on the motion of surfactant-laden bubbles · 2017. 3. 7. · clouds of droplets or bubbles used in...

On the Motion of Surfactant-Laden Bubbles

Demetrios T. Papageorgiou

Department of MathematicsImperial College London

D.T. Papageorgiou (Imperial College London) Surfactant-laden bubbles Joe Keller Meeting 2017 1 / 20

Applications

Clouds of droplets or bubbles used in industrial operations.Two-phase reactionsDropwise extractionAeration processes

Basic physical mechanism required is enhanced mass transfer.

Slip surfaces should enhance mass transfer.(Solute boundary layer size: (i) P−1/2

e , slip, (ii) P−1/3e , no-slip.)

An order of magnitude reduction in mass transfer is observed.Reason: rigidification of slip surfaces by surface active agents.Experimental evidence next.

Classical problem that received a lot of attention: Levich, Frumkin,Acrivos, Moore, Harper, ...


Physical ContextClift, Grace & Weber, Bubbles, Drops, and Particles.Eo = g ∆ρ d2

γ , Re = ρU dµ (Eötvös number is the Bond number)


Spherical particles: Classical solutions, Re = 0

solid sphere settling

Us

Ub

droplet rising or falling

1 Solid sphere - Stokes terminal velocity Us = 2ga2∆ρ9µ

2 Liquid sphere - Hadamard-Rybczynski velocityUb = 2ga2∆ρ

3µ

(1+κ

2+3κ

), κ = µin/µ.

Gas bubble, κ = 0, Ub = 32Us.


Do bubbles/drops move as if they are clean - i.e. zeroshear stress surfaces?

Experiments suggest that they typically don’t.

Velocities measured with Us < U < Ub if Re is small.

If Re = O(1), simulations/experiments needed to produce inertialdrag coefficient correlations for solid spheres, e.g.

CD =24Re

(1 + 0.1935 R0.6305

e

)for 20 ≤ Re ≤ 260.

Culprit: trace amounts of surfactants and/or impurities.

Drop/bubble surface behaves rigidly, the drag increases and theterminal velocity decreases.

Experimental evidence next.


Surfactant-induced retardation: ExperimentsSavic (1953) Bel Fdhila & Duinevelt (1996)


Surfactant-induced retardation mechanism

initial stages stagnant-cap regime


Results considered in this talk

Large parameter space and complicated physicochemical effectsWill not show computations on the stagnant-cap regime.Lots in the literature (besides ours); show complete rigidificationas bulk surfactant concentration increases. See figure for a casewith Re = 100.

Will concentrate on surfactants that can remobilize the interface.


Mathematical model for a spherical gas bubble

oncoming flow Ubulk concentration C

surface surfactant concentration Γ

surface tension γ=γ(Γ)

Langmuir kinetics: C to Γ

(r,θ)

r

θ

C=C∞

far away

Axisymmetric flow u = u er + v eθ.


Mathematical model for a spherical gas bubbleEquations in the bulk - fluid viscosity µ, density ρ; bubble radius a.

Navier-Stokes and concentration equations:

ut + (u · ∇)u = −∇p +1

Re∇2u

∇ · u = 0

Ct + u · ∇C =1

Pe∇2C

Re = ρUa/µ - Reynolds numberPe = Ua/D - Peclet number (D diffusion coefficient)Boundary conditions - axisymmetric flowTangential stress balance - brings in Marangoni forces

Ca τrθ|r=1 = Ca r∂

∂r

(ur

)∣∣∣∣r=1

= −∂γ∂θ

Capillary number Ca = µU/γc .D.T. Papageorgiou (Imperial College London) Surfactant-laden bubbles Joe Keller Meeting 2017 10 / 20

Boundary conditions at bubble surface r = 1Surfactant equation of state:

γ = 1 + Ma Ca log(1− Γ) ⇒ ∂γ

∂θ= −Ma Ca (∂Γ/∂θ)

1− Γ

Langmuir kinetic flux:

D∂C∂r

∣∣∣∣r=a

= βCs(Γ∞ − Γ)− αΓ dimensional

χkPe

∂C∂r

∣∣∣∣r=1

= Bi [k C|r=1 (1− Γ)− Γ] dimensionless

χ = αa/(βΓ∞), k = βC∞/α, Bi = αa/U, Ma = RT Γ∞/(µU)

Surface surfactant mass balance:

∂Γ

∂t+

1sin θ

∂

∂θ(v Γ sin θ) =

χkPe

∂C∂r

∣∣∣∣r=1


Physical mechanisms of remobilization

Two crucial dimensionless numbers regarding surfactant activity:

diffusive rateconvective rate

=χkPe,

kinetic desorptionsurface convection

= Bi

If χk/Pe � 1 or Bi � 1 ⇒ insoluble limit ⇒ stagnant caps.(E.g. Bel Fdhila & Duinevelt (1996), McLaughlin (1997) hadχk/Pe � 1 and Bi � 1.)Suggest fast kinetic rates Bi � 1, and study the effect of χk/Pe.Bi � 1 implies surface/sublayer equilibrium. To leading order

k C|r=1 (1− Γ)− Γ = 0 ⇒ Γ =k C

1 + k C

∣∣∣∣r=1

Computational results next (finite volume projection methods).


I. Zero Reynolds number flowsIn this case the flow remains attached: Surfactant immobilizes theinterface.Parameters: Ma = 5, χ = 1

Pe = 10


II. Non-zero Reynolds numbers

Some facts:Solid sphere: wake formation at rear stagnation point at Re ≈ 12.Clean bubbles:

I Almost spherical bubbles (We = ρaU2/γ small) - no separation.Flow attached at all Re - Moore (1963, 1965).

I Large We implies large distortions - separation and wake formationat large enough Re.

Questions for surfactant-laden spherical bubbles:Do wakes form as the surfactant concentration increases (forRe > 12)?For a fixed Re where a wake has formed, what happens as k isincreased?What is the effect of increasing surfactant on the Marangoni force?


Marangoni forces and surface velocities - Re = 50.


Wake formation as Re increasesMa = 5, χ = 1, k = 5, Pe = 100


Remobilization as bulk surfactant concentrationincreases - Re = 50Ma = 5, χ = 1

Pe = 100 Pe = 200


Open questions - future directionsNot easy to find surfactants that conform to the theory presented.Remobilization typically happens above the critical micelleconcentration (CMC). Experiment by A. Taneja (2007).


Joe Keller and bubbles in highly viscous fluids

Could be very useful in several directions including:(i) stagnant cap solutions, (ii) hybrid Stokes/high Pe algorithms, ...


Collaborators

Charles Maldarelli, Levich Institute, CCNYY. Wang, R. Palaparthi, A. Taneja, PhD students

Relevant publications:

Wang, Papageorgiou & Maldarelli, J. Fluid Mech. (1999)Wang, Papageorgiou & Maldarelli, J. Fluid Mech. (2002)Palaparthi, Papageorgiou & Maldarelli, J. Fluid Mech. (2006)A. Taneja, Ph.D. thesis (2007)


on the motion of surfactant-laden bubbles · 2017. 3. 7. · clouds of droplets or bubbles used in...

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