Fluid Mechanics-I (CE-501) - Unit 2 Short Notes

UNIT 2: FLUID MECHANICS FUNDAMENTALS AND APPLICATIONS

1.0 FLUID PROPERTIES AND CHARACTERISTICS

1.1 Fundamental Properties

[!TIP] Exam Focus: Direct formula-based numericals are common. Remember: $$\displaystyle \gamma = \rho g $$, and for water $$\displaystyle \rho_{water} \approx 1000 $$ kg/m³, $$\displaystyle \gamma_{water} \approx 9.81 $$ kN/m³.

1.2 Viscosity

• Dynamic Viscosity ($\mu$): Measure of internal fluid friction. Defined by Newton's Law of Viscosity:

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

where $\tau$ = shear stress, $du/dy$ = velocity gradient.

Units: N·s/m² or Pa·s (1 Poise = 0.1 Pa·s).

• Kinematic Viscosity ($\nu$): Ratio of dynamic viscosity to density.

$$ \nu = \frac{\mu}{\rho} $$

Units: m²/s (1 Stoke = 10⁻⁴ m²/s).

Variation with Temperature:

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

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

[!TIP] Common Pitfall: Confusing units of $\mu$ and $\nu$. Always check if density is given to convert between them.

1.3 Surface Tension and Capillarity

• Surface Tension ($\sigma$): Force per unit length acting tangentially on the surface. Units: N/m.

• Pressure Intensity inside Curved Surfaces:

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

  • Soap Bubble (two surfaces): $$\displaystyle P_{in} - P_{out} = \frac{4\sigma}{R} $$

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

Cohesion vs Adhesion:

• Cohesion: Attraction between like molecules (e.g., water-water). Causes surface tension.

• Adhesion: Attraction between unlike molecules (e.g., water-glass). Causes capillarity.

[!TIP] Numerical Pattern: Given $\sigma$, $R$, find pressure difference. Remember bubble has factor 4, droplet/jet has factor 2 or 1.

1.4 Other Properties

• Bulk Modulus ($K$): Measure of fluid compressibility.

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

Units: Pa or N/m². For water, $$\displaystyle K \approx 2.1 \times 10^9 $$ N/m².

• Compressibility: $$\displaystyle \beta = \frac{1}{K} $$.

• Vapor Pressure & Cavitation: Vapor pressure is pressure at which liquid boils. Cavitation occurs when local pressure drops below vapor pressure, forming vapor bubbles that collapse violently.

2.0 PRESSURE MEASUREMENT AND MANOMETRY

2.1 Pressure Fundamentals

Pascal's Law: Pressure at a point in a fluid at rest is equal in all directions. Applied in hydraulic presses: $$\displaystyle F_1/A_1 = F_2/A_2 $$.

2.2 Manometers

Principle: Balance pressure heads of different fluids in a U-tube.

Simple U-tube Manometer:

• Measures gauge pressure of a fluid relative to atmosphere.

• Formula: $$\displaystyle P_{A} = P_{atm} + \rho_{man} g h $$

  where $$\displaystyle \rho_{man} $$ = density of manometric fluid (e.g., mercury, $$\displaystyle \rho_{Hg} = 13600 $$ kg/m³).

Differential Manometer (U-tube):

• Measures pressure difference between two points.

• General Formula (for two different fluids in pipes):

$$ P_A - P_B = (\rho_m - \rho_1) g h_1 + (\rho_m - \rho_2) g h_2 \quad \text{(if limbs have different fluids)} $$

Often simplified if both pipes contain same fluid: $$\displaystyle P_A - P_B = (\rho_m - \rho) g h $$.

Inverted U-tube Differential Manometer:

• Used for measuring low pressure differences with lighter manometric fluid (e.g., water).

