A Numerical Study of Thin Liquid Film Flow over a Topographically Patterned Rotating Cylinder

National Institute of Technology Rourkela

A Numerical Study of Thin Liquid Film Flow over a Topographically

Patterned Rotating Cylinder

Shaleka Agrawal

Department of Chemical Engineering

National Institute of Technology Rourkela

A Numerical Study of Thin Liquid Film Flow over a Topographically Patterned

Rotating Cylinder

Dissertation submitted in partial fulfillment of the requirements of the degree of

Master of Technology by Research

in

Chemical Engineering

by

Shaleka Agrawal

(Roll Number: 614CH3004)

based on research carried out under the supervision of Dr. Akhilesh Kumar Sahu

Assistant Professor

January, 2017

Department of Chemical Engineering

Department of Chemical Engineering

National Institute of Technology Rourkela

June, 2017

Certificate of Examination

Roll Number: 614CH3004 Name: Shaleka Agrawal

Title of Dissertation: A Numerical Study of Thin Liquid Film Flow over a Topographically Patterned Rotating Cylinder

We the below signed, after checking the dissertation mentioned above and the official record book of the student, hereby state our approval of the dissertation submitted in partial fulfillment of the requirements of the degree of Master of Technology by Research in Chemical Engineering at National Institute of Technology Rourkela. We are satisfied with the volume, quality, correctness, and originality of the work.

Dr. Hara Mohan Jena Dr. Akhilesh Kumar Sahu

Member, MSC Principal Supervisor

Dr. Soumya Sanjeeb Mohapatra Dr. Bibhuti Bhushan Nayak

Member, MSC Member, MSC

Dr. Basudeb Munshi Dr. Gaurav Chairperson, MSC IIT Roorkee

External Examiner

Dr. R. K. Singh Head of the Department

Department of Chemical Engineering

National Institute of Technology Rourkela

Prof. Akihlesh Kumar Sahu Assistant Professor

June, 2017

Supervisors’ Certificate

This is to certify that the work presented in the dissertation entitled A Numerical Study of Thin Liquid Film Flow over a Topographically Patterned Rotating Cylinder submitted by Shaleka Agrawal, Roll Number 614CH3004, is a record of original research carried out by him under our supervision and guidance in partial fulfillment of the requirements of the degree of Master of Technology by Research in Chemical Engineering. Neither this dissertation nor any part of it has been submitted earlier for any degree or diploma to any institute or university in India or abroad.

Dr. Akhilesh Kumar Sahu Assistant Professor

Dedicated to my parents

Declaration of Originality

I, Shaleka Agrawal, Roll Number 614CH3004 hereby declare that this dissertation entitled A Numerical Study of Thin Liquid Film Flow over a Topographically Patterned Rotating Cylinder presents my original work carried out as a masters student of NIT Rourkela and, to the best of my knowledge, contains no material previously published or written by another person, nor any material presented by me for the award of any degree or diploma of NIT Rourkela or any other institution. Any contribution made to this research by others, with whom I have worked at NIT Rourkela or elsewhere, is explicitly acknowledged in the dissertation. Works of other authors cited in this dissertation have been duly acknowledged under the section “Bibliography”. I have also submitted my original research records to the scrutiny committee for evaluation of my dissertation.

I am fully aware that in case of any non-compliance detected in future, the Senate of NIT Rourkela may withdraw the degree awarded to me on the basis of the present dissertation.

June 2017 NIT Rourkela

Shaleka Agrawal

A Numerical Study of Thin Liquid Film Flow over a Topographically Patterned

Rotating Cylinder

Submitted by

SHALEKA AGRAWAL

Roll No-614CH3004

To the department of Chemical Engineering in partial fulfilment of the requirements for the award of degree of Master of Technology by Research in Chemical

Engineering

Under the guidance of

Dr. AKHILESH KUMAR SAHU

Department of Chemical Engineering National Institute of Technology Rourkela

Orissa-769008, India

Acknowledgement

I would like to express my deep and sincere gratitude to my supervisor Prof. (Dr.) A.K. Sahu, Assistant Professor, Department of Chemical Engineering, National In- stitute of Technology, Rourkela, India for his invaluable guidance and continuous invigoration throughout my M.Tech(Research) research work.

I gratefully acknowledge all the members of my Masters Scrutiny Committee (MSC) Dr. Bibhuti Bhusan Nayak, Dr. Basudeb Munshi, Dr. Soumya Sanjeeb Mohapatra, and Dr. Hara Mohan Jena for their many useful comments and discussions.

I earnestly thank Prof. R.K. Singh, H.O.D., Department of Chemical Engineering and Prof. P. Rath, Ex-H.O.D., Department of Chemical Engineering, for giving me all the facilities during the tenure of my course. I am also thankful to all the faculties and staff-members of the Department of Chemical Engineering for helping me in different ways for my work.

I want to acknowledge all my lab mates Swayamprabha Mishra, Sachin Pandey, Bushra Khatoon, Neerja Shukla, and Deepak Kumar for their assistance, incitement and for making a friendly atmosphere in the laboratory.

I wish to thank all my friends of Chemical Engineering Department and other depart- ments of NIT Rourkela.

Finally, I would like to thank my parents for their mental support, for uplifting me and for their good wishes.

Date: June 2017 NIT Rourkela

Shaleka Agrawal

Abstract

The discoveries in the field of thin liquid film flows over flat and non-flat objects have been thriving over the years. The designing and optimization of such process is quite difficult, owing to it being a complex phenomena. In the present work we are be focusing on understanding the fundamentals. We consider a thin-liquid film is flowing over a rotating cylinder with a sinusoidal pattern over the surface. To account for the intermolecular forces a body force term describing the van der Waals attraction is added to the Navier-Stokes equation. The film behaviour is studied by the ap- plication of lubrication approximation. The evolution equation for film thickness as a function of angular position θ and time is derived. This evolution equation stud- ies the film behaviour under the influence of gravity, surface tension, intermolecular forces, viscous forces, surface topography and rotating rate is derived. A semi-implicit technique is used to numerically solve the evolution equation. For a thin-liquid film over the stationary cylinder at low Bond number(Bo) strong surface tension preserves the shape of the liquid film. And as the Bond number increases liquid flows towards θ = ^3π₂ location and a drop-like shape appears at steady state. On inclusion of in- termolecular forces we are able to capture the film rupture at the thinnest region of the liquid film. The thin-liquid film over the patterned cylinder also evolves in the same manner. However, the film rupture time decreases with the increase in the frequency(ω) of the sinusoidal pattern over the surface. In case of rotating cylinder, most of the liquid mass gets accumulated on the rotating side. Also, the liquid film over the patterned cylinder never reaches a steady state. However, the film interface attains a steady state after revolutions in the absence of pattern over the surface. The time to reach zero thickness decreases with increase inBo, ω, and Hamaker constant A in van der Waals force. For rapidly spinning cylinders, continuous film over the patterned cylinder converts into drops(number equal to ω) over the troughs or crests

depending on the magnitude of Weber number(S). The results clearly establish that the flow of thin-liquid films on rotating surfaces is affected by the presence of surface topography.

