FLUID MECHANICS (ME-404) - Unit 2 Short Notes

UNIT 2: FLUID MECHANICS - CORE CONCEPTS & EXAM FOCUS

I. FUNDAMENTAL PROPERTIES OF FLUIDS

1. Ideal vs. Real Fluids

• Ideal Fluid (Perfect Fluid): Incompressible, non-viscous (μ = 0), and no surface tension. Does not exist in reality. Used for theoretical analysis (potential flow).

• Real Fluid: Has viscosity (μ > 0), may be compressible, and exhibits surface tension. All practical fluids are real.

Exam Tip: Questions often ask to distinguish or state assumptions for Bernoulli's/Euler's equation (ideal fluid assumed).

2. Viscosity & Fluid Types

• Newton's Law of Viscosity: Shear stress (τ) is directly proportional to the rate of shear strain (velocity gradient).

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

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

where μ = **Dynamic Viscosity** (N·s/m² or Pa·s), ν = μ/ρ = **Kinematic Viscosity** (m²/s).

• Newtonian Fluid: Follows Newton's law (straight line through origin on τ vs. du/dy graph). e.g., Water, air, most gases, thin oils.

• Non-Newtonian Fluids:

  • Shear Thinning (Pseudoplastic): τ vs. du/dy curve is concave upward (apparent viscosity decreases with increasing shear rate). e.g., Paint, blood, polymer solutions.

  • Shear Thickening (Dilatant): τ vs. du/dy curve is concave downward (apparent viscosity increases with shear rate). e.g., Cornstarch in water, dense suspensions.

  Diagram:

  DiagramSEARCH: "shear stress vs shear rate curve newtonian pseudoplastic dilatant"

3. Other Key Properties

• Vapor Pressure: Pressure exerted by vapor in equilibrium with its liquid at a given temperature. Crucial for cavitation in pumps/propellers.

• Bulk Modulus of Elasticity (K): Measure of fluid's resistance to compression.

$$ K = -V \frac{dP}{dV} \quad \text{or} \quad K = \rho \frac{dP}{d\rho} $$

\boxed{K = -V \frac{dP}{dV}}

*   **Compressibility (β):** β = 1/K.

*   **Numerical Problem:** (Dec 2024) ΔP = 60 N/cm², ΔV/V = -0.15% → Calculate K.

• Surface Tension & Capillarity: Surface tension (σ, N/m) causes capillary rise/fall in narrow tubes. $$\displaystyle h = \frac{4\sigma \cos\theta}{\rho g d} $$ for circular tube.

4. Hydrostatic Paradox

• Statement: The pressure at a point in a static fluid depends only on the depth of fluid above it and its density, not on the total amount, shape, or surface area of the container.

• Implication: The force on the base of a container is not simply (weight of fluid). For a given base area, the force is $$\displaystyle F = P_{base} \times A = \rho g h A $$, which may be greater or less than the actual weight of the fluid.

Exam Tip: Classic question (Jun 2025). Explain with examples of different shaped containers having same base area and height but different fluid weights.

II. FLUID KINEMATICS

1. Flow Classification (Key Distinctions)

Exam Tip: Be prepared to classify given velocity fields (like in Jun 2024: u=3x+y, v=2x-3y).

2. Velocity Field & Material Derivative

• Velocity Field: V(x,y,z,t) = u(i) + v(j) + w(k).

• Material (Substantial) Derivative: Rate of change following a fluid particle.

$$ \frac{D(\cdot)}{Dt} = \frac{\partial(\cdot)}{\partial t} + u\frac{\partial(\cdot)}{\partial x} + v\frac{\partial(\cdot)}{\partial y} + w\frac{\partial(\cdot)}{\partial z} $$

\boxed{\frac{D}{Dt} = \frac{\partial}{\partial t} + (\mathbf{V} \cdot \nabla)}

*   **Local Acceleration:** ∂V/∂t (unsteady effect).

*   **Convective Acceleration:** (V·∇)V (spatial variation).

3. Path, Streak, and Stream Lines

Exam Tip: (Jun 2023) Asked to explain all three. Know the difference: Path = history of one particle; Streak = history at a point; Stream = instantaneous snapshot.

4. Stream Function (ψ) & Velocity Potential (Φ)

• Stream Function (ψ): For 2D incompressible flow.

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

*   **Physical Significance:** Constant ψ lines are **streamlines**. Δψ between two streamlines = **volumetric flow rate per unit width**.

• Velocity Potential (Φ): For irrotational flow.

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

*   **Physical Significance:** Constant Φ lines are **equipotential lines**. ∇Φ = V.

