Fluid Mech. – II (CE-604 (A)) - Unit 3 Short Notes

UNIT 3: FLUID MECHANICS – II (Exam-Focused Short Notes)

1. Introduction and Historical Development

• Fluid Mechanics: Study of fluids (liquids & gases) at rest (fluid statics) and in motion (fluid dynamics), and their interaction with solids.

• Key Milestones:

  • Archimedes (250 BC): Buoyancy principle.

  • Newton (1687): Law of viscosity, fluid resistance.

  • Bernoulli (1738): Hydrodynamica, energy principle.

  • Euler (1755): Equations of motion for inviscid flow.

  • Reynolds (1883): Experiment on laminar/turbulent transition, Reynolds number.

  • Prandtl (1904): Boundary layer theory.

  • Stokes (1851): Flow past spheres, Stokes' law.

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Exam Focus: Historical questions often ask for contributors and their key laws. Link names to equations (e.g., Bernoulli → energy equation, Reynolds → Re).

2. Fluid Properties and Basic Concepts

Definition of Fluid: A substance that deforms continuously under shear stress, no matter how small.

Newton’s Law of Viscosity:

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

Significance: Relates shear stress $\tau$ to velocity gradient. Fluids obeying this are Newtonian (water, air); others are non-Newtonian.

Temperature Effects:

• Water: Viscosity decreases with temperature (molecular bonds weaken).

• Air: Viscosity increases with temperature (molecular momentum transfer rises).

Surface Tension & Capillarity:

• Surface Tension ($\sigma$): Force per unit length (N/m). Arises from cohesion.

• Capillary Rise/Fall:

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

$\theta$: contact angle; $d$: tube diameter.

• Laplace Equation (droplet/bubble):

$$ \Delta p = \frac{4\sigma}{d} \quad \text{(droplet)} $$

$$ \Delta p = \frac{8\sigma}{d} \quad \text{(soap bubble, two surfaces)} $$

Pascal’s Law:

Pressure change in confined fluid transmits undiminished in all directions.

Proof: Consider equilibrium of fluid element; force balance gives $$\displaystyle dp = \gamma \, dh $$; for horizontal element, $$\displaystyle dp = 0 $$ → pressure same at all points on same horizontal plane.

Pressure Head:

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

Height of liquid column equivalent to pressure $p$.

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Common Pitfalls:

• Capillary rise: $$\displaystyle \cos\theta > 0 $$ for wetting fluids (water-glass), $$\displaystyle \cos\theta < 0 $$ for non-wetting (mercury-glass).

• Laplace for bubble: double surface → factor 8, not 4.

• Pascal’s law applies only to static fluids.

3. Kinematics of Fluid Flow

Flow Classification:

Reynolds Experiment:

• Dye filament in glass tube:

  • Low velocity → filament remains distinct (laminar).

  • High velocity → filament diffuses (turbulent).

• Critical Reynolds Number:

  • Pipe flow: $$\displaystyle Re_{crit} \approx 2000 $$–$2300$.

  • Flat plate: $$\displaystyle Re_{crit} \approx 5 \times 10^5 $$ (based on distance from leading edge).

Eulerian vs Lagrangian:

• Eulerian: Field description; observe fixed points in space. Used in most engineering problems.

• Lagrangian: Track individual fluid particles. Complex, used in particle dynamics.

Continuity Equation (3D Cartesian, compressible):

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

For incompressible ($$\displaystyle \rho = \text{constant} $$):

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

Application: Given $u$ and $v$, find $w$: $$\displaystyle w = \int \left( -\frac{\partial u}{\partial x} - \frac{\partial v}{\partial y} \right) dz + f(x,y) $$.

Velocity Potential ($\phi$) & Stream Function ($\psi$):

• $\phi$: For irrotational flow, $$\displaystyle \vec{V} = \nabla \phi $$.

  $$\displaystyle u = \frac{\partial \phi}{\partial x},\; v = \frac{\partial \phi}{\partial y} $$.

• $\psi$: For 2D incompressible flow,

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

• Cauchy-Riemann Equations (for $\phi$ & $\psi$ to be conjugate):

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

• Orthogonal Trajectories: $$\displaystyle \phi = \text{constant} $$ (equipotentials) and $$\displaystyle \psi = \text{constant} $$ (streamlines) intersect at $$\displaystyle 90^\circ $$.

Streamlines, Pathlines, Streaklines:

• Streamline: Instantaneous line tangent to velocity field.

  $$\displaystyle \frac{dx}{u} = \frac{dy}{v} = \frac{dz}{w} $$.

