On non‐Newtonian incompressible fluids with phase transitions

Abstract

Abstract A modified model for a binary fluid is analysed mathematically. The governing equations of the motion consists of a Cahn–Hilliard equation coupled with a system describing a class of non‐Newtonian incompressible fluid with p ‐structure. The existence of weak solutions for the evolution problems is shown for the space dimension d =2 with p ⩾ 2 and for d =3 with p ⩾ 11/5. The existence of measure‐valued solutions is obtained for d =3 in the case 2⩽ p < 11/5. Similar existence results are obtained for the case of nondifferentiable free energy, corresponding to the density constraint |ψ| ⩽ 1. We also give regularity and uniqueness results for the solutions and characterize stable stationary solutions. Copyright © 2006 John Wiley & Sons, Ltd.