Question
A major artery in human body, with radius 0.4 cm, carries blood at a flow rate of 5.0 cubic centimeters per second. The pressure difference of blood per meter length of the artery is nearly : [Given that the coefficient of viscosity (η) of blood at body temperature is 4.0 × 10−3 Pa s and the density of mercury is 13.6 g/cm3]
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Solution
[Key Concept: Poiseuille's law gives the pressure gradient ΔP/L = 8ηQ/(πr4) for steady laminar flow.]
Rate of liquid flow is given by poiseuilles equation
$$\begin{matrix} \frac{dQ}{dt} & \ = \left( \frac{\pi}{8} \right)\frac{(\Delta P)r^{4}}{\eta\text{ }L} \\ 5 \times 10^{- 6} & \ = \frac{\pi}{8} \times \frac{\Delta P \times \left( 4 \times 10^{- 3} \right)^{4}}{4 \times 10^{- 3} \times \mathcal{l}} \\ \frac{\Delta P}{\mathcal{l}} & \ = \frac{5 \times 10^{- 6} \times 4 \times 10^{- 3}}{\left( 4 \times 10^{- 3} \right)^{4}} \times \frac{8}{\pi} \\ & \ = 0.2 \times 10^{3}\text{ }Pa/m \\ & \ = 1.5\text{ }mm\text{~of~}Hg/m \end{matrix}$$
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