• Cauchy-Riemann Equations (for 2D, Irrotational, Incompressible):

$$ \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \quad \text{and} \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} $$

In terms of ψ and Φ:

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

• Orthogonality of Streamlines & Equipotentials:

  • Slope of streamline (from ψ): $$\displaystyle \frac{dy}{dx} = \frac{u}{v} $$

  • Slope of equipotential (from Φ): $$\displaystyle \frac{dy}{dx} = -\frac{u}{v} $$

  • Product of slopes = -1 → They intersect at 90°.

  Exam Tip: (Dec 2024) Given ψ=2xy, find Φ. Use Cauchy-Riemann: ∂Φ/∂x = ∂ψ/∂y = 2y → Φ = 2xy + f(y); ∂Φ/∂y = 2x + f'(y) = -∂ψ/∂x = -2x → f'(y)= -4x? Contradiction! Check: For ψ=2xy, u=∂ψ/∂y=2y, v=-∂ψ/∂x=-2x. Then ∂u/∂y=2, ∂v/∂x=-2 → ∂u/∂y ≠ ∂v/∂x → Flow is rotational! So Φ does not exist. Common pitfall.

5. Vorticity & Circulation

• Vorticity (ζ): Twice the local rotation. ζ = ∇ × V.

  • 2D (z-component): $$\displaystyle \zeta_z = \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y} $$

  • If ζ = 0 → Irrotational.

• Circulation (Γ): Line integral of velocity around a closed curve C.

$$ \Gamma = \oint_C \mathbf{V} \cdot d\mathbf{s} $$

*   By Stokes' theorem: $$\displaystyle \Gamma = \iint_S (\nabla \times \mathbf{V}) \cdot d\mathbf{S} = \iint_S \zeta \, dS $$

*   For irrotational flow, Γ = 0 around any closed curve.

Exam Tip: (Jun 2025) Asked to explain rotation, vorticity, circulation. Relate: Rotation = ½ ζ; Circulation = area integral of vorticity.

6. Stagnation Point & Properties

• Stagnation Point: Location where fluid velocity V = 0.

• Stagnation Properties: (From Bernoulli along a streamline from a point to stagnation point, assuming adiabatic, no-work, steady, incompressible, inviscid flow)

  • Stagnation Pressure: $$\displaystyle P_0 = P + \frac{1}{2} \rho V^2 $$

  • Stagnation Temperature: $$\displaystyle T_0 = T + \frac{V^2}{2 c_p} $$ (for compressible, calorically perfect gas).

  • Stagnation Density: $$\displaystyle \rho_0 = \rho \left(1 + \frac{\gamma-1}{2} M^2\right)^{1/(\gamma-1)} $$

Exam Tip: (Jun 2025, Nov 2023) Asked for short notes. Stagnation pressure is measured by Pitot tube.

III. FLUID DYNAMICS

1. Euler's Equation of Motion

• Derivation (along a streamline): Apply Newton's 2nd law to a fluid element along a streamline, considering pressure and body forces (like gravity). For steady flow:

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

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

*   **Assumptions:** Steady, inviscid (μ=0), incompressible (ρ=const), along a streamline.

Exam Tip: Be able to derive this from the general momentum equation by neglecting viscous forces.

2. Bernoulli's Theorem

• Statement: For steady, incompressible, inviscid, along-a-streamline flow, the total mechanical energy per unit volume is constant.

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

\boxed{\frac{P}{\rho} + \frac{V^2}{2} + gz = H = \text{constant}}

*   **Terms:** Pressure head (P/ρg), Velocity head (V²/2g), Elevation head (z). Sum = **Total Head (H)**.

• Physical Interpretation: Conservation of mechanical energy (work done by pressure forces = change in kinetic + potential energy).

• Assumptions: Steady, incompressible (ρ=const), inviscid (no friction), along a streamline, no shaft work, no heat transfer.

• Modification for Real Flows (with Head Loss hₗ):

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

\boxed{H_1 = H_2 + h_L}

where hₗ accounts for major (friction) and minor (fittings) losses.

Exam Tip: (Dec 2024, Nov 2023, Jun 2023) Very frequent. State assumptions clearly. Derivation from Euler's equation by integration is key.

3. Navier-Stokes Equations (N-S)

• General Form (for incompressible, Newtonian fluid):

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

\boxed{\rho \frac{D\mathbf{V}}{Dt} = -\nabla P + \mu \nabla^2 \mathbf{V} + \rho \mathbf{g}}

*   **Significance:** Most general governing equations for viscous fluid motion. Includes inertial (left), pressure, viscous, and body force terms.

