flows in liquid foams - school of mathematics : trinity...

FLOWS IN LIQUID FOAMSFLOWS IN LIQUID FOAMS

A finite element approachA finite element approach

A. SAUGEY, E. JANIAUD, W. DRENCKHAN, A. SAUGEY, E. JANIAUD, W. DRENCKHAN,

S. HUTZLER, D. WEAIRES. HUTZLER, D. WEAIRE

Physics DepartmentPhysics Department, TRINITY COLLEGE DUBLIN, TRINITY COLLEGE DUBLIN

22/22/22

Facing a problem of fluid flow in a physical system, Facing a problem of fluid flow in a physical system,

we can either :we can either :

•• calculate the full solution of the Naviercalculate the full solution of the Navier--Stokes Stokes

equations and associated boundary conditionsequations and associated boundary conditions

–– Poiseuille flow in pipesPoiseuille flow in pipes

–– Couette flow under shearCouette flow under shear

•• determine an approximate solution determine an approximate solution

–– by scaling argumentsby scaling arguments

–– by appropriate approximations (lubrification,…)by appropriate approximations (lubrification,…)

–– by by FINITE ELEMENT modelsFINITE ELEMENT models

33/22/22

About the Bretherton problem :About the Bretherton problem :

Bretherton power law comes from a lubrication Bretherton power law comes from a lubrication

approximation. What about tackling the problem with a approximation. What about tackling the problem with a

Finite ElementFinite Element calculation ?calculation ?

«« … … Generations of postdocs and postgrads have failed Generations of postdocs and postgrads have failed to do this ! Good luck !to do this ! Good luck ! » »

D. W. from AustraliaD. W. from Australia

Let’s give the Finite Element technique his bestLet’s give the Finite Element technique his best

44/22/22

OUTLINEOUTLINE

PART I : INTRODUCTION TOPART I : INTRODUCTION TO FINITE ELEMENTFINITE ELEMENT

PART II : SOME APPLICATIONS IN FOAM PHYSICSPART II : SOME APPLICATIONS IN FOAM PHYSICS

PART III : SOME TECHNICAL FEATURESPART III : SOME TECHNICAL FEATURES

1 1 –– NavierNavier--Stokes equation and the strong formulationStokes equation and the strong formulation

Virtual Power Principle and the weak formulationVirtual Power Principle and the weak formulation

2 2 –– Implementation of boundary conditionsImplementation of boundary conditions

3 3 –– Moving meshesMoving meshes

4 4 –– Implementation of the surface tensionImplementation of the surface tension

TUTORIAL : INTRODUCTION TO COMSOL MultiphysicsTUTORIAL : INTRODUCTION TO COMSOL Multiphysics

55/22/22

GENERAL OVERVIEWGENERAL OVERVIEW

FINITE ELEMENT FINITE ELEMENT

= =

Resolution of a Resolution of a

Partial Differential ProblemPartial Differential Problem

APPLICATION FIELDSAPPLICATION FIELDS

Fluid MechanicsFluid Mechanics

Structural MechanicsStructural Mechanics

ElectromagnetismElectromagnetism

AcousticsAcoustics

Heat transferHeat transfer

ChemistryChemistry

DEDICATED SOFTWARES :DEDICATED SOFTWARES :

FEMLAB (COMSOL Multiphysics)FEMLAB (COMSOL Multiphysics)

QUICKFIELDSQUICKFIELDS

Fluent Fluent

Nastran, Ansys,…Nastran, Ansys,…

t

∂ + ∇∂

∇

.Γ = F

.Γ = F

++Boundary

Conditions

I Intro to Finite Element

II Applications in Foams

III Technical Features

1 - Strong Formulation /

Weak Formulation

2 – Boundary conditions

3 - Moving mesh

4 - Surface tension

66/22/22

NUMERICAL ASPECTS OF NUMERICAL ASPECTS OF

FINITE ELEMENT TECHNIQUESFINITE ELEMENT TECHNIQUES

FINITE ELEMENT = FINITE ELEMENT =

Numerical estimation of the solution of the Numerical estimation of the solution of the

Partial Differential Equation (PDE)Partial Differential Equation (PDE)

NUMERICAL ESTIMATE :NUMERICAL ESTIMATE :

Discretization of the geometry in elementsDiscretization of the geometry in elements

Estimation at node and interpolation (degrees of freedom)Estimation at node and interpolation (degrees of freedom)

RESOLUTION OF THE PDERESOLUTION OF THE PDE

Write the PDE in a matrix format :Write the PDE in a matrix format :

