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The system we consider consists of a regularized family of Navier-Stokes equations (including the Navier Stokes-$\\alpha $-like equation, the Leray-$\\alpha $ equation, the Modified Leray-$\\alpha $ equation, the Simplified Bardina model, the Navier Stokes-Voigt model and the Navier-Stokes equation) for the fluid velocity $u$ suitably coupled with a parabolic equation for the director field $d$. 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