Fluid Mech. – II (CE-604 (A)) Unit 5 Notes

1.0 FUNDAMENTAL CONCEPTS & PROPERTIES OF FLUIDS

1.1 Definition and Distinction
- Fluid: A substance that deforms continuously under the application of a shear stress, no matter how small.
- Solid: Deforms a finite amount under shear and can sustain shear stress in equilibrium.
- Compressible Fluid: Density changes significantly with pressure (e.g., gases).
- Incompressible Fluid: Density is essentially constant (good approximation for liquids).
1.2 Basic Properties
- Mass Density (ρ): Mass per unit volume, $$\displaystyle \rho = \frac{m}{V} $$ (kg/m³).
- Specific Weight (γ): Weight per unit volume, $$\displaystyle \gamma = \rho g $$ (N/m³).
- Specific Gravity (S): Ratio of density of a fluid to density of standard fluid (water at 4°C for liquids, air for gases). Dimensionless.
1.3 Viscosity & Newton's Law of Viscosity
- Viscosity (μ): Property of a fluid to resist flow due to internal friction.
- 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}}

*   **Dynamic Viscosity (μ):** N·s/m² or Pa·s.

*   **Kinematic Viscosity (ν):** $$\displaystyle \nu = \frac{\mu}{\rho} $$ (m²/s).

*   **Effect of Temperature:**

    *   **Water:** Viscosity **decreases** with increase in temperature.

    *   **Air:** Viscosity **increases** with increase in temperature.

1.4 Surface Tension and Capillarity
- Surface Tension (σ): Force per unit length acting tangentially on the surface of a liquid, due to cohesion.
- Capillarity: Rise or fall of a liquid in a small-diameter tube due to combined effects of cohesion and adhesion.
- Capillary Rise/Fall:

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

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

    (h = rise if θ < 90°, fall if θ > 90°; d = tube diameter)

*   **Pressure Difference Across Interface:**

    *   **Droplet (1 surface):** $$\displaystyle \Delta P = \frac{2\sigma}{R} $$

    *   **Soap Bubble (2 surfaces):** $$\displaystyle \Delta P = \frac{4\sigma}{R} $$

    \boxed{\Delta P = \frac{2\sigma}{R} \text{ (Droplet)}, \quad \Delta P = \frac{4\sigma}{R} \text{ (Bubble)}}

1.5 Historical Development
- Archimedes (250 BC): Principle of buoyancy.
- Stevin (1586): Hydrostatic paradox.
- Torricelli (1643): Barometer.
- Newton (1687): Law of viscosity.
- Bernoulli (1738): Hydrodynamica, Bernoulli's theorem.
- Euler (1755): Equations of motion for inviscid fluid.
- Reynolds (1883): Experiment on transition to turbulence, Reynolds number.

[!TIP] Exam Focus: Capillary rise formula and pressure difference across curved interfaces are frequently tested. Remember the factor of 2 for droplet and 4 for bubble.

