water-benzene and benzene-water systems. In the dimensionless orm the steady low 2.2) is y [ H] : U 1) y) = Re ν Fr 2 V 1) y) = P 1) y) = Q 1) = H U 1) dy = y [H 1] : U 2) y) = Re Fr 2 V 2) y) = P 2) y) = Q = 1 H U 2) dy ) a 1) y y2 2.5) 2 Re [ 1 2 2ν Fr 2 H2 + ρ a 2) + y y2 2 ) 3 1 ρ ) ] H = Re [ H) Fr ) 1 H + ρ ν ) ] H 2 2ν ρ ν where the coeicients have been introduced: a 1) = 1 ) + 1 1ρ H ρ 1 + a 2) ν = 1 2ν ρ ν ) H 2 + ) 1 1 H. ρ ν 18

35 The total low rate in the ilm is Q 2) = Q 1) + Q = φ Re Fr 2 where [ φρ ν H) = H ν ρ 3 1 ) ρ [ H) ρ ν 3 ] H ) H ) ] H 2. 2ν ρ ν I the velocity scale U c is chosen as the average velocity then Q 2) = 1 and thus φ Re 2 = ) Fr That relation allows us to calculate the velocity scale U c = φgh2 c ν 2) Parameters and basic solution Formula 2.6) allows us to eliminate the Froude number Fr 2 = φre. On the other hand low depends on the irst layer thickness H determining low rates in the layers. Thus the problem 2.4) contains the ollowing 6 similarity parameters: Re = U ch c ν 2) ρ = ρ1) ρ 2) We = ρ2) U 2 c H c σ 2) ν = ν1) ν 2) σ = σ1) H. σ 2) 19

36 We can also use another set o the parameters: Ka σ 2) ρ 2) g ν 2) ) 4) 1 3 δ 1 45 ν 2) ) 2 ) ρ 2) g 4 Hc σ 2) ρ = ρ1) ρ 2) ν = ν1) ν 2) σ = σ1) H σ 2) where the Reynolds number Re and the Froude number Fr have been replaced or the Kapitza number Ka depending on gravity and the properties o the second liquid only and δ is the ilm parameter. Values o the Kapitza number or benzene and water are given in Table 2.2. Then we can calculate the dimensional scales: H c = U c = φ [ 45δ) 3 ν 2)) 6 σ 2) ρ 2) g 4 ] 1 11 [45δ) 6 g3 ν 2) σ 2)) 2 ρ 2) ) 2 Hence the Reynolds number and the Weber number are expressed by the δ and Ka as ollows: ] 1 11 Re = φgh3 c ν 2) ) 2 = φ 45δ) 9 11 Ka 3 11 We = φ2 ρ 2) g 2 H 5 c σ 2) ν 2) ) 2 = φ2 45δ) Ka 6 11 Let s calculate the inverse transormation rom Re and We to δ and Ka. First Ka 3 Re 11 = φ 45δ)

37 and so the Weber number is expressed as δ) 11 We = φ Ka 6 11 = φ 2 45δ) φ 45δ) 9 11 Re ) 2 = φ4 45δ) 3 Re 2. Thereore we calculate δ and then Ka: δ = 1 45 Re 2 We φ 4 ) 1 3 Ka = Re 5 φ ) 1 3. We The dimensionless orm o the steady low 2.5) is rewritten as y [ H] : U 1) y) = 1 φν V 1) y) = P 1) y) = Q 1) = H y [H 1] : U 2) y) = 1 φ V 2) y) = P 2) y) = Q = 1 ) a 1) y y2 2.7) 2 U 1) dy = H2 2φν a 2) + y y2 2 U 2) dy [ ρ ) 3 1 ρ ) ] H H = 1 H φ [ ) 1 H + ρ ν ) ] H 2. 2ν ρ ν To analyse lows in the layers we can also introduce the Reynolds numbers in each layer as ollows: Re 1) Q 1) ν 1) Re Q1) = Re 2) Q = Re Q. ν ν 2) 21