• Formula: $$\displaystyle P_A - P_B = \rho_1 g h_1 - \rho_2 g h_2 - \rho_m g h $$

[!TIP] Problem-Solving Strategy:

1. Draw clear schematic with all heights ($h$) and fluids labeled.

2. Write pressure at each interface moving from one side to the other, adding/subtracting $\rho g h$.

3. Equate pressures at a common horizontal level.

4. Use specific gravity ($SG$) to avoid unit errors: $$\displaystyle \rho = SG \times 1000 $$ kg/m³.

Sample Problem (from Jun 2025):

A differential manometer is connected at points A (sp.gr. 1.5) and B (sp.gr. 0.9). $$\displaystyle P_A = 1 $$ kgf/cm², $$\displaystyle P_B = 1.8 $$ kgf/cm². Find mercury level difference (h).

Solution:

Convert pressures to consistent units (e.g., Pa or N/m²). Use formula:

$$ P_A + \rho_1 g (h_1) = P_B + \rho_m g (h - h_2) + \rho_2 g h_2 $$

Assuming limbs connected directly to pipes, $$\displaystyle h_1 $$ and $$\displaystyle h_2 $$ are heights of light fluids above mercury interface. Solve for $h$.

2.3 Pressure on Submerged Surfaces

Total Force on Plane Surface:

$$ F = \bar{p} \times A = \rho g \bar{h} A $$

where $\bar{h}$ = depth to centroid. Center of Pressure (CEP):

$$ h_{cp} = \bar{h} + \frac{I_{xx}}{A \bar{h}} $$

where $$\displaystyle I_{xx} $$ = second moment of area about centroidal axis.

For Curved Surfaces: Resolve into horizontal ($$\displaystyle F_H $$) and vertical ($$\displaystyle F_V $$) components.

• $$\displaystyle F_H $$ = force on projected vertical area.

• $$\displaystyle F_V $$ = weight of fluid above the curved surface (if submerged) or $$\displaystyle F_V = \gamma \times \text{volume of fluid} $$.

[!TIP] Key Distinction: Plane surface force acts at CEP. Curved surface force components act at specific points (H for $$\displaystyle F_H $$, centroid of volume for $$\displaystyle F_V $$).

3.0 KINEMATICS OF FLUID FLOW

3.1 Flow Visualization

[!TIP] Distinction: In steady flow, streamlines = pathlines = streaklines. In unsteady flow, they differ.

3.2 Potential and Stream Functions

Velocity Potential Function ($\phi$):

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

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

• Orthogonal to streamlines ($$\displaystyle \phi = \text{constant} $$ are streamlines).

Stream Function ($\psi$):

• Defined for 2D incompressible flows: $$\displaystyle u = \frac{\partial \psi}{\partial y},\; v = -\frac{\partial \psi}{\partial x} $$.

• Properties: $$\displaystyle \psi = \text{constant} $$ are streamlines. Automatically satisfies continuity.

• Flow between streamlines: $$\displaystyle Q = \psi_2 - \psi_1 $$.

Cauchy-Riemann Equations: For $\phi$ and $\psi$ to be conjugate (flow both irrotational and incompressible):

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

3.3 Flow Nets

• Definition: Grid formed by intersecting families of $\phi$ and $\psi$ lines (orthogonal).

• Significance: Visualizes flow patterns; each "square" represents equal discharge.

• Method: Sketch by hand ensuring orthogonality and roughly equal-sized cells. Used in seepage analysis (e.g., flow through soils).

3.4 Continuity Equation

Derivation (3D Cartesian): For a control volume, mass inflow = mass outflow + accumulation.

For steady flow:

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

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

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

Application (Discharge between streamlines):

Given stream function $\psi(x,y)$, discharge between $$\displaystyle \psi = \psi_1 $$ and $$\displaystyle \psi = \psi_2 $$ is:

$$ Q = |\psi_2 - \psi_1| $$

[!TIP] Common Problem: Given $u$ and $v$ (or $\psi$), find missing component using continuity. For $\psi$, $$\displaystyle u = \partial \psi/\partial y $$, $$\displaystyle v = -\partial \psi/\partial x $$.