*   **Scope:** Basis for all CFD. Simplifies to Euler's (μ=0), Stokes (low Re), Laplace (irrotational).

*   **Simplification for Pipe Flow (Laminar, Fully Developed):** Assumptions: steady, incompressible, axisymmetric, ∂/∂θ=0, u=u(r) only, ∂P/∂z = const, ∂u/∂z=0. N-S reduces to:

$$ \frac{1}{r} \frac{d}{dr} \left( r \frac{du}{dr} \right) = \frac{1}{\mu} \frac{dP}{dz} $$

    → Leads to **Hagen-Poiseuille** parabolic profile.

4. Control Volume (CV) vs. System Approach

• System (Lagrangian): Follows a specific, identifiable mass of fluid. Difficult to apply due to changing boundaries.

• Control Volume (Eulerian): Fixed region in space through which fluid flows. More important in engineering because:

  1. Easier to define boundaries (e.g., pipe section, around a blade).

  2. Directly relates to measurements (pressure taps, flow meters).

  3. Forms basis for Reynolds Transport Theorem (RTT):

$$ \frac{dB_{sys}}{dt} = \frac{d}{dt} \int_{CV} b \rho \, dV + \int_{CS} b \rho (\mathbf{V} \cdot \mathbf{n}) \, dA $$

    where B = extensive property, b = B/m (intensive).

*   **Significance:** Allows application of conservation laws (mass, momentum, energy) to a fixed CV, which is how real engineering systems are analyzed.

Exam Tip: (Jun 2024, Jun 2023) Asked why CV is more important. Emphasize fixed boundaries and practical measurement.

IV. PIPE FLOW & FLOW MEASUREMENT

1. Laminar Flow in Circular Pipes (Hagen-Poiseuille)

• Velocity Distribution (Parabolic):

$$ u(r) = \frac{\Delta P}{4\mu L} (R^2 - r^2) = u_{max} \left(1 - \frac{r^2}{R^2}\right) $$

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

• Velocity Relationships:

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

  • Average velocity: $$\displaystyle \bar{u} = \frac{Q}{\pi R^2} = \frac{\Delta P R^2}{8\mu L} $$

  • Ratio: $$\displaystyle u_{max} = 2\bar{u} $$

• Shear Stress Distribution (Linear):

$$ \tau(r) = -\frac{\Delta P}{2L} r \quad \text{(max at wall: } \tau_w = -\frac{\Delta P R}{2L} \text{)} $$

• Pressure Drop & Pumping Power:

$$ \Delta P = \frac{8\mu L \bar{u}}{R^2} = \frac{128 \mu L Q}{\pi D^4} \quad \text{( Hagen-Poiseuille Law)} $$

\boxed{\Delta P = \frac{128 \mu L Q}{\pi D^4}}

Power required: $$\displaystyle P = Q \Delta P $$

Numerical Problem: (Dec 2024) Given D=400mm, u_max=3 m/s → find u_avg, r at u_max, u at 6cm from wall. Use u_avg = u_max/2, u(r) = u_max(1 - r²/R²).

2. Turbulent Flow in Pipes

• Velocity Distribution: Flatter than laminar ("plug-like").

  • 1/7th Power Law (Empirical): $$\displaystyle \frac{u}{u_{max}} = \left(\frac{y}{R}\right)^{1/7} $$ where y = R - r.

  • Log-Law (Theoretical): $$\displaystyle \frac{u}{u_*} = \frac{1}{\kappa} \ln \frac{y}{y_0} $$ (κ≈0.4, von Kármán constant).

• Darcy-Weisbach Equation (Major Losses):

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

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

*   **f = Friction Factor.** Function of Re = ρVD/μ and relative roughness ε/D.

• Moody Chart: Graph of f vs. Re for various ε/D.

• Hazen-Williams Formula (Empirical, for water):

$$ V = 0.849 \, C \, R^{0.63} \, S^{0.54} $$

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

3. Minor Losses in Fittings

• Loss Coefficient (K): $$\displaystyle h_{minor} = K \frac{V^2}{2g} $$

  • Entrance (sharp): K≈0.5; (rounded): K≈0.04-0.15.

  • Exit: K=1.0.

  • Bend, valve, contraction, expansion → specific K values from tables.

• Equivalent Length (Lₑ): A fitting's loss is equivalent to additional pipe length.

$$ K = f \frac{L_e}{D} \quad \Rightarrow \quad L_e = \frac{K D}{f} $$

4. Total Energy Line (TEL) & Hydraulic Gradient Line (HGL)

• Total Head (H): $$\displaystyle H = \frac{P}{\rho g} + \frac{V^2}{2g} + z $$

• TEL: Line representing total head H along the pipe. Slopes downward due to friction losses.