• Pathline: Actual trajectory of a fluid particle.

• Streakline: Line joining all particles that passed through a fixed point.

• Steady flow: All three coincide. Unsteady: They differ.

Flow Nets: Grid of orthogonal streamlines and equipotentials. Used in groundwater flow, lubrication problems.

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Exam Tricks:

• Given $\phi$, find $u,v$ by differentiation; then integrate to get $\psi$ using Cauchy-Riemann.

• Continuity for 2D: $$\displaystyle \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0 $$ is a quick check.

• Orthogonality proof: $$\displaystyle \nabla \phi \cdot \nabla \psi = 0 $$ from Cauchy-Riemann.

4. Dynamics of Inviscid Flow

Forces in Fluids:

• Surface Forces: Pressure (normal), viscous (tangential).

• Body Forces: Gravity, electromagnetic.

Euler’s Equation (along streamline, inviscid, steady):

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

Derivation: Apply Newton’s 2nd law to fluid element along streamline; pressure and gravity forces only.

Bernoulli’s Theorem (from Euler integration):

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

Assumptions:

1. Inviscid ($$\displaystyle \mu = 0 $$)

2. Steady flow

3. Incompressible ($$\displaystyle \rho = \text{constant} $$)

4. Along a streamline (or entire flow if irrotational)

Total Energy Line (TEL) & Hydraulic Gradient Line (HGL):

• TEL: Represents total head ($$\displaystyle \frac{p}{\rho g} + \frac{V^2}{2g} + z $$). Horizontal for ideal flow; slopes downward with losses.

• HGL: Represents pressure head ($$\displaystyle \frac{p}{\rho g} + z $$). Always below TEL by $$\displaystyle \frac{V^2}{2g} $$.

• Difference: TEL – HGL = velocity head.

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Common Errors:

• Bernoulli applies only along streamlines unless flow is irrotational.

• Do not mix units: use consistent (SI: m, kg, s).

• TEL slope = head loss per unit length; HGL slope = pressure gradient.

5. Viscous Flow and Boundary Layer Theory

Laminar Flow in Circular Pipes (Hagen–Poiseuille):

• Velocity Profile (parabolic):

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

where $$\displaystyle u_{max} = \frac{\Delta p \, R^2}{4\mu L} $$.

• Shear Stress:

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

• At wall ($$\displaystyle r = R $$): $$\displaystyle \tau_w = \frac{\Delta p \, R}{2L} = \frac{4\mu u_{max}}{R} $$.

• At center ($$\displaystyle r = 0 $$): $$\displaystyle \tau = 0 $$.

Boundary Layer Theory:

• Concept: Thin region near solid surface where velocity from 0 to free-stream $U$ due to viscosity.

• Development over Flat Plate:

  • Laminar: $$\displaystyle \delta \approx \frac{5x}{\sqrt{Re_x}} $$ (Blasius solution).

  • Turbulent: $$\displaystyle \delta \approx \frac{0.37x}{Re_x^{1/5}} $$.

• Thicknesses:

  • Displacement Thickness $$\displaystyle \delta^* $$:

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

Represents mass flow deficit.

• Momentum Thickness $\theta$:

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

Used in momentum integral equation.

• Skin Friction Drag (flat plate):

$$ D_f = \tau_w \times \text{area} $$

For laminar: $$\displaystyle \tau_w = 0.332 \rho U^2 / \sqrt{Re_x} $$.

Turbulence:

• Characteristics: Random fluctuations, enhanced mixing, energy cascade.

• Effect on Velocity Profile: Fuller (flatter) than laminar; increased momentum transfer.

• Energy Dissipation: Viscous dissipation of turbulent kinetic energy into heat.

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Key Formulas:

• Pipe flow shear: $$\displaystyle \tau_w = \frac{\Delta p D}{4L} $$.

• Boundary layer thickness: $$\displaystyle \delta \propto x / \sqrt{Re_x} $$ (lam), $$\displaystyle \delta \propto x / Re_x^{1/5} $$ (turb).

• Drag reduction: Delay transition (smooth surface) or use turbulent boundary layer (delayed separation).

6. Flow Measurement and Instrumentation

Pitot–Static Tube:

• Construction: Forward-facing pitot (stagnation) + side static ports.

• Working:

  $$\displaystyle p_0 = p + \frac{1}{2}\rho V^2 $$ (stagnation pressure)

  $p$ = static pressure.

  Velocity:

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

$C$: calibration coefficient (~0.98–1.0).

