Simulation and validation of surfactant-covered droplets 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 is well-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 in time is described by a convection-diffusion equation. Aim: In this project we simulate the deformation of surfactant-covered drops. We handle both single and multiple 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 = ⇒ R min R max • 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 = R max -R min R max +R min , with errors around time-step tolerance, Figure 1. 0 0.1 0.2 Capillary number Q 0 0.5 1 Deformation D 0.1 0.2 Capillary number Q 10 -7 10 -6 Deformation error 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 Δu = ∇p , ∇· u = 0. • We use a boundary integral method (BIE), u = Z δΩ T 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 )+ Z δΩ F μ(τ), 1 τ - z ! = g ( z , Γ ), z ∈ δΩ. • The solution μ( z ) depends on the surfactant concentra- tion Γ . • T and F become nearly singular close to drop interfaces. Drop interaction Adding surfactants radically changes the behaviour of the droplets through time. • When drops get close to each other, we get nearly singular integrals. • To obtain high accuracy, we apply a special quadrature [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 stresses change the stress balance over the interface. Solving the surfactant equation • The equation for surfactant concentration Γ is solved with an FFT-based method with periodic boundary conditions, with spectral accuracy in space. • We get a system of ODEs to solve for the Fourier coeffi- cients; d b Γ k dt = b f convection | {z } treat explicitly b Γ k , z , u + treat implicitly z }| { b f diffusion b Γ k , z . Stepping through time The system of interface evolution and surfactant con- centration is coupled: dz dt = BIE( z , Γ , u ), δΓ δ t = f convection (Γ , z , u)+ f diffusion (Γ , z ) . • Use explicit and implicit-explicit Runge-Kutta schemes of 2nd order for drop derformation and surfactant 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 for validating multiple surfactant-covered bubbles in strain flow. Furthermore, solid boundaries will be included in the general framework, to simulate for example pipe flow. Acknowledgements and references This work has been supported by the Göran Gustafsson Foundation for Research in Natural Sciences and Medicine. [1] Kropinski, MCA. An efficient numerical method for studying interfacial motion in two-dimensional creeping flows. Journal of Computational Physics , 2001. [2] Ojala, R. and Tornberg, A-K. An accurate integral equation method for simulating multi-phase Stokes flow. Journal of Computational Physics, 2015. [3] Siegel, M. Cusp formation for time-evolving bubbles in two-dimensional Stokes flow. Journal of Fluid Mechanics, 2000.