Bistability in a simple fluid network due to viscosity contrast

Brian Storey, John Geddes, David GardnerFranklin W. Olin College of Engineering

Russell CarrUniversity of New Hampshire

Problem and model

Fluids in inlet 1 and 2 have different viscosities, but are otherwise simple Newtonian fluids.

node aat 0

loop a around 0

flow)(laminar 128

4

i

i

Q

P

D

LR

QRP

One non-linearity – Arrhenius Law

Set flow Q1 and Q2– 2 statesQ1

Q2

Qc

Q1

Q2

Qc

Viscosity ratio = 1Q1

Q2

Qc

Q1

Q2

Qc

Viscosity ratio = 2Q1

Q2

Qc

Q1

Q2

Qc

Viscosity ratio = 10Q1

Q2

Qc

Q1

Q2

Qc

Viscosity ratio = 1,3,5,10,20,200

Pressure driven- 4 statesP1

P2

Qc

P1

P2

Qc

P1

P2

Qc

P1

P2

Qc

Viscosity ratio =1

Q2=0

QC=0

Q1=0

Viscosity ratio=10

Q2=0

QC=0

Q1=0

Viscosity ratio=200

Experimental setup

P1

P2

Qc

Water

Water+Sugar

P=0

Experimental procedure

Experimental data

of sugar in inlet 2 (μ2)

Criterion for existence of bistability

Arrhenius viscosity law

General viscosity law

Conclusions

• This work could have been done ~100 years ago.• We predict and observe bistability in a simple network with

laminar flow of Newtonian fluids. Flow direction depends on history.

• Perhaps the simplest example of bistability in (micro)fluidics?Quake

Prakash and GershenfeldGroisman et al

Stratified flow – effective viscosity

Immiscible

Fully mixed

Miscible, diffuse

Stratified flow experiment

Viscosity ratio = 5Q1

Q2

Qc

Q1

Q2

Qc

Viscosity ratio=5

Q2=0

QC=0

Q1=0

Q1

Q2

Qc

Q1

Q2

Qc

Q1

Q2

Qc

Q1

Q2

Qc