Fluid Mech. – II (CE-604 (A)) - Unit 4 Short Notes

UNIT 4: FLUID MECHANICS – II

High-Impact Exam Notes Based on CE-604(A) Past Papers (May 2024 & May 2022)

1. Introduction & Fundamental Properties

Viscosity (High Frequency)

Definition: Measure of a fluid's resistance to flow or deformation due to internal friction.
Newton's Law of Viscosity:

The shear stress (\(\tau\)) in a fluid is directly proportional to the rate of shear strain (velocity gradient).

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

where \(\mu\) = dynamic viscosity (Pa·s or N·s/m²).

Importance in Fluid Motion:

• Governs laminar flow characteristics.

• Determines energy loss due to friction in pipes (major head loss).

• Influences boundary layer development and flow separation.

Temperature Effect:

• Liquids: Viscosity ↓ with temperature ↑ (weaker intermolecular forces).

• Gases: Viscosity ↑ with temperature ↑ (increased molecular momentum transfer).

[!TIP]

Exam Focus: May 2024 asked for temperature effects on water (liquid) vs. air (gas). Remember: water viscosity decreases with heating; air viscosity increases.

Surface Tension (Medium Frequency)

Definition: Contractile tendency of a liquid surface due to cohesive forces, measured as force per unit length (N/m).

Jurin’s Law (Capillary Rise/Fall):

$$h = \frac{2\sigma \cos\theta}{\rho g r}$$

where \(h\) = rise/fall height, \(\sigma\) = surface tension, \(\theta\) = contact angle, \(\rho\) = density, \(r\) = capillary radius.

Pressure Difference in Droplets/Bubbles:

• Droplet (single surface): \(\Delta p = \frac{2\sigma}{R}\)

• Soap bubble (two surfaces): \(\Delta p = \frac{4\sigma}{R}\)

[!TIP]

Common Pitfall: In manometer problems (May 2024), capillary rise causes error in pressure reading. Use Jurin’s law to correct if tube diameter is small.

2. Fluid Statics

Pascal’s Law (Medium Frequency)

Statement: Pressure applied to an enclosed fluid is transmitted undiminished to every point in the fluid and the container walls.

Proof (Consider a small fluid element):

Applications:

• Hydraulic press/jack

• Hydraulic brakes

• Hydraulic lifts

Buoyancy & Flotation (Medium Frequency)

Archimedes’ Principle: A body immersed in fluid experiences an upward buoyant force equal to weight of displaced fluid.

Problem-Solving (Multiple Fluids):

For a block floating at interface of two immiscible fluids:

$$\rho_{\text{block}} = \frac{\rho_1 V_1 + \rho_2 V_2}{V_{\text{total}}}$$

where \(V_1, V_2\) = volumes in each fluid.

[!TIP]

May 2024 Question: Wooden block floats differently in water vs. glycerin. Use relative density and volume fractions to solve.

3. Fluid Kinematics

Eulerian vs. Lagrangian Description (Medium Frequency)

Continuity Equation (High Frequency)

Derivation (3D Cartesian, Steady/Unsteady):

Apply conservation of mass to a control volume:

$$\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec{V}) = 0$$

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

$$\nabla \cdot \vec{V} = 0 \quad \text{or} \quad \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0$$

Finding Missing Velocity Component (May 2022):

Given \(u(x,y,z)\) and \(v(x,y,z)\), integrate \(\partial w/\partial z = -(\partial u/\partial x + \partial v/\partial y)\) to find \(w(z)\).

Stream Function & Velocity Potential (High Frequency)

Stream Function (\(\psi\)):

• Defined for 2D incompressible flow: \(u = \frac{\partial \psi}{\partial y},\; v = -\frac{\partial \psi}{\partial x}\)

• Automatically satisfies continuity.

• Lines of constant \(\psi\) = streamlines.

Velocity Potential (\(\phi\)):

• Defined for irrotational flow: \(u = \frac{\partial \phi}{\partial x},\; v = \frac{\partial \phi}{\partial y}\)

• Lines of constant \(\phi\) = equipotential lines.

Orthogonal Trajectories:

Streamlines and equipotentials intersect at 90° because:

$$\nabla \phi \cdot \nabla \psi = u \cdot u + v \cdot (-v) = u^2 - v^2 \quad \text{? Wait, correction:}$$

Actually:

$$\nabla \phi = (u, v), \quad \nabla \psi = (v, -u) \quad \Rightarrow \quad \nabla \phi \cdot \nabla \psi = uv - vu = 0$$

[!TIP]

May 2024: Given \(\phi = x(2y-1)\), find velocity at P(4,5) and \(\psi\) at P.

