Viscosity: Flow, Resistance & Applications

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What is Viscosity?

Viscosity is the internal friction in fluids (liquids and gases) that resists their flow. When a fluid flows, its layers slide over each other, and viscosity opposes this motion.

Types of Viscosity

1. Dynamic Viscosity (η or μ)

   • Measures the resistance to flow between two adjacent layers of fluid.
   • Unit: Pascal-second (Pa·s) or Poise (P) (1 P = 0.1 Pa·s).
   • Example: Honey has higher dynamic viscosity than water.

2. Kinematic Viscosity (ν)

   • Ratio of dynamic viscosity to fluid density.
   • Formula: $\nu =\frac{\eta }{\rho }$
   • Unit: m²/s or Stokes (St) (1 St = 10⁻⁴ m²/s).

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Factors Affecting Viscosity

Viscosity depends on:

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Stokes’ Law: Viscous Drag on a Sphere

When a small spherical object (like a raindrop or dust particle) moves through a viscous fluid, it experiences a viscous drag force (F) given by:

$$F=6\pi \eta rv$$

Where:

• $F$ = Viscous drag force (N)
• $\eta$ = Dynamic viscosity (Pa·s)
• $r$ = Radius of the sphere (m)
• $v$ = Velocity of the sphere (m/s)

Derivation (Simplified)

1. Consider a sphere moving slowly in a fluid.
2. The fluid layers near the sphere stick to it (no-slip condition).
3. The velocity gradient ($\frac{dv}{dx}$) causes shear stress.
4. Integrating shear stress over the sphere’s surface gives Stokes’ law.

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Terminal Velocity

When a falling object (e.g., a raindrop) reaches a constant speed, the viscous drag force (F) balances the gravitational force (mg).

At terminal velocity ($v_t$): $$6\pi \eta r{v}_t=\frac{4}{3}\pi r^3(\rho -\sigma )g$$ Where:

• $\rho$ = Density of the sphere
• $\sigma$ = Density of the fluid
• $g$ = Acceleration due to gravity

Solved Example

A small steel ball (radius = 0.5 mm, density = 7800 kg/m³) falls through glycerin (η = 1.5 Pa·s, density = 1260 kg/m³). Find its terminal velocity.

Solution:

1. Write the balance equation: $$6\pi \eta r{v}_t=\frac{4}{3}\pi r^3(\rho -\sigma )g$$
2. Plug in values: $$6\pi (1.5)(0.0005)v_t=\frac{4}{3}\pi (0.0005{)}^3(7800-1260)(9.8)$$
3. Solve for $v_t$: $$v_t=\frac{2}{9}\frac{(\rho -\sigma )g{r}^2}{\eta }=0.0021\text{m/s}$$

Answer: The terminal velocity is 2.1 mm/s.

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Applications of Viscosity

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Measurement of Viscosity

1. Capillary Tube Method (Ostwald Viscometer)

• A liquid flows through a narrow tube under gravity.
• Time taken ($t$) is measured.
• Viscosity is calculated using: $$\eta =\frac{\pi r^4\rho gt}{8LV}$$ Where:
  • $r$ = Radius of the tube
  • $L$ = Length of the tube
  • $V$ = Volume of the liquid

2. Falling Sphere Method

• A ball is dropped in a viscous liquid.
• Terminal velocity is measured and used in Stokes’ law to find $\eta$.

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NEB-Style Questions (Practice)

Short Answer (5 marks)

1. Define dynamic viscosity and kinematic viscosity. How are they related?
2. A raindrop (radius = 0.1 mm) falls through air (η = 1.8 × 10⁻⁵ Pa·s). If its terminal velocity is 2 m/s, find the density of the raindrop.

Long Answer (10 marks)

3. Explain Stokes’ law with a labeled diagram. How does terminal velocity depend on:
   • Radius of the falling object?
   • Viscosity of the fluid?
4. Describe the Ostwald viscometer method for measuring viscosity. Why is a narrow tube used?

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Exam Tip

✅ Key Focus Areas:

• Stokes’ law and terminal velocity (most common in NEB exams).
• Factors affecting viscosity (temperature, pressure, fluid type).
• Applications (lubrication, blood flow, viscometers).
• Units (Pa·s, Poise, Stokes).

❌ Common Mistakes to Avoid:

• Confusing dynamic and kinematic viscosity.
• Forgetting that gases behave differently from liquids with temperature.
• Misapplying Stokes’ law (only for small, spherical objects moving at low speeds).

💡 Pro Tip:

• Memorize the formula for terminal velocity—it appears in almost every problem.
• Practice numericals on raindrops, oil droplets, and viscometer calculations.

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