Manometers:

• Principle: Balance pressure by liquid column height.

• Simple Manometer (U-tube):

  $$\displaystyle p_1 - p_2 = (\rho_m - \rho) g h $$ (if one side open to atmosphere).

• Differential Manometer: Measures $$\displaystyle p_1 - p_2 $$ directly with two connections.

• Inclined Manometer: Amplifies deflection for low pressure; $$\displaystyle h = L \sin\theta $$, $$\displaystyle p_1 - p_2 = \rho g L \sin\theta $$.

Capillary Effects in Manometers:

• Error: Meniscus curvature in small-diameter tubes gives false reading.

• Correction:

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

Subtract from observed $h$ if wetting ($$\displaystyle \theta < 90^\circ $$).

• Max Allowable Diameter for error $$\displaystyle < \epsilon $$:

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

Other Flow Meters:

• Venturi Meter:

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

(Theoretical, apply discharge coefficient $$\displaystyle C_d $$).

• Orifice Meter: Similar but with higher losses; $$\displaystyle C_d \approx 0.6 $$–$0.7$.

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Pitot Tube: Ensure alignment with flow; use for velocity only.

Manometer: Always write pressure balance equation; include all fluid densities.

Capillary: Correction adds to measured $h$ for wetting fluids (water in glass).

7. Applications of Fluid Mechanics

Reciprocating Pump:

• Components: Cylinder, piston, suction valve, delivery valve, crank.

• Working:

  1. Suction stroke: piston moves left, suction valve opens, water enters.

  2. Delivery stroke: piston moves right, suction valve closes, delivery valve opens, water discharged.

• Discharge: $$\displaystyle Q = A \times L \times N \times \eta_v $$ (volumetric efficiency accounts for slip).

Reaction Turbine (e.g., Francis):

• Principle: Both pressure and kinetic energy of water convert to rotational energy.

• Construction: Spiral casing, guide vanes, runner (rotor), draft tube.

• Working: Water enters spiral casing → guide vanes direct onto runner blades at optimal angle → pressure drop occurs both in casing and over blades → runner rotates → water exits via draft tube (recovers kinetic energy).

Biofluid Mechanics:

• Blood Flow: Non-Newtonian (shear-thinning), pulsatile.

  Reynolds number in aorta ~ 2000 (transition risk).

• Heart: Acts as pulsatile pump; pressure–volume loop describes cardiac cycle.

• Veins: Valves prevent backflow; flow aided by muscle pump.

Buoyancy & Flotation:

• Archimedes’ Principle:

$$ F_b = \gamma_f V_{sub} = \rho_f g V_{sub} $$

(Buoyant force equals weight of displaced fluid).

• Floating Body:

$$ \rho_{body} = \rho_f \times \frac{V_{sub}}{V_{total}} $$

Weight = Buoyancy: $$\displaystyle \rho_{body} g V_{total} = \rho_f g V_{sub} $$.

Hydrostatic Pressure:

• Barometer: $$\displaystyle p_{atm} = \rho_{Hg} g h $$ (mercury column).

• Pressure at Depth:

$$ p = p_{atm} + \rho g h $$

(Gauge pressure: $$\displaystyle p_{gauge} = \rho g h $$).

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Pump Discharge: Remember slip $$\displaystyle s = \frac{\text{theoretical} - \text{actual}}{\text{theoretical}} $$.

Turbine vs Pump: Turbine extracts energy; pump adds energy.

Buoyancy: For floating, $$\displaystyle V_{sub} $$ adjusts so weight = buoyancy.

Barometer: Use $$\displaystyle \rho_{Hg} = 13600 $$ kg/m³, $$\displaystyle g = 9.81 $$ m/s².

Quick Reference Table: Exam-Favorite Formulas

Final Advice:

• Definitions: Memorize exact wording (e.g., fluid, viscosity, Bernoulli assumptions).

• Derivations: Practice continuity, Euler, Bernoulli – they are 7-mark questions.

• Numericals: Focus on pipe flow (shear stress), boundary layer thickness, manometers with capillary correction, buoyancy.

• Diagrams: Sketch Pitot tube, boundary layer development, flow net, reciprocating pump cycle.

• Units: Always check SI consistency (Pa, m, kg, s).

Past Paper Pattern:

• 7-mark: Definitions, explanations, derivations, simple numerics.

• 14-mark: Combined problems (e.g., continuity + potential, Bernoulli + manometer).

• Frequent: Viscosity calculations, Reynolds experiment, Pitot, capillary, boundary layer, buoyancy, Bernoulli applications.