UNIT 1: FLUID MECHANICS - EXAM-FOCUSED SHORT NOTES

I. FUNDAMENTAL CONCEPTS & FLUID PROPERTIES

A. Fluid Classification

[!TIP] Non-Newtonian Classification (Past Paper Favorite)

• Shear Thinning (Pseudoplastic): μ decreases with ↑ shear rate (e.g., Paint, Blood).

• Shear Thickening (Dilatant): μ increases with ↑ shear rate (e.g., Cornstarch-water mix).

• Bingham Plastic: Behaves as solid until τ > yield stress τ₀, then Newtonian (e.g., Toothpaste, Mud).

• Thixotropic: μ decreases with time under constant shear (e.g., Some paints, gels).

• Rheopectic: μ increases with time under constant shear (rare).

Graph: Plot τ vs. (du/dy). Newtonian is straight line through origin. Bingham has intercept τ₀.

B. Key Properties

1. Density (ρ): Mass per unit volume. SI: kg/m³.

2. Specific Weight (γ): Weight per unit volume. γ = ρg. SI: N/m³.

3. Specific Gravity (SG): Ratio of density to water density at 4°C. Dimensionless.

4. Viscosity (μ): Measure of internal friction/resistance to flow.

   • Dynamic Viscosity (μ): τ = μ(du/dy). SI: Pa·s (N·s/m²) or Poise (1 P = 0.1 Pa·s).

   • Kinematic Viscosity (ν): ν = μ/ρ. SI: m²/s or Stokes (1 St = 10⁻⁴ m²/s).

   Newton's Law of Viscosity: τ = μ (du/dy) for simple shear.

5. Bulk Modulus (K): Measure of compressibility. K = -V (dP/dV). SI: Pa.

$$K = \frac{\Delta P}{\Delta V / V}$$

> **High K → Low compressibility.** Water has very high K (~2.2 GPa).

6. Vapor Pressure & Cavitation:

   • Vapor Pressure: Pressure exerted by vapor in equilibrium with liquid at given T.

   • Cavitation: Formation & collapse of vapor bubbles when local P < Vapor Pressure. Causes damage.

7. Surface Tension (σ): Force per unit length acting tangentially on surface. SI: N/m.

   • Capillarity: Rise/fall of liquid in narrow tube due to surface tension & wetting.

$$h = \frac{4\sigma \cos\theta}{\rho g d}$$

(for circular tube)

II. FLUID STATICS (HYDROSTATICS)

A. Pressure Fundamentals

• Pascal's Law: Pressure at a point in static fluid is same in all directions.

• Pressure Variation: For incompressible fluid, P = P₀ + ρgh (h measured vertically downward).

• Manometers: Measure pressure difference.

  • U-tube Manometer: ΔP = (ρₘ - ρ)gh (for lighter fluid in pipe).

  • Inverted V-tube (U-tube with one side enlarged): More sensitive. ΔP = ρgh [1 + (A₁/A₂)].

B. Forces on Submerged Surfaces

1. Plane Surface (Inclined):

   • Total Force:

$$F = P_c A = \rho g \bar{h} A$$

  where $\bar{h}$ = depth to centroid.

- **Center of Pressure (CP):** Point of action of F.

$$y_p = \bar{y} + \frac{I_{xx}}{A \bar{y}}$$

  where $$\displaystyle I_{xx} $$ = 2nd moment of area about centroidal axis parallel to surface.

  > **For vertical rectangular plate:** $$\displaystyle y_p = \bar{y} + \frac{b^2}{12\bar{y}} $$ (b=height).

2. Curved Surface:

   • Horizontal Force (Fₕ): Force on projected vertical area.

   • Vertical Force (Fᵥ): Weight of fluid above curved surface (+ if upward, - if downward).

   • Resultant: R = √(Fₕ² + Fᵥ²), acts at intersection of line of action of Fₕ & Fᵥ.

