Fluid Mech. – II (CE-604 (A)) - Unit 2 Short Notes

UNIT 2: FLUID MECHANICS - II

Short Notes (Based on CE-604 (A) Past Exam Analysis: May 2024 & May 2022)

I. FUNDAMENTAL CONCEPTS & PROPERTIES

1.1 Definition and Classification of Fluids

• Fluid: A substance that deforms continuously under shear stress (no fixed shape).

• Solid vs. Fluid: Solids resist shear with finite deformation; fluids flow indefinitely.

• Continuum Hypothesis: Fluid properties are continuous functions of space and time (ignores molecular scale).

• Compressible vs. Incompressible:

  • Compressible: Density changes significantly with pressure (e.g., gases).

  • Incompressible: Density constant (liquids, low-speed gas flows).

1.2 Fundamental Fluid Properties

• Cohesion: Intermolecular attraction within fluid (causes surface tension).

• Adhesion: Attraction between fluid and solid surface (causes capillarity).

• Pressure difference across curved interface (droplet/bubble):

$$ \Delta P = \frac{2\sigma}{r} \quad \text{(droplet, single interface)} $$

$$ \Delta P = \frac{4\sigma}{r} \quad \text{(soap bubble, two interfaces)} $$

1.3 Viscosity

• Newton’s Law of Viscosity: Shear stress (τ) proportional to velocity gradient.

$$ \tau = \mu \frac{du}{dy} $$

where μ = dynamic viscosity (Pa·s).

• Kinematic Viscosity: ν = μ/ρ (m²/s).

• Importance in Fluid Motion:

  • Governs laminar flow resistance.

  • Causes energy dissipation (viscous heating).

  • Provides damping in oscillatory flows.

• Temperature Effect:

  • Liquids: μ ↓ with T ↑ (weaker intermolecular bonds).

  • Gases: μ ↑ with T ↑ (increased molecular momentum transfer).

[!TIP]

Exam Focus: May 2024 & 2022 asked viscosity definitions, temperature effects, and shear stress calculations in pipes.

II. FLUID STATICS

2.1 Pressure Fundamentals

• Pressure at a Point: Force per unit area, scalar quantity.

• Pascal’s Law: Pressure applied to confined fluid transmits undiminished in all directions.

  • Proof: Consider equilibrium of fluid element → pressure same on all faces.

  • Applications: Hydraulic press, brakes, lifts.

• Pressure Head: Height of liquid column producing pressure p.

$$ h = \frac{p}{\rho g} \quad \Rightarrow \quad p = \rho g h $$

2.2 Manometry

• Simple Manometer: Measures gauge pressure.

$$ p_{\text{gauge}} = (\rho_m - \rho_f) g h $$

where ρ_m = manometric liquid density, ρ_f = fluid density.

• Differential Manometer: Measures pressure difference between two points.

$$ p_1 - p_2 = (\rho_m - \rho_f) g h \quad \text{(if both legs contain same fluid)} $$

• Capillary Error in Manometers:

  • Meniscus curvature causes height error.

  • To limit error < 5%, select tube diameter D such that:

$$ D \geq \frac{4\sigma}{(\rho_m - \rho_f)g \cdot \text{error} \cdot h} $$

> *Example (May 2024)*: Find smallest *D* for water manometer with Δp = 100 N/m², error < 5%.  

2.3 Buoyancy and Flotation

• Archimedes’ Principle: Buoyant force = weight of displaced fluid.

$$ F_B = \rho_{\text{fluid}} g V_{\text{sub}} $$

• Floating Body:

$$ \frac{V_{\text{sub}}}{V_{\text{total}}} = \frac{\rho_{\text{body}}}{\rho_{\text{fluid}}} $$

Example (May 2024): Wooden block floats in water (5 cm above) and glycerin (7.5 cm above). Find block height & density.

III. FLUID KINEMATICS

3.1 Flow Description Methods

3.2 Flow Visualization

• Streamline: Tangent to velocity field at instant t (no flow across).

• Pathline: Actual path traced by a particle over time.

• Streakline: Locus of particles passing through a fixed point.

  • Steady flow: All three coincide.

