Fluid Mechanics-I (CE-501) - Unit 1 Short Notes

Unit 1: Fluid Mechanics-I Short Notes (RGPV Focus)

I. Fundamental Fluid Properties

1.1 Basic Definitions

Exam Tip: Specific gravity is dimensionless. Always use $$\displaystyle \rho_{water} = 1000 \, \text{kg/m}^3 $$ or $$\displaystyle 62.4 \, \text{lbf/ft}^3 $$ for conversions.

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.

*   Unit: N·s/m² (Pa·s) or poise (1 P = 0.1 Pa·s).

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

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

*   Unit: m²/s or stoke (1 St = 10⁻⁴ m²/s).

Key Concept: Viscosity is a transport property—it quantifies momentum transfer.

Temperature Variation:

• Liquids: Viscosity decreases with temperature (molecular cohesion weakens).

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

1.3 Surface Tension & Capillarity

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

• Pressure Inside Curved Surfaces:

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

  • Bubble (liquid film): $$\displaystyle p_{in} - p_{out} = \frac{4\sigma}{R} $$

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

  where $R$ = radius of curvature.

• Capillarity: Rise/fall of liquid in a narrow tube due to surface tension & adhesion/cohesion.

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

$\theta$ = contact angle, $r$ = tube radius.

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

1.4 Bulk Modulus & Compressibility

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

$$ K = -V \frac{dp}{dV} = \rho \frac{dp}{d\rho} $$

*   For isentropic process: $$\displaystyle K = \rho c^2 $$, where $c$ = speed of sound.

• Compressibility ($\beta$): Reciprocal of bulk modulus.

$$ \beta = -\frac{1}{V} \frac{dV}{dp} = \frac{1}{K} $$

1.5 Cohesion vs Adhesion

II. Fluid Statics

2.1 Pressure Fundamentals

• Pressure Intensity ($p$): Normal force per unit area.

• Pascal's Law: Pressure at a point in a static fluid is equal in all directions and is transmitted undiminished throughout the fluid.

• Pressure Types:

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

  • Gauge Pressure ($$\displaystyle p_{g} $$): Measured relative to local atmospheric pressure ($$\displaystyle p_{atm} $$).

  • Vacuum Pressure ($$\displaystyle p_{vac} $$): Pressure below atmospheric.

$$ p_{abs} = p_{g} + p_{atm} \quad \text{or} \quad p_{vac} = p_{atm} - p_{abs} $$

2.2 Manometers

Principle: Balance pressure heads of different fluids in connected tubes.

• U-tube Manometer: Measures pressure difference between two points.

  General Equation: $$\displaystyle p_1 + \rho_1 g h_1 = p_2 + \rho_2 g h_2 + \rho_m g h $$

  where $$\displaystyle \rho_m $$ = manometric fluid density.

• Differential Manometer: Connected to two different pipes. Solve by writing pressure equation from a common horizontal line.

• Inverted U-tube Manometer: Used for measuring low pressure differences of light liquids. Contains air/vapor pocket.

Numerical Approach (Differential Manometer):

1. Start from one side, write pressure at the bottom of one limb.

2. Equate to pressure at same horizontal level in other limb.

3. Solve for unknown height $h$.

\boxed{p_A + \rho_A g h_A = p_B + \rho_B g h_B + \rho_m g h}

2.3 Hydrostatic Pressure Distribution

• For incompressible fluid: $$\displaystyle p = p_0 + \rho g h $$

  where $h$ = vertical depth from free surface.

• Pressure Variation in Compressible Fluid (Isothermal Ideal Gas):

$$ p = p_0 e^{-\frac{\rho_0 g z}{p_0}} = p_0 e^{-\frac{z}{H}} \quad (H = \frac{p_0}{\rho_0 g} = \text{scale height}) $$

2.4 Forces on Submerged Surfaces

• Plane Surface:

  • Total Force: $$\displaystyle F_R = p_c A = \rho g \bar{h} A $$

    where $$\displaystyle p_c $$ = pressure at centroid, $\bar{h}$ = depth of centroid.

