Critical phenomena in randomly stirred fluids: Correlation functions, equation of motion, and crossover behavior

A field-theoretic dynamic renormalization-group calculation is carried out to two-loop order for a model of a critical binary liquid undergoing (passive) random stirring. In addition to the mean-field and dynamic ${\phi }^4$ fixed points, a new stable fixed point is found which has a nonanalytic, ${\epsilon }^{1/2}$, ε expansion, where ε≡4-d. The dynamic and static critical exponents for this new fixed point are found to have O(${\epsilon }^{1/2}$) terms. For small stirring Reynolds number, the system will cross over from the dynamic ${\phi }^4$ to the new ${\epsilon }^{1/2}$ fixed point, and we have calculated the effective critical exponents, the static and dynamic correlation function, and the equation of motion, all in crossover form, to one-loop order. The exponent governing the lowering of the critical temperature with stirring Reynolds number is found to be 1.74, which is in agreement with recent experiments of Pine et al. [Phys. Rev. A 29, 308 (1984)].