Viscosity & Surface Tension
Program – IIT‑JEE (Main + Advanced)
Subject – Physics
These notes are written to give you a quick, clear, and problem‑oriented understanding of Viscosity and Surface Tension – two concepts that frequently appear in JEE papers.
1. Viscosity
| Concept | Key Points |
|---|---|
| Definition | Measure of a fluid’s resistance to deformation under shear stress. |
| Types | • Dynamic (Absolute) viscosity (\mu) (Pa·s or N·s/m²) – resistance to shear. • Kinematic viscosity (\nu = \mu/\rho) (m²/s) – resistance per unit density. |
| Units | SI: Pa·s (dynamic), m²/s (kinematic). In cgs: Poise (P) = 0.1 Pa·s. |
| Newtonian Fluid | (\tau = \mu,\dot{\gamma}) (linear relation). Water, air, most gases. |
| Non‑Newtonian Fluid | Shear‑rate dependent viscosity: shear‑thinning, shear‑thickening, Bingham plastics. |
| Shear Stress | (\tau = \mu,\frac{du}{dy}) – shear stress = dynamic viscosity × shear rate. |
| Poiseuille’s Law | For laminar flow in a circular pipe: |
| [ | |
| Q = \frac{\pi r^{4},\Delta P}{8,\mu,L} | |
| ] | |
| where (Q) = volumetric flow rate, (r) = pipe radius, (L) = length, (\Delta P) = pressure drop. | |
| Reynolds Number | (\displaystyle Re = \frac{VD}{\nu}) (or (Re = \frac{\rho V D}{\mu})). |
- (Re < 2000) → laminar flow.
- (Re > 4000) → turbulent flow.
- (2000 < Re < 4000) → transition. | | Viscous Dissipation | (\displaystyle P_{\text{diss}} = \int \tau,\dot{\gamma},dV = \mu \int \left(\frac{du}{dy}\right)^2 dV). |
2. Surface Tension
| Concept | Key Points |
|---|---|
| Definition | Free energy per unit area of the liquid–gas interface. It is the force that minimizes surface area. |
| Units | N/m (SI), dyn/cm (cgs). For water at 20 °C: (γ \approx 0.072,\text{N/m}). |
| Young–Laplace Equation | Pressure jump across a curved interface: |
| [ | |
| ΔP = γ\left(\frac{1}{R_1} + \frac{1}{R_2}\right) | |
| ] | |
| For a spherical droplet (R_1 = R_2 = R), so (ΔP = 2γ/R). | |
| Capillary Rise / Depression | In a vertical capillary tube of radius (r): |
| [ | |
| h = \frac{2γ \cos θ}{\rho g r} | |
| ] | |
| where (θ) = contact angle, (ρ) = fluid density. | |
| Contact Angle | Angle between liquid surface and solid wall. |
- (θ < 90°) → wetting (liquid spreads).
- (θ > 90°) → non‑wetting (liquid beads up). | | Surface Tension Measurement | Du Noüy ring (force needed to detach ring), Wilhelmy plate (force on plate). | | Applications | Bubble formation, droplet formation, capillary action, adhesion, surface energy. |
3. Worked Examples
Example 1 – Pressure Drop in a Pipe (Poiseuille’s Law)
Problem
A water pipe of diameter (D = 0.02,\text{m}) and length (L = 10,\text{m}) carries a steady laminar flow at rate (Q = 5\times10^{-4},\text{m}^3/\text{s}).
Dynamic viscosity of water at (20^\circ\text{C}) is (\mu = 1.0\times10^{-3},\text{Pa·s}).
Find the pressure drop (\Delta P) across the pipe.
Solution
- Radius (r = D/2 = 0.01,\text{m}).
- Use Poiseuille’s law:
[ Q = \frac{\pi r^{4},\Delta P}{8,\mu,L} ] Solve for (\Delta P): [ \Delta P = \frac{8,\mu,L,Q}{\pi r^{4}} ]