C. Buoyancy & Stability

• Archimedes' Principle: Buoyant force (F_B) = weight of displaced fluid = ρ_fluid * V_displaced * g.

• Stability of Floating Bodies:

  • Metacentric Height (GM): Distance between Center of Gravity (G) & Metacenter (M).

$$GM = BM - BG$$

  where $$\displaystyle BM = \frac{I}{V} $$ (I = 2nd moment of waterline area about axis, V = displaced volume).

- **Condition:**

    - **Stable:** GM > 0 (M above G). Body returns to equilibrium.

    - **Neutral:** GM = 0.

    - **Unstable:** GM < 0.

> **Hydrostatic Paradox:** Pressure force on base depends only on height of fluid & area, NOT on shape/volume of container. Explained by Pascal's law & vertical force equilibrium.

III. FLUID KINEMATICS

A. Flow Classifications

B. Visualization Tools

C. Mathematical Tools (2D, Incompressible)

1. Stream Function (ψ):

   • Defined by: $$\displaystyle u = \frac{\partial \psi}{\partial y} $$, $$\displaystyle v = -\frac{\partial \psi}{\partial x} $$.

   • Properties:

     • Constant ψ → Streamline.

     • Satisfies continuity automatically.

     • Δψ between streamlines = Discharge per unit depth.

2. Velocity Potential (φ):

   • Defined by: $$\displaystyle u = \frac{\partial \phi}{\partial x} $$, $$\displaystyle v = \frac{\partial \phi}{\partial y} $$.

   • Properties:

     • Constant φ → Equipotential line.

     • Requires irrotational flow: $$\displaystyle \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y} = 0 $$.

     • Satisfies continuity for incompressible flow (Laplace equation: ∇²φ=0).

3. Orthogonality: Streamlines & equipotential lines are perpendicular.

   Proof: Slope of streamline = dy/dx = v/u. Slope of equipotential = dy/dx = -u/v. Product = -1.

D. Vorticity & Rotation

• Vorticity (ζ): ζ = ∇×V. For 2D: ζ = (∂v/∂x - ∂u/∂y) * k.

• Rotation (ω): Half of vorticity. ω = ζ/2.

$$\omega = \frac{1}{2} \left( \frac{\partial v}{\partial x} - \frac{\frac{\partial u}{\partial y}} \right)$$

• Irrotational Flow: ω = 0 → ζ = 0.

[!TIP] Common Problem: Given u(x,y), v(x,y):

1. Check continuity: ∂u/∂x + ∂v/∂y = 0?

2. Check rotation: ω = (∂v/∂x - ∂u/∂y)/2. If ω≠0 → rotational.

3. If irrotational & continuous → find φ by integration.

4. Find ψ by integration (even if rotational, if continuous).

IV. FLUID DYNAMICS - ENERGY PRINCIPLE

A. Euler's Equation of Motion

Derivation for a fluid element along a streamline (for inviscid, steady flow):

$$\frac{dP}{\rho} + g dz + V dV = 0$$

where V = speed.

B. Bernoulli's Theorem (Very High Frequency)

Statement: For steady, incompressible, inviscid (ideal), along a streamline flow, the total energy (per unit weight) is constant.

$$\frac{P}{\rho g} + \frac{V^2}{2g} + z = \text{Constant}$$

Terms: Pressure head (P/ρg), Velocity head (V²/2g), Datum head (z).

Assumptions:

1. Steady flow

2. Incompressible fluid (ρ=const)

3. Inviscid (μ=0, no friction)

4. Flow along a streamline (or irrotational flow, then holds throughout)

Modifications for Real Flow:

$$\frac{P_1}{\rho g} + \frac{V_1^2}{2g} + z_1 = \frac{P_2}{\rho g} + \frac{V_2^2}{2g} + z_2 + h_L$$

where $$\displaystyle h_L $$ = head loss (major + minor).

Stagnation Properties:

• At stagnation point (V=0), Stagnation Pressure $$\displaystyle P_0 = P + \frac{1}{2}\rho V^2 $$ (from Bernoulli).