2.0 FLUID KINEMATICS (FLOW DESCRIPTION)

2.1 Eulerian vs. Lagrangian Approach
- Eulerian: Describes fluid properties (velocity, pressure) as a function of space and time at a fixed point (field approach). Common in engineering.
- Lagrangian: Follows the history of individual fluid particles as they move. (Less common).
2.2 Classification of Flow
- Steady: Properties do not change with time at any point (∂/∂t = 0).
- Unsteady: Properties change with time.
- Uniform: Properties do not change with distance in the direction of flow.
- Non-uniform: Properties change with distance.
- Laminar: Fluid particles move in smooth, orderly paths (Re < Critical).
- Turbulent: Fluid particles move in irregular, chaotic paths, causing mixing (Re > Critical).
- Rotational: Fluid particles have finite angular velocity (vorticity ≠ 0).
- Irrotational: Fluid particles have zero angular velocity (vorticity = 0).
2.3 Visualizing Flow
- Streamline: Line tangent to velocity vector at every point at a given instant. No flow across streamlines.
- Pathline: Actual path traced by a single fluid particle over time.
- Streakline: Line connecting particles that have passed through a common point (e.g., smoke from a chimney).
- In steady flow: Streamlines = Pathlines = Streaklines.
2.4 Flow Nets
- A grid formed by a family of streamlines and equipotential lines (lines of constant velocity potential φ).
- Orthogonal Intersection Proof: For 2D incompressible flow, $$\displaystyle u = \frac{\partial \phi}{\partial x} = \frac{\partial \psi}{\partial y} $$ and $$\displaystyle v = \frac{\partial \phi}{\partial y} = -\frac{\partial \psi}{\partial x} $$. The slopes are $$\displaystyle \frac{dy}{dx} = \frac{u}{v} $$ (streamline) and $$\displaystyle \frac{dy}{dx} = -\frac{v}{u} $$ (equipotential). Product of slopes = -1 → orthogonal.
2.5 Velocity Potential (φ) & Stream Function (ψ)
- Velocity Potential (φ): Defined for irrotational flow. $$\displaystyle u = \frac{\partial \phi}{\partial x}, v = \frac{\partial \phi}{\partial y} $$. Satisfies Laplace equation: $$\displaystyle \nabla^2 \phi = 0 $$.
- Stream Function (ψ): Defined for incompressible flow (2D). $$\displaystyle u = \frac{\partial \psi}{\partial y}, v = -\frac{\partial \psi}{\partial x} $$. Automatically satisfies continuity: $$\displaystyle \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0 $$.
- Given φ, find ψ: Integrate $$\displaystyle u = \frac{\partial \phi}{\partial x} = \frac{\partial \psi}{\partial y} $$ w.r.t. y, and $$\displaystyle v = \frac{\partial \phi}{\partial y} = -\frac{\partial \psi}{\partial x} $$ w.r.t. x, and combine.
- Given ψ, find φ: Similar process.
2.6 Continuity Equation
- 3D Cartesian (Steady & Unsteady):

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

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

*   **For Incompressible Flow (ρ = constant):**

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

    \boxed{\nabla \cdot \vec{V} = 0}

*   **Application:** Given u(x,y,z) and v(x,y,z), find w(x,y,z) by integrating the continuity equation.

[!TIP] Common Pitfall: Forgetting the unsteady term $ \partial \rho / \partial t $ in the general continuity equation. For incompressible flow, it simplifies to divergence-free condition.

3.0 FLUID DYNAMICS (FORCES & ENERGY)

3.1 Forces in Fluid Flow
- Body Forces: Act throughout the volume (e.g., gravity, electromagnetic).
- Surface Forces: Act on the surface (e.g., pressure, viscous shear stress).
3.2 Euler's Equation of Motion (Along a Streamline)
- Assumptions: Inviscid (μ=0), steady, incompressible, along a streamline.
- Derivation: Apply Newton's 2nd law to a fluid element along a streamline, considering pressure and gravity forces.
- Form:

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

*   **Limitation:** Not valid in boundary layers or for viscous flows.

3.3 Bernoulli's Theorem
- Derived from Euler's equation by integrating along a streamline for steady, incompressible, inviscid flow.
- Statement: Total mechanical energy per unit weight is constant along a streamline.
- Equation:

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

    \boxed{\frac{P}{\gamma} + \frac{V^2}{2g} + z = H}

    (H = Total Head)

*   **Assumptions:** Steady, incompressible, inviscid, along a streamline, no shaft work.

*   **Physical Interpretation:**

    *   $ z $: Elevation Head (potential energy)

    *   $ P/\gamma $: Pressure Head (pressure energy)

    *   $$\displaystyle V^2/2g $$: Velocity Head (kinetic energy)

*   **Applications:** Venturi meter, nozzle, orifice meter, flow measurement.

3.4 Pascal's Law
- Statement: Pressure applied to an enclosed fluid is transmitted undiminished to every point in the fluid and to the walls of the container.
- Proof: Consider a fluid element in equilibrium. Pressure forces on opposite faces must balance. For a prism, equilibrium in x-direction gives $$\displaystyle P_1 A_1 = P_2 A_2 \sin\theta $$, and in z-direction gives $$\displaystyle W = P_3 A_3 - P_2 A_2 \cos\theta $$. Combining shows $$\displaystyle P_1 = P_2 = P_3 $$ if the fluid is at rest.
- Applications: Hydraulic press, hydraulic brakes, hydraulic jacks.
3.5 Pressure Measurement
- Pressure Head: Expressing pressure as the height of a liquid column that would produce that pressure. $$\displaystyle P = \gamma h $$.
- Manometers:
  - Principle: Balance pressure difference with weight of liquid column.
  - Simple Manometer: Measures gauge pressure at a point relative to atmosphere.
  - Differential Manometer: Measures pressure difference between two points.

$$ P_1 - P_2 = (\rho_m - \rho) g h $$

(for U-tube with lighter fluid above)

    \boxed{P_1 - P_2 = (\rho_m - \rho) g h}

[!TIP] Exam Tip: In manometer problems, always draw a sketch, label pressures, and move systematically from one point to another, adding/subtracting ρgh terms.

