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Bistability in a simple fluid network due to viscosity contrast Brian Storey, John Geddes, David Gardner Franklin W. Olin College of Engineering Russell Carr University of New Hampshire

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

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One non-linearity – Arrhenius Law

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Set flow Q1 and Q2– 2 states Q1 Q2 Qc Q1 Q2 Qc

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Viscosity ratio = 1 Q1 Q2 Qc Q1 Q2 Qc

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Viscosity ratio = 2 Q1 Q2 Qc Q1 Q2 Qc

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Viscosity ratio = 10 Q1 Q2 Qc Q1 Q2 Qc

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

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Pressure driven- 4 states P1 P2 Qc P1 P2 Qc P1 P2 Qc P1 P2 Qc

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Viscosity ratio =1 Q2=0 QC=0 Q1=0

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Viscosity ratio=10 Q2=0 QC=0 Q1=0

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Viscosity ratio=200

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Experimental setup P1 P2 Qc Water + Sugar P=0

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Experimental procedure

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Experimental data of sugar in inlet 2 (μ 2 )

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Criterion for existence of bistability Arrhenius viscosity law General viscosity law

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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 Gershenfeld Groisman et al

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Stratified flow – effective viscosity Immiscible Fully mixed Miscible, diffuse

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Stratified flow experiment

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Viscosity ratio = 5 Q1 Q2 Qc Q1 Q2 Qc

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Viscosity ratio=5 Q2=0 QC=0 Q1=0

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Q1 Q2 Qc

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Q1 Q2 Qc

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Q1 Q2 Qc

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Q1 Q2 Qc

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