Study of Viscosity

Viscosity fluid mechanics ka ek bohot hi important concept hai, jo fluid ki internal friction aur resistance to flow ko describe karta hai. Ye property liquids aur gases dono me exist karti hai aur engineering, hydraulics, chemical processes, aur biological systems me critical role play karti hai.

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1. Introduction to Viscosity

Viscosity:

Measure of a fluid’s resistance to gradual deformation by shear stress or tensile stress.

• Commonly, internal friction of fluid is called viscosity.
• Example: Honey flows slowly (high viscosity), water flows quickly (low viscosity).

Importance of studying viscosity:

1. Design of pipelines and pumps
2. Lubrication of machinery
3. Blood flow in arteries and veins
4. Flow of oils and industrial liquids
5. Aerodynamics and hydrodynamics

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2. Nature of Viscosity

• Viscosity arises due to molecular interactions in fluid.
• In liquids, viscosity decreases with increasing temperature.
• In gases, viscosity increases with increasing temperature.
• Newtonian fluids: Viscosity remains constant with shear rate (e.g., water, air).
• Non-Newtonian fluids: Viscosity varies with shear rate (e.g., ketchup, blood).

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3. Units of Viscosity

3.1 Dynamic Viscosity (η)

• SI Unit: Pascal-second (Pa·s)
• CGS Unit: poise (P) → 1 P = 0.1 Pa·s
• Definition: Shear stress per unit velocity gradient

η=Shear stressRate of shear strain\eta = \frac{\text{Shear stress}}{\text{Rate of shear strain}}η=Rate of shear strainShear stress​

3.2 Kinematic Viscosity (ν)

• SI Unit: m²/s
• Relation with density:

ν=ηρ\nu = \frac{\eta}{\rho}ν=ρη​

• Important in fluid flow calculations and Reynolds number:

Re=VLνRe = \frac{V L}{\nu}Re=νVL​

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4. Mathematical Definition (Newton’s Law of Viscosity)

For a Newtonian fluid, the shear stress τ is proportional to velocity gradient: τ=ηdudy\tau = \eta \frac{du}{dy}τ=ηdydu​

Where:

• τ = shear stress (N/m²)
• η = dynamic viscosity
• du/dy = velocity gradient perpendicular to flow
• Graphical representation: Shear stress vs velocity gradient → straight line for Newtonian fluids

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5. Types of Viscosity

1. Absolute (Dynamic) Viscosity (η): Resistance due to internal friction
2. Kinematic Viscosity (ν): Dynamic viscosity per unit density
3. Apparent Viscosity: Measured in non-Newtonian fluids
4. Bulk Viscosity: Resistance to compression

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6. Factors Affecting Viscosity

1. Temperature:
   • Liquids: Viscosity decreases with temperature
   • Gases: Viscosity increases with temperature
2. Pressure:
   • High pressure → slight increase in viscosity for liquids
3. Composition:
   • Impurities can change viscosity
4. Shear rate:
   • Non-Newtonian fluids show shear-thinning or shear-thickening

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7. Experimental Methods for Measuring Viscosity

7.1 Capillary Tube Method (Poiseuille’s Method)

• Based on Hagen-Poiseuille law
• Apparatus: Vertical capillary, liquid reservoir, stopwatch
• Flow rate measured through tube of radius r and length L under pressure difference ΔP:

Q=πr4ΔP8ηLQ = \frac{\pi r^4 \Delta P}{8 \eta L}Q=8ηLπr4ΔP​

• Rearranging to find η:

η=πr4ΔP8QL\eta = \frac{\pi r^4 \Delta P}{8 Q L}η=8QLπr4ΔP​

• Applications: Lab determination of liquid viscosity

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7.2 Falling Sphere Method (Stokes’ Method)

• Sphere of radius r falls through liquid of density ρ
• Terminal velocity v achieved when gravity = viscous drag + buoyant force

v=2r2g(ρs−ρf)9ηv = \frac{2 r^2 g (\rho_s – \rho_f)}{9 \eta}v=9η2r2g(ρs​−ρf​)​

Where:

• ρ_s = density of sphere
• ρ_f = density of fluid
• Solve for dynamic viscosity:

η=2r2g(ρs−ρf)9v\eta = \frac{2 r^2 g (\rho_s – \rho_f)}{9 v}η=9v2r2g(ρs​−ρf​)​

• Advantages: Simple, accurate for low viscosity fluids

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7.3 Rotational Viscometer

• Cylinder or disk rotates in fluid
• Torque required to rotate proportional to viscosity
• Useful for non-Newtonian fluids
• Applications: Paint, polymers, food industry