Matrix inversion and time integrationMatrix inversion and time integration

M +C +X X XK = F

=XK F

ɺɺ ɺ





Weak Formulation


3 - Moving mesh

4 - Surface tension

77/22/22

NUMERICAL ASPECTS OF NUMERICAL ASPECTS OF

FINITE ELEMENT TECHNIQUESFINITE ELEMENT TECHNIQUES

From the Partial Differential Equation to the Matrix Equation :From the Partial Differential Equation to the Matrix Equation :

Vibrating String :Vibrating String :





Weak Formulation


3 - Moving mesh

4 - Surface tension

xx11 xx33 xx44 xx55 xx66xx22

yy11 yy33yy44

yy55 yy66yy22

2 2

2 20

t

∂ ∂⇒−

∂ ∂y y

=x

Y +KY = 0ɺɺ

∂∇

∂

2 1

12 1 2 1 2 1

23 2

33 2 3 2 3 2

y - y 1 1- 0 0

yx - x x - x x - x

yyy =

y - y 1 1= 0 -

yx - x x -x -

0

=x x x

⋱

⋮⋱ ⋱ ⋱⋮

( )( ) ( )( )( )( )( )2

2

∂∆∂

2 1 3 2 3 2 2 1 1

2 1 3 2 3 1 2

3

y - y x - x - y - y x - x y* * * 0 0

x - x x - x x - x y= 0 * *=

yy =

x* 0

y0 0 ⋱ ⋱ ⋱⋮

⋮

xx

y(x,t)y(x,t)

88/22/22

HOW TO USE FEMLABHOW TO USE FEMLAB

DIRECT IMPLEMENTATION OF THE PDE DIRECT IMPLEMENTATION OF THE PDE

NUMERICAL RESOLUTION AS A BLACK BOXNUMERICAL RESOLUTION AS A BLACK BOX

Discretization (elements, node)Discretization (elements, node)

Computation of the matricesComputation of the matrices

Matrix inversion and time integration Matrix inversion and time integration

t

∂ + ∇∂

.Γ = F ++Boundary

Conditions





Weak Formulation


3 - Moving mesh

4 - Surface tension

99/22/22

APPLICATIONS IN FOAMSAPPLICATIONS IN FOAMS





Weak Formulation


3 - Moving mesh

4 - Surface tension

•• Flow in Plateau borders with LiquidFlow in Plateau borders with Liquid--Vapor surfaces Vapor surfaces

subjected to surface viscositysubjected to surface viscosity

•• Flow in vertices with LiquidFlow in vertices with Liquid--Vapor surfaces Vapor surfaces

subjected to surface viscositysubjected to surface viscosity

•• Drainage in a 2D foamDrainage in a 2D foam

•• Wall slip of bubbles : “The Bretherton Problem”Wall slip of bubbles : “The Bretherton Problem”

1010/22/22

DRAINAGE IN LIQUID FOAMS DRAINAGE IN LIQUID FOAMS –– PLATEAU BORDERSPLATEAU BORDERS

Viscous Flow in Bη ρ

Ω∆U = g Surface viscosity on

S Bη η∂Ω

∂∂S

U∆ U=

nNo flow in filmsU= 0





Weak Formulation


3 - Moving mesh

4 - Surface tension

1111/22/22

DRAINAGE IN LIQUID FOAMS DRAINAGE IN LIQUID FOAMS –– VERTICESVERTICES

Viscous Flow in B pη

Ω−∇∆U = f Surface viscosity on

S Bη η∂Ω

∂∂S

U∆ U=

nNo flow in films

U = 0





Weak Formulation


3 - Moving mesh

4 - Surface tension

1212/22/22

DRAINAGE IN LIQUID FOAMSDRAINAGE IN LIQUID FOAMS





Weak Formulation


3 - Moving mesh

4 - Surface tension

1313/22/22

DRAINAGE IN 2D FOAMSDRAINAGE IN 2D FOAMS





Weak Formulation


3 - Moving mesh

4 - Surface tension

Low flow Low flow rate rate High Flow High Flow raterate

( )

( )

0

( , )

( )

B

f tt

kp

p pr

ε ε

ρη

γε

∂ + ∇ =∂

= −∇ +

= −

v r

v g

i

( )3

2

( , )f tt

ε

ε ε

∂ − ∇ =∂

= ∇ + z

Γ r

Γ e

i

CONSTITUTIVE LAWS CONSTITUTIVE LAWS

PDE SYSTEM PDE SYSTEM

++Boundary

Conditions

1414/22/22

WALL SLIP OF BUBBLESWALL SLIP OF BUBBLES

THE BRETHERTON PROBLEMTHE BRETHERTON PROBLEM

Rheometer experiments Train of bubblesRheometer experiments Train of bubbles





Weak Formulation


3 - Moving mesh

4 - Surface tension

1515/22/22

WALL SLIP OF BUBBLESWALL SLIP OF BUBBLES

RESULTSRESULTS

FlowFlow

Dissipation fieldDissipation field

Force lawForce law





Weak Formulation


3 - Moving mesh

4 - Surface tension

STILL TO BE DONE :STILL TO BE DONE :