• HGL: Line representing ** piezometric head** $$\displaystyle \frac{P}{\rho g} + z $$. Lies below TEL by velocity head (V²/2g). Slopes downward due to major + minor losses.

Diagram:

DiagramCANVAS: "Sketch of pipe with TEL and HGL, showing drop at fittings"

5. Pipes in Series & Parallel

• Series: Same flow rate Q through all pipes. Total head loss = sum of individual losses.

$$ h_{L,total} = \sum_{i=1}^n \left( f_i \frac{L_i}{D_i} \frac{V_i^2}{2g} + \sum K_i \frac{V_i^2}{2g} \right) $$

Solve for Q using continuity and head loss.

• Parallel: Same head loss across each pipe. Total flow Q = ΣQᵢ.

$$ h_{f1} = h_{f2} = ... = h_{fn} \quad \Rightarrow \quad f_1 \frac{L_1}{D_1} \frac{Q_1^2}{2g A_1^2} = f_2 \frac{L_2}{D_2} \frac{Q_2^2}{2g A_2^2} = ... $$

Solve with Q = ΣQᵢ.

6. Flow Measurement Devices

• Venturi Meter:

  • Derivation (Ideal, no loss): Apply Bernoulli between sections (1) and (2), and continuity (A₁V₁ = A₂V₂).

$$ Q = A_1 A_2 \sqrt{\frac{2(P_1 - P_2)}{\rho (A_1^2 - A_2^2)}} $$

    \boxed{Q_{ideal} = \frac{A_1 A_2}{\sqrt{A_1^2 - A_2^2}} \sqrt{\frac{2(P_1 - P_2)}{\rho}}}

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

• Orifice Meter: Similar but with higher losses → lower C_d. $$\displaystyle Q = C_d A_o \sqrt{\frac{2(P_1 - P_2)}{\rho (1 - \beta^4)}} $$, β = d/D.

• Pitot-Static Tube:

  • Measures stagnation pressure (Pitot tube, facing flow) and static pressure (static ports, on side).

  • Velocity: $$\displaystyle V = \sqrt{\frac{2(P_0 - P_s)}{\rho}} = \sqrt{2g \Delta h} $$ (if connected to manometer with fluid density ρₘ).

  \boxed{V = \sqrt{\frac{2(P_0 - P_s)}{\rho}}}

7. Forces on Pipe Bends & Fittings

• Apply Linear Momentum Equation (LME) to Control Volume:

$$ \sum \mathbf{F} = \frac{d}{dt} \int_{CV} \rho \mathbf{V} \, dV + \int_{CS} \rho \mathbf{V} (\mathbf{V} \cdot \mathbf{n}) \, dA $$

For steady flow: $$\displaystyle \sum \mathbf{F} = \sum_{out} \dot{m} \mathbf{V} - \sum_{in} \dot{m} \mathbf{V} $$

• Procedure:

  1. Draw CV, cut through inlet, outlet, and walls.

  2. Identify forces: Pressure forces on inlet/outlet, reaction force from bend (Rₓ, Rᵧ), weight (if vertical component matters).

  3. Write LME in x and y directions separately.

  4. Solve for Rₓ, Rᵧ. Resultant $$\displaystyle R = \sqrt{R_x^2 + R_y^2} $$, direction $$\displaystyle \theta = \tan^{-1}(R_y/R_x) $$.

Numerical Problem: (Jun 2025) 90° horizontal bend, given D, Q, ρ, P at inlet. Find force on bend. Use LME in x and y.

V. BOUNDARY LAYER THEORY

1. Concept & Development over Flat Plate

• Boundary Layer (BL): Thin region near solid surface where velocity gradients are large and viscous effects are significant. Outside BL, flow is inviscid.

• Development: Laminar → Transition (Reₓ ≈ 5×10⁵) → Turbulent.

• Boundary Layer Thickness (δ): Distance from wall where u ≈ 0.99U (U = free stream velocity).

2. Boundary Layer Separation

• Cause: Adverse Pressure Gradient (dP/dx > 0, pressure increasing in flow direction).

  • In BL, fluid particles decelerate (kinetic energy used to overcome pressure rise).

  • Near wall, low momentum particles cannot overcome APG → reverse flow → separation.

• Effects: Increased drag (pressure drag), flow losses, stall on airfoils.

• Prevention Methods:

  • Streamlining (reduce APG).

  • Suction through porous wall.