4.0 VISCOUS FLOW & BOUNDARY LAYER

4.1 Laminar Flow in Circular Pipes (Hagen-Poiseuille Flow)
- Velocity Distribution (Parabolic):

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

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

*   **Shear Stress Distribution:** $$\displaystyle \tau = \mu \frac{du}{dr} = -\frac{\Delta P}{2L} r $$. Linear from 0 at center to max at wall.

$$ \tau(r) = -\frac{\Delta P}{2L} r $$

    \boxed{\tau_w = \frac{\Delta P R}{2L}}

*   **Mean vs. Maximum Velocity:** $$\displaystyle V_{mean} = \frac{1}{2} u_{max} $$.

*   **Volumetric Flow Rate (Hagen-Poiseuille Equation):**

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

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

4.2 Boundary Layer Theory
- Concept: Thin region near a solid boundary where velocity changes from zero (at wall, no-slip) to free-stream value $$\displaystyle U_\infty $$.
- Development over Flat Plate:
  - Laminar Region: Smooth, ordered motion (Re_x < 5×10⁵).
  - Transition Region: Unstable, intermittent turbulence.
  - Turbulent Region: Chaotic, mixing motion (Re_x > 5×10⁵).
- Boundary Layer Thickness (δ): Distance from wall where velocity ≈ 0.99$$\displaystyle U_\infty $$.

$$ \delta \approx \frac{5.0 x}{\sqrt{Re_x}} \quad \text{(Laminar)} $$

    \boxed{\delta_{lam} \approx \frac{5.0 x}{\sqrt{Re_x}}}

*   **Displacement Thickness (δ*):** Distance by which the external potential flow is displaced outward due to boundary layer.

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

*   **Momentum Thickness (θ):** Measure of boundary layer's effect on momentum.

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

*   **Shear Stress & Drag (Friction Drag):** Wall shear stress $$\displaystyle \tau_w = \mu \left( \frac{\partial u}{\partial y} \right)_{y=0} $$. Total drag force $$\displaystyle F_D = \tau_w \times \text{Area} $$.

4.3 Turbulence
- Definition: Irregular, chaotic, three-dimensional fluid motion with rapid mixing and fluctuating velocity.
- Characteristics:
  - Velocity, pressure, and other properties fluctuate randomly about a mean value.
  - Enhanced momentum, heat, and mass transfer.
  - Higher energy dissipation (greater frictional losses).
- Reynolds Experiment (1883): Demonstrated transition from laminar to turbulent flow in a glass pipe with dye. Introduced Reynolds Number (Re) as criterion.
- Critical Reynolds Number: $$\displaystyle Re_{crit} \approx 2000 $$ for pipe flow (laminar-turbulent transition).

[!TIP] Problem-Solving: For flat plate boundary layer, remember the formulas for δ, δ*, θ are for laminar flow only. Turbulent formulas are different and usually given.

