Non-Newtonian flows in domains with non-compact boundaries

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Non-Newtonian flow of a viscous fluid in unbounded domains with cylindrical outlets is considered. The viscosity is assumed to be dependent on the shear rate. Applying the Banach fixed point theorem and the Hilbert spaces with exponential weight we prove the existence and uniqueness of solutions with an exponential stabilization to the quasi-Poiseuille flows at the outlets if the right hand side decays exponentially. These results may be used for the matching technique for flows in thin tube structures.