Blood flows at volume rate \(Q\) in a circular tube of radius \(R\). The blood cells concentrate and flow near the center of the tube, while the cell-free fluid (plasma) flows in the outer region. The center core of radius \(R_{c}\) has a viscosity \(\mu_{c}\), and the plasma has a viscosity \(\mu_{p}\). Assume laminar, fully developed flow for both the core and plasma flows and show that an "apparent" viscosity is defined by

\[ \mu_{\mathrm{app}} \equiv \frac{\pi R^{4} \Delta p}{8 L Q} \]

is given by

\[ \mu_{\mathrm{app}}=\frac{\mu_{p}}{1-\left(R_{c} / R\right)^{4}\left(1-\mu_{p} / \mu_{c}\right)} \]