Fingering Instability in Hele-Shaw cell

Lucas Amoudruz

Computational Science and Engineering Laboratory

with:

Sergey LitvinovPetros Koumoutsakos

OUTLINE

• Fingering instabilities in Hele-Shaw cells • Linear stability analysis in 2D • Simulations with Dissipative Particle Dynamics • Results

Fingering instabilities

- low viscosity fluid flowing inside high viscosity - same densities

Credits: Nagel Group, University of Chicago

- oil recovery - sugar refining - fundamental question: instabilities

Hele Shaw flow

b

L

∇p

Re ≪ 1 ∇ ⋅ u = 0Stokes approx: μΔu = ∇p

Darcy + continuity 2D potential flow: Δϕ = 0

μviscosity

bL

≪ 1 Darcy’s law: u =κμ

∇p = ∇ϕ

porosity κ =b2

12

Stability analysis, 2D

perturbate interface: x = α exp (iky + σt)

Saffman, P. G. & Taylor, G. The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid, 1958

no surface tension: p− = p+ (at interface)

σkκ (μ+ + μ−) = (μ+ − μ−) U always unstable for μ− < μ+

x

y

μ− μ+

U

suggests 3D effectscontradicts experiments

Available numerical models?

Tryggvason, G. & Aref, H. Numerical experiments on Hele Shaw flow with a sharp interface

• Continuum methods: Boundary Integral, Level sets • 2D simulations • Previous work did not consider zero surface tension limit

Li, S.; Lowengrub, J. S. & Leo, P. H. A rescaling scheme with application to the long-time simulation of viscous fingering in a Hele--Shaw cell

Dissipative Particle Dynamics (DPD)

• linear and angular momentum conserved • intrinsic fluctuations, suitable for stability analysis • efficient implementation: uDeviceX (GB15 finalist)

Dissipative Particle Dynamics

Fi = ∑j≠i

(FCij + FD

ij + FRij ) eij

∂ri

∂t= vi

∂vi

∂t=

1m

Fi

FCij = awC(rij)

FDij = − γwD(rij)(rij ⋅ vij)

FRij = σwR (rij) ζij

σ2 = 2kBTγ

rc

rij ij

Solvent Wall

fluctuation-dissipation theorem:

Groot, R. D.; Warren, P. B. & others: Dissipative particle dynamics: Bridging the gap between atomistic and mesoscopic simulation, 1997

⟨ξij(t)ξkl(t′�)⟩ = δ(t − t′�)(δikδjl + δilδjk)⟨ξij(t)⟩ = 0

repulsion: pressure

dissipation: viscosity

fluctuations

wD = (wR)2

Boundary conditions

Groot, R. D.; Warren, P. B. & others: Dissipative particle dynamics: Bridging the gap between atomistic and mesoscopic simulation, 1997

• Interactions with frozen particles • Bounce back on wall surface

SolventWall

(r0, v0)

(r1, v1)

(rn, vn)

(rw, vw)

12 (v0 + vn) = vw

vn = 2vW − v0rn = rw + (dt − τ)vn

Collision time τ ∈ [0, dt]

new velocity such that

DPD: validation

4 roller mill Poiseuille Couette Startup

∇p

U0

∇p

High performance software: uDeviceX

• High performance and High throughput solver for N-body, short range interactions • MPI + CUDA implementation

• 2’097'152 particles per node • ~34 neighbours per particle • 7.6 ms per time step

Rossinelli, et al., 2015 The in-silico lab-on-a-chip: petascale and high-throughput simulations of microfluidics at cell resolution (GB 15 Finalist)

Computational Setup

∇p

Solvent 1

Solvent 2

• periodic boundary conditions • label particles crossing outflow boundary 2 → 1

DPD: multiple solvents

a11 = a12 = a22

γ12 =γ11 + γ22

2

no interfacial tension:

solvent 1

solvent 2

two solvents two kinds of particles

dissipative force parameter

Results: fingering patterns

μ−

μ+= 0.5

μ−

μ+= 0.0625

μ−

μ+= 0.125

μ−

μ+= 0.05

Always stable for μ−

μ+≥ 0.33

μ−

μ+≥ 1Recall: stability analysis in 2D: stable only for

Simulation results

RfRi

μ−μ+

Bischofberger, I.; Ramachandran, R. & Nagel, S. R., Fingering versus stability in the limit of zero interfacial tension, 2014

SUMMARY

• DPD able to reproduce fingering instabilities • First simulation to capture phase transition as in experiments • Difficulties for very small viscosity ratios: density fluctuations too large • First step to understand this phenomenon