Fluid Mechanics-I (CE-501) - Unit 4 Short Notes

1.0 FUNDAMENTAL PROPERTIES & PRESSURE MEASUREMENT

1.1 Fluid Properties

• Density ($\rho$): Mass per unit volume, $$\displaystyle \rho = \frac{m}{V} $$, SI: kg/m³.

• Specific Weight ($\gamma$): Weight per unit volume, $$\displaystyle \gamma = \rho g $$, SI: N/m³.

• Specific Volume ($v$): Volume per unit mass, $$\displaystyle v = \frac{1}{\rho} $$, SI: m³/kg.

• Specific Gravity (SG): Ratio of fluid density to water density at 4°C, dimensionless.

• Viscosity:

  • Dynamic Viscosity ($\mu$): Measure of internal friction, Newton's Law: $$\displaystyle \tau = \mu \frac{du}{dy} $$, SI: N·s/m² (Pa·s).

  • Kinematic Viscosity ($\nu$): $$\displaystyle \nu = \frac{\mu}{\rho} $$, SI: m²/s (Stoke: 1 St = 10⁻⁴ m²/s).

  • Temperature Variation: For liquids, $\mu$ ↓ with T ↑; for gases, $\mu$ ↑ with T ↑.

• Surface Tension ($\sigma$): Force per unit length acting tangentially on surface.

  • Pressure in Curved Surfaces:

    • Droplet (gas inside): $$\displaystyle P_{in} - P_{out} = \frac{2\sigma}{R} $$

    • Bubble (liquid inside & outside): $$\displaystyle P_{in} - P_{out} = \frac{4\sigma}{R} $$

    • Liquid Jet (cylindrical): $$\displaystyle P_{in} - P_{out} = \frac{\sigma}{R} $$

[!TIP] Common Pitfall: For a soap bubble, there are two interfaces (inner & outer), hence factor of 4.

1.2 Pressure Concepts

• Absolute Pressure ($$\displaystyle P_{abs} $$): Measured relative to perfect vacuum.

• Gauge Pressure ($$\displaystyle P_{gauge} $$): Measured relative to atmospheric pressure, $$\displaystyle P_{gauge} = P_{abs} - P_{atm} $$.

• Vacuum Pressure ($$\displaystyle P_{vac} $$): When $$\displaystyle P_{abs} < P_{atm} $$, $$\displaystyle P_{vac} = P_{atm} - P_{abs} $$.

• Pascal's Law: Pressure at a point in a fluid at rest is equal in all directions.

• Intensity of Pressure ($p$): Normal force per unit area, $$\displaystyle p = \frac{dF}{dA} $$.

1.3 Pressure Measurement Devices

• Manometers: Use liquid column height difference.

  • U-tube Manometer: Simple, measures $$\displaystyle P_1 - P_2 $$.

  • Differential Manometer: Measures $$\displaystyle P_1 - P_2 $$ when connected to two pipes. General equation:

$$P_1 + \rho_1 g h_1 = P_2 + \rho_2 g h_2 + \rho_m g h$$

    where $$\displaystyle \rho_m $$ is manometric liquid density.

*   **Inverted U-tube Manometer**: Used for low pressure differences, contains air/gas.

• Mechanical Gauges: Bourdon tube, diaphragm, bellows (for high pressures).

[!EXAMPLE] Differential Manometer Problem (Past Paper Pattern):

Two pipes A (sp.gr. 1.5) and B (sp.gr. 0.9) connected to a differential manometer with Hg (sp.gr. 13.6). $$\displaystyle P_A = 1 $$ kgf/cm², $$\displaystyle P_B = 1.8 $$ kgf/cm². Find $h$.

Solution: Convert pressures to consistent units (e.g., m of water or N/m²). Apply balance:

$$P_A + \rho_A g h_A = P_B + \rho_{Hg} g h + \rho_B g h_B$$

Solve for $h$. \boxed{h = \text{calculated value}}.

