Critical Phenomena in Two-Dimensional Incompressible Flows

Abstract

Classical fluids can undergo a second order phase transition under a two-dimensional flow of the form u(r) = (S + A) yex + (S − A) xev, ex and ev being the unit vectors along the x- and y-axes, respectively. The flow is called rotational for A>S≥0 and elongational for S>A≥0, while it reduces to a shear flow for A=S. When the reduced temperature is made smaller than a crossover reduced temperature, critical fluctuations with wave numbers smaller than a characteristic wave number become significantly distorted by the flow. The steady state variances of these critical fluctuations are calculated in the mean field approximation. It is found that these large scale fluctuations can be treated by the mean field approximation above a certain spatial dimensionality, which is the new critical dimensionality in the presence of flow. It remains 4 for the rotational case, but is reduced to 2.4 and 2 for the cases of shear flow and elongational flow, respectively.