Bound on viscosity and the generalized second law of thermodynamics

We describe a new paradox for ideal fluids. It arises in the accretion of an ideal fluid onto a black hole, where, under suitable boundary conditions, the flow can violate the generalized second law of thermodynamics. The paradox indicates that there is in fact a lower bound to the correlation length of any real fluid, the value of which is determined by the thermodynamic properties of that fluid. We observe that the universal bound on entropy, itself suggested by the generalized second law, puts a lower bound on the correlation length of any fluid in terms of its specific entropy. With the help of a new, efficient estimate for the viscosity of liquids, we argue that this also means that viscosity is bounded from below in a way reminiscent of the conjectured Kovtun-Son-Starinets lower bound on the ratio of viscosity to entropy density. We conclude that much light may be shed on the Kovtun-Son-Starinets bound by suitable arguments based on the generalized second law.

Figure 1

Plot of ${log}_{10}$ of experimental viscosities of 11 pure liquids (data taken mostly from Ref. 38) vs ${log}_{10}$ of the estimates from Eq. (33) with $l$ identified with the average intermolecular separation. Both viscosities and estimates are in SI units (mPa s). Nonstandard symbols are “M” for methane, “P” for propane, “A” for acetone, “E” for ethanol, “U” for undecane (${\mathrm{C}}_{11}{\mathrm{H}}_{24}$), “Ni” for nitrobenzene (phenil-${\mathrm{NO}}_2$) and “G” for glycerol (${\mathrm{C}}_3{\mathrm{H}}_8{\mathrm{O}}_3$). A repeated symbol correspond to viscosities at several pressures and temperatures. The solid line is the locus of $\eta =\rho al$; the dotted lines demarcate the region where the viscosity lies within a factor of 3 of $\rho al$.