  • Center of Pressure (CoP): Point of action of $$\displaystyle F_R $$.

$$ \bar{y}_{cp} = \bar{y} + \frac{I_{xx,c}}{A \bar{y}^2} $$

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

• Curved Surface: 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 (up to free surface).

  • $$\displaystyle F_R = \sqrt{F_H^2 + F_V^2} $$; CoP found by moments.

2.5 Buoyancy & Stability

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

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

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

• Stability of Floating Bodies:

  • Metacentric Height (GM): Key parameter.

$$ GM = BM - BG $$

    where $$\displaystyle BM = \frac{I_{waterline}}{V_{sub}} $$ (for small angles), $BG$ = distance between CoG and CoB.

*   **Condition for Stability:** $$\displaystyle GM > 0 $$ (stable), $$\displaystyle GM = 0 $$ (neutral), $$\displaystyle GM < 0 $$ (unstable).

III. Fluid Kinematics

3.1 Flow Visualization & Classification

• Rotational Flow: Fluid particles have net angular velocity (vorticity $\zeta \neq 0$).

• Irrotational Flow: $$\displaystyle \zeta = 0 $$. Often simplifies analysis (potential flow exists).

• Laminar Flow: Smooth, orderly motion in layers.

• Turbulent Flow: Chaotic, fluctuating motion with mixing.

3.2 Velocity Potential Function ($\phi$) & Stream Function ($\psi$)

Orthogonality: In irrotational, incompressible flow, $\phi$ and $\psi$ families are orthogonal.

3.3 Flow Nets

• Construction: Grid of intersecting equipotential lines ($\phi$) and streamlines ($\psi$).

• Properties: Curves are orthogonal, form squares/rectangles (if scaled properly).

• Application: Visualizing 2D potential flows, calculating flow rates ($$\displaystyle Q = \Delta \phi \cdot \text{width} = \Delta \psi \cdot \text{width} $$), determining pressure distribution.

3.4 Continuity Equation

• General 3D (Cartesian):

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

*   For incompressible flow: $$\displaystyle \nabla \cdot \vec{V} = 0 $$.

• Cylindrical Coordinates $(r, \theta, z)$:

$$ \frac{1}{r} \frac{\partial (r v_r)}{\partial r} + \frac{1}{r} \frac{\partial v_\theta}{\partial \theta} + \frac{\partial v_z}{\partial z} = 0 $$

• Spherical Coordinates $(r, \theta, \phi)$:

$$ \frac{1}{r^2} \frac{\partial (r^2 v_r)}{\partial r} + \frac{1}{r \sin\theta} \frac{\partial (v_\theta \sin\theta)}{\partial \theta} + \frac{1}{r \sin\theta} \frac{\partial v_\phi}{\partial \phi} = 0 $$

3.5 Velocity & Acceleration

Given velocity vector $$\displaystyle \vec{V} = (u, v, w) $$:

• Local Acceleration: $$\displaystyle \frac{\partial \vec{V}}{\partial t} $$

• Convective Acceleration: $(\vec{V} \cdot \nabla) \vec{V}$

• Total/Material Acceleration: $$\displaystyle \vec{a} = \frac{D\vec{V}}{Dt} = \frac{\partial \vec{V}}{\partial t} + (\vec{V} \cdot \nabla) \vec{V} $$

  • 2D: $$\displaystyle a_x = \frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} $$

  • $$\displaystyle a_y = \frac{\partial v}{\partial t} + u\frac{\partial v}{\partial x} + v\frac{\partial v}{\partial y} $$

IV. Fluid Dynamics

4.1 Euler's Equation of Motion

For inviscid (ideal) fluid along a streamline:

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

• Derived by applying Newton's 2nd law to a fluid element along a streamline.

• Assumes: Inviscid, steady, incompressible, along a streamline.

