Publication 19-CNA-019
Phase Separation In The Advective Cahn-Hilliard Equation
Yu Feng
Department of Mathematics
University of Wisconsin – Madison
Madison, WI 53706
feng65@wisc.edu
Yuanyuan Feng
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
yuanyuaf@andrew.cmu.edu
Gautam Iyer
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
gautam@math.cmu.edu
Jean-Luc Thiffeault
Department of Mathematics
University of Wisconsin – Madison
Madison, WI 53706
jeanluc@math.wisc.edu
Abstract: The Cahn–Hilliard equation is a classic model of phase separation in binary mixtures that exhibits spontaneous coarsening of the phases. We study the Cahn–Hilliard equation with an imposed advection term in order to model the stirring and eventual mixing of the phases. The main result is that if the imposed advection is sufficiently mixing then no phase separation occurs, and the solution instead converges exponentially to a homogeneous mixed state. The mixing effectiveness of the imposed drift is quantified in terms of the dissipation time of the associated advection-hyperdiffusion equation, and we produce examples of velocity fields with a small dissipation time. We also study the relationship between this quantity and the dissipation time of the standard advection-diffusion equation.
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