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A study on the flow of moderate and high viscosity ratio emulsion through a cylindrical tube

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Abstract

This work deals with asymptotic and numerical solutions for emulsion flowing driven by a pressure gradient. The average macroscopic description of a homogeneous continuous emulsion of high viscosity drops is modeled. A parameter involving the product of the squares of the capillary number and the aspect ratio is the key parameter for developing a new asymptotic solution. Explicit expressions of the velocity profile, the flow rate correction due to the drops stress contribution, drop deformation, and the relative viscosity of the emulsion are shown as function of the capillary number ranging from 0 to 10 and emulsion viscosity ratio ranging from 2 to 20. The theoretical predictions by asymptotic theories developed in this work are compared with those computed results by boundary integral method (BIM) for different viscosity ratios of a dilute emulsion undergoing both pressure-driven flow and linear shear flow. Some discrepancies observed for moderate viscosity ratio are identified and discussed. The present study for emulsion with moderate and high viscosity ratio and arbitrary capillary number are still few explored in the current literature.

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Notes

  1. In Taylor’s original work, \( {D}_T=\frac{19\lambda +16}{16\lambda +16} Ca \), such that, for λ ≫ 1, \( {D}_T=\frac{19\lambda }{16}C{a}_{\lambda }+\mathcal{O}\left(1/{\lambda}^2\right) \).

  2. In terms of dimensional variables \( {u}^{\infty }=2{U}_c\left[1-{\left(\frac{r}{R}\right)}^2\right] \), thus \( \frac{d{u}^{\infty }}{dr}=-4{U}_c\frac{r}{R^2} \), where \( {U}_c=\frac{G{R}^2}{8{\mu}_B} \).

  3. For simple shear flows, at the \( \mathcal{O}\left(\phi /\lambda \right) \) limit, Eq.(39) of Oliveira and Cunha (2015) gives that \( {\mu}_{\phi }={\mu}_B+\frac{5 c\phi}{\lambda \left({c}^2+C{a}_{\lambda}^2\right)} \).

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Funding

The work was supported in part by the Brazilian funding agencies CNPq- Ministry of Science, Technology and Innovation of Brazil, and by the CAPES Foundation of Education of Brazil (Grant Nos. 552221/2009-0 and 142303/2015-1).

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Authors and Affiliations

  1. Laboratory Fluid Mechanics of Complex Flows, Department of Mechanical Engineering - FT, University of Brasília, Campus Universitário Darcy Ribeiro, Brasília-DF, 70910 900, Brazil

    F. R. Cunha

  2. Laboratory of Energy and Environment, Department of Mechanical Engineering - FT, University of Brasília, Campus Universitário Darcy Ribeiro, Brasília-DF, 70910 900, Brazil

    T. F. Oliveira

Authors
  1. F. R. Cunha
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  2. T. F. Oliveira
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Correspondence to T. F. Oliveira.

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Cunha, F.R., Oliveira, T.F. A study on the flow of moderate and high viscosity ratio emulsion through a cylindrical tube. Rheol Acta 58, 63–77 (2019). https://doi.org/10.1007/s00397-018-01124-w

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