Simulation and validation of surfactant-covereddroplets in two-dimensional Stokes flow

Sara Pålsson∗, Anna-Karin Tornberg∗,∗ Department of Mathematics, KTH Royal Institute of Technology, Stockholm, Sweden

On the micro scale

• In micro-scale bubble and drop dynamics, the flow iswell-described by the linear Stokes equations.• Surface tension governs the flow, and can be modi-

fied with the addition of surface reacting agents (sur-factants, Γ ). The change of surfactant concentration intime is described by a convection-diffusion equation.

Aim: In this project we simulate the deformation ofsurfactant-covered drops. We handle both single andmultiple drops in close interaction, and validate our re-sults against analytic results [3].

Validation

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

Initial state Steady state

=⇒ Rmin

Rmax

• Analytical expressions for interface, z , and Γ exist for in-viscid bubbles in linear strain flow, u=Q (x ,−y ).• Results hold for all surfactant covered viscous drops, in

steady state.• For a given Q , we obtain steady-state surfactant con-

centration and deformation, D = Rma x−Rmi nRma x+Rmi n

, with errorsaround time-step tolerance, Figure 1.

0 0.1 0.2

Capillary number Q

0

0.5

1

Deform

ationD

0.1 0.2

Capillary number Q

10-7

10-6

Deform

ationerror

Figure 1: Steady state validation. Left: Capillary number vs deforma-tion. Right: Deformation error for different viscosity ratios λ of drops.

Solving Stokes equations

•Compute the flow velocity through Stokes equations

∆u=∇p , ∇·u= 0.

•We use a boundary integral method (BIE),

u=∫δΩT(µ(τ),

1

τ− z

), ∀z ∈Ω∪δΩ.

•We first need to solve an integral equation over the fluid-drop interface [1],

Solve for µ: µ(z ) +∫δΩF(µ(τ),

1

τ− z

)= g (z ,Γ ), z ∈δΩ.

• The solutionµ(z ) depends on the surfactant concentra-tion Γ .•T andFbecome nearly singular close to drop interfaces.

Drop interaction

Adding surfactants radically changes the behaviour ofthe droplets through time.

•When drops get close toeach other, we get nearlysingular integrals.• To obtain high accuracy,

we apply a specialquadrature [2].

• Left: drop deformation in shear flow, u = G (0,−x ).Colour contour represents concentration of surfac-tants, blue drops surfactant-free.• Surfactans move towards the tips of the droplets, creat-

ing gradients of surface tension =⇒Marangoni stresseschange the stress balance over the interface.

Solving the surfactant equation

• The equation for surfactant concentration Γ is solvedwith an FFT-based method with periodic boundaryconditions, with spectral accuracy in space.•We get a system of ODEs to solve for the Fourier coeffi-

cients;

d Γkd t= fconvection︸ ︷︷ ︸

treat explicitly

(Γk , z , u

)+

treat implicitly︷ ︸︸ ︷fdiffusion

(Γk , z

).

Stepping through time

The system of interface evolution and surfactant con-centration is coupled:

d zd t =BIE(z ,Γ , u ),δΓδt = fconvection (Γ , z , u) + fdiffusion (Γ , z ) .

•Use explicit and implicit-explicit Runge-Kuttaschemes of 2nd order for drop derformation andsurfactant concentration respectively.• Adaptivity in both z and Γ is achieved through

comparison with corresponding first order schemes.

Outlook

The aim is to develop a semi-analytical framework forvalidating multiple surfactant-covered bubbles in strainflow. Furthermore, solid boundaries will be included inthe general framework, to simulate for example pipe flow.

Acknowledgements and references

This work has been supported by the Göran Gustafsson Foundation for Researchin Natural Sciences and Medicine.

[1] Kropinski, MCA. An efficient numerical method for studying interfacialmotion in two-dimensional creeping flows. Journal of Computational Physics, 2001.

[2] Ojala, R. and Tornberg, A-K. An accurate integral equation method forsimulating multi-phase Stokes flow. Journal of Computational Physics, 2015.

[3] Siegel, M. Cusp formation for time-evolving bubbles in two-dimensionalStokes flow. Journal of Fluid Mechanics, 2000.