The viscosity of suspensions has been the subject of much study. Knowing its value is critical to the processing of items such as paints, ketchup, cosmetics, concrete, and pharmaceuticals. One of the most famous relationships between the viscosity of a suspension and the volume fraction of particles in the suspension was developed by Einstein. His original equation is given by:

\[\mu_{r}=\frac{\mu}{\mu_{o}}=\frac{1+0.5 \phi}{(1-\phi)^{2}} \quad \mu_{o}=\text { viscosity of pure fluid }\]

Use a one term, Taylor series expansion for small \(\phi\) to show that:

\[\mu_{r} \approx 1+\left(\frac{\partial \mu_{r}}{\partial \phi}\right)_{\phi=0} \phi=1+2.5 \phi\]