# Hagen-Poiseuille flow-or called Poiseuille flow through a circular pipe The Poiseuille flow demonstrations is the profound influence of pipe geometry on fluid dynamics: The volumetric flow rate ( $$Q$$ ) is critically dependent on the pipe's internal radius ( $$R$$ ) raised to the fourth power ( $$R^4$$ ). This means that while factors like the pressure gradient ( $$\Delta P / L$$ ) and fluid viscosity ( $$\mu$$ ) exhibit a linear relationship with flow, even minor changes in the radius-such as those caused by plaque buildup in arteries or subtle variations in manufactured pipe size-result in disproportionately large impacts on the resulting flow, demanding significantly greater pumping power to compensate for reduced capacity. ### Brief audio {% embed url="" %} ### Condensed Notes {% embed url="" %}