4.0 DYNAMICS OF FLUID FLOW

4.1 Euler's Equation of Motion

Derivation along a streamline for an inviscid fluid:

$$ \frac{dP}{\rho} + g dz + u du = 0 $$

where $u$ = velocity magnitude.

4.2 Bernoulli's Equation

From Euler: Integrate Euler's equation assuming steady, incompressible, inviscid flow along a streamline:

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

Multiply by $\rho$:

$$ P + \frac{1}{2} \rho V^2 + \rho g z = \text{constant} $$

Terms:

• $P$: Pressure energy (static pressure)

• $$\displaystyle \frac{1}{2}\rho V^2 $$: Kinetic energy (dynamic pressure)

• $\rho g z$: Potential energy (elevation head)

Assumptions:

1. Fluid is ideal (inviscid, no friction).

2. Flow is steady.

3. Fluid is incompressible.

4. Flow is along a streamline.

5. No shaft work (pump/turbine) between sections.

Limitations & Modifications:

• Real fluids have friction losses ($$\displaystyle h_f $$).

• With pumps/turbines: add head $$\displaystyle h_p $$ (pump) or subtract $$\displaystyle h_t $$ (turbine).

• Extended Bernoulli (Energy Equation):

$$ \frac{P_1}{\rho g} + \frac{V_1^2}{2g} + z_1 + h_p = \frac{P_2}{\rho g} + \frac{V_2^2}{2g} + z_2 + h_f $$

4.3 Momentum Equation

Linear Momentum (Control Volume):

$$ \sum \vec{F} = \frac{d}{dt} \int_{CV} \rho \vec{V} dV + \int_{CS} \rho \vec{V} (\vec{V} \cdot \vec{n}) dA $$

For steady flow, first term zero:

$$ \sum \vec{F} = \sum (\text{outflow momentum} - \text{inflow momentum}) $$

Applications:

• Force on pipe bends, reducers, nozzles.

• Force on vanes (jets).

[!TIP] Sign Convention: Forces exerted by fluid on solid are positive in direction of flow. Reaction force on solid is opposite.

4.4 Energy Equation (Extended Bernoulli)

As above. Pump Power Calculation:

$$ \text{Power input} = \rho g Q h_p $$

where $$\displaystyle h_p $$ = total head added by pump (from energy equation).

5.0 VISCOUS FLOW AND PIPE FRICTION

5.1 Reynolds Number

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

• Significance: Ratio of inertial to viscous forces.

• Critical $Re$:

  • Pipe flow: $$\displaystyle Re_{cr} \approx 2300 $$ (laminar-turbulent transition).

  • Flow over plate: $$\displaystyle Re_{cr} \approx 5 \times 10^5 $$.

Laminar vs Turbulent:

Smooth vs Rough Surfaces (Turbulent):

• Smooth: Viscous sublayer covers roughness; friction factor depends only on $Re$.

• Rough: Roughness protrudes through sublayer; friction factor depends on relative roughness ($\varepsilon/D$) and $Re$.

5.2 Laminar Flow Analysis

Flow between Parallel Plates (Fixed, distance $h$):

• Velocity Distribution (parabolic):

$$ u(y) = \frac{1}{2\mu} \frac{dp}{dx} (y^2 - hy) $$

where $$\displaystyle y=0 $$ at bottom plate.

• Shear Stress Distribution: Linear, $$\displaystyle \tau = \mu \frac{du}{dy} = \frac{y}{h} \tau_{wall} $$.

Hagen-Poiseuille Flow (Circular Pipe):

• Velocity Distribution:

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

• Max Velocity: $$\displaystyle u_{max} = \frac{\Delta P R^2}{4\mu L} $$

• Mean Velocity: $$\displaystyle V_{avg} = \frac{u_{max}}{2} $$

• Discharge (Poiseuille's Law):

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

• Shear Stress: $$\displaystyle \tau(r) = \frac{r}{2} \frac{\Delta P}{L} $$, max at wall: $$\displaystyle \tau_w = \frac{R}{2} \frac{\Delta P}{L} $$

[!TIP] Numerical: Given velocity at a point $r$, find $$\displaystyle u_{max} $$, $$\displaystyle V_{avg} $$, $Q$. Use $$\displaystyle u(r) = u_{max}(1 - r^2/R^2) $$.

