^{1,a)}and L. Gary Leal

^{2}

### Abstract

We present the constitutive equation for the volume-averaged hydrodynamic stress for a dilute emulsion in a linear ambient flow, when there is slip at the liquid-liquid interface between the Newtonian drop and the suspending fluids. Slip is modeled using a simple Navier slip boundary condition. We provide analytical solutions in the limit of small capillary numbers for the shape deformation, viscosity, and normal stresses. Slip moderates these quantities, with changes from the no-slip case being more pronounced for large viscosity ratios of the drop relative to the suspending fluid. It has been suggested in the past that slip can explain the anomalously low viscosities of certain polymeric blends. Our analysis indicates that slip can only partially account for these deviations, and that other mechanisms should be explored to explain the residual discrepancy.

Dr. Ramachandran is grateful to the Department of Chemical Engineering and Applied Chemistry at University of Toronto for funding. The authors thank Professor Ronald Larson for providing the reference to his work describing cavitation phenomena in polymers sheared at high stresses.

I. INTRODUCTION

II. THEORY

A. Zeroth order solution

B. First order solution

C. Rheology of a dilute emulsion of drops with interfacial slip

III. DROP DEFORMATION AND EMULSION RHEOLOGY FOR SOME COMMON AMBIENT FLOWS

A. Uniaxial extensional flow

B. Simple shear flow

C. Small-amplitude oscillatory shear flow

IV. DISCUSSION

V. CONCLUSIONS

### Key Topics

- Viscosity
- 133.0
- Fluid drops
- 46.0
- Emulsions
- 37.0
- Shear rate dependent viscosity
- 29.0
- Polymers
- 25.0

## Figures

An initially spherical drop (dotted line) of unit radius deformed by a linear ambient flow (a simple shear flow is shown as an example). The vectors **n** and **t** are the local normal and tangent vectors on the surface of the deformed drop.

An initially spherical drop (dotted line) of unit radius deformed by a linear ambient flow (a simple shear flow is shown as an example). The vectors **n** and **t** are the local normal and tangent vectors on the surface of the deformed drop.

(a) The time constant in Eq. (61) and (b) the dimensionless drop deformation in Eq. (60) as functions of the dimensionless slip coefficient and the viscosity ratio .

(a) The time constant in Eq. (61) and (b) the dimensionless drop deformation in Eq. (60) as functions of the dimensionless slip coefficient and the viscosity ratio .

The deformation parameter as a function of the capillary number for different values of the slip coefficient and for (a), (b), and (c). The square and brace brackets at the right edges of (a), (b), and (c) denote the linear and quadratic results, respectively. The solid blue, dashed green, and dash-dotted red lines represent the results for , , and , respectively. In (d), we have shown the steady-state shape of the drop for and shapes for different slip coefficients.

The deformation parameter as a function of the capillary number for different values of the slip coefficient and for (a), (b), and (c). The square and brace brackets at the right edges of (a), (b), and (c) denote the linear and quadratic results, respectively. The solid blue, dashed green, and dash-dotted red lines represent the results for , , and , respectively. In (d), we have shown the steady-state shape of the drop for and shapes for different slip coefficients.

The Trouton viscosity as a function of the viscosity ratio for different slip coefficients.

The Trouton viscosity as a function of the viscosity ratio for different slip coefficients.

Drop shapes in the flow-gradient (a), and flow-vorticity (b) planes passing through the origin for and . The solid blue lines, the dashed green lines, and the dash-dotted red lines are the results for 0, 0.1, and 1 respectively.

Drop shapes in the flow-gradient (a), and flow-vorticity (b) planes passing through the origin for and . The solid blue lines, the dashed green lines, and the dash-dotted red lines are the results for 0, 0.1, and 1 respectively.

The deformation in the flow-velocity gradient plane defined in Eq. (81) for different values of the slip coefficient and for (a), (b), and (c). The color code is the same as for prior figures. For the case in (c), we have chosen a smaller range of . This is because the range of for which our perturbation expansion is valid reduces for large viscosity ratios. In this case, with and , we observe cusps and kinks in the shape of the drop, which confirm the breakdown of our shape approximations. In each subfigure, the solid blue lines, the dashed green lines, and the dash-dotted red lines are the results for 0, 0.1, and 1, respectively.

The deformation in the flow-velocity gradient plane defined in Eq. (81) for different values of the slip coefficient and for (a), (b), and (c). The color code is the same as for prior figures. For the case in (c), we have chosen a smaller range of . This is because the range of for which our perturbation expansion is valid reduces for large viscosity ratios. In this case, with and , we observe cusps and kinks in the shape of the drop, which confirm the breakdown of our shape approximations. In each subfigure, the solid blue lines, the dashed green lines, and the dash-dotted red lines are the results for 0, 0.1, and 1, respectively.

(a) The effective emulsion viscosity ; (b) first normal stress difference, , normalized by ; (c) second normal stress difference, , normalized by ; and (d) the ratio of the second normal stress difference to the first, , as functions of the viscosity ratio, , and the slip coefficient, .

(a) The effective emulsion viscosity ; (b) first normal stress difference, , normalized by ; (c) second normal stress difference, , normalized by ; and (d) the ratio of the second normal stress difference to the first, , as functions of the viscosity ratio, , and the slip coefficient, .

The emulsion effective viscosity as a function of the viscosity ratio and slip coefficient for spherical drops at the instant of initiating shear when the contribution of interfacial to the stress is negligible. The black horizontal line denotes .

The emulsion effective viscosity as a function of the viscosity ratio and slip coefficient for spherical drops at the instant of initiating shear when the contribution of interfacial to the stress is negligible. The black horizontal line denotes .

The contribution of the drops to the storage modulus (a) and loss modulus (b) of the dilute emulsion for in an oscillatory shear flow experiment.

The contribution of the drops to the storage modulus (a) and loss modulus (b) of the dilute emulsion for in an oscillatory shear flow experiment.

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