  • Vortex Generators (add momentum to near-wall flow).

3. Boundary Layer Thickness Definitions

Numerical Problem: (Nov 2023) Given u/U = 2(y/δ - (y/δ)²), find δ* and δ**. Integrate from 0 to δ.

4. Velocity Profiles & Skin Friction Coefficient (C_f)

• Laminar (Polynomial): $$\displaystyle \frac{u}{U} = 2\left(\frac{y}{\delta}\right) - \left(\frac{y}{\delta}\right)^2 $$ (satisfies u=0 at y=0, u=U at y=δ, du/dy=0 at y=δ).

• Turbulent (Power Law): $$\displaystyle \frac{u}{U} = \left(\frac{y}{\delta}\right)^{1/n} $$, n≈7 (1/7th power law).

• Skin Friction Coefficient:

  • Local: $$\displaystyle C_f(x) = \frac{\tau_w}{\frac{1}{2} \rho U^2} $$

  • Average: $$\displaystyle \bar{C}_f = \frac{1}{L} \int_0^L C_f(x) dx $$

  • From Profile: $$\displaystyle \tau_w = \mu \left. \frac{du}{dy} \right|_{y=0} $$. For polynomial: $$\displaystyle \frac{du}{dy} = \frac{2U}{\delta}\left(1 - \frac{y}{\delta}\right) $$ → at y=0, $$\displaystyle \frac{du}{dy} = \frac{2U}{\delta} $$ → $$\displaystyle \tau_w = \frac{2\mu U}{\delta} $$.

  • For Blasius (exact laminar solution): $$\displaystyle \delta = \frac{5.0x}{\sqrt{Re_x}} $$, $$\displaystyle C_f = \frac{0.664}{\sqrt{Re_x}} $$.

5. von Kármán Momentum Integral Equation

• Statement: For steady, 2D BL, the momentum thickness growth rate is related to wall shear stress.

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

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

*   **Use:** Approximate solution for BL when exact N-S solution is difficult. Assume a velocity profile (e.g., polynomial), compute θ, plug into LHS, solve for τ_w and δ(x).

Exam Tip: (Jun 2023) Asked to evaluate δ and C_f for a given profile. Use momentum integral: 1) Find θ from profile, 2) dθ/dx → τ_w, 3) τ_w = μ(du/dy)|w → find δ in terms of x, 4) C_f = τ_w/(½ρU²).

VI. HYDROSTATICS & BUOYANCY

1. Pressure Distribution in Static Fluids

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

• Variation with Depth: $$\displaystyle P = P_0 + \rho g h $$ (for incompressible fluid, h measured vertically downward from free surface).

2. Hydrostatic Force on Surfaces

• Plane Surface:

  • Total Force (Magnitude): $$\displaystyle F_R = P_c A = \rho g h_c A $$

    where $$\displaystyle h_c $$ = depth to centroid.

  • Center of Pressure (CP): Point of action of F_R. Below centroid.

$$ y_{CP} = \bar{y} + \frac{I_{xx,c}}{A \bar{y}^2} \quad \text{(for vertical plane surface, y measured from surface)} $$

    \boxed{y_{CP} = \bar{y} + \frac{I_{xx,c}}{A \bar{y}}}

    where $$\displaystyle I_{xx,c} $$ = 2nd moment of area about centroidal axis.

*   **Derivation:** From moment equilibrium: $$\displaystyle F_R y_{CP} = \int_A p y \, dA $$.

Numerical Problem: (Nov 2023) Isosceles triangular plate, base at surface. Find F_R and CP. Compute A, ȳ, Iₓₓ,c.

• Curved Surface:

  • Horizontal Force Component (F_H): Equal to force on vertical projection of curved surface.

    $$\displaystyle F_H = \text{Pressure at centroid of vertical projection} \times \text{Area of projection} $$.

  • Vertical Force Component (F_V): Equal to weight of fluid above the curved surface (plus atmospheric pressure effect if applicable).

    $$\displaystyle F_V = \text{Weight of fluid column} \pm \text{Pressure force on projected area} $$.

  • Resultant: $$\displaystyle F_R = \sqrt{F_H^2 + F_V^2} $$, acts at angle $$\displaystyle \theta = \tan^{-1}(F_V/F_H) $$.

3. Stability of Floating Bodies

• Key Points:

  • Center of Buoyancy (B): Centroid of displaced volume. Always on centerline of submerged part.

  • Center of Gravity (G): Centroid of total weight.

  • Metacenter (M): Intersection point of buoyant force line (through B) for slightly heeled position with original vertical centerline.

