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Determination of the shear and extensional rheology of bubbly liquids with a shear-thinning continuous phase

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Abstract

The effect of bubble volume fraction, ϕ, and bubble size on the rheology of rheology of κ/ι-hybrid carrageenan gum solutions and bubbly liquids with ϕ ≤ 0.25 was studied in steady shear using a parallel plate configuration and in extension using a filament stretching device. Three different bubble size distributions were prepared using (1) planetary mixing (mean bubble size 90–130 μm), (2) mixing followed by pumping through a symmetric syringe (25–38 μm), and (3) as (2) but using an asymmetric syringe (12–20 μm). Gum concentrations of 10 and 20 g/L, lying in the semi-dilute regime, were studied. The unaerated gum solutions are viscoelastic and could be modelled very well as a single-mode Giesekus fluid. The bubbly liquid behaviour in both shear and extension could also be modelled as a single-mode Giesekus fluid. The Giesekus parameters for the bubbly liquids were related to those of the unaerated solutions by monatonic increasing dependencies on ϕ. The low shear rate apparent viscosity followed the trend predicted by the Choi and Schowalter (Phys Fluids 18: 420, 1975) model for emulsions, while the mobility parameter and relaxation time were linearly dependent on ϕ. In these tests bubble size only affected filament extension noticeably, influencing the mobility parameter. Further work is required to explain the observations in terms of micromechanical models.

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Abbreviations

a :

Giesekus mobility parameter

Bo :

Bond number

C :

Gum concentration, in gram per liter

Ca:

Capillary number

Ca G :

Elastic capillary number

D b :

Diameter at break-up, in micro meter

D :

Needle diameter, in meter

D mid :

Diameter of the filament at midpoint, in micrometer

D 0 :

Initial sample diameter, in micrometer

D 1 :

Diameter of the filament when first formed, in micrometer

d a :

Number average measured bubble diameter, in meter

d max :

Largest measured bubble diameter, in meter

d min :

Smallest measured bubble diameter, in meter

d vol :

Average bubble diameter based on the volume, in meter

F :

Normal force correction, in Newton

F normal :

Normal force generated by the flow between plates, in Newton

g :

Gravitational constant, in meter per square second

I :

Parameter from Choi and Schowalter model

k :

Parameter from Choi and Schowalter model

M n :

Number average molecular weight, in gram per mole

M w :

Weight average molar mass, in gram per mole

M z :

Higher average molecular weight, in gram per mole

n b :

Number of bubbles

n 2 :

Parameter in shear expression

N c :

Number of classes of bubble size

N 1 :

First normal stress difference, in Pascal

N 2 :

Second normal stress difference, in Pascal

p :

Probability

Q :

Volumetric flow rate, in cubic meter per second

R :

Bubble radius, in meter

R pp :

Radius of parallel plate geometry, in meter

R 2 :

Square of the correlation coefficient

t :

Time, in second

t F :

Time to capillary break-up, in millisecond (or second)

T :

Torque, in Newton meter

w :

Class interval width, in meter

Wi :

Weissenberg number

x :

Scaled variable Table 1

X :

Filament shape factor Eq. (13)

α :

Surface tension between liquid phase and the air, in Newton meter

ε :

Hencky strain

ϕ :

Bubble volume fraction

\( \dot{\gamma} \) :

Shear rate, in per second

\( {\dot{\gamma}}_R \) :

Shear rate experienced at the rim of the parallel plates, in per second

η app :

Apparent viscosity, in Pascal second

η e :

Estimated apparent extensional viscosity, in Pascal second

η 0 :

Zero-shear-rate viscosity, in Pascal second

η r0 :

Relative viscosity

η ∞ :

Infinite shear rate viscosity, in Pascal second

Λ:

Group used within shear expression

λ :

Relaxation time, in second

ρ s :

Density, in kilogram per cubic meter

ρ us :

Density of unaerated sample, in kilogram per cubic meter

σ :

Standard deviation

τ :

Shear stress, in Pascal

τ * :

Dimensionless shear stress

ψ :

Parameter from Choi and Schowalter model

Ω :

Angular velocity, in radian per second

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Acknowledgments

The authors acknowledge the financial support (POS-A/2012/116) from Xunta de Galicia’s Consellería de Cultura, Educación e Ordenación Universitaria of Spain and the European Union’s European Social Fund. Helpful conversations with Professors Malcolm Mackley and Paco Chinesta as well as Dr Damien Vadillo are all gratefully acknowledged.

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Authors and Affiliations

  1. Department of Chemical Engineering and Biotechnology, New Museums Site, University of Cambridge, Pembroke St, Cambridge, CB2 3RA, UK

    Maria D. Torres, Bart Hallmark & D. Ian Wilson

  2. Department of Chemical Engineering, University of Santiago de Compostela, Lope Gómez de Marzoa St, Santiago de Compostela, E-15782, Spain

    Maria D. Torres

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  1. Maria D. Torres
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Correspondence to D. Ian Wilson.

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Torres, M.D., Hallmark, B. & Wilson, D.I. Determination of the shear and extensional rheology of bubbly liquids with a shear-thinning continuous phase. Rheol Acta 54, 461–478 (2015). https://doi.org/10.1007/s00397-014-0832-9

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  • DOI: https://doi.org/10.1007/s00397-014-0832-9

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