•• Different boundary conditions (slip, no slip, surface viscosityDifferent boundary conditions (slip, no slip, surface viscosity))

•• Volume constraint (constant static pressure, constant volume)Volume constraint (constant static pressure, constant volume)

•• 2D models, 3D models2D models, 3D models

•• Correlation with experimentsCorrelation with experiments

1616/22/22

TROU NORMANDTROU NORMAND

POSSIBILITIES OF FINITE ELEMENT TECHNIQUES :POSSIBILITIES OF FINITE ELEMENT TECHNIQUES :

Numeric solution for Numeric solution for

•• Non linear Partial Differential EquationsNon linear Partial Differential Equations

•• PDE with specific boundary conditionsPDE with specific boundary conditions

•• PDE with moving geometriesPDE with moving geometries





Weak Formulation


3 - Moving mesh

4 - Surface tension

1717/22/22

TECHNICAL FEATURESTECHNICAL FEATURES





Weak Formulation


3 - Moving mesh

4 - Surface tension

THE BRETHERTON PROBLEM step by step :THE BRETHERTON PROBLEM step by step :

1. 1. Bulk liquid FlowBulk liquid Flow

2. 2. Flow boundary conditionsFlow boundary conditions

3. 3. Moving geometriesMoving geometries4. Surface tension4. Surface tension

1818/22/22

INCOMPRESSIBLE FLUID FLOWINCOMPRESSIBLE FLUID FLOW

MASS CONSERVATIONMASS CONSERVATION

0∇ =Ui





Weak Formulation


3 - Moving mesh

4 - Surface tension

MOMENTUM EQUATION : TWO EQUIVALENT FUNDAMENTAL LAWSMOMENTUM EQUATION : TWO EQUIVALENT FUNDAMENTAL LAWS

Navier Stokes Equation => Strong FormulationNavier Stokes Equation => Strong Formulation

boundary conditionsB pη∇ = −∇σ ∆U = f + i

kinematically admissible : : ( ) . . .Ω Ω ∂Ω

∀ = +∫ ∫ ∫U σ ε U f U nσUɶ ɶ ɶ ɶ

Virtual Power Law => Weak FormulationVirtual Power Law => Weak Formulation

( ) ( )1

2

t t

Bp η= − + ∇ + ∇ = ∇ + ∇σ 1 U U ε U U

1919/22/22

BOUNDARY CONDITIONSBOUNDARY CONDITIONS





Weak Formulation


3 - Moving mesh

4 - Surface tension

DIRICHLET BC :DIRICHLET BC :

( )00, R 0 = = ⇒ =U UU U

( )0, - . G δ

∂ ∂ = = ⇒ ∂ ∂ =U U U

n nnσ U

NEUMANN BC :NEUMANN BC :

MIXED NEUMANNMIXED NEUMANN--DIRICHLET DIRICHLET

BOUNDARY CONDITIONS :BOUNDARY CONDITIONS :

( )

( )

R 0

R- . G

T

µ

=

∂= + ∂

U

nσ UU

PDE BC :PDE BC :

, =f S Bη η σ∂ ∇ ⇒ ∂ S

U∆ U =

ni

Modification of Modification of the weak the weak formulation formulation

of¨Navier Stokes of¨Navier Stokes equationsequations

2020/22/22

MOVING MESHMOVING MESH





Weak Formulation


3 - Moving mesh

4 - Surface tension

( )

( , ) 0

( , ) 0

NS x y

NS X Yφ

=

⇓

=

U

U

2121/22/22

SURFACE TENSIONSURFACE TENSION





Weak Formulation


3 - Moving mesh

4 - Surface tension

CURVATURE EQUATIONCURVATURE EQUATION

pγ∇ =ni

WEAK FORMULATIONWEAK FORMULATION

Minimization of the bulk + surface energy for a virtual Minimization of the bulk + surface energy for a virtual

displacement of the interface :displacement of the interface :

Non Non linearlinear solutionsolutionCorrection of Correction of the the Navier Stokes Navier Stokes weak weak formulationformulation

2222/22/22

SUMMARYSUMMARY





Weak Formulation


3 - Moving mesh

4 - Surface tension

Strong FormStrong Form

Weak FormWeak Form

Mixed BCMixed BC

Weak Weak BCBC

Moving GeometryMoving Geometry

Surface TensionSurface Tension

Technical Technical

RequirementRequirement

Physical Physical

Aspects Aspects

2323/22/22

flows in liquid foams - school of mathematics : trinity...

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