Potential Flow Theory

Module 2

2.1 Flow Kinematics

2.2 Stream Function

2.3 Elemental Flow

2.4 Numerical Exercise

2.1 Flow Kinematics

Basic Fluid Properties

Density (ρ)

Mass per unit volume: ρ = m / V. Units: kg·m⁻³. Water (25°C) ≈ 1000 kg/m³, Air (25°C) ≈ 1.184 kg/m³.

Specific weight & specific gravity

Specific weight γ = ρ g (N/m³).
Specific gravity SG = ρ_fluid / ρ_water (dimensionless).

Viscosity (μ, ν)

Viscosity is internal resistance to shear. Dynamic μ relates shear stress to velocity gradient: τ = μ du/dy. Kinematic ν = μ/ρ. Liquids: μ ↓ with T; gases: μ ↑ with T.

Compressibility & Bulk Modulus

Bulk modulus K = −V dP/dV. Liquids have high K (low compressibility), gases low K. Speed of sound: a = √(K/ρ).

Vapor pressure & cavitation

Vapor pressure is the pressure at which liquid vaporizes. High vapor pressure → easier evaporation; important for cavitation in pumps and propellers.

Surface tension & capillarity

Surface tension σ causes liquids to minimize surface area; capillary rise formula:

h = (2 σ cos θ) / (ρ g r)

θ = contact angle, r = tube radius.

2.2 Stream Function

Fluid Flow

Newtonian Law of Viscosity

The Newtonian law of viscosity states that the shear stress between adjacent fluid layers is directly proportional to the velocity gradient (rate of shear strain). Formula: τ = μ (du/dy)

Non-Newtonian Fluids

Shear-Thinning: Paint, blood, ketchup

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Shear-Thickening: Cornflour mixture (oobleck)

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Bingham Plastic: Toothpaste

Thixotropic: Gels, yogurt

Rheopectic: Gypsum paste

Newtonian vs Non-Newtonian Fluids

Category	Newtonian Fluids	Non-Newtonian Fluids
Nature / Behavior	Viscosity remains constant, independent of shear rate.	Viscosity changes with shear rate or time.
Mathematical Model	τ = μ (du/dy)	τ = k (du/dy)ⁿ or τ = τ₀ + μₚ (du/dy)
Shear Stress vs Shear Rate	Linear relationship	Non-linear relationship
Viscosity Dependence	Depends only on temperature	Depends on shear rate, time, and molecular structure
Flow Curve	Straight line	Curved / non-linear
Examples	Water, air, kerosene, alcohol, mercury	Honey, ketchup, blood, toothpaste, paints, oobleck
Types	— (No types; simple behavior)	Pseudoplastic, Dilatant, Bingham Plastic, Thixotropic, Rheopectic
Applications	Aerodynamics, lubrication, pipelines, hydraulic systems	Food processing, cosmetics, drilling mud, biomedical fluids, paints
Industries Used	Aviation, aerospace, chemical plants, HVAC	FMCG, oil & gas, biomedical, pharmaceuticals, construction
Behavior Under Force	Predictable, easy to model	Complex, may require rheometers for analysis
Temperature Sensitivity	Moderate	High (strong dependence on structure)

Fluid Flow

1. Steady & Unsteady Flow

Steady flow properties do not change with time.

Steady: ∂(property)/∂t = 0

Example (Steady): Water flowing through a constant pipe

Unsteady flow properties vary with time. ∂(property)/∂t ≠ 0
Example (Unsteady): Pump start-up flow

2. Uniform & Non-uniform Flow

Uniform: Velocity same at every point.

∂u/∂x = 0
Example: Long straight pipe flow

Non-uniform: Velocity varies along path.

∂u/∂x ≠ 0
Example: Flow through a nozzle

3. Laminar & Turbulent Flow

Re = (ρUD)/μ
Laminar: Smooth, orderly motion of fluid layers.E.g., Honey or oil flow (Re < 2000)
Turbulent:Chaotic, mixing flow. E.g., Water from a fast-open tap (Re > 4000)

4. Compressible & Incompressible Flow

Compressible: Density changes significantly, dρ/dt = 0
or, ∇·V = 0
Example: High-speed air flow.
Incompressible: Constant density, dρ/dt ≠ 0
Example: Water at low speeds.

5. Rotational & Irrotational Flow

ω = ∇ × V ≠ 0
Rotational: Vortex near drain
∇ × V = 0
Irrotational: Potential flow over airfoils

6. 1D, 2D & 3D Flow

V = V(x) 1D: Simple pipe flow
V = V(x, y) 2D: Flow over a flat plate
V = V(x, y, z) 3D: Turbulent flow in real world