We identify the leading term describing the behavior at large distances of the steady state solutions of the Navier-Stokes equations in 3D exterior domains with vanishing velocity at the spatial infinity.

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In this work, we report the results of comprehensive experimental and theoretical study of magnetic properties of TiO 2 nanoparticles (20 nm) doped with Fe at various concentrations ranging from 0.1 to 4.6 at. %. Our electron paramagnetic resonance and magnetic measurements data evidence the mixed magnetic state, where paramagnetic Fe 3+ ions coexist with short-range antiferromagnetic correlations caused by negative exchange interaction between neighboring Fe 3+ ions. A quantum mechanical model of the Fe-based magnetic cluster represented as a set of dimers with strong ∼(100−300) K and weak (∼1 K) exchange interactions has been proposed. Our model was found to provide a good description of magnetic properties of TiO 2 :Fe nanopowders. Density-functional theory (DFT) calculations revealed Fe 3+ oxidation state of the iron center in the vicinity of an oxygen vacancy in the crystal structure of anatase. DFT calculations confirmed that two types of Fe 3+ spin-pairs with weak and strong exchange interactions can be formed in the vicinity of an oxygen vacancy. Accumulation of magnetic moment carriers and formation of magnetic clusters in TiO 2 nanoparticles with anatase structure were found to be a general tendency for all studied TiO 2 :Fe nanopowders.

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