FLUID MECHANICS (ME-404) - Unit 4 Short Notes

UNIT 4: FLUID MECHANICS (Comprehensive Short Notes)

1.0 FUNDAMENTAL FLUID PROPERTIES & CLASSIFICATION

1.1 Newtonian vs. Non-Newtonian Fluids

• Newton's Law of Viscosity: The shear stress ($\tau$) on a fluid element is directly proportional to the rate of shear strain (velocity gradient).

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

where $\mu$ is the **dynamic viscosity**.

• Newtonian Fluid: Follows Newton's law. $\mu$ is constant at a given $T,P$. Linear $\tau$ vs. $du/dy$ curve passing through origin. Examples: Water, air, most common oils.

• Non-Newtonian Fluids: Do not follow Newton's law. $\mu$ is not constant.

  • Shear-Thinning (Pseudoplastic): $\mu$ decreases with increasing $du/dy$. Curve is concave downward. Examples: Paint, blood, ketchup.

  • Shear-Thickening (Dilatant): $\mu$ increases with increasing $du/dy$. Curve is concave upward. Example: Cornstarch in water (oobleck).

  • Bingham Plastic: Behaves as a solid until $\tau$ exceeds a yield value $$\displaystyle \tau_y $$, then flows like a Newtonian fluid. Example: Toothpaste, mud.

[!TIP] Exam often asks to sketch $\tau$ vs. $du/dy$ for all types. Remember Bingham plastics have a finite intercept on the $\tau$-axis.

1.2 Viscosity

• Dynamic Viscosity ($\mu$): Measure of a fluid's internal resistance to flow. Units: Pa·s (N·s/m²), poise (1 poise = 0.1 Pa·s).

• Kinematic Viscosity ($\nu$): Ratio of dynamic viscosity to density. $$\displaystyle \nu = \mu / \rho $$. Represents momentum diffusivity. Units: m²/s, stokes (1 stokes = 10⁻⁴ m²/s).

• Variation:

  • Liquids: $\mu$ decreases with temperature, $\nu$ increases.

  • Gases: $\mu$ increases with temperature, $\nu$ also increases (since $\rho$ decreases more slowly).

1.3 Other Fluid Properties

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

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

For finite changes: $$\displaystyle K = \frac{\Delta P}{\Delta V / V} $$. **Units:** Pa.

> **Numerical:** *If pressure increases from 70 N/cm² to 130 N/cm² and volume decreases by 0.15%, find K.*

> $$\displaystyle \Delta P = 60 \times 10^4 $$ N/m², $$\displaystyle \Delta V/V = -0.0015 $$. $$\displaystyle K = \frac{60 \times 10^4}{0.0015} = 4 \times 10^8 $$ N/m².

• Vapor Pressure: Pressure exerted by vapor in equilibrium with its liquid at a given temperature. Significance: If local pressure in a flowing fluid drops below vapor pressure, cavitation occurs (formation and collapse of vapor bubbles), causing damage.

• Surface Tension & Capillarity: Surface tension ($\sigma$) is force per unit length along an interface. Capillarity is the rise/fall of liquid in a narrow tube due to surface tension and wetting.

2.0 FLOW KINEMATICS & DESCRIPTION

2.1 Flow Classification & Patterns

2.2 Visualizing Flow: Lines and Tubes

2.3 Mathematical Description of Flow

• Stream Function ($\psi$): Exists for incompressible flow. Defined such that:

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

*   **Properties:** Lines of constant $\psi$ are **streamlines**. Volume flow rate between two streamlines = $\Delta \psi$. For irrotational flow, $\psi$ satisfies **Laplace equation**: $$\displaystyle \nabla^2 \psi = 0 $$.

• Velocity Potential Function ($\phi$): Exists for irrotational flow ($$\displaystyle \nabla \times \mathbf{V} = 0 $$). Defined such that:

$$u = \frac{\partial \phi}{\partial x}, \quad v = \frac{\partial \phi}{\partial y}$$

*   **Properties:** Lines of constant $\phi$ are **equipotential lines**. For incompressible flow, $\phi$ satisfies **Laplace equation**: $$\displaystyle \nabla^2 \phi = 0 $$.

