We elucidate the universal properties of the nonequilibrium steady states (NESS) in a driven symmetric binary fluid mixture, an example of active advection, in its miscible phase. We use the symmetries of the equations of motion to establish the appropriate form of the structure functions which characterise the statistical properties of the NESS of a driven symmetric binary fluid mixture. We elucidate the universal properties described by the scaling exponents and the amplitude ratios. Our results suggest that these exponents and amplitude ratios vary continuously with the degree of crosscorrelations between the velocity and the gradient of the concentration fields. Furthermore, we demonstrate, in agreement with Celani et al, [Phys. Rev. Lett., 89, 234502 (2002)], that the conventional structure functions as used in passive scalar turbulence studies exhibit only simple scaling in the problem of symmetric binary fluid mixture even in the weak concentration limit. We also discuss possible experimental verifications of our results.
PACS numbers:Critical dynamics of equilibrium systems show universal scaling properties for thermodynamic functions and correlation functions characterised by universal scaling exponents that depend on the spatial dimension d and the symmetry of the order parameter (e.g., Ising, XY etc.) and conservation laws [1, 2], but not on the parameters that specify the details of the Hamiltonian. A different situation arises in driven, dissipative, nonequilibrium systems with nonequilibrium statistical steady states (NESS). In this letter we demonstrate that in the NESS of externally driven turbulent homogeneous and isotropic symmetric binary fluid mixtures (50-50 concentration of the two components) universal properties are characterised by certain structure functions, determined by the symmetries of the equations of motion, and which are different from those used in pasive scalar turbulence studies. We find that the universal properties vary continuously with the degree of crosscorrelations between the velocity and the gradient of the concentration fields, a situation which has no analogue in equilibrium.The turbulent hydrodynamics of an incompressible symmetric binary fluid is described by the coupled evolution of the velocity field u(x, t) and the concentration field gradient ∇ψ(x, t) with ψ(x, t) = [ρ A (x, t) − ρ B (x, t)]/ρ 0 where ρ 0 is the mean density, and ρ A and ρ B are the densities of the two components. In a symmetric binary fluid ψ(x, t) = 0. Experiments performed on such systems have been typically concerned with the measurements of effective transport coefficients [3]. In a symmetric binary mixture, ψ is not simply advected passively by the u-field, but is active, i.e., concentration gradient ∇ψ reacts back on u to produce flows. In ordinary fluid turbulence the NESS is characterised by the multiscaling exponents of the structure functions of the velocity fields (see below) [4,5]. Compared to the studies of ordinary fluid turbulence the study of turbulent binary mixtures is in it...