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Background: The near-equilibrium properties of a QCD plasma can be encodedinto transport coefficients such as bulk and shear viscosity. In QCD, the ratioof these transport coefficients to entropy density, $\zeta/s$ and $\eta/s$,depends non-trivially on the plasma's temperature.  Purpose: We show that in a 0+1D boost-invariant fluid, atemperature-dependent $\zeta/s(T)$ or $\eta/s(T)$ can be described by anequivalent effective viscosity $\left\langle \zeta/s\right\rangle_{\textrm{eff}}$ or $\left\langle \eta/s\right\rangle_{\textrm{eff}}$. We extend the concept of effective viscosity insystems with transverse expansion, and discuss how effective viscosities can beused to identify families of $\zeta/s(T)$ and $\eta/s(T)$ that lead to similarhydrodynamic evolution.  Results: In 0+1D, the effective viscosity is expressed as a simple integralof $\zeta/s(T)$ or $\eta/s(T)$ over temperature, with a weight determined bythe speed of sound of the fluid. The result is general for any equation ofstate with a moderate temperature dependence of the speed of sound, includingthe QCD equation of state. In 1+1D, a similar definition of effective viscosityis obtained in terms of characteristic trajectories in time and transversedirection. This leads to an infinite number of constraints on an infinitefunctional space for $\zeta/s(T)$ and $\eta/s(T)$.  Conclusions: The definition of effective viscosity in a 0+1D system clarifieshow infinite families of $\zeta/s(T)$ and $\eta/s(T)$ can result in nearlyidentical hydrodynamic temperature profiles. By extending the study to aboost-invariant cylindrical (1+1D) fluid, we identify an approximate but moregeneral definition of effective viscosity that highlight the potential andlimits of the concept of effective viscosity in fluids with limited symmetries.

Effective viscosities in a hydrodynamically expanding boost-invariant QCD plasma