UNIT 3: FLUID MECHANICS - EXAM-FOCUSED SHORT NOTES

I. FLUID PROPERTIES & RHEOLOGY (High Frequency)

A. Fundamental Properties

• Density (ρ): Mass per unit volume. Units: kg/m³.

• Specific Weight (γ): Weight per unit volume. γ = ρg. Units: N/m³.

• Specific Gravity (SG): Ratio of density to a reference (usually water at 4°C). Dimensionless.

• Viscosity (μ): Measure of a fluid's resistance to flow (internal friction).

  • Dynamic Viscosity (μ): Units: Pa·s or N·s/m² (1 Poise = 0.1 Pa·s).

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

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

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

\boxed{\tau = \mu \frac{du}{dy}} (For 1D flow)

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

$$ K = -V \frac{dP}{dV} \approx \frac{\Delta P}{\Delta V / V} $$

\boxed{K = -V \frac{dP}{dV}} (Negative sign indicates volume decreases with increasing pressure).

**Compressibility (β):** β = 1/K.

B. Fluid Behaviour

• Newtonian Fluid: Follows Newton's law of viscosity (τ ∝ du/dy). Straight line through origin on τ vs. du/dy graph. Examples: Water, air, most gases, thin oils.

• Non-Newtonian Fluids:

  • Shear Thinning (Pseudoplastic): Viscosity decreases with increasing shear rate. Examples: Paints, blood, ketchup.

  • Shear Thickening (Dilatant): Viscosity increases with increasing shear rate. Example: Cornstarch in water (oobleck).

  • Bingham Plastic: Behaves as a solid until yield stress is exceeded, then flows like a Newtonian fluid. Example: Toothpaste, mud.

• Ideal Fluid: Inviscid (μ=0) and incompressible. Theoretical concept.

• Real Fluid: Has viscosity (μ > 0).

** [!TIP] Exam Focus:** Be prepared to calculate K from given ΔP and % volume change (Dec 2024 Q). Differentiate fluid types with sketches of τ vs. (du/dy).

II. FLOW KINEMATICS & VISCOUS FLOW (Very High Frequency)

A. Flow Descriptions & Patterns

• Reynolds Number (Re): $$\displaystyle Re = \frac{\rho V D}{\mu} = \frac{VD}{\nu} $$. Ratio of inertial to viscous forces. Governs transition.

B. Mathematical Description of Flow

• Velocity Field: $$\displaystyle \vec{V}(x,y,z,t) = u(x,y,z,t)\hat{i} + v(x,y,z,t)\hat{j} + w(x,y,z,t)\hat{k} $$.

• Acceleration (Material Derivative):

$$ \vec{a} = \frac{D\vec{V}}{Dt} = \frac{\partial \vec{V}}{\partial t} + (\vec{V} \cdot \nabla)\vec{V} $$

\boxed{\frac{D}{Dt} = \frac{\partial}{\partial t} + u\frac{\partial}{\partial x} + v\frac{\partial}{\partial y} + w\frac{\partial}{\partial z}}

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

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

  • Properties: Constant ψ = streamline. Satisfies continuity automatically.

• Velocity Potential Function (Φ): Defined for irrotational flow.

  • $$\displaystyle u = \frac{\partial \phi}{\partial x} $$, $$\displaystyle v = \frac{\partial \phi}{\partial y} $$, $$\displaystyle w = \frac{\partial \phi}{\partial z} $$.

  • Properties: Constant Φ = equipotential line. Laplace equation: $$\displaystyle \nabla^2 \phi = 0 $$.

• Orthogonality of Streamlines & Equipotential Lines:

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

  • Slope of equipotential: $$\displaystyle dy/dx = -u/v $$ (from Cauchy-Riemann conditions for ψ & Φ).

  • Product of slopes = -1. Hence, they intersect at 90°.

  \boxed{\text{Streamlines & Equipotentials are Orthogonal}}

C. Flow Characteristics

• Stagnation Point: Point where velocity is zero ($$\displaystyle \vec{V}=0 $$). Found by solving $$\displaystyle u=0, v=0 $$ simultaneously.