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7.4 Oscillating Viscometer

• Measures damping of oscillations due to viscous drag
• Accurate for very low viscosity fluids
• Used in labs and calibration

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8. Stokes’ Law and Derivation

• For small sphere falling slowly, viscous force:

Fv=6πηrvF_v = 6 \pi \eta r vFv​=6πηrv

• Equilibrium at terminal velocity:

mg−Vρfg=6πηrvmg – V \rho_f g = 6 \pi \eta r vmg−Vρf​g=6πηrv

• Solve for η:

η=2r2g(ρs−ρf)9v\eta = \frac{2 r^2 g (\rho_s – \rho_f)}{9 v}η=9v2r2g(ρs​−ρf​)​

• Assumptions: Laminar flow, small Reynolds number

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9. Flow and Viscosity Relationship

9.1 Laminar Flow

• Smooth layers
• Hagen-Poiseuille law applies:

Q=πr4ΔP8ηLQ = \frac{\pi r^4 \Delta P}{8 \eta L}Q=8ηLπr4ΔP​

9.2 Turbulent Flow

• Chaotic mixing
• Viscosity less dominant, inertial effects dominate

9.3 Reynolds Number and Viscosity

Re=ρVDη=VDνRe = \frac{\rho V D}{\eta} = \frac{V D}{\nu}Re=ηρVD​=νVD​

• Re < 2000 → laminar
• Re > 4000 → turbulent

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10. Temperature Dependence

• Liquids: Arrhenius-type relation

η=η0eERT\eta = \eta_0 e^{\frac{E}{RT}}η=η0​eRTE​

• Gases: Sutherland formula

η=η0(TT0)3/2T0+ST+S\eta = \eta_0 \left(\frac{T}{T_0}\right)^{3/2} \frac{T_0 + S}{T + S}η=η0​(T0​T​)3/2T+ST0​+S​

Where S = Sutherland constant

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11. Applications of Viscosity

1. Engineering and Machinery: Lubricants, hydraulic fluids
2. Biomedical: Blood flow, IV fluids
3. Industrial: Paints, food, chemicals
4. Aerodynamics and Hydrodynamics: Air and water resistance calculations
5. Geophysics: Magma flow, glacier movement

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12. Viscosity in Everyday Life

• Honey, oil, water
• Ketchup (shear-thinning fluid)
• Toothpaste (non-Newtonian)
• Engine oils (viscosity grades SAE 30, 40)

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13. Experimental Errors in Viscosity Measurement

1. Temperature fluctuations
2. Impurities in fluid
3. Misalignment in apparatus
4. Surface effects (wall effects in capillary method)
5. Timing errors (human reaction in falling sphere method)

• Reduce errors: Calibration, repeated trials, temperature control

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14. Numerical Examples

Example 1: Falling Sphere

• Sphere radius r = 0.005 m
• Density sphere ρ_s = 7800 kg/m³
• Density fluid ρ_f = 1000 kg/m³
• Terminal velocity v = 0.02 m/s

η=2r2g(ρs−ρf)9v=2(0.005)2(9.81)(6800)9(0.02)≈18.5 Pa\cdotps\eta = \frac{2 r^2 g (\rho_s – \rho_f)}{9 v} = \frac{2 (0.005)^2 (9.81)(6800)}{9(0.02)} \approx 18.5 \text{ Pa·s}η=9v2r2g(ρs​−ρf​)​=9(0.02)2(0.005)2(9.81)(6800)​≈18.5 Pa\cdotps

Example 2: Capillary Flow

• Radius r = 0.001 m, L = 0.1 m, ΔP = 500 Pa, Q = 1×10⁻⁷ m³/s

η=πr4ΔP8QL≈1.5 Pa\cdotps\eta = \frac{\pi r^4 \Delta P}{8 Q L} \approx 1.5 \text{ Pa·s}η=8QLπr4ΔP​≈1.5 Pa\cdotps

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15. Summary Table

Concept	Formula / Relation	Notes
Shear stress	τ = η du/dy	Newtonian fluids
Dynamic viscosity	η = F / (A du/dy)	N·s/m²
Kinematic viscosity	ν = η / ρ	m²/s
Falling sphere	η = 2 r² g (ρ_s – ρ_f)/9v	Stokes’ law, terminal velocity
Capillary flow	η = π r⁴ ΔP / 8 Q L	Poiseuille law
Reynolds number	Re = V D / ν	Laminar / turbulent flow