Keywords: Thin liquid films;lubrication approximation;evolution equation;van der Waals potential;rotating cylinder.

Contents

1 Introduction 1

1.1 Motivation . . . 1

1.2 Review of prior Work . . . 2

1.3 Objectives of the Present Work . . . 4

2 Basic Concept and Theory 5 2.1 Fundamental concepts of Fluid mechanics . . . 5

2.1.1 Conservation of momentum . . . 5

2.1.2 Conservation of mass . . . 6

2.1.3 Constitutive equations . . . 7

2.2 Rotating frame of reference . . . 7

2.3 Lubrication Approximation . . . 9

2.4 Boundary conditions . . . 10

2.4.1 Kinematic boundary condition . . . 10

2.4.2 No-slip condition . . . 10

2.5 Forces influencing the fluid system . . . 11

2.5.1 Gravitational force . . . 11

2.5.2 Surface tension . . . 11

2.5.3 London-Van der waal force . . . 12

3 Mathematical Formulation 13 3.1 Mathematical Modelling . . . 13

3.2 Numerical Modelling . . . 22

3.3 Validation of the Solver . . . 28

3.3.1 One-dimensional biharmonic equation . . . 28

3.3.2 Bounded film . . . 29

4 Results and Discussion 32 4.1 Stationary Cylinder . . . 32

4.2 Rotating Cylinder . . . 49

4.2.1 Slow Spinning Cylinder . . . 49

4.2.2 Rapidly Spinning Cylinder . . . 66

5 Conclusions 77 5.1 Summary of results . . . 77

5.2 Direction for future Work . . . 79 Bibliography

List of Figures

3.1 Rotating topographically patterned cylinder[38]. . . 13 3.2 Solution of the biharmonic equation on the circle with sinusoidal initial

condition. Computed solution is shown at points t=0.0, 0.2, 0.4, 0.6, 0.8 and 1.0, and compared with exact solution. . . 29 3.3 Schematics of Bounded liquid film flow[28]. . . 30 3.4 Effect of surface tension for a thin film on a flat plate, with t. . . 31 4.1 Unsteady film profiles for a stationary cylinder withA= 0 andBo= 1

att = 0, t= 28000, t = 56100, t= 112200, t = 224400 and t= 448800. 33 4.2 Steady state film profiles for a stationary cylinder at A = 0, to study

the effect of Bond number. (a)Bo= 0.001 , (b) Bo= 0.1, (c)Bo= 1, (d) Bo= 100. Here Bois non-dimensional. . . 34 4.3 Unsteady state film profiles for a stationary cylinder withA= 10⁻⁸ and

Bo= 1, for various time-intervals. With t = 0, t = 11500, t = 23100, t= 46300, t = 69400 andt= 926000 . . . 35 4.4 Effect ofA on a steady state film profiles over the stationary cylinder.

(a)A= 10⁻⁸ andt= 87500, (b)A= 10⁻⁷ andt= 27500, (c)A= 10⁻⁶ and t= 12500. . . 36 4.5 Variation of h atθ_rupture with t atBo= 1 andBo= 0.1. . . 37 4.6 Effect of Bond number on a steady state film profiles for a stationary

cylinder at A = 10⁻⁸. (a) Bo = 0.001 and t = 118900, (b) Bo = 0.1 andt = 105900, (c)Bo= 1 andt= 92600, (d)Bo= 100 andt = 37500 . Here Boand t are non-dimensional. . . 38 4.7 Film profiles for a stationary cylinder at A = 0. (a) Bo = 0.04, (b)

Bo= 1. Here Boand t are non-dimensional. . . 39

4.8 Location of local areas of film thinning as a function ofBo, for A= 0. 39 4.9 Film profiles for a stationary cylinder at A = 10⁻⁸. (b) Bo = 0.2

and t = 99860, (b) Bo = 0.02 and t = 110000. Here Bo and t are non-dimensional. . . 40 4.10 Location of local areas of film thinning as a function of Bo, forA = 10⁻⁸. 40 4.11 Unsteady film profiles for a patterned cylinder with ω = 11, A = 0

and Bo= 1. At t = 0, t = 14000, t = 28000, t = 56100, t = 112200, t= 224400, t= 336600 and t= 448800 . . . 41 4.12 Effect of Bond number on a steady state film profiles for a patterned

stationary cylinder at δ = 0.01R_mean, ω = 11 and A = 0. (a) Bo = 0.001, (b) Bo = 0.1, (c) Bo = 1, (d) Bo = 100. Here Bo, δ, ω and t are non-dimensional. . . 42 4.13 Effect of pattern frequency ω on a steady state film profiles for a pat-

terned cylinder withA= 0 and Bo= 10. . . 43 4.14 Unsteady film profiles for a patterned cylinder with ω= 11, A = 10⁻⁸

and Bo = 1 at various time-intervals. At t = 0, t = 5700, t = 11500, t= 23100, t = 46300 andt= 92460 . . . 44 4.15 Effect ofA on a steady state film profiles at Bo= 10 over a stationary

cylinder. (a) A= 10⁻⁸ and t= 85000, (b) A= 10⁻⁶ and t= 10200. . 45 4.16 Effect of Bond numberBowithA= 10⁻⁸on a steady state film profiles

for a patterned stationary cylinder at δ = 0.01Rmean and ω = 11. (a) Bo= 0.001 and t = 118700, (b) Bo= 0.1 and t= 105750, (c) Bo= 1 and t = 92460, (d) Bo = 100and t = 37500 . Here Bo, δ, ω and t are non-dimensional. . . 46 4.17 Variation of film thickness at θrupture with time. . . 47 4.18 Effect of pattern frequency ω on a steady state film profiles for a pat-

terned cylinder withA= 10⁻⁸ and Bo= 1. . . 48 4.19 Film dynamics of a rotating cylinder withBo= 1, W = 0.002 andA=

0 at various time-intervals. t = 0, t = 28000, t = 56100, t = 112200, t= 224400 and t = 448800. . . 50

4.20 Effect of Bond number Bowith A= 0 on a steady state film profile of a rotating cylinder at W = 0.002. (a) Bo = 0.01 , (b) Bo = 0.1, (c) Bo= 1, (d) Bo= 100. Here Bo, W and A are all non-dimensional. . 51 4.21 Effect of increasing rotation rate W on a steady state profiles for a

rotating cylinder at Bo = 10 and A = 0. (a) W = 0.002, (b) W = 0.004, and (c) W = 0.008. Here Bo, W and A are non-dimensional. . 52 4.22 Film dynamics of a rotating cylinder with Bo = 1, W = 0.002 and

A = 10⁻⁸ at various time-intervals. At t = 0, t = 11300, t = 22700, t= 45400, t = 68100 andt= 90650 . . . 53 4.23 Effect of Bond number atW = 0.002 on a steady state film profile of a

smooth rotating cylinder withA= 10⁻⁸. (a)Bo= 0.001 att= 112100, (b) Bo= 0.01 att= 107300 , (c)Bo= 1 att= 90650, (d) Bo= 10 at t= 62320. Here Bo, W, t and A are all non-dimensional. . . 54 4.24 Steady state film profiles for different A over a rotating cylinder. (a)

A= 10⁻⁸ and t= 84500, (b) A= 10⁻⁷ and t = 22000. . . 55 4.25 Variation of film thickness at θ_rupture with non-dimensional time for a

rotating cylinder with A= 10⁻⁸. . . 56 4.26 Steady state film profile to observe the effect ofW on a rotating cylinder

at Bo = 10 and A = 10⁻⁸. (a) W = 0.002 and t = 62320, (b) W = 0.004 and t = 63350,and (c) W = 0.008 and t= 64100. Here Bo, ω, t and δ are non-dimensional. . . 57 4.27 Steady state film profile of a patterned rotating cylinder with δ =

0.01Rmean, ω = 11, A = 0 and W = 0.002 at different Bond number.

