Fluid Mech. – II (CE-604 (A)) - Unit 1 Short Notes

UNIT 1: Fluid Mechanics Fundamentals

1. Basic Fluid Properties and Characteristics

Definition & Classification:

• Fluid: A substance that deforms continuously under the application of a shear stress, no matter how small.

• Compressible Fluid: Density changes significantly with pressure (e.g., gases). $$\displaystyle \rho = f(P) $$.

• Incompressible Fluid: Density is constant. $$\displaystyle \rho = \text{constant} $$. (Liquids are often approximated as incompressible).

Key Properties:

Viscosity & Newton's Law of Viscosity:

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

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

Where $$\displaystyle \frac{du}{dy} $$ is the velocity gradient perpendicular to the flow direction.

• Newtonian Fluids: Follow Newton's law (e.g., water, air, most common oils). $\mu$ is constant.

• Non-Newtonian Fluids: Do not follow the law (e.g., blood, paint, ketchup).

Temperature Effect on Viscosity:

• Liquids (e.g., Water): Viscosity decreases with increase in temperature.

• Gases (e.g., Air): Viscosity increases with increase in temperature.

Surface Tension & Capillarity:

• Pressure Difference Across a Curved Interface (Spherical Droplet/Bubble):

$$\Delta P = P_{\text{inside}} - P_{\text{outside}} = \frac{2\sigma}{r} \quad \text{(Droplet)}$$

$$\Delta P = \frac{4\sigma}{r} \quad \text{(Soap Bubble)}$$

• Capillary Rise/Depression in a Tube:

$$h = \frac{4\sigma \cos\theta}{\rho g d}$$

Where $\theta$ is the angle of contact, $d$ is the tube diameter.

> **Result:** Rise for $$\displaystyle \theta < 90^\circ $$ (e.g., water in glass), Depression for $$\displaystyle \theta > 90^\circ $$ (e.g., mercury in glass).

[!TIP] Exam Focus: Be prepared to calculate shear stress in pipes (laminar flow) and capillary rise in manometers. Remember the sign convention for $\cos\theta$.

2. Fluid Kinematics

Description of Motion:

• Lagrangian Approach: Follows individual fluid particles (tracking a "parcel"). Complex for continuous flows.

• Eulerian Approach: Observes flow properties at fixed points in space. Standard method in fluid mechanics.

Flow Classification:

• Steady Flow: Flow properties ($u,v,w,P,\rho$) at any point do not change with time. $$\displaystyle \frac{\partial(\cdot)}{\partial t} = 0 $$.

• Unsteady Flow: Flow properties change with time.

• Uniform Flow: Flow properties are the same at all points in the flow field.

• Laminar vs. Turbulent: Distinguished by Reynolds Number ($Re$).

Streamlines, Equipotential Lines & Flow Nets:

• Streamline: A line everywhere tangent to the velocity vector at a given instant. For steady flow, streamlines = pathlines.

• Equipotential Line ($$\displaystyle \phi = \text{constant} $$): A line along which the velocity potential is constant.

• Flow Net: A system of intersecting streamlines and equipotential lines forming curvilinear squares.

• Orthogonality: Streamlines and equipotential lines always intersect at 90°.

  • Proof: For $$\displaystyle \phi = \text{constant} $$, $\nabla\phi \perp d\vec{s}$. But $$\displaystyle \vec{V} = \nabla\phi $$. Hence, $\vec{V} \perp$ equipotential line. By definition, $\vec{V}$ is tangent to streamline. Therefore, streamline $\perp$ equipotential line.

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

Continuity Equation (Cartesian Coordinates):

Based on conservation of mass for a differential fluid element.

• General (Unsteady, Compressible):

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

• For Steady, Incompressible Flow ($$\displaystyle \rho = \text{constant} $$):

$$\nabla \cdot \vec{V} = 0 \quad \Rightarrow \quad \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0$$

Application: Given two velocity components, use the continuity equation to find the third. For 2D incompressible flow: $$\displaystyle \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0 $$.

[!TIP] Exam Pitfall: Forgetting the sign in the definition of stream function ($$\displaystyle v = -\partial\psi/\partial x $$). Always verify orthogonality using Cauchy-Riemann equations.

