Natural convection of a two-dimensional Boussinesq fluid does not maximize entropy production

Rayleigh-Bénard convection is a canonical example of spontaneous pattern formation in a nonequilibrium system. It has been the subject of considerable theoretical and experimental study, primarily for systems with constant (temperature or heat flux) boundary conditions. In this investigation, we have explored the behavior of a convecting fluid system with negative feedback boundary conditions. At the upper and lower system boundaries, the inward heat flux is defined such that it is a decreasing function of the boundary temperature. Thus the system's heat transport is not constrained in the same manner that it is in the constant temperature or constant flux cases. It has been suggested that the entropy production rate (which has a characteristic peak at intermediate heat flux values) might apply as a selection rule for such a system. In this work, we demonstrate with Lattice Boltzmann simulations that entropy production maximization does not dictate the steady state of this system, despite its success in other, somewhat similar scenarios. Instead, we will show that the same scaling law of dimensionless variables found for constant boundary conditions also applies to this system.

Figure 1

Model system diagram showing the various heat fluxes, which comprise the boundary energy balances. The solid walls enforce the no-slip velocity condition.

Figure 2

Macroscopic, steady-state properties of the model system. Boundary temperatures $T_a$ (red dot-dashed line) and $T_b$ (blue dashed line) and dimensionless entropy production (solid black line) are plotted as a function of the normalized heat flux.

Figure 3

Ratio of total to diffusive heat flux Nu as a function of dimensionless thermal driving force Ra for NC systems with (a) fixed temperature and fixed heat flux BCs, and (b) negative feedback BCs. Red asterisks correspond to fixed temperature BCs, blue circles to fixed flux BCs, red circles to negative feedback systems with uniform boundary temperature profiles, and blue triangles to negative feedback systems with a variable boundary temperature profile. The solid black line shows the empirical scaling law $\mathrm{Nu}\approx 0.138{\mathrm{Ra}}^{0.285}$ due to [5].

Figure 4

Steady-state temperature difference as a function of total heat flux for TLBM simulations with negative feedback BCs. Red circles and blue triangles show results from the uni and vari simulations, respectively. The black dotted line shows the heat flux value corresponding to a state of maximum entropy production, and the black asterisk shows the corresponding value of the temperature difference.

Figure 5

Temperature field of a natural convection fluid system with negative feedback BCs. This example is a uni system with $\mathrm{Ra}\approx {10}^8$. Streamlines show the two principal convection rolls.