Critical behaviour of a fluid in a random shear flow: Renormalization group analysis of a simplified model

Critical behaviour of a fluid (binary mixture or liquid crystal), subjected to strongly anisotropic turbulent mixing, is studied by means of the field theoretic renormalization group. As a simplified model, relaxational stochastic dynamics of a non-conserved scalar order parameter, coupled to a random velocity field with prescribed statistics, is considered. The velocity is taken Gaussian, white in time, with correlation function of the form ∝ δ(t − t′)/|k⊥| , where k⊥ is the component of the wave vector, perpendicular to the distinguished direction (“direction of the flow”) — the d-dimensional generalization of the ensemble introduced by Avellaneda and Majda [Commun. Math. Phys. 131 381] within the context of passive scalar advection. It is shown that, depending on the relation between the exponent ξ and the space dimensionality d, the system exhibits various types of large-scale self-similar behaviour, associated with different infrared attractive fixed points of the renormalization-group equations. In addition to well known asymptotic regimes (model A of equilibrium critical dynamics and passively advected scalar with no self-interaction), existence of a new, non-equilibrium and strongly anisotropic, type of critical behaviour (universality class) is established, and the corresponding critical dimensions are calculated to the second order of the double expansion in ξ and ε = 4 − d (two-loop approximation). The most realistic values of the model parameters (for example, d = 3 and the Kolmogorov exponent ξ = 4/3) belong to this class. The scaling behaviour appears anisotropic in the sense that the critical dimensions related to the directions parallel and perpendicular to the flow are essentially different. The results are in qualitative agreement with the results, obtained in experiments and simulations of fluid systems subjected to various kinds of regular and chaotic anisotropic flows. PACS numbers: 64.75.+g, 05.10.Cc, 64.60.Ht, 05.40−a Critical behaviour of a fluid in a random shear flow 2