The effect of interfacial slip on the rheology of a dilute emulsion of drops for small capillary numbers
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 .
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.
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.
(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 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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