• Orthogonality of $\psi$ and $\phi$:

$$\nabla \phi \cdot \nabla \psi = \left( \frac{\partial \phi}{\partial x}\frac{\partial \psi}{\partial x} + \frac{\partial \phi}{\partial y}\frac{\partial \psi}{\partial y} \right) = (u)(-v) + (v)(u) = 0$$

Hence, families of $\phi$=const and $\psi$=const are orthogonal.

• Circulation ($\Gamma$) & Vorticity ($\zeta$):

$$\Gamma = \oint_C \mathbf{V} \cdot d\mathbf{s} \quad \text{(Line integral around a closed curve C)}$$

$$\zeta = \nabla \times \mathbf{V} = \left( \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y} \right) \mathbf{k} \quad \text{(2D, out of plane)}$$

Relationship: $$\displaystyle \zeta = \lim_{dA \to 0} \frac{\Gamma}{dA} $$.

2.4 Special Flow Features

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

• Stagnation Properties (for compressible flow):

  • Stagnation Pressure ($$\displaystyle P_0 $$): Pressure when fluid is brought to rest isentropically.

  • Stagnation Temperature ($$\displaystyle T_0 $$): Temperature when fluid is brought to rest isentropically. For perfect gas, $$\displaystyle T_0 = T(1 + \frac{\gamma-1}{2}M^2) $$.

3.0 GOVERNING EQUATIONS & PRINCIPLES

3.1 System vs. Control Volume

• System (Lagrangian): Follows a specific, identifiable mass of fluid. Difficult for engineering analysis.

• Control Volume (Eulerian): Fixed region in space through which fluid flows. More important for most engineering problems (pumps, pipes, turbines) as it deals with fluxes across boundaries.

3.2 Continuity Equation

• General 3D (Integral Form): $$\displaystyle \frac{d}{dt} \int_{CV} \rho dV + \int_{CS} \rho \mathbf{V} \cdot d\mathbf{A} = 0 $$

• Differential Form: $$\displaystyle \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{V}) = 0 $$

• For Incompressible Flow ($$\displaystyle \rho = \text{constant} $$): $$\displaystyle \nabla \cdot \mathbf{V} = 0 $$ or $$\displaystyle \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0 $$.

• Application (1D, Steady): $$\displaystyle A_1 V_1 = A_2 V_2 = Q $$ (Discharge).

3.3 Euler’s Equation of Motion

• Derivation: For a steady, inviscid flow along a streamline, applying Newton's 2nd law to a fluid element.

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

> **Assumptions:** Steady, inviscid ($$\displaystyle \mu=0 $$), incompressible ($\rho$=const), along a streamline.

3.4 Bernoulli’s Theorem

• Statement: For steady, incompressible, inviscid, along-a-streamline flow, the total mechanical energy per unit weight is constant.

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

where $$\displaystyle \frac{P}{\gamma} $$ = pressure head, $$\displaystyle \frac{V^2}{2g} $$ = velocity head, $z$ = elevation head.

• Derivation: Direct integration of Euler's equation along a streamline.

• Physical Significance: Conservation of Mechanical Energy (work done by pressure forces = change in kinetic + potential energy).

• Modification for Real Flows: Include head loss ($$\displaystyle h_L $$) due to viscous effects.

$$\frac{P_1}{\gamma} + \frac{V_1^2}{2g} + z_1 = \frac{P_2}{\gamma} + \frac{V_2^2}{2g} + z_2 + h_L$$

> [!TIP] $$\displaystyle h_L $$ is **always positive**. Bernoulli's equation is applied **between two points on the same streamline** for inviscid flow, or between any two points in a region of **irrotational flow**.

3.5 Navier-Stokes Equations (N-S)

• General 3D Cartesian Form (for incompressible, Newtonian fluid):

$$\rho \left( \frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} + w\frac{\partial u}{\partial z} \right) = -\frac{\partial P}{\partial x} + \mu \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} \right) + \rho g_x$$

(Similar equations for $v$ and $w$ components).

• Explanation of Terms (x-momentum):

  1. $$\displaystyle \rho \frac{\partial u}{\partial t} $$: Local inertia term.

  2. $$\displaystyle \rho (u\frac{\partial u}{\partial x} + ...) $$: Convective inertia term.

  3. $$\displaystyle -\frac{\partial P}{\partial x} $$: Pressure force.

  4. $$\displaystyle \mu (\frac{\partial^2 u}{\partial x^2} + ...) $$: Viscous (diffusion) term.

