Nineteenth-century physician Jean Louis Marie Poiseuille...

Nineteenth-century physician Jean Louis Marie Poiseuille discovered that the rate of change of the velocity of blood $r \mathrm{~cm}$ from the central axis of an artery (in centimeters/second/centimeter) is given by $$ a(r)=-k r $$ where $k$ is a constant. If the radius of an artery is $R \mathrm{~cm}$, find an expression for the velocity of blood as a function of $y$ (see the accompanying figure).

Nineteenth-century physician Jean Louis Marie Poiseuille discovered that the rate of change of the velocity of blood $r \mathrm{~cm}$ from the central axis of an artery (in centimeters/second/centimeter) is given by
$$
a(r)=-k r
$$
where $k$ is a constant. If the radius of an artery is $R \mathrm{~cm}$, find an expression for the velocity of blood as a function of $y$ (see the accompanying figure).

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Key Concepts

Ordinary Differential Equations

An ordinary differential equation (ODE) relates a function with its derivative(s). In this problem, the rate of change of blood velocity with respect to the radial distance forms a simple linear ODE that must be integrated to retrieve the velocity profile.

Integration

Integration is the inverse process of differentiation and is used here to recover the velocity function from its derivative. This step involves determining the antiderivative and incorporating an arbitrary constant that is later fixed by applying physical constraints.

Boundary Conditions

Boundary conditions provide necessary additional information to uniquely determine a solution to a differential equation. In the context of fluid flow in arteries, applying a condition such as the velocity being zero at the artery wall (no-slip condition) is essential to determine the integration constant and ensure the solution is physically valid.

Parabolic Velocity Profile

A parabolic velocity profile is characteristic of laminar flow in cylindrical conduits. When a linear relation between the derivative of velocity and the radial position is integrated, the resulting quadratic (parabolic) expression describes how the velocity varies across the radius, reflecting classical behavior as predicted by Poiseuille's law.

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