Area Tensors for Modeling Microstructure During Laminar Liquid-Liquid Mixing

Fluid-fluid mixtures often possess a fine structure, or morphology, whose length scale is much smaller than the length scale over which the flow field and morphology vary. We define a microstructural variable called the area tensor, which describes the local morphology of such mixtures through volume-averaged size, shape, and orientation characteristics. The area tensor is equivalent to the interface tensor of the Doi and Ohta (1991) rheological model, and is closely related to the general microstructural tensors of Hinch and Leal (1976). The evolution equation for the area tensor during laminar mixing is derived for the case of equal component viscosities and negligible surface tension. Solution of this evolution equation requires a closure approximation for estimating higher-order microstructural statistics. A closure approximation is generated based on exact area tensor relations for ellipsoidal shapes, and is shown to provide highly accurate evolutions of the area tensor. Area tensor histories are calculated in homogeneous elongational and shearing flows, as well as in temporally and spatially varying flows. The results are shown to be consistent with well-known mixing principles.