Viscosity, Stokes' Law, Terminal Velocity & Poiseuille Flow

Just as solid surfaces in contact experience mechanical friction opposing relative sliding, adjacent layers of a real fluid experience an internal friction opposing relative motion. This internal property of fluids is termed viscosity. Viscous forces dissipate kinetic energy into heat as fluids flow through pipes or past solid obstacles.

1. Velocity Gradient & Newton's Law of Viscosity

Consider a liquid flowing laminar-style over a horizontal fixed solid plate. The layer in direct contact with the solid boundary is stationary (v = 0, the "no-slip" condition). As perpendicular distance x from the fixed plate increases, layer velocity increases continuously up to v:

   Top Layer:      ~~~~~~~~~~~~~~~~==================> Velocity (v + dv)
                   |  Shear Stress tau = F / A       |
                   |  Velocity Gradient = dv / dx    |
   Intermediate:   ~~~~~~~~~~~~~~~~============>       Velocity v
                   |                                 |
   Bottom Plate:   =================================== Fixed Solid Wall (v = 0)
      

The rate of change of flow velocity with perpendicular distance is called the velocity gradient (dv/dx), measured in s^-1.

According to Sir Isaac Newton, the backward tangential viscous drag force F acting between two adjacent fluid layers of contact area A is:

where η is the coefficient of viscosity of the fluid. The negative sign signifies that viscous drag opposes relative fluid motion.

Units and Dimensions of Viscosity

• SI Unit: N·s/m² = Pa·s (also called Decapoise or Poiseuille, Pl).
• CGS Unit: dyne·s/cm² = Poise (P).
• Conversion: 1 Pa·s = 10 Poise = 1000 cP (centipoise). (Water at 20^°C has η_w ≈ 10^-3 Pa·s = 1 cP).
• Dimensional Formula: [η] = [M¹ L^-1 T^-1].

2. Stokes' Law & Terminal Velocity Derivation

Derivation of Terminal Velocity (v_t)

Consider a small sphere of radius r and solid density ρ falling vertically under gravity through a viscous medium of density σ (ρ > σ):

                               +-------------------+
                               |   Upward Forces:  |
                               |   FB (Buoyancy)  |
                               |   Fv (Viscous)   |
                               +------------------+
                                         |
                                      ( O ) Sphere (Radius r, Density rho)
                                         |
                               +---------v---------+
                               |  Downward Force:  |
                               |  W = mg (Gravity) |
                               +-------------------+
             
             At Terminal Velocity: W = FB + Fv (Net Acceleration a = 0)
      

Three concurrent forces act on the descending sphere:

1. Downward Gravitational Weight: W = mg = 4/3π r³ ρ g.
2. Upward Buoyant Upthrust: F_B = 4/3π r³ σ g.
3. Upward Stokes' Viscous Drag: F_v = 6π η r v.

Net downward equation of motion: m a = W - F_B - F_v.

As the velocity increases, F_v increases proportionally until the net force becomes zero (a = 0). Thereafter, the sphere falls with a constant maximum steady speed termed terminal velocity (v_t):

3. Poiseuille's Law of Capillary Flow

For steady, streamline laminar flow of a viscous liquid through a horizontal cylindrical capillary pipe of radius r and length l under a driving hydrostatic pressure difference P:

• Enormous Fourth-Power Sensitivity (Q ∝ r⁴): If a capillary tube's radius is halved (r → r/2) while keeping driving pressure constant, the discharge rate drops by a staggering factor of 2⁴ = 16! This is of monumental clinical importance in vascular hemodynamics: minor arterial plaque buildup severely impairs blood flow.
• Fluid Resistance Analogy: Analogous to Ohm's Electrical Law (I = V / R):
  Q = P/R_fluid where R_fluid = (8η l)/(π r⁴)

4. Critical Velocity & Reynolds Number

The transition from smooth laminar flow to turbulent, vortex-shedding chaotic flow is governed by the dimensionless Reynolds Number (R_e), introduced by Osborne Reynolds:

where ρ is fluid density,vis flow speed,dis pipe diameter, andη is viscosity.

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