(a) Bo= 0.01, (b) Bo= 0.1, (c) Bo= 1, (d) Bo= 100. Here Bo, W, t and A are all non-dimensional. . . 59 4.28 Film dynamics of a patterned rotating cylinder with δ = 0.01R_mean,

ω = 11, W = 0.002, A = 10⁻⁸ and Bo= 10 at various time-intervals.

The time intervals aret= 0,t= 7800,t= 15600,t= 24000,t = 31200, t= 42000 and t= 62270. . . 60

4.29 Steady state film profile of a patterned rotating cylinder with W = 0.002 and A = 10⁻⁸ at different Bond number. (a) Bo = 0.01 at t = 122300, (b) Bo = 1 at t = 90540, (c) Bo = 10 at t = 62270, (d) Bo= 100 at t= 37500. Here Bo, W, t and A are all non-dimensional. 61 4.30 Variation of film thickness at θ_rupture with non-dimensional time for a

patterned rotating cylinder with A= 10⁻⁸. . . 62 4.31 Steady state film profiles for different A over a rotating cylinder with

patterns(ω = 11). (a) A = 10⁻⁸ and t = 81500, (b) A = 10⁻⁶ and t= 8500. . . 63 4.32 Steady state fluid dynamics for a patterned rotating cylinder at δ =

0.01Rmean, ω = 11, Bo = 10 and A = 10⁻⁸. (a) W = 0.002 and t= 62270, (b)W = 0.004 andt= 62950, (c)W = 0.008 andt = 63900.

HereBo, W,t,ω and δ are non-dimensional. . . 64 4.33 Steady state fluid dynamics for a rotating cylinder atδ = 0 andA= 0.

(a) S = 50, (b) S = 100. Here S and δ are non-dimensional. . . 68 4.34 Steady state fluid dynamics for a rotating cylinder at δ = 0 and S =

100. (a) A= 10⁻⁸ and t= 59400, (b)A = 10⁻⁶ and t = 1500. HereS, δ and t are non-dimensional. . . 69 4.35 Variation of film thickness with θ. . . 69 4.36 Steady-state profiles for a rotating cylinder in absence of gravity at

A= 10⁻⁸. ForS = 100 andt = 59400. HereS andtare non-dimensional. 70 4.37 Steady state fluid dynamics for a patterned rotating cylinder at δ =

0.01Rmean, ω = 11 and A = 0. (a)S = 30 (b) S = 80. Here S, ω and δ are non-dimensional. . . 71 4.38 Steady state fluid dynamics for a patterned rotating cylinder at δ =

0.01R_mean and S = 30. (a) A= 10⁻⁸ and t = 1000, (b) A= 10⁻⁶ and t= 70. Here S, ω and δ are non-dimensional. . . 72 4.39 Steady-state profiles for a patterned rotating cylinder in absence of

gravity at A = 10⁻⁸ with ω = 11 and δ = 0.01Rmean. (a) S = 30 and t = 1000, (b) S = 80 and t = 1144. Here S, ω, δ and t are non-dimensional. . . 73 4.40 Variation of film thickness with θ. . . 74

4.41 Steady state fluid dynamics for a patterned rotating cylinder at δ = 0.01R_mean and A = 10⁻⁸. (a) S = 30 and ω = 11, (b) S = 16 and ω= 8, (c) S = 7 andω = 5. HereS, ω and δ are non-dimensional. . . 75

List of Tables

3.1 Typical values of mean cylinder radius R_mean, rotation rate Ω, and liquid property values(like viscosity µ, density ρ, and surface tension σ) used in previous exterior coating studies. . . 15 3.2 Values of dimensionless numbers for previous exterior coating studies.

Bo= ^ρgR2mean_σ , M = ^µ

ρ√

gR³_mean, W = √ ^Ω_g

Rmean

, A= _6πHν^A0 2,and = _R^H

mean. 17 4.1 Range of values of different dimensionless numbers used in the present

work. . . 33 4.2 Location of θ at which film rupture occurs for A= 10⁻⁸ and different

Bo, on a smooth stationary cylinder. . . 37 4.3 Location of film rupture points for A = 10⁻⁸ for different Bo, on a

patterned cylinder. . . 47 4.4 Time of rupture at frequency ω. . . 49 4.5 Variation in A for a smooth stationary cylinder and a patterned sta-

tionary cylinder. . . 49 4.6 θ at rupture for W = 0.002 and W = 0.004 at different Bo, on a

rotating cylinder. . . 58 4.7 Rupture location(θ) at Bo= 10 for different ω. . . 58 4.8 Rupture time at A= 10⁻⁸ for different W, on a patterned cylinder. . 59 4.9 Variation of θ_rupture with W and Bo, on a patterned cylinder. . . 61 4.10 Time of rupture for different ω. . . 65 4.11 Variation in A for a smooth stationary cylinder and a patterned sta-

tionary cylinder at Bo= 10. . . 65 4.12 Critical values of S for δ= 0.01 and different values of ω. . . 75 4.13 Time of rupture at different S and ω values. . . 76

Nomenclature

p Pressure of the liquid on the solid β Slip co-efficient

τ Viscous stress tensor Γ Rate-of-strain tensor Ω Angular rotation Π Normal stress Ψ Body force

f₀ Acceleration of inertial frame of reference f₁ Acceleration of the fluid element

δ Amplitude of the pattern Aspect ratio

κ Free surface curvature µ Dynamic viscosity ν Kinematic viscosity ω Frequency of the pattern φ Potential function

ρ Density of the liquid ρ⁰ Density of solid

σ Surface tension b Body force

n Unit normal to the surface

r Position vector of the fluid element u Velocity of fluid

u⁰ Velocity of solid

v Velocity at which the liquid is flowing

θ Azimuthal angle in rotating frame of reference θf Azimuthal angle in stationary frame of reference A⁰ Dimensional Hamaker constant

