Summary of "Fluid Mechanics | Module 1 | Viscosity (Lecture 4)"

This lecture, delivered by Mehboobe Gopal Sharma from Gate Academy Plus, covers the fundamental concepts of Viscosity in Fluid Mechanics, focusing on its causes, behavior, mathematical description, and dependence on temperature. The lecture also introduces Newton’s law of Viscosity and distinguishes between dynamic and kinematic Viscosity.

Main Ideas and Concepts

1. Introduction to Viscosity

Definition: Viscosity is the resistance a fluid offers to slow flow; it is analogous to friction between solid bodies but occurs between fluid molecules.
Physical Interpretation: Viscosity arises due to internal friction between fluid layers moving at different velocities.
Two Main Causes of Viscosity:
- Cohesive forces (intermolecular attraction) between molecules.
- Intermolecular momentum transfer (especially significant in gases where molecules collide and exchange momentum).

2. Molecular Explanation of Viscosity

In liquids, Viscosity is mainly due to strong intermolecular attractions that resist flow.
In gases, Viscosity arises primarily from momentum transfer during molecular collisions.
Gas molecules move freely with higher velocity, leading to momentum transfer when they collide, causing viscous forces.
The viscous force in gases is generally smaller than in liquids but increases with temperature.

3. Velocity Profile and Relative Motion in Fluids

Fluid flow near a solid boundary exhibits a velocity gradient: the fluid layer in direct contact with the solid surface sticks to it (no slip condition), having zero velocity relative to the surface.
Velocity increases with distance from the solid surface, forming a velocity profile.
Different fluid layers moving at different velocities experience relative motion, causing viscous resistance.
Example: Two bulls tied together running at different speeds feel resistance analogous to viscous forces between fluid layers.

4. Newton’s Law of Viscosity

Shear stress (τ) in a fluid is directly proportional to the rate of shear strain (velocity gradient).
Mathematically: τ = μ (du/dy) where: - τ = shear stress - μ = dynamic Viscosity (proportionality constant) - du/dy = velocity gradient (rate of shear strain)
Dynamic Viscosity (μ) is a measure of a fluid’s internal resistance to flow.
Shear stress causes continuous deformation in fluids under applied force, unlike solids which deform up to a limit.

5. Units of Dynamic Viscosity

SI unit: Pascal-second (Pa·s) or equivalently Newton-second per square meter (N·s/m²)
CGS unit: Poise (P), where 1 Poise = 0.1 Pa·s
Conversion between units is important for practical problems.

6. Kinematic Viscosity

Defined as the ratio of dynamic Viscosity to fluid density: ν = μ / ρ
Represents the fluid’s ability to diffuse momentum.
Unit in SI: m²/s
Unit in CGS: Stokes (St), where 1 St = 1 cm²/s

7. Effect of Temperature on Viscosity

Liquids:
- Increasing temperature decreases Viscosity because molecular attraction weakens as molecules move apart.
- Dynamic Viscosity of liquids decreases with temperature.
Gases:
- Increasing temperature increases Viscosity because molecular momentum and collision frequency increase.
- Dynamic Viscosity of gases increases with temperature.
Empirical formulas for temperature dependence of Viscosity for both liquids and gases were presented, involving constants α and β.

Methodology / Key Points in Bullet Format

Understanding Viscosity:
- Viscosity = internal friction or resistance to flow in fluids.
- Caused by cohesive forces and molecular momentum transfer.
Velocity Profile Near Solid Boundaries:
- Fluid layer adjacent to solid surface has zero velocity (no slip).
- Velocity increases with distance from the surface.
- Relative motion between layers causes viscous resistance.
Newton’s Law of Viscosity:
- Shear stress proportional to velocity gradient.
- Introduce dynamic Viscosity as proportionality constant.
- Express mathematically and understand physical meaning.
Units:
- SI: Pascal-second (Pa·s)
- CGS: Poise (P)
- Kinematic Viscosity units: m²/s (Stokes in CGS)
Temperature Effects:
- Liquids: Viscosity decreases with temperature.
- Gases: Viscosity increases