Transport between two fluids across their mutual flow interface: the streakline approach. Sanjeeva Balasuriya

Transport between two fluids across their mutual flow interface: the streakline approach Sanjeeva Balasuriya

Steady (autonomous) flows Particle trajectories: dx dt = ẋ = u(x), x R2 Flow is steady (autonomous); velocities at each point remain the same in time

Steady (autonomous) flows Particle trajectories: dx dt = ẋ = u(x), x R2 Flow is steady (autonomous); velocities at each point remain the same in time Stagnation (fixed) points can be determined by u(a) = 0 Their stable and unstable manifolds locally determined by eigensystem of Du(a)

Steady (autonomous) flows Taylor-Green cellular flow An eddy Oceanic jet Hill s spherical vortex

Unsteady flow barriers Experimental vorticity field Nick Ouellette (Stanford) Gulf Stream rings Darryn Waugh (Johns Hopkins) Jupiter s Great Red Spot NASA/JPL/Caltech (NASA photo #PIA00343) Drifters and forbidden zone Josefina Olascoaga (Miami)

Unsteady flow barriers Flow is ẋ = u(x, t) nonautonomous

Unsteady flow barriers ( u(x, t)= sin 4t 2 + cos 4t 2 + cos 4t sin 4t ) x Haller (J Fluid Mech, 2005; Ann Rev Fluid Mech, 2015) Flow is ẋ = u(x, t) nonautonomous Fixed points Eigenvalues/vectors Eulerian structures Stable/unstable manifolds applicable, but may intersect in complicated ways or not at all

Unsteady flow barriers Diagnostic ways of defining flow barriers [Lagrangian Coherent Structures (LCSs)] Places where fluid piles up after advection Tél, Speetjens, Pierrehumbert Ridges of FTLEs/FSLEs and extensions Shadden, Tang, Peacock, Weinkauf, Haller, Karrasch, Peikert, Lapeyre, d Ovidio Eigen/Singular vectors associated with transfer operators Dellnitz, Junge, Froyland, Padberg-Gehle Clipped stable/unstable manifolds SB, Padberg-Gehle Averages along trajectories Mezić, Mancho, Rypina, Haller Curves/Surfaces of maximal stretching or shearing Haller, Beron-Vera of minimal flux Mackay, Polterovich, SB, Froyland, Santitissadeekorn of extremal attraction/repulsion Haller, Farazmand, Blazevski Ergodic/stochastic measures Mezić, Budisić, Tang, Rowley Topological entropy of trajectories Thiffeault, Allshouse Clustering Froyland, Padberg-Gehle, Haller, Karrasch, Dabiri Etc, etc Bollt, Rom-Kedar,...

Unsteady flow barriers 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.2 Variational LCSs George Haller (ETH), Mohammad Farazmand (MIT) Transfer operator / ocean separation Gary Froyland (UNSW), Erik van Sebille (Imperial) 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 FTLEs for double gyre SB (Adelaide), Nick Ouellette (Stanford) Clustering in ABC flow George Haller (ETH)

Unsteady flow barriers 1 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 0.5 1 1.5 Nearby steady flow I Interface splits into stable and unstable manifolds I Time-varying mixing area I Causes transport between gyres 2 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 FTLEs for double gyre

My related work unsteady LCSs 1. Locating stable/unstable manifolds SB (SIAM J Appl Dyn Sys, 2011; Phys Fluids, 2012) 2. Defining and quantifying flux in 2D SB (Nonlinearity, 2006; SIAM J Appl Dyn Sys, 2005; in: Erg Theor Open Dyn & Coherent Struct, Springer, 2014) 3. Optimising flux across LCS interfaces, and design of micromixers SB (Phys D, 2005; Phys Fluids, 2005; Phys Rev Lett, 2010, Phys Fluids, 2015), SB & Finn (Phys Rev Lett, 2012), SB (J Micromech Microeng, 2015) 4. Controlling stable and unstable manifolds SB & Padberg-Gehle (Phys D, 2014; SIAM J Appl Math, 2013; Phys Rev E, 2014), SB (J Nonlin Sci, 2016) 5. Controlling structures in turbulent experimental data Ouellette, SB & Kalampattel (sbmt, 2016), Ouellette & SB (ongoing) 6. Manifolds under impulses (e.g., tapping) SB (sbmt, 2016) 7. LCSs and ocean transport SB (Nonlin Proc Geophys, 2001; Geophys Astrophys Fluid Dyn, 2004; in progress), SB & Jones (Nonlin Proc Geophys, 2001) 8. Manifold locations used to determine front speeds and profiles SB et al (SIAM J Appl Math, 2007), SB & Volpert (Combust Theor Model, 2008), SB & Gottwald (J Math Biol, 2010), SB (J Theor Biol, 2010), SB & Binder (Phys D, 2014), Binder, Blyth & SB (J Fluid Mech, 2015) 9. Book: SB, Melnikov methods for barriers and transport in unsteady flows, SIAM Press, in press (2016)

My related work unsteady LCSs All these flow barrier definitions: Analyse the unsteady velocity field by itself Using LCS (stable/unstable manifold) ideas Just think of one fluid

Mixing of two fluids I Mixing of a sample and a reagent is a typical problem in microfluidics (DNA synthesis, biochemical assay,...)

