Home › Blog › Physics › Viscosity Guide
Viscosity & Fluid Dynamics in Physics: Stokes' Law & Terminal Velocity Guide
Viscosity is the internal friction of a fluid that resists relative motion between its adjacent flowing layers. From motor oil lubrication in car engines to blood circulation through human capillaries and raindrops falling at safe terminal speeds, viscosity governs real-world fluid dynamics.
This 4,000+ word comprehensive exam guide covers **Newton's Law of Viscosity ($F = -\eta A \frac{dv}{dx}$)**, **Coefficient of Viscosity ($\eta$) & Units (Poise, $\text{Pa}\cdot\text{s}$)**, **Temperature Effects (Liquids vs Gases)**, **Poiseuille's Formula for Pipe Flow**, **Stokes' Viscous Drag Law ($F_v = 6\pi \eta r v$)**, **Terminal Velocity Derivation ($v_t \propto r^2$)**, **Raindrop Mechanics & Parachutes**, **Reynolds Number ($R_e$) & Laminar vs Turbulent Flow**, and solved numerical problems for SSC CGL, RRB NTPC, and UPSC Prelims.
Table of Contents
- 1. Definition of Viscosity & Newton's Viscous Force Law ($F = -\eta A \frac{dv}{dx}$)
- 2. Units of Viscosity (Pascal-Second vs Poise)
- 3. Effect of Temperature & Pressure on Viscosity (Liquids vs Gases)
- 4. Stokes' Drag Law ($F_v = 6\pi \eta r v$)
- 5. Terminal Velocity Derivation ($v_t \propto r^2$) & Raindrop Physics
- 6. Poiseuille's Law of Fluid Flow in Capillary Tubes
- 7. Reynolds Number ($R_e$) & Laminar vs Turbulent Flow
- 8. Solved Numerical Examples for Competitive Exams
- 9. Must Remember Points for Quick Revision
- 10. Frequently Asked Questions (FAQ)
Key Takeaways & Core Highlights
- Newton's Viscosity Law: Tangential viscous force $F = -\eta A \frac{dv}{dx}$, where $\frac{dv}{dx}$ is the velocity gradient.
- SI Unit of Viscosity: Pascal-second ($\text{Pa}\cdot\text{s} = \text{N}\cdot\text{s/m}^2 = \text{kg/m}\cdot\text{s}$). CGS Unit: **Poise** ($1 \text{ Pa}\cdot\text{s} = 10 \text{ Poise}$).
- Temperature Dependence (Opposite for Liquids vs Gases!):
- Liquids: Viscosity **DECREASES as temperature rises** (cohesive forces weaken).
- Gases: Viscosity **INCREASES as temperature rises** (molecular momentum exchange collisions increase).
- Stokes' Drag Law: Retarding viscous force on a sphere moving through fluid: $F_v = 6 \pi \eta r v$.
- Terminal Velocity Formula: $$v_t = \frac{2 r^2 (\rho - \sigma) g}{9 \eta} \implies v_t \propto r^2$$ A raindrop with double the radius falls **4 times faster**!
- Reynolds Number ($R_e$):
- $R_e < 1000 \rightarrow$ **Streamline / Laminar Flow**.
- $R_e > 2000 \rightarrow$ **Turbulent Flow**.
1. Definition of Viscosity & Newton's Viscous Force Law ($F = -\eta A \frac{dv}{dx}$)
Viscosity is the internal fluid friction resisting liquid or gas flow:
$$F = -\eta \cdot A \cdot \frac{dv}{dx}$$Where $\eta$ is the **Coefficient of Viscosity**, $A$ is layer area, and $\frac{dv}{dx}$ is the velocity gradient.
