Fluid Mechanics-I (CE-501) - Unit 5 Short Notes

UNIT 5: Fluid Mechanics-I - Short Notes

I. Fluid Properties

Viscosity

• Dynamic Viscosity ($\mu$): Measure of a fluid's internal resistance to shear or angular deformation. Newton's Law of Viscosity: $$\displaystyle \tau = \mu \frac{du}{dy} $$, where $\tau$ is shear stress, $$\displaystyle \frac{du}{dy} $$ is velocity gradient.

• Kinematic Viscosity ($\nu$): Ratio of dynamic viscosity to density. $$\displaystyle \nu = \frac{\mu}{\rho} $$. Represents momentum diffusivity.

• Temperature Effect:

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

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

[!TIP] Common Pitfall: Do not confuse dynamic viscosity ($\mu$, units: Ns/m² or Pa·s) with kinematic viscosity ($\nu$, units: m²/s or Stokes).

Surface Tension

• Cohesive force at the liquid's surface, measured as force per unit length ($\sigma$, N/m).

• Pressure inside a curved surface (due to surface tension):

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

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

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

Compressibility & Bulk Modulus

• Compressibility: Measure of change in volume under pressure.

• Bulk Modulus ($K$): Ratio of increase in pressure to relative decrease in volume.

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

For liquids, $K$ is very large (nearly incompressible). For gases, $$\displaystyle K = \gamma P $$ (isentropic).

II. Fluid Statics

Pressure

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

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

• Vacuum Pressure: Pressure below atmospheric ($$\displaystyle P_{vac} = P_{atm} - P_{abs} $$).

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

Manometers

Devices to measure pressure difference using liquid columns.

[!TIP] In manometers, always equate pressure heads along a continuous fluid column. Convert all pressures to head of a common fluid (usually water or mercury) for clarity.

Hydrostatic Forces on Submerged Surfaces

• Total Force ($F$): $$\displaystyle F = \gamma \bar{h} A $$, where $\bar{h}$ is depth to centroid.

• Center of Pressure ($$\displaystyle h_{cp} $$): Point of action of total force.

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

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

• Pressure Distribution Diagram: Linear variation with depth. Triangular for vertical surfaces, trapezoidal for inclined.

Buoyant Force & Archimedes' Principle

• Buoyant Force ($$\displaystyle F_B $$): Upward force exerted by fluid on a submerged body.

$$\boxed{F_B = \gamma \cdot V_{disp}}$$

where $$\displaystyle V_{disp} $$ is volume of fluid displaced.

• Archimedes' Principle: Buoyant force equals weight of displaced fluid and acts through the centroid of displaced volume (Center of Buoyancy).

III. Fluid Kinematics

Flow Visualization & Classification

[!TIP] In steady flow, streamlines, path lines, and streak lines coincide.

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

Flow Net

• Definition: A grid formed by a family of streamlines ($$\displaystyle \psi = \text{const.} $$) and equipotential lines ($$\displaystyle \phi = \text{const.} $$) that are orthogonal to each other.

• Method of Construction:

  1. Graphical Method: Sketch streamlines and equipotentials satisfying boundary conditions and orthogonality.

  2. Analytical Method: Solve Laplace equation $$\displaystyle \nabla^2 \phi = 0 $$ or $$\displaystyle \nabla^2 \psi = 0 $$ for $\phi$ or $\psi$.

• Application: Determines flow patterns, calculates discharge between streamlines ($$\displaystyle Q = \Delta \psi $$), and finds velocity magnitude ($$\displaystyle |\vec{V}| = \frac{d\phi}{ds} = \frac{d\psi}{dn} $$).

Continuity Equation

• Statement: Mass is conserved for a fluid in motion.

• Derivation (3D Cartesian) for a differential fluid element ($dx\,dy\,dz$):

  Rate of mass in - Rate of mass out = Rate of accumulation.

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

• For steady, incompressible flow ($$\displaystyle \rho = \text{const.}, \frac{\partial \rho}{\partial t}=0 $$):

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

• 1D Flow: $$\displaystyle A_1 V_1 = A_2 V_2 = Q $$ (Discharge constant).

Velocity & Acceleration (Material Derivative)

• Material (Total) Derivative: Follows a fluid particle.

$$\frac{D}{Dt} = \frac{\partial}{\partial t} + u\frac{\partial}{\partial x} + v\frac{\partial}{\partial y} + w\frac{\partial}{\partial z}$$

• Acceleration Components:

$$a_x = \frac{Du}{Dt} = \frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} + w\frac{\partial u}{\partial z}$$

(Similarly for $$\displaystyle a_y, a_z $$).

IV. Fluid Dynamics

Euler's Equation of Motion

For an inviscid (ideal) fluid along a streamline:

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

Assumptions: Inviscid, steady, incompressible, along a streamline.

Bernoulli's Theorem

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

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

(Each term is a **head**: Pressure head, Velocity head, Datum head).