Mixing of two fluids I I Mixing of a sample and a reagent is a typical problem in microfluidics (DNA synthesis, biochemical assay,...) Low Re flow, so no turbulent mixing

Mixing of two fluids I I I Mixing of a sample and a reagent is a typical problem in microfluidics (DNA synthesis, biochemical assay,...) Low Re flow, so no turbulent mixing Agitations may lead to chaotic mixing Aref (J Fluid Mech, 1984) Cordero et al (New J Phys, 2009) I I Experiment of mixing of two fluids within a microdroplet Mixing across flow interface achieved using thermocapillary forces (lasers)

Flow interface Velocity? 1 a b 2 a b 1 2 2 fluids in channel Anomalous fluid in vortex

Flow interface Velocity? 1 a b 2 a b 1 2 2 fluids in channel Anomalous fluid in vortex Flow interface is a physical entity

Flow interface Velocity? 1 a b 2 a b 1 2 2 fluids in channel Anomalous fluid in vortex Flow interface is a physical entity Methods of LCSs, FTLEs, etc, obtained from Lagrangian advection under the velocity field, cannot obtain these interfaces

Flow interface Velocity? 1 a b 2 a b 1 2 2 fluids in channel Anomalous fluid in vortex Under a weak unsteady velocity agitation, can we Determine unsteady flow interface? Quantify transport across it?

Why streaklines? 1 2 3 n a x p 1 x p 2 x p 3 x p n U b 2 d 1 2 d 2 2 d 3 2 d n

Mathematical setup a x p u n p ẋ = u(x) x p b x p d Γ = { x(p), p R} { } Γ = x(p), p [p u, p d ]

Mathematical setup a x p u n p ẋ = u(x) x p b x p d Γ = { x(p), p R} { } Γ = x(p), p [p u, p d ] u := ( 0 1 1 0 ) u, ˆn(p) := u ( x(p)) u ( x(p))

Mathematical setup a x p u n p ẋ = u(x) x p b x p d Γ = { x(p), p R} { } Γ = x(p), p [p u, p d ] upstream streakline Γ u 0(t) = p R { } x0 u (p, t) solving with x0 u (p, t p+p u ) = a downstream streakline Γ d 0 (t) = p R { } x0 d (p, t) solving with x0 d (p, t p+p d ) = b

Mathematical setup a x Ε u p,t x p Ε u t b ẋ = u(x) + v(x, t) v(x, t) < ε u(x) Ε d t v = 0 on Γ \ Γ

Mathematical setup a x Ε u p,t x p Ε u t b ẋ = u(x) + v(x, t) v(x, t) < ε u(x) Ε d t v = 0 on Γ \ Γ upstream streakline Γ u ε(t) = p [ P,P] downstream streakline Γ d ε (t) = p [ P,P] { } xε u (p, t) solving with xε u (p, t p+p u ) = a { } xε d (p, t) solving with xε d (p, t p+p d ) = b

Streakline expressions a x Ε u p,t x p Ε u t b upstream streakline Ε d t [x u ε (p, t) x(p)] ˆn(p) = Mu (p, t) u ( x(p)) + O(ε2 ) M u (p, t) = {p > p u} min{p,p d } p e τ [ u]( x(ξ))dξ u ( x(τ)) v ( x(τ), τ +t p) dτ p u SB (Phys Rev Fluids, submitted)

Streakline expressions a x Ε u p,t x p Ε u t b downstream streakline Ε d t [ ] xε d (p, t) x(p) ˆn(p) = Md (p, t) u ( x(p)) + O(ε2 ) M d (p, t) = {p < p d} p d p e τ [ u]( x(ξ))dξ u ( x(τ)) v ( x(τ), τ +t p) dτ max{p,p u } SB (Phys Rev Fluids, submitted)

