FLUID MECHANICS (ME-404) Unit 5 Notes - RGPV

I. FLUID PROPERTIES AND CHARACTERISTICS

A. Newtonian vs. Non-Newtonian Fluids

Newton's Law of Viscosity: Shear stress $ \tau $ is proportional to the rate of strain (velocity gradient):

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

where $ \mu $ = dynamic viscosity (constant for Newtonian fluids).

Newtonian Fluids: Linear $ \tau $ vs. $ du/dy $ relationship; $ \mu $ independent of strain rate.

Examples: Water, air, most gases, low-molecular-weight liquids.
Non-Newtonian Fluids: $ \mu $ varies with strain rate.
- Shear-thinning (Pseudoplastic): $ \mu $ decreases with increasing $ du/dy $.
  
  Examples: Paints, blood, polymer solutions.
  
  DiagramSEARCH: shear stress vs shear rate pseudoplastic curve
- Shear-thickening (Dilatant): $ \mu $ increases with increasing $ du/dy $.
  
  Examples: Cornstarch-water mixture, some suspensions.
  
  DiagramSEARCH: shear stress vs shear rate dilatant curve
Ideal Fluid: Inviscid ($ \mu = 0 $)—used for theoretical analysis.

B. Viscosity

Dynamic Viscosity ($ \mu $): Measure of internal friction.

Units: N·s/m² (SI), Poise (P) where 1 P = 0.1 N·s/m².
Kinematic Viscosity ($ \nu $): $ \nu = \mu / \rho $.

Units: m²/s (SI), Stokes (St) where 1 St = 10⁻⁴ m²/s.
Temperature Effect: For liquids, $ \mu $ decreases with temperature; for gases, $ \mu $ increases with temperature.

C. Other Thermodynamic Properties

Vapor Pressure: Pressure exerted by vapor in equilibrium with its liquid at a given temperature.

Significance: Causes cavitation in pumps and turbines when local pressure falls below vapor pressure.
Bulk Modulus of Elasticity ($ K $): Measure of fluid compressibility:

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

Units: Pa (N/m²). For water, $ K \approx 2.2 \times 10^9 $ Pa (nearly incompressible).

[!TIP]

Common Problem: Given initial/final pressure and density change, compute average $ K $:

$$ K \approx \frac{\Delta P}{\Delta \rho / \rho} $$

Example (Dec 2024): $ P_1 = 70 \, \text{N/cm}^2 $, $ P_2 = 130 \, \text{N/cm}^2 $, volume decrease 0.15% → $ \Delta \rho / \rho = 0.0015 $, $ \Delta P = 60 \times 10^4 \, \text{N/m}^2 $ → $ K = \boxed{4 \times 10^9 \, \text{N/m}^2} $.

Density ($ \rho $): Mass per unit volume. Specific Gravity: $ SG = \rho / \rho_{\text{water}} $ (dimensionless).

II. FLUID KINEMATICS

A. Flow Classification

Steady vs. Unsteady: $ \partial / \partial t = 0 $ or not.
Uniform vs. Non-uniform: No spatial change in velocity (magnitude/direction) or not.
Laminar vs. Turbulent: Determined by Reynolds number $ Re = \rho V D / \mu $. Laminar: $ Re < 2000 $ (pipe flow), smooth paths; turbulent: $ Re > 4000 $, chaotic mixing.
Compressible vs. Incompressible: Density constant or variable (significant when $ M > 0.3 $).
Rotational vs. Irrotational: Vorticity $ \zeta = \frac{1}{2} (\nabla \times \mathbf{V}) \neq 0 $ or $ = 0 $.

B. Mathematical Tools: Divergence and Curl

Divergence ($ \nabla \cdot \mathbf{V} $): Volumetric strain rate. For incompressible flow: $ \nabla \cdot \mathbf{V} = 0 $ (continuity).
Curl ($ \nabla \times \mathbf{V} $): Vorticity $ (2\zeta) $. Non-zero indicates rotation.
Circulation ($ \Gamma $): $ \Gamma = \oint \mathbf{V} \cdot d\mathbf{l} $. By Stokes' theorem, $ \Gamma = \iint (\nabla \times \mathbf{V}) \cdot d\mathbf{S} $.

