université lille 1films-lab.univ-lille1.fr/michael/michael/teaching_files/...while bernoulli evokes...

Université Lille 1

Microfluidics

Michael Baudoin ([email protected])

Contents

1 Introduction to interfacial flows 1

1.1 A brief history of Navier-Stokes equations . . . . . . . . . . . . . . . 1

1.2 Navier-Stokes bulk equations and equilibrium conditions at interfaces 6

1.2.1 Navier-Stokes equations . . . . . . . . . . . . . . . . . . . . . 6

1.2.2 Simplified expressions . . . . . . . . . . . . . . . . . . . . . . 13

1.2.3 Mass and momentum balance at interfaces . . . . . . . . . . . 23

1.3 Surface tension, Laplace pressure and Marangoni effect . . . . . . . . 25

1.3.1 Where does surface tension come from ? . . . . . . . . . . . . 25

1.3.2 Laplace pressure . . . . . . . . . . . . . . . . . . . . . . . . . 27

1.3.3 Marangoni effect . . . . . . . . . . . . . . . . . . . . . . . . . 29

1.3.4 Momentum equilibrium at interfaces with surface tension . . 30

1.4 Useful dimensionless numbers and illustrations . . . . . . . . . . . . 31

1.4.1 The Reynolds number . . . . . . . . . . . . . . . . . . . . . . 31

1.4.2 The Capillary number . . . . . . . . . . . . . . . . . . . . . . 33

1.4.3 The Weber number . . . . . . . . . . . . . . . . . . . . . . . . 33

1.4.4 The Bond Number . . . . . . . . . . . . . . . . . . . . . . . . 34

2 Interfaces and vibrations 37

2.1 Bubbles and the Rayleigh-Plesset equation . . . . . . . . . . . . . . . 38

2.1.1 Why are bubbles outstanding resonators ? . . . . . . . . . . . 39

2.1.2 Static equilibrium of bubbles . . . . . . . . . . . . . . . . . . 42

2.1.3 Modeling of the liquid phase: the Rayleigh-Plesset equation . 45

2.1.4 Modeling of the gas phase . . . . . . . . . . . . . . . . . . . . 53

2.1.5 From bubbles to metamaterials . . . . . . . . . . . . . . . . . 60

2.2 Inertio-capillary “Rayleigh-Lamb” modes of vibration . . . . . . . . . 61

2.2.1 Dimensional analysis . . . . . . . . . . . . . . . . . . . . . . . 61

2.2.2 Complete solution of the problem . . . . . . . . . . . . . . . . 61

2.2.3 Sessile droplets and possible use of these surface vibrations. . 65

3 Dynamics of bubbles or plugs in confined geometries 67

3.1 Two phase flow at small scales . . . . . . . . . . . . . . . . . . . . . . 67

3.2 Bretherton law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.2.1 Semi infinite bubble . . . . . . . . . . . . . . . . . . . . . . . 69

3.2.2 Long bubble motion . . . . . . . . . . . . . . . . . . . . . . . 78

3.3 Large capillary number and rectangular channels . . . . . . . . . . . 80

3.3.1 Large capillary numbers . . . . . . . . . . . . . . . . . . . . . 80

3.3.2 Rectangular channels . . . . . . . . . . . . . . . . . . . . . . . 81

Bibliography 83

Chapter 1

Introduction to interfacial flows

Contents

1.1 A brief history of Navier-Stokes equations . . . . . . . . . . 1

1.2 Navier-Stokes bulk equations and equilibrium conditions

at interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2.1 Navier-Stokes equations . . . . . . . . . . . . . . . . . . . . . 6

1.2.2 Simplified expressions . . . . . . . . . . . . . . . . . . . . . . 13

1.2.3 Mass and momentum balance at interfaces . . . . . . . . . . 23

1.3 Surface tension, Laplace pressure and Marangoni effect . . 25

1.3.1 Where does surface tension come from ? . . . . . . . . . . . . 25

1.3.2 Laplace pressure . . . . . . . . . . . . . . . . . . . . . . . . . 27

1.3.3 Marangoni effect . . . . . . . . . . . . . . . . . . . . . . . . . 29

1.3.4 Momentum equilibrium at interfaces with surface tension . . 30

1.4 Useful dimensionless numbers and illustrations . . . . . . . 31

1.4.1 The Reynolds number . . . . . . . . . . . . . . . . . . . . . . 31

1.4.2 The Capillary number . . . . . . . . . . . . . . . . . . . . . . 33

1.4.3 The Weber number . . . . . . . . . . . . . . . . . . . . . . . . 33

1.4.4 The Bond Number . . . . . . . . . . . . . . . . . . . . . . . . 34

The main objective of this workshop is to display basic knowledge of the physics

of interfaces along with real time experiments and numerical simulations. The range

of topics is expected to promote an exchange between students and scientists working

on contemporary issues of interfacial phenomena.

1.1 A brief history of Navier-Stokes equations

Interest in the behavior of fluids flow has been raised in early civilizations by the

fundamental need of transporting water for irrigation and supply. During the 3rd

century BC, Archimedes (287-212 BC) formulates “Archimedes principle”, which is

at the origin of hydrostatics (the study of fluids at rest). Afterwards, many tech-

nical treatises have been written during Roman empire describing the hydraulics

of aqueducts. One can cite the extensive work (“De aquaeductu”) of Sextus Julius

Frontinus (40-103 AD), a Roman aristocrate, whose treatise aims at describing the

water-supply of Rome. In 1643, Evangelista Toricelli (1608-1647), an Italian mathe-

matician sets out Toricelli’s law relating the speed of fluid flowing out of a container

2 Chapter 1. Introduction to interfacial flows

Figure 1.1: From left to right: Daniel Bernoulli, Jean le Rond d’Alembert, Leonhard

Euler, Henri Navier, George Gabriel Stokes

to the square root of the height of the fluid. Despite this early work on the dynamics

of fluids, Jean le Rond D’Alembert (1717-1783) dates back the origin of hydrodynam-

ics to the publication of “Hydrodynamica” in 1738 by Daniel Bernoulli (1700-1782)

[3] (Fig. 1.2). The work of Daniel Bernoulli differs from the one of his predecessors

since he establishes the relation between the velocity of a flow and gravity effects

from a general mechanics principle: the conservation of “vis viva” (living forces) .

This concept, proposed by Gottfried Leibniz (1646-1716), served as an elementary

formulation of the principle of conservation of energy. While Bernoulli evokes some

pressing [nisus], compression [compressio] and pressure [pressio], the concept of in-

ternal pressure does not appear directly in his equations and Bernoulli’s principle

is not formalized yet in its modern form.

Figure 1.2: First page of Daniel Bernoulli’s book “Hydrodynamica”

Using differential calculus introduced by Gottfried Leibniz and Isaac Newton

(1643-1727) and applying Newton’s laws of classical mechanics to a small amount of

fluid, Jean le Rond d’Alembert and Leonhard Euler (1707-1783) derive the so-called

Euler equations for an inviscid flow (frictionless fluids). Indeed, d’Alembert intro-

1.1. A brief history of Navier-Stokes equations 3

duces in 1749 the notions of “Eulerian” velocity field and partial derivatives [6]. He

also derives the incompressibility condition in an axially symmetric case. However

he fails to identify the role of internal pressure in the dynamics of incompressible

fluids. Between 1750 and 1755, Euler clarifies and generalizes the notions introduced

by d’Alembert, and includes both the notion of internal pressure and compressibility

in the equations. He obtains the current form of Euler equations [9] (see Fig. 1.3)

and also formalizes Bernoulli’s law for potential flow. It is interesting to see how

quickly new mathematical and physical concepts (differential calculus, Newton’s

laws of mechanics) have been assimilated by d’Alembert and Euler. The difficulties

at that time were to adapt them to a continuous medium with no reference state,

and to identify the role played by internal pressure.

Figure 1.3: Euler equations as written by Euler in 1755. In these equations, q is

the density, u, v, and w are the projections of the velocity on x, y and z axes, t is

the time, p the internal pressure, and P , Q, R the components of an external force

field (such as gravity) applied to the fluid.

However, Euler’s formulation of the equations suffers a contradiction with ex-

periments, the so-called d’Alembert paradox raised by this latter in 1752: the drag

force applied to an elliptical body moving with constant velocity in an inviscid flow

is zero. To solve this paradox, Henri Navier (1786-1836), a French scientist from

Ecole Polytechnique, introduces for the first time the “viscosity” of the flow in 1822

[24] (while the term “viscosity” does not appear in Navier’s manuscript). Navier

postulates the existence of a force between adjacent molecules which depends on

both the distance between them (repulsion force decreasing with the distance) and

their relative velocity (friction force). From this basic principle and by integrating

momentum on all molecules, he obtains the equations predicting the evolution of a

viscous incompressible fluid (Fig. 1.4). He also expresses the condition of adhesion

to a wall and the friction appearing between two fluid layers. Because of the use

of molecular theory and its initial hypotheses, some contemporary scientists remain

skeptical about Navier’s equations. In fact, Navier attributes to molecules a prop-

erty that should be attributed to fluid particles (that is to say fictitious particles


containing a large amount of molecules): the stress between adjacent fluid particles

depends linearly on their relative velocity and thus on the flow velocity gradient.

d~PCd~-i-u~y+~d~wdz~âx'~'a~+âz'

-n ~p <~ < /M'ddw.

dw dw

dw daw

daw

d2wRidz-PCdt+udx+vdy-I-dz~ E.~dxx+dy~+dz'

En second lieu à l'égard desconditions qui se rapportent

aux points de la surface du fluide, si l'on désigne., comme

on l'a fait plus haut, par 7,/M,Kles angles que le plan tan-Figure 1.4: Navier’s equations for a viscous incompressible flow obtained by Navier

in 1822. The notations are similar to Fig. 1.3, with ε the fluid viscosity.

A few years later (1834), Adhémar Barré de Saint-Venant derive again these

equations without the help of molecular theory, by considering the friction between

adjacent “fluid particles” (in the modern sense of continuum mechanics). While

this work is achieved in 1834, it is only published in 1843. In the meanwhile,

Georges Gabriel Stokes [34] (1819-1913) develops a continuum mechanics theory of

compressible viscous flow. He identifies the rate of dilatation, shifting and rotation,

and the corresponding tensors (of course without tensorial notation). Then, by

postulating a linear relation between stress and elongation and with the help of

symmetry considerations, he obtains the “almost” modern formulation of Navier-

Stokes equations (Fig. 1.5).

Figure 1.5: Navier-Stokes equations for a viscous compressible flow obtained by

Stokes in 1845. Here, ρ is the density, X the x-component of the external force field

and µ the shear viscosity.

The only difference between Stokes mass and momentum equations and current

formulation of these equations is the existence of a second coefficient of viscosity

in analogy with the two Lamé coefficients in solid mechanics. While some early

references to this coefficient appear in Lamb’s Hydrodynamics, it will be properly

introduced much later, in 1942, by L. Tisza from Massachusetts Institute of Technol-

ogy. Stokes theory predicts that this second viscosity coefficient is equal to −2µ/3,

where µ is the shear viscosity. At that time, Stokes is well aware of the deficien-

1.1. A brief history of Navier-Stokes equations 5

cies of his argument: he writes in 1851 “... I have always felt that the last term of

this equation (the term which includes the dilatational viscosity) does not rest on

as firm a basis as the correctness of the equation of motion of an incompressible

fluid”. However, since the second coefficient of viscosity plays no role in incom-

pressible hydrodynamics, no experiment contradicts this theory before the study of

the attenuation of acoustical waves. Indeed, this latter depends on both shear and

dilatational viscosity. At the beginning of the 20th Century, measured attenuation

coefficients in liquid differ from Stokes theory by factors 3 to 1000 (Stokes theory

is indeed only appropriate for ideal monoatomic gases). Different theories are ad-

vanced to explain this discrepancy (see Karim and Rosenhead [15]). Following solid

mechanics, Tisza introduces a second coefficient of viscosity [37].

Finally, to complete compressible Navier-Stokes equations, it is necessary

to introduce some law of energy conservation and state equations to establish

relations between pressure, density and temperature. But these laws inherited

from equilibrium thermodynamics are another part of history, which will not be

developed here.

Although Navier-Stokes equations have been formulated in current actual form

by Euler, Navier and Stokes in the 18th and 19th centuries, the existence and

smoothness of their solution has not been demonstrated yet in 3 dimensions. It is

one of the 7 Millenium problems proposed by Clay Mathematics Foundation (see

http://www.claymath.org/millennium/). “The challenge is to make substantial

progress towards a mathematical theory, which will unlock the secrets hidden in the

Navier-Stokes equations”. In the following manuscript, we will show that interfaces

can still add some complexity to these “mysterious” equations.

All the historical manuscripts evoked in this course can be down-

loaded from http://films-lab.univ-lille1.fr/michael/michael/History_of_

Navier_Stokes_equations.html.


1.2 Navier-Stokes bulk equations and equilibrium con-

ditions at interfaces

In this section, we will recall the different formulations of Navier-Stokes equations,

the physical interpretation of the various terms and possible simplifications of the

equations. This manuscript adopts a continuum mechanics point of view. Thus we

will often refer to the notion of “fluid particles”. A fluid particle is a volume of fluid

containing a large number of molecules, but whose size remains small compared to

the characteristic size of flow variations.

1.2.1 Navier-Stokes equations

Historical non-conservative form of the equations

From the historical derivation described in the first section, the following equations

have been obtained:

Mass conservation∂ρ

∂t+ div(ρ−→v ) = 0 (1.1)

Momentum conservation

ρ

(

∂−→v∂t

+−→v .−→∇(−→v ))

= −−→∇p+ µ∆−→v + (µ + µ′)−→∇(div(−→v )) +−→

fv (1.2)

with ddt =

∂∂t +

−→v .−→∇() the material derivative.

In these equations, ρ, −→v , p designate respectively the density, velocity and

internal pressure of the flow, µ and µ′ the shear dynamic viscosity and the second

coefficient of viscosity. We can note that the so-called “bulk viscosity” ξ = µ+2µ′/3

is sometimes introduced instead of the second coefficient of viscosity. We will now

try to understand the physical meaning of the different terms which appear in the

momentum conservation equation.

Unsteady term ∂−→v∂t

This term corresponds to the temporal variation of the flow. It appears only if

the velocity field evolves in time and is equal to zero otherwise.

Convective term −→v .−→∇(−→v )

To understand the physical meaning of the term −→v .−→∇(−→v ), we can consider a 1D

field represented by a function f(x, t), which maintains its shape while translating

along x-axis at a given velocity c. The equation describing the evolution in space

and time of this function can be obtained by considering two times separated by an

increment dt (see Fig. 1.6). Since this function propagates without deformation, we

1.2. Navier-Stokes bulk equations and equilibrium conditions atinterfaces 7

Am

plitu

de f(x,t) f(x+dx,t+dt)

x x+dx

dx = c dt

Space

Figure 1.6: Function f(x, t) translating at a velocity c.

have:

f(x+ dx, t+ dt) = f(x, t) as long as dx = c dt

Thus:

f(x+ dx, t+ dt)− f(x, t) =∂f

∂xdx+

∂f

∂tdt = 0

Since, dx = c dt, we obtain:∂f

∂t+ c

∂f

∂x= 0

Now, if we consider a 1D velocity field (f(x,t) = v(x,t)) and suppose that it is

convected by its own velocity (c = v(x,t)), we obtain the so-called inviscid Burgers’

equation:∂v

∂t+ v

∂v

∂x= 0

Finally, if we generalize this formula to a 3D velocity field, we obtain:

∂−→v∂t

+−→v .−→∇(−→v ) = 0

Thus the term −→v .−→∇(−→v ) corresponds to the convection of the velocity field by

itself. A fundamental issue is that this term is nonlinear since it is quadratic with

the velocity field −→v . The complexity of turbulent flow is intimately related to the

nonlinearity of this term.