4.2 Bernoulli's Theorem

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

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

• Terms: Pressure head ($p/\rho g$), Velocity head ($$\displaystyle V^2/2g $$), Elevation head ($z$).

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

• Limitations: Not valid for viscous (real) flows with significant friction or across streamlines in rotational flow.

• Modifications:

  • For turbulent flow: Add loss term $$\displaystyle h_L $$.

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

$$ \frac{p_1}{\rho g} + \frac{V_1^2}{2g} + z_1 + h_p = \frac{p_2}{\rho g} + \frac{V_2^2}{2g} + z_2 + h_t + h_L $$

4.3 Momentum Equation

General Form (Integral):

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

For steady flow:

$$ \sum \vec{F} = \sum_{out} \rho Q \vec{V}_{out} - \sum_{in} \rho Q \vec{V}_{in} $$

• Applications: Force on pipe bends, nozzles, vanes. Include pressure forces and reaction force.

• Key: Choose a control volume that cuts through the inlet/outlet and the surface where force is to be found.

4.4 Energy Equation

From 1st Law of Thermodynamics for a control volume:

$$ \dot{Q} - \dot{W}_s = \frac{d}{dt} \int_{CV} \rho \left( e + \frac{V^2}{2} + gz \right) dV + \int_{CS} \rho \left( e + \frac{V^2}{2} + gz + \frac{p}{\rho} \right) (\vec{V} \cdot \hat{n}) dA $$

For incompressible, steady flow with no heat transfer/shaft work (except pumps/turbines), reduces to Bernoulli's equation with losses.

Pump Power Calculation:

$$ \text{Power added} = \rho g Q H_p = \gamma Q H_p $$

where $$\displaystyle H_p $$ = total head added by pump (from energy eqn).

V. Flow Measurement

5.1 Orifice Meter

• Theory: Sharp-edged plate creates a vena contracta (minimum area $$\displaystyle A_c $$).

• Coefficients:

  • Coefficient of Contraction ($$\displaystyle C_c $$): $$\displaystyle C_c = A_c / A_o $$ (typically 0.62-0.65).

  • Coefficient of Velocity ($$\displaystyle C_v $$): $$\displaystyle C_v = V_{actual} / V_{theoretical} $$.

  • Coefficient of Discharge ($$\displaystyle C_d $$): $$\displaystyle C_d = C_c C_v $$ (typically 0.6-0.65).

• Discharge Equation:

$$ Q = C_d A_o \sqrt{\frac{2(p_1 - p_2)}{\rho}} = C_d A_o \sqrt{2g \Delta h} $$

where $\Delta h$ = differential head of manometric fluid.

5.2 Venturi Meter

• Theory: Converging-diverging tube. Novena contracta (flow fills the throat).

• Discharge Equation (with differential manometer):

$$ Q = C_d \frac{A_1 A_2}{\sqrt{A_1^2 - A_2^2}} \sqrt{2g \left( \frac{p_1 - p_2}{\rho} + h \right)} $$

For manometer with fluid of density $$\displaystyle \rho_m $$:

$$ Q = C_d \frac{A_1 A_2}{\sqrt{A_1^2 - A_2^2}} \sqrt{2g \Delta h \left( \frac{\rho_m}{\rho} - 1 \right)} \quad (\text{if } \rho_m > \rho) $$

• Comparison with Orifice:

  | Venturi Meter | Orifice Meter | | :--- | :--- | | Higher $$\displaystyle C_d $$ (0.95-0.99), lower energy loss | Lower $$\displaystyle C_d $$ (0.6-0.65), higher energy loss | | More expensive, larger size | Cheaper, compact | | Used for high flow rates | Used for low to moderate flows |

5.3 Flow Nozzle

• Intermediate between venturi and orifice. Converging nozzle only. $$\displaystyle C_d \approx 0.93-0.98 $$. Less susceptible to wear than orifice.

5.4 Pitot Tube

• Working Principle: Stagnation point where velocity becomes zero.

$$ V = \sqrt{\frac{2(p_{stag} - p_{static})}{\rho}} = \sqrt{2g \Delta h} $$

• Measures point velocity. For pipe flow, must traverse to get average velocity.