Bo Bond number

F Free surface component H Characteristic film thickness k Wave number

l Characteristic length in horizontal direction M Dimensionless viscosity

P Position of material element R Radius of patterned cylinder R_mean Mean radius of cylinder s Growth rate

t Time of rotation

u Azimuthal component of velocity W Dimensionless rotation

w Radial component of velocity z New radial coordinate

t Unit tangent to the surface g Acceleration due to gravity ζ Film thickness

Chapter 1 Introduction

1.1 Motivation

The dynamics of thin liquid film is of significance in a variety of areas like paint, lubrication, printing, pace-makers, cereals, contact lenses, automobiles, foams and most recently in semiconductors, microfluidics and nanofluidics. Rupture occurs at thinnest film locations, Scheludko[40] gave the concept of negative disjoining pressure which turns thinnest film locations into locations of film rupture. Ruckenstein and Jain[36] were the first to study rupture on a planar topography, by adding the van der Waals force term to the Navier-Stokes equation. There work was further elaborated by William and Davis[42] by studing the effects of nonlinearity on film rupture. The works of Scheludko[40], Ruckenstein and Jain[36], Pukhnachev[34][33] and Moffatt[24]

were further used by Reisfeild and Bankoff[35] in 1992 to study the film dynamics on the surface of a cylinder, under the influence of gravity, capillary, thermocapillary and intermolecular forces. Evan et al.[14][13] built upon the efforts of these early researchers to analyse the film behaviour on a rotating cylinder for a two-dimensional flow as well as a three-dimensional flow. He showed the effects of surface tension and gravity.

Even though the field of thin liquid flows was booming with new discovers, one char- acteristic of the flow had remained unnoticed, it was the impact of surface topography.

In recent years it was capaciously reviewed by Craster and Matar[7]. Later, the work of Evans et al.[14][13] was expanded by Sahu and Kumar[38] for study of a topograph- ically patterned rotating cylinder.

In the present work we will study the effects of surface tension and gravity in the presence of intermolecular forces. Because the effect of intermolecular forces has been ignored on rotating substrates. But while coating real life objects as mentioned in the examples we cannot disregard the effect of interaction between the molecules of the liquid and the solid. To resolve this issue we have taken up the model problem of a thin liquid film flowing over a patterned rotating cylinder.

We have built upon the work of Evans et al.[14][13] and Sahu and Kumar[38]. To derive a model equation in the presence of intermolecular forces a van der Waals potential term is added to the Navier-Stokes equation as an extra body force. This body force is inversely proportional to the third power of film thickness. The evolution equation is further obtained by simplifying the Navier-Stokes equation using lubrica- tion approximation. The final model equation thus obtained contains film thickness as a function of angular coordinate and time. Further, the numerical simulations help in obtaining qualitative results and an approximate rupture time, the time required by the film thickness to become zero at some points.

1.2 Review of prior Work

Before proceeding to our problem, we acknowledge that the flow of thin liquid film on planar and non-planar surfaces, using lubrication approximation, has been the subject of a number of studies in the past few decades.

Flow of thin fluid layers spreading on solid surfaces have a distinguished history having been studied since the days of Reynolds, who was among the first to examine lubrica- tion flows[27]. Experiments done by Beauchamp Towers in 1883 and 1884 prompted Reynolds to create what we now call ”lubrication theory, which, subsequently, has been widely used to study thin film flows. It is interesting to note that the great successes of lubrication theory and the technological advances that it would create were not foreseen at that time and indeed were partially dismissed. In fact, the whole approach did not meet with a universal welcome, which can be seen in the short letter to the editor of The Engineer, February 1884. Now-a-days whatever we use is coated by the use of the very mechanism given by the name of Reynolds lubrication theory.

The lubrication theory was further used by Benney[2] for finding the evolution equa-

tion on liquid films. Huppert[18], presented series of fingers emerging when a fluid is released on an inclined plane. Following up, Alekseenko et al.[37], investigated the formation of a wave on a vertically falling liquid film. Fenkel[15][16], studied the flow down a cylinder and showed that if the cylinder radius was large, as compared to the film thickness, the long-wave perturbation approach yielded a rather simple evolution equation. His nonlinear equation was similar to the well-known Benney’s equation[2]

of planar films.

In 1959, Dzyaloshinskii and Pitaevskii[11] developed the molecular formula for the chemical potential of the liquid film between two solid surfaces with very small dis- tance between them. Ruckenstein and Jain[36], studied spontaneous rupture of a liquid film on a planar solid. Evaluated range of wavelengths of the perturbation for which instability occurs is established and the time of rupture. Prevost[32] investi- gating the stability of a viscous free film to finite amplitude disturbances. It was shown that the nonlinearities significantly accelerate the rupture process. Rupture in thin films was studied by Yiantsios and Davis[17]. His analysis yielded an evolution equation for the film thickness which reflected the effects of disjoining pressure and interfacial tension, as well as the dynamics of the two fluid phases. Numerical stud- ies obtained qualitative features of the instability that cannot be captured by linear theory, and quantitative information about film rupture times. Erneux and Davis[12]

used a bifurcation technique leading to an estimate for the nonlinear rupture time.

Ida and Miksis[19] in 1996, evolved a system of equations which described the dy- namics of a thin two-dimensional free film and analyzed it in the vicinity of rupture.

These investigations were reviewed by Oron and Bankoff[28] in 1997.

Moffatt[24], extended the work of Yih[43], Pedley[30] and Pucknachev[33], discussing the problem of the dynamics of a thin fluid film on the outer surface of a thin rotating roller. He did some experiments and found out that with zero rotation the film drains off the cylinder, and with non-zero rotation speed a drop is formed at the steady state which can be unstable as the rotation speed is increased.

Reisfeld and Bankoff[35] considered the flow of a viscous liquid film on the surface of a cylinder under the influence of gravity, capillary, thermo-capillary, and intermolecular forces. They used analytical and numerical techniques and concluded that given some amount of time the film drains off the cylinder. Moreover, in presence of intermolecu-

lar forces, film rupture can be at a single point or at two points, depending upon the magnitude of surface tension and gravity.

In 2004, Evans et al.[14][13] evolved the model for a thin liquid coating on a circular cylinder, undergoing uniform rotation. He included the effects of cylinder rotation, gravity, surface tension and flow along the cylinder axis, in his study. The numerical scheme showed a transition from pendant drop hanging beneath the cylinder to a nearly uniform coating wrapped around it as rotation rate was increased.

One aspect of these flows that appears to have been overlooked is the influence of film rupture on varying surface topography. For planar substrates, the influence of surface topography and film rupture on thin-liquid-film flow has been extensively investigated in recent years, as reviewed by Craster and and Matar[7]. Sahu and Kumar[38] con- sidered a model problem in which a thin liquid film flows over a rotating cylinder patterned with a sinusoidal surface topography. They applied lubrication theory to develop a partial differential equation that governs the film thickness as a function of time and the angular coordinate. They demonstrated that the flow of thin liquid films on rotating surfaces can be very sensitive to the presence of surface topography.

Thus, surface topography will be an important factor to account for when developing accurate models for the coating of discrete objects, particularly those that are rotated at high speeds.