C. Pathlines, Streaklines, and Streamlines

Streamline: Curve tangent to velocity vector at an instant: $ \frac{dx}{u} = \frac{dy}{v} = \frac{dz}{w} $.
Pathline: Actual path of a fluid particle over time.
Streakline: Locus of particles passing through a fixed point.
Steady Flow: All three coincide. Unsteady Flow: They differ.
Stream Tube: Bundle of streamlines; no flow crosses its boundary.

D. Velocity Potential ($ \phi $) and Stream Function ($ \psi $)

Velocity Potential ($ \phi $): Defined for irrotational flows: $ \mathbf{V} = \nabla \phi $. Satisfies Laplace's equation $ \nabla^2 \phi = 0 $.
Stream Function ($ \psi $): For incompressible 2D flows:

$ u = \frac{\partial \psi}{\partial y}, \; v = -\frac{\partial \psi}{\partial x} $. Automatically satisfies continuity $ \nabla \cdot \mathbf{V} = 0 $.
Cauchy-Riemann Equations: For $ \phi $ and $ \psi $ to coexist (irrotational + incompressible):

$ \frac{\partial \phi}{\partial x} = \frac{\partial \psi}{\partial y}, \; \frac{\partial \phi}{\partial y} = -\frac{\partial \psi}{\partial x} $.
Orthogonality: $ \nabla \phi \cdot \nabla \psi = 0 $ → streamlines ($ \psi = \text{const} $) and equipotentials ($ \phi = \text{const} $) intersect at 90°.
Given $ \psi $, find $ \phi $:
- Example 1: $ \psi = 2xy $
  
  $ u = \frac{\partial \psi}{\partial y} = 2x, \; v = -\frac{\partial \psi}{\partial x} = -2y $
  
  Integrate: $ \frac{\partial \phi}{\partial x} = 2x \Rightarrow \phi = x^2 + f(y) $; $ \frac{\partial \phi}{\partial y} = -2y \Rightarrow f'(y) = -2y \Rightarrow f(y) = -y^2 $
  
  $ \therefore \phi = x^2 - y^2 $.
- Example 2: $ \psi = x^2 - y^2 $
  
  $ u = 2y, \; v = -2x $ → $ \phi = -2xy $.
Elementary Flows:
- Uniform flow: $ \phi = Ux, \; \psi = Uy $.
- Source/sink: $ \phi = \frac{m}{2\pi} \ln r, \; \psi = \frac{m}{2\pi} \theta $.
- Doublet: $ \phi = -\frac{\kappa \cos \theta}{r}, \; \psi = -\frac{\kappa \sin \theta}{r} $.
- Flow around cylinder: Doublet + uniform flow.

E. Stagnation Points and Stagnation Properties

Stagnation Point: Where velocity $ \mathbf{V} = 0 $. Found by solving $ u = 0, v = 0 $ from velocity field.

Example (Jun 2025): $ u = x + 2y + 2, \; v = 2x - y - 3.5 $ → solve: $ x + 2y = -2, \; 2x - y = 3.5 $ → $ (x,y) = (0.5, -1.25) $.
Stagnation Pressure (incompressible, inviscid):

$$ p_0 = p + \frac{1}{2} \rho V^2 $$

Measured by Pitot tube.

Stagnation Temperature (compressible):

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

III. FLUID DYNAMICS

A. Euler’s Equation

Derivation along a streamline (inviscid, steady):

Consider a fluid element along streamline:

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

Assumptions: Inviscid, steady, along streamline.

Application: Basis for Bernoulli’s theorem.

B. Bernoulli’s Theorem

Statement: For steady, incompressible, inviscid flow along a streamline:

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

Assumptions: Inviscid, steady, incompressible, along streamline, no shaft work.
Derivation: Integrate Euler’s equation along streamline.
Modification for Real Flows: Include head loss $ h_L $:

$$ \frac{p_1}{\rho g} + \frac{V_1^2}{2g} + z_1 = \frac{p_2}{\rho g} + \frac{V_2^2}{2g} + z_2 + h_L $$

Applications:
- Venturi Meter:

$$ Q = C_d \frac{A_1 A_2}{\sqrt{A_1^2 - A_2^2}} \sqrt{2g \Delta h} $$

where \( \Delta h \) = manometer reading.

