Interest for thin falling films . . . . 1

Contents

Notations v

Introduction 1

Interest for thin falling films . . . . 1

Phenomenology . . . . 2

Marangoni effect . . . . 2

Surface wave instability . . . . 4

Surface wave instability and Marangoni effect . . . . 8

Inhomogeneous heating . . . . 8

Modelling . . . . 9

Structure of this work . . . . 11

Original contributions . . . . 14

1 Definition of the general problem 15 1.1 Governing equations and boundary conditions . . . . 16

1.2 Dimensionless equations and parameters . . . . 21

1.3 Base state and discussion on the Biot number . . . . 25

1.3.1 Temperature condition (TC) . . . . 26

1.3.2 Heat flux condition (HFC) . . . . 27

1.4 Linear stability analysis . . . . 27

1.4.1 Pure transverse perturbations: k

= 0, k = k

. . . . 31

1.4.2 Pure longitudinal perturbations: k = k

, k

= 0 . . . . 33

1.5 Boundary layer equations . . . . 37

1.5.1 Two-dimensional boundary layer equations . . . . 39

1.5.2 Pure longitudinal linear stability analysis . . . . 40

1.6 Sets of parameters . . . . 40

1.7 Reduction of the governing equations . . . . 42

I Low Reynold number: the Benney equation 45 2 Benney equation including Marangoni effect 47 2.1 The Benney equation . . . . 47

2.2 Long-wave asymptotic expansion . . . . 50

2.2.1 Evolution equation for the film thickness . . . . 53

i

ii CONTENTS

2.2.2 Higher-order terms in the Benney equation . . . . 54

2.2.3 Weakly nonlinear models . . . . 55

2.2.4 Primary instability . . . . 56

2.3 Search for stationary wave solutions . . . . 57

2.4 Closed and open flow conditions . . . . 59

2.5 Blow-up for closed flows . . . . 61

2.5.1 Families of stationary solutions . . . . 61

2.5.2 Blow-up versus wavenumber . . . . 64

2.6 Parametric study for closed and open flows . . . . 67

2.6.1 Reduced systems and parameters . . . . 67

2.6.2 Vertical and isothermal films . . . . 68

2.6.3 Influence of the inclination . . . . 70

2.6.4 Influence of the Marangoni effect . . . . 72

2.7 Stability of stationary solutions . . . . 74

2.8 Further discussion about the Benney equation . . . . 76

3 Periodic heating 79 3.1 Non-uniform heating . . . . 80

3.2 Stationary solutions . . . . 83

3.2.1 Moving reference frame - Uniform heating . . . . 84

3.2.2 Fixed reference frame - Non-uniform heating . . . . 86

3.3 2D computations . . . . 86

3.3.1 Numerical method . . . . 86

3.3.2 Influence of the imposed temperature gradient . . . . 88

3.3.3 Non-uniform versus uniform heating. . . . 93

3.4 Heat transfer . . . . 94

4 Local heating 99 4.1 Experimental test section . . . . 100

4.2 Base state: comparison with experiments . . . . 103

4.3 Linear stability analysis . . . . 106

4.4 Linear stability: comparison with experiments . . . . 113

4.5 3D simulations . . . . 117

II Moderate Reynolds number: the weighted integral bound- ary layer models 127 5 The weighted residual method 129 5.1 Weighted residuals approach . . . . 130

5.2 Formulation at first order . . . . 132

5.3 Formulation at second order . . . . 137

5.4 Reduced models . . . . 141

5.5 Discussion . . . . 146

CONTENTS iii

6 Linear stability and nonlinear waves 149

6.1 Linear stability results . . . . 150

6.2 Solitary waves . . . . 159

6.3 Interactions between H and S-modes . . . . 163

7 Three-dimensional dynamics 171 7.1 Transition from 2D to 3D waves . . . . 171

7.2 Three-dimensional model . . . . 173

7.3 Three-dimensional simulations . . . . 175

7.3.1 Isothermal simulations with periodic forcing . . . . 176

7.3.2 Isothermal simulations with natural noise . . . . 183

7.3.3 Non-isothermal simulations with periodic forcing . . . . 187

8 Further topics 191 8.1 Heat flux condition . . . . 191

8.1.1 Formulation at first-order . . . . 192

8.1.2 Formulation at second-order . . . . 193

8.2 Small Biot number limit . . . . 195

8.2.1 Analogy with forced convection . . . . 195

8.2.2 Temperature condition (TC) . . . . 196

8.2.3 Heat fluc condition (HFC) . . . . 197

Conclusions and perspectives 201 A Falling film applications 209 B Systems of 2D boundary layer equations 213 C Typical parameter values 215 D Full second-order models 217 D.1 Two-dimensional with non-isothermal conditions . . . . 217

D.2 Three-dimensional with isothermal conditions . . . . 219

Bibliography 223

Summary 235

R´ esum´ e 236