Viscosity |

**In General** > s.a. dark
energy; fluids; Kinetic
Theory; Stokes'
Law; Transport.

* __Idea__: Viscosity is a fluid's resistance to gradual deformation by shear stress or tensile stress; It measures how local disturbances in the system are transmitted to the rest of
the system through interactions, and depends on the strength of the interactions between its constituents;
If those interactions are strong, neighboring parts of the fluid more readily
transmit the disturbances through the system; Thus low shear
viscosities, as in a perfect fluid, indicate significant interaction strength,
while no interactions, as in an ideal gas, correspond to an infinite viscosity.

* __Coefficient of viscosity__:
The constant of proportionality between the stress and the derivatives of the
velocity in a fluid (when the velocity gradient is not too large, in terms of the mean free path),

*P*_{ij} = *η* ∂*u ^{ j}*/∂

* __Temperature dependence__: Molecular theory tells us that the
viscosity of a gas tends to increase with temperature, that of a liquid tends to decrease; However, at low temperature a gas condenses to a liquid and then can become a solid, so viscosity increases with decreasing temperature and ne expects t find a minimum at some intermediate temperature; The same behavior is expected in QCD.

@ __General references__: Desloge AJP(62)dec
[coefficient of viscosity for a gas]; Viscardy cm/06/SHPMP [history]; letter PT(10)oct.

@ __Related topics__: news pn(08)may [shear thinning – as in ketchup]; Behroozi et al AJP(10)nov [measurement from the attenuation of capillary waves].

**In Specific Systems** > s.a. phenomenology of gravity [gravitational viscosity].

* __Extreme viscosity__: Vigorously stirring a suspension of micrometer-sized particles, such as cornstarch in water, can make the flowing fluid so viscous it behaves like a solid (shear thickening); Experiments indicate that the reason is increased friction between the particles forced into contact.

@ __General references__: Enss et al AP(11) [unitary Fermi gas]; Torrieri PRD(12) [ideal relativistic quantum fluid]; Lin et al PRL(15) [shear thickening].

@ __In cosmology__: Disconzi et al PRD(15) + news tel(15)jul [and big rip].

**Viscosity Bound**

* __Idea__: We do not know of any fluid in nature that violates the bound
*η*/*s* ≥ \(\hbar\)/4π*k*_{B} (and see temperature dependence above).

* __Kovtun-Starinets-Son bound__:
The conjecture, based on gauge-gravity duality, that there is a universal bound for the shear-viscosity-to-entropy-density ratio, given by *η*/*s*
≥ \(\hbar\)/4π*k*_{B},
at least for black holes in theories with a holographic interpretation; In general relativity, results by Thorne & Price
on the membrane paradigm imply that for Kerr black holes *η*/*s* = \(\hbar\)/4π*k*_{B}, and measurements on quark-gluon plasmas assumed to be modelled by the dual theories on the brane give results close to \(\hbar\)/4π*k*_{B}; But the bound was shown to be violated in Gauss-Bonnet gravity, where *η*/*s* = \(\hbar\)/4π*k*_{B} (1–8*λ*) in the IR limit (*λ* is the Gauss-Bonnet coupling constant), and for black holes in *f*(*R*) gravity results show that *η*/*s* = *f *'(*R*)\(\hbar\)/4π*k*_{B}.

@ __References__:
Kovtun et al PRL(05)ht/04 [proposal]; Cherman et al JHEP(08)-a0708 [re
viability of conjecture]; Fouxon et al PRD(08)-a0710 [and
generalized second law]; Brigante et al PRL(08)-a0802 [in Gauss-Bonnet gravity, and
inconsistency from microcausality
violation], PRD(08);
Hod NPB(09)-a0907 [from
thermodynamics], GRG(09)-a0905-GRF;
Brustein & Medved PLB(10)-a0908 [proof];
Pal PRD(10)-a0910 [at
finite coupling]; Schäfer Phy(09)
[rev]; Shu PLB(10)
[in Lovelock gravity]; Jakovac PRD(10)
[of shear-viscosity to entropy-density
ratio]; Johnson & Steinberg PT(10)may;
Chirco et al PRD(10)-a1005 [from
entanglement entropy in Rindler spacetime]; Kovtun et al PRD(11)-a1104 [absolute lower limit]; Cremonini MPLB(11)-a1108 [rev]; Rebhan & Steineder PRL(12) [violation in a strongly coupled anisotropic plasma].

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