1.3 Objectives of the Present Work

The present work aims to investigate the following aspects of thin-film flow over a topographically patterned cylinder:

1. Effect of surface tension, gravity, rotation speed and surface topography on the evolution of the film.

2. Rupture dynamics on stationary and rotating cylinder.

Chapter 2

Basic Concept and Theory

2.1 Fundamental concepts of Fluid mechanics

Since fluids, like all matter, are comprised of enormous number of molecules, consid- ering individual motion of these many molecules would be extremely complex. The convenient method is to think of the fluid as a continuous distribution of matter or a continuum. On the basis of continuum hypothesis, the value of a property at any point is defined as averaged over a neighbourhood of the point, the neighbourhood being large enough to contain large number of molecules (so that the average is independent of the size of the neighbourhood) but very small compared to the macroscopic dimen- sions of the system. As such, it is assumed that all physical properties are continuous function of position and hence their spatial derivatives exist.

General balance equations for the quantity of interest are considered along with the constitutive equation for fluxes and are put into differential form. These equations are then solved for the desired property, by applying boundary conditions appropri- ate to the problem. The fundamental concepts in fluid mechanics leading to these differential equations are discussed below[3][9][23][29].

2.1.1 Conservation of momentum

The fundamental physical law on which fluid-flow analyses is based is Newton’s second law of motion, which states that, the time rate of change of momentum of a system is equal to the net force acting on the system and takes place in the direction of the net

force. The Newton’s second law of motion in differential form helps to arrive at the Navier-Stokes equation[6][1]. Under the assumptions of incompressible flow, constant viscosity and laminar flow, Navier-Stokes equation is written as:

ρDv

Dt =−∇p+µ∇²v+ρb (2.1.1)

where

ρ^Dv_Dt is the rate of change of momentum per unit volume.

−∇p is the net force per unit volume due to pressure.

µ∇²vis the net force per unit volume due to viscous stress.

ρb is the body force per unit volume.

Body forces are those which act without physical contact and exert a force like grav- itational force, intermolecular force, centrifugal force, and coriolis force.

2.1.2 Conservation of mass

The law of conservation of mass states that mass may be neither created nor destroyed.

With respect to a control volume, the law of conservation of mass may be simply stated as:

Rate of mass efflux from control volume - Rate of mass flow into control volume + Rate of accumulation of mass within control volume = 0

The law of conservation of mass expressed in differential form (by letting the control volume shrink to a point) is the continuity equation:

∂ρ

∂t =−∇ ·(ρv) (2.1.2)

The vectorρvrepresents mass flux. The equation states that the rate of accumulation of mass per unit volume ^∂ρ_∂t

at a fixed point is equal to the net inward flow rate of mass towards a neighbourhood of a point per unit volume of the neighbourhood, in the limit that the volume tends to be zero. This condition should be satisfied at every point in the system.

The continuity equation can be written for incompressible fluid flow as:

∇ ·v= 0 (2.1.3)

i.e., the net volumetric flux of an incompressible fluid to neighbourhood of any point is zero.

2.1.3 Constitutive equations

The equations discussed so far are applied to any fluid in motion. In order to complete the governing equations, these deformations have to be related to internal forces and this relation is provided by constitutive laws.

For a Newtonian fluid, constitutive equation for the viscous stress tensor should be a function only of the rate-of-strain tensor, or τ =τ(Γ). Viscous stresses are frictional in character, in that they reflect the molecular-level resistance to attempt to move one part of a fluid relative to another. Thus, the type of fluid motion which matters in this regard is deformation (the pulling apart or pushing together of material points) and not uniform translation or rigid-body rotation. The fact that τ must depend on Γ, and not simplyv, can be proven formally by considering what is necessary to ensure material objectivity. This is the principle that the form of a constitutive equation must not depend on the position or velocity of an observer, and therefore must be invariant to time dependent translations or rotations of the coordinate system. For an incompressible Newtonian fluid, the viscous stress is simply

τ = 2µΓ =µ[∇v + (∇v)^t] (2.1.4)

where,

τ is the viscous stress tensor.

µis the dynamic viscosity.

Γ is the rate-of-strain tensor.

2.2 Rotating frame of reference

A rotating frame of reference is a special case of a non-inertial reference frame that is rotating relative to an inertial reference frame. All non-inertial reference frames exhibit fictitious forces. Here we will be discussing only uniform rotating reference frame. Uniform rotating reference frames are characterized by two fictitious forces.

1. the Centrifugal force, 2. the Coriolis force,

Coriolis forces is an inertial force (also called a fictitious force) that acts on objects that are in motion relative to a rotating reference frame. In a reference frame with

clockwise rotation, the force acts to the left of the motion of the object. In one with anticlockwise rotation, the force acts to the right[1][6].

Centrifugal forces is used to refer to an inertial force (also called a ’fictitious’ force) directed away from the axis of rotation that appears to act on all objects when viewed in a rotating reference frame.

Newton’s laws of motion describe the motion of an object in an inertial (non-accelerating) frame of reference. When Newton’s laws are transformed to a rotating frame of refer- ence, the Coriolis force and centrifugal force appear. Both forces are proportional to the mass of the object. The Coriolis force is proportional to the rotation rate and the centrifugal force is proportional to its square. The Coriolis force acts in a direction perpendicular to the rotation axis and to the velocity of the body in the rotating frame and is proportional to the object’s speed in the rotating frame. The centrifugal force acts outwards in the radial direction and is proportional to the distance of the body from the axis of the rotating frame.

To find out these forces we can choose a frame of reference relative to which the boundaries are at rest. The acceleration of the fluid element relative to the rotating frame is then different from the absolute acceleration in the inertial frame of reference.

We assume that instantaneously the moving frame of reference is rotating with angu- lar velocityω about a point O which itself is moving relative to the Newtonian frame of reference with an acceleration of f₀. The absolute acceleration of the elementO is

f =f₀+f₁ (2.2.1)

where f₁ is the acceleration of the fluid element relative to the point O.

The relation between f₁ and the acceleration of the element relative to the rotating frame is determining by taking a material element located at P, where P can be written as,

P =P₁i+P₂j+P₃k (2.2.2)

Noting the change of P with time t dP

dt + Ω×P (2.2.3)

where Ω is the rotation rate.

If we take P as the position of the material element of the fluid r relative to z = 0.

Then the v₁ velocity relative to the non-rotating frame moving with f₀ is:

v₁ = dr

dt + Ω×r (2.2.4)

and acceleration of f₁ is:

d²r

dt² + 2Ω× dr

dt +r× dΩ

dt + Ω×(Ω×r) (2.2.5)

When the frame of reference is not moving f₀ = 0 and r× ^dΩ_dt = 0.

So,

f =−2Ω×u−Ω×(Ω×r) (2.2.6)

where

−2Ω×u is the coriolis term.

−Ω×(Ω×r) is the centrifugal term.

These terms are directly added to the Navier-Stokes equation.

2.3 Lubrication Approximation

One of the most convenient models to study thin film flows is the lubrication approxi- mation (thin film approximation). A prototypical geometry consists of a spread-out of liquid on a solid substrate, with the free surface of the liquid exposed to air, another liquid or bounded by another solid. The thickness of the film (h) is much smaller than the characteristic length (l) in the horizontal direction. An aspect ratio = ^h_l could be defined which is essentially small. i.e., 1. The existence of this small aspect ratio is made useful by taking a perturbation series in power of , and sub- stituting it into the governing equations. By proceeding so, one can reduce the full set of governing equations and boundary conditions into a simplified set of evolution equations. The mathematical system thus obtained is not as complex as the original problem but preserves the important features of its physics[26].