Solution: \(u = \partial\phi/\partial x = 2y-1 = 9\), \(v = \partial\phi/\partial y = 2x = 8\). Then \(\psi\) from \(d\psi = u\,dy - v\,dx\) → integrate.

Flow Nets (Medium Frequency)

Construction: Orthogonal grid of streamlines and equipotentials.
Applications:

• Groundwater flow through soils

• Seepage under dams

• Laplace equation solutions (\(\nabla^2 \phi = 0, \nabla^2 \psi = 0\))

Properties:

• Each "square" represents equal flow rate and equal head drop.

• Density of lines indicates velocity magnitude.

4. Fluid Dynamics

Euler’s Equation of Motion (High Frequency)

Derivation along a Streamline:

Consider a fluid element of length \(ds\) along streamline. Forces: pressure, weight, inertia.

$$-dp - \rho g\,dz = \rho \frac{DV}{Dt} ds$$

For steady flow: \(\frac{DV}{Dt} = V\frac{dV}{ds}\)

$$\boxed{-dp = \rho V\,dV + \rho g\,dz}$$

Bernoulli’s Theorem (High Frequency)

Derivation: Integrate Euler’s equation for steady, incompressible, inviscid, along a streamline:

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

or

$$\boxed{p + \frac{1}{2}\rho V^2 + \rho gz = \text{constant}}$$

Assumptions:

1. Steady flow

2. Incompressible fluid

3. Inviscid (zero viscosity)

4. Along a streamline

5. No shaft work/heat transfer

Limitations:

• Not valid in viscous boundary layers, wakes, or highly turbulent regions.

• Does not account for friction losses (use with correction factors).

Applications:

• Venturi meter, orifice meter

• Pitot tube (velocity measurement)

• Flow through pipes with varying area

Total Energy Line (TEL) & Hydraulic Gradient Line (HGL) (High Frequency)

Representation in Piping Systems:

[!TIP]

Exam Distinction: TEL includes kinetic energy; HGL does not. In a horizontal pipe with constant diameter, HGL is parallel to TEL but lower by constant velocity head.

5. Pipe Flow & Boundary Layer Theory

Laminar Flow in Circular Pipes (High Frequency)

Velocity Distribution (Parabolic):

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

where \(u_{\text{max}} = 2V_{\text{avg}}\).

Shear Stress:

• At wall (\(r=R\)): \(\tau_w = \mu \left.\frac{du}{dr}\right|_{r=R} = \frac{\Delta p \cdot R}{2L}\)

• At any radius \(r\): \(\tau(r) = \tau_w \cdot \frac{r}{R}\)

Volumetric Flow Rate (Hagen-Poiseuille):

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

Reynolds Experiment (Medium Frequency)

• Introduced dimensionless Reynolds number: \(Re = \frac{\rho V D}{\mu}\)

• Critical \(Re\):

  • For pipe flow: \(Re_{\text{crit}} \approx 2000-2300\) (transition range)

  • Laminar: \(Re < 2000\)

  • Turbulent: \(Re > 4000\)

Turbulence (Medium Frequency)

Characteristics:

• Irregular, chaotic fluctuations in velocity/pressure.

• Enhanced momentum/heat/mass transfer.

• Increased energy dissipation (higher friction factor).

Effects:

• Velocity profile flatter (more uniform).

• Pressure drop increases (Darcy-Weisbach \(f\) higher).

• Noise and vibration.

Boundary Layer Theory (High Frequency)

Development on Flat Plate:

Key Thicknesses:

1. Displacement thickness (\(\delta^*\)):

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

Represents outward shift of streamline due to slowing.

2. Momentum thickness (\(\theta\)):

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

Represents momentum deficit.

3. Boundary layer thickness (\(\delta\)):

   \(u(\delta) \approx 0.99U\) (arbitrary but standard).

Laminar vs. Turbulent BL:

Shear Stress Distribution:

• Laminar: \(\tau = \mu \frac{du}{dy}\), max at wall.

• Turbulent: \(\tau = \mu \frac{du}{dy} - \rho \overline{u'v'}\), includes Reynolds stress.

[!TIP]

May 2024 Question: Calculate friction drag, boundary layer thickness, shear stress at trailing edge for a plate in oil flow. Use Blasius solution for laminar BL: \(\delta \approx 5.0 x / \sqrt{Re_x}\), \(C_f = 0.664 / \sqrt{Re_x}\).

6. Flow Measurement & Instrumentation

Manometers (High Frequency)

Principle: Pressure difference balanced by liquid column height difference.

$$\Delta p = (\rho_m - \rho_f) g h$$

for inclined/differential manometers.

Simple vs. Differential Manometer:

• Simple: One limb open to atmosphere → measures gauge pressure.