• Rotation (ω) & Vorticity (ζ):

  • Rotation (scalar): $$\displaystyle \omega = \frac{1}{2}\left(\frac{\partial v}{\partial x} - \frac{\partial u}{\partial y}\right) $$ (2D).

  • Vorticity (vector): $$\displaystyle \vec{\zeta} = \nabla \times \vec{V} $$.

  • Irrotational Flow: $$\displaystyle \vec{\zeta} = 0 $$ everywhere.

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

$$ \Gamma = \oint_C \vec{V} \cdot d\vec{l} $$

For irrotational flow, Γ = 0 for any closed curve not enclosing singularities.

D. Viscous Flow Fundamentals

• Couette Flow: Viscous flow between two infinite parallel plates, where one plate moves with constant velocity $U$ and the other is stationary. Velocity profile: Linear, $$\displaystyle u = \frac{U}{h}y $$ (if bottom at y=0, top at y=h moving).

• Flow between Concentric Cylinders (Couette Flow - Rotational): Inner cylinder radius $$\displaystyle R_i $$, outer $$\displaystyle R_o $$, inner rotates at $$\displaystyle \Omega_i $$, outer at $$\displaystyle \Omega_o $$. Velocity: $$\displaystyle u_\theta(r) = Ar + B/r $$.

• Navier-Stokes Equations (N-S): Full equations of motion for a viscous, Newtonian fluid.

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

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

**Use:** Fundamental governing equations. Solve for velocity/pressure fields in viscous flows (requires boundary conditions).

** [!TIP] Exam Focus:** Given ψ, find u,v & Φ (Dec 2024 Q3). Find stagnation point from u,v (Jun 2025 Q3). Prove orthogonality (Dec 2024 Q3). Understand Couette flow setup (Jun 2025 Q2).

III. FLUID DYNAMICS - ENERGY & MOMENTUM (Very High Frequency)

A. Euler & Bernoulli Equations

• Euler's Equation: For inviscid flow along a streamline.

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

\boxed{\frac{dP}{\rho} + V dV + g dz = 0} (Along a streamline)

• Bernoulli's Theorem: For steady, incompressible, inviscid flow along a streamline.

  Statement: Total mechanical energy per unit weight is constant.

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

  Derivation: Integrate Euler's equation.

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

\boxed{\frac{P}{\rho g} + \frac{V^2}{2g} + z = H} (Total Head)

*   $P/\rho g$ = Pressure Head

*   $$\displaystyle V^2/2g $$ = Velocity/Kinetic Head

*   $z$ = Potential/Datum Head

• Modifications for Real Flow:

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

$$\displaystyle H_{pump} $$ = Head added, $$\displaystyle H_{loss} $$ = Major + Minor losses.

• Stagnation Properties:

  • Stagnation Pressure (P₀): Pressure when fluid is brought to rest isentropically.

$$ P_0 = P + \frac{1}{2}\rho V^2 $$

(Incompressible)

*   **Stagnation Temperature (T₀):** Temperature at stagnation point (for compressible flow, includes recovery factor).

B. Momentum Principle & Control Volume Analysis

• Linear Momentum Equation (General Form):

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

For **steady flow**, first term (system momentum) = 0.

$$ \sum \vec{F} = \int_{CS} \rho \vec{V} (\vec{V} \cdot \hat{n}) dA \approx \sum (\rho Q \vec{V})_{out} - \sum (\rho Q \vec{V})_{in} $$

\boxed{\sum F_x = \dot{m}(V_{out,x} - V_{in,x})} (Scalar form for 1D)

**Force on CV:** Includes **reaction force** from solid boundaries + **net pressure force**.

• Application Steps:

  1. Draw CV, show all forces (pressure, reaction, weight if significant).

  2. Apply momentum equation in x, y, z directions separately.

  3. Calculate net momentum flux.

  4. Solve for unknown force (usually reaction force from boundary).

• Torque on Rotating Systems (Sprinkler, Turbine):

$$ T = \dot{m} [r_2 V_{\theta 2} - r_1 V_{\theta 1}] $$

(For radial flow devices, angular momentum equation).