2.0 FLOW KINEMATICS & VISUALIZATION

2.1 Flow Descriptions

2.2 Flow Lines & Functions

• Pathline: Actual path traced by a fluid particle over time.

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

• Streamline: Line tangent to velocity vector at a given instant. For steady flow, pathline = streamline = streakline.

• Stream Function ($\psi$):

  • Defined for 2D incompressible flow.

  • Properties: (1) Constant on a streamline, (2) Flow between two streamlines = $$\displaystyle \psi_2 - \psi_1 $$.

  • Velocity Components: $$\displaystyle u = \frac{\partial \psi}{\partial y} $$, $$\displaystyle v = -\frac{\partial \psi}{\partial x} $$.

  • Satisfies continuity automatically: $$\displaystyle \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0 $$.

• Velocity Potential Function ($\phi$):

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

  • Properties: (1) Constant on an equipotential line, (2) Lines of $\phi$ and $\psi$ are orthogonal.

  • Velocity Components: $$\displaystyle u = \frac{\partial \phi}{\partial x} $$, $$\displaystyle v = \frac{\partial \phi}{\partial y} $$.

  • Satisfies Laplace equation: $$\displaystyle \nabla^2 \phi = 0 $$.

• Flow Net: Grid formed by intersecting streamlines and equipotential lines.

  • Significance: Visualizes flow pattern; used in seepage analysis, groundwater flow.

  • Method: Graphical ( sketching), Analytical (solving $\psi$ and $\phi$), or numerical.

2.3 Continuity Equation

• Derivation (3D Cartesian): For a differential fluid element, mass inflow = mass outflow.

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

For steady, incompressible flow ($\rho$ = constant):

$$\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0 \quad \boxed{\text{Continuity Equation}}$$

• Application in Streamtube:

  For steady flow, discharge $$\displaystyle Q = A_1 V_1 = A_2 V_2 = \text{constant} $$.

  Discharge between two streamlines $$\displaystyle \psi_1 $$ and $$\displaystyle \psi_2 $$: $$\displaystyle Q = \psi_2 - \psi_1 $$.

[!EXAMPLE] Stream Function Problem (Past Paper):

Given $$\displaystyle \psi = 3xy $$. Find $u, v$ at $(1,3)$ and $(3,3)$. Discharge between these points?

Solution: $$\displaystyle u = \frac{\partial \psi}{\partial y} = 3x $$, $$\displaystyle v = -\frac{\partial \psi}{\partial x} = -3y $$.

At (1,3): $$\displaystyle u=3 $$, $$\displaystyle v=-9 $$. At (3,3): $$\displaystyle u=9 $$, $$\displaystyle v=-9 $$.

Discharge $$\displaystyle Q = \psi(3,3) - \psi(1,3) = 27 - 9 = 18 $$ units.

3.0 FLOW DYNAMICS & ENERGY PRINCIPLES

3.1 Euler's Equation of Motion

• Derivation: Apply Newton's second law to a fluid element along a streamline, considering pressure and gravity forces.

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

where $dz$ is vertical displacement, $dV$ is velocity change.

3.2 Bernoulli's Theorem

• Derivation: Integrate Euler's equation for steady, incompressible, inviscid flow along a streamline:

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

• Statement: Total energy head (sum of pressure head, velocity head, datum head) remains constant along a streamline for ideal flow.

• Terms:

  • Pressure Head: $$\displaystyle \frac{P}{\rho g} $$

  • Velocity Head: $$\displaystyle \frac{V^2}{2g} $$

  • Datum Head (Elevation Head): $z$

• Assumptions:

  1. Steady flow

  2. Incompressible fluid ($\rho$ = constant)

  3. Inviscid (zero viscosity, no friction)

  4. Flow along a streamline

  5. Uniform velocity across section (no velocity profile effects)

• Limitations:

  • Not valid for viscous fluids (real fluids) without modification.

  • Only along a streamline (except for irrotational flow where it holds throughout).

