12.6 Motion of an Object in a Viscous Fluid

Objects moving through fluids experience complex interactions. The Reynolds number helps predict flow behavior, distinguishing between smooth laminar flow and chaotic turbulent flow. Understanding these concepts is crucial for analyzing real-world scenarios.

Terminal speed, where drag balances gravitational and buoyant forces, is a key concept in fluid dynamics. This equilibrium state occurs in both laminar and turbulent flows, affecting everything from falling raindrops to descending skydivers.

Motion of an Object in a Viscous Fluid

Reynolds number calculation and interpretation

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• Reynolds number ($Re$) dimensionless quantity characterizes flow behavior of fluid
  • Ratio of inertial forces to viscous forces: $Re = \frac{\rho vD}{\mu}$
    • $\rho$ fluid density (water, air)
    • $v$ object's velocity relative to fluid (m/s)
    • $D$ characteristic length (diameter for sphere or cylinder) (m)
    • $\mu$ fluid's dynamic viscosity (Pa·s)
  • Can also be expressed using kinematic viscosity: $Re = \frac{vD}{\nu}$, where $\nu$ is kinematic viscosity
• Low Reynolds numbers ($Re < 2300$) indicate laminar flow
  • Viscous forces dominate fluid flow is smooth and predictable (honey, motor oil)
• High Reynolds numbers ($Re > 4000$) indicate turbulent flow
  • Inertial forces dominate fluid flow is chaotic and unpredictable (fast-moving river, wind around buildings)
• Transitional flow occurs between laminar and turbulent flow ($2300 < Re < 4000$)

Laminar vs turbulent flow

• Laminar flow
  • Fluid layers slide smoothly past each other without mixing
  • Velocity profile parabolic, highest velocity at center and zero at boundaries
  • Drag force proportional to velocity: $F_D = 6\pi\mu Rv$
    • $\mu$ fluid's dynamic viscosity (Pa·s)
    • $R$ object's radius (m)
    • $v$ object's velocity relative to fluid (m/s)
  • Examples: blood flow in capillaries, ink flowing from pen
• Turbulent flow
  • Fluid layers mix chaotically, forming eddies and vortices
  • Velocity profile more uniform, thin boundary layer near object's surface
  • Drag force proportional to square of velocity: $F_D = \frac{1}{2}C_D\rho Av^2$
    • $C_D$ drag coefficient (depends on object shape and Reynolds number)
    • $\rho$ fluid density (kg/m³)
    • $A$ object's cross-sectional area (m²)
    • $v$ object's velocity relative to fluid (m/s)
  • Examples: air flow around cars, water flow in pipes

Terminal speed in viscous fluids

• Terminal speed is constant velocity object reaches when net force is zero
  • Gravitational force ($F_g$) balanced by drag force ($F_D$) and buoyant force ($F_B$)
    • $F_g = mg$, $m$ object's mass (kg), $g$ acceleration due to gravity (m/s²)
    • $F_B = \rho gV$, $\rho$ fluid density (kg/m³), $V$ object's volume (m³)
• For laminar flow, terminal speed reached when: $6\pi\mu Rv_t = (\rho_o - \rho)gV$
  • $\mu$ fluid's dynamic viscosity (Pa·s)
  • $R$ object's radius (m)
  • $v_t$ terminal speed (m/s)
  • $\rho_o$ object's density (kg/m³)
  • $\rho$ fluid density (kg/m³)
  • $g$ acceleration due to gravity (m/s²)
  • $V$ object's volume (m³)
• For turbulent flow, terminal speed reached when: $\frac{1}{2}C_D\rho Av_t^2 = (\rho_o - \rho)gV$
  • $C_D$ drag coefficient
  • $\rho$ fluid density (kg/m³)
  • $A$ object's cross-sectional area (m²)
  • $v_t$ terminal speed (m/s)
  • $\rho_o$ object's density (kg/m³)
  • $g$ acceleration due to gravity (m/s²)
  • $V$ object's volume (m³)
• Examples: skydiver reaching terminal velocity, dust particle settling in air

Fluid Dynamics Principles

• Shear stress: force per unit area exerted parallel to the surface of a fluid
  • Responsible for the development of the boundary layer in fluid flow
• Bernoulli's principle: increase in fluid velocity corresponds to a decrease in pressure
  • Explains lift generation in airplane wings and the curve of a baseball