Fluid Mechanics - Chapter 1 & 2 Notes

Conversion Factors (BG to SI)

To convert from British Gravitational (BG) units to International System of Units (SI), you multiply by the following factors:

• Acceleration: ft/s² to m/s² multiply by $0.3048$
• Area: ft² to m² multiply by $9.2903 \times 10^{-2}$; mi² to m² multiply by $2.5900 \times 10^{6}$; acres to m² multiply by $4.0469 \times 10^{3}$
• Density: slug/ft³ to kg/m³ multiply by $5.1538 \times 10^{2}$; lbm/ft³ to kg/m³ multiply by $1.6019 \times 10^{1}$
• Energy: ft-lbf to J multiply by $1.3558$; Btu to J multiply by $1.0551 \times 10^{3}$; cal to J multiply by $4.1868$
• Force: lbf to N multiply by $4.4482$; kgf to N multiply by $9.8067$
• Length: ft to m multiply by $0.3048$; in to m multiply by $2.5400 \times 10^{-2}$; mi (statute) to m multiply by $1.6093 \times 10^{3}$; nmi (nautical) to m multiply by $1.8520 \times 10^{3}$
• Mass: slug to kg multiply by $1.4594 \times 10^{1}$; lbm to kg multiply by $4.5359 \times 10^{-1}$
• Mass flow: slug/s to kg/s multiply by $1.4594 \times 10^{1}$; lbm/s to kg/s multiply by $4.5359 \times 10^{-1}$
• Power: ft⋅lbf/s to W multiply by $1.3558$; hp to W multiply by $7.4570 \times 10^{2}$
• Pressure: lbf/ft² to Pa multiply by $4.7880 \times 10^{1}$; lbf/in² to Pa multiply by $6.8948 \times 10^{3}$; atm to Pa multiply by $1.0133 \times 10^{5}$; mm Hg to Pa multiply by $1.3332 \times 10^{2}$
• Specific weight: lbf/ft³ to N/m³ multiply by $1.5709 \times 10^{2}$
• Specific heat: ft²/(s²⋅°R) to m²/(s²⋅K) multiply by $1.6723 \times 10^{-1}$
• Surface tension: lbf/ft to N/m multiply by $1.4594 \times 10^{1}$
• Temperature: °F to °C use $t<em>C = (t</em>F - 32) / 1.8$; °R to K multiply by $0.5556$
• Velocity: ft/s to m/s multiply by $0.3048$; mi/h to m/s multiply by $4.4704 \times 10^{-1}$; knot to m/s multiply by $5.1444 \times 10^{-1}$
• Viscosity: lbf⋅s/ft² to N⋅s/m² multiply by $4.7880 \times 10^{1}$; g/(cm⋅s) to N⋅s/m² multiply by $0.1$
• Volume: ft³ to m³ multiply by $2.8317 \times 10^{-2}$; L to m³ multiply by $1.0000 \times 10^{-3}$; gal (U.S.) to m³ multiply by $3.7854 \times 10^{-3}$; fluid ounce (U.S.) to m³ multiply by $2.9574 \times 10^{-5}$
• Volume flow: ft³/s to m³/s multiply by $2.8317 \times 10^{-2}$; gal/min to m³/s multiply by $6.3090 \times 10^{-5}$