Viscous diffusion term µ∆−→v

To understand the meaning of the viscous diffusion term, we will follow its

original derivation. Let’s consider two adjacent incompressible fluid particles flowing

along x-axis and separated by a distance dy along y-axis (see Figure below). If

the fluid particles have the same velocity (vx(y + dy) = vx(y)), then there is no

friction between them. However if they slide one over another, some friction (viscous


x−axisy+dy

y

v (y+dy)

v (y)

x

x

z−axis

y−axis

resistance) will appear at their interface. The shear stress (force per surface unit)

τxy exerted on the xz plane between the two fluid particles will thus depend on their

relative velocity δv = vx(y + dy) − vx(y) = ∂v∂ydy. If we suppose a linear relation

between these two quantities, we obtain:

τxy = µ∂vx∂y

(1.3)

We can note that viscous friction may have different microscopic origins depending

on the considered fluid. In gases, it is mainly due to the exchange of momentum

through “collisions” between molecules moving at different velocities. Indeed the

molecules in a fluid particle do not have the same velocity because of molecular

agitation (whose temperature is a measure). Thus, there are collisions between

molecules and some momentum is transmitted from regions where the average ve-

locity (and thus momentum) of the molecules is higher to regions where it is lower

(see Fig 1.7). An increase in the temperature contributes to raise the agitation of the

fluid and thus momentum exchanges, explaining why the viscosity of gas increases

with temperature. Sutherland’s law (which is valid for a large range of gases and

temperatures) indeed predicts this increase in viscosity µ with the temperature T :

µ(T )

µ(To)=

(

T

To

)3/2 To + S

T + S,

with To = 273.15K a reference temperature and S = 110.4K the Sutherland tem-

perature.

In liquids however, molecules are closer to each other and the attractive forces

between them (Van der Waals forces and hydrogen bounds), play a more important

role (see Fig 1.7). This attraction is at the origin of the friction which appears

between two fluid particles. In this case, when the temperature is raised, the

agitation of the molecules is increased and thus the molecules move more freely.


Thus for most liquids, an increase in temperature results in a decrease in its

viscosity (see oil in a warm pan).

vx(y+dy)

vx(y)

vx(y+dy)

vx(y)

Figure 1.7: Left: exchange of momentum between molecules in gas through collisions

leading to viscous friction between two fluid particles. Right: friction in liquid due

to attractive Van der Waals forces and hydrogen bounds (thick blue lines on the

graph).

Now, from relation 1.3, we can compute the volume force fµ applied to an

infinitesimal volume δV and resulting from the viscous friction which appears at its

top and bottom surfaces (see Fig. 1.2.1):

−→fµ = [(τxz(y + dy)dxdy − τxz(y)dxdy)]/dxdydz = µ

∂2vx∂y2

(y)

τ xz(y+dy)

y−axis

z−axis

y+dy

y τ xz

If we generalize this formula to a 3D velocity field depending on all directions

(−→v = vx(x, y, z)−→x + vy(x, y, z)

−→y + vz(x, y, z)−→z ), we obtain:

−→fµ = ∆−→v

An interesting point, is that, if we consider only the unsteady and diffusion terms

of Navier-Stokes momentum equation, we get:

∂−→v∂t

= ν∆−→v


with ν = µ/ρ, the kinematic viscosity. This equation is the 3D analogous of

Fick’s law of molecular diffusion ∂φ∂t = D∆φ or heat diffusion equation ∂T

∂t = κ∆T ,

where φ and T are respectively the concentration of a given component, T the

temperature, and finally D and κ the diffusion coefficient of species and heat. All

these phenomena share the same diffusion equation and all refer (at least for gases)

to the transmission of momentum between molecules.

Viscous dilatation term (µ+ µ′)−→∇(div(−→v ))

This term represents the viscous dissipation due to the dilatation of fluid par-

ticles. Indeed, it depends on the divergence of the velocity div(−→v ), which vanishes

when the flow is incompressible (see section 1.2.2). The velocity divergence div(−→v )gives a measure of the local variation of fluid particles volume.

An interesting point is that the viscous dilatation force is proportional to the

gradient of this local change of volume. It means that if all the fluid particles were

dilating the same way, there will be no resulting volume force.

External force field−→fv

This term represents any volume force deriving from an external force field. In

most cases, this term simply corresponds to the effect of gravity ρ−→g . However,

other force fields can play a role in the flow dynamics such as magnetic or electric

ones (see e.g the behavior of a ferrofluid submitted to a magnetic field or in

astrophysics, the study of flows at the surface of the sun, Fig. 1.2.1).

Figure 1.8: Left: Ferrofluid moved by a magnetic field. Right: NASA image of the

flow at the sun surface during two simultaneous solar eruptions.

General considerations

Equations 1.1 and 1.2 do not form a closed set of equations since there are 5


unknown variables (the density ρ, the 3 projections of the velocity field ~v and the

pressure p) and only 4 equations (one equation of continuity and 3 projections of

the equation of momentum conservation). One equation is missing; a state equation

giving the relation between the pressure and the density of the fluid is required.

However, according to Gibb’s phase rule (introduced by Josiah Williard Gibbs in

the 1870s), the number of degrees of freedom of a single component, in a single

state (phase) and in absence of chemical reaction is equal to 2. It means that the

pressure depends on both the density ρ and another thermodynamical parameter

such as the temperature or the entropy: p = p(ρ, s). Thus another equations (the

equation of energy conservation) is required to close the system. In this course, we

will not develop further these thermodynamic considerations.

Conservative form of the equations

Navier-Stokes equations can be re-written under the following form by combining

properly the mass and momentum conservation equation and introduce the stress

tensor⇒

σ= −p⇒

I +2µ⇒

D +µ′div(−→v )⇒

I , with⇒

D= 1/2(

⇒

∇ −→v +T⇒

∇ −→v)

the rate of

strain tensor:


∂t+ div(ρ−→v ) = 0 (1.4)

Momentum conservation∂ρ−→v∂t

+ div(ρ−→v ⊗−→v ) = div(⇒

σ ) +−→fv (1.5)

with ⊗ the tensor product.

Demonstration

We will demonstrate this equation by using Einstein notation (summation con-

vention):

∂ρ−→v∂t

+ div(ρ−→v ⊗−→v ) = ∂ρvi∂t

+∂ρvivj∂xj

= vi

[

∂ρ

∂t+∂ρvj∂xj

]

+ ρ

[

∂vi∂t

+ vj∂vi∂xj

]

= −→v[

∂ρ

∂t+ div(ρ−→v )

]

+ ρ

[

∂−→v∂t

+−→v .−→∇(−→v )]

The first term in square bracket is null because of mass conservation and we therefore

obtain:

∂ρ−→v∂t

+ div(ρ−→v ⊗−→v ) = ρ

[

∂−→v∂t

+−→v .−→∇(−→v )]


For the second part of the equations, we have:

div(⇒

σ ) =∂

∂xj

[

−pδij + 2µ∂

∂xj

[

1

2

(

∂vi∂xj

+∂vj∂xi

)]

+∂

∂xj

[

µ′∂vk∂xk

δij

]]

= − ∂p

∂xi+ µ

∂vi∂x2j

+ (µ′ + µ)∂

∂xi

∂vk∂xk

= −−→∇p+ µ∆−→v + (µ′ + µ)−→∇(div(−→v ))

where δij is Kronecker delta equal to 1 when i = j and 0 otherwise.


While this form of the momentum equation, called conservative form, has

been obtained (here and historically) after the non-conservative form, it derives

directly from momentum conservation principle. The non-conservative form of the

momentum equation requires the mass conservation to be fulfilled. Of course, this

condition is most of the time verified. But for complex two-phase flow with phase

change or fluids with nuclear transformation, this momentum equation can no

longer be used. In addition, it plays a fundamental role in numerical simulations:

numerical schemes are generally obtained from this formulation of the equations.

Integral form of the equations

By integrating the conservative form of the equations over a volume Ω following the

movement of the fluid, we obtain:

Mass conservationd

dt

∫∫∫

ΩρdV = 0 (1.6)

Momentum conservationd

dt

∫∫∫

Ωρ−→v =

∫∫

∂Ω

⇒

σ .−→n dS +

∫∫∫

Ω

−→fvdV (1.7)

These equations describe balance of mass and momentum over an arbitrary volume

Ω.

Demonstration

To obtain this result, we first integrate the conservative form of local Navier-

Stokes equations and then use Green-Ostrogradski formula:∫∫∫

Ωdiv(

−→ψ )dV =

∫∫

∂Ω

−→ψ .−→n dS,

which relates the integral over a volume Ω of the divergence of any continuous

vectorial field−→ψ to an integral over the surrounding surface ∂Ω, −→n being the normal


Ω

−→n

∂Ω

vector. In that way, we obtain the following formula:

Mass conservation∫∫∫

Ω

∂ρ

∂t+ div(ρ−→v )dV = 0 (1.8)

Momentum conservation∫∫∫

Ω

∂ρ−→v∂t

+ div(ρ−→v ⊗−→v ) =∫∫

∂Ω

⇒

σ .−→n dS +

∫∫∫

Ω

−→fvdV (1.9)

Then, we use the expression of the material derivative ddt of the integral of a field

ψ over a volume Ω following the motion (that is to say that the volume Ω moves

according to the velocity field −→v ):

d

dt

∫∫∫

ΩψdV =

∫∫∫

Ω

∂ψ

∂t+ div(ψ−→v )dV


In classical mechanics of continuum, these equations are in fact the starting

point since they derive directly from balance laws (mass and momentum) and do

not require the knowledge of constitutive equations, which are specific to a medium.

1.2.2 Simplified expressions

Incompressible flow: Navier equations

Derivation of the incompressibility condition

There are different ways to demonstrate the mathematical expression of this

condition. A first way is to consider an infinitesimal volume of fluid δV and state that

during a time interval dt, the same volume of fluid enters and exits this infinitesimal

volume. Through an infinitesimal surface dS, the amount of fluid which crosses the

surface is the quantity −→v .−→n dSdt. Thus, if we consider all the infinitesimal surfaces

surrounding δV , we get:


n=z

n=y

n=−y

n=−z

v(x+dx,y,z)v(x,y,z)

v(x,y,z)

v(x,y+dy,z)

n=xn=−xv(x,y,z)

v(x,y,z+dz)

[vx(x+ dx, y, z) − vx(x, y, z)] dydzdt+ [vy(x, y + dy, z) − vy(x, y, z)] dxdzdt

+ [vz(x, y, z + dz)− vz(x, y, z)] dxdydt =

[

∂vy∂x

+∂vy∂y

+∂vz∂z

]

dxdydzdt =

div(−→v )δV dt = 0

that is to say:

div(−→v ) = 0 (1.10)

This equation can also be obtained by noticing that, since the flow is incom-

pressible, any domain of fluid Ω keeps its volume constant when it is transported

by the flow, that is to say if we use the material derivative of integrals:

d

dt

∫∫∫

Ω1 dV =

∫∫

∂Ω

∂1

∂t+ div(−→v )dV =

∫∫

∂Ωdiv(−→v )dV = 0

Since it is valid for any domain Ω, we find again div(−→v ) = 0.

Incompressible Navier-Stokes equations

As a consequence, the conservation equations adopt the following form:

Mass conservation

div(−→v ) = 0 (1.11)


ρ

(

∂−→v∂t

+−→v .−→∇(−→v ))

= −−→∇p+ µ∆−→v +−→fv (1.12)

In this case, since the density remains constant on fluid particle trajectories

(and therefore is no more an unknown variable), the system is closed. It means that

for an incompressible flow, the equations of energy conservation and the equations

of dynamics (mass+momentum) are uncoupled.


When can a flow be considered as incompressible ?

While the incompressibility condition is rather simple to obtain mathematically,

it is harder to define which flow can be considered as incompressible. Several situa-

tions can lead to an influence of the compressibility of the flow on its dynamics.

• Supersonic flow:

When the characteristic speed of the flow U is larger than the sound speed

co, that is to say when the Mach number M = U/co ≥ 1, the fluid can

experience some brutal variations of density, temperature and velocity. In

this case, the compressibility of the fluid cannot be neglected. The first

question that one can ask is: why is the sound speed the reference velocity ?

The sound speed can be viewed as the speed of propagation of information in

a fluid. For example, when you turn off the tap at home, the movement of

the fluid does not stop instantaneously in the whole pipe. The information

that the pipe is closed at its extremity propagates in the pipe at sound speed.

Thus imagine the following situation: a plane is moving in the air at a speed

smaller than the sound speed. In this case, the information that the plane is

arriving travels more rapidly than the plane. Thus fluid particles at rest are

“aware” that the plane is arriving and the flow can “adapt” to the arrival of

the plane. But if the plane is traveling at a larger velocity than the sound

speed, then fluid particles learn about its arrival at the last moment and the

flow will adapt brutally to the boundary conditions imposed by the plane.

These brutal variations of temperature, pressure and velocity are called a

shock and have been observed for the first time around a projectile by P.

Salcher and S.Riegler in 1887 [31], following E. Mach suggestion (see Fig. 1.9).

Figure 1.9: Left: first observation of shock waves around a projectile by P. Salcher

and S. Riegler following Mach’s suggestion. The shock is visible since the large

variations of density and temperature modify the light refraction index. Right:

shock wave around an airplane made visible by the condensation of water induced

by the variations of temperature and density.


• Natural convection:

Natural convection is a flow produced by variations of temperature in a fluid,

leading to variations of density and hence buoyancy forces. Indeed, the hottest

regions of the fluid are less dense, and thus buoyancy forces induce a rise of

these regions. Natural convection is commonly observed in a room heated in

winter. In this case, heating leads to recirculation of air inside the room. An

academic example is the appearance of Benard cells in a fluid with a sufficient

viscosity submitted to up and down temperature gradients (see Fig. 1.10).

When natural convection is involved, the change of density due to tempera-

Figure 1.10: Left: natural convection flow due to heaters in an house. Right: Benard

cells appearing in gold paint due to the cooling of the top surface by evaporation of

a solvent (acetone).

ture gradients is at the origin of the fluid motion and thus the compressibility

cannot be neglected. In some cases however (Boussinesq approximation for

weak variations of density), the incompressibility condition div(−→v ) = 0 re-

mains valid to describe mathematically natural convection at first order. The

compressibility of the fluid appears only through a buoyancy force in the mo-

mentum equation. It does not mean that the compressibility plays no role in

the equations but only that at first order, the compressible terms appearing

in the mass balance can be neglected.

• Acoustic waves

A last example in which the fluid compressibility plays a fundamental role

is the propagation of acoustic waves. In fluid, acoustic waves propagate as

disturbance of pressure and thus density. These pressure perturbations can be

extremely weak (the threshold of human hearing is about 20 µPa). To give

a point of comparison, this corresponds to the variation of pressure (due to

gravity field) that you experience when you climb a ladder of ... “20 microns”.

However, these variations can be extremely rapid since the audible frequency

range (for average people) lies between 20 Hz to 20 kHz.


Incompressible creeping flow: Stokes equations

Stokes equations are valid for a steady incompressible creeping flow, that is to say a

flow with a low Reynolds number. The Reynolds number compares the convective

term to the diffusive one (see definition in section 1.4.1). When the Reynolds number

is small, convection can be neglected compared to viscous diffusion and the equations

become linear:

Mass conservation

div(−→v ) = 0 (1.13)

Momentum conservation−→∇p = µ∆−→v (1.14)

The linearity of these equations implies many physical and mathematical interesting

properties:

• Instantaneity: There is no time derivative in the Stokes equations. Thus

the flow depends only on instantaneous boundary conditions and there is no

memory of previous velocity fields.

• Superposition principle: If the fields (−→v 1, p1) and (−→v 2, p2) are solutions of

Stokes equations and satisfy respectively the boundary conditions BC1 and

BC2, then the fields (λ1−→v 1+λ2

−→v 2, λ1p1+λ2p

2) (with λ1, λ2 two constants)

are solutions of Stokes equations with the boundary conditions λ1BC1 +

λ2BC2.

• Reversibility: If the boundary conditions are reversed, then the inverse flow

−−→v ,−−→∇P is solution of the problem.