  • Unsteady flow: They differ.

• Equipotential Lines: Contours of constant velocity potential φ.

• Flow Net: Orthogonal network of streamlines (ψ = const) and equipotentials (φ = const).

3.3 Orthogonality of Streamlines & Equipotential Lines

• Proof:

  For 2D incompressible flow:

$$ u = \frac{\partial \phi}{\partial x} = \frac{\partial \psi}{\partial y}, \quad v = \frac{\partial \phi}{\partial y} = -\frac{\partial \psi}{\partial x} $$

Slope of streamline: $$\displaystyle dy/dx = v/u $$.

Slope of equipotential: $$\displaystyle dy/dx = -u/v $$.

Product = -1 → orthogonal.

3.4 Mathematical Tools

• Velocity Potential (φ): Scalar function where $$\displaystyle \vec{V} = \nabla \phi $$. Exists only for irrotational flows.

• Stream Function (ψ): Scalar function for 2D flows:

$$ u = \frac{\partial \psi}{\partial y}, \quad v = -\frac{\partial \psi}{\partial x} $$

• ψ = constant → streamline.

• Continuity automatically satisfied.

• Cauchy-Riemann Equations (for φ, ψ compatibility):

$$ \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} = 0 \quad \text{(Laplace’s equation)} $$

Example (May 2024): Given φ = x(2y – 1), find velocity at (4,5) and ψ at that point.

Solution: $$\displaystyle u = \partial\phi/\partial x = 2y-1 = 9 $$, $$\displaystyle v = \partial\phi/\partial y = 2x = 8 $$.

Then $$\displaystyle \psi = \int u \, dy = \int (2y-1) dy = y^2 - y + f(x) $$, match $$\displaystyle v = -\partial\psi/\partial x = -f'(x) = 8 $$ → $$\displaystyle f(x) = -8x $$. So ψ = y² – y – 8x. At (4,5): ψ = 25 – 5 – 32 = -12.

3.5 Continuity Equation

• Derivation (3D, Cartesian):

  Mass flow into control volume = rate of mass increase.

$$ \frac{\partial \rho}{\partial t} + \frac{\partial (\rho u)}{\partial x} + \frac{\partial (\rho v)}{\partial y} + \frac{\partial (\rho w)}{\partial z} = 0 $$

• For steady, incompressible flow:

$$ \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0 $$

Example (May 2022): Given u and v, find w such that continuity holds.

IV. FLUID DYNAMICS

4.1 Euler’s Equation of Motion

• Derivation along streamline (force balance on fluid element):

$$ \rho \frac{DV}{Dt} = -\frac{\partial p}{\partial s} + \rho g_s $$

For steady flow: $$\displaystyle \rho V dV = -dp + \rho g \, dz $$.

4.2 Bernoulli’s Theorem

• Derivation: Integrate Euler’s equation along streamline (steady, inviscid, incompressible).

$$ \frac{p}{\rho} + \frac{V^2}{2} + gz = \text{constant} $$

• Assumptions:

  1. Steady flow

  2. Inviscid (μ = 0)

  3. Incompressible (ρ = constant)

  4. Along streamline (or irrotational flow for whole field)

• Energy Interpretation:

  • $ p/\rho $: pressure head (m)

  • $$\displaystyle V^2/2g $$: velocity head (m)

  • $ z $: elevation head (m)

  • Sum = total head (constant).

4.3 Energy Lines in Pipe Flow

• Total Energy Line (TEL) / Energy Gradient Line (EGL):

  Represents total head $$\displaystyle H = z + \frac{p}{\rho g} + \frac{V^2}{2g} $$.

  • Slopes downward due to friction losses.

• Hydraulic Gradient Line (HGL):

  Represents piezometric head $$\displaystyle z + \frac{p}{\rho g} $$.

  • Lies below TEL by $$\displaystyle \frac{V^2}{2g} $$.

• Interpretation:

  • For series pipes: same Q → HGL drop proportional to friction loss.

  • For parallel pipes: same Δp → different HGL slopes.

