= Einstein viscosity formula for a dilute suspension
{c}
{title2=$\mu_{\mathrm{eff}}=\mu(1+5\phi/2)$}
= Einstein suspension viscosity
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{synonym}
A dilute suspension of identical rigid spheres raises the <dynamic viscosity> by a fraction $5\phi/2$, where $\phi$ is the particle volume fraction. At imposed trace-free <rate-of-strain tensor> $E$, a sphere of radius $a$ adds <viscous dissipation> $20\pi\mu a^3(E:E)/3$. Multiplication by the number density $3\phi/(4\pi a^3)$ gives $5\mu\phi E:E$, which added to $2\mu E:E$ proves the formula. This is the leading noninteracting-sphere term; <hydrodynamic interactions> enter at higher concentration.
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