Past exam of the soft condensed matter course of the University of Cambridge / 2018 / Section A / 1

Assume steady fully developed laminar flow of a Newtonian fluid in a long circular tube, with no slip at a rigid wall. The Hagen-Poiseuille equation gives the volume flow ratewhere $R$ is the lumen radius, $\ell$ the tube length, $\eta$ the dynamic viscosity and $\Delta p$ the pressure drop. If $\Delta p$, $\ell$ and $\eta$ remain fixed, thenThus doubling the flow requiresThis estimate neglects pulsation, arterial compliance, entrance losses, the detailed shape of a stenosis and changes in the rest of the circulatory resistance. It treats blood as a homogeneous Newtonian fluid and assumes that enlarging the artery does not change its applied pressure drop. These assumptions supply the intended fourth-power scaling; an actual circulation need not keep that pressure drop fixed.

At terminal speed, the downward excess weight is balanced by Stokes's law drag:The subtraction of the displaced water mass includes buoyancy. Hence the Stokes settling velocity isA density is needed but is not supplied. For the estimate, take a dense iron oxide represented by magnetite, with ${\rho }_p\simeq 5.2\times 10^3\mathrm{kg}{\mathrm{m}}^{-3}$, and water with ${\rho }_w\simeq 10^3\mathrm{kg}{\mathrm{m}}^{-3}$. This givesUsing the diameter for the particle Reynolds number gives $\mathrm{Re}\simeq {\rho }_w{v}_t(2a)/\eta \simeq 1.8\times 10^{-5}$, which supports the creeping-flow assumption. The model assumes a dilute isolated sphere far from walls and no imposed flow. The precise numerical speed depends on the unspecified oxide phase and particle density; the estimate is of order $10\mu \mathrm{m}{\mathrm{s}}^{-1}$.

The polymer overlap concentration is reached when there is roughly one chain per coil volume. If $M$ is the chain molar mass, choose the explicit geometric convention $V_{\mathrm{coil}}=4\pi R_g^3/3$, where $R_g$ is the radius of gyration. ThenHere $c^{\ast }$ is mass per solution volume and $N_A$ is the Avogadro constant. A good solvent gives $R_g\propto N^{\nu }$ with $\nu \simeq 0.588$, approximated by $3/5$ in the Flory approximation. With a polystyrene repeat-unit molar mass of about $104.15\mathrm{g}\mathrm{mo}{\mathrm{l}}^{-1}$, the chain has approximately $N=960$ repeat units. This fixes the chain length but not the size amplitude: the problem supplies no statistical segment length or measured coil size.

For a numerical estimate, use the good-solvent size calibration for polystyrene,This empirical calibration is quoted for benzene or toluene near room temperature at molar masses above $2\times 10^5\mathrm{g}\mathrm{mo}{\mathrm{l}}^{-1}$; using it here is a modest extrapolation to $M=10^5\mathrm{g}\mathrm{mo}{\mathrm{l}}^{-1}$. It gives $R_g\simeq 11.3\mathrm{nm}$. ThusThe unrounded calibration gives about $27\mathrm{g}{\mathrm{L}}^{-1}$. This is an estimate under a stated extra material input, not a unique number determined solely by molecular weight and the phrase good solvent. Choosing a different coil-volume convention changes its prefactor. The robust chain-length dependence is $c^{\ast }\propto M^{1-3\nu }\simeq M^{-4/5}$ for a flexible good-solvent coil.