Evolution of a Bubble in Extensional Flow Academic Participants: Elyse Fosse, Nicholas Gewecke, Alvaro Guevara, Saleem Ahmed, Kuan Xu, Burt Tilley, Linda Cummings, Don Schwendeman, Colin Please, Ellis Cumberbatch, Chris Breward, Giles Richardson Industrial Presenter: John Abbott, Corning, Inc. MPI 2008 at WPI June 20, 2008 The Bubble Group (MPI 2008) Evolution of a Bubble in Extensional Flow June 20, 2008 1 / 25
The Problem temperature glass bubble z w draw The Bubble Group (MPI 2008) Evolution of a Bubble in Extensional Flow June 20, 2008 2 / 25
Questions What shape is the final bubble? Fore/Aft symmetric? Same as uniform stretching? Longer/shorter than expected? What controls final bubble shape? Temperature profile Glass viscosity Surface tension Gas pressure Initial bubble shape Are previous analyses of bubbles in shear flow relevant? The Bubble Group (MPI 2008) Evolution of a Bubble in Extensional Flow June 20, 2008 3 / 25
Model Assumptions Newtonian viscous flow Variable viscosity due to temperature (prescribed) Constant surface tension Axisymmetric flow Ideal gas with spatially uniform properties in the bubble The Bubble Group (MPI 2008) Evolution of a Bubble in Extensional Flow June 20, 2008 4 / 25
Previous Work ”The Mathematical Modelling of Capillary Drawing for Holey Fibre Manufacture,” Fitt, et.al. Optical fibers that contain air holes, inner and outer radii of air hole is taken into consideration Gave three governing equations: Conservation of mass Normal stress condition on each surface Our model is obtained by making the inner radius much less than the outer radius The Bubble Group (MPI 2008) Evolution of a Bubble in Extensional Flow June 20, 2008 5 / 25
Equations Glass Behavior ∂ wR 2 � � = 0 ∂ z � w ∂ w � = ∂ � 3 R 2 µ ( T ( z )) ∂ w � R 2 Re ∂ z + σ R ∂ z ∂ z Bubble Behavior ∂ h 2 ∂ t + ∂ 1 h 2 P − σ h h 2 w � � � � = ∂ z µ ( T ( z )) � z max π h 2 ( z , t ) P dz = M . R g T ( z ) z min The Bubble Group (MPI 2008) Evolution of a Bubble in Extensional Flow June 20, 2008 6 / 25
Characteristic Scales Parameter Range Length 1 m 10 − 1 − 10 − 4 m Glass Radius R ( z ) 10 − 5 − 10 − 8 m Bubble Radius h ( z , t ) 10 5 − 10 6 kgm − 1 s − 1 Viscosity µ ( T ( z )) 10 − 3 − 10 1 ms − 1 Axial Velocity w ( z ) . 25 Nm − 1 Surface Tension σ The Bubble Group (MPI 2008) Evolution of a Bubble in Extensional Flow June 20, 2008 7 / 25
Temperature and Viscosity profiles The Bubble Group (MPI 2008) Evolution of a Bubble in Extensional Flow June 20, 2008 8 / 25
Data fit Question At steady state, do the equations which govern the velocity w result in a velocity profile which fits the provided data? We try to answer this question using numerical methods, by focusing on the first two equations. The Bubble Group (MPI 2008) Evolution of a Bubble in Extensional Flow June 20, 2008 9 / 25
Simplest approximation Assume Reynolds number, σ is small. Method Equations First equation ⇒ R 2 w = Q (constant) ∂ wR 2 � � = 0 Set R 2 = Q / w ⇒ Second ∂ z equation is DE in terms of w Integrate twice to get expression ∂ � 3 R 2 µ ( T ( z )) ∂ w � for w = 0 Apply BCs numerically and solve ∂ z ∂ z for w numerically The Bubble Group (MPI 2008) Evolution of a Bubble in Extensional Flow June 20, 2008 10 / 25
Basic approximation The Bubble Group (MPI 2008) Evolution of a Bubble in Extensional Flow June 20, 2008 11 / 25
Second approximation Neglect only inertia Equations Method Equation (1) ⇒ R 2 = Q (constant) ∂ Set R 2 = Q / w ⇒ Second wR 2 � � = 0 ∂ z equation is DE in terms of w Integrate once, then apply a shooting method on constant of ∂ � 3 R 2 µ ( T ( z )) ∂ w � integration (binary search and ∂ z − σ R = 0 ∂ z forward Euler) to find w The Bubble Group (MPI 2008) Evolution of a Bubble in Extensional Flow June 20, 2008 12 / 25
Second approximation The Bubble Group (MPI 2008) Evolution of a Bubble in Extensional Flow June 20, 2008 13 / 25
Third approximation Surface tension and inertia effects are included. Rewrite the second equation with R 2 = Q / w : Equations ∂ wR 2 � � = 0 ∂ z � � � ∂ 3 Q w µ ( T ( z )) ∂ w Q ∂ z + σ w − Q Re w = 0 ∂ z Follow same procedure as with Second approximation Results are similar The Bubble Group (MPI 2008) Evolution of a Bubble in Extensional Flow June 20, 2008 14 / 25
Comparison of Fits A tanh fit to the data was used for other work on this problem The Bubble Group (MPI 2008) Evolution of a Bubble in Extensional Flow June 20, 2008 15 / 25
