\babel@toc {english}{} \boolfalse {citerequest}\boolfalse {citetracker}\boolfalse {pagetracker}\boolfalse {backtracker}\relax \babel@toc {english}{} \contentsline {chapter}{Contents}{2}{section*.1} \contentsline {chapter}{Syllabus}{5}{section*.2} \contentsline {chapter}{\numberline {1}Basic flow quantities}{9}{chapter.1} \contentsline {section}{\numberline {1.1}Concept of a fluid}{9}{section.1.1} \contentsline {section}{\numberline {1.2}Fluid mechanics}{9}{section.1.2} \contentsline {subsection}{\numberline {1.2.1}Solution of a flow}{9}{subsection.1.2.1} \contentsline {subsection}{\numberline {1.2.2}Modeling of fluids}{10}{subsection.1.2.2} \contentsline {subsection}{\numberline {1.2.3}Theory, numerics, and experiment}{11}{subsection.1.2.3} \contentsline {section}{\numberline {1.3}Important concepts in mechanics}{11}{section.1.3} \contentsline {subsection}{\numberline {1.3.1}Position, velocity, acceleration}{11}{subsection.1.3.1} \contentsline {subsection}{\numberline {1.3.2}Forces and moments}{12}{subsection.1.3.2} \contentsline {subsection}{\numberline {1.3.3}Energy}{13}{subsection.1.3.3} \contentsline {section}{\numberline {1.4}Properties of fluids}{14}{section.1.4} \contentsline {subsection}{\numberline {1.4.1}Density}{14}{subsection.1.4.1} \contentsline {subsection}{\numberline {1.4.2}Phase}{14}{subsection.1.4.2} \contentsline {subsection}{\numberline {1.4.3}Temperature}{14}{subsection.1.4.3} \contentsline {subsection}{\numberline {1.4.4}Perfect gas model}{15}{subsection.1.4.4} \contentsline {subsection}{\numberline {1.4.5}Speed of sound}{15}{subsection.1.4.5} \contentsline {subsection}{\numberline {1.4.6}Viscosity}{16}{subsection.1.4.6} \contentsline {section}{\numberline {1.5}Forces on fluids}{17}{section.1.5} \contentsline {subsection}{\numberline {1.5.1}Gravity}{17}{subsection.1.5.1} \contentsline {subsection}{\numberline {1.5.2}Pressure}{17}{subsection.1.5.2} \contentsline {subsection}{\numberline {1.5.3}Shear}{17}{subsection.1.5.3} \contentsline {section}{\numberline {1.6}Basic flow quantities}{18}{section.1.6} \contentsline {section}{\numberline {1.7}Four balance equations}{19}{section.1.7} \contentsline {section}{\numberline {1.8}Classification of fluid flows}{20}{section.1.8} \contentsline {section}{\numberline {1.9}Limits of fluid mechanics}{22}{section.1.9} \contentsline {section}{\numberline {1.10}Problems}{23}{section.1.10} \contentsline {subsubsection}{\numberline {1.1}Compressibility effects}{23}{subsubsection.1.10.0.1} \contentsline {subsubsection}{\numberline {1.2}Pressure-induced force}{23}{subsubsection.1.10.0.2} \contentsline {subsubsection}{\numberline {1.3}Shear-induced force}{24}{subsubsection.1.10.0.3} \contentsline {subsubsection}{\numberline {1.4}Speed of sound}{24}{subsubsection.1.10.0.4} \contentsline {subsubsection}{\numberline {1.5}Power lost to drag}{24}{subsubsection.1.10.0.5} \contentsline {subsubsection}{\numberline {1.6}Go-faster exhaust pipe}{25}{subsubsection.1.10.0.6} \contentsline {subsubsection}{\numberline {1.7}Acceleration of a particle}{26}{subsubsection.1.10.0.7} \contentsline {subsubsection}{\numberline {1.8}Flow classifications}{26}{subsubsection.1.10.0.8} \contentsline {chapter}{\numberline {2}Analysis of existing flows with\nobreakspace {}one\nobreakspace {}dimension}{29}{chapter.2} \contentsline {section}{\numberline {2.1}Motivation}{29}{section.2.1} \contentsline {section}{\numberline {2.2}One-dimensional flow problems}{29}{section.2.2} \contentsline {section}{\numberline {2.3}Balance of mass}{31}{section.2.3} \contentsline {section}{\numberline {2.4}Balance of momentum}{32}{section.2.4} \contentsline {section}{\numberline {2.5}Balance of energy}{34}{section.2.5} \contentsline {section}{\numberline {2.6}The Bernoulli equation}{36}{section.2.6} \contentsline {subsection}{\numberline {2.6.1}Theory}{36}{subsection.2.6.1} \contentsline {subsection}{\numberline {2.6.2}Reality}{37}{subsection.2.6.2} \contentsline {section}{\numberline {2.7}Problems}{39}{section.2.7} \contentsline {subsubsection}{\numberline {2.1}Pipe expansion without losses}{39}{subsubsection.2.7.0.1} \contentsline {subsubsection}{\numberline {2.2}Pipe flow with losses}{40}{subsubsection.2.7.0.2} \contentsline {subsubsection}{\numberline {2.3}Combustor from a jet engine}{40}{subsubsection.2.7.0.3} \contentsline {subsubsection}{\numberline {2.4}Water jet on a wall}{41}{subsubsection.2.7.0.4} \contentsline {subsubsection}{\numberline {2.5}High-speed gas flow}{42}{subsubsection.2.7.0.5} \contentsline {chapter}{\numberline {3}Analysis of existing flows with\nobreakspace {}three\nobreakspace {}dimensions}{47}{chapter.3} \contentsline {section}{\numberline {3.1}Motivation}{47}{section.3.1} \contentsline {section}{\numberline {3.2}The Reynolds transport theorem}{47}{section.3.2} \contentsline {subsection}{\numberline {3.2.1}Control volume}{47}{subsection.3.2.1} \contentsline {subsection}{\numberline {3.2.2}Rate of change of an additive property}{48}{subsection.3.2.2} \contentsline {section}{\numberline {3.3}Balance of mass}{51}{section.3.3} \contentsline {section}{\numberline {3.4}Balance of momentum}{52}{section.3.4} \contentsline {section}{\numberline {3.5}Balance of angular momentum}{54}{section.3.5} \contentsline {section}{\numberline {3.6}Balance of energy}{55}{section.3.6} \contentsline {section}{\numberline {3.7}Limits of integral analysis}{55}{section.3.7} \contentsline {section}{\numberline {3.8}Problems}{57}{section.3.8} \contentsline {subsubsection}{\numberline {3.1}Pipe bend}{58}{subsubsection.3.8.0.1} \contentsline {subsubsection}{\numberline {3.2}Exhaust gas deflector}{58}{subsubsection.3.8.0.2} \contentsline {subsubsection}{\numberline {3.3}Pelton water turbine}{59}{subsubsection.3.8.0.3} \contentsline {subsubsection}{\numberline {3.4}Snow plow}{60}{subsubsection.3.8.0.4} \contentsline {subsubsection}{\numberline {3.5}Drag on a cylindrical profile}{60}{subsubsection.3.8.0.5} \contentsline {subsubsection}{\numberline {3.6}Drag on a flat plate}{61}{subsubsection.3.8.0.6} \contentsline {subsubsection}{\numberline {3.7}Drag measurements in a wind tunnel}{62}{subsubsection.3.8.0.7} \contentsline {subsubsection}{\numberline {3.8}Moment on gas deflector}{63}{subsubsection.3.8.0.8} \contentsline {subsubsection}{\numberline {3.9}Helicopter tail moment}{64}{subsubsection.3.8.0.9} \contentsline {subsubsection}{\numberline {3.10}Pressure losses in a pipe flow}{65}{subsubsection.3.8.0.10} \contentsline {subsubsection}{\numberline {3.11}Thrust reverser}{65}{subsubsection.3.8.0.11} \contentsline {chapter}{\numberline {4}Effects of pressure}{69}{chapter.4} \contentsline {section}{\numberline {4.1}Motivation}{69}{section.4.1} \contentsline {section}{\numberline {4.2}Pressure forces on walls}{69}{section.4.2} \contentsline {subsection}{\numberline {4.2.1}Magnitude of the pressure force}{69}{subsection.4.2.1} \contentsline {subsection}{\numberline {4.2.2}Position of the pressure force}{70}{subsection.4.2.2} \contentsline {section}{\numberline {4.3}Pressure fields in fluids}{71}{section.4.3} \contentsline {subsection}{\numberline {4.3.1}The direction of pressure}{71}{subsection.4.3.1} \contentsline {subsection}{\numberline {4.3.2}Pressure on an infinitesimal volume}{72}{subsection.4.3.2} \contentsline {section}{\numberline {4.4}Special case: pressure in