5.0 ENERGY CONCEPTS & FLOW VISUALIZATION TOOLS

5.1 Total Energy Line (TEL) / Energy Grade Line (EGL)
- Definition: Graphical representation of the total head (Total Energy per unit weight) along a flow system.
- Expression: $$\displaystyle TEL = z + \frac{P}{\gamma} + \frac{V^2}{2g} = H $$
- Slope: Equals the head loss (hf) per unit length for uniform pipe flow. Always decreases in the direction of flow due to friction.
- With Pumps/Turbines: Jumps up by head added (pump) or drops by head extracted (turbine).
5.2 Hydraulic Grade Line (HGL)
- Definition: Graphical representation of the pressure head + elevation head.
- Expression: $$\displaystyle HGL = z + \frac{P}{\gamma} $$
- Relationship with TEL: $$\displaystyle TEL = HGL + \frac{V^2}{2g} $$. The vertical distance between TEL and HGL equals the velocity head.
- Interpretation:
  - For pipes of constant diameter, HGL is parallel to TEL (since V constant).
  - HGL can rise or fall above the pipe centerline (e.g., in a vacuum).
  - Never intersects the TEL (except at a point where V=0).
5.3 Pitot-Static Tube (Pilot Tube)
- Construction: Two concentric tubes. Impact port (facing flow) for stagnation pressure, static ports (on side, perpendicular to flow) for static pressure.
- Working Principle:
  1. Stagnation Pressure (P₀): Measured at the impact port where flow comes to rest (V=0). $$\displaystyle P_0 = P + \frac{1}{2}\rho V^2 $$.
  2. Static Pressure (P): Measured at static ports, equal to the static pressure of the flow.
- Velocity Determination:

$$ V = \sqrt{\frac{2(P_0 - P)}{\rho}} = \sqrt{2g \Delta h} $$

(if using manometer with differential height Δh of liquid of density ρ_m).

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

[!TIP] Common Mistake: Confusing Pitot tube (measures stagnation pressure only) with Pitot-static tube (measures both stagnation and static). The latter gives direct velocity.

6.0 APPLICATIONS OF FLUID MECHANICS PRINCIPLES

6.1 Pumps - Reciprocating Pump (Single Acting)
- Working Principle: Positive displacement pump. A piston moves back and forth in a cylinder.
- Suction Stroke: Piston moves back, suction valve opens, atmospheric pressure pushes liquid into cylinder.
- Delivery Stroke: Piston moves forward, suction valve closes, delivery valve opens, liquid is forced out.
- Flow: Pulsating (not continuous). Requires air vessel for smoothing.
- Sketch: Show cylinder, piston, suction valve, delivery valve, crank mechanism.
6.2 Turbines - Reaction Turbine (Francis Turbine)
- Working Principle: Pressure energy of water is converted to kinetic energy in the stationary guide mechanism (wicket gates). Water then strikes the moving runner blades, changing momentum. Pressure drop occurs both in guide vanes and on runner blades.
- Flow: Radial inward (water enters at outer periphery, exits at center).
- Sketch: Show spiral casing, guide vanes, runner, draft tube (recovers pressure).
6.3 Buoyancy & Floatation - Archimedes' Principle
- Statement: A body submerged in a fluid experiences an upward buoyant force equal to the weight of the fluid displaced by the body.

$$ F_B = \gamma \cdot \text{Volume Displaced} = \rho_f g V_d $$

*   **Floating Body:** $$\displaystyle F_B = \text{Weight of body} $$. $$\displaystyle \rho_f V_d = \rho_b V_b $$.

*   **Problem Solving (Given heights in two fluids):** For a floating block, weight = buoyancy in each fluid. Let total height = H, projected height in fluid 1 = h₁, in fluid 2 = h₂. Then:

$$ \rho_b H = \rho_f h_1 \quad \text{and} \quad \rho_b H = \rho_f' h_2 $$

    Solve for H and ρ_b.

6.4 Capillary Effects in Measurement
- Capillary Rise/Fall in Manometers: In narrow tubes, meniscus curvature causes pressure error.

$$ h_{cap} = \frac{4\sigma \cos\theta}{\rho_m g d} $$

(for manometric liquid of density ρ_m in tube of diameter d).

*   **Error in Pressure Measurement:** True pressure difference $$\displaystyle \Delta P_{true} = \rho_m g (h_{man} - h_{cap}) $$ if capillary rise occurs.

*   **Minimum Tube Diameter for Limited Error:**

    Given allowable error % in ΔP, i.e., $$\displaystyle \frac{h_{cap}}{h_{man}} \leq \text{fraction} $$. Solve for d.

$$ d \geq \frac{4\sigma \cos\theta}{\rho_m g h_{man} \cdot (\text{allowable fraction})} $$

    \boxed{d_{min} = \frac{4\sigma \cos\theta}{\rho_m g h_{man} \cdot \epsilon}}

6.5 Bio-Fluid Mechanics
- Blood Flow in Veins (Laminar Flow): Assumed laminar in most vessels. Follows Poiseuille's Law for flow in a cylindrical vessel: