The interactions between globular proteins in the presence of poly (ethylene glycol) (PEG) are probed through the measurement of the protein solution second virial coefficient (B2). The solution properties of PEG are characterized for four molecular weights (400, 1000, 6000, and 12 000), providing an opportunity for quantitative comparison of measurements and theoretical predictions of B2. PEG displays a buffer and molecular weight-dependent lower critical solution temperature. As the polymer solution approaches phase separation, the consequences of depletion attractions increase significantly. For lysozyme and bovine serum albumin in sulfate buffers with PEG, B2 is not well described by standard depletion models. This failure is accentuated in acetate buffers where B2 is a nonmonotonic function of polymer concentration. The attractive minima in B2 are closely associated with the proximity of the heating-induced phase separation of aqueous PEG solutions. The experimental data for both proteins in the presence of PEG are well captured by the thermal polymer reference interaction site model for depletion interactions where the polymer density fluctuation correlation length is treated as a function of temperature, polymer concentration, and molecular weight.

Using high-resolution direct numerical simulation and arguments based on the kinetic energy flux Π(u), we demonstrate that, for stably stratified flows, the kinetic energy spectrum E(u)(k)∼k(-11/5), the potential energy spectrum E(θ)(k)∼k(-7/5), and Π(u)(k)∼k(-4/5) are consistent with the Bolgiano-Obukhov scaling. This scaling arises due to the conversion of kinetic energy to the potential energy by buoyancy. For weaker buoyancy, this conversion is weak, hence E(u)(k) follows Kolmogorov's spectrum with a constant energy flux. For Rayleigh-Bénard convection, we show that the energy supply rate by buoyancy is positive, which leads to an increasing Π(u)(k) with k, thus ruling out Bolgiano-Obukhov scaling for the convective turbulence. Our numerical results show that convective turbulence for unit Prandt number exhibits a constant Π(u)(k) and E(u)(k)∼k(-5/3) for a narrow band of wave numbers.

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We develop an analytic integral equation theory for treating polymer-induced effects on the structure and thermodynamics of dilute suspensions of hard spheres. Results are presented for the potential of mean force, free energy of insertion per particle into a polymer solution, and the second virial coefficient between spheres. The theory makes predictions for all size ratios between the spheres and the polymer coil dimension. Based on the Percus-Yevick ͑PY͒ closure, the attractive polymer-induced depletion interaction is predicted to be too weak under athermal conditions to induce a negative value for the second virial coefficient, B 2 cc , between spheres in the colloidal limit when the spheres are much larger than the coil size. A nonmonotonic dependence of the second virial coefficient on polymer concentration occurs for small enough particles, with the largest polymer-mediated attractions and most negative B 2 cc occurring near the dilute-semidilute crossover concentration. Predictions for the polymer-mediated force between spheres are compared to the results of computer simulations and scaling theory.

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