5.3 Turbulent Flow and Friction

Darcy-Weisbach Equation (Major Losses):

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

where $f$ = Darcy friction factor.

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

• For turbulent flow: $$\displaystyle f = \phi(Re, \varepsilon/D) $$. Use Moody Chart or Colebrook equation:

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

Minor Losses: $$\displaystyle h_m = K \frac{V^2}{2g} $$, where $K$ depends on fitting (e.g., entrance $$\displaystyle K=0.5 $$, exit $$\displaystyle K=1 $$, 90° elbow $$\displaystyle K=0.75 $$).

5.4 Special Cases

• Stokes' Law (Drag on Sphere, laminar flow, $$\displaystyle Re < 0.1 $$):

$$ F_D = 3\pi \mu D V $$

• Torque on Rotating Cylinders (Viscosity Determination):

  For concentric cylinders (inner radius $$\displaystyle R_i $$, outer $$\displaystyle R_o $$, length $L$, angular velocity $\omega$):

$$ T = \frac{4\pi \mu \omega R_i^2 R_o^2 L}{R_o^2 - R_i^2} $$

6.0 FLOW MEASUREMENT DEVICES

6.1 Orifice Meter

• Construction: Thin plate with sharp-edged orifice.

• Vena Contracta: Jet contracts downstream; coefficient of contraction $$\displaystyle C_c = A_j / A_o $$.

• Discharge Equation:

$$ Q_{act} = C_d A_o \sqrt{\frac{2(P_1 - P_2)}{\rho}} $$

where $$\displaystyle C_d = C_v C_c $$ (coefficient of discharge).

• Coefficient of Velocity ($$\displaystyle C_v $$): Accounts for friction; $$\displaystyle C_v = V_{act}/V_{th} $$.

• Typical $$\displaystyle C_d $$: 0.6–0.7 (lower than venturi due to losses).

6.2 Venturi Meter

• Construction: Converging cone, throat, diverging cone.

• Discharge Equation (theoretical):

$$ Q_{th} = \frac{A_1 A_2}{\sqrt{A_1^2 - A_2^2}} \sqrt{\frac{2(P_1 - P_2)}{\rho}} $$

• Actual: $$\displaystyle Q = C_d Q_{th} $$, with $$\displaystyle C_d \approx 0.95–0.99 $$ (higher than orifice).

• Advantages over Orifice: Lower permanent pressure loss, less wear, higher $$\displaystyle C_d $$.

• Disadvantages: Larger size, more expensive.

6.3 Flow Nozzle

• Intermediate between orifice and venturi. Higher $$\displaystyle C_d $$ than orifice (~0.95) but lower than venturi. Used for high-velocity flows.

6.4 Pitot Tube

• Measures stagnation pressure (total pressure) at tip.

• Velocity: $$\displaystyle V = \sqrt{\frac{2(P_0 - P_s)}{\rho}} $$, where $$\displaystyle P_0 $$ = stagnation, $$\displaystyle P_s $$ = static pressure.

• Prandtl Pitot: Includes static pressure tap; measures $V$ directly.

6.5 Notches and Weirs

Rectangular Notch:

• Theoretical Discharge:

$$ Q_{th} = \frac{2}{3} C_d b \sqrt{2g} H^{3/2} $$

where $b$ = width, $H$ = head over notch.

• Coefficient of Discharge ($$\displaystyle C_d $$): Accounts for contraction and velocity; typically 0.6–0.62.

Triangular Notch (e.g., 90°):

$$ Q = \frac{8}{15} C_d \sqrt{2g} \tan(\theta/2) H^{5/2} $$

7.0 DIMENSIONAL ANALYSIS AND SIMILITUDE

7.1 Dimensional Homogeneity

• All terms in an equation must have same dimensions (M, L, T).