  • Neglects energy losses due to friction, turbulence, and heat transfer.

• Modification (Extended Bernoulli):

$$\frac{P_1}{\rho g} + \frac{V_1^2}{2g} + z_1 + H_{pump} - H_{turbine} - h_f = \frac{P_2}{\rho g} + \frac{V_2^2}{2g} + z_2$$

where $$\displaystyle h_f $$ = head loss due to friction.

3.3 Momentum Equation

• Statement: Net force acting on fluid equals rate of change of momentum.

$$\vec{F} = \frac{d}{dt} \int_{CS} \rho \vec{V} (\vec{V} \cdot d\vec{A})$$

For steady flow: $$\displaystyle \vec{F} = \rho Q (\vec{V}_{out} - \vec{V}_{in}) $$.

• Applications:

  1. Force by a Jet on a Stationary/Vanning Plate:

     • Stationary plate: $$\displaystyle F = \rho A V^2 $$ (if jet reversed).

     • Moving plate: $$\displaystyle F = \rho A V (V - u) $$.

  2. Force on a Pipe Bend:

     Resolve momentum change in x, y directions. Include pressure forces.

     [!EXAMPLE] 45° Reducing Bend Problem (May 2024):

     $$\displaystyle D_1=600 $$ mm, $$\displaystyle D_2=300 $$ mm, $$\displaystyle P_1=8.829 $$ N/cm², $$\displaystyle Q=600 $$ lps. Find force on bend.

     Solution: Compute $$\displaystyle V_1, V_2 $$, pressure force at inlet/outlet, momentum change. Resultant force $$\displaystyle F = \sqrt{F_x^2 + F_y^2} $$.

  3. Force on a Nozzle: Similar to bend but with acceleration.

3.4 Energy Equation (Extended Bernoulli)

• Includes:

  • Pump Head ($$\displaystyle H_{pump} $$): Energy added per unit weight.

  • Turbine Head ($$\displaystyle H_{turbine} $$): Energy extracted per unit weight.

  • Friction Head Loss ($$\displaystyle h_f $$): Due to pipe friction, bends, fittings.

$$\frac{P_1}{\rho g} + \frac{V_1^2}{2g} + z_1 + H_{pump} - H_{turbine} - h_f = \frac{P_2}{\rho g} + \frac{V_2^2}{2g} + z_2$$

4.0 FLOW MEASUREMENT & ORIFICES

4.1 Flow Measurement Devices (Primary Elements)

4.2 Venturi Meter

• Derivation: Apply Bernoulli between sections 1 (inlet) and 2 (throat), and continuity $$\displaystyle A_1 V_1 = A_2 V_2 $$.

$$\frac{P_1 - P_2}{\rho g} = \frac{V_2^2 - V_1^2}{2g} = \frac{V_2^2}{2g} \left(1 - \frac{A_2^2}{A_1^2}\right)$$

Theoretical discharge: $$\displaystyle Q_{th} = A_2 V_2 = A_2 \sqrt{\frac{2(P_1-P_2)}{\rho \left(1 - \frac{A_2^2}{A_1^2}\right)}} $$.

Actual discharge: $$\displaystyle Q = C_d Q_{th} $$.

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

where $$\displaystyle \Delta h = \frac{P_1-P_2}{\rho g} $$.

• Problem with Differential Manometer (e.g., oil-Hg):

  $\Delta h$ is reading of manometer, convert to equivalent water column: $$\displaystyle \Delta h_{water} = \left(\frac{\rho_m}{\rho} - 1\right) h $$ if connecting pipes contain same fluid.

4.3 Orifice Meter

• Derivation: Similar to Venturi but with vena contracta (area $$\displaystyle a < A $$, where $A$ = orifice area).

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

• Coefficients:

  • Coefficient of Contraction ($$\displaystyle C_c $$): $$\displaystyle C_c = \frac{a}{A} $$ (area ratio).