y y y y Validation: cross-channel micromixer Stills from the previous streakline movie (ε = 0.1) 0.05 0.04 t=2.3091 0.05 0.04 t=4.0055 0.05 0.04 t=5.702 0.05 0.04 t=7.854 0.03 0.03 0.03 0.03 0.02 0.02 0.02 0.02 0.01 0.01 0.01 0.01 0 0 0 0-0.01-0.01-0.01-0.01-0.02-0.02-0.02-0.02-0.03-0.03-0.03-0.03-0.04-0.04-0.04-0.04-0.05 1 2 3 4 5 6 7 x -0.05 1 2 3 4 5 6 7 x -0.05 1 2 3 4 5 6 7 x -0.05 1 2 3 4 5 6 7 x

y y y y y y y y Validation: cross-channel micromixer Stills from the previous streakline movie (ε = 0.1) 0.05 0.04 t=2.3091 0.05 0.04 t=4.0055 0.05 0.04 t=5.702 0.05 0.04 t=7.854 0.03 0.03 0.03 0.03 0.02 0.02 0.02 0.02 0.01 0.01 0.01 0.01 0 0 0 0-0.01-0.01-0.01-0.01-0.02-0.02-0.02-0.02-0.03-0.03-0.03-0.03-0.04-0.04-0.04-0.04-0.05 1 2 3 4 5 6 7 x -0.05 1 2 3 4 5 6 7 x -0.05 1 2 3 4 5 6 7 x -0.05 1 2 3 4 5 6 7 x Computations using the theory (ε = 0.1) 0.04 t 2.3091 0.04 t 4.0055 0.04 t 5.702 0.04 t 7.854 0.02 0.02 0.02 0.02 0.00 0.00 0.00 0.00 0.02 0.02 0.02 0.02 0.04 0.04 0.04 0.04 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7

Validation: cross-channel micromixer y 0.04 0.03 0.02 0.01 0.01 1 2 3 4 5 6 7 x 0.02 ε = 0.1, t = 7.854

Validation: cross-channel micromixer y 0.04 0.03 0.03 0.02 0.02 0.01 0.01 0.02 1 2 3 4 5 6 7 x 0.01 0-0.01-0.02 1 2 3 4 5 6 ε = 0.1, t = 7.854 Backwards FTLE

Validation: cross-channel micromixer y 0.04 0.03 0.03 0.02 0.02 0.01 0.01 0.02 1 2 3 4 5 6 7 x 0.01 0-0.01-0.02 1 2 3 4 5 6 ε = 0.1, t = 7.854 ln E 0 Backwards FTLE 2 4 6 ln E 2.44 2.93 ln Ε 8 3.5 3.0 2.5 2.0 1.5 1.0 L 2 -error ln Ε

Transport across flow interface a x p u x p n p b x p d Γ separates the two fluids (upstream and downstream)

Transport across flow interface Ε u t a n p x u Ε p,t x p x Ε d p,t p,t Ε d t b Γ u ε separates from upstream Γ d ε separates downstream

Transport across flow interface Ε u t a n p x u Ε p,t x p x Ε d p,t p,t Ε d t b 1 2 a n p Φ p,t b 1' 2' Uses idea of a gate (cf. Haller & Poje (Phys D, 1998))

Transport across flow interface Ε u t a n p x u Ε p,t x p x Ε d p,t p,t Ε d t b 1 2 a n p Φ p,t b 1' 2' Transport of fluid from side 2 to 1 at time t will be defined as the instantaneous transport across the pseudo-streakline. For p [p u, p d ] φ(p, t) = M(p, t) + O(ε 2 ) M(p, t) = p d p u p e τ [ u]( x(ξ))dξ u ( x(τ)) v ( x(τ), τ + t p) dτ SB (Phys Rev Fluids, submitted)

y y y y Transport in cross-channel mixer Pseudo-streaklines constructed by numerical streaklines 0.05 0.04 t=-1 0.05 0.04 t=1 0.03 0.03 0.02 0.02 0.01 0.01 0 0-0.01-0.01-0.02-0.02-0.03-0.03-0.04-0.04-0.05 1 2 3 4 5 6 x -0.05 1 2 3 4 5 6 x 0.05 0.04 t=1.62 0.05 0.04 t=2 0.03 0.03 0.02 0.02 0.01 0.01 0 0-0.01-0.01-0.02-0.02-0.03-0.03-0.04-0.04-0.05 1 2 3 4 5 6 x -0.05 1 2 3 4 5 6 x

Transport in cross-channel mixer Comparison between numerical flux (obtained by integrating across gate) [red] and theory [blue] 0.03 ǫ=0.1 0.02 0.01 φ 0-0.01-0.02-0.03-1 -0.5 0 0.5 1 1.5 2 t ε = 0.1

Transport in cross-channel mixer Comparison between numerical flux (obtained by integrating across gate) [red] and theory [blue] 0.03 ǫ=0.1 1.2 0.02 1 0.8 ǫ=1 0.01 0.6 φ 0 φ 0.4 0.2-0.01 0-0.02-0.2-0.4-0.03-0.6-1 -1-0.5 0 0.5 1 1.5 2-0.5 0 0.5 1 1.5 2 t t ε = 0.1 ε = 1