Orifice Meter:

$$ Q = C_d A_o \sqrt{2g \Delta h} $$

\( C_d \) includes contraction (\( C_c \)) and velocity (\( C_v \)) coefficients.

Pitot Tube:

$$ V = \sqrt{\frac{2(p_0 - p)}{\rho}} $$

\( p_0 \) = stagnation pressure, \( p \) = static pressure.

[!TIP]

Common Pitfall: In Venturi/orifice, ensure $ \Delta h $ is converted to pressure difference correctly if manometric fluid differs.

C. Momentum Equation and Applications

Control Volume Formulation (steady flow):

$$ \sum \mathbf{F} = \dot{m} (\mathbf{V}_{\text{out}} - \mathbf{V}_{\text{in}}) $$

where $ \sum \mathbf{F} $ includes pressure forces, weight, reaction forces.

Applications:
- Force on Pipe Bend: Compute axial and radial components.
- Force on Nozzle: $ F_x = \dot{m} V_{\text{exit}} + (p_{\text{exit}} - p_{\text{atm}}) A_{\text{exit}} - p_{\text{inlet}} A_{\text{inlet}} $.
- Sudden Enlargement: Force on fluid = $ \dot{m}(V_1 - V_2) $, reaction force on pipe.
- Torque Calculation (e.g., lawn sprinkler):

$$ T = \dot{m} r (V_{\theta,\text{out}} - V_{\theta,\text{in}}) $$

For friction torque equilibrium, net torque = 0.

IV. LAMINAR FLOW EXACT SOLUTIONS

A. Flow Between Parallel Plates

Couette Flow (moving plate, no pressure gradient):

Velocity profile: $ u = \frac{\tau_w}{\mu} y $ (linear), where $ y $ from stationary plate.

Flow rate per unit width: $ q = \frac{1}{2} \frac{\tau_w}{\mu} h^2 $.
Poiseuille Flow (pressure-driven, both plates stationary):

Parabolic profile: $ u(y) = \frac{1}{2\mu} \left( -\frac{dp}{dx} \right) (y^2 - hy) $ for plates at $ y=0, h $.

Flow rate: $ q = -\frac{h^3}{12\mu} \frac{dp}{dx} $.

B. Hagen–Poiseuille Flow (Circular Pipe)

Derivation: From Navier-Stokes in cylindrical coordinates (axisymmetric, fully developed).
Velocity Distribution:

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

where $ \Delta p = p_{\text{in}} - p_{\text{out}} $, $ L $ = length.

Maximum Velocity: $ u_{\text{max}} = \frac{\Delta p R^2}{4\mu L} $ at $ r=0 $.
Mean Velocity:

$$ U_{\text{avg}} = \frac{1}{\pi R^2} \int_0^R u(r) 2\pi r dr = \frac{u_{\text{max}}}{2} $$

\boxed{U_{\text{avg}} = \frac{\Delta p R^2}{8\mu L}}

Shear Stress:

$$ \tau(r) = \mu \frac{du}{dr} = -\frac{\Delta p}{2L} r $$

Wall shear stress: $ \tau_w = \frac{\Delta p R}{2L} $.

Typical Problem (Jun 2024, Nov 2023): Given $ D=400 \, \text{mm} $, $ u_{\text{max}}=3 \, \text{m/s} $:

$ U_{\text{avg}} = u_{\text{max}}/2 = 1.5 \, \text{m/s} $.

Radius where $ u = u_{\text{max}} $? At $ r=0 $.

Velocity at $ y=6 \, \text{cm} $ from wall: $ r = R - 0.06 = 0.2 - 0.06 = 0.14 \, \text{m} $,

$ u = u_{\text{max}} (1 - (r/R)^2) = 3(1 - (0.14/0.2)^2) = 1.53 \, \text{m/s} $.

C. Viscous Flow in Rotating Systems

Rotating Discs/Cones: Viscous torque due to shear in fluid film.
Parallel Discs (gap $ h \ll R $):

$$ T = \frac{\pi \mu \omega R^4}{2h} $$

Power: $ P = T \omega $.