Figure 1.11: Motion of microorganisms using cilia (left) and flagella (right)

This last property has many important implications. For example, it is very com-

plicated to mix some fluids using creeping flow since “complicated patterns” do not

appear naturally like in turbulent flows. Second, some specific strategies must be


developed by microorganism for their locomotion, since the Reynolds number is gen-

erally small at these scales. Because of the reversibility property, symmetric motions

as birds wings flap does not produce any propulsion. Thus the symmetry must be

broken see e.g. the collective motion of cilia or the rotative motion of flagella (Fig.

1.11).

To conclude, we can underline that analytical solutions of Stokes equations exist

since the pressure field verifies Laplace equation :

∆p = 0,

whose analytical solutions are called harmonic functions. Thus, creeping flow

with complex boundary conditions can be computed very rapidly by using fast

summation of harmonic functions (see Fast Multipole Method [11]).

Demonstration:

Laplace equation is obtained by taking the divergence of equation (1.14) and

using the double-curl formula:−→∇ × (

−→∇ × (−→v )) = −→∇div(−→v )−∆−→v :

∆p = µdiv(∆−→v ) = µ div(−→∇ × (

−→∇ × (−→v ))− div(−→∇(div−→v )))

Since div(−→∇ × (ψ)) = 0 for any field ψ and div(−→v ) = 0, we get:

∆p = 0.

Inviscid flow: Euler equations

Euler’s equations describe the motion of a fluid with no viscosity. Of course, all

fluids have a viscosity, except superfluids (like superfluid helium-4). However, these

equations remain valid when the magnitude of the convection term largely overcomes

the magnitude of the viscous diffusion term, that is to say when Re ≫ 1. The

equations become:


∂t+ div(ρ−→v ) = 0 (1.15)


ρ

(

∂−→v∂t

+−→v .−→∇(−→v ))

= −−→∇p+−→fv (1.16)

Since Euler equations are valid for compressible flows, the density ρ is not con-

stant, and as for the Navier-Stokes equations, state and energy conservation equa-

tions are required to close the system.

Euler’s equations are widely used to describe the dynamics of flow in supersonic

(1 < M = Uco< 5) and hypersonic regimes (5 < M = U

co< 10), see e.g. Fig. 1.12.

Acoustic waves equations


Figure 1.12: Hypersonic flow around a space shuttle simulated with the commercial

code ANSYS. One can see the shock waves (in green) appearing in front of the

different parts of the shuttle.

From Euler’s equations, one can easily obtain the equations describing the prop-

agation of acoustic waves by considering an adiabatic evolution of a small pertur-

bation field (p1,−→v 1) around a steady state (po,

−→v o =−→0 ):

p = po + εp1−→v = 0 + ε−→v 1

ρ = ρo + ερ1 (1.17)

with ε≪ 1. If we replace these expressions in equations (1.15) and (1.16), we obtain

at first order:

Mass conservation∂ρ1∂t

+ ρodiv(−→v 1) = 0 (1.18)


ρo∂−→v∂t

= −−→∇p1 (1.19)

Since the evolution is adiabatic, we have c2o = ∂p∂ρ = p1

ρ1, with co the sound speed.

Now if we subtract ∂∂t (1.18) - div(1.19) and replace p1 by c2oρ1, we obtain the wave

equation:∂2ρ1∂t2

− c2o∆ρ1 = 0.

Inviscid incompressible flow and Bernoulli’s principle

Bernoulli’s principle expresses a conservation of kinetic energy in the absence

of viscous friction: the variation of kinetic energy is balanced by a variation of


potential energy and the work of pressure forces.

Original Bernoulli’s principle

Bernoulli’s original theorem is applicable if:

• the flow is incompressible div(−→v ) = 0 and hence ∂ρ∂t = 0

• the external force field is conservative: ∃ U,−→fv = −−→∇U , where U is the scalar

potential associated with fv

• the flow is inviscid (µ = 0)

• the flow is steady ∂−→v∂t =

−→0

Then, along a streamline:

p

ρ+v2

2+ U = constant

Demonstration

To demonstrate this theorem, the (steady) momentum conservation of Euler

equation (1.16) can be rewritten under the following form from previous hypotheses

and the relation −→v .∇(−→v ) = −→∇(

v2

2

)

−−→v ×−→∇ × (−→v ):

−→∇(

p

ρ+v2

2+ U

)

= −→v ×−→∇ × (−→v ) (1.20)

Since streamlines are by definition parallel at any point to the velocity field, any

infinitesimal displacement−→dl along a streamline is parallel to −→v . The vector −→v ×−→∇ × (−→v ) is therefore perpendicular to −→v and the dot product of equation (1.20)

with the displacement−→dl gives:

−→∇(

p

ρ+v2

2+ U

)

.−→dl = 0 (1.21)

Thus the function H =(

pρ + v2

2 + U)

is constant.

Unsteady Bernoulli’s principle

The last hypothesis (steady flow) can be released if the flow is irrotational (−→∇×

(−→v ) = 0). In this case, the velocity field is such that ∃ φ,−→v =−→∇φ. Bernouilli’s

principle becomes (at any point of the flow):

∂φ

∂t+p

ρ+v2

2+ U = K(t) (1.22)


where K is a time-dependent function.

Demonstration

The demonstration is similar to the previous one but this time unsteady Euler

equation can be written under the form:

−→∇(

∂φ

∂t+p

ρ+v2

2+ U

)

= 0 (1.23)

Thus the function K =(

∂φ∂t +

pρ + v2

2 + U)

only experiences times variations.


Figure 1.13: Boundary layer appearing on the walls of a projectile flowing at super-

sonic speed. The boundary layer perturbs the projectile wake.

As already mentioned in the historical introduction of this manuscript, these

inviscid incompressible equations suffer d’Alembert’s paradox: the drag force ap-

plied on a body moving with constant velocity relative to the fluid is zero. Indeed,

since the viscosity of the fluid is neglected, the fluid particles slip on the walls and

there is no transmission of tangential momentum. This paradox has been solved

by the discovery and description of boundary layers by Ludwig Prandtl in 1904.

The boundary layer is a thin layer around moving bodies where viscosity cannot be

neglected anymore (see Fig. 1.13). Indeed, the Reynolds Number Re = ρULµ , which

compares convective effects to viscous diffusion is generally computed by taking as

reference length scale L, the average size of the considered object (see definition

in section 1.4.1). If the body moves fast enough this Reynolds number is large.

However if we zoom on a thin layer around this body, the characteristic length


scale becomes smaller and smaller and, at a given point, the Reynolds number is

no more ≫ 1. In the corresponding layer, viscous diffusion cannot be neglected

anymore as compared to convection. This thin layer introduces some friction drag

Figure 1.14: Flow separation behind a wing tilted with a large angle compared to

the direction of the flow

and for bluff bodies results in flow separation (see Fig. 1.14) and low pressure wake

behind the considered object.

Another surprise when we look at inviscid incompressible flow is how much the

flow is regular (see Fig. 1.15) despite the magnitude of the the nonlinear convective

term. Turbulence cannot be obtained in inviscid flows. Indeed, while turbulence

appears at high Reynolds number it requires a hint of viscosity to develop.

Figure 1.15: Inviscid incompressible flow around a cylinder


1.2.3 Mass and momentum balance at interfaces

Bulk equations have been established and simplified for different physical cases.

Some boundary conditions are now required for the problem to be well-posed. It is

at this stage that the complexity induced by rigid or deformable interfaces appears.

An interface is a discontinuity of the flow properties (density, velocity, pressure,

viscosity). Such discontinuity can appear between two different phases (a liquid and

a gas, two immiscible liquids, a liquid and a solid, ...) but also in the same phase

for compressible flow. In this last case, they are called shocks. In this section, we

will establish the balance equations at these discontinuities. While these equations

are valid for both interfaces between two phases and shocks, this second case will

not be developed further in this manuscript.

Continuity condition

dS

−→nρ2, ~v2, p2, µ2, µ

′2

ρ1, ~v1, p1, µ1, µ′1

The continuity condition expresses the mass conservation across an interface.

Let’s consider an infinitesimal interface dS between two phases 1 and 2, −→n being

the normal to the surface oriented from fluid 1 to fluid 2. To establish the balance

equations, the mass of fluid crossing the surface during an infinitesimal time dt must

be estimated. As a starting point, we will suppose that the interface is motionless.

The fluid particles move in phase 1 with a velocity −→v 1. If they move tangentially

to the surface, they will never cross the interface. Thus only the normal velocity−→v 1.

−→n contribute to the balance. Now, if the fluid particles are too far away from

the surface, they will never reach it during the infinitesimal time dt. So we must

determine how far the fluid particles can lie from the interface to cross it during

dt. Since they are moving towards the interface with a velocity −→v 1.−→n , this distance

is naturally, −→v 1.−→n dt. Finally, to get the volume of fluid particles crossing the

interface, we have to multiply this distance by the infinitesimal surface dS, and to

get their mass dm1, to multiply it by ρ1: dm1 = ρ1−→v 1.

−→n dtdS. This mass which

comes from phase 1 to cross the interface is equal to the mass of fluid which comes

from the interface and moves into phase 2, that is to say dm2 = ρ2−→v 2.

−→n dtdS. Thus

we get:

ρ1−→v 1.

−→n dtdS = ρ2−→v 2.

−→n dtdS.

If the interface is moving with a velocity −→u , the balance is the same but in the


referential frame of the interface, that is to say (if we simplify dtdS on both sides):

ρ1(−→v 1 −−→u ).−→n = ρ2(

−→v 2 −−→u ).−→n (1.24)

For interfacial flow, this condition expresses phase change, that is to say for a

fluid/gas interface, evaporation or condensation. In the absence of such phase

change, (and in the absence of shock), there is no mass crossing the interface and

the continuity condition becomes:

−→v 1.−→n = −→v 2.

−→n = −→u .−→n (1.25)

Momentum condition

Momentum transfers through an interface can occur in two different ways. First,

the fluid which crosses the interface (phase change) from phase 1 to 2 carries with

it some momentum ρ1−→v 1. Since, the volume of fluid which crosses an infinitesimal

surface dS during an infinitesimal time dt is (−→v 1−−→u ).−→n dtdS, then the momentum

transferred by phase change from phase 1 to 2 is: ρ1−→v 1(

−→v 1 − −→u ).−→n dSdt. Some

momentum is also exchanged because of the stresses⇒

σ1 and⇒

σ2 applied on each side

of the interface. Since the surface normal is directed from fluid 1 to fluid 2, the

stress applied by phase 1 on phase 2 on dS during dt is: −σ1.−→n dSdt. Thus the

momentum balance can be written under the form:

(ρ1−→v 1(

−→v 1 −−→u )− ⇒

σ 1).−→n = (ρ2

−→v 2(−→v 2 −−→u )− ⇒

σ 2).−→n (1.26)

It is important to underline that we suppose here that equilibrium is reached at

the interface and that the interface has no inner mass, momentum or density. This

hypothesis will be discussed further in section 1.3.

No phase change

In absence of phase change, −→v 1.−→n = −→v 2.

−→n = −→u .−→n , and thus the momentum

conservation at the interface reduces to the stress equilibrium condition:

⇒

σ1 .−→n =

⇒

σ2 .−→n

Rigid body

In the case of an interface with a rigid body the conditions at the interface reduce

to a continuity of the velocity field:

−→v 1 =−→v 2

This condition of continuity must be used with care and is inappropriate in specific

cases, e.g. the study of flow in superhydrophobic microchannels, or the study of gas

flow around objects of the same size as the mean free path of molecules.

1.3. Surface tension, Laplace pressure and Marangoni effect 25

1.3 Surface tension, Laplace pressure and Marangoni ef-

fect

1.3.1 Where does surface tension come from ?

Interactions between molecules

Surface tension appears at the interface between two immiscible fluids. To ex-

plain it, one must first understand the nature of molecular interactions in fluids.

A molecule is made of dense positive protons and uncharged neutrons surrounded

by orbits of negatively charged electrons. Positive and negative charges compen-

sate one another resulting in a global neutral charge for the molecule. However

the heterogenous distribution of these charges induces some interactions between

neighboring molecules. Following Pauli’s exclusion principle, two electrons can-

not occupy the same orbit (or more rigorously, the same quantum state). Thus,

when two molecules are brought close to one another, a short range repulsive force

appears since the negatively charged electronic clouds cannot overlap one another.

Moreover, molecules behave as permanent and induced dipoles and some attractive

intermolecular forces can also appear: the Van der Waals forces. They can be

classified into three categories:

• Keesom forces between two permanent dipoles

• Debye forces between a permanent dipole and the corresponding induced

dipole

• London dispersion force between instantaneously induced dipoles.

1 1.5 2 2.5−2

−1

0

1

2

3

4

5

Distance r/σ

Pot

entia

l V(r

/σ)

ε

Figure 1.16: Lennard-Jones potential


The interaction between pair of neutral atoms or molecules is commonly approxi-

mated by the so-called Lennard-Jones potential which takes into account both the

short range Pauli repulsive term and the long range Van der Waals attractive term

(see Fig. 1.16):

VLJ = 4ε

[

(σ

r

)12−(σ

r

)6]

with ε the depth of the potential well, σ the distance at which the inter-particle

interaction is zero and r the distance between the particles. It is important to note

that while a comprehensive view of these interactions can be given through simple

electrostatic analogies, the correct explanations of the existence of Van der Waals

forces lies in quantum mechanics and is well beyond the objective of this course.

Finally, in some common liquids (such as water), so called attractive hydrogen

bonds can appear between hydrogen atoms and electronegative atoms such as

nitrogen, oxygen or fluorine. These bonds are stronger than Van der Waals

interactions but weaker than covalent or ionic bond.

Interface between a condensed and a gas phase:

GAS

Figure 1.17: Interactions of molecules in the bulk (left) and at a liquid-gas interface

(right). The green attractive interactions disappear when an interface is created.

Fluids molecules experience constant thermal agitation, whose temperature is a

measure. When the attractive forces (described in the previous section) dominate

thermal agitation, the molecules condensate and form a liquid phase. Thus, there

are some strong interactions between neighboring molecules in a liquid and an “equi-

librium” is reached between them. In gases however, the molecules are generally too

far away to feel the attractive and repulsive porentials and they only interact when


they collide. Thus, at the interface between a liquid and a gas, liquid molecules

lose half of their attractive forces and are not in equilibrium. The existence of this

interface “costs” some energy. To reach a minimum energy state, a volume of liquid

surrounded by a gas will deform to reduce its interface. That is why a drop (in

levitation) adopts a spherical shape which minimizes its surface over volume ratio.

The surface tension γ is simply the coefficient which relates the work δW which is

required to stretch a surface to the corresponding increase of this surface dA:

δW = γdA

Interface between two immiscible liquids

For the same reason, a surface tension exists between two immiscible liquids 1

and 2 since the attractive interactions lost by phase 1 because of the presence of the

interface are not compensated by the interactions with phase 2.

1.3.2 Laplace pressure

Origin of Laplace pressure

dR

pp

1

2

R

Figure 1.18: Pressure jump through a spherical interface due to surface tension.

Since the surface tension tends to reduce the interface of a fluid 1 surrounded by

a fluid 2, an excess of pressure appears in fluid 1: p1 > p2. It can be computed by

considering a spherical drop of fluid 1 surrounded by an immiscible fluid 2. Then,

the virtual work which is required to increase its radius by dR is:

δW = −p1dV1 − p2dV2 + γ12dA

with dA = d(4πR2)dR dR = 8πRdR the increase of surface, dV1 = −dV2 = d(4/3πR3)

dR =

4πR2dR the increase of volume, and γ12 is the surface tension between fluid 1 and


2. With the equilibrium condition δW = 0, we obtain the following condition:

∆p = p1 − p2 =2γ12R

= γ12C

with C the curvature of the surface.

Expression for any surface

Intersection lines

R

R’

P

mardi 24 mai 2011

Figure 1.19: Calculation of the curvature of a surface.

