Isogeometric analysis of the advective Cahn–Hilliard equation: Spinodal decomposition under shear flow

We present a numerical study of the spinodal decomposition of a binary fluid undergoing shear flow using the advective Cahn–Hilliard equation, a stiff, nonlinear, parabolic equation characterized by the presence of fourth-order spatial derivatives. Our numerical solution procedure is based on isogeometric analysis, an approximation technique for which basis functions of high-order continuity are employed. These basis functions allow us to directly discretize the advective Cahn–Hilliard equation without resorting to a mixed formulation. We present steady state solutions for rectangular domains in two-dimensions and, for the first time, in three-dimensions. We also present steady state solutions for the two-dimensional Taylor–Couette cell. To enforce periodic boundary conditions in this curved domain, we derive and utilize a new periodic Bézier extraction operator. We present an extensive numerical study showing the effects of shear rate, surface tension, and the geometry of the domain on the phase evolution of the binary fluid. Theoretical and experimental results are compared with our simulations.