38 2.2 Linear stability o the waveless low In this section we ollow [62]. To analyse the linear stability o the waveless low the eigenvalue problem is ormulated. Then the numerical method in order to solve the eigenvalue problem is examined and numerical results are presented at the end o this section. To investigate linear stability o the steady low 2.7) a non-steady solution o 2.4) is looked or in the orm ux y t) = Uy) + ûx y t) vx y t) = ˆvx y t) px y t) = ˆpx y t) h 1) x t) = H + ĥ1) x t) h 2) x t) = 1 + ĥ2) x t) where cups denote small perturbations. Ater substituting in 2.4) and linearising we arrive at û x + ˆv y = û t + U û x + U ˆv = 1 ˆp ρ j) ˆv t + U ˆv x = 1 ρ j) ˆp y + νj) Re y = : û = ˆv = x + νj) Re 2û x + 2 û 2 y 2 2ˆv x + 2ˆv 2 y 2 ) ) y = H : ĥ1) + U ĥ1) t x = ˆv [ ] 2 U ĥ 1) + û = [ˆv] 2 1 = 1 [ ˆp + 2ρj) ν j) ˆv Re y U ĥ1) x [ ρ j) ν j) û y + ˆv x + U ĥ 1) )] 2 1 )] 2 = 1 + σ 2 ĥ 1) = We x 2 22

39 y = 1 : ĥ2) + U ĥ2) t x = ˆv ) ˆp + 2 ˆv Re y U ĥ2) x û y + ˆv x + U ĥ 2) =. 1 2 ĥ 2) = We x 2 where the primes denote derivatives with respect to y. Then solution in the orm o normal mode is used ) ûx y t) ˆvx y t) ˆpx y t) ĥ1) x t) ĥ2) x t) = ŭy) vy) py) h 1) h ) 2) exp iα x ct)) where α is the x-wavenumber and c = c r + ic i here and below the lower index r denotes the real part o the corresponding variable and the lower index i denotes its imaginary part) is the complex wave velocity to be ound and the sign o the imaginary part c i indicates the stability c i < or instability c i > o the steady waveless low. System or the amplitude unctions is iαŭ + v = iαu c)ŭ + U v = iα iαu c) v = 1 ρ j) ρ j) y = : ŭ = v = p + νj) Re p + νj) Re v α 2 v) ŭ α 2 ŭ ) 23

40 y = H : y = 1 : iαu c) h 1) = v [ ] 2 U h1) + ŭ = [ v] 2 1 = 1 [ p + 2ρj) ν j) Re [ ρ j) ν j) ) ] 2 v iαu h1) ŭ + iα v + U h1) 1 )] 2 = iαu c) h 2) = v p + 2 ) v iαu h2) + α2 Re We h 2) = ŭ + iα v + U h2) =. 1 α2 σ We h 1) = Eliminating perturbation o the longitudinal velocity rom the continuity equation ŭ = i α v and the pressure perturbation rom x-axis momentum equation [ p = ρj) ν j) α 2 Re v ρ j) i α ] U c) + νj) Re v + iρj) α U v and perturbation o the interace thickness rom kinematic condition at the interace iα U H) c) h 1) = v H) and perturbation o ilm thickness rom the kinematic condition at the ree surace iαu1) c) h 2) = v1) leads to the ollowing eigenvalue problem or perturbation o the y-axis velocity compo- 24

41 nent: v 2α 2 v + α 4 v iαre [ U c) v α 2 v) U v ] = 2.8) ν j) y = : v = v = y = H : [ v] 2 1 [ v = U ] 2 U c v = 1 [ { ) ρ j) ν j) v 3α 2 + iαre U c) v + ν j) ) } ] iαreu + 2α2 U 2 v ν j) U c 1 [ { ρ j) ν j) v + α 2 U ) }] 2 v = U c 1 y = 1 : v + α 2 U ) v = U c v [ 3α 2 + iαreu c) ] v [ iα α 2 Re + U c We + iαre + 2α2 U c iσ α 3 Re v = WeU c) ) U ] v =. The problem 2.8) is a generalized Orr-Sommereld problem studied in [62]. To solve 2.8) numerically the dierential sweep method is applied. I new variables w 1 = v w 2 = v w 3 = v w 4 = v are introduced then the problem is rewritten in the orm dw 1 dy = w dw 2 2 dy = w dw 3 3 dy = w 4 2.9) dw 4 dy = 2α2 w 3 α 4 w 1 + iαre [ ] U c)w3 α 2 w 1 ) U w 1 ν j) y = : w 1 = w 2 = 25

