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),
\[ P^~_{ij} = \eta\,\partial u^{\,j}/\partial x^i\;.\]

* __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 one expects to 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];
Niethammer & Schubert a1903 [suspensions, Einstein's formula].

@ __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 \(\eta/s \ge \hbar/4\pi k^~_{\rm B}\) (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 \(\eta/s \ge \hbar/4\pi k^~_{\rm B}\), 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 \(\eta/s = \hbar/4\pi k^~_{\rm B}\), and measurements on quark-gluon plasmas assumed
to be modelled by the dual theories on the brane give results close to \(\hbar/4\pi k^~_{\rm B}\); But the
bound was shown to be violated in Gauss-Bonnet gravity, where \(\eta/s = \hbar/4\pi k^~_{\rm B}\, (1-8\lambda)\)
in the IR limit (*λ* is the Gauss-Bonnet coupling constant), and for black holes in \(f(R)\)
gravity results show that \(\eta/s = f'(R)\,\hbar/4\pi k^~_{\rm B}\).

@ __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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