FLOWS IN LIQUID FOAMS

Transcription

1 FLOWS IN LIQUID FOAMS A finite element approach A. SAUGEY, E. JANIAUD, W. DRENCKHAN, S. HUTZLER, D. WEAIRE Physics Department,, TRINITY COLLEGE DUBLIN

2 Facing a problem of fluid flow in a physical system, we can either : calculate the full solution of the Navier-Stokes equations and associated boundary conditions Poiseuille flow in pipes Couette flow under shear determine an approximate solution by scaling arguments by appropriate approximations (lubrification, ) by FINITE ELEMENT models 2/22

3 About the Bretherton problem : Bretherton power law comes from a lubrication approximation. What about tackling the problem with a Finite Element calculation? «Generations of postdocs and postgrads have failed to do this! Good luck!» D. W. from Australia Let s give the Finite Element technique his best 3/22

4 OUTLINE PART I : INTRODUCTION TO FINITE ELEMENT PART II : SOME APPLICATIONS IN FOAM PHYSICS PART III : SOME TECHNICAL FEATURES 1 Navier-Stokes equation and the strong formulation Virtual Power Principle and the weak formulation 2 Implementation of boundary conditions 3 Moving meshes 4 Implementation of the surface tension TUTORIAL : INTRODUCTION TO COMSOL Multiphysics 4/22

5 GENERAL OVERVIEW FINITE ELEMENT = Resolution of a Partial Differential Problem +.Γ = F t.γ = F + Boundary Conditions APPLICATION FIELDS Fluid Mechanics Structural Mechanics Electromagnetism Acoustics Heat transfer Chemistry DEDICATED SOFTWARES : FEMLAB (COMSOL Multiphysics) QUICKFIELDS Fluent Nastran, Ansys, 5/22

6 NUMERICAL ASPECTS OF FINITE ELEMENT TECHNIQUES FINITE ELEMENT = Numerical estimation of the solution of the Partial Differential Equation (PDE) NUMERICAL ESTIMATE : Discretization of the geometry in elements Estimation at node and interpolation (degrees of freedom) RESOLUTION OF THE PDE Write the PDE in a matrix format : M Xɺɺ + C Xɺ + K X = F KX = F Matrix inversion and time integration 6/22

7 NUMERICAL ASPECTS OF FINITE ELEMENT TECHNIQUES From the Partial Differential Equation to the Matrix Equation : Vibrating String : y y = t x Y ɺɺ + KY = 0 y(x,t) y 2 y 1 y 3 y 4 y 5 y 6 x 1 x 2 x 3 x 4 x 5 x 6 x y 2 - y x2 - x1 x2 - x1 x2 - x y 1 1 y y3 - y2 1 1 y2 y = = = 0 - x x3 - x2 x3 - x2 x3 - x2 y3 0 ( )( ) ( )( ) ( )( )( ) y 2 - y1 x3 - x2 - y3 - y2 x2 - x1 y1 * * * y x2 - x1 x3 - x2 x3 - x1 y 2 y = = 2 = 0 * * * 0 x y /22

8 HOW TO USE FEMLAB DIRECT IMPLEMENTATION OF THE PDE +.Γ = F + t Boundary Conditions NUMERICAL RESOLUTION AS A BLACK BOX Discretization (elements, node) Computation of the matrices Matrix inversion and time integration 8/22

9 APPLICATIONS IN FOAMS Flow in Plateau borders with Liquid-Vapor surfaces subjected to surface viscosity Flow in vertices with Liquid-Vapor surfaces subjected to surface viscosity Drainage in a 2D foam Wall slip of bubbles : The Bretherton Problem 9/22

10 DRAINAGE IN LIQUID FOAMS PLATEAU BORDERS Viscous Flow in Ω ηb U = ρg Surface viscosity on U ηs SU = ηb No flow in films n U = 0 Ω 10/22

11 DRAINAGE IN LIQUID FOAMS VERTICES Viscous Flow in Ω η B U p = f Surface viscosity on U No flow in films ηs SU = ηb U = 0 n Ω 11/22

12 DRAINAGE IN LIQUID FOAMS 12/22

13 CONSTITUTIVE LAWS DRAINAGE IN 2D FOAMS ε + i( ε v) = f ( r, t) t k v = + η B p = p 0 ( p ρg) γ r( ε ) PDE SYSTEM ε i( Γ) = t f ( r, t) 3 2 Γ = ε + ε ez + Boundary Conditions Low flow rate High Flow rate 13/22

14 WALL SLIP OF BUBBLES THE BRETHERTON PROBLEM Rheometer experiments Train of bubbles 14/22

15 RESULTS WALL SLIP OF BUBBLES Flow Dissipation field Force law STILL TO BE DONE : Different boundary conditions (slip, no slip, surface viscosity) Volume constraint (constant static pressure, constant volume) 2D models, 3D models Correlation with experiments 15/22

16 TROU NORMAND POSSIBILITIES OF FINITE ELEMENT TECHNIQUES : Numeric solution for Non linear Partial Differential Equations PDE with specific boundary conditions PDE with moving geometries 16/22

17 TECHNICAL FEATURES THE BRETHERTON PROBLEM step by step : 4. Surface tension 3. Moving geometries 1. Bulk liquid Flow 2. Flow boundary conditions 17/22

18 INCOMPRESSIBLE FLUID FLOW MASS CONSERVATION iu = 0 MOMENTUM EQUATION : TWO EQUIVALENT FUNDAMENTAL LAWS Navier Stokes Equation => Strong Formulation iσ = η B U p = f + boundary conditions Virtual Power Law => Uɶ kinematically admissible : σ : ε( Uɶ ) = f. Uɶ + nσ.. Uɶ Ω Ω Ω 1 σ = p1 + η B U + U ε = U + U 2 ( t ) ( t ) 18/22

19 BOUNDARY CONDITIONS DIRICHLET BC : { U U U } 0 ( U) = 0, = R = 0 NEUMANN BC : U U U = = n n δ 0, - nσ. = G( U) MIXED NEUMANN-DIRICHLET BOUNDARY CONDITIONS : ( U) R = 0 R - nσ. = G( U) + U T µ PDE BC : U ηs SU = ηb, iσ =f n Modification of the weak formulation of Navier Stokes equations 19/22

20 MOVING MESH NS U( x, y) = 0 { } NS { φ ( U X Y )} (, ) = 0 20/22

21 SURFACE TENSION CURVATURE EQUATION γ in = p WEAK FORMULATION Minimization of the bulk + surface energy for a virtual displacement of the interface : Non linear solution Correction of the Navier Stokes weak formulation 21/22

22 SUMMARY Strong Form Weak Form Mixed BC Weak BC Physical Aspects Moving Geometry Surface Tension Technical Requirement 22/22

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