42 Figure 2.1: Wave velocities a) and ampliication actors b) in the case o the one-layer ilm at Re = We =.26 when δ =.2. The results do not depend on the value o H. y = H : [w 1 ] 2 1 [w = 2 U ] 2 U c w 1 = 1 [ { ) ρ j) ν j) w 4 3α 2 + iαre U c) w 2 + [ ρ j) ν j) iαreu ν j) { w 3 + y = 1 : w 3 + α 2 U U c ν j) ) } ] + 2α2 U 2 w 1 U c 1 α 2 U U c ) w 1 = w 4 [ 3α 2 + iαreu c) ] w 2 [ iα α 2 Re + U c We + iα3 σ Re WeU c) w 1 = ) w 1 }] 2 1 = iαre + 2α2 U c ) U ] w 1 =. The numerical algorithm to solve the eigenvalue problem 2.9) is given in Appendix A. To test the code calculations have been carried out in the case o a one-layer ilm see Fig. 2.1 showing the wave velocities and ampliication actors. In particular results do not depend on value o the irst layer thickness as expected. Numerical solutions o 2.9) are used in Section below to veriy an approximate model derived in Section

43 2.3 Evolution equations In this section the evolution equations approximating the ull Navier-Stokes problem are derived. To do this irstly an approximate long-wave model is ormulated. Then the integral method is applied to state the evolution equations New variables To describe the long wave motions new variables are introduced as x κ = κx t κ = κt v κ = v κ 2.1) where κ is a stretching parameter deined below. Then the system 2.4) is rewritten in the orm u x κ + v κ y = u + u u u + v κ t κ x κ y = 1 κ 2 vκ + u v κ v κ + v κ t κ x κ y y = : u = v κ = y = h 1) x κ t κ ) : y = h 2) x κ t κ ) : h 1) t κ ρ j) ) p + νj) x κ κre κ 2 2 u x 2 κ = 1 p ρ j) y + κ2 ν j) κre ) + 2 u + 1 y 2 κφre ) κ 2 2 v κ + 2 v κ x 2 κ y 2 + u h1) x κ = v κ [p nn ] κ2 σ We ς1) κ = [p nτ ] 2 1 = [u] 2 1 = [v κ] 2 1 = h 2) t κ + u h2) x κ = v κ p nn κ2 We ς2) κ = p nτ = 2.11) 27

44 where [ p nn = p + 2κ2 ρ j) ν j) ) ] 2 1 h 1 + κ 2 κre x κ [ ) ) 2 h 1 κ 2 v κ x κ y h u x κ y + v ) ] κ2 κ x κ [ p nτ = ρj) ν j) ) ] 2 1 h 1 + κ 2 ς κ = Re [ [ 1 κ 2 h x κ 1 + κ 2 h x κ x κ ) ) 2 u ) 2 ] 3 2 y + κ2 v κ x κ 2 h x 2 κ ) ] + 4κ 2 h v κ x κ y with h = h 1) at j = 1 and h = h 2) at j = 2. The capillary waves in liquid ilms appear in the case o a balance o gravity viscosity and capillarity that can be written in the orm as 1 κφre = κ2 We. Here the let-hand side is the gravity term in the irst momentum equation. At the same time this term express the balance o the viscosity and gravity leading to the existence o the waveless low. The right-hand side shows the scale o capillarity at the ilm surace. This equation allows us to ind the stretching parameter κ: κ = ρ 2) gh 2 c σ 2) ) 1 3 ) 45δ) = Ka 3. It is worthy to note that the parameter κ can be written by using the capillary number 28