static fluids}{73}{section.4.4} \contentsline {subsection}{\numberline {4.4.1}Forces in static fluids}{73}{subsection.4.4.1} \contentsline {subsection}{\numberline {4.4.2}Pressure and depth}{74}{subsection.4.4.2} \contentsline {subsection}{\numberline {4.4.3}Buoyancy}{76}{subsection.4.4.3} \contentsline {section}{\numberline {4.5}Problems}{79}{section.4.5} \contentsline {subsubsection}{\numberline {4.1}Pressure in a static fluid}{79}{subsubsection.4.5.0.1} \contentsline {subsubsection}{\numberline {4.2}Pressure measurement}{79}{subsubsection.4.5.0.2} \contentsline {subsubsection}{\numberline {4.3}Straight water tank door}{80}{subsubsection.4.5.0.3} \contentsline {subsubsection}{\numberline {4.4}Water canal access door}{80}{subsubsection.4.5.0.4} \contentsline {subsubsection}{\numberline {4.5}Pressure force on a cylinder}{81}{subsubsection.4.5.0.5} \contentsline {subsubsection}{\numberline {4.6}Buoyancy of a barge}{81}{subsubsection.4.5.0.6} \contentsline {subsubsection}{\numberline {4.7}Atmospheric pressure distribution}{82}{subsubsection.4.5.0.7} \contentsline {chapter}{\numberline {5}Effects of shear}{85}{chapter.5} \contentsline {section}{\numberline {5.1}Motivation}{85}{section.5.1} \contentsline {section}{\numberline {5.2}Shear forces on walls}{85}{section.5.2} \contentsline {subsection}{\numberline {5.2.1}Magnitude of the shear force}{85}{subsection.5.2.1} \contentsline {subsection}{\numberline {5.2.2}Direction and position of the shear force}{86}{subsection.5.2.2} \contentsline {section}{\numberline {5.3}Shear fields in fluids}{86}{section.5.3} \contentsline {subsection}{\numberline {5.3.1}The direction of shear}{86}{subsection.5.3.1} \contentsline {subsection}{\numberline {5.3.2}Shear on an infinitesimal volume}{87}{subsection.5.3.2} \contentsline {subsection}{\numberline {5.3.3}The no-slip condition}{89}{subsection.5.3.3} \contentsline {subsection}{\numberline {5.3.4}Viscosity}{90}{subsection.5.3.4} \contentsline {subsection}{\numberline {5.3.5}Newtonian Fluid}{91}{subsection.5.3.5} \contentsline {section}{\numberline {5.4}Special case: shear in simple laminar flows}{92}{section.5.4} \contentsline {section}{\numberline {5.5}Problems}{93}{section.5.5} \contentsline {subsubsection}{\numberline {5.1}Flow in between two plates}{93}{subsubsection.5.5.0.1} \contentsline {subsubsection}{\numberline {5.2}Friction on a plate}{94}{subsubsection.5.5.0.2} \contentsline {subsubsection}{\numberline {5.3}Viscometer}{94}{subsubsection.5.5.0.3} \contentsline {subsubsection}{\numberline {5.4}Boundary layer}{94}{subsubsection.5.5.0.4} \contentsline {subsubsection}{\numberline {5.5}Clutch}{96}{subsubsection.5.5.0.5} \contentsline {chapter}{\numberline {6}Prediction of fluid flows}{99}{chapter.6} \contentsline {section}{\numberline {6.1}Motivation}{99}{section.6.1} \contentsline {section}{\numberline {6.2}Organizing calculations}{99}{section.6.2} \contentsline {subsection}{\numberline {6.2.1}Problem description}{99}{subsection.6.2.1} \contentsline {subsection}{\numberline {6.2.2}The total time derivative}{100}{subsection.6.2.2} \contentsline {section}{\numberline {6.3}Equations for all flows}{103}{section.6.3} \contentsline {subsection}{\numberline {6.3.1}Balance of mass}{103}{subsection.6.3.1} \contentsline {subsection}{\numberline {6.3.2}Balance of linear momentum}{105}{subsection.6.3.2} \contentsline {subsection}{\numberline {6.3.3}Balance of energy}{107}{subsection.6.3.3} \contentsline {subsection}{\numberline {6.3.4}Other terms and equations}{108}{subsection.6.3.4} \contentsline {subsection}{\numberline {6.3.5}Interlude: where does this leave us?