The sphere is a specific case since its curvature is constant and its two

principal radii of curvature are equal to the radius of the sphere. In a general

case, the mean curvature of a surface at a given point P can be computed by

plotting the normal vector −→n at this point and then considering the intersection

lines between the considered surface and some planes containing −→n . These

intersection lines all have a radius of curvature in P. The mean curvature is

the sum of the minimum and maximum radius of curvatures R and R′. The

corresponding planes are called planes of principal curvatures. The radius of curva-

ture is positive when the circle (in green) is inside the object and negative otherwise.


Thus in the general case, Laplace law can be written:

∆P = γC = γ

(

1

R+

1

R′

)

with C the mean curvature.

1.3.3 Marangoni effect

Marangoni effect is the mass transfer along an interface due to surface tension gra-

dients. Such gradients of surface tension can appear because of the dependence

of surface tension on temperature, or on the concentration of chemicals in a fluid

mixture. An originally plane interface will be deformed by these surface tension

gradients.

Figure 1.20: Wine tears at the surface of a glass.

A well known consequence of Marangoni effect is the wine tears. The five basic

steps for wine testing are “see, swirl, sniff, sip and savor”, the famous “five S”. When

the wine is swirled, a thin film of liquid is deposited on the walls. This film rises

progressively and thickens at the top. At a given point, some droplets form and

flow slowly down the inside of the glass. We will not develop the instability process


of drop formation and will focus on the issue of the film thickening at the top

despite gravity effects. Wine is mainly made of alcohol and water. Alcohol has a

smaller surface tension than water and is more volatile. Since the liquid film is thin,

and its surface/volume ratio important, the alcohol evaporates rapidly. Thus the

concentration of alcohol decreases and the surface tension increases. This excess of

surface tension pulls the film to the top and causes more liquid to be drawn up from

the bulk. When the film becomes too thick it becomes unstable and some liquid

drop form and flow down due to gravity.

1.3.4 Momentum equilibrium at interfaces with surface tension

If we take into account Laplace pressure and Marangoni effect in the momentum

condition at the interface (eq. 1.26), we get the following expression:

(ρ1−→v 1(

−→v 1 −−→u )− ⇒

σ 1).−→n = (ρ2

−→v 2(−→v 2 −−→u )− ⇒

σ 2).−→n (1.27)

+ γ12div(−→n )−→n +

(

⇒

I −−→n ⊗−→n)

.−→∇γ12

with −→n the normal vector oriented from fluid 1 to 2. The two additional terms

on the second line correspond respectively to Laplace pressure and Marangoni ef-

fects. Indeed, the curvature C can be expressed as a function of the normal vector

C = div(−→n ). Since, the surface tension introduces a pressure jump, the associated

momentum will be oriented towards −→n . On the opposite, Marangoni effect induces

a tangential force in the interface plane and it is proportional to surface tension

gradient−→∇γ12.

1.4. Useful dimensionless numbers and illustrations 31

1.4 Useful dimensionless numbers and illustrations

1.4.1 The Reynolds number

Reynolds number whose terminology has been introduced by Sommerfeld in 1908

[30] is named after Osborne Reynolds (1842-1912), an Irish scientist working at

the University of Manchester. Reynolds studied the transition from laminar to

turbulent flow in pipes (see setup on Fig. 1.21) and introduced similarity parameters

by dimensional analysis. It is interesting to note that the subcritical transition in

pipes is still a controversial issue [8], while considerable progress has been made in

the understanding of underlying mechanisms. The Reynolds number compares the

Figure 1.21: Sketch of Reynolds experiments studying the transition from laminar

to turbulent flow in pipes. The original setup is still available at the University of

Manchester and it has been reproduced recently in the same condition. Since the

transition is subcritical and depends on the level of perturbation, a much smaller

critical Reynolds number of transition from laminar to turbulent flow has been

measured, ... a consequence of the large increase in ambient noise.

magnitude of convective terms to the one of diffusive terms. If U is the characteristic

velocity of the flow, L the characteristic size of its variations, ρ the density and µ


the dynamic viscosity, we get:

Re =Convection

Viscous diffusion=

[ρ−→v .∇(−→v )][µ∆−→v ] =

ρUL

µ(1.28)

Re = 10 Re = 26

Re = 300 Re = 2000

Re = 10000

Figure 1.22: Flow over a cylinder at different Reynolds number.

A classical example illustrating the role played by the Reynolds number is the

flow around a cylinder. According to its value, the flow is more or less complex

(see Fig. 1.22). At low Reynolds numbers (Re<10), the flow is laminar. When

the Reynolds number increases (Re ∼ 26), some vortex appear in the wake of the

cylinder. At Re ∼ 300, they are shed from each side of the body, forming rows of


vortices in its wake, the so-called Karman vortex street. At Re ∼ 2000, there is

flow separation and the wake behind the body becomes turbulent at Re ∼ 10000.

Flow separation is a primary issue in Aerodynamics, since it results in increased

drag because of the difference of pressure between the front and the rear surfaces of

the object.

1.4.2 The Capillary number

The capillary number compares viscous effects to surface tension:

Ca =Viscous diffusion

Surface tension=µU

γ

with U the characteristic speed of the flow, µ the viscosity and γ the surface

tension. This number is widely used in two-phase microfluidics, where the Reynolds

number is small and therefore viscous effects are dominant compared to convective

effects.

The capillary number gives a measure of the ability of a viscous flow to deform

an interface. For example, let’s consider a droplet of a liquid 1 moving in an

immiscible liquid 2. As long as the capillary number is small, it keeps its spherical

shape due to surface energy minimization. However, if the capillary number

becomes large enough, the flow deforms the droplet.

1.4.3 The Weber number

The Weber number compares inertial effects to surface tension:

We =Inertial effects

Surface tension=µU2L

γ

with µ the dynamic viscosity, U the characteristic speed of the flow, L the charac-

teristic size of the interface and γ the surface tension.

This number is widely used for the atomization of jets or bubble breakup in the

inertial regime. It gives a measure of the ability of an inertial flow to deform an

interface. Fig. 1.24 shows the deformation of a pulsed 2D liquid film at different

Weber numbers.


We = 1

We = 10

We = 100

We = 1000

We = 10000

Figure 1.23: Pulsed 2D liquid film at different Weber numbers. The black line

corresponds to the interface between the liquid (beneath) and the air (above). The

colors correspond to the norm of the velocity field. At weak Weber number, the

interface remains unaffected by the pulsed flow while at large Weber number, the

interface is deformed leading to its atomization into small droplet.

1.4.4 The Bond Number

The Bond number (also called Eötvös number) compares the effect of buoyancy

force to surface tension:

Bo =Buoyancy force

Surface tension force=

∆ρgL2

γ


with ∆ρ the difference of density between the considered immisible fluids, g the

gravitational acceleration, L the characteristic size of the considered interface, and

γ the surface tension.

samedi 28 mai 2011

Figure 1.24: Sessile droplets of different volumes lying on a solid substrate. When

the drop is larger than the capillary length, it is flattened by gravity.

For example let’s consider a drop lying on the surface. If the Bond number

is small, the drop keeps its hemi-spherical shape due to surface tension. If the

Bond number is larger, the drop is flattened by gravity effects (see Fig. 1.24). The

bond number can also be seen as the ratio between the square of the characteristic

length of the considered surface and the square of the so-called capillary length

Lc = γ/∆ρg:

Bo =L2

L2c

The capillary length is about 2 mm for water surrounded by air, which means that

gravity effects play no role on the dynamics of liquid drops smaller than this length.

Chapter 2

Interfaces and vibrations

Contents

2.1 Bubbles and the Rayleigh-Plesset equation . . . . . . . . . . 38

2.1.1 Why are bubbles outstanding resonators ? . . . . . . . . . . . 39

2.1.2 Static equilibrium of bubbles . . . . . . . . . . . . . . . . . . 42

2.1.3 Modeling of the liquid phase: the Rayleigh-Plesset equation . 45

2.1.4 Modeling of the gas phase . . . . . . . . . . . . . . . . . . . . 53

2.1.5 From bubbles to metamaterials . . . . . . . . . . . . . . . . . 60

2.2 Inertio-capillary “Rayleigh-Lamb” modes of vibration . . . 61

2.2.1 Dimensional analysis . . . . . . . . . . . . . . . . . . . . . . . 61

2.2.2 Complete solution of the problem . . . . . . . . . . . . . . . . 61

2.2.3 Sessile droplets and possible use of these surface vibrations. . 65

The vibration of interfaces is a wide subject, ranging from Faraday waves

patterns appearing at the surface of a vibrated liquid layer (see Fig. 2.1), to the

multiple scattering of acoustic waves in metamaterials. In this chapter, we will

focus on monopolar vibration of bubbles and the different modes of vibration of an

incompressible liquid droplet.

Figure 2.1: Faraday waves appearing at the surface of a vibrated liquid layer. Source:

http://www.ia.csic.es/Temas.aspx?Lang=EN&Id=8

38 Chapter 2. Interfaces and vibrations

NB: the surface tension will be called σ in this chapter to avoid confusion with the

heat capacity ratio.

2.1 Bubbles and the Rayleigh-Plesset equation

Figure 2.2: Champagne bubbles bursting and associated mist.

The dynamics of bubbles is of primary importance in physics, chemistry, biology

and engineering. Indeed, bubbles create a mist that wafts champagne’s aroma to the

drinker (Fig. 2.2), they are used by pistol shrimps to kill their prey (Fig. 2.3), by

doctors as contrast agents for ultrasonic imaging, by dentists for scaling, by chemists

for sonochemistry. They are also feared by engineers when they design a boat

because of the damages to propellers induced by cavitation (Fig. 2.4). More recently,

bubbles are under consideration to create invisible objects ... toward acoustic waves.

clawbubble

Figure 2.3: Cavitation bubble created by a snapping shrimp [39, 19].

2.1. Bubbles and the Rayleigh-Plesset equation 39

2.1.1 Why are bubbles outstanding resonators ?

Bubbles have this outstanding property that small (but optimized) variations of

pressure or temperature of the hosting liquid can lead to an “explosive” dynamics,

with speeds of its interface larger than the sound speed or temperatures in the

bubble three times larger than the temperature at the surface of the sun. Bubbles

can even emit some light, the so-called sonoluminescence (Fig. 2.5). Thus, even at

small concentrations, they can drastically modify the behavior of the hosting liquid.

For example, the speed of sound in bubbly media decreases at values much smaller

than the one in the corresponding liquid or gas. The response of the suspension

can become highly nonlinear, with the generation of harmonics when an acoustic

wave travels in it. The absorption of ultrasonic waves is also largely increased by

the presence of bubbles.

Figure 2.4: Cavitation induced by the rotation of a propeller (left) and associated

damages (right).

Early studies of the behavior of bubbles have been motivated by some of these

astonishing properties, such as the study of “The damping of sound in Frothy

liquids” by Mallock in 1920 [20] or the study of the “Music of air-bubbles and the

sound of running water” by Minaert in 1933 [23]. Then, more detailed studies of

these phenomena have been performed in connection with the understanding of

underwater explosions [13, 16].

So what is the secret behind bubbles ? While it will be difficult (and disappoint-

ing) to reveal all the secrets of bubble in an introduction, many of its properties

result from the following consideration. Bubbles are resonators with a weak string

(the compressibility of air) and a large mass (the mass of the liquid). Indeed, when

we want to compress a bubble, the resistance to the deformation is due to the gas

pressure change, while the mass that must be moved comes mainly from the liquid.


Figure 2.5: Sonoluminescence induced by acoustic excitation. Left: photo of the

experimental setup. Right: zoom on the bubbles (the white dots correspond to

light emitted by the bubbles.)

Minnaert frequency

From this simple analogy, the resonance frequency of bubbles can be estimated.

Indeed the characteristic frequency of a spring-mass oscillator is given by the well

known formula ωo =√

k/m, with k the stiffness of the spring and m the considered

mass. Since we consider a continuum medium, these quantities will be replaced

by their volume counterparts kv and ρ. As mentioned in the previous section, the

density of the gas ρg can be neglected compared to the one of the liquid ρl and thus

ρ = ρl. In the absence of surface tension, the bubble is prevented from collapsing by

the compression of the gas phase. Therefore, simple dimensional analysis shows that

the stiffness per unit volume kv is given by kv = 1/χgR2, with χg =

[

1ρg

∂ρg∂pg

]

sthe

adiabatic compressibility of the considered medium and R the radius of the bubble.

For an ideal gas in adiabatic evolution, the product pgρ−γg is constant, with γ =

Cpg

Cvg

the heat capacity ratio and thus χg = 1/γpg. If we combine all these equations, we

obtain:

ωo ∼1

R

√

γpgρl

(2.1)

The proportionality coefficient in this expression will be determined later in section

2.1.4. From this formula, we can deduce one of the astonishing properties of bubbles.

Usually, an object is seen by a wave (that is to say it will diffuse this wave) when

its characteristic size is of about or larger than the wavelength. This is why milk

(emulsion of cream in water) or clouds (aerosol of droplets in air) are “white”. Here

if we compare the wavelength of acoustic waves in water λ at resonance to the size

of the bubble R, we have:

λ

R=

clωoR

=cl√ρl√

γpg∼ 102 for air bubbles in water

with cl the speed of sound in the liquid phase. Thus resonance appears when the

wavelength is much larger than the size of the bubble.


Speed of sound in bubbly media

Another astonishing property is the speed of sound in bubbly liquids (at low fre-

quency compared to Minnaert’s resonance frequency). To compute it, we can go

back to the definition of the sound speed c:

c2 =

[

∂p

∂ρ

]

s

(2.2)

with p the pressure of the medium, ρ the density of the medium and the subscript

s is used to designate an isentropic evolution. Since the suspension is a mixture of

gas (bubbles) and liquid, we have:

ρ = αρg + (1− α)ρl (2.3)

with α the volume concentration of gas bubbles. If we combine equations (2.2) and

(2.3), we get:1

c2=

[

∂α

∂p

]

s

(ρg − ρl) +α

c2g+

(1− α)

c2l(2.4)

If we assume that there is no dissolution of gas or phase change (evapora-

tion/condensation) during the propagation of the acoustic wave, we have: αρg/(1−α)ρl = K, with K a constant. If we differentiate equation αρg = K(1 − α)ρl with

respect to p, we obtain:

[

∂α

∂p

]

s

ρg +α

c2g= K

([

∂α

∂p

]

s

ρl +(1− α)

c2l

)

=αρg

(1− α)ρl

([

∂α

∂p

]

s

ρl +(1− α)

c2l

)

If we recombine this expression properly, we obtain:

[

∂α

∂p

]

s

= α(1 − α)

(

1

ρlc2l

− 1

ρgc2g

)

(2.5)

Finally replacing equation (2.5) in equation (2.4), gives:

c2 =

[

α

c2g+

(1− α)

c2l+ α(1− α)(ρg − ρl)

(

1

ρlc2l

− 1

ρgc2g

)]

−1

(2.6)

The sound speed calculated from this formula is plotted on Fig. 2.6 for an

air-water suspension. This figure shows the dramatic decrease in the sound speed

(down to c = 20ms−1 at α ∼ 0.5 compared to its value in water (cl ∼ 1500 ms−1

or in air cg = 320 ms−1). The right plot in Fig. 2.6 shows a zoom on small values

of the concentration α. One can see that for concentrations as small as 0.01%, the

sound speed is already smaller than 100 ms−1.... Conclusion: don’t forget removing

bubbles before measuring the sound speed in liquids !


0 0.25 0.5 0.75 10

500

1000

1500

Concentration α

Sou

nd s

peed

c (

m s

−1 )

c = 20 ms−1

cl = 1500 ms−1

cg = 320 ms−1

0 0.2 0.4 0.6 0.8 1x 10

−3

0

500

1000

1500

Concentration α

Sou

nd s

peed

c (

m s

−1 )

Figure 2.6: Sound speed in bubbly media at frequencies smaller than Minnaert’s

frequency ωo. Right: zoom on low concentrations.

2.1.2 Static equilibrium of bubbles

Henri’s law and dissolved gas

Most of the liquids that surround us contain some dissolved gas (mostly air). Henri’s

law stipulate that, at constant temperature, the amount of a given gas that dissolves

in a given type and volume of liquid is directly proportional to the partial pressure

of that gas in equilibrium with that liquid, namely:

pg = kHβdg

where pg is the partial pressure of the considered gas, kH Henri’s variable and βdgthe concentration of dissolved gas.

A classical illustration of this law is given by carbonated soft drink. Carbonated

drinks contain some dissolved carbon dioxide in higher concentration that the

value given by Henri’s law at atmospheric pressure. Thus the pressure of the gas

in the bottle above the carbonated drink is higher than atmospheric pressure po.

When the bottle is opened, this pressure drops down and thus the concentration of

gas has to decrease. Thus some bubbles of carbon dioxide form and rise because

of buoyancy force (see Fig. 2.7). If we leave the bottle at atmospheric pressure,

the dissolved gas will reach a new equilibrium and the liquid will become “flat”.

This process takes some time since it is limited by the diffusion of the dissolved

gas through the liquid. Its speed depends on the difference between the actual

concentration of gas βdg(t) and the equilibrium value βe(pa) explaining why it

is quick at the beginning (with the release of bubbles), and becomes slower and

slower. But when the equilibrium value is reached, it does not mean that there

is no more dissolved gas in the liquid. It can be simply demonstrated by putting

“flat” water in a vacuum chamber. In this case, we will see the same phenomenon

as the one previously observed when the bottle was opened.


Figure 2.7: Illustration of Henry’s law. When the bottle is closed (right), the pres-

sure in the bottle is higher than atmospheric pressure and thus there is a large

quantity of dissolved gas. When the bottle is opened, the pressure is now equal to

atmospheric pressure and thus the quantity of dissolved gas decreases through the

formation and expulsion of bubbles (left).

An interesting property of Henry’s constant is that it increases with temper-

ature (see Van’t Hoff equations) and thus the solubility of gases decreases with

temperature. Indeed, when we heat water, air bubbles form long before the boiling

crisis (at temperature smaller than 100C) and the formation of vapor bubbles.

It is important to note that the times associated with this diffusion of gas through

the liquid are “long”, and therefore this mass diffusion process can generally be

neglected if we consider quick oscillations of bubbles. They can nevertheless play a

significant role on a large number of oscillations (the so-called rectified diffusion).

Single bubble in a “flat” liquid

Laplace law and dissolution of bubbles

Now we will consider some “flat” liquid, and create a bubble by injecting some

gas inside it. Some liquid evaporates in the bubble until the partial pressure of

vapor reaches its equilibrium value at the considered temperature pv = pv(T ) given

by Clausius-Clapeyron relation. The gas in the bubble is therefore a mixture of

vapor and added gas and the value of the pressure of the gas phase in the bubble is

the sum of the partial pressures of added gas pa and vapor pv according to Dalton’s

law: pg = pa + pv. The static equilibrium with the liquid phase is given by Laplace

law:

pl(R) = pg(R)−2σ

R= pa(R) + pv(R)−

2σ

R(2.7)


Far from the bubbleLiquid

γ

Vapor+gas

R(t)

pg = pa + pvρg = ρa + ρv

Tg

pl(r), vl(r), Tl(r)p∞, v∞ = 0, T∞

Figure 2.8: Sketch of a spherical bubble in an infinite liquid

The vapor pressure pv(R) only depends on the temperature at the bubble

surface TΣ: pv(R) = pv(TΣ). Since the liquid is flat, it contains a quantity of

dissolved gas given by Henri’s law at atmospheric pressure cdg = po/kH . In the

bubble however, surface tension induces an excess of pressure compared to po. This

excess of pressure will inevitably lead to the dissolution of the gas contained in the

bubble according to Henri’s law. Thus, any bubble should dissolve after a sufficient

amount of time. However, experience shows that bubble nuclei remain stable for

times much longer than the time required for mass diffusion. This stability of

bubble nuclei can be explained by the impurities present in the solution. The

impact of these impurities can be seen in a glass of Champagne. A careful look

at the glass shows that bubbles always form at the same locations. In fact, they

appear where the dirts cover the glass. When the sommelier “cleans” the glasses

with a rag before serving Champagne, he does not try to remove the dust but

instead to leave more dust inside the glass so that bubbles can form ...

Blake pressure threshold and cavitation.

In the following, we consider the evolution of bubbles on characteristic times

much shorter than the time associated with gas dissolution. We will see that, even

in this case, the stability of bubbles is not granted. To demonstrate it, let’s consider

that the evolution of the gas in the bubble is isothermal (constant temperature To).

In this case, we have the following relation between the pressure of the added gas

pa and the radius of the bubble R:

paR3 = K = paoR

3o (2.8)

with K a constant and finally Ro and pao the radius and added gas pressure in the

bubble at rest.

If we combine equations (2.7) and (2.8), and assume that the vapor pressure


remains at its equilibrium value pv = pvo (see the discussion about this hypothesis

in section 2.1.4), we obtain:

pl(R) = pao

(

Ro

R

)3

+ pvo −2σ

R(2.9)

Now we can study the stability of the system if a pressure perturbation is applied:

∂R

∂pl=

[

∂pl∂R

]

−1

= R

[

2σ

R− 3pao

(

Ro

R

)3]

−1

If the bubble radius reaches a critical value Rb such that:

RB =

√

3paoR3o

2σ=

√

9mgKgTo8πσ

(2.10)

with mg and Kg the mass of gas in the bubble and the gas constant respectively, the

bubble growth rate becomes “infinite” (explosive growth). The associated pressure

perturbation, called the Blake threshold, is equal to: pB = pL(R = RB). Thus if a

negative pressure variation larger than the Blake threshold is applied to a bubble

nuclei, it cavitates. Indeed, the stiffness of bubbles’ spring comes both from the gas

pressure variations and surface tension. When the bubble radius increases, Laplace

pressure jump decreases, leading to a reduction of the bubble stiffness due to surface

tension. At a given point (Blake’s threshold), the pressure reduction in the bubble

is no more sufficient to prevent the bubble from growing. This principle is used in

ultrasonic cleaners. A more rigorous derivation of this condition will be given in

section 2.1.4.

2.1.3 Modeling of the liquid phase: the Rayleigh-Plesset equation

In this section, we will focus on the dynamical evolution of bubbles.

Inviscid Rayleigh-Plesset equation

To derive Rayleigh-Plesset equation, we will compute the flow around a spherical

bubble whose radius depends on time: R(t). We consider in this first paragraph that

the flow is inviscid, incompressible and keeps a spherical symmetry: pl = pl(r, t),−→vl = vl(r, t)−→er . From the continuity equation, we have:

div(−→vl ) = 0 =⇒ 1

r2∂

∂r(r2vl) = 0 =⇒ vl =

K(t)

r2

with K a time-dependent function. The boundary condition at the surface of the

bubble is (according to equation (1.24) in chapter 1):

ρl(vl(R)− R) = ρg(vg(R)− R)


with R the derivative of R with respect to t. Since the density of the gas phase ρgis much smaller than the density of the liquid phase ρl, the condition at the bubble

surface becomes vl(R) = R and thus K(t) = RR2:

vl(r, t) =RR2

r2(2.11)

From this equation, we can deduce the value of the velocity potential φ defined by−→vl =

−→∇φ:∂φ

∂r=RR2

r2=⇒ φ =

−RR2

r(2.12)

Now if we apply unsteady Bernouilli’s principle (equation (1.22) in chapter 1) and

neglect gravity effects, we obtain:

∂φ

∂t+v2l2

+plρl

= C(t) =⇒ − RR2

r− 2

R2R

r+RR4

2r4+pl(r, t)

ρl= C(t) (2.13)

with C(t) a time-dependent function. C(t) can be computed from the boundary

condition at the bubble surface:

C(t) = −RR− 3

2R2R+

pl(R, t)

ρl(2.14)

If we combine equations (2.13) and (2.14), we obtain:

RR

(

1− R

r

)

+RR2

2

(

1 +R4

3r− 4R

3r

)

=pl(R, t)− pl(r, t)

ρl

If we estimate this equation at a distance r ≫ R, the pressure pl(r, t) tends to p∞(t)

(see Fig. 2.8) and R/r → 0:

RR+3

2R2 =

pl(R, t)− p∞(t)

ρl(2.15)

Finally, since the liquid is supposed to be inviscid, Laplace equation can be used to

express the liquid pressure at the interface: pl(R, t) = pg(R, t)−2σ/R and equation

(2.15) becomes:

RR+3

2R2 =

pg(R, t)− p∞(t)

ρl− 2σ

ρlR(2.16)

This is the so-called Rayleigh-Plesset equation. To close it, the pressure in the

bubble pg(R, t) must be expressed as a function of the radius. This issue will be

thoroughly discussed in section 2.1.4. Nevertheless, to get a first insight of the

behavior predicted by Rayleigh-Plesset equations, we can perform some simulations

when the gas phase follows an adiabatic evolution: pg = pgo (Ro/R)3γ . Fig. 2.9 and

2.10 illustrate the bubble response to a driving pulse of about 105Pa. These figures

show the large departure from linear response. First, we see that when the bubble

collapses, the stiffness of the spring increases due to the compression of the gas. This

entails the bubble bouncing with a quick velocity drop from -100 ms−1 to 100 ms−1.


0 2 4 6

x 10−6

−1

0

1

x 105 Driving Pulse

Time [s]

Pre

ssur

e [P

a]

0 2 4 6

x 10−6

0

1

2

3x 10

−6 Bubble Radius

Time [s]

Rad

ius

[m]

0 2 4 6

x 10−6

−100

0

100

Bubble Wall Velocity

Time [s]

Vel

ocity

[m/s

]

Figure 2.9: Adiabatic evolution of the radius (amplitude and speed) of a 1µm bubble

submitted to a driving pulse of ∼ 105 Pa computed from inviscid Rayleigh-Plesset

equation. The blue line corresponds to the driving pulse, red lines to the bubble

response predicted from Rayleigh-Plesset equation and black ones to the bubble

linear response estimated from the linearized version of Rayleigh-Plesset (equation

(2.29)).


Moreover nonlinear terms lead to a large increase in the amplitude and velocity of

bubble oscillations (compared to linear response). Finally, the natural frequency of

the drop is shifted to lower values. We can also notice that, at this stage, there is

no dissipation term, and following this equation a bubble would oscillate endlessly.

In the next sections, we will consider the sources of dissipation.

2.2 2.4 2.6 2.8

x 10−6

0

0.5

1

1.5

2

2.5x 10

−6Bubble Radius

Time [s]

Rad

ius

[m]

2 2.2 2.4 2.6

x 10−6

−100

−50

0

50

100


Time [s]

Vel

ocity

[m/s

]

Figure 2.10: Same graph as Fig. 2.9 but zoomed on two oscillation periods.

Viscous dissipation

To take into account viscous dissipation, one must start with Navier incompressible

equation instead of Bernoulli’s principle. Since the continuity condition still holds,

the expression of the velocity field remains unchanged:

vl(r, t) =RR2

r2(2.17)

The momentum equation in spherical coordinates takes the following form:

ρl

(

∂vl∂t

+ vl∂vl∂r

)

= −∂pl∂r

+ µl

(

∂2vl∂r2

+2

r

∂vl∂r

− 2vlr2

)

(2.18)

If we replace equation (2.17) in equation (2.18), we obtain:

ρl

(

2RR2 +R2R

r2− 2

R4R2

r5

)

= −∂pl∂r

+ µlR2R

(

6

r4− 4

r4− 2

r4

)

(2.19)

We can see that the viscous term is zero and thus that the viscosity plays no role

in the bulk momentum equation. Thus equation (2.15) remains valid:

RR+3

2R2 =

pl(R, t)− p∞(t)

ρl(2.20)


Indeed, viscosity only plays a role in the boundary condition. If we neglect the

density and viscosity of vapor compared to their counterparts in the liquid and as-

sume that the surface tension is uniform, the momentum equilibrium at the interface

(equation (1.27)) is:

−pl(R, t) + 2µl

[

∂vl∂r

]

r=R

= −pv(R, t) + 2σ

R

Thus, if we replace vl by its expression given by equation (2.17) in this equation, we

obtain:

pl(R, t) = pv(R, t)− 2σ

R− 4µl

R

R(2.21)

Finally, if by combining equations (2.18) and (2.21) together, we get:

ρl

(

RR+3

2R2

)

+ 4µlR

R= pg(R, t)− p∞(t)− 2σ

R(2.22)

From the comparison of the order of magnitude of the two terms on the left

hand side of this equation, one can estimate the characteristic time τv associated

with viscous attenuation:

τv =ρlR

2o

4µl

This time is about 2.5 × 10−7s for a micrometric bubble oscillating in water. Ef-

fectively, if we solve numerically equation (2.22), we see that bubble oscillations are

completely damped after a time ∼ 10τv (see Fig. 2.11). Thus, to determine whether

viscous dissipation plays a significant role, one must compare the characteristic time

associated with the bubble dynamics, τ , to τv. If τ ≪ τv, viscous damping can be

neglected.

Thermal dissipation

Some thermal dissipation can also occur due to the large temperature variations

experienced by the gas in the bubble. This heat can spread in the liquid leading

to an attenuation of bubble oscillations. Since the liquid phase is assumed to be

incompressible, the Heat equation and Navier equations are uncoupled, and thus

the heat equation must be solved separately to take into account thermal effects.

However, since the thermal conductivity of the liquid is much larger than the one

of air, the heat diffusion process is not limited by the time required for diffusion

through the liquid but by the one required for diffusion through the gas. Thermal

dissipation will therefore be discussed again later in the section dedicated to the

modeling of the gas behaviour.

Radiation damping

On the incompressibility condition


0 2 4 6

x 10−6

−1

0

1

x 105 Driving Pulse

Time [s]

Pre

ssur

e [P

a]

0 2 4 6

x 10−6

0

1

2x 10

−6Bubble Radius

Time [s]

Rad

ius

[m]

0 2 4 6

x 10−6

−50

0

50


Time [s]

Vel

ocity

[m/s

]

0 2 4 6 8

x 106

−180

−160

−140

Power Spectra

Frequency [Hz]

Am

plitu

de [d

B]

Figure 2.11: Adiabatic evolution of the radius (amplitude and speed) of a 1µm

bubble submitted to a driving pulse of ∼ 105 Pa computed from Rayleigh Plesset

equation including viscous dissipation. Blue lines correspond to the driving pulse,

red lines to the bubble response predicted from Rayleigh-Plesset equation and black

ones to the bubble linear response estimated from the linearized version of Rayleigh-

Plesset (equation (2.29)).


Rayleigh-Plesset equation have been derived assuming an incompressible liquid

phase. A counter-intuitive result is that Rayleigh-Plesset equation still holds for

the description (at first order) of bubbles oscillations induced by acoustic waves.

It seems surprising since the liquid compressibility is required for acoustic waves to

propagate. To understand it, the linearized mass conservation equation in the liquid

can be written in a dimensionless form:

ωδρl∂ρl

∂t+ρloδvlRo

div(−→v ) = 0 (2.23)

with ω the frequency of the acoustic wave, δρl and δvl the orders of magnitude of

density and velocity variations induced by the acoustic wave, and Ro the bubble

radius at rest. The characteristic length scale is Ro and the characteristic time

scale is 1/ω. For acoustic waves, the relation between the amplitude of density and

velocity variations is: δρl = ρloclδvl, with cl the sound speed in the liquid. Thus

equation (2.23) becomes:Ro

λ

∂ρl

∂t+ ˜div(−→v ) = 0 (2.24)

with λ = ω/cl the acoustic wavelength. As demonstrated in section 2.1.1, when

a bubble is excited around its resonance frequency, the wavelength is much larger

than the radius of the bubble (Roλ ≪ 1). Thus, the first term of the equation

can be neglected compared to the second one and the liquid can be considered as

incompressible close to the bubble.

div(−→v ) ∼ 0

Dissipation by acoustic radiation

However, while the compressibility of the liquid can be neglected over a single

period of oscillation (the characteristic time scale was chosen as the inverse of the

frequency), it can play a significant role on a large number of oscillations. Indeed,

the bubble oscillations induce the emission of acoustic waves, and thus some energy

is radiated inside the liquid. This energy loss entail a damping of the bubble oscil-

lations. This effect has been introduced by Keller and Kolodner in 1956 [16], and

Keller and Miksis in 1980 [17]. We will not demonstrate how the radiation damping

can be accounted for and will just give the form of the modified Rayleigh-Plesset

equation:

RR

(

1− R

cl

)

+3

2R

(

1− R

3cl

)

(2.25)

=

(

1− R

cl

)

pl(R, t)− p∞(t)

ρl− R

ρlcl

d

dt(pl(R, t) + p∞(t))

When the liquid becomes incompressible, cl → ∞ and Rayleigh-Plesset equation is

recovered. Fig. 2.12 shows the evolution of a bubble in the presence of radiation

damping (and in the absence of any other source of dissipation). The acoustic

radiation indeed leads to a damping of the bubble oscillations, although the effect

is small compared to viscous dissipation in the considered regime.