5.5 Weirs & Notches

• Rectangular Weir:

$$ Q = C_d \cdot \frac{2}{3} \sqrt{2g} \cdot L \cdot H^{3/2} $$

$L$ = length, $H$ = head over crest.

• Triangular (V-notch):

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

$\theta$ = notch angle.

• Trapezoidal (Cipolletti): $$\displaystyle Q = C_d \cdot \frac{8}{15} \sqrt{2g} \left( H + \frac{K H}{2} \right) H^{3/2} $$ (approx. rectangular with side slopes).

VI. Dimensional Analysis & Similitude

6.1 Dimensional Homogeneity

All terms in a physically meaningful equation must have identical dimensions.

6.2 Buckingham Pi Theorem

Steps:

1. Identify n variables involved in the problem.

2. Identify k fundamental dimensions (M, L, T, Θ, etc.) among them.

3. Number of dimensionless $\Pi$ terms = $n - k$.

4. Choose k repeating variables (must include all fundamental dimensions, not form a dimensionless group themselves).

5. Form each $\Pi$ term by combining repeating variables with the remaining (n-k) variables, making it dimensionless.

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

Example: Drag Force on Partially Submerged Body

Variables: $R$ (drag), $\rho$, $\mu$, $V$, $l$, $g$. ($$\displaystyle n=6 $$, $$\displaystyle k=3 $$: M, L, T)

Repeating: $\rho$, $V$, $l$.

$$\displaystyle \Pi_1 = \frac{R}{\rho V^2 l^2} $$ (Reynolds number not involved? Wait, check: $R$ has MLT⁻², $\rho$ ML⁻³, $V$ LT⁻¹, $l$ L. So $$\displaystyle \Pi_1 = R/(\rho V^2 l^2) $$ is dimensionless. But we also have $\mu$ and $g$.)

Actually, standard form: $$\displaystyle R = f(V, l, \rho, \mu, g) $$.

$$\displaystyle \Pi_1 = \frac{R}{\rho V^2 l^2} $$ (Force coefficient)

$$\displaystyle \Pi_2 = \frac{\rho V l}{\mu} = Re $$ (Reynolds)

$$\displaystyle \Pi_3 = \frac{V^2}{g l} = Fr $$ (Froude)

So: $$\displaystyle \frac{R}{\rho V^2 l^2} = \phi\left(Re, Fr\right) $$.

6.3 Similarity

• Attainability in Practice: Dynamic similarity is hardest. Often, only one or two dimensionless groups can be matched simultaneously (e.g., match $Re$ or $Fr$, but not both if $$\displaystyle L_r \neq 1 $$). Geometric and kinematic are easier to achieve.

VII. Internal Pipe Flow

7.1 Reynolds Number ($Re$)

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

• Critical $Re$: For pipe flow, $$\displaystyle Re_{crit} \approx 2300 $$.

  • $$\displaystyle Re < 2300 $$: Laminar

  • $$\displaystyle Re > 4000 $$: Turbulent

  • $$\displaystyle 2300 < Re < 4000 $$: Transition (unstable)

• Significance: Predicts flow regime, governs velocity profile and friction factor.

7.2 Laminar Flow (Hagen-Poiseuille Flow)

• Velocity Distribution (Circular Pipe):

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

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

• Mean Velocity: $$\displaystyle V = \frac{1}{2} u_{max} $$.

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

• Pressure Drop (Darcy-Weisbach for laminar):

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

Or from Hagen-Poiseuille:

$$ \Delta p = \frac{128 \mu L Q}{\pi D^4} \quad \text{or} \quad \Delta p = \frac{32 \mu L V}{D^2} $$

7.3 Turbulent Flow

• Velocity Profile: Flatter ("plug-like") than laminar. Described by power law or log law.