• Assumptions:

  1. Fluid is ideal (inviscid).

  2. Flow is steady.

  3. Flow is incompressible.

  4. Flow is along a streamline.

• Limitations: Cannot account for viscous losses (friction) or energy addition (pump/turbine).

• Modifications (Extended Bernoulli):

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

where $$\displaystyle H_{pump} $$ is head added, $$\displaystyle H_{loss} $$ is head loss due to friction.

Momentum Equation

• Statement: Net force on a fluid mass 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: $$\displaystyle \vec{F} = \int_{CS} \rho \vec{V} (\vec{V} \cdot d\vec{A}) $$

• Applications: Calculate forces on bends, nozzles, vanes, and pipe fittings. Control Volume Approach is essential.

V. Viscous Flow

Reynolds Number ($Re$)

• Definition: Ratio of inertial forces to viscous forces.

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

• Significance: Predicts flow regime.

  • Pipe flow: $$\displaystyle Re < 2000 $$: Laminar; $$\displaystyle Re > 4000 $$: Turbulent; $$\displaystyle 2000 < Re < 4000 $$: Transition.

  • Critical $Re$ depends on geometry and upstream conditions.

Laminar Flow

1. Between Parallel Plates (Fixed, separated by distance $h$):

   • Velocity Profile (plane Poiseuille flow): Parabolic.

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

*   **Max Velocity**: $$\displaystyle u_{max} = \frac{h^2}{8\mu} \left(-\frac{dp}{dx}\right) $$

*   **Mean Velocity**: $$\displaystyle V_{avg} = \frac{2}{3} u_{max} $$

*   **Shear Stress**: $$\displaystyle \tau = \mu \frac{du}{dy} $$, linear distribution ($$\displaystyle \tau=0 $$ at center, max at walls).

2. In Circular Pipes (Hagen-Poiseuille Flow):

   • Velocity Profile: Parabolic.

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

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

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

*   **Discharge (Q)**:

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

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

Stokes Law

• Drag Force ($$\displaystyle F_D $$) on a small, smooth sphere moving at low $Re$ ($$\displaystyle < 0.1 $$) in an infinite fluid:

$$\boxed{F_D = 6\pi \mu R V}$$

where $R$ is sphere radius, $V$ is velocity.

• Application: Settling velocity of particles, viscometers.

Turbulent Flow & Pipe Friction

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

• Darcy-Weisbach Equation for major (friction) loss:

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

where $f$ = **friction factor** (dimensionless).

• Friction Factor ($f$):

  • Laminar ($$\displaystyle Re < 2000 $$): $$\displaystyle f = \frac{64}{Re} $$

  • Turbulent: $$\displaystyle f = \phi(Re, \frac{\varepsilon}{D}) $$. Determined from Moody Chart or Colebrook equation.

• Minor Losses (due to fittings, bends, valves):

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

where $K$ = loss coefficient (tabulated).

Pipes in Series & Parallel

• Series: Same discharge $Q$, total head loss $$\displaystyle h_{f, total} = \sum h_f $$.

  Solve using $$\displaystyle h_f = f \frac{L}{D} \frac{V^2}{2g} $$ and $$\displaystyle Q = A V $$ for each pipe.

• Parallel: Same head loss $$\displaystyle h_f $$ for all pipes, total discharge $$\displaystyle Q = \sum Q_i $$.

  Solve $$\displaystyle h_{f1} = h_{f2} = ... $$ and $$\displaystyle Q = \sum \frac{\pi D_i^4}{128 \mu L_i} \Delta P $$ (laminar) or using $f$ for turbulent.

VI. Flow Measurement

Orifice Meter

• Principle: Orifice plate causes a sudden contraction, creating a vena contracta.

• Discharge Equation:

$$Q = C_d A_o \sqrt{\frac{2(P_1 - P_2)}{\rho (1 - \beta^4)}}$$

where $$\displaystyle C_d $$ = coefficient of discharge, $$\displaystyle A_o $$ = orifice area, $$\displaystyle \beta = d/D $$.

• Coefficients:

  • $$\displaystyle C_c $$ (Contraction coefficient): $$\displaystyle \frac{A_{vc}}{A_o} \approx 0.62 $$

  • $$\displaystyle C_v $$ (Velocity coefficient): $$\displaystyle \frac{V_{vc}}{V_{th}} < 1 $$

  • $$\displaystyle C_d = C_c C_v \approx 0.6 - 0.7 $$

• High Energy Loss due to abrupt contraction and turbulence.

Venturi Meter

• Principle: Gradual converging-diverging section minimizes energy loss.

• Discharge Equation:

$$Q = C_d \frac{\pi d^2}{4} \sqrt{\frac{2(P_1 - P_2)}{\rho (1 - \beta^4)}}$$

$$\displaystyle C_d \approx 0.95 - 0.99 $$ (much higher than orifice).