0 2 4 6

x 10−6

−1

0

1

x 105 Driving Pulse

Time [s]

Pre

ssur

e [P

a]

0 2 4 6

x 10−6

0

1

2

3x 10

−6Bubble Radius

Time [s]

Rad

ius

[m]

0 2 4 6

x 10−6

−100

0

100


Time [s]

Vel

ocity

[m/s

]

0 2 4 6 8

x 106

−160

−140

−120

Power Spectra

Frequency [Hz]

Am

plitu

de [d

B]


bubble submitted to a driving pulse of ∼ 105 Pa computed from Rayleigh Plesset

equation including radiation damping. Blue lines correspond to the driving pulse,

red lines to the bubble response predicted from Rayleigh-Plesset equation and black

ones to the bubble linear response estimated from the linearized version of Rayleigh-

Plesset (equation (2.29)).


2.1.4 Modeling of the gas phase

Inertially controlled dynamics

To close Rayleigh-Plesset equations the pressure difference pg(R, t)−p∞(t) must be

expressed as a function of the bubble radius R. If we consider the partial pressures

of added gas pa and vapor pv, this expression becomes:

pg(R, t)− p∞(t) = pa(R, t) + pv(TΣ)− p∞(t)

since the vapor pressure only depends on the temperature at the surface of the

bubble TΣ. Following C.E Brennen textbook [4], this expression can be decomposed

into:

pa(R, t) + [pv(T∞)− p∞(t)] + [pv(TΣ)− pv(T∞)]

The first term simply corresponds to the pressure variation in the added gas, the

second to the driving term (thermal and inertial) and the third one to the varia-

tion of vapor pressure induced by the departure of bubble temperature from the

remote liquid one. In the following, we consider that no variations of temperature

are imposed (T∞(t) = To), and we neglect the effects of phase change (evapora-

tion/condensation at the bubble surface) induced by temperature increase in the

bubble: [pv(TΣ)− pv(T∞)] = 0. This behavior is called inertially controlled dy-

namics, to distinguish from a dynamics controlled by phase change. Obviously, the

inertially controlled dynamics is not appropriate to describe events such as the boil-

ing crisis and more suitable for dynamics induced by acoustic waves. In some cases,

phase change can however play a fundamental role in bubbles dynamics induced by

pressure variations. The validity range of the different approximations is thoroughly

discussed by C.E Brennen in its textbook [4].

Heat exchange and temperature variations in the gas

The heat diffusion equation must be solved in the gas phase to determine its tem-

perature variations:

ρgCpg

dTgdt

=∂pg∂t

+ kg∆Tg (2.26)

with ddt =

∂∂t +

−→vg .−→∇ the material derivative, and ρg, Tg, C

pg , pg and kg the density,

temperature, heat capacity, pressure and heat conductivity of the gaseous phase

respectively.

Heat transfers from the bubble to the liquid are responsible for a damping of

bubbles oscillations. The characteristic time associated with heat transfers is τT ∼ρloR

2o/ρgoκg, with κg the thermal diffusivity. For air bubbles in water, this time if

about 10−4 s. On Fig. 2.13, we can see indeed that thermal dissipation induces a

slower damping of bubble oscillations than viscous effects.


0 2 4 6

x 10−6

−1

0

1

x 105 Driving Pulse

Time [s]

Pre

ssur

e [P

a]

0 2 4 6

x 10−6

0

1

2

3x 10

−6Bubble Radius

Time [s]

Rad

ius

[m]

0 2 4 6

x 10−6

−2

0

2x 10

4Bubble Wall Velocity

Time [s]

Vel

ocity

[m/s

]

0 2 4 6 8

x 106

−160

−140

−120

Power Spectra

Frequency [Hz]

Am

plitu

de [d

B]

Figure 2.13: Evolution of the radius (amplitude and speed) of a 1µm bubble sub-

mitted to a driving pulse of ∼ 105 Pa computed from Rayleigh-Plesset equation

including heat transfers. Blue lines correspond to the driving pulse, red lines to

the bubble response predicted from Rayleigh-Plesset equation and black ones to

the bubble linear response estimated from the linearized version of Rayleigh-Plesset

(equation (2.29)).


Simple cases: adiabatic and isothermal behaviors

When can an evolution be considered as adiabatic or isothermal ?

Since we have introduced a characteristic time associated with thermal ex-

changes, we can define properly in which cases bubbles behavior can be considered

as adiabatic or isothermal. The evolution of the gas phase is adiabatic when there

is no heat exchange between the bubble and the liquid. This happens, when the

characteristic time associated with the bubble dynamics τ ≪ τT . In this case, the

bubble dynamics is too fast for heat transfers to occur. On the opposite, when

τ ≫ τT , the temperature in the bubble is at equilibrium with the one in the liquid.

As long as no temperature variations is imposed, the bubble behavior is isothermal

(T∞(t) = To). In these simple cases, the relation between the added gas pressure

and the radius is:

paR3k = K

with K a constant, k = 1 when the evolution is isothermal and k = γ when the

evolution is adiabatic.

Linearized version of Rayleigh-Plesset equation and Minaert’s frequency

If we consider an adiabatic or isothermal evolution and neglect phase changes

pv = pvo, Rayleigh-Plesset equation becomes:

ρl

(

RR+3

2R2

)

+ 4µlR

R− pao

(

Ro

R

)3k

+2σ

R= pvo − p∞(t) (2.27)

with k = 1 if the evolution is isothermal and k = γ if it is adiabatic.

Considering small radius variations around its equilibrium value: R(t) = Ro(1+

ε(t)), with ε≪ 1, Rayleigh-Plesset equation becomes:

ρlR2oε+ 4µlε− pao(1− 3kε) +

2σ

Ro(1− ε) = pvo − p∞(t) (2.28)

After reorganization of the terms of this equation, we finally obtain:

ε+4µlρlR2

o

ε+

(

3kpaoρlR2

o

− 2σ

ρlR3o

)

ε = po − p∞(t) (2.29)

with po = pao+pvo− 2σRo

the equilibrium pressure before the perturbation. This equa-

tion corresponds to the one of a damped linear oscillator as long as the coefficient

in front of ε is positive. From this expression, we can compute the characteristic

frequency of this oscillator, the so called Minnaert’s frequency:

ωo =1

Ro

√

3kpaoρl

− 2σ

ρlRo(2.30)


Compared to equation (2.1) previously obtained from dimensional analysis, the

influence of surface tension has been added.

Rigorous derivation of Blake’s threshold

However when the coefficient in front of ε in equation 2.29 becomes negative:

3kpaoρl

<2σ

ρlRo(2.31)

the solution of this equation is no more the sum of harmonic functions but the sum

of two exponential functions, one with a positive coefficient and one with a negative

coefficient. As long as the imposed pressure variation p∞−po is positive, the radius

decreases exponentially and the bubble is stable. Conversely, when the pressure

variation p∞ − po is negative, the radius increases exponentially, that is to say the

bubble nuclei cavitate.

If the added gas behaves as an ideal gas, pao =mgKgTg

4/3πR3o

, with mg the mass of

added gas, Kg the gas constant and Tg the gas temperature. Thus from equation

(2.31), we obtain the expression of Blake’s critical radius:

Rb =

√

9kmgTg8πσ

This expression is similar to the one previously obtained in section 2.1.2 when the

evolution is isothermal (k = 1). Fig. 2.14 shows the difference of behavior when

a bubble is submitted to a negative or a positive driving pulse with an amplitude

higher than Blake’s threshold. We can see how much the pressure threshold is

increased when the imposed pressure variation is negative.

Nonlinear response of a bubble to a sinusoidal excitation

Fig. 2.15 illustrates the response of a bubble to a periodic sinusoidal excitation

at Minnaert’s frequency (∼ 2.78 MHz for a micrometric bubble). The power spectra

shows clearly the appearance of harmonics and subharmonics due to the nonlinear

response of the bubble.

Outstanding characteristics of bubbles oscillations

The evolution of a 1µm bubble submitted to a pulse of ∼ 2 × 105 Pa can be

considered at first order as adiabatic since the characteristic time associated with

the bubble dynamics is much larger than one associated with heat exchange, τT .

Thus the temperature inside the bubble can be estimated simply from the evolution

of the radius: T/To = (Ro/R)3(γ−1). We see on Fig. 2.16 that the radius is divided

by ∼ 7, which means that the temperature is increased by 10 if the added gas is air

(γ = 1.4). Thus if the initial temperature is 20oC (293 K) the final temperature

is about 2930 K, half the temperature at the sun surface (for a pressure


0 0.5 1 1.5

x 10−6

−5

0

5x 10

5 Driving Pulse

Time [s]

Pre

ssur

e [P

a]

0 0.5 1 1.5

x 10−6

0

2

4x 10

−6Bubble Radius

Time [s]

Rad

ius

[m]

0 0.5 1 1.5

x 10−6

−500

0

500


Time [s]

Vel

ocity

[m/s

]

0 2 4 6 8

x 106

−180

−160

−140

−120Power Spectra

Frequency [Hz]

Am

plitu

de [d

B]

Figure 2.14: Evolution of the radius (amplitude and speed) of a 1µm bubble sub-

mitted to a driving pulse either negative (blue) or positive (red) with an ampli-

tude higher than Blake’s threshold (∼ 5× 105 Pa) computed from Rayleigh-Plesset

equation. Black line correspond to the bubble linear response estimated from the

linearized version of Rayleigh-Plesset (equation (2.29)).


0 2 4 6 8

x 10−6

−1

0

1x 10

5 Driving Pulse

Time [s]

Pre

ssur

e [P

a]

0 2 4 6 8

x 10−6

0

1

2x 10

−6Bubble Radius

Time [s]

Rad

ius

[m]

0 2 4 6 8

x 10−6

−40

−20

0

20

40Bubble Wall Velocity

Time [s]

Vel

ocity

[m/s

]

0 2 4 6 8

x 106

−180

−160

−140

−120Power Spectra

Frequency [Hz]

Am

plitu

de [d

B]


bubble submitted to a sinusoidal pressure excitation of ∼ 105 Pa and frequency

2.78 MHz computed from Rayleigh-Plesset equation including viscous dissipation

and radiation damping. Blue lines correspond to the driving wave, red lines to

the bubble response predicted from Rayleigh-Plesset equation and black ones to

the bubble linear response estimated from the linearized version of Rayleigh-Plesset

(equation (2.29)).


wave of only 2 bar). The wall speed also reaches some values close to the

sound speed.

0 2 4 6

x 10−6

−2

0

2

x 105 Driving Pulse

Time [s]

Pre

ssur

e [P

a]

0 2 4 6

x 10−6

0

1

2

3x 10

−6Bubble Radius

Time [s]

Rad

ius

[m]

0 2 4 6

x 10−6

−200

0

200


Time [s]

Vel

ocity

[m/s

]

0 2 4 6 8

x 106

−180

−160

−140

Power Spectra

Frequency [Hz]

Am

plitu

de [d

B]


bubble submitted to a pulse of ∼ 2× 105 Pa computed from Rayleigh-Plesset equa-

tion including viscous dissipation and radiation damping. Blue lines correspond to

the driving wave, red lines to the bubble response predicted from Rayleigh-Plesset

equation and black ones to the bubble linear response estimated from the linearized

version of Rayleigh-Plesset (equation (2.29)).

Mass diffusion, inertia and asymmetry effects.

For the sake of completeness, we will just mention the hypotheses made in this

lecture and whose validity is questionable in some cases:

1. Mass diffusion induced by phase change or dissolution of gas has been ne-

glected.

2. Bubbles are assumed to remain spherical.

3. The liquid phase is assumed to remain incompressible around the bubble

(which is false if the walls speed reaches the liquid sound speed).

4. There is no chemical reaction or plasma formation in the bubble.

5. The flow in the bubble has not been considered.

Consideration of these phenomena would increase the model complexity.


2.1.5 From bubbles to metamaterials

We will conclude this chapter with an exciting contemporary application of bubbles:

metamaterials. Metamaterials are artificial materials engineered to have properties

that cannot be found in nature. These properties are obtained by inserting small

inhomogeneities in a matrix to create effective macroscopic behavior. One of the

primary aims of metamaterial is to investigate materials with negative refractive

index [40] or with gradient index materials. The former would allow the conception

of superlenses, with subwavelength resolution while the latter would allow the design

of “invisibility cloaks” which make object invisible toward a given type of wave

(electromagnetic, acoustic, water surface).

To understand it, we must first define what the expression “invisible object”

means. The detection of an object comes from the reflection or scattering of a given

wave by this object. Thus, if one can design a cloak which allows the wave to follow

the same path as it would in the absence of the object (see Fig. 2.17), the object

becomes invisible. This does not seem a priori so complicated: the wave ray must

be curved to follow the object. However mathematical transformations show that to

enable the curvature of rays in the appropriate way, one must have materials with

large variations of refractive index. In acoustic, the refractive index n is given by:

n2 = ρχ

with ρ the density of the medium and χ the compressibility of the medium. With

Figure 2.17: Trajectories of rays around an object covered by an invisibility cloak

(left: sketch, right: simulation): “any radiation attempting to penetrate the secure

volume is smoothly guided around the cloak to emerge traveling in the same direction

as if it has passed through an empty volume of space.” [27]

conventional materials, it is hard to obtain the adequate properties to distort the

rays around an object. It is even harder to get a negative refractive index, since

in this case the material must have both a negative density and a nega-

tive compressibility ! This latter means that when the pressure is increased, the

material expands. Of course, it cannot be achieved as a static property. But ma-

terials can behave as if they had effective negative density or compressibility in the

2.2. Inertio-capillary “Rayleigh-Lamb” modes of vibration 61

frequency domain. That is where bubbles come into play. If we consider a bubble

cloud surrounded by a soft matrix and make the bubble oscillate in antiresonance

compared to the excitation, the effective medium expands when the pressure in-

creases. Thus the material behaves as if its compressibility were negative [14]. To

obtain a negative density, one can consider some heavy particles in a soft matrix.

When the wave propagates, if the particles move in the opposite way as the sur-

rounded matrix, and their density exceeds the matrix density, the medium behaves

as if its density were negative [21]. If both properties are obtained in the same

frequency range, the metamaterial has a negative refractive index.

2.2 Inertio-capillary “Rayleigh-Lamb” modes of vibra-

tion

In the previous section, we have only considered spherical deformations of bubbles

involving the compressibility of the fluid. Here, we will show that incompressible

modes of deformation can also appear at the surface of droplets or bubbles. These

vibrations come from a competition between inertia and surface tension.

2.2.1 Dimensional analysis

The characteristic frequency ωo of these surface vibrations can be established from

dimensional analysis for a droplet of liquid with a density ρ surrounded by a gas

with a density ρ′ ≪ ρ. In this case, inertial effects correspond to the term ρ∂−→v∂t in

Navier-Stokes equation. Since Ro is the characteristic length associated with the

flow in the droplet, the magnitude of the inertial term is about:

fi = ρ∂−→v∂t

∼ ρRoω2o

This term is homogeneous to a force per volume unit. Since Laplace pressure jump

is equal to 2σ/Ro, the force per unit volume associated with surface tension is:

fs ∼ σ/R2o

If these two terms have the same order of magnitude, the characteristic frequency

scales as:

ωo =

√

σ

ρR3o

2.2.2 Complete solution of the problem

In this second section, we solve the complete problem by considering a droplet

or bubble of density ρ, initially spherical with a radius Ro, and surrounded by an

infinite amount of immiscible fluid of density ρ′ [18] . We will study small vibrations

appearing at the interface between these two fluids. Thus the radius r at any point

of the interface verifies:

r = Ro + ξ with ξ ≪ Ro


The flow is assumed to be inviscid, incompressible, and following vorticity equation

irrotational. The surface tension between the two fluids is σ.