• Smooth vs Rough Pipes:

  • Smooth: Roughness height $$\displaystyle k_s $$ negligible compared to viscous sublayer thickness.

  • Rough: Roughness protrudes through viscous sublayer, dominates friction.

• Friction Factor ($f$): Determined from Moody Chart or empirical equations.

  • Blasius (smooth pipes, $$\displaystyle 4000 < Re < 10^5 $$): $$\displaystyle f = 0.316 / Re^{0.25} $$.

7.4 Darcy-Weisbach Equation

General Head Loss Equation:

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

• Derivation: From momentum balance on a cylindrical fluid element, using shear stress at wall.

$$ \Delta p = f \frac{L}{D} \frac{\rho V^2}{2} \quad \text{or} \quad \tau_w = f \frac{\rho V^2}{8} $$

• Application: Calculate major losses in any pipe (laminar or turbulent) if $f$ is known from Moody chart or Colebrook equation.

7.5 Minor Losses ($$\displaystyle h_m $$)

Due to fittings, valves, bends, etc. Expressed as:

$$ h_m = K \frac{V^2}{2g} $$

where $K$ = loss coefficient (empirical).

• Common $K$ values:

  • Entrance (sharp): $K \approx 0.5$

  • Exit: $K \approx 1.0$

  • 90° standard elbow: $K \approx 0.75$

7.6 Pipes in Series & Parallel

• Series: Same flow rate $Q$. Total head loss = sum of losses in each pipe.

$$ H = \sum h_{f_i} + \sum h_{m_i} = \sum \left( f_i \frac{L_i}{D_i} + \sum K_i \right) \frac{V_i^2}{2g} $$

Solve iteratively (since $$\displaystyle V_i $$ and $$\displaystyle f_i $$ depend on $Q$).

• Parallel: Same head loss $H$ in each pipe. Total flow $$\displaystyle Q = \sum Q_i $$.

$$ H = f_1 \frac{L_1}{D_1} \frac{V_1^2}{2g} = f_2 \frac{L_2}{D_2} \frac{V_2^2}{2g} = ... $$

• Equivalent Pipe: Single pipe that gives same total head loss as a compound system.

VIII. Viscous Flow Applications

8.1 Stokes' Law

• Assumptions: Very small particle (low $Re$), creeping flow ($$\displaystyle Re < 0.1 $$), spherical, Newtonian fluid, no wall effects.

• Terminal Velocity ($$\displaystyle V_t $$): When drag force $$\displaystyle F_D $$ balances net weight.

$$ F_D = 6\pi \mu R V_t = (\rho_p - \rho) g \frac{4}{3}\pi R^3 $$

$$ \therefore V_t = \frac{2}{9} \frac{(\rho_p - \rho) g R^2}{\mu} $$

• Applications: Settling of fine particles, viscosity measurement.

8.2 Flow between Rotating Cylinders (Couette Flow)

• Setup: Inner cylinder radius $$\displaystyle R_i $$, outer $$\displaystyle R_o $$, length $L$, rotates at $\omega$. Fluid in annular gap.

• Torque ($T$) on inner cylinder:

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

• Viscosity Determination:

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

8.3 Lubrication (Basic Theory)

• Model: Thin viscous film between a moving plate (velocity $U$) and a stationary inclined plane (angle $\alpha$).

• Assumptions: Steady, laminar, 1D, pressure $p$ constant across film, inertia negligible.

• Velocity Profile (from $$\displaystyle \frac{d^2 u}{dy^2} = \frac{1}{\mu} \frac{dp}{dx} $$):

$$ u(y) = \frac{1}{2\mu} \frac{dp}{dx} y^2 + C_1 y + C_2 $$

Apply no-slip BCs to get profile.

• Viscosity from Force Balance:

  Weight component down plane = Viscous shear force.

$$ W \sin\alpha = \tau_w A = \mu \left( \frac{du}{dy} \right)_{y=h} A $$

Solve for $\mu$.

Exam Focus: This setup is classic for determining $\mu$ from sliding plate experiment.