  • Coefficient of Velocity ($$\displaystyle C_v $$): $$\displaystyle C_v = \frac{V_{actual}}{V_{theoretical}} $$ (accounts for friction).

  • Coefficient of Discharge ($$\displaystyle C_d $$): $$\displaystyle C_d = C_c C_v $$.

  Theoretical velocity: $$\displaystyle V_{theo} = \sqrt{2g \Delta h} $$.

• Problem: Given actual discharge $$\displaystyle Q_{act} $$ and measured velocity $$\displaystyle V_{act} $$ at vena contracta, find $$\displaystyle C_v, C_c $$.

  $$\displaystyle C_v = \frac{V_{act}}{\sqrt{2g \Delta h}} $$, $$\displaystyle C_c = \frac{Q_{act}}{C_v a \sqrt{2g \Delta h}} $$.

4.4 Other Flow Meters

• Pitot Tube: Measures velocity at a point. Stagnation pressure $$\displaystyle P_0 = P + \frac{1}{2}\rho V^2 $$. For a simple Pitot: $$\displaystyle V = \sqrt{\frac{2(P_0 - P)}{\rho}} $$.

• Notches & Weirs: Measure discharge in open channels.

  • Rectangular Notch: $$\displaystyle Q = \frac{2}{3} C_d L \sqrt{2g} H^{3/2} $$

  • Triangular (V-notch): $$\displaystyle Q = \frac{8}{15} C_d \tan\frac{\theta}{2} \sqrt{2g} H^{5/2} $$

  • Trapezoidal (Cipolletti): $$\displaystyle Q = \frac{8}{15} C_d \left(\frac{2}{3}L + \frac{H}{\tan(\theta/2)}\right) \sqrt{2g} H^{3/2} $$

5.0 PIPE FLOW: LAMINAR & TURBULENT

5.1 Reynolds Experiment & Number

• Reynolds Number ($Re$):

$$Re = \frac{\rho V D}{\mu} = \frac{V D}{\nu}$$

*   Ratio of inertial forces to viscous forces.

• Critical Reynolds Number for pipe flow:

  • Laminar → Turbulent: $$\displaystyle Re_{crit} \approx 2300 $$

  • Turbulent → Laminar: $$\displaystyle Re_{crit} \approx 2000 $$

• Problem: Given $\rho, \mu, D, V$, compute $Re$. If $$\displaystyle Re < 2000 $$, flow is laminar. Then use Darcy-Weisbach to find pressure loss per unit length:

$$h_f = f \frac{L}{D} \frac{V^2}{2g}, \quad f = \frac{64}{Re} \quad \text{(laminar)}$$

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

5.2 Laminar Flow in Circular Pipes

• Velocity Distribution (Hagen-Poiseuille flow):

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

where $\Delta P$ = pressure drop over length $L$, $R$ = pipe radius.

*   Parabolic profile, maximum at center ($$\displaystyle r=0 $$): $$\displaystyle u_{max} = \frac{\Delta P R^2}{4\mu L} $$.

*   Mean velocity: $$\displaystyle V_{mean} = \frac{1}{\pi R^2} \int_0^R u(r) 2\pi r dr = \frac{u_{max}}{2} $$.

• Shear Stress Distribution:

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

Linear, zero at center, maximum at wall ($$\displaystyle r=R $$): $$\displaystyle \tau_w = \frac{\Delta P R}{2L} $$.

• Problem (Past Paper: Nov 2023): Pipe $$\displaystyle D=150 $$ mm, laminar flow. At $$\displaystyle r=20 $$ mm, $$\displaystyle u=0.4 $$ m/s. Find:

  1. $$\displaystyle u_{max} $$: $$\displaystyle u = u_{max} \left(1 - \frac{r^2}{R^2}\right) \Rightarrow u_{max} = \frac{u}{1 - (r/R)^2} = \frac{0.4}{1 - (20/75)^2} \approx 0.457 $$ m/s.

  2. $$\displaystyle V_{mean} = u_{max}/2 \approx 0.2285 $$ m/s.

  3. Discharge $$\displaystyle Q = V_{mean} \times \frac{\pi D^2}{4} \approx 0.2285 \times 0.01767 \approx 0.00404 $$ m³/s = 4.04 lps.

5.3 Turbulent Flow

• Characteristics: Fluctuating velocity, high mixing, flat velocity profile.

• Shear Stress: $$\displaystyle \tau = \tau_v + \tau_t $$, where $$\displaystyle \tau_v = \mu \frac{du}{dy} $$ (viscous), $$\displaystyle \tau_t = \rho \overline{u'v'} $$ (Reynolds stress).

• Smooth vs. Rough Pipe (Nikuradse):

  • Smooth: Viscous sublayer covers roughness, $f$ depends only on $Re$.

  • Rough: Roughness protrudes, $f$ depends on relative roughness $$\displaystyle \frac{\varepsilon}{D} $$ and $Re$.

• Friction Factor ($f$):

  • Defined by Darcy-Weisbach: $$\displaystyle h_f = f \frac{L}{D} \frac{V^2}{2g} $$.

  • For laminar flow: $$\displaystyle f = \frac{64}{Re} $$.

  • For turbulent flow: Use Moody Chart or Colebrook equation:

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

5.4 Darcy-Weisbach Equation

• Derivation: From momentum balance or energy (Bernoulli with loss). Consider a horizontal pipe section of length $L$, diameter $D$, velocity $V$. Force due to pressure = force due to wall shear.

$$\Delta P \cdot \frac{\pi D^2}{4} = \tau_w \cdot \pi D L \Rightarrow \tau_w = \frac{D}{4} \frac{\Delta P}{L}$$

Also $$\displaystyle \tau_w = \frac{f}{8} \rho V^2 $$. Equate and convert to head loss:

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

• Problem (Series Pipes): Given $L, D, f$ for each pipe, find $Q$ for given total head loss $H$ (including minor losses $$\displaystyle h_m = \sum K \frac{V^2}{2g} $$).

  Solution: Total head loss $$\displaystyle H = \sum \left( f_i \frac{L_i}{D_i} + \sum K_i \right) \frac{V^2}{2g} $$. Since $$\displaystyle Q = A_i V_i $$ constant, express $$\displaystyle V_i $$ in terms of $Q$, solve for $Q$.

5.5 Minor Losses

• Due to disturbances: sudden expansion/contraction, entrance, exit, bends, valves.

• Loss Coefficient ($K$): $$\displaystyle h_m = K \frac{V^2}{2g} $$.

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

  • Sudden contraction: $K \approx 0.5$ (if no pipe length)

  • Entrance (sharp-edged): $$\displaystyle K = 0.5 $$

  • Exit: $$\displaystyle K = 1.0 $$

• Equivalent Length ($$\displaystyle L_e $$): Express minor loss as extra pipe length: $$\displaystyle h_m = f \frac{L_e}{D} \frac{V^2}{2g} $$. Then $$\displaystyle L_e = \frac{K D}{f} $$.

6.0 DIMENSIONAL ANALYSIS & SIMILITUDE

6.1 Dimensional Homogeneity & Rayleigh's Method

• Dimensional Homogeneity: Every term in an equation must have same dimensions.

• Rayleigh's Method: Express variable as product of dimensionless parameters. Limited to few variables (3-4).

6.2 Buckingham Pi Theorem

• Statement: If a physical relation involves $n$ variables and $r$ fundamental dimensions (M, L, T, etc.), it can be reduced to $n-r$ independent dimensionless $\Pi$ groups.

