6.10 Viscosity and Drag

Viscosity is a fundamental property of fluids that measures their resistance to flow. It's what makes honey thick and slow-moving, while water is thin and flows easily. When an object moves through a fluid, this viscosity creates a resistive force known as drag, which opposes the object's motion.

Viscosity

Viscosity can be thought of as the internal friction between the layers of a fluid. A fluid with high internal friction (high viscosity) resists motion, while a fluid with low internal friction (low viscosity) flows readily. Real fluids are viscous because intermolecular forces between fluid layers resist relative sliding motion between those layers.

• Coefficient of Viscosity ($\eta$): This is the quantitative measure of viscosity.
  • SI Units: Pascal-second ($\text{Pa}\cdot \text{s}$), also written as $N\cdot s/m^2$ or $kg\cdot m^{-1}\cdot s^{-1}$.
  • Dimensions: $[M{L}^{-1}{T}^{-1}]$
• Temperature Dependence: The effect of temperature on viscosity differs for liquids and gases.
  • Liquids: Viscosity decreases as temperature increases (e.g., warm honey flows more easily than cold honey).
  • Gases: Viscosity increases as temperature increases.

Dimensions→

Drag Force

Drag is the resistive force exerted by a fluid on an object moving through it. It is the practical consequence of the fluid's viscosity.

• Types of Drag:
  • Aerodynamic Drag: Drag force in a gas (e.g., air resistance on a moving car).
  • Hydrodynamic Drag: Drag force in a liquid (e.g., water resistance on a swimmer).
• Factors Affecting Drag Force: The magnitude of the drag force depends on:
  • The velocity of the object relative to the fluid.
  • The size and shape of the object (streamlined shapes experience less drag).
  • The viscosity and density of the fluid.

Stokes' Law

Stokes' Law provides a formula to calculate the drag force ($F_d$) on a small, spherical object moving at a low speed through a viscous fluid (laminar/streamline flow).

$F_d=6\pi \eta rv$

Where:

• $F_d$ = Drag force (N)
• $\eta$ = Coefficient of viscosity of the fluid ($\text{Pa}\cdot \text{s}$)
• $r$ = Radius of the spherical object (m)
• $v$ = Velocity of the object relative to the fluid ($\text{m/s}$)

Conditions for validity: Stokes' Law applies only to small spherical objects at low speeds where flow is laminar (not turbulent).

Terminal Velocity

When an object falls through a fluid, it initially accelerates due to gravity. As its velocity increases, the drag force also increases (according to Stokes' Law). Eventually, the drag force plus the buoyant (upthrust) force equals the weight of the object. At this point, the net force is zero, and the object falls at a constant speed called terminal velocity.

Condition for terminal velocity: $W=F_d+F_u$ $mg=6\pi \eta r{v}_t+{\rho }_{f}Vg$

For a solid sphere of radius $r$ and density ${\rho }_s$ falling through a fluid of density ${\rho }_f$ and viscosity $\eta$, substituting $m=\frac{4}{3}\pi r^3{\rho }_s$ and $V=\frac{4}{3}\pi r^3$:

$v_t=\frac{2r^2({\rho }_s-{\rho }_f)g}{9\eta }$

This shows that terminal velocity is directly proportional to $r^2$ — a larger sphere reaches a higher terminal velocity.

Equilibrium→