2.4 Boundary conditions

2.4.1 Kinematic boundary condition

Kinematic boundary condition, is derived from the principle of mass conservation at any boundary of the flow domain. Let the bulk-phase densities on the two sides of the interface be denoted as ρand ρ⁰ and the fluid velocities as u and u⁰. The orientation of surface S is specified in terms of a unit normaln. In general, the surface S is not a material surface. However, the surface S is not a source or sink for mass, and thus mass conservation requires that the net flux of mass to (or from) the surface must be zero.

No phase transformation occurs. S is in fact a material surface separating a viscous fluid and a second medium that may either be solid or fluid[23][5]. Hence, in the absence of phase change at S, the normal component of velocity must be continuous across it and equal to the normal velocity of the surface:

u·n =u⁰ ·n at S (2.4.1)

If the second phase is a solid wall, then u⁰ =U_solid, which is assumed to be known.

In a frame of reference fixed to a solid wall, u⁰·n= 0, and in this frame of reference

u·n= 0 at S, (2.4.2)

A generalization of the condition,

ρ(u−u⁰)·n =ρ⁰(u−u⁰)·n at S, (2.4.3) where the velocity components are all still measured with respect to fixed, ”labora- tory” coordinates. The term on the left-hand side is just the net mass flux of material from the first fluid to (or from) the interface, and because mass cannot accumulate on the surface S, this is balanced by an equal flux of mass away from (or to) S on the other side. It will be noted in this case that the normal velocity componentsu·n and u⁰·n are no longer equal.

2.4.2 No-slip condition

The boundary condition at the surface S involving the bulk-phase velocities is known as the dynamic condition[23]. It specifies a relationship between the tangential com- ponents of velocity, [u−(u·n)n] and [u⁰−(u⁰·n)n]. The most common assumption

is that the tangential velocities are continuous across S, i.e.,

u−(u·n)n=u⁰ −(u⁰ ·n)n at S, (2.4.4) and this is known as the no-slip condition. If the second medium is a solid, then again u⁰ =U_solidwhich is assumed to be known, and the condition prescribes a specific value for the tangential velocity of the fluid. A convenient frame of reference in this latter case is one fixed to the solid wall, so that

u−(u·n)n= 0 at S. (2.4.5) This is the most common form of the no-slip condition[39].

2.5 Forces influencing the fluid system

2.5.1 Gravitational force

Gravity or gravitation is a natural phenomenon by which all things with energy are brought toward (or gravitate toward) one another. It is a body force which acts with- out physical contact. The thin film on the cylinder will be affected by the gravitational force. For the cylinder the gravity term can be written as:

g= (−gsinθ)er+ (−gcosθ)eθ (2.5.1)

where g is the constant acceleration due to gravity.

2.5.2 Surface tension

Surface tension is the elastic tendency of a fluid surface which makes it acquire the least surface area possible. At liquid-air interfaces, surface tension results from the greater attraction of liquid molecules to each other (due to cohesion) than to the molecules in the air (due to adhesion). The net effect is an inward force at its surface that causes the liquid to behave as if its surface were covered with a stretched elastic membrane. Thus, the surface becomes under tension from the imbalanced forces, leading to the formation of the ”surface tension” term.

2.5.3 London-Van der waal force

The interaction between the molecules of the liquid and that of the solid form inter- molecular force term.

We follow Ruckenstein and Jain[36] and write the potential function as φ= A

h³ (2.5.2)

The dimensional Hamaker constant A⁰ is related to A by A= A⁰

6πHν² (2.5.3)

where H is the characteristic film thickness and ν = ^µ_ρ is the kinematic viscosity of the fluid. When A⁰ > 0, usually called the case of negative disjoining pressure[10]

(Deryagin and Kusakov 1937), intermolecular forces are destabilizing; when A⁰ < 0 (Visser 1972), these forces are stabilizing[41].

Chapter 3

Mathematical Formulation

3.1 Mathematical Modelling

A Newtonian liquid with constant material properties, is flowing over the outer surface of a topographically patterned cylinder of radius R. The pattern over the cylinder is sinusoidal with frequency ω and amplitude δ.

Figure 3.1: Rotating topographically patterned cylinder[38].

The cylinder is rotating about its central axis at constant angular speed Ω in the anticlockwise direction. We will be restricting our attention to two-dimensional flow with no axial variations.

R =R_mean+δsin(ωθ) (3.1)

We use polar coordinate system withr= 0 as the location of the center of the cylinder and θ the azimuthal angle being measured horizontally from the right. We consider a frame of reference rotating with the cylinder. For convenience, we define a new radial coordinate z =r−(R_mean+δsin(ωθ) such that the solid-liquid interface is at z = 0

and the liquid-gas interface is at z =h(θ, t). The liquid velocity is written as,

u(r, θ) = we_r+ue_θ (3.2) where er and eθ are the coordinate unit vectors.

The following assumptions are made while modelling the flow:

1. Incompressible flow 2. Newtonian fluid 3. 2-D flow

We consider the possibility of intermolecular forces in very thin films by adding an intermolecular potential function, φ in the Navier-Stokes equation. The governing equation in the rotating frame of reference is

∂u

∂t +u· ∇u =−1

ρ∇(p+φ) + µ

ρ∇²u−Ω×(Ω×r)−2Ω×u+g (3.3)

∇ ·u= 0 (3.4)

where p is the pressure of the liquid, ρ is the density, µ is the dynamic viscosity, Ω = Ωe_y with e_y being the unit vector along the cylinder axis, g = (−gsinθ_f)e_r+ (−gcosθf)eθ with g being the constant gravitational acceleration. φ being the poten- tial function. Described as,

φ= A

h³ (3.5)

The dimensional Hamaker constant A⁰ is related to A by A= A⁰

6πHν² (3.6)

where H is the characteristic film thickness and ν = ^µ_ρ is the kinematic viscosity of the fluid. The relation between the angular position of stationary frame of reference (θ_f) and rotating frame (θ) is θ_f =θ+ Ωt.

Values of various dimensional parameters for some previous experiments are given in Table 3.1. The resulting values for the dimensionless quantities M, W and A are given in Table 3.2.

The governing equations are solved using the following boundary conditions,

Table 3.1: Typical values of mean cylinder radius R_mean, rotation rate Ω, and liquid property values(like viscosity µ, density ρ, and surface tension σ) used in previous exterior coating studies.