  5. $$\displaystyle \rho g_x $$: Body force (gravity).

• Uses & Significance: The fundamental governing equations for viscous fluid flow. They are nonlinear PDEs. Exact solutions exist only for simple geometries (Couette, Poiseuille). They form the basis for all Computational Fluid Dynamics (CFD).

4.0 VISCOUS FLOWS & BOUNDARY LAYER THEORY

4.1 Exact Viscous Flow Solutions

• Flow between Parallel Plates (Couette Flow): One plate moves at $U$, other stationary, no pressure gradient.

  Velocity Profile: $$\displaystyle u = \frac{U}{h} y $$ (Linear). Shear stress: $$\displaystyle \tau = \mu \frac{U}{h} $$ (constant).

• Flow between Parallel Plates (Plane Poiseuille Flow): Both plates stationary, flow driven by pressure gradient $$\displaystyle (-\frac{dP}{dx}) $$.

  Velocity Profile: $$\displaystyle u = \frac{1}{2\mu} \left( -\frac{dP}{dx} \right) (hy - y^2) $$ (Parabolic). Max at center ($$\displaystyle y=h $$).

• Flow through Circular Pipes (Hagen-Poiseuille Flow):

  • Governing: N-S in cylindrical coords, simplifies to $$\displaystyle \frac{1}{r}\frac{d}{dr}\left(r \frac{du}{dr}\right) = \frac{1}{\mu}\frac{dP}{dx} $$.

  • Velocity Distribution (Parabolic):

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

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

*   **Mean Velocity:** $$\displaystyle U_{avg} = \frac{Q}{\pi R^2} = \frac{\Delta P R^2}{8\mu L} $$. **Relationship:** $$\displaystyle u_{max} = 2 U_{avg} $$.

*   **Shear Stress:** $$\displaystyle \tau(r) = \frac{r}{2} \frac{\Delta P}{L} $$. Max at wall ($$\displaystyle r=R $$): $$\displaystyle \tau_w = \frac{R}{2} \frac{\Delta P}{L} $$.

*   **Volumetric Flow Rate (Poiseuille's Law):**

$$Q = \frac{\pi \Delta P R^4}{8\mu L} = \frac{\pi d^4 \Delta P}{128 \mu L}$$

    \boxed{Q = \frac{\pi d^4 \Delta P}{128 \mu L}}

4.2 Boundary Layer Fundamentals

• Concept: Region near a solid boundary where velocity gradients are large and viscous effects are significant. Outside is potential flow (inviscid).

• Thicknesses:

  • Boundary Layer Thickness ($\delta$): $$\displaystyle u(x, \delta) = 0.99 U_\infty $$ (arbitrary but standard).

  • Displacement Thickness ($$\displaystyle \delta^* $$): Distance by which the external streamlines are displaced outward due to boundary layer formation.

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

*   **Momentum Thickness ($\theta$):** Represents the "missing" momentum flux due to boundary layer.

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

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

• Laminar Boundary Layer (Flat Plate):

  • Blasius Solution (Exact): $$\displaystyle \frac{u}{U_\infty} = f'(\eta) $$, $$\displaystyle \eta = y \sqrt{\frac{U_\infty}{\nu x}} $$. $$\displaystyle \delta \propto \sqrt{\frac{\nu x}{U_\infty}} $$.

  • Approximate Profiles & Results:

    | Profile | $$\displaystyle \frac{u}{U_\infty} $$ | $\delta$ | $$\displaystyle \delta^* $$ | $\theta$ | $$\displaystyle C_f $$ (Skin Friction) | | :--- | :--- | :--- | :--- | :--- | :--- | | Linear | $y/\delta$ | $$\displaystyle \frac{5.0 x}{\sqrt{Re_x}} $$ | $$\displaystyle \frac{2}{3}\delta $$ | $$\displaystyle \frac{1}{6}\delta $$ | $$\displaystyle \frac{1.33}{\sqrt{Re_x}} $$ | | Parabolic | $$\displaystyle 2(y/\delta - (y/\delta)^2) $$ | $$\displaystyle \frac{5.0 x}{\sqrt{Re_x}} $$ | $$\displaystyle \frac{2}{3}\delta $$ | $$\displaystyle \frac{2}{15}\delta $$ | $$\displaystyle \frac{0.773}{\sqrt{Re_x}} $$ |

    Numerical: For $$\displaystyle u/U = 2(y/\delta - (y/\delta)^2) $$, find $$\displaystyle \delta^* $$ and $\theta$.

    $$\displaystyle \delta^* = \int_0^\delta (1 - 2\eta + \eta^2) \delta d\eta = \delta \left[\eta - \eta^2 + \frac{\eta^3}{3}\right]_0^1 = \frac{2}{3}\delta $$.

    $$\displaystyle \theta = \int_0^\delta (2\eta - 2\eta^2)(1 - 2\eta + \eta^2) \delta d\eta = \frac{2}{15}\delta $$.