Equation Sheet

• Ideal-gas law: $p = \rho RT$, $R_{air} = 287 \frac{J}{kg \cdot K}$
• Hydrostatics, constant density: $p<em>2 = p</em>1 + \rho g (z<em>2 - z</em>1)$, $\gamma = \rho g$
• Buoyant force: $F<em>B = \gamma \cdot V</em>{\text{displaced volume}}$
• CV momentum: $\sum<em>{out} (\rho A V) V - \sum</em>{in} (\rho A V) V = \sum F$
• Steady flow energy: $\left(\frac{p}{\rho g} + \frac{V^2}{2g} + z\right)<em>{in} = \left(\frac{p}{\rho g} + \frac{V^2}{2g} + z\right)</em>{out} + h<em>{\text{friction}} + h</em>{\text{pump}} - h_{\text{turbine}}$
• Incompressible continuity: $\oint \vec{V} \cdot d\vec{A} = 0$
• Incompressible stream function: $u = \frac{\partial \psi}{\partial y}; v = -\frac{\partial \psi}{\partial x}$, $\psi(x, y): \nabla^2 \psi = 0$
• Bernoulli unsteady irrotational flow: $\frac{\partial \phi}{\partial t} + \int \frac{dp}{\rho} + \frac{V^2}{2} + gz = \text{Const}$
• Pipe head loss: $h_f = f \frac{L}{d} \frac{V^2}{2g}$, where f is the Moody chart friction factor
• Laminar flat plate flow: $C<em>D = \frac{\text{Drag}}{\frac{1}{2} \rho V^2 A}$, $C</em>L = \frac{\text{Lift}}{\frac{1}{2} \rho V^2 A}$, $C<em>D = \frac{1.328}{\sqrt{Re</em>L}}$, $c<em>f = \frac{0.664}{\sqrt{Re</em>x}}$, $\frac{\delta}{x} = \frac{5.0}{\sqrt{Re_x}}$
• Isentropic flow: $\frac{p<em>0}{p} = \left(\frac{T</em>0}{T}\right)^{\frac{k}{k-1}}$, $\frac{\rho<em>0}{\rho} = \left(\frac{T</em>0}{T}\right)^{\frac{1}{k-1}}$, $\frac{T_0}{T} = 1 + \frac{k-1}{2} Ma^2$
• Prandtl-Meyer expansion: $K = \frac{k-1}{k+1}$, $\nu = \sqrt{\frac{K+1}{K}} \tan^{-1}\left[ \sqrt{\frac{Ma^2 - 1}{K}} \right] - \tan^{-1}\left[ \sqrt{Ma^2 - 1 } \right]$
• Gradually varied channel flow: $\frac{dy}{dx} = \frac{S<em>0 - S}{1 - Fr^2}$, $Fr = \frac{V}{V</em>{crit}}$
• Turbulent friction factor: $\frac{1}{\sqrt{f}} = -2.0 \log_{10} \left( \frac{\epsilon}{3.7d} + \frac{2.51}{Re \sqrt{f}} \right)$
• Turbulent flat plate flow: $c<em>f = \frac{0.027}{(Re</em>x)^{\frac{1}{7}}}$, $C<em>D = \frac{0.031}{(Re</em>L)^{\frac{1}{7}}}$, $\frac{\delta}{x} = \frac{0.16}{(Re_x)^{\frac{1}{7}}}$
• Orifice, nozzle, venturi flow: $Q = C<em>d A</em>{throat} \sqrt{\frac{2 \Delta p}{\rho (1 - (d/D)^4)}}$
• Uniform flow, Manning’s n, SI units: $V<em>0(m/s) = (\frac{1.0}{n}) R</em>h(m)^{\frac{2}{3}} S_0^{\frac{1}{2}}$
• Euler turbine formula: $Power = \rho Q (u<em>2 V</em>{t2} - u<em>1 V</em>{t1})$, $u = r \omega$
• Surface tension: $\Delta p = \gamma (\frac{1}{R<em>1} + \frac{1}{R</em>2})$
• Hydrostatic panel force: $\vec{F} = - \hat{n} \gamma h<em>{CGA} A$ , $y</em>{CP} = - \frac{I<em>{xx} \sin \theta}{h</em>{CGA} A}$, $x<em>{CP} = - \frac{I</em>{xy} \sin \theta}{h_{CGA} A}$
• CV mass: $\frac{d}{dt} \left( \int<em>{CV} \rho dV \right) + \sum</em>{out} (\rho A V) - \sum_{in} (\rho A V) = 0$
• CV angular momentum: $\sum M<em>0 = \frac{d}{dt} \left( \int</em>{CV} (r \times V) \rho dV \right) + \sum<em>{out} (r \times V) (\rho A V) - \sum</em>{in} (r \times V) (\rho A V)$
• Acceleration: $\frac{dV}{dt} = \frac{\partial V}{\partial t} + u \frac{\partial V}{\partial x} + v \frac{\partial V}{\partial y} + w \frac{\partial V}{\partial z}$
• Navier-Stokes: $\rho \frac{dV}{dt} = - \nabla p + \mu \nabla^2 V + \rho g$
• Velocity potential: $u = \frac{\partial \phi}{\partial x}; v = \frac{\partial \phi}{\partial y}; w = \frac{\partial \phi}{\partial z}$, $\nabla^2 \phi = 0$