Experiment Rmean

(cm)

Ω(rad.s⁻¹) µ(g.(cms)⁻¹) ρ(g.cm⁻³) σ(dyn/cm) Liquid

Yih[8] 11.1 9.4-19.2 1.0-1.3 1.2 60 Glycerine

11.1 10.5-25.5 0.071-0.080 1.1 59 Water-glycerine

11.1 9.0-29.4 0.0086-

0.0088

1.0 72 Water

Moffatt[24] 2.0 2.2-8.1 80 - - Gold syrup

Kovac and Balmer[22]

2.2-5.4 0.7-2.4 1.0-2.8 1.1-1.2 35-61 Glycerine-water

Preziosi and Joseph[31]

0.72-2 0.7-31 0.2-1000 0.95-0.98 21 Glycerine

Kelmanson[21] 0.6 1.0-5.2 2.0 0.89 - SAE 30 engine oil

0.6 1.0-5.2 5.7 0.90 - SAE 30 engine oil

0.6 1.0-5.2 3.4 0.89 - SAE 15/50 engine oil

Reisfeld and 0.01 0.0 1 x 10⁻² 1.0 72 Water

Bankoff[35] 0.01 0.0 1.6 x 10⁻² 13.6 475 Mercury

1. No-slip boundary condition is applied at z = 0

u=w= 0 (3.7)

2. The boundary conditions at z =h

(a) The shear stresses are zero, at the free surface, so n·τ ·t= 0

(3.8)

whereτ is the rate-of-strain tensor, andnandtare the unit vectors normal and tangential to the free surface, respectively. The unit vectors are,

n= 1 N

er+1

r

−δωcos(ωθ)− ∂h

∂θ

eθ

(3.9)

t= 1 N

1 r

δωcos(ωθ) + ∂h

∂θ

e_r+e_θ

(3.10) whereN =

q

1 + _r¹2 δωcos(ωθ) + ^∂h_∂θ2

.

(b) At the free surface, normal stress is balanced by pressure and surface ten- sion effects, so

−p+µ(n·τ ·n) = −σκ

(3.11)

where surface tension σ is constant and κ is the free surface curvature, denoted by,

κ= 1 N³

1

r 1 + 2 r²

δωcos(ωθ) + ∂h

∂θ

2!!

− 1 N³

1 r²

∂²h

∂θ² −δω²sin(ωθ)

(3.12) whereN =

q

1 + _r¹2 δωcos(ωθ) + ^∂h_∂θ2

.

3. F(θ, z, t) = z−h(θ, t) = 0, at the free surface we suppose the kinematic bound- ary condition to be

DF

Dt = 0 (3.13)

The ambient atmospheric pressure is taken zero, without loss of generality.

In our case, the film is very thin, with a characteristic film thicknessH that is much small in comparison to the mean cylinder radius (= _R^H

mean 1). This difference is thickness is exploited by applying lubrication approximation theory, to simplify the problem. In doing so, we scale δ the pattern amplitude, h the film thickness, and z the new radial component by the characteristic film thickness H. The characteristic speed is taken as U = ^ρgH_µ². The governing equations and boundary conditions are made dimensionless by introducing the following quantities, denoted by bars:

¯ u= u

U,w¯= w

U,r¯= r

R_mean,δ¯= δ

R_mean,p¯= p ρgH, φ¯= φ

ρgH,t¯= t

Rmean

U

,¯h= h

R_mean,z¯= z

R_mean (3.14)

The scaled radius can be written as ¯r= 1 +¯z.

The scaled governing equations after neglecting small terms ofO(²) and dropping off the bars are,

−∂p

∂z +∂²w

∂z² −sin

θ+ M W t ²

=−W²(1 +z) (3.15)

Table 3.2: Values of dimensionless numbers for previous exterior coating studies.

Bo= ^ρgR_σ^2mean, M = ^µ

ρ√

gR³_mean, W = √ ^Ω_g

Rmean

, A= _6πHν^A0 2, and = _R^H

mean.

Experiment Bo M W A Liquid

Yih[8] 2450 (7.4-9.7)x10⁻⁴ 1.0-2.0 3.5x10⁻¹⁴ 0.0096-0.018 Glycerine 2340 (5.3-6.0)x10⁻⁵ 1.1-2.7 5.5x10⁻¹⁴ 0.0043-0.0016 Water-glycerine 1660 (7.4-7.5)x10⁻⁶ 1.0-3.1 0.8x10⁻⁹ 0.0030-0.019 Water

Moffatt[24] 80 0.31 0.14-0.52 - 0.061 Gold syrup

Kovac and Balmer[22]

90-1000 0.0021-0.025 0.051-0.11 8.0x10⁻¹⁵ 0.025-0.068 Glycerine-water

Preziosi and Joseph[31]

14-23 0.011-54 0.002-1.4 9.4x10⁻³ 0.3-2 Glycerine

Kelmanson[21] 9 0.16 0.026-0.13 2.6x10⁻¹⁴ 0.044-0.088 SAE 30 engine oil

9 0.44 0.026-0.13 2.6x10⁻¹⁵ 0.066-0.11 SAE 30 engine oil

9 0.26 0.026-0.13 6.5x10⁻¹⁵ 0.066-0.15 SAE 15/50 engine oil

Reisfeld and 0.001 0.319 0 10⁻⁸ 0.01 Water

Bankoff[35] 0.001 0.0376 0 10⁻⁷ 0.01 Mercury

−

∂(p+φ)

∂θ −∂(p+φ)

∂z δωcos(ωθ)

+ ∂²u

∂z² +

∂u

∂z

−cos

θ+M W t ²

= 0 (3.16)

∂(rw)

∂z +∂u

∂θ − ∂u

∂zδωcos(ωθ) = 0 (3.17)

Here, the dimensionless viscosity M, and the dimensionless rotation W are given by,

M = µ

ρp

gR³_mean, W = Ω q g

Rmean

(3.18) Note that the r-component of the equation of motion, Eq. (3.15), is unaffected by intermolecular forces. However, the θ-component, Eq. (3.16), has an additional po- tential term due to intermolecular forces.

The scaled boundary conditions are 1. No-slip at the substrate

u=w= 0 at z = 0 (3.19)

2. The shear stress boundary condition from, Eq. (3.8) become

∂u

∂z −u+O(²) = 0 (3.20)

3. The normal stress boundary condition from, Eq. (3.11) becomes

−p+(n·τ ·n) =−_Bo¹ κ

(3.21)

where bond number Bo relating the gravity force to mean surface tension is defined as

Bo= ρgR²_mean

σ (3.22)

The components of rate-of-strain tensor (τ) and the curvature (κ) are scaled by

U

Rmean and _R ¹

mean respectively. The scaled components are τ_rr = ∂w

∂z

τ_θθ = 1 r

∂u

∂θ −δωcos(ωθ)∂u

∂z

+w r

τ_rθ =τ_rθ = r

∂

∂z u

r

+ r

∂w

∂θ −δωcos(ωθ)∂w

∂z

(3.23)

κ= 1−(h+δsin(ωθ))− ∂²h

∂θ² −δω²sin(ωθ)

+O(²) (3.24)

n=e_r+ r

−δωcos(ωθ)−∂h

∂θ

e_θ+O(²) (3.25) 4. The kinematic boundary condition takes the following form

∂h

∂t + u r

∂h

∂θ + u

rδωcos(ωθ) = w (3.26)

Applying the boundary condition of z = h to Eq. (3.24), and substituting in Eq.

(3.21) we obtain

p0 = 1

Bo (3.27)

The variables are expanded in powers of as u=u⁽⁰⁾+u⁽¹⁾+...

w=w⁽⁰⁾+w⁽¹⁾+... (3.28)

φ=⁻¹φ₀+...

p=⁻¹p₀+p⁽⁰⁾+p⁽¹⁾+...