Since the flow is incompressible, the velocity in both fluids can be expressed

as a function of a velocity potential: −→v =−→∇φ and −→v ′ =

−→∇φ′. Since it is also

irrotational, these two potentials verify Laplace equation:

∆φ = 0 and ∆φ′ = 0

The solution of Laplace equation in spherical polar coordinates (r, θ, ϕ) are called

solid harmonics. They can be divided into regular solid harmonics φn which vanish

at origin an irregular solid harmonics φ−n−1 which are singular at origin, with n the

degree of the considered harmonic:

φn = rnSn(θ, ϕ) and φ−n−1 = r−n−1Sn(θ, ϕ)

where Sn(θ, ϕ) is a surface harmonic of degree n. The expression of surface harmon-

ics can be calculated from Laplace equation in spherical coordinates by applying the

separation of variables. We will not develop this theory further here and refer to

the book of Lamb [18] or Gumerov and Duraiswami [11] for a complete overview

of spherical harmonics theory. We just mention here how surface harmonics can be

calculated:

Sn = AoTn(ν) +

n∑

s=1

(Ascos(sϕ) +Bssin(sϕ))Psn(ν)

P sn(ν) = (1− ν)s/2

dsTn(ν)

dνs

Tn(µ) =1

2nn!

dn

dνn(ν2 − 1)n

ν = cos(θ)

with As, 2n + 1 coefficients. The functions P sn are called associated Legendre

polynomials.

The small deformations at the sphere surface can be expressed in terms of surface

harmonics:

r = Ro + ξ = Ro + Sn sin(ωt+ ε)

with ω the frequency of the vibration, t the time and ε the vibration phase. Fig.

2.19 shows different spherical harmonics. The colors correspond to the intensity of

the sphere surface deformation.

In this case, the velocity potentials inside and outside the drop take the form:

φ =ωRo

n

rn

Rno

Sn cos(ωt+ ε) and φ′ =ωRo

n+ 1

Rn+1o

rn+1Sn cos(ωt+ ε)

since:


Figure 2.18: Sketch representing the different modes of deformation of a sphere

(spherical harmonics). The colors correspond to the intensity of the sphere surface

deformation. Source: http://principles.ou.edu/earth_figure_gravity/index.html

1. the potential inside the spherical inclusion cannot be singular at r = 0 and

the potential outside the drop cannot become infinite when r → ∞.

2. the potential must fulfill the boundary condition (continuity of the normal

velocity) at the sphere surface (r = Ro):

∂ξ

∂t=∂φ

∂r=∂φ′

∂r

From unsteady Bernoulli’s principle (equation 1.22 in chapter 1), we can deduce the

expression of internal and external pressures at the interface:

p = po +ρω2Ro

nSn sin(ωt+ ε) and p = p′o −

ρ′ω2Ro

n+ 1Sn sin(ωt+ ε) (2.32)

with po and p′o the pressures inside and outside the spherical inclusion at rest. Since

the flow inside or and outside the sphere are inviscid, the momentum equilibrium

at the interface (r ∼ Ro) is simply Laplace equation:

∆p = p− p′ = σC = σdiv(−→n ) (2.33)

with C = div(−→n ) the interface curvature and −→n the surface normal. The theorem

of solid geometry stipulates that the normal −→n at any point of a surface defined by

F (x, y, z) = 0 is given by:−→n =

−→∇F (x, y, z)


Thus, the curvature is given by:

C = div(−→∇F ) = ∆F

Here, the equation of the surface is given by:

F = r −Ro − ξ = 0

Since the deformations of the interface are small, the function ξ = Sn sin(ωt + ε)

can be expanded as a function of ξn = rn/RnoSn sin(ωt+ ε) according to:

ξ = ξn(r)− (r −Ro)∂ξn∂r

(Ro)−(r −Ro)

2

2

∂ξn∂r

(Ro) + ...

= ξn(r)− (r −Ro)n

Roξn(Ro)−

(r −Ro)2

2

n(n− 1)

R2o

ξn(Ro) + ...

If (x, y, z) are the Cartesian coordinates, the radius is equal to r =√

x2 + y2 + z2

and the derivatives of r with respect to x, y and z are given by:

∂r

∂x=x

r,∂r

∂y=y

r, and

∂r

∂z=z

r

Thus, at first order

ξ = ξn(r)

∂ξ

∂x=∂ξn∂x

− ∂r

∂x

n

Roξn(Ro) + ... =

∂ξn∂x

− nx

rRoξn(Ro) + ...

∂2ξ

∂x2=∂2ξn∂x2

−[

1

r− x2

r3

]

n

Roξn(Ro)−

n(n− 1)

R2o

[

Rox2

r3+

(r −Ro)

r

]

ξn(Ro) + ...

The same formula can be obtained for the derivatives with respect to y and z.

∂2ξ

∂y2=∂2ξn∂y2

−[

1

r− y2

r3

]

n

Roξn(Ro)−

n(n− 1)

R2o

[

Roy2

r3+

(r −Ro)

r

]

ξn(Ro) + ...

∂2ξ

∂z2=∂2ξn∂z2

−[

1

r− z2

r3

]

n

Roξn(Ro)−

n(n− 1)

R2o

[

Roz2

r3+

(r −Ro)

r

]

ξn(Ro) + ...

As a consequence,

∂2F

∂x2=∂2r

∂x2−∂

2ξ

∂x2=

[

1

r− x2

r3

]

−∂2ξn∂x2

+

[

1

r− x2

r3

]

n

Roξn(Ro)+

n(n− 1)

R2o

[

Rox2

r3+

(r −Ro)

r

]

ξn(Ro)

Since x2+ y2+ z2 = r2, and ∂2ξn∂x2 + ∂2ξn

∂y2 + ∂2ξn∂z2 = 0 (ξn is a solid harmonic and thus

a solution of Laplace equation), we get:

∂2F

∂x2+∂2F

∂y2+∂2F

∂z2=

[

2

r

]

+[O]+

[

2n

Ror

]

ξn(Ro)+

[

n(n− 1)

(

3(r −Ro)

r+Ro

r

)]

ξn(Ro)

R2o


Finally, if we estimate this expression in r = Ro and since 2/r ∼ 2/Ro(1 − ξn), we

obtain:

C =

[

∂2F

∂x2+∂2F

∂y2+∂2F

∂z2

]

r=Ro

=2

Ro+

(n− 1)(n + 2)

R2o

Snsin(ωt+ ε) (2.34)

Now if we combine equation (2.32), (2.33) and (2.34), we obtain the expression of

the characteristic frequency of a Rayleigh-Lamb mode of order n:

ωo =

[

(n+ 2)(n+ 1)n(n − 1)σ

[(n+ 1)ρ+ nρ′]R3o

]1/2

(2.35)

When the density ρ of the fluid inside the sphere is much larger than the one of the

surrounding fluid, we obtain:

ωo =

√

(n+ 2)n(n− 1)σ

ρR3o

which is the same as the previous formula obtained from dimensional analysis, but

with the appropriate proportionality coefficient. This expression has been obtained

by Rayleigh in 1879 [29] from energy considerations and Lagrange’s method. Then,

it has been extended by Lamb in 1932 [18] for two inviscid fluids of density (ρ) and

(ρ′), equation (2.34). The viscosity of the fluids has been introduced by Miller and

Scriven (1968) [22] and Prosperetti (1980) [28]. Finally, larger (nonlinear) vibrations

have been considered by Foote (1971) [10], Tsamopoulos and Brown (1983) [38], and

more recently by Smith [33]. An interesting result is that when droplet oscillations

become larger, there is a decrease in the characteristic frequency ωo due to nonlinear

effects.

2.2.3 Sessile droplets and possible use of these surface vibrations.

To observe Rayleigh-Lamb vibrations, one must start with an initially spherical

drop (levitating drop or drop lying on a superhydrophobic surface) and create an

excitation of the drop at the appropriate frequency. For example, a drop of mercury

lying on a Teflon plate (contact angle of ∼ 157o) can be excited through the

vibration of the underlying plate (see Fig. 2.19). Levitating drop can be obtained

acoustically with the use of radiation pressure [32]. In this case, the excitation of

the drop can be achieved by a modulation of the acoustic signal. Levitating drop

can also be obtained through the deposition of a drop on a hot plate (the drop lies

on an air cushion): it is the so-called Leidenfrost effect. However, the size of the

drop is not constant and the characteristic frequency is time-dependent.

In most practical situation, drops do not levitate and lie on substrates with

contact angles different from 0. In this case, Rayleigh-Lamb oscillations are affected

by the value of the contact angle [35] and the pinning or free motion of the contact

line [26]. An interesting application of Rayleigh-Lamb oscillations is their use in

order to unpin the contact line and induce [25] or enhance the motion of droplets.


Mode n=2

Mode n=3

Mode n=4

Mode n=5

Figure 2.19: Different modes of vibrations of a mercury drop lying on a Teflon

plate (contact angle ∼ 157o). Source: http://www.youtube.com/watch?v=

MperC7ySjSU by L. Floc’h, A. Hubert, S. Place and S. Remadi

Chapter 3

Dynamics of bubbles or plugs in

confined geometries

The dynamics of bubbles in capillary tubes is a wide subject, ranging from

Marangoni flow to boiling crisis [1]. In this chapter, we will focus on the “pressure-

driven” motion of bubbles in confined geometries.

NB: In this chapter, the surface tension will be called γ.

3.1 Two phase flow at small scales

On the relative importance of surface effects

Figure 3.1: Pygmy shrew picture.

Let’s start this section by drawing a parallel between the physics at small scales

and the “Pygmy Shrew”. The Pygmy Shrew is the smallest mammal in the world.

68 Chapter 3. Dynamics of bubbles or plugs in confined geometries

It has an average weight of 4 grams and its body measures about 50 mm (see Fig.

3.1). A fundamental question is: why is there a critical size for mammals? If we

compare the surface to volume ratio of a body, it scales as ∼ L2/L3 = 1/L. Thus

the smaller the considered length scale is, the stronger the surface effects compared

to volume ones are. To regulate their temperature, mammals must eat. The smaller

a mammal is, the larger its surface to volume ratio and thus heat exchanges with

the surrounding atmosphere are. Thus the Pygmy Shrew has one of the highest

metabolic rates of any animals and it spends most of its time eating to keep its

temperature constant.

Figure 3.2: Picture of an ant trying to drink the water contained in a rain droplet.

Top picture: the ant tries to pierce the drop surface. Bottom picture: the ant is

trapped in a drop by capillary forces. Source: http://www.greenwala.com/channels/

nature/blog/14735-Cool-Nature-Picture-of-the-Day-Ant-Stuck-in-a-Rain-Drop

One must therefore keep in mind that the small scales world is ruled by com-

pletely different laws from our usual environment. A good illustration of the aston-

ishing physics at small scales is the study of an ant fall from the 4th floor of a building

(see http://www.youtube.com/watch?v=RmgIsk19RSM&feature=related). While

3.2. Bretherton law 69

if you try the same experiment (I strongly advise you not doing it), you would have

little chance to survive, the ant is in perfectly good health after such an incredible

jump. Indeed, since the ant is small, gravity (bulk) effects are rapidly counterbal-

anced by drag (surface) effects. Thus, it reaches its maximum velocity after only

a few centimeters of fall. The little impact of gravity on ants is also obvious when

we see them walking on ceilings). On the other hand, surface effects, like surface

tension, can strongly affect it: Fig. 3.2 shows a picture of an ant trapped in a drop

by surface tension.

Singularities

A key idea is therefore that, at small scales, surface forces are “generally” domi-

nant compared to bulk forces. As a consequence, Capillary and Bond numbers are

generally small. So why taking precautions when stating that surface effects are

dominant? We have seen in a previous chapter that viscosity plays a fundamental

role on the dynamics of the triple line, while the characteristic length involved is

small. This comes from the singularity of the flow close to the triple line, which

results in large velocity gradient and thus viscous dissipation. Thus bulk effects can

play a fundamental role close to walls or more generally singularities.

3.2 Bretherton law

3.2.1 Semi infinite bubble

Description of the problem

R

A

C

Liquid

B

y

x

U

U

Liquid

z

y

x

Air

Air

Figure 3.3: Sketch of the motion of a semi-infinite bubble in a cylindrical tube at a

constant velocity U .

In this section, we will demonstrate a surprising result: the harder you blow air

in a capillary tube filled with liquid, the more liquid you leave on the walls. For this


purpose, we will study the flow of a semi-infinite bubble moving at a constant speed

U in a cylindrical capillary tube (see Fig. 3.3) filled with a wetting liquid [36, 5].

The Bond number ρlgR2/γ, the Reynolds number Re = ρlUR/µl and the Capillary

number Ca = µlU/γ are all supposed to be small, with ρl the density of the liquid,

g the gravity constant, R the radius of the tube, µl the viscosity of the liquid and

γ the surface tension between the liquid and the gas. When the liquid-gas interface

moves, it is deformed close to the wall, resulting in the deposition of a thin liquid

film of height H. Indeed, away from the walls, the characteristic length scale of

both viscous effects and surface tension is the radius R. Since the capillary number

is small, viscous stresses are too weak to deform the interface. This corresponds to

the region AB on Fig. 3.3, which will be called the static meniscus. But closer to

the wall (region BC on Fig. 3.3), velocity gradients are high since the velocity of

the fluid is equal to 0 on the walls and U on the moving interface (see triple line

dynamics). Thus, viscous effects are strong enough to deform the interface. This

region will be called the dynamic meniscus. Finally, away from the interface (region

following point C on Fig. 3.3) a thin film of constant thickness H lies on the walls.

Dimensional analysis

From dimensional analysis, one can estimate the evolution of the thickness of the

liquid filmH as a function of the Capillary number [2]. Indeed, in the region BC, the

film thickness is determined by the equilibrium between viscous force and pressure

gradients:

µlU

H2∼ γ

LR(3.1)

with L the (unknown) length of the meniscus. Now the matching between dynamic

and static meniscus (equilibrium of Laplace pressure) gives:

− γ

R− γH

L2∼ −−2γ

R(3.2)

since one of the radius of curvature is equal to R, while the other is equal to[

d2h(x)/dx2]

−1 ∼ L2/H, with h(x) the function defining the shape of the interface

along x-axis. From this equation we get, L ∼√HR, and combined with equation

(3.1):

H/R ∼ Ca2/3 (3.3)

The lubrication approximation

Here we assume that the transition region BC can be considered as plane and not

annular. This approximation is discussed by Bretherton [5] and is valid as long as the

Capillary number is small. The equation is written in the bubble frame of reference,

which is moving at constant speed U . Incompressible steady Navier-Stokes equation


H

x

y

h(x)v(y)

x

B

C

L

Figure 3.4: Lubrication approximation in region BC.

at small Reynolds, Bond and Capillary numbers can be written under the form:

∂vx∂x

+∂vy∂y

= 0

µl

[

∂2vx∂x2

+∂2vx∂y2

]

=∂p

∂x

µl

[

∂2vy∂x2

+∂2vy∂y2

]

=∂p

∂y

with vx an vy the velocity projections along x-axis and y-axis respectively, p the

pressure and µl the viscosity. If we introduce the characteristic scales of the problem,

L, H, U, V and P, such that:

x = Lx

y = Hy

vx = Uvx

vy = V vy

p = P p

we obtain the following dimensionless equations:

U

L

∂vx∂x

+V

H

∂vy∂y

= 0 (3.4)

µl

[

U

L2

∂2vx∂x2

+U

H2

∂2vx∂y2

]

=P

L

∂p

∂x(3.5)

µl

[

V

L2

∂2vy∂x2

+V

H2

∂2vy∂y2

]

=P

H

∂p

∂y(3.6)

From the first equation we obtain:

U

L∼ V

H

In the second and third equations, the first term can be neglected compared to the

second one since:H2

L2∼ H

R∼ Ca2/3 ≪ 1


according to equation (3.3). Finally, from equation (3.5), we have:

P ∼ µlUL

H2∼ µlV L

2

H3

and thus the second term on left hand side of equation (3.6) can be neglected

compared to the term on the right hand side. The equations (3.4) to (3.6) become:

∂vx∂x

+∂vy∂y

= 0 (3.7)

∂2vx∂y2

=∂p

∂x(3.8)

∂p

∂y= 0 =⇒ p = p(x) (3.9)

Thus the lubrication equation describing the fluid motion in region BC (with dimen-

sion) is:

∂2vx∂y2

(x, y) =1

µl

dp

dx(x) (3.10)

The integration of this equation gives:

vx =1

2µl

dp

dx(x)y2 +K1(x)y +K2(x) (3.11)

Boundary conditions

To determine the different functions appearing in this expression, we must apply

the boundary conditions.