4.4 Applications of Bernoulli

• Venturi Meter:

$$ Q = A_1 A_2 \sqrt{\frac{2(p_1 - p_2)}{\rho (A_1^2 - A_2^2)}} $$

• Pitot Tube: Measures stagnation pressure → velocity:

$$ V = \sqrt{\frac{2(p_{\text{stag}} - p_{\text{static}})}{\rho}} $$

• Orifice Meter:

$$ Q = C_d A_o \sqrt{\frac{2(p_1 - p_2)}{\rho (1 - \beta^4)}} $$

where β = d/D, C_d = discharge coefficient.

V. VISCOUS FLOW & TURBULENCE

5.1 Laminar Flow Analysis (Hagen-Poiseuille Flow)

• Velocity Distribution (circular pipe, fully developed):

$$ u(r) = u_{\text{max}} \left(1 - \frac{r^2}{R^2}\right) $$

where $$\displaystyle u_{\text{max}} = 2V_{\text{avg}} $$.

• Shear Stress:

  • At wall: $$\displaystyle \tau_w = \mu \left. \frac{du}{dr} \right|_{r=R} = \frac{4\mu V_{\text{avg}}}{R} = \frac{\mu u_{\text{max}}}{R} $$

  • At radius r: $$\displaystyle \tau(r) = \frac{r}{R} \tau_w $$

  Example (May 2024 & 2022): Given μ, D, u_max, find τ_w and τ at specified r.

  Solution: Compute $$\displaystyle u_{\text{max}} $$, then $$\displaystyle \tau_w = \mu u_{\text{max}}/R $$, $$\displaystyle \tau = \tau_w \cdot (r/R) $$.

5.2 Transition to Turbulence

• Reynolds Experiment:

  • Dye filament in glass tube:

    • Low Re: Dye remains distinct (laminar).

    • High Re: Dye mixes (turbulent).

  • Critical Reynolds Number:

    • Pipe flow: $$\displaystyle Re_{\text{crit}} \approx 2300 $$

    • $$\displaystyle Re = \frac{\rho V D}{\mu} $$

5.3 Turbulence

• Definition: Irregular, chaotic flow with random fluctuations.

• Characteristics:

  1. Irregularity (stochastic).

  2. Diffusivity (enhanced mixing).

  3. Vorticity (3D rotational fluctuations).

  4. Energy cascade (large → small eddies → dissipation).

• Effects:

  • Increased momentum/heat/mass transfer.

  • Higher frictional losses.

  • Fluctuating velocity components: $$\displaystyle u = \bar{u} + u' $$.

VI. BOUNDARY LAYER THEORY

6.1 Boundary Layer Concept

• Definition: Thin region near solid surface where velocity changes from 0 (no-slip) to free-stream value.

• Development over Flat Plate:

  • Laminar (Blasius solution) → Transition (Re_x ~ 5×10⁵) → Turbulent (1/7th power law).

• Thickness Definitions:

  • δ: Distance where u = 0.99U_∞.

  • δ* (displacement thickness): $$\displaystyle \delta^* = \int_0^\delta \left(1 - \frac{u}{U_\infty}\right) dy $$

  • θ (momentum thickness): $$\displaystyle \theta = \int_0^\delta \frac{u}{U_\infty} \left(1 - \frac{u}{U_\infty}\right) dy $$

6.2 Boundary Layer Characteristics

• Velocity Profiles:

  • Laminar (Blasius): $$\displaystyle u/U_\infty = f'(\eta) $$, $$\displaystyle \eta = y \sqrt{U_\infty/(\nu x)} $$.

  • Turbulent (1/7th power law): $$\displaystyle u/U_\infty = (y/\delta)^{1/7} $$.

• Shear Stress:

  • Wall shear stress: $$\displaystyle \tau_w = \mu \left. \frac{\partial u}{\partial y} \right|_{y=0} $$.

  • Within layer: τ varies linearly (laminar) or nonlinearly (turbulent).

• Boundary Layer Separation:

  • Cause: Adverse pressure gradient ($$\displaystyle dp/dx > 0 $$) → flow reversal near wall.

  • Consequences: Increased drag (pressure drag), flow losses, stall.