• Stagnation Temperature $$\displaystyle T_0 = T + \frac{V^2}{2c_p} $$ (for compressible isentropic flow).

C. Energy Lines

1. Total Energy Line (TEL): Represents total head (P/ρg + V²/2g + z). Slopes downward due to $$\displaystyle h_L $$.

2. Hydraulic Grade Line (HGL): Represents piezometric head (P/ρg + z). Lies below TEL by V²/2g.

   Note: HGL can rise in a convergent section (V↑ → P/ρg↓ but V²/2g↑ more? Check Bernoulli).

V. FLOW THROUGH PIPES

A. Losses

1. Major Losses (Friction Loss):

   • Darcy-Weisbach Equation:

$$h_f = f \frac{L}{D} \frac{V^2}{2g}$$

  where f = friction factor (function of Re & ε/D).

- **Hazen-Williams Formula (empirical for water):** 

$$V = 0.85 C R^{0.63} S^{0.54}$$

  where C = Hazen-Williams coefficient, R = hydraulic radius, S = slope (h_f/L).

2. Minor Losses (Local Losses): $$\displaystyle h_m = K \frac{V^2}{2g} $$, where K = loss coefficient.

   • Sudden Enlargement: $$\displaystyle K = (1 - A_1/A_2)^2 $$

   • Sudden Contraction: $K \approx 0.5$ (for sharp-edged)

   • Entrance: $$\displaystyle K = 0.5 $$ (sharp), 0.04-0.08 (rounded)

   • Exit: $$\displaystyle K = 1.0 $$

   • Bends, Valves: Given in tables.

B. Laminar Pipe Flow (Hagen-Poiseuille)

• Velocity Profile (Parabolic):

$$u(r) = \frac{\Delta P}{4\mu L} (R^2 - r^2)$$

• Max Velocity: $$\displaystyle u_{max} = \frac{\Delta P R^2}{4\mu L} $$

• Mean Velocity: $$\displaystyle \bar{u} = \frac{u_{max}}{2} $$

• Discharge:

$$Q = \frac{\pi R^4 \Delta P}{8\mu L}$$

• Friction Factor: $$\displaystyle f = \frac{64}{Re} $$ (for laminar, Re<2000).

C. Flow Measurement - Venturimeter

• Derivation: Apply Bernoulli between sections (1) & (2) & Continuity.

$$\frac{P_1}{\rho} + \frac{V_1^2}{2} = \frac{P_2}{\rho} + \frac{V_2^2}{2} + h_L$$

$$A_1 V_1 = A_2 V_2$$

Ideal discharge: $$\displaystyle Q_{ideal} = A_2 \sqrt{\frac{2(P_1-P_2)/\rho}{1-(A_2/A_1)^2}} $$

• Actual Discharge: $$\displaystyle Q = C_d Q_{ideal} $$, where $$\displaystyle C_d $$ = coefficient of discharge (<1).

VI. BOUNDARY LAYER THEORY

A. Thickness Definitions

B. Analysis with Polynomial Profiles (Past Paper Pattern)

Given: $$\displaystyle \frac{u}{U} = f(y/\delta) $$

1. Check: u(0)=0, u(δ)=U.

2. Displacement Thickness: $$\displaystyle \delta^* = \delta \int_0^1 (1 - f(\eta)) d\eta $$, where η=y/δ.

3. Momentum Thickness: $$\displaystyle \theta = \delta \int_0^1 f(\eta)(1 - f(\eta)) d\eta $$.

4. Skin Friction Coefficient: $$\displaystyle C_f = \frac{\tau_w}{\frac{1}{2}\rho U^2} = \frac{2\mu (\partial u/\partial y)_{y=0}}{\rho U^2} $$.