Chapter 1: Introduction to Fluid Mechanics

• Fluid mechanics is the study of fluids either in motion (fluid dynamics) or at rest (fluid statics).
• Fluids include both gases and liquids.
• Fluid engineering applications are vast.
• Fluid flow analysis is a compromise between theory and experiment.
• Theory is limited by geometry and viscosity.
• Computational Fluid Dynamics (CFD) approximates solutions.
• Viscosity complicates the basic equations.
• Turbulence is a disorderly, random phenomenon.
• Experimental data provide information about specific flows.
• Theory and experiment should go hand in hand.

1.2 History and Scope of Fluid Mechanics

• Ancient civilizations used knowledge to solve flow problems.
• Archimedes (285-212 B.C.) formulated laws of buoyancy.
• Romans built aqueduct systems.
• Leonardo da Vinci (1452-1519) stated conservation of mass in one-dimensional steady flow.
• Edme Mariotte (1620-1684) built the first wind tunnel.
• Isaac Newton (1642-1727) postulated laws of motion and viscosity.
• Eighteenth-century mathematicians solved frictionless-flow problems.
• Engineers developed hydraulics, relying on experiment.
• William Froude (1810-1879) and son Robert (1846-1924) developed laws of model testing.
• Lord Rayleigh (1842-1919) proposed dimensional analysis.
• Osborne Reynolds (1842-1912) showed the importance of the Reynolds number.
• Navier (1785-1836) and Stokes (1819-1903) added viscous terms to the equations of motion.
• Ludwig Prandtl (1875-1953) developed boundary layer theory.
• Computational Fluid Dynamics (CFD) emerged with the advent of digital computers.

1.3 Problem-Solving Techniques

1. Read the problem and restate desired results.
2. Gather property data.
3. Understand the question being asked.
4. Make a detailed sketch.
5. List assumptions (steady/unsteady, compressible/incompressible, viscous/inviscid).
6. Find an algebraic solution.
7. Report the solution with proper units and significant figures.

1.4 The Concept of a Fluid

• Fluid and solid are the two states of matter.
• Solid resists shear stress by static deflection.
• Fluid cannot resist shear stress without motion.
• Fluid at rest is in a state of zero shear stress.
• Liquids retain volume and form a free surface.
• Gases expand until encountering confining walls.
• Some substances exhibit both solid and fluid behavior (rheology).
• Liquids and gases can coexist in two-phase mixtures.
• The distinction between liquid and gas blurs at temperatures and pressures above the critical point.

1.5 The Fluid as a Continuum

• Fluid density is mass per unit volume.
• The density of a fluid is defined as the limit of the ratio of mass to volume as the volume approaches a limiting value:
  $\rho=\lim_{\Delta V \to \Delta V^*} \frac{\Delta m}{\Delta V}$
• A continuum means that properties vary smoothly; differential calculus can be used.

1.6 Dimensions and Units

• A dimension is a measure of a physical variable.
• A unit is a way of attaching a number to the quantitative dimension.
• The International System of Units (SI) standardizes the metric system.
• There are four primary dimensions: mass {M}, length {L}, time {T}, and temperature {\Theta}.
• Force is related to mass, length, and time by Newton's second law: $F = ma$.
• The principle of dimensional homogeneity states that each additive term in an equation must have the same dimensions.
• Use consistent units; each additive term must have the same units.