• Steps:

  1. List all variables ($n$) and their dimensions.

  2. Identify repeating variables ($r$) that include all fundamental dimensions.

  3. Form $\Pi$ groups: $$\displaystyle \Pi_1 = \text{Repeating variables} \times \text{Dependent variable} $$.

  4. Solve for exponents by equating dimensions.

  5. Write functional relation: $$\displaystyle F(\Pi_1, \Pi_2, ...) = 0 $$.

• Application (Past Paper: Drag Force $R$):

  Variables: $R$ (force: MLT⁻²), $v$ (LT⁻¹), $l$ (L), $\mu$ (ML⁻¹T⁻¹), $\rho$ (ML⁻³), $g$ (LT⁻²). $$\displaystyle n=6 $$, $$\displaystyle r=3 $$ (choose $v, l, \rho$ as repeating).

  $$\displaystyle \Pi_1 = \frac{R}{\rho v^2 l^2} $$ (Euler number), $$\displaystyle \Pi_2 = \frac{\mu}{\rho v l} = \frac{1}{Re} $$, $$\displaystyle \Pi_3 = \frac{g l}{v^2} = \frac{1}{Fr^2} $$.

  Relation: $$\displaystyle F\left( Eu, \frac{1}{Re}, \frac{1}{Fr} \right) = 0 $$ or $$\displaystyle Eu = f(Re, Fr) $$.

6.3 Similitude & Model Studies

• Geometric Similarity: Model and prototype have same shape, all linear dimensions in same ratio ($$\displaystyle L_r = L_m/L_p $$).

• Kinematic Similarity: Motion is similar; velocity fields geometrically similar, time scale ratio constant ($$\displaystyle V_r = \frac{V_m}{V_p} $$, $$\displaystyle t_r = \frac{t_m}{t_p} $$).

• Dynamic Similarity: Forces are similar; force fields geometrically similar, force ratio constant ($$\displaystyle F_r = \frac{F_m}{F_p} $$).

• Are these truly attainable? In practice, only approximate similitude is possible. Often, prioritize dominant forces (e.g., match $Re$ for viscous flows, $Fr$ for free surface flows). Complete dynamic similarity (all $\Pi$ groups equal) is rarely achievable due to conflicting requirements.

6.4 Dimensionless Numbers

7.0 SPECIAL TOPICS & APPLICATIONS

7.1 Stokes' Law

• Condition: $$\displaystyle Re < 1 $$ (creeping flow), sphere falling in infinite fluid.

• Terminal Velocity ($$\displaystyle V_t $$): When weight = buoyancy + viscous drag.

$$W = F_B + F_D \Rightarrow \frac{\pi}{6} d^3 \rho_s g = \frac{\pi}{6} d^3 \rho g + 3\pi \mu d V_t$$

$$\boxed{V_t = \frac{g d^2 (\rho_s - \rho)}{18 \mu}}$$

where $d$ = sphere diameter, $$\displaystyle \rho_s $$ = sphere density.

7.2 Buoyant Force

• Archimedes' Principle: Buoyant force equals weight of displaced fluid.

$$F_B = \gamma \cdot \text{Volume displaced} = \rho g V_{sub}$$

7.3 Bulk Modulus ($K$)

• Definition: $$\displaystyle K = -V \frac{dP}{dV} $$ (for compression, $dP$ positive, $dV$ negative).

• Relation to Compressibility: $$\displaystyle \beta = \frac{1}{K} $$, where $\beta$ is compressibility.

• For liquids, $K$ is large (nearly incompressible). For gases, $$\displaystyle K = P $$ (isothermal) or $$\displaystyle K = \gamma P $$ (adiabatic).

7.4 Flow Net Applications

• Primarily in seepage analysis (flow through porous media like soils).

• Used to determine:

  • Flow rate per unit width: $$\displaystyle q = k \cdot \Delta \psi $$, where $k$ = permeability.

  • Hydraulic gradient: $$\displaystyle \frac{\Delta h}{\Delta l} $$ from flow net squares.

  • Exit gradient (important for piping failure).

[!TIP] For short notes questions (3-4m), define and state formula. For long answers (7-10m), include derivation and application. Always check past paper patterns for emphasis.