12.4 Viscosity and Laminar Flow; Poiseuille’s Law

Fluid dynamics explores how liquids and gases move. It distinguishes between smooth laminar flow and chaotic turbulent flow, which depend on factors like velocity and viscosity. Understanding these concepts is crucial for designing efficient fluid systems.

Poiseuille's law is a key principle in fluid dynamics. It relates flow rate to pressure difference, pipe dimensions, and fluid viscosity. This law helps engineers design pipelines, analyze blood flow, and optimize fluid transport in various applications.

Fluid Dynamics

Laminar vs turbulent flow

• Laminar flow exhibits smooth, orderly motion in parallel layers without mixing between layers (honey flowing slowly)
• Occurs at low velocities, high viscosities, and low Reynolds numbers (Re < 2300)
• Turbulent flow is chaotic and irregular with mixing between layers, forming eddies and vortices (fast-flowing river)
• Happens at high velocities, low viscosities, and high Reynolds numbers (Re > 4000)
• Transitional flow is an intermediate state between laminar and turbulent flow with Reynolds numbers between 2300 and 4000
• Streamlines, which represent the paths of fluid particles, are parallel in laminar flow but irregular in turbulent flow

Viscosity and fluid behavior

• Viscosity measures a fluid's resistance to flow or shear stress due to intermolecular forces and friction between layers
• Higher viscosity leads to slower flow and more resistance (molasses), while lower viscosity results in faster flow and less resistance (water)
• Liquid viscosity decreases with increasing temperature, while gas viscosity increases with temperature
• Newtonian fluids have constant viscosity independent of shear stress (water, air), while non-Newtonian fluids have varying viscosity with shear stress (blood, ketchup)
• Dynamic viscosity (η) is the ratio of shear stress to shear rate, measuring the fluid's internal resistance to flow
• Kinematic viscosity (ν) is the ratio of dynamic viscosity to fluid density, often used in fluid dynamics calculations

Shear rate and viscosity

• Shear rate is the rate of change of velocity between adjacent layers of fluid
• It affects the behavior of non-Newtonian fluids, causing changes in their viscosity
• In laminar flow, the shear rate is highest near the pipe walls and lowest at the center

Poiseuille's Law

Poiseuille's law applications

• Relates flow rate ($Q$), pressure difference ($\Delta P$), pipe dimensions ($r$, $L$), and fluid viscosity ($\eta$) as $Q = \frac{\pi r^4 \Delta P}{8 \eta L}$
• Calculates resistance to flow ($R$) in a pipe using $R = \frac{8 \eta L}{\pi r^4}$, which increases with length and viscosity but decreases with pipe radius to the fourth power
• Applies to laminar flow of Newtonian fluids in cylindrical pipes with constant cross-section
• Used in designing fluid transport systems (oil pipelines, blood vessels) and understanding fluid behavior in various applications (microfluidics, hydraulic systems)

Pressure changes in pipes

• Pressure decreases linearly along the length of the pipe due to fluid resistance
• Pressure drop ($\Delta P$) is proportional to flow rate ($Q$) and resistance ($R$) as $\Delta P = Q \cdot R$
• Higher flow rates, longer pipes, smaller radii, and more viscous fluids lead to greater pressure drops
• Pumps must overcome pressure drops to maintain desired flow rates in fluid transport systems (water distribution networks)
• Pipe dimensions and materials should be selected to minimize resistance and optimize flow (large-diameter, smooth-walled pipes for long-distance transport)