Conical Thrust Bearing (vertex angle $ 2\theta $, mean radius $ R $, gap $ h $):

$$ T = \frac{\pi \mu \omega R^4}{2h \sin(\theta)} $$

Example (Jun 2023): $ 2\theta=60^\circ \Rightarrow \theta=30^\circ $, $ R=0.1 \, \text{m} $, $ h=0.001 \, \text{m} $, $ \mu=0.1 \, \text{Pa·s} $ (1 P), $ \omega = 600 \times 2\pi/60 = 62.83 \, \text{rad/s} $ →

$ T = \frac{\pi \times 0.1 \times 62.83 \times 0.1^4}{2 \times 0.001 \times \sin 30^\circ} = 19.74 \, \text{N·m} $,

$ P = T \omega = 1240 \, \text{W} $.

V. PIPE FLOW AND LOSSES

A. Major Losses (Friction)

Darcy–Weisbach Equation:

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

where $ f $ = Darcy friction factor.

Friction Factor:
- Laminar ($ Re < 2000 $): $ f = \frac{64}{Re} $.
- Turbulent: Moody chart or Colebrook Equation:

$$ \frac{1}{\sqrt{f}} = -2 \log_{10} \left( \frac{\varepsilon/D}{3.7} + \frac{2.51}{Re \sqrt{f}} \right) $$

Hazen–Williams Formula (empirical, water only):

$$ h_f = 10.67 \, L \, \frac{Q^{1.852}}{C^{1.852} D^{4.871}} \quad \text{(SI units)} $$

$ C $ = Hazen-Williams coefficient (depends on pipe roughness/age).

B. Minor Losses

Loss Coefficient ($ K $): $ h_L = K \frac{V^2}{2g} $.
Sudden Enlargement (diameters $ D_1 \to D_2 $):

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

Sudden Contraction: $ h_L = K_c \frac{V_2^2}{2g} $, $ K_c \approx 0.5 $ (sharp-edged).
Entrance Loss: $ K \approx 0.5 $ (square edge), $ \approx 0.04 $ (well-rounded).
Exit Loss: $ K \approx 1.0 $.

C. Pipe Systems

Series Pipes: Same flow rate $ Q $, total head loss $ h_{f,\text{total}} = \sum h_{f,i} $.
Parallel Pipes: Same head loss $ h_f $, total $ Q = \sum Q_i $.
Branching Pipes: Apply continuity and head loss balance at junctions.
Hydraulic Gradient Line (HGL): $ \frac{p}{\rho g} + z $ (pressure head + elevation head).
Total Energy Line (TEL): $ \frac{p}{\rho g} + \frac{V^2}{2g} + z $. HGL lies below TEL by velocity head.

D. Problem-Solving

Given pressures, diameters, discharge → find head loss (Jun 2025 Q5): Apply Bernoulli with losses between sections A and B.

Example: Valve closed → static pressure difference; valve open → dynamic pressure difference + losses.
Pumping Power (Jun 2023 Q21):

$$ P_{\text{pump}} = \frac{\rho g Q H_{\text{total}}}{\eta_{\text{pump}}} $$

where $ H_{\text{total}} $ includes elevation, pressure, and friction heads.

VI. BOUNDARY LAYER THEORY

A. Concepts and Definitions

Boundary Layer Thickness ($ \delta $): Distance from wall where $ u = 0.99 U $ (or $ \tau \approx 0 $).
Displacement Thickness ($ \delta^* $):

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

Represents reduction in mass flow due to boundary layer.

Momentum Thickness ($ \theta $):

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

Related to momentum deficit.

Energy Thickness ($ \delta^{**} $):

$$ \delta^{**} = \int_0^\delta \frac{u}{U} \left(1 - \left(\frac{u}{U}\right)^2\right) dy $$

B. Velocity Profiles

Laminar:
- Linear: $ \frac{u}{U} = \frac{y}{\delta} $ → $ \delta^* = \frac{\delta}{2}, \; \theta = \frac{\delta}{6}, \; \delta^{**} = \frac{2\delta}{3} $.
- Parabolic: $ \frac{u}{U} = 2\left(\frac{y}{\delta} - \left(\frac{y}{\delta}\right)^2\right) $ →
  
  $ \delta^* = \frac{\delta}{3}, \; \theta = \frac{2\delta}{15}, \; \delta^{**} = \frac{3\delta}{10} $.
  