4. Stokes' Drag Law ($F_v = 6\pi \eta r v$)
Formulated by Sir George Gabriel Stokes in 1851, the retarding viscous force experienced by a small rigid sphere of radius $r$ moving through a viscous fluid at velocity $v$ is:
$$F_v = 6 \pi \cdot \eta \cdot r \cdot v$$5. Terminal Velocity Derivation ($v_t \propto r^2$) & Raindrop Physics
When a body falls under gravity in a viscous fluid, downward gravity ($W$) is opposed by upward buoyancy ($F_b$) and viscous drag ($F_v$). When net force reaches zero, the body falls at constant **Terminal Velocity ($v_t$)**:
$$v_t = \frac{2 r^2 (\rho - \sigma) g}{9 \eta}$$Why Raindrops Don't Kill People:
If raindrops fell in a vacuum from a $2000\text{-meter}$ cloud, gravity would accelerate them to over $700\text{ km/h}$, making them fatal! Due to air viscosity, raindrops reach a safe terminal speed of only $7\text{--}9\text{ m/s}$ ($25\text{ km/h}$).
7. Reynolds Number ($R_e$) & Laminar vs Turbulent Flow
$$R_e = \frac{\text{Inertial Force}}{\text{Viscous Force}} = \frac{\rho \cdot v \cdot D}{\eta}$$8. Solved Numerical Examples for Competitive Exams
Numerical Problem 1 (Terminal Velocity Ratio):
Question: Two rain droplets of radii ratio $1:2$ fall through air. Calculate the ratio of their terminal velocities.
Solution:
$$v_t \propto r^2 \implies \frac{v_{t1}}{v_{t2}} = \left(\frac{r_1}{r_2}\right)^2 = \left(\frac{1}{2}\right)^2 = \frac{1}{4}$$Answer: The ratio of terminal velocities is $1:4$.
Numerical Problem 2 (Stokes' Force Calculation):
Question: A sphere of radius $r = 1\text{ mm}$ ($10^{-3}\text{ m}$) moves at speed $v = 2\text{ m/s}$ in oil ($\eta = 0.1\text{ Pa}\cdot\text{s}$). Calculate the viscous drag force.
Solution:
$$F_v = 6 \pi \eta r v = 6 \times 3.1416 \times 0.1 \times 10^{-3} \times 2 = 3.77 \times 10^{-3} \text{ N}$$Answer: Viscous force is $3.77 \times 10^{-3}\text{ N}$.
9. Must Remember Points for Quick Revision
Exam Revision Cheat Sheet:
- Newton's Viscosity Law: $F = -\eta A \frac{dv}{dx}$.
- Units: SI Unit: $\text{Pa}\cdot\text{s}$ ($\text{kg/m}\cdot\text{s}$). CGS Unit: Poise. $1\text{ Pa}\cdot\text{s} = 10\text{ Poise}$.
- Temp Effects: Liquids $\eta \downarrow$ with $T \uparrow$; Gases $\eta \uparrow$ with $T \uparrow$.
- Stokes' Law: $F_v = 6 \pi \eta r v$.
- Terminal Velocity: $v_t \propto r^2$. Double radius $\rightarrow 4\times$ terminal speed.
- Reynolds Number ($R_e$): $R_e < 1000$ Laminar; $R_e > 2000$ Turbulent.
10. Frequently Asked Questions (FAQ)
What is Newton's Law of Viscosity and the formula for viscous force?
Newton's Law of Viscosity states that the tangential viscous force (F) between two adjacent liquid layers is directly proportional to the surface area (A) and velocity gradient (dv/dx): F = -η A (dv/dx), where η is the Coefficient of Viscosity.
What is Terminal Velocity and how does it depend on object radius?
Terminal Velocity (v_t) is the constant maximum velocity attained by a body falling through a viscous fluid when upward buoyant force and viscous drag balance downward gravity: v_t = [2 r² (ρ - σ) g] / (9 η). Crucially, Terminal Velocity is directly proportional to the SQUARE of the radius (v_t ∝ r²).
How does temperature affect the viscosity of liquids versus gases?
For Liquids: Viscosity DECREASES with rising temperature because thermal energy weakens intermolecular cohesive forces. For Gases: Viscosity INCREASES with rising temperature because higher molecular speeds increase momentum transfer collisions between gas layers!
Related Physics & Science Guides
Continue your exam preparation with our comprehensive, deep-dive physics modules:
Master General Science & Physics on RRBCONTENTS
Practice PYQs, read formulas cheat sheets, and explore complete exam study modules.
Solve Physics PYQs → Explore Science Notes →Join our official Telegram channel: @rrbcontents