Boundary condition on the wall

On the wall, since the frame of reference is linked to the moving bubble, we have:

vx(y = 0) = −U (3.12)

Boundary condition at the liquid-gas interface

At the interface between the liquid and the gas (y = h(x)) and in the absence

of phase change, the momentum conservation (equation (1.27) in chapter 1) is:

⇒

σ l .−→n =

⇒

σ g .−→n − γ C−→n (3.13)

with⇒

σ l= −p⇒

I +2µl⇒

D and⇒

σ g= −pg⇒

I +2µg⇒

Dg the stress tensors in the liquid

and gas phase respectively, γ the surface tension, and −→n the surface normal oriented

from the liquid phase to the gas phase. The expression of the rate of strain tensor

in the liquid phase is:

⇒

D=

∂vx∂x

12∂vx∂y 0

12∂vx∂y

∂vy∂y 0

0 0 0


Since∂vy∂x ,

∂vx∂x , and

∂vy∂y ≪ ∂vy

∂x , the rate of stress tensor becomes at first order:

⇒

D=

0 12∂vx∂y 0

12∂vx∂y 0 0

0 0 0

Because of the small interface curvature (H/L ≪ 1), the normal vector −→n and

tangential vector−→t can be approximated by:

−→n ∼ −→y , −→t ∼ −→x

Thus the projection of equation (3.13) onto−→t gives:

(3.13).−→t =⇒

(

⇒

D .−→n)

.−→t =

µgµl

(

⇒

D .−→n)

.−→t ≈ 0 since

µgµl

≪ 1

that is to say:

∂vx∂y

(y = h(x)) = 0 (3.14)

Finally the projection of equation (3.13) onto−→t gives (see demonstration below):

p∗ = p− pg = − γ

R− γ

d2h(x)

dx2(3.15)

with p∗ the difference of pressure between the liquid and gas phases.

Expression of the curvature of a graph defined by y = y(x)

The mathematical definition of the curvature of a graph is:

κ =

∣

∣

∣

∣

∣

d−→t

ds

∣

∣

∣

∣

∣

with−→t the tangential vector and s the arc length defined by ds =

√

dx2 + dy2. The

tangential vector can be written as:

−→t =

1√

1 + y′2

[

1

y′(x)

]

with y′ = dy/dx the derivative of y with respect to x. The curvature can be expressed

as:

κ =

∣

∣

∣

∣

∣

d−→t

ds

∣

∣

∣

∣

∣

=

∣

∣

∣

∣

∣

d−→t

dx

dx

ds

∣

∣

∣

∣

∣

(3.16)

with:d−→t

dx=

1

(1 + y′2)3/2

[ −y′′y′y′′

]

(3.17)


and:ds

dx=√

1 + y′2 (3.18)

Finally, if we replace equation (3.18) and the norm of equation (3.17) in equation

(3.16), we obtain:

κ =

∣

∣y′′2∣

∣

(1 + y′2)3/2(3.19)

Demonstration of equation (3.15)

The curvature C in equation (3.13) is the sum of transverse and axial curvatures

κt and κa. Both curvatures are negative. The transverse curvature is obviously

equal to κt = − 1R at first order (that is to say if we neglect H compared to R). The

axial curvature can be computed from formula (3.19):

κa = − d2h/dx2

(1 + (dh/dx)2)3/2that is to say at first order κa = −d

2h

dx2

Mass conservation

Equations (3.12), (3.14) and (3.15) express momentum conservation on the wall

and at the gas-liquid interface. An additional equation is required to express the

mass conservation in the liquid layer. The conservation of the volumetric flow rate

gives:

φ =

∫ h(x)

ovx(y)dy = −UH (3.20)

since a constant layer of height H is left behind the meniscus.

Complete solution

Equation of the free surface h(x)

From boundary conditions (3.12), (3.14), we can establish the expressions of the

two functions K1(x) and K2(x) appearing in equation (3.11):

K1(x) = − U

K2(x) = − 1

µl

dp∗

dxh(x)

with p∗ = p− pg. Thus the velocity field takes the form:

vx(x, y) =1

2µl

dp∗

dx

(

y2 − 2h(x)y)

− U

From mass conservation (equation (3.20)), we obtain:

φ =

∫ h(x)

ovx(y)dy = − 1

µl

dp∗

dx

h3

3− Uh(x) = −UH


that is to say:dp∗

dx=

3µlU(H − h)

h3(3.21)

If we differentiate equation (3.15) with respect to x, we obtain a second equation:

dp∗

dx= −γ d

3h(x)

dx3(3.22)

Combining these two equations gives:

d3h

dx3= 3Ca

h(x) −H

h3

with Ca = µlU/γ the Capillary number. Finally, if we introduce the variables:

ψ =h(x)

Hand ξ = 31/3Ca1/3

x

H

and substitute them in previous equation, we get:

ψ3 d3ψ

dξ3= ψ − 1 (3.23)

Asymptotic matching

3Ψζd 3

=Ψ−1Ψ d2

h/H>>1Ψ=

dd3Ψ

ζ3= 0

B

C

L

H

Ψ=h/H~13dΨ

dζ3=Ψ−1

ζ<<−1 ζ>>1ζ=Ο(1)

1 2

h/H~1

3

Ψ=

Figure 3.5: Asymptotic matching in region BC.

As long as 31/3Ca1/3L/H ≫ 1, the dynamic meniscus can be separated into

three asymptotic region (see Fig. 3.5):


1. A first region (ξ ≪ −1), in red on Fig. 3.5, where ψ = h/H ∼ 1 and thus

equation (3.23) can be approximated by:

d3ψ

dξ3= ψ − 1

2. A second region (|ξ| = O(1)), in blue on Fig. 3.5, where no approximation

can be made and the full equation:

ψ3 d3ψ

dξ3= ψ − 1

must be solved

3. A third region (ξ ≫ 1), in green on Fig. 3.5, where ψ = h/H ≫ 1 and thus

equation (3.23) can be approximated by:

d3ψ

dξ3= 0

The solution of the equation in the third region is:

ψ3 =1

2P1ξ

2 +Q1ξ +R1

with P1, Q1 and R1 three constants. If we go back to the definition of ξ and ψ, we

obtain:

h3(x) =1

2P1

(3Ca)2/3

Hx2 +Q1(3Ca)

1/3x+R1H

The matching between the curvature of the dynamic meniscus in this region and

the static meniscus gives:

d2h3(x)

dx2+

1

R=P1

H(3Ca)2/3 +

1

R≈ 2

R

From this equation we find the expression of the thickness of the film H as a function

of the Capillary number:H

R≈ P1 (3Ca)

2/3

Inly the value of P1 has not been established yet. This expression matches the one

previously obtained from dimensional analysis (equation (3.3)).

The solution of the equation in the first region is:

ψ1 = 1 + α1eξ + e−1/2ξ

[

β1 cos(

√3

2ξ) + η1sin(

√3

2ξ)

]

with α1, β1 and η1 three constants. Since in the first region ξ ≪ −1 and the height

ψ1 cannot become infinite when ξ → −∞, thus β1 = η1 = 0. The solution in this

region is therefore:

ψ1 = 1 + α1eξ


The constant α1 can be made unity by a suitable change of origin of ξ, and thus the

solution in this region is unique:

ψ1 = 1 + eξ

Complete solution of the problem

To get the complete solution of the problem, the values of P1, Q1 and R1 must

be calculated. They can be computed numerically through stepwise integration of

equation (3.23) from the first region where the solution is known to the third region.

Bretherton in his paper obtains:

P1 ≈ 0.643, Q1 = 0 and R1 ≈ 2.79

Thus we have:

h1(x) = H(

1 + exp(31/3Ca1/3x/H))

(3.24)

h3(x) =1

2

x2

R+ 1.79(3Ca)2/3R (3.25)

The final expression of the layer thickness as a function of the capillary number

is:

H

R≈ 0.643 (3Ca)2/3 (3.26)

Drift velocity of the bubble

Because of the liquid layer left on the walls, the speed of the bubble U exceeds

the average speed of the fluid V in the tube by an amount WU such as:

W =U − V

U≈ 2

H

R≈ 1.29(3Ca)2/3 (3.27)

since the flow rate conservation gives:

U × π(R−H)2 = V πR2

Thus, one must be careful when approximating the velocity of a liquid finger by

the velocity of a bubble moving in it.

Pressure drop at the front interface

Finally, because of the liquid film left on the walls, the curvature in region AB

is larger than 2/R. It is equal to 2/(R −H∗), with H∗ the height when the sphere

tangent becomes parallel to the walls (See Fig. 3.6). Since equation (3.24) describes


Air

H H*

Figure 3.6: Definition of H∗.

a continuation of the spherical region AB, we can compute the height H∗ from this

formula:

H∗ = 1.79(3Ca)2/3R

Thus the curvature is:

2

R−H∗≈ 2

R(1 +

H∗

R) ≈ 2

R

(

1 + 1.79(3Ca)2/3)

Therefore, the bubble motion induces a dynamic pressure drop ∆P through the

front meniscus in addition to the static value (2γ/R):

∆P =2γ

R× 1.79(3Ca)2/3 (3.28)


In this section, we have solved the problem of a semi-infinite bubble moving at a

constant speed in a capillary tube filled with a wetting liquid. The liquid layer left

on the walls can be seen as a way to remove the singularity at the contact line.

Indeed, in the absence of such a layer, the same singularity as the one appearing at

the meniscus of a triple line advancing on a dry substrate would appear. Now, why

do we leave more liquid when we blow harder in the tube (i.e. U increases) ? Simply

because, if we blow harder, viscous stress increases, and thus the interface can be

deformed more easily. Or, in other words, since viscous stresses in the dynamic

meniscus are of the order of µlU/H2, the higher U is, the higher H must be to

equilibrate surface tension.

3.2.2 Long bubble motion

In this section, we consider a long bubble moving in a cylindrical tube at constant

speed U in a capillary tube filled with a wetting liquid (see Fig. 3.7). This problem

is similar to the previous one but in addition, the solution for the rear interface

must match with the solution for the front interface. As the front meniscus, the

rear meniscus can be decomposed into a first region (region EF on Fig. 3.7) where


A

BC

Liquid

D

U

y

xE

F Air

Figure 3.7: Sketch of the motion of a bubble in a cylindrical tube at a constant

velocity U .

viscous stress is not sufficient to deform the interface (the static meniscus) and a

second region DE where the lubrication approximation can be used to compute

the deformation (the dynamic meniscus). Again, the dynamic meniscus DE can be

devided into three asymptotic regions (regions 4, 5 and 6 on Fig. 3.8) with the same

equations as in regions 1, 2 and 3.

H

L

6 5 4

E

D

Figure 3.8: Asymptotic matching in region DE.

Thus, in region 6 the solution of the equations is:

ψ6 =1

2P2ξ

2 +Q2ξ +R2

while in region 4, the solution becomes:

ψ4 = 1 + e−1/2ξ

[

β2 cos(

√3

2ξ) + η2sin(

√3

2ξ)

]

The solution in region 4 differs from the one in region 1 since, this time, the film

thickness cannot become infinite when ξ → +∞ instead of −∞. We see that this

solution has an oscillating part, and indeed some “waves” are observed experimen-

tally in this part of the meniscus. We also observe that a simple change of origin is

not sufficient to determine both β2 and η2. This is why the solution in the region

DE must be matched with the solution in the region BC to obtain the complete

solution of the problem.


3.3 Large capillary number and rectangular channels

In this section, we will describe briefly what happens when the geometry is changed,

or the capillary number increased.

3.3.1 Large capillary numbers

B.

FIG. 2. Normalized film thickness as a function of the capillary number for

viscous liquids. The full circles are Taylor’s data, which are compared withFigure 3.9: Comparison of the evolution of the normalized film thickness predicted

by equation (3.31) to experimental data of Taylor (full circle) and Aussilous & Quéré

(open squares).

When the capillary number is increased, viscous stresses induce larger deforma-

tion of the interface and thus more liquid is left on the walls. However above a

certain threshold, this layer thickness reaches a saturation regime. Following Aus-

silous and Quéré [2], this saturation can be explained through dimensional analysis.

When the capillary number is increased, the layer thickness H cannot be neglected

anymore compared to the radius R. Thus this latter must be replaced by R−H in

equations (3.1) and (3.2):µlU

H2∼ γ

L(R−H)(3.29)

with L the (unknown) length of the meniscus given by

− γ

R−H− γH

L2∼ − −2γ

R−H(3.30)

Thus dimensional analysis, gives:

H

R∼ Ca2/3

1 + Ca2/3

3.3. Large capillary number and rectangular channels 81

Indeed, with appropriate coefficients:

H

R=

1.34Ca2/3

1 + 1.34 × 2.5 Ca2/3(3.31)

Aussilous and Quéré obtain a good fit with experimental data (see Fig. 3.9). These

authors also studied the influence of inertia (when the Weber number is not small

anymore). They showed that inertia effects tends to thicken the film.

3.3.2 Rectangular channels

Most of the channels used in microfluidics systems have rectangular cross section,

since they are produced with soft lithography technique. The main difference in

rectangular channels, is that a large amount of liquid is left in the corner of the

channels since the bubble aims at reaching a cylindrical shape to reduce transverse

curvature (see Fig. 3.11). Bubbles motion in rectangular channels has been investi-

Air Air Air

Liquid Liquid Liquid

Ca<<1 Ca<1 Ca>1

Figure 3.10: Section of a semi-infinite bubble moving in a rectangular channel at

different Capillary numbers.

gated both experimentally and numerically [42, 41, 12]. Recently, de Lózar et al. [7]

have shown that the “coating film thickness” in a rectangular channel of aspect ratio

α = Ly/Lz can be inferred from the one in a square channel if the Capillary num-

ber Ca is replaced by a modified one Ca =[

1 + 0.12(α − 1) + 0.018(α − 1)2]

Ca

(see Fig. 3.12). Of course in a rectangular channel, the “film thickness” H is not

appropriate to determine how much liquid is left on the wall since it is not constant

over the whole section. Instead the wet fraction m = (S −Sg)/S is used with S the

surface of the channel section and Sg the surface occupied by air in a section of the

semi-infinite bubble (behind the meniscus), see Fig. 3.11. The data of de Lozár et

al. can be fitted by the following law:

m = 1−[

1 + 2.631Ca2/3

1 + 4.385Ca2/3

]2

which is nothing but the law proposed by Aussilous and Quéré but expressed in

terms of wet fraction (with appropriate coefficients).


y zS = L Lm = (S−S )/Sg

L

L

y

z Sg

Figure 3.11: Section of a semi-infinite bubble moving in a rectangular channel:

definition of the wet fraction

Figure 3.12: Wet fraction m as a function of the modified capillary number Ca

for different aspect ratio α [7]. The divergence of the curves for the low capillary

numbers is due to the appearance of gravity effects.

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