• Metacentric Height (GM): Distance between G and M.

$$ GM = BM - BG $$

where $$\displaystyle BM = \frac{I}{\nabla} $$ (I = moment of inertia of waterline plane about axis of tilting, ∇ = volume of displaced fluid).

• Conditions of Equilibrium:

  • Stable: M above G (GM > 0). Restoring moment.

  • Unstable: M below G (GM < 0). Overturning moment.

  • Neutral: M coincides with G (GM = 0).

• Period of Oscillation (for small angles):

$$ T = 2\pi \sqrt{\frac{K^2}{g GM}} $$

where K = radius of gyration about horizontal axis through G.

Exam Tip: (Jun 2025) "Why relative position of CG and CB cannot decide stability?" Because CB moves when heeled. Stability depends on relative position of G and M, not G and B. M depends on geometry (I/∇).

4. Buoyancy & Floatation

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

$$ F_B = \rho_{fluid} g \nabla $$

• Condition for Floatation: Weight of body = Buoyant force → $$\displaystyle \rho_{body} V_{total} = \rho_{fluid} V_{sub} $$.

• Metacenter Role: For a floating body, stability is determined by GM, not just whether B is above/below G. A body can have B above G but still be unstable if M is below G (e.g., wide, flat barge with heavy top).

VII. ROTATIONAL FLOWS & VORTICES

1. Forced Vortex Flow

• Definition: Fluid rotates as a rigid body with constant angular velocity ω. (e.g., liquid in a rotating cylindrical tank).

• Velocity: $$\displaystyle V_\theta = \omega r $$, V_r = V_z = 0.

• Pressure Distribution (from Euler or Bernoulli along radius):

$$ \frac{\partial P}{\partial r} = \rho \frac{V_\theta^2}{r} = \rho \omega^2 r $$

Integrate: $$\displaystyle P = P_0 + \frac{1}{2} \rho \omega^2 r^2 $$ (P₀ = pressure at r=0).

• Free Surface: Parabolic profile.

$$ z = z_0 + \frac{\omega^2}{2g} r^2 $$

\boxed{z = z_0 + \frac{\omega^2}{2g} r^2}

*   **Derivation:** (Jun 2025) "Prove curve is parabola." Set P = atmospheric on surface, integrate radial pressure gradient, relate z to P via hydrostatic.

Numerical Problem: (Jun 2024) Rotating cylinder, initial height 45cm, find ω for spill (when surface at rim). Also find pressure at given (r,z).

2. Free Vortex Flow

• Definition: Irrotational flow with circular streamlines. Circulation Γ constant.

$$ V_\theta = \frac{C}{r} \quad \text{(C = constant)} $$

*   **Check Irrotational:** ζ_z = 0.

• Pressure Variation: $$\displaystyle P = P_0 - \frac{1}{2} \rho \frac{C^2}{r^2} $$ (P₀ = pressure at large r or axis).

  • Cyclonic: Low pressure at center (like tornado).

  • Anticyclonic: High pressure at center (like bathtub drain in Southern Hemisphere? Actually, vortex direction depends on hemisphere, but pressure is low at center for free vortex).

VIII. DIMENSIONAL ANALYSIS & COMPRESSIBLE FLOW

1. Important Dimensionless Numbers

• Reynolds Number (Re): $$\displaystyle Re = \frac{\rho V L}{\mu} = \frac{V L}{\nu} $$

  • Significance: Ratio of inertial to viscous forces. Criterion for flow regime:

    • Re < 2000 (pipes): Laminar

    • 2000 < Re < 4000: Transition

    • Re > 4000: Turbulent

• Mach Number (Ma): $$\displaystyle Ma = \frac{V}{c} $$, c = speed of sound.

  • Significance in Compressible Flow:

    • Ma < 0.3: Incompressible assumption OK.

    • 0.3 < Ma < 0.8: Subsonic, compressible effects start.

    • Ma ≈ 1: Sonic, choked flow in nozzles.

    • Ma > 1: Supersonic, shock waves possible.

2. Compressible Flow Basics

• Stagnation Properties (for adiabatic, isentropic flow of ideal gas):

  • Temperature: $$\displaystyle T_0 = T \left(1 + \frac{\gamma-1}{2} M^2\right) $$

  • Pressure: $$\displaystyle P_0 = P \left(1 + \frac{\gamma-1}{2} M^2\right)^{\gamma/(\gamma-1)} $$

  • Density: $$\displaystyle \rho_0 = \rho \left(1 + \frac{\gamma-1}{2} M^2\right)^{1/(\gamma-1)} $$

• Fanno Lines: Model adiabatic flow with friction in a constant area duct.