• Comparison:

  | Feature | Venturi Meter | Orifice Meter | Flow Nozzle | | :--- | :--- | :--- | :--- | | $$\displaystyle C_d $$ | Highest (0.95-0.99) | Lowest (0.6-0.7) | Intermediate (0.9-0.95) | | Energy Loss | Minimal | High | Moderate | | Cost | High | Low | Moderate | | Application | Clean fluids, high accuracy | Dirty fluids, cost-sensitive | Steam, high-velocity flows |

Pitot Tube

• Principle: Stagnation point where velocity becomes zero. Measures stagnation (total) pressure.

$$P_{total} = P_{static} + \frac{1}{2}\rho V^2$$

• Velocity Measurement: $$\displaystyle V = \sqrt{\frac{2(P_{total} - P_{static})}{\rho}} $$

• Types: Simple Pitot (measures $$\displaystyle P_{total} $$), Pitot-Static (measures both $$\displaystyle P_{total} $$ and $$\displaystyle P_{static} $$).

Weirs & Notches

• Used to measure open channel flow.

• Discharge General Form: $$\displaystyle Q = C_d \cdot \text{Area of flow} \cdot \sqrt{2g h} $$

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

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

• $H$ = head over crest, $L$ = length of weir, $\theta$ = notch angle.

VII. Dimensional Analysis & Similitude

Buckingham Pi Theorem

1. Identify all variables ($n$) and fundamental dimensions ($k$: M, L, T, etc.).

2. Form $(n - k)$ independent dimensionless $\Pi$ groups.

3. Select $k$ repeating variables (must include all fundamental dimensions, not form a dimensionless group themselves).

4. Express each remaining variable as a product of repeating variables raised to powers to form a $\Pi$ group.

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

Example: Pipe Friction Factor $f$

Variables: $f, \rho, V, D, \mu, \varepsilon$ (6 vars, 3 dims: M, L, T) → 3 $\Pi$ groups.

Repeating vars: $\rho, V, D$.

1. $$\displaystyle \Pi_1 = f \cdot \rho^a V^b D^c $$ → $$\displaystyle f = \Pi_1(\rho, V, D) $$ → $$\displaystyle \Pi_1 = f $$.

2. $$\displaystyle \Pi_2 = \mu \cdot \rho^a V^b D^c $$ → $$\displaystyle \Pi_2 = \frac{\rho V D}{\mu} = Re $$.

3. $$\displaystyle \Pi_3 = \varepsilon \cdot \rho^a V^b D^c $$ → $$\displaystyle \Pi_3 = \frac{\varepsilon}{D} $$.

Result: $$\displaystyle f = \phi(Re, \varepsilon/D) $$.

Similitude

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

• Kinematic Similarity: Similar motion, velocity fields similar ($$\displaystyle V_r = \sqrt{L_r} $$ for Froude scaling).

• Dynamic Similarity: Similar forces, force fields similar ($$\displaystyle F_r = \rho_r L_r^2 V_r^2 $$).

• Scale Ratios: Must satisfy all similarity requirements. Often achieved by equating key dimensionless numbers (e.g., $$\displaystyle Re_m = Re_p $$, $$\displaystyle Fr_m = Fr_p $$).

Important Dimensionless Numbers

VIII. Key Definitions (Exam-Focused)

• Bulk Modulus ($K$): Measure of fluid's resistance to compression. $$\displaystyle K = -V \frac{dP}{dV} $$. For water, $$\displaystyle K \approx 2.2 \times 10^9 $$ N/m².

• Friction Factor ($f$): Dimensionless number in Darcy-Weisbach equation representing pipe friction resistance. $$\displaystyle f = \frac{\tau_w}{\frac{1}{2}\rho V^2} $$.

• Flow Net: Orthogonal grid of streamlines and equipotential lines used to visualize and analyze 2D potential flow. Discharge between streamlines $$\displaystyle Q = \Delta \psi $$.

• Stream Function ($\psi$): Scalar function for 2D incompressible flow where $$\displaystyle u = \frac{\partial \psi}{\partial y} $$, $$\displaystyle v = -\frac{\partial \psi}{\partial x} $$. Lines of constant $\psi$ are streamlines.

• Pitot Tube: Device to measure fluid velocity by sensing stagnation pressure. $$\displaystyle V = \sqrt{2(P_{stag} - P_{static})/\rho} $$.

• Stokes Law: Drag force on a sphere at very low $Re$: $$\displaystyle F_D = 6\pi \mu R V $$. Basis for falling sphere viscometer.

• Buoyant Force ($$\displaystyle F_B $$): Net upward force on a submerged body. $$\displaystyle F_B = \gamma \cdot V_{disp} $$. Equal to weight of displaced fluid (Archimedes' principle).

• Buckingham's Pi Theorem: A method for dimensional analysis stating that a problem with $n$ variables and $k$ fundamental dimensions can be reduced to $(n-k)$ independent dimensionless $\Pi$ groups.