}{109}{subsection.6.3.5} \contentsline {section}{\numberline {6.4}Equations for incompressible flow}{109}{section.6.4} \contentsline {subsection}{\numberline {6.4.1}Balance of mass}{109}{subsection.6.4.1} \contentsline {subsection}{\numberline {6.4.2}Balance of linear momentum}{110}{subsection.6.4.2} \contentsline {subsection}{\numberline {6.4.3}The Bernoulli equation (again)}{112}{subsection.6.4.3} \contentsline {section}{\numberline {6.5}CFD: the Navier-Stokes equations in practice}{112}{section.6.5} \contentsline {section}{\numberline {6.6}Problems}{115}{section.6.6} \contentsline {subsubsection}{\numberline {6.1}Revision questions}{115}{subsubsection.6.6.0.1} \contentsline {subsubsection}{\numberline {6.2}Acceleration field}{115}{subsubsection.6.6.0.2} \contentsline {subsubsection}{\numberline {6.3}Volumetric dilatation rate}{116}{subsubsection.6.6.0.3} \contentsline {subsubsection}{\numberline {6.4}Incompressibility}{116}{subsubsection.6.6.0.4} \contentsline {subsubsection}{\numberline {6.5}Missing components}{116}{subsubsection.6.6.0.5} \contentsline {subsubsection}{\numberline {6.6}Another acceleration field}{116}{subsubsection.6.6.0.6} \contentsline {subsubsection}{\numberline {6.7}Vortex}{116}{subsubsection.6.6.0.7} \contentsline {subsubsection}{\numberline {6.8}Pressure fields}{117}{subsubsection.6.6.0.8} \contentsline {chapter}{\numberline {7}Pipe flows}{119}{chapter.7} \contentsline {section}{\numberline {7.1}Motivation}{119}{section.7.1} \contentsline {section}{\numberline {7.2}Friction-less flow in pipes}{119}{section.7.2} \contentsline {section}{\numberline {7.3}Quantifying losses in pipes}{120}{section.7.3} \contentsline {section}{\numberline {7.4}Laminar flow in pipes}{121}{section.7.4} \contentsline {subsection}{\numberline {7.4.1}Laminar flow between plates}{121}{subsection.7.4.1} \contentsline {subsection}{\numberline {7.4.2}Laminar flow in pipes}{123}{subsection.7.4.2} \contentsline {section}{\numberline {7.5}Turbulent flow in pipes}{126}{section.7.5} \contentsline {subsection}{\numberline {7.5.1}When is a pipe flow turbulent?}{126}{subsection.7.5.1} \contentsline {subsection}{\numberline {7.5.2}Characteristics of turbulent flow}{127}{subsection.7.5.2} \contentsline {subsection}{\numberline {7.5.3}Velocity profile in turbulent pipe flow}{128}{subsection.7.5.3} \contentsline {subsection}{\numberline {7.5.4}Pressure losses in turbulent pipe flow}{129}{subsection.7.5.4} \contentsline {section}{\numberline {7.6}Summary}{130}{section.7.6} \contentsline {section}{\numberline {7.7}Problems}{131}{section.7.7} \contentsline {subsubsection}{\numberline {7.1}Revision questions}{131}{subsubsection.7.7.0.1} \contentsline {subsubsection}{\numberline {7.2}Air flow in a small pipe}{133}{subsubsection.7.7.0.2} \contentsline {subsubsection}{\numberline {7.3}Water piping}{133}{subsubsection.7.7.0.3} \contentsline {subsubsection}{\numberline {7.4}Design of a water piping system}{134}{subsubsection.7.7.0.4} \contentsline {subsubsection}{\numberline {7.5}Major oil pipeline}{135}{subsubsection.7.7.0.5} \contentsline {subsubsection}{\numberline {7.6}Pump with pipe expansion}{136}{subsubsection.7.7.0.6} \contentsline {subsubsection}{\numberline {7.7}Kugel fountain}{136}{subsubsection.7.7.0.7} \contentsline {subsubsection}{\numberline {7.8}A more complex ducted flow}{138}{subsubsection.7.7.0.8} \contentsline {subsubsection}{\numberline {7.9}Politically incorrect fluid mechanics}{138}{subsubsection.7.7.0.9} \contentsline {chapter}{\numberline {8}Dealing with\nobreakspace {}turbulence}{141}{chapter.8} \contentsline {chapter}{\numberline {9}Engineering models}{143}{chapter.9} \contentsline {section}{\numberline {9.1}Motivation}{143}{section.9.1} \contentsline {section}{\numberline {9.2}Comparing influences: the weighted momentum balance}{143}{section.9.2} \contentsline {subsection}{\numberline {9.2.1}Principle}{143}{subsection.9.2.1} \contentsline {subsection}{\numberline {9.2.2}The non-dimensional Navier-Stokes equation}{144}{subsection.9.2.2} \contentsline {subsection}{\numberline {9.2.3}The flow parameters of Navier-Stokes}{146}{subsection.9.2.3} \contentsline {subsection}{\numberline {9.2.4}Flow parameters obtained as force ratios}{147}{subsection.9.2.4} \contentsline {subsection}{\numberline {9.2.5}The Reynolds number in practice}{147}{subsection.9.2.5} \contentsline {section}{\numberline {9.3}Making models}{149}{section.9.3} \contentsline {section}{\numberline {9.4}Comparing results: coefficients}{150}{section.9.4} \contentsline {subsection}{\numberline {9.4.1}Principle}{150}{subsection.9.4.1} \contentsline {subsection}{\numberline {9.4.2}Force coefficients}{150}{subsection.9.4.2} \contentsline {subsection}{\numberline {9.4.3}Power coefficient, and other coefficients}{151}{subsection.9.4.3} \contentsline {subsection}{\numberline {9.4.4}The systematic non-dimensionalization of problems}{152}{subsection.9.4.4} \contentsline {section}{\numberline {9.5}Problems}{153}{section.9.5} \contentsline {subsubsection}{\numberline {9.1}Scaling with the Reynolds number}{153}{subsubsection.9.5.0.1} \contentsline {subsubsection}{\numberline {9.2}Fluid mechanics of a giant airliner}{154}{subsubsection.9.5.0.2} \contentsline {subsubsection}{\numberline {9.3}Scale effects on a dragonfly}{155}{subsubsection.9.5.0.3} \contentsline {subsubsection}{\numberline {9.4}Formula One testing}{155}{subsubsection.9.5.0.4} \contentsline {chapter}{\numberline {10}Flow near walls}{159}{chapter.10} \contentsline {section}{\numberline {10.1}Motivation}{159}{section.10.1} \contentsline {section}{\numberline {10.2}The concept of boundary layer}{159}{section.10.2} \contentsline {subsection}{\numberline {10.2.1}Rationale}{159}{subsection.10.2.1} \contentsline {subsection}{\numberline {10.2.2}Why do we study the boundary layer?}{161}{subsection.10.2.2} \contentsline {subsection}{\numberline {10.2.3}Characterization of the boundary layer}{161}{subsection.10.2.3} \contentsline {section}{\numberline {10.3}Laminar boundary layers}{163}{section.10.3} \contentsline {subsection}{\numberline {10.3.1}Governing equations}{163}{subsection.10.3.1} \contentsline {subsection}{\numberline {10.3.2}Blasius\IeC {\textquoteright } solution}{164}{subsection.10.3.2} \contentsline {section}{\numberline {10.4}Boundary layer transition}{165}{section.10.4} \contentsline {section}{\numberline {10.5}Turbulent boundary layers}{166}{section.10.5} \contentsline {section}{\numberline {10.6}Flow separation}{167}{section.10.6} \contentsline {section}{\numberline {10.7}Problems}{171}{section.10.7} \contentsline {subsubsection}{\numberline {10.1}Water and air flow}{171}{subsubsection.10.7.0.1} \contentsline {subsubsection}{\numberline {10.2}Boundary layer sketches}{172}{subsubsection.10.7.0.2} \contentsline {subsubsection}{\numberline {10.3}Shear force due to boundary layer}{173}{subsubsection.10.7.0.3} \contentsline {subsubsection}{\numberline {10.4}Wright Flyer I}{173}{subsubsection.10.7.0.4} \contentsline {subsubsection}{\numberline {10.5}Power lost to shear on an airliner fuselage}{174}{subsubsection.10.7.0.5} \contentsline {subsubsection}{\numberline {10.6}Laminar wing profile}{174}{subsubsection.10.7.0.6} \contentsline {subsubsection}{\numberline {10.7}Separation mechanism}{174}{subsubsection.10.7.0.7} \contentsline {chapter}{\numberline {11}Large- and small-scale\nobreakspace {}flows}{179}{chapter.11} \contentsline {section}{\numberline {11.1}Motivation}{179}{section.11.1} \contentsline {section}{\numberline {11.2}Flow at large scales}{179}{section.11.2} \contentsline {subsection}{\numberline {11.2.1}Problem statement}{179}{subsection.11.2.1} \contentsline {subsection}{\numberline {11.2.2}Investigation of inviscid flows}{180}{subsection.11.2.2} \contentsline {section}{\numberline {11.3}Plotting velocity with functions}{181}{section.11.3} \contentsline {subsection}{\numberline {11.3.1}Kinematics that fit conservation laws}{181}{subsection.11.3.1} \contentsline {subsection}{\numberline {11.3.2}Strengths and weaknesses of potential flow}{182}{subsection.11.3.2} \contentsline {subsection}{\numberline {11.3.3}Superposition: the lifting cylinder}{183}{subsection.11.3.3} \contentsline {subsection}{\numberline {11.3.4}Circulating cylinder}{185}{subsection.11.3.4} \contentsline {subsection}{\numberline {11.3.5}Modeling lift with circulation}{187}{subsection.11.3.5} \contentsline {section}{\numberline {11.4}Flow at very small scales}{190}{section.11.4} \contentsline {section}{\numberline {11.5}Problems}{193}{section.11.5} \contentsline {subsubsection}{\numberline {11.1}Volcanic ash from the Eyjafjallaj\IeC {\"o}kull}{193}{subsubsection.11.5.0.1} \contentsline {subsubsection}{\numberline {11.2}Water drop}{193}{subsubsection.11.5.0.2} \contentsline {subsubsection}{\numberline {11.3}Idealized flow over a hangar roof}{193}{subsubsection.11.5.0.3} \contentsline {subsubsection}{\numberline {11.4}Cabling of the Wright Flyer}{195}{subsubsection.11.5.0.4} \contentsline {subsubsection}{\numberline {11.5}Ping pong ball}{195}{subsubsection.11.5.0.5} \contentsline {subsubsection}{\numberline {11.6}Flow field of a tornado}{198}{subsubsection.11.5.0.6} \contentsline {subsubsection}{\numberline {11.7}Lift on a symmetrical object}{200}{subsubsection.11.5.0.7} \contentsline {subsubsection}{\numberline {11.8}Air flow over a wing profile}{200}{subsubsection.11.5.0.8} \contentsline {chapter}{Appendix}{203}{section*.124} \contentsline {section}{\numberline {A1}Notation}{204}{section.Alph0.1} \contentsline {section}{\numberline {A2}Field operators}{205}{section.Alph0.2} \contentsline {subsection}{\numberline {A2.1}Gradient}{205}{subsection.Alph0.2.1} \contentsline {subsection}{\numberline {A2.2}Divergent}{205}{subsection.Alph0.2.2} \contentsline {subsection}{\numberline {A2.3}Laplacian}{206}{subsection.Alph0.2.3} \contentsline {subsection}{\numberline {A2.4}Curl}{206}{subsection.Alph0.2.4} \contentsline {section}{\numberline {A3}Derivations of the Bernoulli equation}{207}{section.Alph0.3} \contentsline {subsection}{\numberline {A3.1}The Bernoulli equation from the energy equation}{207}{subsection.Alph0.3.1} \contentsline {subsection}{\numberline {A3.2}The Bernoulli equation from the integral momentum equation}{207}{subsection.Alph0.3.2} \contentsline {subsection}{\numberline {A3.3}The Bernoulli equation from the Navier-Stokes equation}{207}{subsection.Alph0.3.3} \contentsline {section}{\numberline {A4}Flow parameters as force ratios}{210}{section.Alph0.4} \contentsline {subsection}{\numberline {A4.1}Acceleration vs. viscous forces: the Reynolds number}{210}{subsection.Alph0.4.1} \contentsline {subsection}{\numberline {A4.2}Acceleration vs. gravity force: the Froude number}{211}{subsection.Alph0.4.2} \contentsline {subsection}{\numberline {A4.3}Acceleration vs. elastic forces: the Mach number}{211}{subsection.Alph0.4.3} \contentsline {subsection}{\numberline {A4.4}Other force ratios}{212}{subsection.Alph0.4.4} \contentsline {section}{\numberline {A5}List of references}{213}{section.Alph0.5}