• Rayleigh's Method: Express variable as product of repeating variables raised to powers; determine exponents by dimensional homogeneity. Limited to few variables.

7.2 Buckingham Pi Theorem

• Statement: If $n$ variables in a problem, with $r$ fundamental dimensions, then $(n-r)$ independent dimensionless pi terms exist.

• Procedure:

  1. List all variables ($n$).

  2. Identify fundamental dimensions (M, L, T) among them ($r$).

  3. Choose $(n-r)$ repeating variables (must include all fundamental dimensions, be independent, and not include dependent variable).

  4. Form pi terms: $$\displaystyle \pi_i = (\text{repeating})^a (\text{repeating})^b \cdots (\text{dependent}) $$.

  5. Solve for exponents by setting dimensions to zero.

  6. Write functional relation: $$\displaystyle F(\pi_1, \pi_2, ...) = 0 $$.

Example (Drag Force $R$ on body):

Variables: $R, V, l, \mu, \rho, g$. Dimensions: $$\displaystyle [R]=MLT^{-2}, [V]=LT^{-1}, [l]=L, [\mu]=ML^{-1}T^{-1}, [\rho]=ML^{-3}, [g]=LT^{-2} $$. $$\displaystyle n=6 $$, $$\displaystyle r=3 $$ (M,L,T) → 3 pi terms.

Choose repeating: $V, l, \rho$. $$\displaystyle \pi_1 = R/(\rho V^2 l^2) $$ (Euler number), $$\displaystyle \pi_2 = \mu/(\rho V l) = 1/Re $$, $$\displaystyle \pi_3 = gl/V^2 = 1/Fr^2 $$.

7.3 Similitude

• Geometric Similarity: All linear dimensions in model and prototype are in same ratio ($$\displaystyle \lambda_L $$).

• Kinematic Similarity: Motion patterns similar; velocity ratios same ($$\displaystyle \lambda_V = \lambda_L / \lambda_T $$).

• Dynamic Similarity: Force ratios same; requires similarity of forces (e.g., Reynolds similarity, Froude similarity).

Attainability: All three rarely fully attainable simultaneously. Choose dominant forces (e.g., Reynolds for viscous, Froude for gravity).

7.4 Dimensionless Numbers (Contextual)

• Reynolds ($Re$): $$\displaystyle \frac{\rho V L}{\mu} $$ (inertial/viscous).

• Froude ($Fr$): $$\displaystyle \frac{V}{\sqrt{gL}} $$ (inertial/gravity).

• Euler ($Eu$): $$\displaystyle \frac{\Delta P}{\rho V^2} $$ (pressure/inertial).

• Mach ($Ma$): $$\displaystyle \frac{V}{c} $$ (inertial/elastic).

8.0 APPLICATIONS AND COMPREHENSIVE PROBLEMS

8.1 Combined Manometer Problems

• Invert manometer readings to find pressure differences.

• Account for multiple fluids with different SGs.

• Example: Differential manometer connecting two pipes with different fluids (Jun 2025).

8.2 Force Calculations (Momentum Equation)

General Steps:

1. Draw CV, label inlet/outlet areas, velocities, pressures.

2. Compute net force in x, y, z: $$\displaystyle \sum F_x = \dot{m}(V_{x,out} - V_{x,in}) $$, etc.

3. Include pressure forces: $$\displaystyle F_P = P_1 A_1 - P_2 A_2 \cos\theta $$ (if outlet at angle).

4. Include reaction force from supports (if any).

5. Solve for unknown force.

Example (45° Reducing Bend - May 2024):

• Given: $$\displaystyle D_1=600 $$ mm, $$\displaystyle D_2=300 $$ mm, $$\displaystyle P_1 $$, $Q$.

• Steps: Find $$\displaystyle V_1, V_2 $$, $$\displaystyle \dot{m} = \rho Q $$. Resolve momentum change into x,y components. Include pressure forces. Find resultant force on bend.