  Common exam problem (Dec 2024, Nov 2023).
Turbulent (1/9th power law):

$ \frac{u}{U} = \left(\frac{y}{\delta}\right)^{1/9} $ →

$ \delta^* = \frac{\delta}{10}, \; \theta = \frac{7\delta}{72} $, ratio $ \delta^*/\delta = 0.1 $.

C. Skin Friction Coefficient ($ C_f $)

Definition: $ C_f = \frac{\tau_w}{\frac{1}{2} \rho U^2} $.
From Velocity Profile:

$ \tau_w = \mu \left( \frac{du}{dy} \right)_{y=0} $.

Example: Parabolic profile $ u = U \cdot 2(y/\delta - (y/\delta)^2) $ →

$ \frac{du}{dy} = \frac{2U}{\delta} (1 - 2y/\delta) $ → at $ y=0 $, $ \tau_w = \mu \cdot \frac{2U}{\delta} $ →

$ C_f = \frac{2\mu U}{\delta \cdot \frac{1}{2} \rho U^2} = \frac{4\mu}{\rho U \delta} $.

D. Boundary Layer Separation

Adverse Pressure Gradient ($ dp/dx > 0 $): Decelerates flow, velocity gradient near wall decreases.
Separation Criterion: $ \left( \frac{\partial u}{\partial y} \right)_{y=0} = 0 $ → $ \tau_w = 0 $, flow reverses near wall.
Prevention Methods:
- Streamlining (delay separation).
- Suction through porous walls.
- Vortex generators (create turbulent mixing).
[!TIP]

Key Point: Separation occurs only with adverse pressure gradient in viscous flows. For inviscid flow, no separation.

VII. FORCES ON IMMERSED BODIES AND BUOYANCY

A. Hydrostatic Forces

Plane Surface (inclined/vertical):

Total force $ F = p_{\text{cg}} A = \rho g h_{\text{cg}} A $.

Center of pressure (CP) below centroid:

$$ y_{\text{cp}} = \frac{I_{\text{cg}}}{y_{\text{cg}} A} + y_{\text{cg}} $$

where $ I_{\text{cg}} $ = second moment of area about horizontal axis through centroid.

Curved Surface:
- Horizontal component $ F_H $ = force on vertical projection of area.
- Vertical component $ F_V $ = weight of fluid above + pressure on projection (if any).
Hydrostatic Paradox: Force on bottom depends only on depth and area, not on volume of fluid or shape of container (Pascal’s law).

B. Buoyancy and Stability

Archimedes’ Principle: Buoyant force $ F_b = \rho_{\text{fluid}} g \nabla $, where $ \nabla $ = displaced volume.
Center of Buoyancy (CB): Centroid of displaced volume.
Metacentric Height ($ GM $):

$$ GM = BM - BG $$

where $ BM = \frac{I}{\nabla} $, $ I $ = waterplane inertia about axis through CG, $ \nabla $ = displaced volume.

$ BG $ = distance between CG and CB (positive if CG above CB).

Stability Conditions:
- Stable: $ GM > 0 $ (metacenter above CG).
- Neutral: $ GM = 0 $.
- Unstable: $ GM < 0 $.
Why CG and CB Alone Insufficient? Stability depends on metacenter position, not just relative positions of CG and CB. For small angles, $ GM $ determines righting moment.
Periodic Time of Oscillation (for rolling):

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

$ k $ = radius of gyration about horizontal axis through CG.

Floating Body at Interface (Dec 2024 Q13):

Weight = buoyancy from mercury + buoyancy from water.

Example: Cylinder floats at Hg-water interface, 40% in Hg, 60% in water →

$ \rho_{\text{body}} = 0.4 \rho_{\text{Hg}} + 0.6 \rho_{\text{water}} $.