3. Fluid Dynamics

Euler's Equation of Motion:

Derived by applying Newton's Second Law (F=ma) to a fluid element, considering only pressure and body forces (neglecting viscosity).

• Along a Streamline (for steady flow):

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

Where $dz$ is upward positive.

Bernoulli's Theorem:

• Statement: For an incompressible, inviscid (non-viscous), steady, along a streamline flow, the total energy per unit volume is constant.

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

*   $P/\rho$: Pressure Energy (Pressure Head, $P/\rho g$)

*   $$\displaystyle V^2/2 $$: Kinetic Energy (Velocity Head, $$\displaystyle V^2/2g $$)

*   $gz$: Potential Energy (Datum Head, $z$)

• Assumptions: Steady, Incompressible, Inviscid, Along a streamline (or irrotational flow for whole field).

• Limitations: Cannot account for viscous losses (friction), energy addition (pump), or energy extraction (turbine). For real flows, include head loss $$\displaystyle h_L $$:

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

Pascal's Law:

• Statement: The pressure intensity at any point in a static fluid is the same in all directions.

• Proof: Consider a small triangular fluid element. Resolving forces horizontally and vertically shows $$\displaystyle P_x = P_y = P_z $$.

• Application: Hydraulic Systems (car lifts, hydraulic brakes). Force multiplication: $$\displaystyle F_2 = F_1 \cdot (A_2/A_1) $$.

Pressure Measurement:

• Principle: Use a manometric fluid to balance the pressure difference. $$\displaystyle P_A - P_B = (\rho_m - \rho_f)gh $$ for inclined/differential manometers.

• Manometers:

  • Simple Manometer: Measures pressure at a single point relative to atmosphere.

  • Differential Manometer: Measures pressure difference between two points.

• Pitot Tube: Measures stagnation pressure ($$\displaystyle P_0 = P + \frac{1}{2}\rho V^2 $$). Used with a static pressure tap to find dynamic pressure and hence velocity:

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

• Capillary Effect: Important for small-diameter tubes. Causes error in manometer readings. Formula given above.

[!TIP] Exam Key: Derivation of Bernoulli from Euler is a must. Know how to apply Bernoulli with head loss. For manometers, always draw a sketch and label heights carefully.

4. Boundary Layer and Flow Resistance

Boundary Layer Concept (Prandtl):

• Definition: A thin region near a solid boundary where the flow velocity increases from zero (at the wall, no-slip condition) to the free-stream velocity $$\displaystyle U_\infty $$.

• Development over a Flat Plate:

  1. Leading Edge: Boundary layer starts at zero thickness.

  2. Laminar Region: Smooth, orderly flow. Velocity profile parabolic.

  3. Transition Point: Laminar flow becomes unstable.

  4. Turbulent Region: Chaotic, mixing flow. Velocity profile fuller (flatter).

  5. Boundary Layer Thickness ($\delta$): Distance from wall where $$\displaystyle u \approx 0.99 U_\infty $$.

  • Laminar: $\delta \propto \sqrt{x}$ (Blasius solution: $$\displaystyle \delta \approx \frac{5x}{\sqrt{Re_x}} $$)

  • Turbulent: $$\displaystyle \delta \propto x^{4/5} $$

Shear Stress & Friction Drag:

• Wall Shear Stress ($$\displaystyle \tau_w $$):

  • Laminar: $$\displaystyle \tau_w = \mu \left( \frac{\partial u}{\partial y} \right)_{y=0} $$

  • Turbulent (empirical): $$\displaystyle \tau_w = \frac{1}{2} C_f \rho U_\infty^2 $$

• Friction Drag on a Flat Plate:

$$D_f = \int_0^L \tau_w(x) \cdot b \, dx$$

*   **Laminar Plate (Length $L$):** $$\displaystyle D_{f,\text{lam}} = \frac{1}{2} C_{f,\text{lam}} \rho U_\infty^2 (bL) $$

    where $$\displaystyle C_{f,\text{lam}} = \frac{1.328}{\sqrt{Re_L}} $$ (for local $$\displaystyle Re_x $$, use $$\displaystyle Re_x $$).