C. Applications of Bernoulli & Momentum

• Venturi Meter:

$$ Q = C_d \frac{A_1 A_2}{\sqrt{A_1^2 - A_2^2}} \sqrt{2g(h_1 - h_2)} $$

\boxed{Q = C_d \frac{A_t}{\sqrt{1 - \beta^4}} \sqrt{2g \Delta h}} ($$\displaystyle \beta = d/D $$, $$\displaystyle A_t $$=throat area)

$$\displaystyle C_d $$ = Coefficient of discharge (~0.98 for Venturi).

• Orifice & Nozzle Meter: Similar but with vena contracta. $$\displaystyle C_d $$ lower (~0.6-0.7). Include contraction coefficient (Cc).

• Pitot-Static Tube:

  • Stagnation pressure at impact port (facing flow).

  • Static pressure at side ports.

  • Velocity: $$\displaystyle V = \sqrt{\frac{2(P_0 - P_s)}{\rho}} = C_d \sqrt{\frac{2 \Delta P}{\rho}} $$

• Airfoil & Magnus Effect:

  • Lift (L): $$\displaystyle L = \rho V \Gamma $$ (Kutta-Joukowski theorem for 2D). Due to circulation Γ.

  • Magnus Effect: Lift on a spinning cylinder/ball due to circulation induced by rotation.

** [!TIP] Exam Focus:** Momentum equation for pipe bend (Jun 2025 Q13), sprinkler torque (Jun 2025 Q11). Derive discharge for Venturi (Dec 2024 Q14). Stagnation properties (Jun 2025 Q10). Bernoulli derivation & assumptions (frequent).

IV. INTERNAL FLOWS - PIPE FLOW (Very High Frequency)

A. Laminar Flow in Pipes (Hagen-Poiseuille Flow)

• Velocity Distribution (Parabolic):

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

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

• Mean Velocity: $$\displaystyle \bar{u} = \frac{1}{\pi R^2} \int_0^R u(r) 2\pi r dr = \frac{u_{max}}{2} $$

  \boxed{\bar{u} = \frac{u_{max}}{2}}

• Wall Shear Stress: $$\displaystyle \tau_w = \mu \left.\frac{du}{dr}\right|_{r=R} = \frac{\Delta P R}{2L} $$

• Hagen-Poiseuille Equation (Discharge):

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

\boxed{Q = \frac{\pi R^4 \Delta P}{8\mu L}} (Major loss for laminar flow)

• Friction Factor (Laminar): $$\displaystyle f = \frac{64}{Re} $$ (Moody Chart).

B. Turbulent Flow in Pipes

• Velocity Profile: Flatter ("plug-like") than laminar. Law of the wall: $$\displaystyle u^+ = \frac{1}{\kappa} \ln y^+ + C $$.

• 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 & relative roughness, from Moody chart).

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

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

$C$ = Hazen-Williams coefficient, $$\displaystyle S = h_f/L $$.

C. Minor Losses

Due to fittings, bends, entrances, exits, expansions, contractions.

$$ h_m = K \frac{V^2}{2g} $$

\boxed{h_m = K \frac{V^2}{2g}} ($K$ = loss coefficient, tabulated).

• Sudden Enlargement: $$\displaystyle K = \left(1 - \frac{A_1}{A_2}\right)^2 $$

• Sudden Contraction: $K \approx 0.5$ (for sharp-edged, depends on area ratio).

• Entrance (from reservoir): $$\displaystyle K = 0.5 $$ (sharp), $$\displaystyle K = 0.04 $$ (well-rounded).

D. Pipe Systems & Network

• Pipes in Series: Same flow rate Q. Total head loss: $$\displaystyle h_{L,total} = \sum h_{f,i} + \sum h_{m,i} $$.

