Chapter 1 Notes - ME321: Introduction to Fluid Mechanics

Definition of Fluid

• A substance that deforms continuously when acted on by a shearing stress of any magnitude.
• A shearing stress (force per unit area) is created whenever a tangential force acts on a surface.

Continuum Theory

• We assume that all the fluid characteristics we are interested in (pressure, velocity, etc.) vary continuously throughout the fluid.
• We treat the fluid as a continuum and refer to very small volumes as a point in the flow.

Dimensions and Units

• Dimensional Homogeneity: All theoretically derived equations are dimensionally homogeneous; the dimensions of the left side of the equation must be the same as those on the right side, and all additive terms must have the same dimensions.
• Primary Units
  • Basic Dimension: L (length), T (time), M (mass), Q (temperature)
  • SI (Systems International):
  • Length: L = m
  • Time: T = s
  • Mass: M = kg
  • Temperature: Q = °C (K) or °F (°R)
  • British Gravitational System (BG):
  • Length: ft
  • Time: s
  • Mass: slug
  • Temperature: °F (°R)
• Secondary Units (derived quantities)
  • Area: $L^2$
  • Velocity: $LT^{-1}$
  • Acceleration: $LT^{-2}$
  • Force: $MLT^{-2}$ (N in SI or lbf in BG)
  • Density: $ML^{-3}$
  • Pressure: $ML^{-1}T^{-2}$ (Pa or lbf/ft^2)
  • Energy: $ML^2T^{-2}$
• Conversion factors between BG and SI are provided in Tables 1.3 and 1.4 (not reproduced here).
• Temperature conversions:
  • $K = °C + 273.15$
  • $^{\circ}R = ^{\circ}F + 459.67$

Weight and Mass

• Mass and force units:
  • In SI: mass unit is kilogram (kg); force unit is newton (N); $1\text{ N} = 1\text{ kg} \cdot 1\text{ m/s}^2$
  • In BG: mass unit is slug; force unit is pound-force (lbf); $1\text{ lbf} = 1\text{ slug} \cdot 1\text{ ft/s}^2$
• Weight: $W = m g$
  • In SI: $W (N) = m (kg) \cdot g$ with $g = 9.81\ \text{m/s}^2$
  • In BG: $W (lbf) = m (slug) \cdot g$ with $g = 32.2\ \text{ft/s}^2$
• English Engineering (EE) System:
  • $W(\text{lbf}) = m(\text{lbm}) \, g_7 \cdot 32.2\ \text{ft/s}$
  • where g_7 = 32.2\ \text{lbm\,ft/(lbf\,s^2)}

Density, Specific Volume, Specific Weight, Specific Gravity, and Ideal Gas Law

• Density: $\rho = \frac{m}{V}$
  • Liquids: density variations with pressure/temperature are generally small.
  • Gases: density is strongly influenced by both pressure and temperature (i.e., $\rho\to p, T$ changes matter).
• Specific Volume: $\nu = \frac{1}{\rho}$
• Specific Weight: $\gamma = \rho g$ (N/m^3 or lbf/ft^3)
• Specific Gravity: $SG = \frac{\rho}{\rho_f}$ (ratio relative to water at a specified temperature, usually water at 4°C)
• Ideal Gas Law: $p = \rho R T$
  • $p$: pressure (Pa or psf)
  • $\rho$: density (kg/m^3 or slug/ft^3)
  • $T$: absolute temperature (K or °R)
  • $R$: gas constant (J/kg·K or ft·lb·slug^{-1}·K)

Viscosity and Fluid Resistance

• Consider a material between two wide parallel plates: bottom plate fixed, top plate moving under load $P$. A shear strain $\gamma$ develops.
• Equilibrium requires shear resistance $t$ at the plate–material interface: $P = t A$ where $A$ is the plate area.
• Solid response: $\tau \propto \delta \beta$; linear relation $\tau = G \delta \beta$ (shear modulus $G$, so $\tau = G\gamma$ if $\gamma = \delta \beta$).
• Fluids: no-slip condition at the fluid–plate surface; the fluid in contact with the upper plate moves with velocity $U$, while the fluid at the bottom plate has zero velocity.
• For small time and small angles, the shear strain rate is
  • $\dot{\gamma} = \frac{du}{dy}$ and tangent of the angle gives $\gamma \approx \dot{\beta} = \frac{U \; \delta t}{b}$
• Viscosity relation: $\tau \propto \dot{\gamma}$; for Newtonian fluids, $\tau = \mu \dot{\gamma}$; where $\mu$ is the dynamic (absolute) viscosity. Its value depends on the fluid and temperature.