*   **Turbulent Plate:** $$\displaystyle C_{f,\text{tur}} = \frac{0.074}{Re_L^{1/5}} $$ (for $$\displaystyle 5 \times 10^5 < Re_L < 10^7 $$).

*   **Mixed (Transition):** Calculate laminar drag up to $$\displaystyle x_{tr} $$, turbulent drag from $$\displaystyle x_{tr} $$ to $L$, sum them.

Turbulence:

• Characteristics: Chaotic, irregular, 3D, diffusive (enhances mixing/momentum transfer), fluctuating velocity components ($u'$, $v'$, $w'$).

• Effect on Flow Properties: Increases skin friction drag and pressure drop compared to laminar flow, but can delay separation.

Reynolds Experiment & Critical Reynolds Number:

• Experiment: Dye injected into flow in a glass pipe.

  • Low Velocity (Laminar): Dye forms a straight, smooth line.

  • High Velocity (Turbulent): Dye mixes and spreads across the pipe.

• Critical Reynolds Number ($$\displaystyle Re_{cr} $$): The $Re$ at which flow transitions from laminar to turbulent.

  • For pipe flow: $$\displaystyle Re_{cr} \approx 2000 - 2300 $$.

  • For flow over a flat plate: $$\displaystyle Re_{cr} \approx 5 \times 10^5 $$.

$$Re = \frac{\rho U D}{\mu} = \frac{U D}{\nu} \quad \text{(Pipe)} \quad \text{or} \quad Re_x = \frac{\rho U x}{\mu} \quad \text{(Plate)}$$

[!TIP] Exam Focus: Be able to calculate drag force for a plate with given dimensions, viscosity, and velocity. Identify flow regime using $Re$. Know the formulas for $\delta$ and $$\displaystyle C_f $$ for both laminar and turbulent cases.

5. Applications in Hydraulic Systems

Reciprocating Pump:

• Working Principle: Positive displacement pump. A piston/plunger reciprocates (back-and-forth) inside a cylinder.

  1. Suction Stroke: Piston moves back, suction valve opens, atmospheric pressure pushes liquid into cylinder.

  2. Delivery Stroke: Piston moves forward, suction valve closes, delivery valve opens, liquid is forced out.

• Components: Cylinder, piston/plunger, suction & delivery valves, crank mechanism.

• Flow: Pulsating (not uniform). Requires air vessel to reduce pulsation.

Reaction Turbine:

• Working Principle: Converts pressure energy and kinetic energy of fluid into mechanical energy. The runner is entirely immersed in water.

• Key Feature: Pressure drop occurs both in the stationary blades (nozzles/guide vanes) and on the moving blades (runner).

• Common Type: Francis Turbine (Radial flow).

• Schematic:

  DiagramSEARCH: "Francis turbine schematic diagram labeled"
  • Water enters spiral casing → Guide vanes (stationary, direct flow) → Runner (rotating, reaction) → Draft tube (recovers pressure).

Total Energy Line (TEL) & Hydraulic Gradient Line (HGL):

• Total Energy Line (TEL): Represents the total energy per unit weight along the pipe.

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

*   **Slope** = head loss per unit length ($$\displaystyle h_L/L $$). **Always slopes downward** in the direction of flow.

• Hydraulic Gradient Line (HGL): Represents the pressure head + datum head.

$$\text{HGL} = \frac{P}{\rho g} + z$$

*   **Slope** = frictional head loss gradient. Lies **below TEL** by a vertical distance of velocity head ($$\displaystyle V^2/2g $$).

*   At a **reservoir** (large $A$, $V \approx 0$), HGL coincides with TEL.

*   **Interpretation:** If HGL dips below the pipe, pressure becomes negative (risk of cavitation).

[!TIP] Exam Application: Always sketch TEL and HGL for problems involving pipes, reservoirs, and pumps. Remember: TEL - HGL = $$\displaystyle V^2/2g $$.