• Pipes in Parallel: Same head loss $$\displaystyle h_L $$. Total discharge: $$\displaystyle Q = Q_1 + Q_2 + ... $$

$$ h_L = f_1 \frac{L_1}{D_1} \frac{V_1^2}{2g} = f_2 \frac{L_2}{D_2} \frac{V_2^2}{2g} = ... $$

• Hydraulic Gradient Line (HGL): Line representing $P/\rho g + z$ (total head minus velocity head).

• Total Energy Line (TEL): Line representing total head $$\displaystyle H = P/\rho g + V^2/2g + z $$.

  • TEL slopes downward in direction of flow due to losses.

  • HGL = TEL - $$\displaystyle V^2/2g $$.

• Inertia Effects Problem (Valve Closure): When valve closes suddenly, pressure rise due to momentum change (water hammer concept). Use momentum eq. on CV between sections A & B with valve open/closed (Jun 2025 Q5).

** [!TIP] Exam Focus:** Derive velocity profile for laminar pipe flow from N-S (Jun 2024 Q11). Hagen-Poiseuille derivation (Dec 2024 Q7). Pipe series/parallel problems (Jun 2023 Q22). HGL & TEL plotting (Jun 2025 Q8). Inertia effect with valve (Jun 2025 Q5).

V. BOUNDARY LAYER THEORY (Very High Frequency)

A. Fundamentals & Development

• Boundary Layer (δ): Thin region near solid surface where velocity changes from 0 (no-slip) to free stream $$\displaystyle U_\infty $$.

• Displacement Thickness (δ):* Distance by which outer streamline is displaced outward due to boundary layer growth.

$$ \delta^* = \int_0^\delta \left(1 - \frac{u}{U_\infty}\right) dy $$

\boxed{\delta^* = \int_0^\delta \left(1 - \frac{u}{U_\infty}\right) dy}

• Momentum Thickness (θ): Represents momentum deficit due to boundary layer.

$$ \theta = \int_0^\delta \frac{u}{U_\infty} \left(1 - \frac{u}{U_\infty}\right) dy $$

\boxed{\theta = \int_0^\delta \frac{u}{U_\infty} \left(1 - \frac{u}{U_\infty}\right) dy}

• Energy Thickness (δ):** $$\displaystyle \delta^{**} = \int_0^\delta \frac{u}{U_\infty} \left(1 - \frac{u^2}{U_\infty^2}\right) dy $$

B. Boundary Layer Characteristics

• Laminar BL: Smooth, thin, velocity profile parabolic-like. $\delta \propto \sqrt{x}$.

• Turbulent BL: Thicker, flatter profile, high mixing. $$\displaystyle \delta \propto x^{4/5} $$.

• Boundary Layer Separation: Occurs when adverse pressure gradient ($$\displaystyle dP/dx > 0 $$) causes flow reversal near wall. Causes: Sharp corners, high angle of attack, diffusers.

• Effect of Pressure Gradient:

  • Favorable ($$\displaystyle dP/dx < 0 $$): Accelerates flow, delays separation.

  • Adverse ($$\displaystyle dP/dx > 0 $$): Decelerates flow, promotes separation.

• Methods to Prevent/Delay Separation:

  1. Streamlining (reduce adverse gradient).

  2. Boundary layer suction (remove low-momentum fluid).

  3. Boundary layer blowing (add high-momentum fluid).

  4. Vortex generators (induce mixing).

C. Integral Methods & Coefficients

• Von Kármán Momentum Integral Equation (for flat plate, zero pressure gradient):

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

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

**Use:** Given assumed velocity profile $$\displaystyle u/U_\infty = f(y/\delta) $$, find $\delta(x)$, $$\displaystyle \tau_w $$, $$\displaystyle C_f $$.

• Skin Friction Coefficient (Cf):

  • Local: $$\displaystyle C_f = \frac{\tau_w}{\frac{1}{2}\rho U_\infty^2} $$

  • Average: $$\displaystyle C_{f,avg} = \frac{1}{L} \int_0^L C_f dx = \frac{2\theta}{L} $$ (for flat plate).

• Drag & Lift:

  • Friction Drag: Due to shear stress (viscous). Dominant in streamlined bodies.

  • Pressure Drag (Form Drag): Due to pressure distribution (separation). Dominant in bluff bodies.