• Turbulent Boundary Layer (1/7th Power Law):

$$\frac{u}{U_\infty} = \left(\frac{y}{\delta}\right)^{1/7}$$

$$\displaystyle \delta^* \approx 0.125 \delta $$, $\theta \approx 0.036 \delta$.

4.3 Boundary Layer Separation

• Cause: Adverse Pressure Gradient ($$\displaystyle dP/dx > 0 $$ or $$\displaystyle dU/dx < 0 $$). Viscous shear stress near wall becomes insufficient to overcome the increasing pressure, causing flow reversal.

• Effect of Pressure Gradient:

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

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

• Consequences: Large pressure drag (form drag), flow losses, stall on airfoils.

• Methods to Delay Separation:

  1. Streamlining: Reduce adverse pressure gradient.

  2. Suction: Remove low-momentum fluid from near wall.

  3. Favorable Pressure Gradient: Design with gradual area increase.

  4. Boundary Layer Tripping: Induce early transition to turbulent (turbulent has more momentum near wall, better at overcoming APG).

5.0 PIPE FLOW & NETWORK ANALYSIS

5.1 Losses in Pipe Flow

• Major Losses (Frictional): Due to shear stress at the wall over the pipe length.

  Darcy-Weisbach Equation:

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

where $f$ = **friction factor** (function of $Re$ and relative roughness $\epsilon/D$). Found from **Moody Chart**.

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

$$V = 0.849 C R^{0.63} S^{0.54}$$

($C$ = Hazen-Williams coefficient, $R$ = hydraulic radius, $$\displaystyle S = h_f/L $$).

• Minor Losses (Local): Due to disturbances (fittings, valves, entrances, exits, area changes).

  General Form: $$\displaystyle h_m = K \frac{V^2}{2g} $$, where $K$ = loss coefficient (experimental).

  • Sudden Contraction: $K \approx 0.5$ (for sharp-edged).

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

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

  • Exit: $$\displaystyle K = 1.0 $$.

  • Bends, Valves, etc.: Given in tables.

5.2 Hydraulic and Energy Lines

• Total Energy Line (TEL): Represents total head ($$\displaystyle \frac{P}{\gamma} + \frac{V^2}{2g} + z $$) along the pipe. Slopes downward in direction of flow due to losses.

• Hydraulic Gradient Line (HGL): Represents piezometric head ($$\displaystyle \frac{P}{\gamma} + z $$). Lies below TEL by $$\displaystyle \frac{V^2}{2g} $$.

  Interpretation: Distance between TEL and HGL = velocity head. Vertical distance between HGL and pipe centerline = pressure head (if pipe is full).

5.3 Pipe Systems

• Pipes in Series: Same flow rate $Q$. Total head loss: $$\displaystyle h_{L,total} = \sum h_{f,i} + \sum h_{m,i} $$. Can use equivalent length method for minor losses.

• Pipes in Parallel: Same head loss $$\displaystyle h_L $$ across each pipe. Total discharge: $$\displaystyle Q = Q_1 + Q_2 + ... $$. Equivalent resistance: $$\displaystyle \frac{1}{\sqrt{f_1 L_1/D_1}} = \frac{1}{\sqrt{f_2 L_2/D_2}} $$ for same $D$? No, solve using $$\displaystyle h_L = f \frac{L}{D} \frac{V^2}{2g} $$ and $$\displaystyle Q = AV $$.

  Problem: Given two pipes (300mm & 150mm, both 300m long) for $$\displaystyle Q=0.085 $$ m³/s. Find ratio of head loss (series) to (parallel).

  Series: $$\displaystyle h_s = h_1 + h_2 = f_1 \frac{L_1}{D_1} \frac{V_1^2}{2g} + f_2 \frac{L_2}{D_2} \frac{V_2^2}{2g} $$.

