Topic 6.213 min read
Viscosity, Viscous Drag and Stokes' Law
Viscosity is a fluid's resistance to flow. Viscous drag opposes relative motion through a fluid, and for a small sphere in laminar flow Stokes' law gives F_d = 6πηrv.
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Beginner Level
Honey flows more slowly than water because honey has greater viscosity. An object moving through a liquid or gas experiences a force opposing its motion. This resistance is called viscous drag.
- F_d
- drag force (N)
- η
- coefficient of viscosity (Pa s)
- r
- sphere radius (m)
- v
- relative speed (m s⁻¹)
When to use: Small sphere moving slowly through a viscous fluid in laminar flow.
Recommended for Class 11
6.2 Viscous Drag and Terminal Velocity
Overview
The opening part of Section 6.2 develops viscosity, drag and Stokes' law. These ideas form one connected explanation: viscosity describes internal fluid resistance, drag is the resulting retarding force on a moving body, and Stokes' law quantifies the drag on a small sphere.
Learning Outcomes
- Define viscosity and viscous drag.
- State the coefficient and units of viscosity.
- Explain temperature effects in liquids and gases.
- Distinguish aerodynamic and hydrodynamic drag.
- State and apply Stokes' law.
- Explain the factors that affect drag.
Viscosity
A fluid with large viscosity flows reluctantly, while a fluid with small viscosity flows easily. Honey and engine oil have larger viscosity than water and air.
| Fluid type | Effect of increasing temperature | Reasoning used at this level |
|---|---|---|
| Most liquids | Viscosity decreases | Molecules move more freely between layers |
| Most gases | Viscosity increases | Faster molecules transfer more momentum between layers |
Because viscosity depends on temperature, the temperature should be stated when viscosity is measured. The textbook lists water at 20°C as about 1.0016 mPa s and honey as roughly 2000-10000 mPa s.
Coefficient of Viscosity
The coefficient of viscosity is represented by the Greek letter η (eta). Its equivalent SI units are:
Intermediate Level
Viscous Drag
Drag always acts opposite the velocity of the object relative to the fluid. Drag in a gas is commonly called aerodynamic drag or air resistance. Drag in a liquid is called hydrodynamic drag.
| Factor | Effect on drag |
|---|---|
| Relative speed | Greater speed generally gives greater drag |
| Object size | Larger effective size gives greater drag |
| Shape | Streamlined shapes reduce drag |
| Orientation | Smaller frontal area usually reduces drag |
| Fluid viscosity | More viscous fluid gives greater drag |
Geese fly in a V formation to reduce drag and energy use. Swimmers and skydivers change body orientation to control their drag, speed and direction.
Stokes' Law
For a small spherical object moving slowly through a viscous fluid with streamline flow, drag is directly proportional to fluid viscosity η, sphere radius r and relative speed v.
- F_d
- viscous drag magnitude (N)
- η
- coefficient of viscosity (Pa s)
- r
- sphere radius (m)
- v
- sphere-fluid relative speed (m s⁻¹)
When to use: Small sphere, low speed, laminar flow.
Common mistake: Convert millimetres to metres and use radius rather than diameter.
Conditions and Limitations
- The body is spherical.
- The sphere is small compared with the fluid region.
- Relative speed is low.
- Flow around the sphere is laminar.
- The fluid is treated as continuous and Newtonian.
- The sphere is sufficiently far from container walls.
Dimensional Check
Solved Numerical Examples
Textbook Example 6.2: Drag on a Sphere
A spherical body of radius 2 mm moves through air at 2 m s⁻¹. Air viscosity is 1.9 × 10⁻⁵ kg m⁻¹ s⁻¹. Find the drag force.
- r = 2 mm = 2 × 10⁻³ m
- v = 2 m s⁻¹
- η = 1.9 × 10⁻⁵ kg m⁻¹ s⁻¹
Fd = 6πηrv
Fd = 6(3.14)(1.9 × 10⁻⁵)(2 × 10⁻³)(2)
Fd = 1.432 × 10⁻⁶ N
Final answer: Fd = 1.432 × 10⁻⁶ N = 143.2 × 10⁻⁸ N
Example 2: Effect of Radius and Speed
A small sphere obeys Stokes' law. Its radius doubles and speed triples in the same fluid. By what factor does drag change?
- r₂ = 2r₁
- v₂ = 3v₁
- η unchanged
F_d ∝ rv
F₂F₁ = (r₂r₁)(v₂v₁)
F₂F₁ = (2)(3)
Final answer: Drag becomes 6 times larger.