• Flow over Flat Plate:

  • Laminar extent: $$\displaystyle x_{cr} = \frac{Re_{cr} \nu}{U_\infty} $$ (with $$\displaystyle Re_{cr} \approx 5 \times 10^5 $$).

  • Total Drag: Sum of laminar ($$\displaystyle x < x_{cr} $$) and turbulent ($$\displaystyle x > x_{cr} $$) contributions.

** [!TIP] Exam Focus:** Calculate δ, θ, Cf* for given $u/U$ profile (Jun 2024 Q12, Q13; Nov 2023 Q11). Common profiles: $$\displaystyle u/U = y/\delta $$, $$\displaystyle u/U = 2(y/\delta - (y/\delta)^2) $$, $$\displaystyle u/U = (y/\delta)^{1/7} $$ (turbulent). Explain separation & pressure gradient effect (Jun 2025 Q9, Nov 2023 Q10).

VI. HYDROSTATICS & BUOYANCY (High Frequency)

A. Pressure Distribution

• Hydrostatic Equation: $$\displaystyle dP = -\rho g dz $$ (for incompressible fluid, z upward).

$$ P = P_0 + \rho g h $$

($h$ = depth below free surface).

\boxed{P = P_{atm} + \rho g h} (Gauge pressure = ρgh)

• Pressure on Surfaces:

  • Horizontal Plane: $$\displaystyle P = \rho g h $$ (constant). Force $$\displaystyle F = P \cdot A $$.

  • Vertical Plane: $$\displaystyle P = \rho g y $$ (varies linearly with depth y). Force $$\displaystyle F = \rho g \bar{y} A $$, where $\bar{y}$ = centroid depth.

  • Inclined Plane: $$\displaystyle F = \rho g \bar{y} A $$, where $\bar{y}$ = depth to centroid. Force acts through Center of Pressure (CP).

• Hydrostatic Paradox: Pressure at a point depends only on depth of fluid above it, not on total volume or shape of container. Example: Different shaped vessels at same base area & liquid height exert same force.

B. Forces on Submerged Surfaces

• Total Hydrostatic Force (Plane Surface):

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

($$\displaystyle P_c $$ = pressure at centroid, $\bar{h}$ = depth to centroid).

• Center of Pressure (CP): Point of application of resultant force.

$$ y_{cp} = \bar{y} + \frac{I_{xx,c}}{A \bar{y}^2} $$

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

$$\displaystyle I_{xx,c} $$ = 2nd moment of area about centroidal axis (parallel to free surface).

*   CP is **always below** centroid ($$\displaystyle y_{cp} > \bar{y} $$).

• Curved Surfaces: Resolve into horizontal (projected area) and vertical (weight of fluid above) components.

  • $$\displaystyle F_H = \text{Force on vertical projection} $$.

  • $$\displaystyle F_V = \text{Weight of fluid above curved surface} + \text{force on bottom projection (if any)} $$.

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

C. Buoyancy & Stability

• Archimedes' Principle: Buoyant force ($$\displaystyle F_B $$) equals weight of displaced fluid.

$$ F_B = \rho_{fluid} g V_{displaced} $$

*   **Center of Buoyancy (CB):** Centroid of displaced volume.

• Stability of Floating Bodies:

  • Metacenter (M): Point about which body starts to oscillate when tilted.

  • Metacentric Height (GM): Distance between Center of Gravity (CG) and Metacenter (M).

$$ GM = BM - BG $$

    $$\displaystyle BM = \frac{I}{V_{displaced}} $$ (For small heel angle, I = 2nd moment of waterline plane area about axis of tilt).

*   **Condition for Stability:**

    *   **Stable:** $$\displaystyle GM > 0 $$ (M above CG). Righting moment develops.

    *   **Neutral:** $$\displaystyle GM = 0 $$ (M coincides with CG). No righting moment.

    *   **Unstable:** $$\displaystyle GM < 0 $$ (M below CG). Overturning moment.

*   **Why CG & CB alone insufficient?** For a floating body, CB moves as body heels. Stability depends on the **relative movement of CG and the new CB position**, which is determined by the location of M.