1.7 Properties of the Velocity Field

• Fluid flow analysis focuses on the space-time distribution of fluid properties.
• The Eulerian method describes the field of flow.
• The Lagrangian description follows an individual particle.
• The velocity field is a vector function of position and time: $\vec{V}(x, y, z, t) = i u(x, y, z, t) + j v(x, y, z, t) + k w(x, y, z, t)$

1.8 Thermodynamic Properties of a Fluid

• Common thermodynamic properties:
  • Pressure p
  • Density (\rho)
  • Temperature T
  • Internal energy (\hat{u})
  • Enthalpy h = (\hat{u}) + p/(\rho)
  • Entropy s
  • Specific heats (cp) and (cv)
  • Viscosity (\mu)
  • Thermal conductivity k
• For a single-phase substance, two basic properties fix the value of all others.
• A perfect gas follows the perfect-gas law: $p = \rho R T$
• The total stored energy e per unit mass is the sum of internal, potential, and kinetic energies:
  $e = \hat{u} + \frac{V^2}{2} + gz$
  where z is upward, so that $\vec{g} \cdot \vec{r} = -gz \vec{k}$
• State relations for gases: R = (\frac{\mathcal{R}}{M})
• Specific heat relations: (cv = \frac{d \hat{u}}{dT}), (cp = \frac{dh}{dT}), (k = \frac{cp}{cv})
• Liquids are nearly incompressible and have a single, reasonably constant specific heat.

1.9 Viscosity and Other Secondary Properties

• Viscosity quantifies resistance to flow.
• Newtonian fluids have a linear relation between shear stress and strain rate: $\tau = \mu \frac{du}{dy}$
• Fluids at a point of contact with a solid take on the velocity of that surface (no-slip condition).
• The Reynolds number characterizes flow regimes: $Re = \frac{\rho V L}{\mu}$
• Kinematic viscosity is the ratio of viscosity to density: $\nu = \frac{\mu}{\rho}$
• Gas viscosity increases with temperature; liquid viscosity decreases.
• Nonnewtonian fluids do not follow the linear law.
• Surface tension is the surface energy per unit area.
• The pressure change across a curved interface is: $\Delta p = \gamma\left( \frac{1}{R<em>1} + \frac{1}{R</em>2} \right)$
• Vapor pressure is the pressure at which a liquid boils.
• Cavitation is flow-induced boiling.
• Speed of sound is the rate of propagation of small-disturbance pressure pulses.
• The no-slip and no-temperature-jump conditions state that Vfluid = Vwall and Tfluid = Twall, respectively.

1.10 Basic Flow Analysis Techniques

• Three basic ways to attack a fluid flow problem:
  1. Control-volume, or integral analysis
  2. Infinitesimal system, or differential analysis
  3. Experimental study, or dimensional analysis
• The flow must satisfy the three basic laws of mechanics, a thermodynamic state relation, and associated boundary conditions.

1.11 Flow Patterns: Streamlines, Streaklines, and Pathlines

• A streamline is a line everywhere tangent to the velocity vector at a given instant.
• A pathline is the actual path traversed by a given fluid particle.
• A streakline is the locus of particles that have earlier passed through a prescribed point.
• Streamlines, pathlines, and streaklines are identical in steady flow.
• To be parallel to V, their respective components must be in proportion (streamline): $\frac{dx}{u} = \frac{dy}{v} = \frac{dz}{w}$
• PathLine: $x = \int u dt ; y = \int v dt ; z = \int w dt$

1.13 Uncertainty in Experimental Data

• Estimating uncertainty: $\frac{\Delta P}{P} \cong \sqrt{\left( n<em>1 \frac{\Delta x</em>1}{x<em>1} \right)^2 + \left( n</em>2 \frac{\Delta x<em>2}{x</em>2} \right)^2 + …}$