(a) What is the largest average velocity of blood flow in an...

(a) What is the largest average velocity of blood flow in an artery of radius $2 \times 10^{-3} \mathrm{~m}$ if the flow must remain lanimar? (b) What is the corresponding flow rate ? (Take viscosity of blood to be $2.084 \times 10^{-3} \mathrm{~Pa} \mathrm{~s}$ ).

Key Concepts

Hagen–Poiseuille Law

The Hagen–Poiseuille law describes the flow of an incompressible and Newtonian fluid in a long, cylindrical pipe under laminar conditions. It relates the flow rate to the pressure drop, the viscosity of the fluid, and the geometric parameters of the pipe. This law is crucial when analyzing how fluid properties and pipe dimensions influence flow behavior in practical applications.

Flow Rate

Flow rate is the volume of fluid passing through a cross-sectional area per unit time, typically expressed as Q = A × v, where A is the cross-sectional area and v is the average fluid velocity. It is essential in quantifying and controlling the movement of fluids in systems such as blood vessels or pipes.

Viscosity

Viscosity is a measure of a fluid's resistance to deformation or flow. It quantifies the internal friction arising when layers of fluid move relative to one another. In problems dealing with fluid flow in pipes, viscosity directly influences the maximum velocity at which a flow can remain laminar and affects the overall flow rate.

Laminar Flow

Laminar flow refers to the smooth, orderly motion of a fluid, where its particles move along well?defined paths or streamlines. This concept is fundamental in fluid dynamics since many elegant equations, such as the Hagen–Poiseuille law, only hold under laminar conditions. The transition of flow from laminar to turbulent is often delineated by a critical Reynolds number.

Reynolds Number

The Reynolds number is a dimensionless quantity used to predict the flow regime in a fluid system by comparing inertial forces to viscous forces. In the context of flow through conduits, it helps determine whether the flow is laminar or turbulent. A critical Reynolds number (usually around 2300 for pipe flow) marks the upper limit for which laminar flow is maintained.

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