We substitute the these expanded powers of in the Eq. (3.15)-(3.17), and then equate equal powers ofto get a sequence of linear equations. Subsequently, we solve the equations for leading-order terms and first-order correction terms, as a function of h.

The leading-order terms are

−∂p⁽⁰⁾

∂z −sin

θ+ M W t ²

=−W² (3.29)

∂²u⁽⁰⁾

∂z² −cos

θ+M W t ²

= 0 (3.30)

∂²w⁽⁰⁾

∂z² +∂u⁽⁰⁾

∂θ − ∂u⁽⁰⁾

∂z δωcos(ωθ) = 0 (3.31) Atz = 0,

u⁽⁰⁾=w⁽⁰⁾ = 0 (3.32)

and Eq. (3.20) at z =h,

∂u⁽⁰⁾

∂z = 0 (3.33)

and Eq. (3.24) at z =h is, p⁽⁰⁾ =− 1

Bo

h+∂²h

∂θ² +δsin(ωθ)−δω²sin(ωθ)

(3.34) Integrating Eq. (3.29)-(3.31), using the boundary conditions of Eq. (3.32)-(3.34), we obtain the leading-order terms p⁽⁰⁾, u⁽⁰⁾,and w⁽⁰⁾ as

p⁽⁰⁾=

W²−sin

θ+M W t ²

(z−h)

− 1 Bo

h+ ∂²h

∂θ² +δsin(ωθ)−δω²sin(ωθ)

(3.35)

u⁽⁰⁾ = z²

2 −hz

cos

θ+M W t ²

(3.36)

w⁽⁰⁾ = z³

6 − hz² 2

sin

θ+ M W t ²

+1

2

∂h

∂θz²cos

θ+M W t ²

+ z²

2 −hz

cos

θ+M W t ²

δωcos(ωθ) (3.37) Note that the leading-order terms are same as that obtain by Sahu and Kumar[38]

for a patterned rotating cylinder.

At next-order equations are

−∂p⁽¹⁾

∂z + ∂²w⁽⁰⁾

∂z² =−W²z (3.38)

−∂p⁽⁰⁾

∂θ − ∂φ₀

∂θ + ∂p⁽⁰⁾

∂z δωcos(ωθ) + ∂²u⁽¹⁾

∂z² + ∂u⁽⁰⁾

∂z = 0 (3.39)

∂w⁽⁰⁾z

∂z +∂w⁽¹⁾

∂z + ∂u⁽¹⁾

∂θ −∂u⁽¹⁾

∂z δωcos(ωθ) = 0 (3.40) Atz = 0,

u⁽¹⁾=w⁽¹⁾ = 0 (3.41)

and at z =h,

∂u⁽¹⁾

∂z −u⁽⁰⁾ = 0 (3.42)

Since first-order terms asw⁽¹⁾ and p⁽¹⁾ are not required in the next step of calculation, we are only interested in u⁽¹⁾.

Solving (3.38)-(3.42) for u⁽¹⁾ we get u⁽¹⁾ =cos

θ+ M W t

² −z³

3 +hz²−3 2h²z

− 1 Bo

∂h

∂θ + ∂³h

∂θ³ +δωcos(ωθ)−δω³cos(ωθ) z² 2 −hz

−

W²−sin

θ+ M W t ²

∂h

∂θ + 2δωcos(ωθ) z² 2 −hz

−3Ah⁻⁴∂h

∂θ z²

2 −hz

(3.43)

Next we combined the continuity equation and the kinematic condition to get the following evolution equation

[1 +(h+δsin(ωθ))]∂h

∂t = ∂

∂θ h³

3 cos

θ+M W t ²

+ ∂

∂θ

h⁴ 2 cos

θ+M W t ²

− ∂

∂θ h³

3Bo ∂h

∂θ +∂³h

∂θ³ +δωcos(ωθ)−δω³cos(ωθ)

− ∂

∂θ h³

3

W²−sin

θ+M W t ²

∂h

∂θ + 2δωcos(ωθ)

+ ∂

∂θ

−A h

∂h

∂θ

(3.44) Following Evans et al.[14] and Sahu and Kumar[38], small terms of order related to cylinder curvature and second term on the left hand side are dropped.

While numerically solving Eq. 3.44 we rescaled all the lengths including film thickness with R_mean and we obtain

∂h

∂t = ∂

∂θ h³

3 cos(θ+M W t)− h³ 3Bo

∂h

∂θ + ∂³h

∂θ³

− ∂

∂θ h³

3 W²−sin(θ+M W t)∂h

∂θ

+ ∂

∂θ

− h³

3Bo δωcos(ωθ)−δω³cos(ωθ)

− ∂

∂θ ∂

∂θ h³

3 W²−sin(θ+M W t)

2δωcos(ωθ)

+ ∂

∂θ

−A h

∂h

∂θ

(3.45) This equation shows the nature of a thin film over a topographically patterned ro- tating cylinder in the rotating frame of reference when surface tension, gravitational force, viscous force, centrifugal force, and intermolecular forces are all taken under consideration. The above equation can be written in stationary frame of reference by applying the relationθ_f =θ+^{M W t}2 . So, the equation in stationary frame of reference is

∂h

∂t =−M W ∂h

∂θ_f

+ ∂

∂θ_f h³

3 cosθf − h³ 3Bo

∂h

∂θ_f + ∂³h

∂θ_f³

!

− h³ 3

W²−sinθf

∂h

∂θ_f !

+ ∂

∂θf

∂

∂θf

− h³

3Bo δωcosω(θ_f −M W t)−δω³cosω(θ_f −M W t)

+ ∂

∂θ_f ∂

∂θ_f

−h³

3 W² −sin(θ_f)

2δωcosω(θ_f −M W t)

+ ∂

∂θf

−A h

∂h

∂θf

(3.46) The terms in the equation contain all the essential physics of the problem including (i) viscous drag produced by the liquid due to rotation, (ii) drainage due to the azimuthal and radial component of gravity, (iii) effect of surface tension, (iv) effect of the surface topography, (v) effect of the intermolecular forces.

The Eq. 3.45 would reduce to the evolution equation given by Sahu and Kumar[38], if written with intermolecular term as negligible (A = 0). This same equation would also reduce to the equation presented by Evans et al.[13][14] if written for a smooth cylinder with negligible intermolecular forces A= 0 and δ = 0.

3.2 Numerical Modelling

A numerical technique would be required to solve Eq. 3.45. This numerical technique should be able to solve a fourth-order partial differential equation. As a first step towards building this numerical scheme we discretised the fourth-order partial differ- ential, using finite-difference method. To discretize this evolution eq. 3.45, which is a fourth-order partial differential equation, a finite-difference method is used[25][13][14].

A partially implicit method is used, in which only the highest derivatives are discre- tised implicitly. This procedure avoids the use of small time steps required to ensure the stability of an explicit numerical scheme. The inclusion of the effect of time(in the discretisation technique) is made sure by approximating the coating thickness by h^k_i, wherek denotes the time level.