   For polynomial: $$\displaystyle (\partial u/\partial y)_{y=0} = \frac{U}{\delta} f'(0) $$ → $$\displaystyle C_f = \frac{2\mu U}{\rho U^2 \delta} f'(0) = \frac{2\nu}{U\delta} f'(0) $$.

5. Von Karman Momentum Integral Equation (Concept):

$$\frac{d\theta}{dx} = \frac{\tau_w}{\rho U^2}$$

Used to find δ(x) & τ_w(x) for approximate profiles.

Example for $$\displaystyle \frac{u}{U} = 2(\frac{y}{\delta} - (\frac{y}{\delta})^2) $$:

• f(η) = 2(η - η²), f'(0)=2.

• $$\displaystyle \delta^* = \delta \int_0^1 (1 - 2η + 2η²) dη = \delta [1 - 1 + 2/3] = \frac{2}{3}\delta $$

• $$\displaystyle \theta = \delta \int_0^1 2(η-η²)(1-2η+2η²) dη = \frac{2}{15}\delta $$

• $$\displaystyle C_f = \frac{2\nu}{U\delta} * 2 = \frac{4\nu}{U\delta} $$

C. Boundary Layer Separation

• Definition: Boundary layer reverses flow (u=0, ∂u/∂y<0) → detaches from surface.

• Cause: Adverse Pressure Gradient (dP/dx > 0). Fluid near wall loses kinetic energy, cannot overcome increasing pressure → reverses.

• Prevention Methods:

  1. Streamlining: Avoid abrupt changes, delay APG.

  2. Boundary Layer Suction: Suck low-momentum fluid through wall.

  3. Vortex Generators: Create streamwise vortices to mix high-momentum fluid.

  4. Trip Wires: Promote early transition to turbulent (higher momentum near wall).

VII. FORCE ANALYSIS & MOMENTUM PRINCIPLE

A. Momentum Equation (Control Volume)

For steady flow, 1D:

$$\sum F = \dot{m} (V_{out} - V_{in})$$

where $$\displaystyle \dot{m} = \rho Q $$, F = external forces on CV (pressure + reaction forces), V = velocity vector.

Procedure:

1. Draw CV, cut through surfaces where pressure known.

2. Identify inlet/outlet, compute $\dot{m}$, V vectors.

3. Compute net momentum flux: $$\displaystyle \sum \dot{m} V_{out} - \sum \dot{m} V_{in} $$.

4. Compute forces: Pressure forces on CV boundaries + unknown reaction force (what we solve for).

5. Apply vector equation: $$\displaystyle F_x = \dot{m}(V_{x,out}-V_{x,in}) $$, etc.

B. Applications

1. Force on Pipe Bend (Horizontal):

$$F_x = P_1 A_1 - P_2 A_2 \cos\theta - \dot{m}(V_2 \cos\theta - V_1)$$

$$F_y = -P_2 A_2 \sin\theta - \dot{m} V_2 \sin\theta$$

Resultant $$\displaystyle R = \sqrt{F_x^2 + F_y^2} $$.

2. Force on Nozzle: $$\displaystyle F = \dot{m} V_{exit} + (P_{exit}A_{exit} - P_{amb}A_{exit}) $$ (if exit pressure ≠ ambient).

3. Torque on Rotating Sprinkler/ Turbine:

$$\tau = \dot{m} r (V_{t,out} - V_{t,in})$$

(t = tangential component).

4. Magnus Effect: Lift force on spinning cylinder/ball due to pressure difference from asymmetric boundary layer separation.

VIII. SPECIAL FLOW PHENOMENA

A. Vortex Flow

Proof for Forced Vortex Free Surface Parabola:

From Euler: $$\displaystyle \frac{dP}{\rho} = \omega^2 r dr - g dz $$. At surface P=const → dz = (ω²r/g)dr → integrate → z = (ω²/(2g))r² + const → parabola.

B. Compressible Flow Basics

• Mach Number (Ma): $$\displaystyle Ma = V/a $$, where a = speed of sound.