Example 3: Finding Viscosity
A 1.0 mm radius sphere moving at 0.50 m s⁻¹ experiences 9.42 × 10⁻³ N Stokes drag. Find η.
- r = 1.0 × 10⁻³ m
- v = 0.50 m s⁻¹
- Fd = 9.42 × 10⁻³ N
η = Ffracd6πrv
η = 9.42 × 10⁻³6(3.14)(1.0 × 10⁻³)(0.50)
Final answer: η = 1.0 Pa s
Real-Life Applications
- Lubricants are selected by viscosity for engines and machines.
- Falling-sphere viscometers determine liquid viscosity.
- Streamlining reduces transport energy use.
- Swimming technique changes hydrodynamic drag.
- Bird formations can reduce aerodynamic drag.
Common Mistakes
- Using Stokes' law for large objects or turbulent flow.
- Using diameter instead of radius.
- Leaving radius in millimetres.
- Forgetting that drag opposes relative motion.
- Assuming viscosity changes with temperature in the same way for liquids and gases.
- Writing η in joules or newtons instead of Pa s.
Short-Question Preparation
Q. Define viscosity.
A. Viscosity is the resistance offered by different layers of a fluid to its flow.
Q. Define viscous drag.
A. Viscous drag is the retarding force on an object moving relative to a fluid.
Q. State Stokes' law.
A. For a small sphere moving slowly in laminar flow, F_d = 6πηrv.
Q. How does temperature affect viscosity?
A. In most liquids viscosity decreases with temperature, while in most gases it increases.
Long-Question Preparation
Q. Explain viscosity, drag and Stokes' law.
A. 1. Define viscosity as internal resistance to flow. 2. Name η and its units Pa s, N s m⁻² and kg m⁻¹ s⁻¹. 3. State the opposite temperature trends for liquids and gases. 4. Define viscous drag and its direction. 5. Distinguish aerodynamic and hydrodynamic drag. 6. List size, shape, orientation, viscosity and relative speed as factors. 7. State F_d = 6πηrv. 8. Give its conditions: small sphere, low speed and laminar flow. 9. Check dimensions [MLT⁻²].
Sticky Notes
- Viscosity = resistance to flow.
- Drag opposes relative motion.
- Gas drag → aerodynamic.
- Liquid drag → hydrodynamic.
- Fd = 6πηrv.
- η unit = Pa s.
- Stokes' law needs a small sphere and laminar flow.
60-Second Revision
- F_d ∝ η.
- F_d ∝ r.
- F_d ∝ v.
- Liquid viscosity usually falls with temperature.
- Gas viscosity usually rises with temperature.
Summary
Viscosity measures resistance within a fluid, while viscous drag opposes an object's relative motion through it. Under low-speed laminar conditions, Stokes' law gives the drag on a small sphere as F_d = 6πηrv.
Ask-the-Tutor Modes
- Explain viscosity simply.
- Compare liquid and gas temperature effects.
- Explain drag direction.
- Check whether Stokes' law applies.
- Check my unit conversion.
- Test me with an MCQ.
Exam Focus
Exam Focus
Important points
- Viscosity definition and unit
- Temperature trends
- Drag definition and types
- Stokes' law
- Conditions
Important formulas
- Fd = 6πηrv
- [η] = [ML⁻¹T⁻¹]
Common exam traps
- Diameter for radius
- mm not converted
- Using law in turbulence
Short question tips
- Give equivalent viscosity units.
Long question structure
Viscosity → temperature → drag → factors → Stokes law → conditions → dimensions.
Common Mistakes
Common Mistakes
- Mistake
Use radius, not diameter.
- Mistake
Convert mm to m.
- Mistake
State Stokes-law conditions.
- Mistake
Drag opposes relative motion.
- Mistake
Do not confuse liquid and gas temperature trends.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
η unit: Pa s.
Drag opposes motion.
F_d ∝ ηrv.
Small sphere and laminar flow.
Viscosity is fluid resistance to flow; Stokes drag is F_d = 6πηrv.
Fd = 6πηrv
- Definition
- Viscosity is fluid resistance to flow; Stokes drag is F_d = 6πηrv.
- Formula
- Fd = 6πηrv
- Example
- For r = 2 mm, v = 2 m s⁻¹ and η = 1.9 × 10⁻⁵ Pa s, Fd = 1.432 × 10⁻⁶ N.
- Critical warning
- Convert the radius to metres.
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Exam Practice
Important Short Question
Q1. Define viscosity and state its unit.
Important Short Question
Q2. State Stokes' law.
Conceptual Question
Q3. How does temperature affect viscosity?
Long Question Preparation
Q4. Explain viscous drag and its factors.