  Parallel: $$\displaystyle h_p = h_1 = h_2 \Rightarrow f_1 \frac{L}{D_1} \frac{V_1^2}{2g} = f_2 \frac{L}{D_2} \frac{V_2^2}{2g} $$. Find $$\displaystyle V_1, V_2 $$ from $$\displaystyle Q=Q_1+Q_2 $$, then $$\displaystyle h_p $$. Compute ratio $$\displaystyle h_s/h_p $$.

6.0 HYDROSTATICS & BUOYANCY

6.1 Pressure Distribution in Static Fluids

• Basic Equation: $$\displaystyle dP = -\rho g dz $$. For incompressible fluid: $$\displaystyle P = P_0 + \rho g h $$ (where $h$ is depth below free surface).

• Pressure on Planes:

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

  • Vertical/Inclined Plane: $P$ varies linearly with depth. Total Force: $$\displaystyle F_R = P_c A $$, where $$\displaystyle P_c $$ = pressure at centroid of the area.

• Manometers: Measure pressure difference by balancing fluid columns.

  • U-tube: $$\displaystyle \Delta P = (\rho_m - \rho) g h $$ (if one side open to atmosphere).

  • Inverted U-tube (Differential): $$\displaystyle \Delta P = \rho g (h_1 - h_2) $$ (if both legs contain same fluid).

  • Inclined Manometer: Increased sensitivity, $$\displaystyle \Delta P = \rho g l \sin\theta $$ ($l$ = length along incline).

6.2 Hydrostatic Force on Surfaces

• Plane Surface:

  • Force: $$\displaystyle F_R = P_c A = \rho g \bar{h} A $$ ($\bar{h}$ = depth of centroid).

  • Center of Pressure (CP): Point of action of $$\displaystyle F_R $$. For vertical plane: $$\displaystyle y_{cp} = \bar{y} + \frac{I_{xx,c}}{A \bar{y} \bar{h}} $$ where $$\displaystyle I_{xx,c} $$ is 2nd moment of area about centroidal axis.

• Curved Surface: Resolve into horizontal ($$\displaystyle F_H $$ = force on projected vertical area) and vertical ($$\displaystyle F_V $$ = weight of fluid above + buoyancy if submerged) components. $$\displaystyle F_R = \sqrt{F_H^2 + F_V^2} $$.

6.3 Hydrostatic Paradox

• Statement: The hydrostatic thrust on a plane surface depends only on the area and the depth of its centroid ($$\displaystyle P_c A $$), not on the total weight or volume of fluid above it.

• Explanation: Consider a tall, narrow tank vs. a short, wide tank with same base area and same depth of fluid. The pressure at the base is identical ($$\displaystyle P = \rho g h $$), hence base force is identical, even though the total fluid weight differs greatly. The sidewalls exert different vertical force components to account for the weight difference.

6.4 Buoyancy & Stability of Floating Bodies

• Archimedes' Principle: A body immersed in a fluid experiences an upward buoyant force equal to the weight of displaced fluid.

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

• Stability: Determined by relative positions of Center of Gravity (G) and Metacenter (M).

  • Center of Buoyancy (B): Centroid of displaced volume.

  • Metacenter (M): Point where the line of action of buoyant force intersects the original vertical line through B for a small angle of heel.

  • Metacentric Height (GM): $$\displaystyle GM = BM - BG $$, where $$\displaystyle BM = \frac{I}{\nabla} $$.

    • $I$ = second moment of area of waterplane about axis of tilting.

    • $\nabla$ = volume of displaced fluid (submerged volume).

  • Condition:

    • $$\displaystyle GM > 0 $$: Stable (M above G, righting moment).

    • $$\displaystyle GM = 0 $$: Neutral (M at G).

    • $$\displaystyle GM < 0 $$: Unstable (M below G, overturning moment).

  • Period of Oscillation: $$\displaystyle T \propto \frac{1}{\sqrt{GM}} $$ (for small angles).

• Why G and B Alone Are Insufficient: For a floating body, as it heels, B moves sideways. The new line of action of $$\displaystyle F_B $$ may pass to the left or right of G. Stability depends on whether this new line of action creates a righting or overturning moment, which is determined by the position of M, not the original B.