  • Constant: Stagnation temperature T₀ (adiabatic), mass flow rate ṁ, area A.

  • Changes: Pressure, density, temperature, Mach number along duct.

  • Choking: Maximum possible ṁ occurs at duct exit when Ma=1. If downstream pressure too low, flow chokes at Ma=1 at throat/narrowest point.

• Rayleigh Lines: Model constant area flow with heat addition/subtraction (but no friction).

  • Constant: Mass flow rate ṁ, area A, stagnation pressure P₀ (no friction, but heat addition changes total pressure? Actually, Rayleigh flow assumes no shaft work and constant area, but heat transfer changes stagnation enthalpy. Stagnation pressure is not constant).

  • Used to analyze combustion chambers, heat exchangers.

IX. SPECIAL APPLICATIONS & PROBLEMS

1. Lawn Sprinkler (Angular Momentum Principle)

• Apply Steady Flow Angular Momentum Equation to CV:

$$ \sum M_{axis} = \sum_{out} \dot{m} (r V_\theta) - \sum_{in} \dot{m} (r V_\theta) $$

*   For sprinkler with jets: $$\displaystyle \sum M = \dot{m} r V_\theta $$ (each jet contributes).

*   Torque due to friction at axis: $$\displaystyle T_{fric} = \sum M_{axis} $$ (if steady, no acceleration).

Numerical Problem: (Jun 2025, Jun 2023) Given: arm radius R, jet diameter d, N rpm, Q. Find torque due to friction. Steps: 1) ṁ = ρQ, 2) V_jet = Q/(n * area_jet), 3) r = R, 4) M = ṁ R V_jet (per jet), 5) Total M = n * M, 6) In steady state, friction torque = Total M (since no net angular acceleration).

2. Flow Between Parallel Plates (Couette Flow)

• Setup: Two parallel plates separated by distance 2h. Bottom plate stationary, top plate moving at U.

• Velocity Distribution (No Pressure Gradient, dp/dx=0):

$$ u(y) = \frac{U}{2h} (y + h) \quad \text{(linear, y=0 at mid-plane)} $$

or $$\displaystyle u = \frac{U}{h} y $$ if y from bottom.

• With Pressure Gradient (dp/dx = constant): Superposition of Couette and Poiseuille flow.

$$ u(y) = \frac{1}{2\mu} \frac{dP}{dx} (y^2 - h^2) + \frac{U}{2h}(y + h) $$

*   **Shear Stress:** $$\displaystyle \tau = \mu \frac{du}{dy} = \frac{y}{h} \tau_w $$ (linear).

*   **Force on Plate:** $$\displaystyle F = \tau_w \times \text{Area} $$.

3. Viscous Flow in Rotating Geometries

• Flow Between Rotating Discs (Cone/Plate):

  • Assumptions: Axisymmetric, thin gap, laminar, negligible inertia.

  • Velocity: Primarily tangential, V_θ ≈ ω r (like solid body rotation in gap).

  • Torque & Power: Shear stress $$\displaystyle \tau = \mu r \frac{d\omega}{dr} \approx \mu \frac{\omega r}{h} $$ (for small h).

    Torque on disc: $$\displaystyle T = \int_0^R \tau \cdot r \cdot dA = \int_0^R \mu \frac{\omega r}{h} \cdot r \cdot 2\pi r dr = \frac{\pi \mu \omega}{2h} R^4 $$

    Power: $$\displaystyle P = T \omega $$

  • Conical Bearing: Similar but area element changes with radius.

Numerical Problem: (Jun 2023) Conical thrust bearing, vertex angle 60°, D_max=200mm, h=1mm, ω=600rpm, μ=1 Poise. Calculate power lost. Find area of cone, use formula.

4. Flow Through Nozzles

• Mass Flow Rate (Incompressible): $$\displaystyle \dot{m} = \rho A V = \rho A \sqrt{2 \Delta P / \rho} = A \sqrt{2 \rho \Delta P} $$

• Compressible (Isentropic, from large tank):

$$ \dot{m} = C_d A_t P_0 \sqrt{\frac{\gamma}{R T_0} \left(\frac{2}{\gamma+1}\right)^{(\gamma+1)/(\gamma-1)}} \quad \text{when choked (Ma_t=1)} $$

\boxed{\dot{m}_{choked} = C_d A_t P_0 \sqrt{\frac{\gamma}{R T_0} \left(\frac{2}{\gamma+1}\right)^{(\gamma+1)/(\gamma-1)}}}

*   **Choked Flow:** Occurs when P_exit / P₀ ≤ (2/(γ+1))^{γ/(γ-1)}. Mass flow rate becomes independent of downstream pressure.