C. Flow Around Bodies

Streamline Body: Tapered ends, separation delayed (e.g., airfoil).
Bluff Body: Separation early, large wake (e.g., cylinder, sphere).
Drag and Lift:

$ D = \frac{1}{2} \rho V^2 A C_D $, $ L = \frac{1}{2} \rho V^2 A C_L $.
Magnus Effect: Lift on spinning cylinder due to circulation (Kutta-Joukowski theorem: $ L' = \rho V \Gamma $ per unit span).
Airfoil: Cambered shape, angle of attack generates lift.

VIII. FLOW MEASUREMENT DEVICES

A. Venturi Meter

Derivation: Bernoulli + continuity:

$$ \frac{p_1}{\rho} + \frac{V_1^2}{2} = \frac{p_2}{\rho} + \frac{V_2^2}{2} $$

$ V_1 A_1 = V_2 A_2 $ →

$$ V_2 = \sqrt{\frac{2(p_1 - p_2)}{\rho \left(1 - (A_2/A_1)^2\right)}} $$

Theoretical discharge $ Q_{\text{th}} = A_2 V_2 $.

Actual $ Q = C_d Q_{\text{th}} $, $ C_d \approx 0.98 $ (well-designed).

Compressible Flow: Include expansion factor $ Y $: $ Q = C_d A_2 Y \sqrt{\frac{2(p_1 - p_2)}{\rho_1 (1 - \beta^4)}} $, $ \beta = D_2/D_1 $.

B. Orifice Meter

Similar to Venturi but with vena contracta (jet narrows to $ A_c < A_o $).

$ Q = C_d A_o \sqrt{\frac{2(p_1 - p_2)}{\rho}} $,

$ C_d = C_c C_v $, $ C_c $ = contraction coefficient (≈0.62), $ C_v $ = velocity coefficient (≈0.98).

C. Pitot-Static Tube

Stagnation Pressure from pitot opening (facing flow).
Static Pressure from side ports (perpendicular to flow).
Velocity: $ V = \sqrt{\frac{2(p_0 - p_s)}{\rho}} $.
Stagnation Properties: $ p_0 = p + \frac{1}{2}\rho V^2 $ (incompressible), $ T_0 = T(1 + \frac{\gamma-1}{2}M^2) $ (compressible).

D. Manometers

U-tube Manometer (light manometric fluid):

$ p_1 + \rho_1 g h_1 = p_2 + \rho_2 g h_2 + \rho_m g h $.
Inclined Manometer: Sensitivity $ \propto 1/\sin\theta $, $ \Delta p = \rho_m g L \sin\theta $.
Inverted V-tube (for low pressure):

$ \Delta p = \rho g h \left( \frac{1}{A_1} - \frac{1}{A_2} \right) \approx \rho g h $ if $ A_1 \gg A_2 $.

IX. ROTATING FLOWS AND TURBOMACHINERY

A. Forced Vortex

Rigid Body Rotation: $ V_\theta = \omega r $, $ \omega $ = constant angular velocity.
Free Surface Shape: From $ \frac{\partial p}{\partial r} = \rho \omega^2 r $, $ \frac{\partial p}{\partial z} = -\rho g $.

Integrate: $ p = \frac{1}{2} \rho \omega^2 r^2 - \rho g z + C $.

At free surface $ p = \text{constant} $ →

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

→ Parabola.

Pressure Distribution: $ p = p_0 + \frac{1}{2} \rho \omega^2 r^2 - \rho g z $.

B. Torque and Power in Rotating Systems

Lawn Sprinkler (Jun 2025 Q11):

Torque due to friction $ T_f $ balances angular momentum change:

$$ T_f = \dot{m} r (V_{\theta,\text{out}} - V_{\theta,\text{in}}) $$

Example: Two jets, diameter $ d=1.5 \, \text{cm} $, arm radius $ R=30 \, \text{cm} $, $ \omega = 210 \, \text{rpm} = 22 \, \text{rad/s} $, $ Q=3 \, \text{L/s}=0.003 \, \text{m}^3/\text{s} $.

$ V_{\text{jet}} = Q/(2 \cdot \pi d^2/4) = 1.7 \, \text{m/s} $, $ V_\theta = V_{\text{jet}} $ (tangential).