6.5 Forced Vortex Flow

• Definition: Rotational flow where fluid rotates as a solid body (angular velocity $\omega$ constant) due to external torque (e.g., rotating tank).

• Free Surface Profile: From $$\displaystyle dP = \rho ( \omega^2 r dr - g dz ) $$ and $$\displaystyle P = \text{const} $$ on free surface ($$\displaystyle P = P_{atm} $$):

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

**Parabolic shape.**

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

7.0 FLOW MEASUREMENT DEVICES

7.1 Venturi Meter

• Principle: Based on Bernoulli's equation and continuity between sections 1 (inlet) and 2 (throat).

• Derivation (with losses):

$$\frac{P_1}{\gamma} + \frac{V_1^2}{2g} = \frac{P_2}{\gamma} + \frac{V_2^2}{2g} + h_L$$

$$\displaystyle h_L = K \frac{(V_1 - V_2)^2}{2g} $$ (often $K \approx 0.05$ for venturi). Let $$\displaystyle \Delta h = \frac{P_1 - P_2}{\gamma_m} $$ (manometer reading of fluid $$\displaystyle \rho_m $$).

After substitution and solving for $Q$:

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

where $$\displaystyle C_d $$ = discharge coefficient (<1, accounts for losses and contraction).

• Without losses ($$\displaystyle C_d=1 $$): $$\displaystyle Q = \frac{A_1 A_2}{\sqrt{A_1^2 - A_2^2}} \sqrt{2g \Delta h} $$.

7.2 Orifice & Nozzle Meters

• Principle: Same as venturi, but with higher energy loss due to abrupt contraction.

• Discharge Equation: $$\displaystyle Q = C_d A_2 \sqrt{2g \Delta h} $$ (orifice, $$\displaystyle A_2 $$ = orifice area). $$\displaystyle C_d $$ is significantly lower than venturi (~0.6-0.7).

7.3 Pitot-Static Tube

• Principle: Measures stagnation pressure ($$\displaystyle P_0 = P + \frac{1}{2}\rho V^2 $$) at the tip (facing flow) and static pressure ($P$) via side ports. Velocity:

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

Used for flow velocity measurement in pipes and open channels.

8.0 FORCES IN FLUID FLOW & APPLICATIONS

8.1 Linear Momentum Equation (Control Volume)

• General (Steady Flow): $$\displaystyle \sum \mathbf{F} = \dot{m} (\mathbf{V}_{out} - \mathbf{V}_{in}) = \rho Q (\mathbf{V}_{out} - \mathbf{V}_{in}) $$

  $$\displaystyle \sum \mathbf{F} $$ includes surface forces (pressure, viscous) and body forces (gravity). Often gravity cancels if inlet/exit at same elevation.

• Application to Pipe Bend:

  • Choose CV around bend.

  • Forces: Pressure forces at inlet/exit, reaction force from bend ($$\displaystyle \mathbf{F}_R $$).

  • $$\displaystyle \mathbf{F}_R + \mathbf{F}_{pressure} = \rho Q (\mathbf{V}_2 - \mathbf{V}_1) $$.

  • Solve for magnitude and direction of $$\displaystyle \mathbf{F}_R $$.

8.2 Angular Momentum Equation (Control Volume)

• General (Steady Flow, about an axis): $$\displaystyle \sum T = \dot{m} (r V_t)_{out} - \dot{m} (r V_t)_{in} = \rho Q (r_2 V_{t2} - r_1 V_{t1}) $$

  $$\displaystyle \sum T $$ includes torque from shaft, blades, etc.

• Application to Lawn Sprinkler:

  • CV around rotating arm.

  • Torque due to fluid ($$\displaystyle T_{fluid} $$) acts on CV. Friction torque ($$\displaystyle T_f $$) acts on arm from axis.

  • For steady rotation: $$\displaystyle T_f = \rho Q r_2 V_{t2} $$ (assuming $$\displaystyle V_{t1}=0 $$ at inlet from pipe).

  • Given $$\displaystyle Q, r_2, V_2 $$ (from $$\displaystyle Q = A_2 V_2 $$), find $$\displaystyle T_f $$.

8.3 Drag and Lift Forces

• Definitions:

  • Drag Force ($$\displaystyle F_D $$): Component parallel to free stream velocity.

  • Lift Force ($$\displaystyle F_L $$): Component perpendicular to free stream velocity.