*   **Period of Oscillation (T):** For small angles, $T \propto 1/\sqrt{GM}$.

** [!TIP] Exam Focus:** CP calculation for triangular/rectangular plate (Nov 2023 Q14, Jun 2022 Q18). Metacentric height for cylinder (Jun 2024 Q4). Explain hydrostatic paradox (Jun 2025 Q6). Buoyancy problems (Nov 2023 Q13: body at mercury-water interface).

VII. FLOW MEASUREMENT & APPLICATIONS (Medium Frequency)

A. Flow Measurement Devices

• Venturi Meter: See Section III.C.

• Orifice Meter: $$\displaystyle Q = C_d A_o \sqrt{\frac{2(P_1 - P_2)}{\rho (1 - \beta^4)}} $$ (with $$\displaystyle \beta = d/D $$). $$\displaystyle C_d $$ lower than Venturi due to vena contracta & higher losses.

• Nozzle Meter: Intermediate between Venturi & orifice. $$\displaystyle C_d \approx 0.98 $$.

• Pitot Tube: $$\displaystyle V = \sqrt{2(P_0 - P_s)/\rho} $$. Measures velocity.

• Weirs:

  • Rectangular Weir: $$\displaystyle Q = C_d \frac{2}{3} L \sqrt{2g} H^{3/2} $$

  • V-Notch (Triangular) Weir: $$\displaystyle Q = C_d \frac{8}{15} \tan(\theta/2) \sqrt{2g} H^{5/2} $$

B. Specific Flow Problems

• Forced Vortex Flow: Fluid rotates as a rigid body ($$\displaystyle r\omega = \text{constant}, \omega = \text{const} $$).

  • Free Surface: Paraboloid of revolution.

  • Proof: From Euler equation radial component: $$\displaystyle \frac{\partial P}{\partial r} = \rho \omega^2 r $$. Integrate: $$\displaystyle P = \frac{1}{2}\rho \omega^2 r^2 + f(z) $$. Use $$\displaystyle P=P_{atm} $$ at free surface $$\displaystyle z = z_0 + \frac{\omega^2 r^2}{2g} $$.

  \boxed{z = z_0 + \frac{\omega^2 r^2}{2g}} (Parabolic free surface)

  • Pressure Distribution: $$\displaystyle P = P_0 + \frac{1}{2}\rho \omega^2 (r^2 - r_0^2) - \rho g (z - z_0) $$.

• Free Vortex Flow: $$\displaystyle r v_\theta = \text{constant} $$ (circulation constant). $$\displaystyle \omega \propto 1/r^2 $$. Pressure: $$\displaystyle P = \frac{\rho \Gamma^2}{2\pi^2 r^2} + f(z) $$.

• Flow through Rotating Container: Combination of forced vortex (due to rotation) and free surface effect. Pressure: $$\displaystyle P = P_{atm} + \frac{1}{2}\rho \omega^2 r^2 - \rho g z $$ (if container open at top). Spill condition: When free surface touches top at outer radius.

• Lawn Sprinkler / Reaction Turbine: Use angular momentum equation to find torque. Torque on axis = rate of change of angular momentum of fluid.

** [!TIP] Exam Focus:** Prove parabola for forced vortex (Jun 2025 Q7). Sprinkler torque (Jun 2025 Q11). Pressure in rotating container (Jun 2024 Q2).

VIII. ADVANCED TOPICS & COMPRESSIBLE FLOW (Low-Medium Frequency)

A. Compressible Flow Fundamentals

• Mach Number (M): $$\displaystyle M = V/a $$, where $$\displaystyle a = \sqrt{\gamma RT} $$ = speed of sound.

  • Subsonic: M < 1

  • Transonic: M ≈ 1

  • Supersonic: M > 1

  • Hypersonic: M >> 1

• Stagnation Properties (Isentropic):

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

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

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

• Fanno Line: Represents adiabatic flow in a constant-area duct with friction. Constant: Stagnation temperature $$\displaystyle T_0 $$. Variables: M, P/P0, ρ/ρ0, etc. Maximum entropy at M=1 (choked flow).

