Effective viscosity of a random mixture of fluids

We propose an estimation of the effective viscosity of a random mixture of Newtonian fluids that ignores capillary effects. The local viscosity of the mixture is assumed to be a random function of the position. Using perturbation expansions up to the second order, the resulting formula can be recast under the form of a simple power averaging mixing low. Numerical tests are used to assess the validity of the formula and the range of its applicability.

Figure 1

Random fields using isolevel surfaces and slices of viscosity (${\sigma }^2=0.1, l_c=3\delta x, 10\delta x$, and $25\delta x$ at resolution ${257}^3$, top to bottom).

Figure 2

Histograms of the values of $ln\left(\eta \right)$ (${\sigma }^2=0.3$) for three correlation lengths.

Figure 3

Empirical and expected properties of generated $\eta$ fields.

Figure 4

Regressions for $l_c=6\delta x$ and $10\delta x$ on ${257}^3$ grid.

Figure 5

Regressions for $l_c=2\delta x$ on ${257}^3$ grid.

Figure 6

Regression slopes of $ln\left({\eta }_{\text{eff}}\right)$ vs ${\left(\text{real}\sigma \right)}^2/2$ for different values of $l_c$ and resolutions. Top : slope estimations; bottom: slope estimations with $95\%$ confidence intervals.