Closed streakline situation Γ is closed (a = b) Θ n Θ x Θ a

Closed streakline situation Γ is closed (a = b) Require v(a, t) 0 Θ n Θ x Θ a

Closed streakline situation Γ is closed (a = b) Require v(a, t) 0 Θ n Θ x Θ a Perturbed streaklines will continue to wrap around

Closed streakline situation Γ is closed (a = b) Require v(a, t) 0 Perturbed streaklines will continue to wrap around Θ n Θ x Θ a Upstream/downstream expressions change slightly: M u (p, t) = M d (p, t) = p p u e p d p p τ [ u]( x(ξ))dξ u ( x(τ)) v ( x(τ), τ +t p) dτ e p τ [ u]( x(ξ))dξ u ( x(τ)) v ( x(τ), τ +t p) dτ

Kirchhoff s elliptic vortex x Θ ẋ 1 = 2x 1 m 2, ẋ 2 = 2x 1 l 2 Θ n Θ a

Kirchhoff s elliptic vortex x Θ ẋ 1 = 2x 1 m 2, ẋ 2 = 2x 1 l 2 Θ n Θ a { x 2 Suppose different fluid inside Γ := 1 l 2 + x 2 2 } m 2 = 1

Kirchhoff s elliptic vortex x Θ ẋ 1 = 2x 1 m 2, ẋ 2 = 2x 1 l 2 Θ n Θ a { x 2 Suppose different fluid inside Γ := 1 l 2 + x 2 2 } m 2 = 1 Examine effect of weak external strain on the vortex Turner (Phys Fluids, 2014) ( 0 Choose v(x 1, x 2, t) = ε sin (x 1 l) 1 ) tanh (t 5)

Kirchhoff s elliptic vortex x Θ Θ n Θ a

y y y y Kirchoff s elliptic vortex Stills from the previous streakline movie (ε = 0.2) 1.5 t=3 1.5 t=5 1.5 t=7 1.5 t=10 1 1 1 1 0.5 0.5 0.5 0.5 0 0 0 0-0.5-0.5-0.5-0.5-1 -1-1 -1-1.5-3 -2-1 0 1 2 3 x -1.5-3 -2-1 0 1 2 3 x -1.5-3 -2-1 0 1 2 3 x -1.5-3 -2-1 0 1 2 3 x Computations using the theory (ε = 0.2) 1.5 t 3 1.5 t 5 1.5 t 7 1.5 t 10 1.0 1.0 1.0 1.0 0.5 0.5 0.5 0.5 0.0 0.0 0.0 0.0 0.5 0.5 0.5 0.5 1.0 1.0 1.0 1.0 1.5 1.5 1.5 1.5 3 3 3 2 1 0 1 2 3 3 2 1 0 1 2 3 3 2 1 0 1 2 2 1 0 1 2 3

y y y Kirchoff s elliptic vortex 1.5 t=3 1.5 t=5 1 1 0.5 0.5 0 0-0.5-0.5-1 -1-1.5-3 -2-1 0 1 2 3 x -1.5-3 -2-1 0 1 2 3 x 1.5 t=8 1 0.5 0-0.5-1 -1.5-3 -2-1 0 1 2 3 x Pseudo-streaklines numerics [solid] theory [dashed]

y y y Kirchoff s elliptic vortex 1.5 t=3 1.5 t=5 1 1 0.5 0.5 0 0-0.5-0.5-1 -1-1.5-3 -2-1 0 1 2 3 x 1.5 t=8 1-1.5-3 -2-1 0 1 2 3 x Mv Π,t,Φv Π,t 5 5 10 t 15 0.5 0.2 0-0.5 0.4-1 -1.5-3 -2-1 0 1 2 3 x 0.6 0.8 Ε 0.2 Pseudo-streaklines numerics [solid] theory [dashed] Flux entering eddy numerics [red] theory [blue]

In summary... Physical flow barrier (interface between two fluids) discussed

In summary... Physical flow barrier (interface between two fluids) discussed These are not associated with standard LCS/FTLE/etc methods obtained purely from the velocity field

In summary... Physical flow barrier (interface between two fluids) discussed These are not associated with standard LCS/FTLE/etc methods obtained purely from the velocity field Streaklines are used to define these Obtained numerically in cross-channel mixer and elliptic vortex examples Theory developed for these entities, and transport across them

Thank You! Floating liquid marble (radius 0.5mm) for robust cell culture Nam-Trung Nguyen (QMNC, Griffith Univ)