5. Sudden Enlargement & Contraction

• Sudden Enlargement (Loss):

$$ h_L = \frac{(V_1 - V_2)^2}{2g} = \left(1 - \frac{A_1}{A_2}\right)^2 \frac{V_1^2}{2g} $$

*   **Loss Coefficient:** $$\displaystyle K = \left(1 - \frac{A_1}{A_2}\right)^2 $$

• Sudden Contraction (Loss):

$$ h_L = K_c \frac{V_2^2}{2g} \quad \text{with } K_c \approx 0.5 \text{ (typical)} $$

*   **Coefficient of Contraction (C_c):** Ratio of jet area at vena contracta to orifice area. C_d = C_c C_v.

Numerical Problem: (Nov 2023) Sudden enlargement from 240mm to 480mm, HGL rise = 10mm → find Q. Use h_L = (1 - A1/A2)² V1²/2g, and Δ(h_f) = HGL rise? Actually, HGL rise means head loss? Clarify: "hydraulic gradient rises by 10 mm" likely means the difference in HGL between sections is 10mm (which equals h_L). So h_L = 0.01 m. Solve for V1, then Q.

6. Kite and Airfoil Mechanics

• Lift & Drag Forces (on kite/airfoil):

$$ L = \frac{1}{2} \rho V^2 A C_L, \quad D = \frac{1}{2} \rho V^2 A C_D $$

*   **Kite Equilibrium:** Resolve forces along string and perpendicular. Tension T has components balancing weight, lift, drag.

$$ T \cos\theta = W + D \sin\theta, \quad T \sin\theta = L - D \cos\theta $$

    (θ = angle of string with horizontal, α = angle of attack).

Numerical Problem: (Nov 2023) Kite weight, area, α=10°, string angle 45°, C_D=0.6, C_L=0.8, ρ_air. Find wind speed V and tension T. Use equilibrium equations.

• Magnus Effect: Lift force on a spinning cylinder/sphere due to circulation. $$\displaystyle L = \rho V \Gamma $$ per unit span (Kutta-Joukowski theorem).

7. Flat Plate Boundary Layer (Specific Calculations)

• Extent of Laminar Region: $$\displaystyle x_{crit} = \frac{Re_{crit} \nu}{U_\infty} $$, with $$\displaystyle Re_{crit} \approx 5 \times 10^5 $$.

• Boundary Layer Thickness (Laminar, Blasius): $$\displaystyle \delta(x) = \frac{5.0 x}{\sqrt{Re_x}} $$

• Wall Shear Stress (Laminar): $$\displaystyle \tau_w = 0.332 \rho U_\infty^2 / \sqrt{Re_x} $$

• At Trailing Edge (x=L): Compute δ(L), τ_w(L).

Numerical Problem: (Jun 2025) Flat plate 1.5m wide, 2m long, U=2 m/s, water properties. Find: i) x_crit, ii) δ at x_crit and at L, iii) τ_w at L. Use formulas above.

X. PRESSURE MEASUREMENT

1. Manometers

• Principle: Balance pressure difference with column of fluid of known density.

$$ P_1 - P_2 = (\rho_{man} - \rho_{fluid}) g h \quad \text{(for inverted U-tube with light fluid)} $$

or $$\displaystyle P_1 - P_2 = \rho_{fluid} g h $$ (for simple U-tube with same fluid).

• Classification by Sensitivity:

  • Simple U-tube: Low sensitivity.

  • Inclined Manometer: Increases sensitivity by factor 1/sinθ (θ = inclination).

  • Micromanometer (e.g., Well-type): Very high sensitivity, uses floating piston or small diameter well.

• Differential Manometer: Measures P₁ - P₂ directly.

• Reading Procedure: Start from one end, add/subtract ρgh terms for each fluid column, moving to other end. Common Pitfall: Forgetting that pressure in a continuous static fluid column is the same at the same horizontal level.

2. Pressure Gauges (Brief)

• Bourdon Tube: Mechanical gauge, measures gauge pressure.

• Mcleod Gauge: For very low pressures, compresses gas and measures height difference (uses Boyle's law).

Final Note for RGPV Exams: Focus on derivations (Bernoulli, Hagen-Poiseuille, Venturi, center of pressure, forced vortex surface), definitions with formulas (viscosity, BL thicknesses, friction factor), and numerical problems applying LME, Bernoulli with losses, pipe networks, and boundary layer integrals. Always state assumptions clearly.