$ T_f = 2 \cdot \dot{m}_{\text{per jet}} \cdot R \cdot V_\theta = 2 \cdot (\rho Q/2) \cdot R \cdot V_\theta = \rho Q R V_\theta $ →

$ T_f = 1000 \times 0.003 \times 0.3 \times 1.7 = 1.53 \, \text{N·m} $.

Rotating Discs/Cones: As in Section IV.C.

X. COMPRESSIBLE FLOW BASICS

A. Mach Number

Definition: $ M = \frac{V}{a} $, where $ a = \sqrt{\gamma R T} $ = speed of sound.
Significance:
- Subsonic: $ M < 1 $ (disturbances propagate upstream).
- Sonic: $ M = 1 $.
- Supersonic: $ M > 1 $ (disturbances confined to Mach cone).
Compressibility Effects: Significant when $ M > 0.3 $. Density change $ \Delta \rho / \rho \approx \frac{\gamma-1}{2} M^2 $.

B. Fanno Lines

Adiabatic flow in constant-area duct with friction.
Stagnation properties constant: $ T_0, p_0 $ constant? Actually, $ T_0 $ constant, $ p_0 $ decreases due to friction (entropy increases).
Maximum possible $ M $: For given inlet conditions, as friction increases, $ M \to 1 $ (choked). Cannot exceed 1 in Fanno flow.
Diagram: Plot in $ h-s $ or $ p-v $ plane: curve from given inlet state to $ M=1 $ (if possible).

C. Rayleigh Lines

Constant-area flow with heat addition/removal.
Stagnation temperature changes: $ T_0 $ increases with heat addition.
Maximum $ M $: At maximum $ T_0 $ for given $ p $, $ M \to 1 $.
Diagram: In $ h-s $ plane, horizontal line? Actually, for constant area, $ h + V^2/2 = \text{constant} $, so $ h $ vs $ s $ curve.

D. Sketches and Applications

Fanno Line: In $ h-s $, concave curve, entropy increases, $ M=1 $ at maximum entropy for given $ h $.
Rayleigh Line: In $ h-s $, also curve, but $ h $ varies; $ M=1 $ at maximum $ T_0 $.
Application: Design of adiabatic ducts (Fanno) and combustors (Rayleigh).

E. Compressible Flow Through Nozzles

Mass Flow Rate: $ \dot{m} = \rho A V $.
Isentropic Relations (for $ \gamma $):

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

$$ \frac{p}{p_0} = \left(1 + \frac{\gamma-1}{2} M^2 \right)^{-\gamma/(\gamma-1)} $$

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

Choked Flow: When $ M=1 $ at throat, mass flow maximized. Condition:

$$ \frac{p_0}{p_{\text{throat}}} \geq \left( \frac{\gamma+1}{2} \right)^{\gamma/(\gamma-1)} $$

For air ($ \gamma=1.4 $), critical pressure ratio ≈ 1.89.

XI. ADVANCED ANALYSIS TOOLS

A. Navier–Stokes Equations

General Form (Cartesian, Newtonian fluid):

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

Physical Significance: Governs viscous fluid motion; includes inertia, pressure, viscous, body forces.
Uses: Analytical/numerical solutions for boundary layers, pipe flow, complex geometries. Basis for CFD.

B. Control Volume vs. System Approach

System (Lagrangian): Follows specific fluid mass. Difficult for engineering due to deforming boundaries.
Control Volume (Eulerian): Fixed region in space. More important because:
- Aligns with engineering observations (e.g., flow through pipe).
- Allows application of Reynolds Transport Theorem:

$$ \frac{d}{dt} \int_{\text{sys}} B \, dm = \frac{d}{dt} \int_{\text{cv}} B \rho \, dV + \int_{\text{cs}} B \rho \mathbf{V} \cdot d\mathbf{A} $$

where \( B \) = extensive property per unit mass.

Enables use of conservation laws (mass, momentum, energy) on fixed regions.

[!TIP]

Exam Focus: Control volume approach used for momentum equation (pipe bends, turbines) and energy equation (Bernoulli derivation).