8.3 Pipes in Series (Equivalent Length Method)

• Same discharge $Q$ through all pipes.

• Total head loss: $$\displaystyle h_f^{total} = \sum \left( f_i \frac{L_i}{D_i} + \sum K_{minor,i} \right) \frac{V_i^2}{2g} $$.

• But $$\displaystyle V_i $$ differ due to $$\displaystyle D_i $$: $$\displaystyle V_i = Q / A_i $$.

• Solve iteratively or use trial $f$ (since $f$ depends on $$\displaystyle Re_i $$ which depends on $$\displaystyle V_i $$).

Example (May 2024): Three pipes with given $L, D, f$, find $Q$ given $$\displaystyle \Delta H = 12 $$ m. Write:

$$ \Delta H = \sum \left( f_i \frac{L_i}{D_i} + K_i \right) \frac{Q^2}{2g A_i^2} $$

Solve for $Q$ (may require iteration if $f$ not given constant).

8.4 Energy Equation Applications (Pump Power)

Steps:

1. Apply energy eq between inlet (1) and outlet (2):

$$ \frac{P_1}{\rho g} + \frac{V_1^2}{2g} + z_1 + h_p = \frac{P_2}{\rho g} + \frac{V_2^2}{2g} + z_2 + h_f $$

2. Compute $$\displaystyle V_1, V_2 $$ from diameters.

3. Compute $$\displaystyle h_f $$ (major + minor) if given $f$ and lengths.

4. Solve for $$\displaystyle h_p $$ (head added).

5. Power: $$\displaystyle P_{in} = \rho g Q h_p / \eta_{pump} $$ (if efficiency given; else assume $$\displaystyle \eta=1 $$ for theoretical).

Example (Nov 2023): Inlet vacuum = 15 cm Hg → $$\displaystyle P_1 = P_{atm} - \rho_{Hg} g h $$. Outlet $$\displaystyle P_2 $$, $$\displaystyle z_2 - z_1 = 1.2 $$ m. Compute $$\displaystyle h_p $$, then power.

8.5 Viscosity Determination

• Rotating Cylinders: Use torque formula (Section 5.4).

• Lubrication Film (Inclined Plane): Plate sliding on inclined plane with oil film. Shear stress balances weight component:

$$ \mu = \frac{W \sin\theta}{A \cdot (du/dy)} = \frac{W \sin\theta}{A \cdot (V/h)} $$

where $h$ = film thickness, $V$ = velocity, $\theta$ = incline angle.

9.0 DEFINITIONS AND DISTINCTIONS (SHORT NOTES)

9.1 Key Definitions (3m/4m)

• Stream Function ($\psi$): Scalar function whose partial derivatives give velocity components in 2D incompressible flow; $$\displaystyle \psi = \text{constant} $$ are streamlines.

• Velocity Potential ($\phi$): Scalar function whose gradient gives velocity vector for irrotational flow; $$\displaystyle \phi = \text{constant} $$ are equipotential lines.

• Flow Net: Orthogonal grid of streamlines and equipotential lines; each cell represents equal discharge.

• Bulk Modulus ($K$): $$\displaystyle K = -V \frac{dP}{dV} $$; measure of fluid compressibility.

• Reynolds Number ($Re$): $$\displaystyle Re = \rho V D / \mu $$; ratio of inertial to viscous forces; predicts flow regime.

• Stokes' Law: $$\displaystyle F_D = 3\pi \mu D V $$ for sphere at very low $Re$ (<0.1).

• Pitot Tube: Device measuring stagnation pressure to determine fluid velocity.

• Friction Factor ($f$): Dimensionless factor in Darcy-Weisbach equation: $$\displaystyle h_f = f (L/D) (V^2/2g) $$.

9.2 Important Distinctions (7m)

[!TIP] Exam Strategy: For distinction questions, use a two-column table in your answer. Define each term briefly, then list 3–4 contrasting points.