  • Coefficients: $$\displaystyle C_D = \frac{F_D}{\frac{1}{2}\rho V^2 A} $$, $$\displaystyle C_L = \frac{F_L}{\frac{1}{2}\rho V^2 A} $$ ($A$ = reference area).

• Magnus Effect: Generation of lift on a rotating cylinder/sphere due to pressure difference created by circulation (Kutta-Joukowski theorem: $$\displaystyle L' = \rho V \Gamma $$ per unit span).

• Airfoil: Streamlined shape. Lift generated by pressure difference between upper (low P, high V) and lower (high P, low V) surfaces, explained by Bernoulli and circulation.

• Streamline Body vs. Bluff Body:

  • Streamline: Flow remains attached, separation delayed. Low pressure drag, significant friction drag. (e.g., airfoil at small AoA).

  • Bluff: Early separation, large wake. High pressure drag (form drag) dominates. (e.g., sphere, cylinder, flat plate normal to flow).

• Friction Drag vs. Pressure Drag:

  • Friction Drag: Due to shear stress on surface. Dominant for streamlined bodies.

  • Pressure Drag (Form Drag): Due to pressure differential between front and rear. Dominant for bluff bodies.

  Numerical: Parachute descent: $$\displaystyle W = F_D = \frac{1}{2}\rho V^2 C_D A $$. Given $$\displaystyle W, \rho, C_D $$, find required area $A$ for terminal velocity $V$.

9.0 COMPRESSIBLE FLOW (INTRODUCTORY)

9.1 Mach Number (M)

$$M = \frac{V}{a} = \frac{V}{\sqrt{\gamma R T}}$$

• Significance:

  • $$\displaystyle M < 0.3 $$: Density changes < 5%, can treat as incompressible.

  • $M \approx 1$: Transonic (choked flow, shock waves possible).

  • $$\displaystyle M > 1 $$: Supersonic (shock waves, expansion fans).

  • $M \ll 1$: Subsonic.

9.2 Fanno Lines

• Model: Adiabatic ($$\displaystyle Q=0 $$), frictional flow in a constant area duct.

• Properties:

  • Stagnation pressure decreases (due to friction).

  • Entropy increases (irreversible).

  • For given $M$, $fL/D$ is fixed. Maximum possible $fL/D$ occurs at $$\displaystyle M=1 $$ (choked condition).

  • Choking: If $$\displaystyle M<1 $$, friction increases $M$. If $$\displaystyle M>1 $$, friction decreases $M$. Flow tends toward $$\displaystyle M=1 $$. If downstream conditions prevent $$\displaystyle M=1 $$, flow is subsonic throughout.

• Sketch: Plot of $M$ vs. $$\displaystyle P_0/P^* $$ or $$\displaystyle T_0/T^* $$ along Fanno line.

9.3 Rayleigh Lines

• Model: Flow with heat addition/rejection ($Q \neq 0$), frictionless, in constant area duct.

• Properties:

  • Stagnation temperature changes ($$\displaystyle T_0 $$ increases with heat addition).

  • Entropy changes (increases with heat addition, decreases with heat rejection).

  • For given $M$, $Q$ is fixed. Maximum possible $Q$ occurs at $$\displaystyle M=1 $$.

  • Choking: Heat addition increases $M$ for subsonic, decreases $M$ for supersonic. Flow tends toward $$\displaystyle M=1 $$.

• Sketch: Plot of $M$ vs. $$\displaystyle T_0/T^* $$ along Rayleigh line.

9.4 Isentropic Flow through Nozzles

• Mass Flow Rate from Large Tank (Stagnation Conditions $$\displaystyle P_0, T_0 $$):

$$\dot{m} = \rho A V = \frac{P_0 A}{\sqrt{T_0}} \sqrt{\frac{\gamma}{R}} M \left(1 + \frac{\gamma-1}{2}M^2\right)^{-\frac{\gamma+1}{2(\gamma-1)}}$$

• Choked Flow Condition: At the throat ($$\displaystyle A_{min} $$), $$\displaystyle M=1 $$ is achieved when $$\displaystyle \frac{P^*}{P_0} = \left(\frac{2}{\gamma+1}\right)^{\gamma/(\gamma-1)} $$. Maximum mass flow rate is attained. Further decrease in exit pressure ($$\displaystyle P_e < P^* $$) does not increase $\dot{m}$; shocks may form in the divergent section.