C. Dimensional Analysis (Implied)

Buckingham π Theorem: $ n $ variables → $ n-k $ dimensionless groups ($ k $ = fundamental dimensions).
Key Numbers:
- Reynolds number $ Re = \frac{\rho V L}{\mu} $ (inertia/viscous).
- Froude number $ Fr = \frac{V}{\sqrt{gL}} $ (inertia/gravity).
- Mach number $ M = V/a $ (inertia/elasticity).
- Weber number $ We = \frac{\rho V^2 L}{\sigma} $ (inertia/surface tension).
Application: Similitude in model testing.

XII. SPECIAL TOPICS FROM PAST PAPERS

A. Hydrostatic Paradox

Force on bottom of container depends only on depth and area, not on volume or shape.

Example: Different shaped containers with same base area and liquid height exert same force on base.

B. Stagnation Properties

Stagnation Pressure: $ p_0 = p + \frac{1}{2} \rho V^2 $ (incompressible).
Stagnation Temperature: $ T_0 = T \left(1 + \frac{\gamma-1}{2} M^2 \right) $ (compressible, adiabatic, no work).
Significance: Used in Pitot tubes, compressible flow measurements.

C. Magnus Effect

Lift force on spinning cylinder/ball due to circulation.

Lift per unit span: $ L' = \rho V \Gamma $, where $ \Gamma $ = circulation.

Application: Curve balls in sports, rotor ships.

D. Flow Around Circular Cylinder (Potential Flow)

Combination of uniform flow + doublet:

Stream function: $ \psi = U r \sin \theta \left(1 - \frac{R^2}{r^2}\right) $.

Stagnation points at $ \theta = 0, \pi $ on cylinder surface ($ r=R $).

Pressure distribution from Bernoulli.

E. Head Losses in Turbulent Flow

Darcy–Weisbach: General, uses Moody chart/Colebrook.
Hazen–Williams: Empirical for water, $ h_f \propto Q^{1.852} $, easier but limited to water, specific temperatures.

F. Boundary Layer Phenomenon

Development: Laminar → transition → turbulent.

Laminar length $ x_{\text{crit}} \approx 0.05 Re_x $ (flat plate).
Separation: Due to adverse pressure gradient; causes drag increase, stall in airfoils.

SUMMARY OF HIGH-FREQUENCY EXAM TOPICS:

Boundary Layer: Calculate $ \delta^*, \theta, \delta^{**} $ from given $ u/U $ profile (frequent 14m questions).
Pipe Flow: Series/parallel networks, head loss with Darcy, pumping power.
Bernoulli’s Theorem: Derivation, assumptions, Venturi/orifice problems.
Stability: Metacentric height calculation for floating bodies.
Rotating Flows: Forced vortex parabola proof, torque on rotating systems.
Stream Function & Potential: Given $ \psi $, find $ \phi $; orthogonality proof.
Viscosity & Non-Newtonian: Definitions, examples, graphs.
Compressible Basics: Mach number, Fanno/Rayleigh sketches.
Forces: Hydrostatic on curved surfaces, buoyancy.
Flow Measurement: Venturi derivation, manometer problems.

[!EXAM TIP]

Common Pitfalls:

In boundary layer integrals, ensure limits are 0 to $ \delta $ and $ u/U $ profile is correct.

In pipe flow, distinguish Darcy $ f $ (4× Fanning $ f $).

In stability, $ GM > 0 $ for stable, but $ BM = I/\nabla $ uses waterplane inertia, not cross-sectional.

In rotating flows, pressure variation includes centrifugal term $ \frac{1}{2} \rho \omega^2 r^2 $.

In compressible flow, stagnation properties differ from static; choked flow condition critical.

Account

FLUID MECHANICS (ME-404) - Unit 5 Short Notes

I. FLUID PROPERTIES AND CHARACTERISTICS

II. FLUID KINEMATICS

III. FLUID DYNAMICS

IV. LAMINAR FLOW EXACT SOLUTIONS

V. PIPE FLOW AND LOSSES

VI. BOUNDARY LAYER THEORY

VII. FORCES ON IMMERSED BODIES AND BUOYANCY

VIII. FLOW MEASUREMENT DEVICES

IX. ROTATING FLOWS AND TURBOMACHINERY

X. COMPRESSIBLE FLOW BASICS

XI. ADVANCED ANALYSIS TOOLS

XII. SPECIAL TOPICS FROM PAST PAPERS