6.3 Drag and Lift

• Skin Friction Drag (flat plate):

$$ D_f = \int_0^L \tau_w(x) \, b \, dx $$

• Laminar: $$\displaystyle D_f = 1.328 \sqrt{\rho \mu U_\infty^3 b L} $$

• Turbulent: $$\displaystyle D_f = 0.074 (\rho U_\infty^2 b L) Re_L^{-1/5} $$

Example (May 2024): Given plate width, length, fluid properties, compute drag and δ at trailing edge.

VII. FLOW MEASUREMENT & TURBOMACHINERY

7.1 Flow Measurement Devices

• Pitot-Static Tube:

  • Stagnation pressure at tip (total pressure).

  • Static pressure on side ports.

  • Velocity: $$\displaystyle V = \sqrt{\frac{2(p_t - p_s)}{\rho}} $$.

  • Sketch: Shows impact tube, static ports, manometer connection.

• Venturi & Orifice Meters:

  • Based on Bernoulli + continuity.

  • Venturi: lower energy loss, higher Cd (~0.98).

  • Orifice: compact, higher loss, Cd ~0.6–0.7.

7.2 Turbomachinery

• Reciprocating Pump:

  • Components: Cylinder, piston, suction/delivery valves.

  • Working Cycle:

    1. Suction stroke: Piston back → suction valve opens → water enters.

    2. Delivery stroke: Piston forward → suction valve closes, delivery valve opens → water discharged.

  • Slip Factor: Actual delivery < theoretical due to valve dynamics and leakage.

• Reaction Turbine (e.g., Francis):

  • Principle: Pressure + kinetic energy change in rotor blades.

  • Classification: Radial, axial, mixed flow.

  • Sketch: Show casing, guide vanes, rotor, draft tube.

VIII. APPLICATIONS & ADVANCED TOPICS

8.1 Biological Fluid Mechanics

• Blood Flow in Arteries:

  • Pulsatile flow (heartbeat-driven).

  • Reynolds number: Typically 500–2000 (laminar in small arteries, transitional/turbulent in aorta).

  • Atherosclerosis: Plaque buildup → narrowed lumen → increased resistance, possible turbulence → risk of thrombosis.

• Heart as a Pump:

  • Generates pressure (~120/80 mmHg) via ventricular contraction.

  • Valves prevent backflow (one-way operation).

  • Cardiac output = stroke volume × heart rate (~5 L/min).

8.2 Dimensional Analysis (Brief)

• Buckingham π Theorem: n variables → (n – k) dimensionless π groups, where k = fundamental dimensions.

• Flow Similitude: Model-prototype similarity requires matching relevant π numbers (e.g., Re, Fr).

[!CAUTION]

Common Pitfalls in Exams:

1. Bernoulli’s Equation: Only applicable along streamline for inviscid flow; do not apply across streamlines in rotational flow.

2. Manometry: Sign convention—higher pressure pushes manometric fluid down on that side.

3. Boundary Layer Thickness: δ ≠ 0.99U_∞ definition only; some texts use δ where u = 0.99U_∞, others where shear stress ≈ 0.

4. Laminar Flow Shear Stress: τ ∝ r (linear), not constant.

5. Pitot Tube: Measures stagnation pressure; static pressure must be measured separately (pitot-static tube).

Key Formulas Boxed:

• Newton’s Law: $$\displaystyle \boxed{\tau = \mu \frac{du}{dy}} $$

• Pressure head: $$\displaystyle \boxed{h = \frac{p}{\rho g}} $$

• Continuity (3D steady incompressible): $$\displaystyle \boxed{\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0} $$

• Bernoulli: $$\displaystyle \boxed{\frac{p}{\rho} + \frac{V^2}{2} + gz = \text{constant}} $$

• Laminar pipe flow τ_w: $$\displaystyle \boxed{\tau_w = \frac{4\mu V_{\text{avg}}}{R}} $$

• Droplet pressure: $$\displaystyle \boxed{\Delta P = \frac{2\sigma}{r}} $$

• Pitot velocity: $$\displaystyle \boxed{V = \sqrt{\frac{2(p_t - p_s)}{\rho}}} $$