Bubble Dynamics Having found a good fit for w and R it is now possible to study the bubble behavior equations. Governing Equations ∂ h 2 ∂ t + ∂ 1 h 2 P − σ h h 2 w � � � � = ∂ z µ ( T ( z )) � z max ρ b ( z , t ) π h 2 ( z , t ) dz = M , P ( t ) = ρ b ( z , t ) R g T ( z ) z min w ( z ) given from previous analysis initial conditions: h ( z , 0 ) for z min ≤ z ≤ z max and P ( 0 ) given Analysis Pinch-off behavior Numerical solution The Bubble Group (MPI 2008) Evolution of a Bubble in Extensional Flow June 20, 2008 16 / 25
Pinch-off Analysis General solution behavior t characteristics h =0 z z min z max h ( z , t ) evolves on characteristics starting between z min and z max If no pinch-off occurs, bubble stretches according to limiting characteristics Bubble may pinch off, so bubble may be shorter than expected The Bubble Group (MPI 2008) Evolution of a Bubble in Extensional Flow June 20, 2008 17 / 25
Pinch-off model Assumptions P , µ and σ taken to be constant Velocity profile taken to be w ( z ) = γ z for a fixed γ . Equation to solve: ∂ h 2 ∂ t + γ z ∂ h 2 � P � h 2 − σ ∂ z = µ − γ µ h subject to h ( z , 0 ) given. Solve using method of characteristics The Bubble Group (MPI 2008) Evolution of a Bubble in Extensional Flow June 20, 2008 18 / 25
Solution Along characteristics, � α t � � � ( α h 0 − β ) exp + β 2 h ( t ) = , α while h > 0, otherwise h = 0, where α = P β = σ µ − γ, µ. Result: For P > γµ , no pinch off occurs provided h 0 > β α . For P < γµ , the bubble pinches off at � β + ( − α ) h 0 � 2 ˜ t = − α ln , β for any positive value of h 0 . The Bubble Group (MPI 2008) Evolution of a Bubble in Extensional Flow June 20, 2008 19 / 25
Numerical Solution of Bubble Dynamics Characteristic form: dh 2 1 on dz = − w z h 2 + Ph 2 − σ h � � , dt = w ( z ) dt µ ( T ( z )) Fixed mass constraint: � z max π h 2 ( z , t ) P dz = M R g T ( z ) z min Basic numerical scheme (assuming w ( z ) is known): Pick mesh points { z j ( 0 ) } ∈ [ z min , z max ] , specify h j ( 0 ) (ICs). Advance z j ( t ) and h j ( t ) to t + ∆ t using the characteristic form with P ( t ) held fixed. Compute P ( t + ∆ t ) using fixed mass constraint. The Bubble Group (MPI 2008) Evolution of a Bubble in Extensional Flow June 20, 2008 20 / 25
Results Initial state: Ellipsoidal Bubble: 1mm length, 5 µ m radius (maximum) Bubble pressure balanced by surface tension 5 180 160 4 height (microns) 140 pressure (kPa) 3 120 2 100 1 80 0 60 0.4 0.6 0.8 1 1.2 0 0.5 1 1.5 z (meters) time (seconds) Observations Bubble radius shrinks due to stretching Smaller bubble radius requires higher pressure to balance surface tension Bubble shrinks more to raise pressure Pressure reduction occurs when stretching ceases and temperature continues to drop The Bubble Group (MPI 2008) Evolution of a Bubble in Extensional Flow June 20, 2008 21 / 25
Further Results Initial state: Dumb-bell shaped Bubble: 1mm length, 5 µ m radius (maximum), 1 µ m radius (minimum near center) Bubble pressure balanced by surface tension 6 6 5 5 height (microns) 4 height (microns) 4 3 3 2 2 1 1 0 0 0.4 0.4002 0.4004 0.4006 0.4008 0.401 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 z (meters) z (meters) Observation Bubbles may pinch off in the middle and break in two The Bubble Group (MPI 2008) Evolution of a Bubble in Extensional Flow June 20, 2008 22 / 25
Model 2: “Leaky bubble” Extensions to previous model: Gas may diffuse from bubble into surrounding glass (locally) Bubble mass no longer conserved Gas flux due to diffusion and bubble elongation Model Equations: ∂ρ ∂ t + ∂ + ∂ = D ∂ r ∂ρ � � � � � � w ρ u ρ ∂ z ∂ r r ∂ r ∂ r where ρ ( z , t ) is density of gas in glass, h ( z , t ) is bubble radius, and u = h ∂ h ∂ z . r Initial data for ρ specified, and the boundary condition ρ at r = h determined from pressure in bubble The Bubble Group (MPI 2008) Evolution of a Bubble in Extensional Flow June 20, 2008 23 / 25
Model 2: “Leaky bubble” Gas conservation of bubble ( ρ b = bubble density) � z max � z max D ∂ρ � dM ρ b π h 2 dz = M , � dt = 2 π h dz . � ∂ r � z min z min r = h Pressure inside bubble related to density by gas law P ( t ) = ρ b ( z , t ) R g T ( z ) and ρ = Λ ρ b on r = h , Λ is the solubility coefficient. This must be coupled to original three equations for w , R , h . To be done! The Bubble Group (MPI 2008) Evolution of a Bubble in Extensional Flow June 20, 2008 24 / 25
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