14.7 Viscosity and Turbulence

Fluid dynamics explores how liquids and gases move and interact. It's crucial for understanding everything from blood flow in our bodies to air movement around airplanes. This topic dives into viscosity, flow types, and how objects behave in fluids.

We'll look at key concepts like ###Poiseuille's_Law_0###, Reynolds number, and terminal speed. These ideas help us grasp how fluids behave in pipes, around objects, and when things fall through them. Understanding these principles is essential for many real-world applications.

Fluid Dynamics

Viscosity in fluid mechanics

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• Measures a fluid's resistance to flow or deformation caused by internal friction between fluid layers as they move past each other
• Represented by the Greek letter $\eta$ (eta) with SI unit pascal-second (Pa·s) or newton-second per square meter (N·s/m²)
• Determines the ease or difficulty with which fluids flow through pipes (water vs honey) or around objects (air vs molasses)
• Affects the velocity profile of fluid flow, with higher viscosity leading to more pronounced velocity gradients
• Influences the pressure drop in fluid systems, as more viscous fluids require greater pressure differences to maintain flow
• Contributes to the formation of boundary layers near surfaces, where fluid velocity approaches zero due to viscous effects
• Kinematic viscosity, defined as the ratio of viscosity to density, is used to characterize fluid flow behavior in certain applications

Application of Poiseuille's law

• Describes the flow of viscous fluids through cylindrical pipes relating volumetric flow rate ($Q$) to pressure difference ($\Delta P$), pipe radius ($r$), length ($L$), and fluid viscosity ($\eta$) using formula $Q = \frac{\pi r^4 \Delta P}{8 \eta L}$
• Calculates resistance to fluid flow in pipes ($R$) as the ratio of pressure difference to volumetric flow rate using formula $R = \frac{\Delta P}{Q} = \frac{8 \eta L}{\pi r^4}$
• Shows that resistance increases with higher viscosity (molasses vs water), longer pipes, and smaller pipe radii (capillaries vs aorta)

Pressure changes from viscous resistance

• Pressure drop ($\Delta P$) occurs along a pipe due to viscous resistance, calculated using Poiseuille's law $\Delta P = \frac{8 \eta L Q}{\pi r^4}$
• Higher viscosity fluids (oil vs water) lead to greater pressure drops for the same flow conditions
• Longer pipes result in larger pressure drops, as fluid experiences more viscous resistance over the increased distance
• Smaller pipe radii (hypodermic needle vs garden hose) cause more significant pressure drops due to increased viscous effects near the walls
• Increased flow rates contribute to higher pressure drops, as fluid layers experience greater shear forces

Flow Characteristics

Reynolds number for moving objects

• Dimensionless quantity that characterizes flow regime as the ratio of inertial forces to viscous forces, calculated using formula $Re = \frac{\rho v D}{\eta}$
  • $\rho$ is fluid density (air vs water)
  • $v$ is fluid velocity (slow vs fast)
  • $D$ is characteristic length (pipe diameter, object size)
  • $\eta$ is fluid viscosity (air vs oil)
• Low Reynolds numbers ($Re < 2300$ for pipe flow) indicate laminar flow dominated by viscous forces with smooth, parallel fluid layers and no mixing
• High Reynolds numbers ($Re > 4000$ for pipe flow) indicate turbulent flow dominated by inertial forces with chaotic, irregular motion and mixing between fluid layers

Laminar vs turbulent flow

• Laminar flow occurs at low Reynolds numbers ($Re < 2300$ for pipe flow) characterized by smooth, parallel fluid layers with no mixing and a parabolic velocity profile (highest at center, zero at walls)
• Turbulent flow occurs at high Reynolds numbers ($Re > 4000$ for pipe flow) characterized by chaotic, irregular motion with mixing between fluid layers and a flatter velocity profile (more uniform across pipe)
• Transitional flow occurs between laminar and turbulent regimes ($2300 < Re < 4000$ for pipe flow) exhibiting characteristics of both with developing flow instabilities
• In laminar flow, fluid particles follow smooth paths called streamlines, which do not intersect

Concept of terminal speed

• Constant velocity reached by an object falling through a fluid (skydiver in air, raindrop in atmosphere) when the drag force equals the object's weight
• At terminal speed, net force on the object is zero and acceleration stops
• Requires presence of a fluid (air, water) that exerts drag on the falling object and sufficient fall distance to reach terminal speed
• Factors affecting terminal speed:
  1. Object's shape and size: Larger cross-sectional area perpendicular to flow leads to higher drag and lower terminal speed (open vs closed parachute)
  2. Fluid density: Higher density results in greater drag and lower terminal speed (water vs air)
  3. Fluid viscosity: Higher viscosity leads to increased drag and lower terminal speed (oil vs water)
  4. Object's weight: Heavier objects have a higher terminal speed, as they require a larger drag force to balance their weight (bowling ball vs feather)
  5. Drag coefficient: A dimensionless quantity that represents the object's resistance to fluid flow, with higher values resulting in lower terminal speeds

Advanced fluid dynamics concepts

• Shear flow occurs when adjacent layers of fluid move relative to each other, causing internal friction and viscous effects
• Flow separation happens when a fluid flowing along a surface detaches from that surface, often leading to increased drag and turbulence
• The drag coefficient is influenced by factors such as object shape, surface roughness, and flow conditions, affecting the overall fluid resistance experienced by moving objects