The Structure Factor of Flowing Complex Liquid Mixtures

7 The Structure Factor of Flowing Complex Liquid Mixtures 

We derive here the governing equations necessary to describe the structure factor, S (q, t), for a complex liquid mixture subject to flow. Since this observable is the Fourier transformation of the spatial correlation of concentration functions, it is first required to develop an equation of motion for 5c (r, t). The approach described here employs a modified Cahn-Hilliard equation and is described in greater detail in the book by Goldenfeld [91]. To describe the physical system, an order parameter, q/ (r, ), is introduced. In a complex mixture, this parameter would simply be /(r, 0 = c(r, r)-(c), where c(r, t) is the local concentration of one of the constituents and c) is its mean concentration. The order parameter has the property of being zero in a disordered, or on phase region, and non-zero in the ordered or two-phase region. The observed structure factor, which is the object of this calculation, is simply 

An expression for the rate of change of the order parameter is required. Since the concentration is conserved, the following equation of continuity applies, 

Applying this operator in equation (7.93), the dynamical expression for the order parameter becomes 

In the presence of flow, the time derivative in equation (7.98) is modified to include a convective term so that 

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