Existence Theory for Generalized Newtonian Fluids

Chapter 10

Stochastic power law fluids

Abstract

We consider the equations of motion for an incompressible non-Newtonian fluid in a bounded Lipschitz domain $G\subset {\mathbb{R}}^d$ during the time interval $(0,T)$ with a stochastic perturbation driven by a Brownian motion W. The balance of momentum reads as

$\mathrm{d}\mathbf{v}=-(\mathrm{\nabla }\mathbf{v})\mathbf{v}\mathrm{d}t+\mathrm{div}\mathbf{S}\mathrm{d}t-\mathrm{\nabla }\pi \mathrm{d}t+\mathbf{f}\mathrm{d}t+\Phi (\mathbf{v})\mathrm{d}{\mathbf{W}}_t,$

where v is the velocity, π the pressure and f an external volume force. ...