Newtonian vs Non-Newtonian Fluids; Apparent Viscosity

• Newtonian fluids: $\tau$ is linearly related to $\dot{\gamma}$ (i.e., $\tau = \mu \dot{\gamma}$).
• Non-Newtonian fluids: the relation is not linear and may depend on shear history.
• Apparent viscosity: $\mu_{\text{ap}} = \text{slope of } (\tau \text{ vs } \dot{\gamma})$ in the linear region.
• Shear thinning: $\mu_{\text{ap}}$ decreases with increasing $\dot{\gamma}$.
• Shear thickening: $\mu_{\text{ap}}$ increases with increasing $\dot{\gamma}$.
• Temperature and pressure effects:
  • Viscosity is mildly dependent on pressure (often neglected).
  • Temperature sensitivity: liquids – viscosity decreases with temperature; gases – viscosity increases with temperature.
• Viscosity units:
  • SI: $[\mu] = \mathrm{Pa\cdot s}$
  • BG: often given as $\mathrm{lb\cdot s/ft^2}$ (unit convention in notes)
  • CGS: $\mathrm{dyne\cdot s/cm^2}$ (poise, P)
• Kinematic viscosity: $\nu = \frac{\mu}{\rho}$
  • Units: SI $\mathrm{m^2/s}$; BG $\mathrm{ft^2/s}$; CGS $\mathrm{St}$ (Stokes)
  • Conversions: $1\ \mathrm{m^2/s} = 10^4\ \mathrm{St}$; $1\ \mathrm{St} = 100\ \mathrm{cSt}$; hence $1\ \mathrm{m^2/s} = 10^6\ \mathrm{cSt}$
• Empirical correlations for viscosity
  • Sutherland equation (gases): $\mu = C \frac{T^{3/2}}{T + S}$
  • Andrade’s equation (liquids): $\mu = A \exp\left(\frac{B}{T}\right)$
  • Constants: $C, S$ for Sutherland; $A, B$ for Andrade

Compressibility of Fluids and Speed of Sound

• Compressibility concept: how a fluid volume changes with pressure.
• Bulk modulus of elasticity (bulk modulus):
  • General form: $E<em>g = -V \frac{\partial p}{\partial V}\Big|</em>T = \rho \frac{\partial p}{\partial \rho}\Big|_T$
  • Units: $\mathrm{N\,m^{-2}}$ (or psi in imperial units)
  • Larger $E_g$ means less compressible; liquids are generally incompressible (example: water ~ 2.15 GN/m^2).
• Gas compressibility: $E_g varies with pressure$; two limiting processes:
  • Isothermal: $p = \rho R T$ ⇒ $\frac{\partial p}{\partial \rho} = \text{const}$ ⇒ $E_g = p$
  • Isentropic (adiabatic, reversible): $p \rho^{-\kappa} = \text{const} \Rightarrow E_g = \kappa p$
• Speed of sound: disturbance propagation speed in a fluid given by$c = \sqrt{\frac{E_g}{\rho}}$
  • For ideal gases: $c = \sqrt{\frac{\gamma p}{\rho}} = \sqrt{\gamma R T}$ where $\gamma$ is the ratio of specific heats.
• Note: heat transfer during sound propagation is very small; process is essentially isentropic.
• The speed of sound in air at various temperatures is listed in reference appendices (Appendix B in notes).

Vapor Pressure and Cavitation

• Vapor pressure $p_v$: pressure at which a completely liquid-filled container has its vapor in equilibrium with the liquid.
• If the end is moved without allowing air, the space becomes filled with vapor at $p_v$. Local pressure drops may cause boiling; bubbles can form in low-pressure regions.
• Bubbles may implode when entering high-pressure regions, releasing energy that can damage structures over time.
• Reference video provided for context: https://www.youtube.com/watch?v=U-uUYCFDTrc

Surface Tension and Capillarity

• Surface tension $\sigma$: cohesive molecular forces act along the surface, creating a contracted “skin” at the interface (liquid–gas or immiscible liquid interfaces).
• Pressure difference across a curved interface (spherical droplet):
  • $\Delta p = p<em>{\text{inside}} - p</em>{\text{outside}} = \frac{2\sigma}{R}$
• Capillary tube phenomena and wetting
  • Wetting: adhesion between wall and liquid is strong enough to pull liquid up the wall (contact angle \theta < 90^{\circ}).
  • Non-wetting: adhesion is weak relative to cohesion; liquid level in tube is depressed (contact angle \theta > 90^{\circ}, e.g., mercury).
• Capillary rise in a tube of radius $R$ (balance of forces):
  • $2 \pi R \sigma \cos \theta = \pi R^2 \rho g h \Rightarrow h = \frac{2 \sigma \cos \theta}{\rho g R}$
  • Alternatively, using the specific weight $\gamma = \rho g$: $h = \frac{2 \sigma \cos \theta}{\gamma R}$
• Note: wetting angle conventions apply: $\theta < 90^{\circ}\Rightarrow\text{wetting}, \theta > 90^{\circ}\Rightarrow\text{non-wetting}$
• Examples: water wets common glasses (low contact angle); mercury does not wet glass (high contact angle).

Additional Notes and References

• The transcript includes a video link for cavitation and mentions Appendix B for speed of sound data.
• Tables for conversion factors (BG↔SI) are provided in the source material (Tables 1.3 and 1.4).
• Equations and constants cited reflect standard fluid mechanics conventions, with some notation variations in the original lecture notes.