• Differential: Both limbs connected to pressure points → measures \(\Delta p\).

Capillary Correction (May 2024):

Error in small tubes due to meniscus curvature.

Given gauge pressure \(p\), tube radius \(r\), surface tension \(\sigma\), contact angle \(\theta\):

Capillary rise \(h_c = \frac{2\sigma \cos\theta}{\rho_m g r}\)

True pressure: \(p_{\text{true}} = p_{\text{measured}} \pm \rho_m g h_c\) (sign depends on meniscus shape).

[!TIP]

May 2024 Problem: Find smallest manometer tube diameter so capillary error < 5% for \(p = 100\,\text{N/m}^2\). Use water as manometric liquid (\(\sigma \approx 0.0728\,\text{N/m}\), \(\theta \approx 0^\circ\)).

Solution: Error \(\% = \frac{\rho_m g h_c}{p} \times 100 < 5\) → solve for \(r\).

Pitot Tube (Medium Frequency)

Working: Stagnation pressure measured at nose; static pressure on side taps.
Velocity Calculation:

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

where \(p_0\) = stagnation pressure, \(p_s\) = static pressure.

Applications:

• Aircraft airspeed measurement

• Flow velocity in pipes/channels

7. Applications of Fluid Mechanics

Reciprocating Pump (Low Frequency)

• Positive displacement pump.

• Piston/plunger reciprocates in cylinder.

• Suction and delivery valves control flow direction.

• Pulsating flow; requires air vessel for smoothing.

Reaction Turbine (Low Frequency)

• Pressure energy → kinetic energy → mechanical work.

• Examples: Francis, Kaplan turbines.

• Water flows over fixed blades ( stator ) then rotating blades ( rotor ), generating torque.

Biofluid Mechanics (Low Frequency)

• Blood flow in veins: Low Reynolds number, often laminar; influenced by gravity, valves.

• Heart pumping: Pulsatile flow; compliance of arteries dampens pulses.

• Reynolds number in aorta: \(Re \sim 2000\), transitional; laminar in capillaries.

8. Advanced Concepts

Fluid Deformation under Sustained Shear Stress

• Newtonian fluids: Constant viscosity, linear \(\tau\) vs. \(\dot{\gamma}\).

• Non-Newtonian:

  • Dilatant: \(\mu \uparrow\) with \(\dot{\gamma} \uparrow\) (e.g., cornstarch-water)

  • Pseudoplastic: \(\mu \downarrow\) with \(\dot{\gamma} \uparrow\) (e.g., paint, blood)

  • Bingham plastic: Yield stress required (e.g., toothpaste).

Key Formulas Boxed

1. Newton’s Law: \(\boxed{\tau = \mu \frac{du}{dy}}\)

2. Continuity (incompressible): \(\boxed{\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0}\)

3. Euler’s Equation: \(\boxed{-dp = \rho V dV + \rho g dz}\)

4. Bernoulli’s Theorem: \(\boxed{\frac{p}{\rho} + \frac{V^2}{2} + gz = \text{constant}}\)

5. Laminar Pipe Flow (Hagen-Poiseuille): \(\boxed{Q = \frac{\pi R^4 \Delta p}{8\mu L}}\)

6. Boundary Layer Shear Stress (laminar): \(\boxed{\tau_w = \mu \left.\frac{du}{dy}\right|_{y=0} = 0.332 \rho U^2 / \sqrt{Re_x}}\)

7. Manometer \(\Delta p\): \(\boxed{\Delta p = (\rho_m - \rho_f) g h}\)

8. Pitot Tube Velocity: \(\boxed{V = \sqrt{\frac{2(p_0 - p_s)}{\rho}}}\)

Diagram References for Study

1. Laminar velocity profile in pipe: Search: "parabolic velocity profile laminar flow pipe"

2. Boundary layer on flat plate: Search: "laminar turbulent boundary layer thickness flat plate"

3. Flow net through earth dam: Search: "flow net seepage under dam"

4. Pitot-static tube: Search: "Pitot tube stagnation static pressure"

5. TEL & HGL in piping system: Search: "energy grade line hydraulic grade line pipe"

Final Exam Strategy:

• May 2024 & 2022 heavily tested viscosity, Bernoulli, continuity, boundary layer, manometers.

• Always state assumptions before applying Bernoulli.

• For continuity problems, write the 3D equation first, then simplify.

• In boundary layer questions, recall \(\delta \propto x/\sqrt{Re_x}\) for laminar.

• For capillary correction in manometers, compute \(h_c\) and adjust pressure.

• Derive Euler → Bernoulli stepwise; it’s a sure 7-mark question.