6. Buoyancy and Flotation

Archimedes' Principle:

A body immersed in a fluid experiences an upward buoyant force equal to the weight of the fluid displaced by the body.

$$F_B = \gamma_f \cdot V_{\text{disp}} = \rho_f g V_{\text{disp}}$$

Where $$\displaystyle V_{\text{disp}} $$ is the volume of fluid displaced.

Condition for Floating:

For a body floating in equilibrium:

• Weight of Body ($W$) = Buoyant Force ($$\displaystyle F_B $$)

$$W = \gamma_f V_{\text{disp}}$$

• Relative Density (Specific Gravity, $SG$):

$$SG = \frac{\rho_{\text{body}}}{\rho_{\text{water}}} = \frac{\text{Weight in air}}{\text{Weight in air} - \text{Weight in water}}$$

• For a floating body, the fraction submerged = $SG$ (if fluid is water).

$$\frac{V_{\text{sub}}}{V_{\text{total}}} = \frac{\rho_{\text{body}}}{\rho_f} = SG_{\text{relative to fluid}}$$

Problem Solving (e.g., block in water & glycerin):

1. Let $H$ = total height, $$\displaystyle h_w $$ = submerged depth in water, $$\displaystyle h_g $$ = submerged depth in glycerin.

2. Weight of block is constant: $$\displaystyle \gamma_b H A = \gamma_w h_w A = \gamma_g h_g A $$.

3. Given heights above surface, so $$\displaystyle h_w = H - (\text{height above water}) $$, etc.

4. Solve the two equations:

$$\frac{H - h_w}{H} = 1 - \frac{\gamma_w}{\gamma_b}, \quad \frac{H - h_g}{H} = 1 - \frac{\gamma_g}{\gamma_b}$$

Eliminate $$\displaystyle \gamma_b/\gamma_w $$ to find $H$ and $SG$.

[!TIP] Exam Trap: Ensure consistent units (use $SG$ or densities). The volume of displaced fluid is the submerged volume, not the total volume.

7. Special Topics and Historical Context

Historical Development (Key Milestones):

• Ancient: Archimedes (Buoyancy, 250 BC), Vitruvius (Pipes, aqueducts).

• Renaissance: da Vinci (flow visualization, turbulence), Torricelli (Barometer, orifice equation, 1643).

• 17th-18th Century: Pascal (Pascal's Law, 1653), Newton (Viscosity, 1687), Bernoulli (Bernoulli's Equation, 1738), Euler (Euler Equations, 1755).

• 19th Century: Navier & Stokes (Navier-Stokes Equations, 1820s), Reynolds (Reynolds Number, transition, 1883), Prandtl (Boundary Layer Theory, 1904).

• 20th Century: Development of turbulence models, computational fluid dynamics (CFD).

Fluid Mechanics in Biological Systems:

• Blood Flow:

  • Blood is a non-Newtonian fluid (shear-thinning, exhibits yield stress).

  • Flow in arteries is pulsatile (unsteady) due to heart pumping.

  • Reynolds Number in aorta is ~2000, so flow can be transitional/turbulent.

  • Poiseuille's Law (for steady laminar flow in rigid tubes) approximates flow in small vessels: $$\displaystyle Q = \frac{\pi R^4 \Delta P}{8\mu L} $$.

• Heart as a Pump:

  • Acts as a positive displacement pump (reciprocating type).

  • Right side pumps deoxygenated blood to lungs (low pressure).

  • Left side pumps oxygenated blood to body (high pressure).

  • Valves ensure unidirectional flow (like check valves).

Pressure Measurement in Barometers & Manometric Fluids:

• Barometer: Measures atmospheric pressure. Simple mercury barometer: $$\displaystyle P_{\text{atm}} = \gamma_{Hg} h $$.

• Manometric Fluids: Should have high density (for compact instrument) and low vapor pressure (to avoid evaporation/column break). Mercury is ideal but toxic. Alternatives: water (low density), oils, alcohols.

• Capillary Error: In small tubes, meniscus curvature causes error. Use large-diameter tubes or apply capillary correction formula.

[!TIP] Exam Short Note: For "Fluid mechanics in biological systems," focus on blood's non-Newtonian nature, pulsatile flow, Reynolds number relevance, and the heart's pumping analogy. For history, know 4-5 key contributors and their primary contribution.