  • Ma < 1: Subsonic

  • Ma = 1: Sonic (choked flow)

  • Ma > 1: Supersonic

• Fanno Line: Adiabatic, constant area flow with friction. On h-s diagram, it's a curve from subsonic to sonic (Ma→1) as friction increases.

• Rayleigh Line: Constant area flow with heat addition/removal. On h-s diagram, maximum entropy at Ma=1.

C. Flow Around Immersed Bodies

• Streamlined Body: Flow remains attached, low pressure drag, high friction drag (e.g., airfoil at small AoA).

• Bluff Body: Early separation, large wake → high pressure/form drag (e.g., sphere, cylinder).

• Drag Components:

  • Skin Friction Drag: Viscous shear on surface.

  • Pressure Drag (Form Drag): Pressure difference due to separation.

• Airfoil: Lift = C_L (1/2 ρ V² A), Drag = C_D (1/2 ρ V² A). Angle of attack (α) affects C_L & C_D.

IX. MATHEMATICAL FOUNDATIONS

A. Continuity Equation (3D Cartesian)

For any fluid (derived from Reynolds Transport Theorem or CV analysis):

$$\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{V}) = 0$$

For steady, incompressible flow: $$\displaystyle \nabla \cdot \mathbf{V} = 0 $$ → $$\displaystyle \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0 $$.

B. Navier-Stokes Equations (Expression & Significance)

For Newtonian fluid, 3D:

$$\rho \left( \frac{\partial \mathbf{V}}{\partial t} + \mathbf{V} \cdot \nabla \mathbf{V} \right) = -\nabla P + \mu \nabla^2 \mathbf{V} + \rho \mathbf{g}$$

Physical Meaning of Terms (LHS → RHS):

1. Inertia (unsteady + convective acceleration)

2. Pressure force

3. Viscous force (μ∇²V)

4. Body force (gravity) Significance: Fundamental equations of motion for viscous flow. Solve with continuity & boundary conditions to get velocity & pressure fields. Complex; often simplified (e.g., boundary layer, pipe flow).

C. Vector Operators

X. PROBLEM-SOLVING METHODOLOGIES (Quick Reference)

1. Finding ψ & φ from u,v:

   • From u = ∂ψ/∂y → ψ = ∫u dy + f(x). Use v = -∂ψ/∂x to find f(x).

   • For φ: u = ∂φ/∂x → φ = ∫u dx + g(y). Use v = ∂φ/∂y to find g(y).

   • Condition for existence: For ψ, need continuity. For φ, need irrotational (∂v/∂x = ∂u/∂y).

2. Stagnation Point: Solve u(x,y)=0 and v(x,y)=0 simultaneously.

3. Laminar Pipe Flow: Use Hagen-Poiseuille. Given ΔP, find Q, u_max, τ_w = (R/2)(ΔP/L).

4. Boundary Layer with Profile:

   • Compute δ*, θ, C_f as shown in Section VI.B.

   • Use Von Karman: dθ/dx = τ_w/(ρU²) to find δ(x) if needed.

5. Force/Torque via Momentum:

   • Choose CV that cuts through inlet/outlet & unknown force.

   • Sum forces (pressure + reaction) = rate of momentum out - in.

   • For torque: Sum moments about axis = rate of angular momentum out - in.

6. Metacentric Height (Floating Body):

   • Find submerged volume V, depth to centroid $\bar{h}$.

   • $$\displaystyle BM = I / V $$, where I = moment of inertia of waterline plane about axis of tilt.

   • $$\displaystyle GM = BM - BG $$, where BG = distance between B (center of buoyancy = centroid of V) and G (given CG).

[!TIP] Exam Alert: Past papers frequently ask for derivation of Bernoulli's (from Euler), proof of orthogonality (ψ & φ), stagnation point from velocity field, displacement/momentum thickness for parabolic profile, force on bend/sprinkler, metacentric height, and explanation of boundary layer separation with pressure gradient. Practice these numerically and theoretically.