• Rayleigh Line: Represents flow in a constant-area duct with heat transfer but no friction. Constant: Stagnation pressure $$\displaystyle P_0 $$. Maximum temperature & entropy at M=1.

B. Flow in Nozzles

• Mass Flow Rate: $$\displaystyle \dot{m} = \rho A V = \frac{A P_0}{\sqrt{T_0}} \sqrt{\frac{\gamma}{R}} M \left(1 + \frac{\gamma-1}{2}M^2 \right)^{-(\gamma+1)/(2(\gamma-1))} $$

• Choked Flow: Occurs at throat when $$\displaystyle M=1 $$. Maximum mass flow rate for given $$\displaystyle P_0, T_0 $$.

$$ \dot{m}_{max} = A^* P_0 \sqrt{\frac{\gamma}{R T_0}} \left(\frac{2}{\gamma+1}\right)^{(\gamma+1)/(2(\gamma-1))} $$

$$\displaystyle A^* $$ = throat area.

C. Dimensional Analysis

• Buckingham Pi Theorem: If n variables, r fundamental dimensions, then (n-r) independent dimensionless Π groups.

• Common Groups:

  • Reynolds (Re) = ρVL/μ (Inertia/Viscous)

  • Froude (Fr) = V/√(gL) (Inertia/Gravity)

  • Euler (Eu) = ΔP/(ρV²) (Pressure/Inertia)

  • Mach (M) = V/a (Inertia/Elastic)

  • Weber (We) = ρV²L/σ (Inertia/Surface tension)

** [!TIP] Exam Focus:** Mach number significance (Nov 2023 Q8). Choked flow condition (Jun 2023 Q19). Sketch Fanno & Rayleigh lines (Jun 2022 Q12).

IX. MISCELLANEOUS & REVISION TOPICS

A. Manometers

• Types: Simple U-tube, Differential U-tube, Inverted U-tube, Micromanometer.

• Sensitivity: Ability to detect small pressure differences. Inclined manometer increases sensitivity (longer column for same Δh).

• Reading: $$\displaystyle \Delta P = (\rho_{mano} - \rho_{fluid}) g h $$ (for lighter manometric fluid).

B. Model Analysis & Similitude

• Geometric Similarity: All linear dimensions in same ratio (scale ratio λ).

• Kinematic Similarity: Similarity of motion (same velocity ratios, time ratios).

• Dynamic Similarity: Similarity of forces (same force ratios). Requires Reynolds number equality for viscous flows, Froude number for gravity-dominated flows.

• Model Laws: Reynolds model law, Froude model law, etc.

C. Key Derivations & Proofs (Must Practice)

1. 3D Continuity Equation (Cartesian): $$\displaystyle \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0 $$ (from conservation of mass for infinitesimal CV).

2. Stream Function from Velocity (2D): $$\displaystyle \psi = \int u dy + f(x) = -\int v dx + g(y) $$.

3. Velocity Potential from Stream Function (2D, irrotational): $$\displaystyle \phi = \int \frac{\partial \psi}{\partial y} dx = -\int \frac{\partial \psi}{\partial x} dy $$.

4. Laminar Pipe Flow Velocity Distribution: Solve $$\displaystyle \frac{1}{r}\frac{d}{dr}(r \tau) = \frac{dP}{dx} $$ with $$\displaystyle \tau = \mu du/dr $$, $$\displaystyle u(R)=0 $$.

5. Displacement & Momentum Thickness: Direct integration of given $u/U$ profile.

6. Forced Vortex Free Surface Parabola: As shown in Section VII.B.

7. Orthogonality of ψ & Φ: As shown in Section II.B.

D. Distinctions (Frequently Asked)

** [!TIP] Exam Focus:** Practice all derivations listed in IX.C—they are repeated verbatim in exams (Jun 2024 Q11, Dec 2024 Q3, Jun 2023 Q17). Be ready to distinguish terms concisely (Jun 2024 Q5, Q6, Q7, Q16).