Lecture-6 - APPH 4200 Physics of Fluids Fluid Equations of...

Info iconThis preview shows pages 1–7. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: APPH 4200 Physics of Fluids Fluid Equations of Motion (Ch. 4) September 22, 2011 1.! ! Conservation of Mass 2.! Navier-Stokes Equation (Force-Momentum) 3.! Mechanical and Thermal Energy 4.! Entropy 5.! Some examples 1 Equations of Fluid Dynamics (Conservation Laws) 2 Newton’s Law 3 Continuity Mass 4 Momentum 5 ~ ¿ ~.. a .~ r- Momentum x-direction 3¡ I) )i .. 1- . V It. ~ - l Q "X 1 ') ¡: l l '. ~ - ---\L "' r/I ~ \ ('b l l \! ~ ) ~ v i: l.u J lL -= .. ~ + (\ G- ~ l~ i r' rN " + '1. rG "\"' ~ "- )- -: .. ~ + ~ :: r= l. C~0 ,~\(~ . ~ .J ~ ~ \J -+ ~ rl ~ X ~ '-Ji ~~ + )I. "J ~ /" L hJ q~' ~ y ~\l ~ '- ~ ~ "' ) ~ 'Vi ~ V) l~ " ~\ .l ~v 33 ~ ~ ~ .. ~ ~~ - o I::r; \ \J '\ .. ~ - I. '4 lt ~~ Q J ~2J .J ~.) V "- ~ 'Á ¡~ .~ .. (~ ii ~ I~ \'t 0- )/ ). -: "1 ri ~ Il r\ ~ I~ )'" + l" I~ l~ '4l:r \ ,¡ (: C: -- ~ IJ \ -j II ~ ~cb L ,. ","X I. ~ ~~ ~~ ~ Vl 6 "' Q -l ~ ~ o Vl P ~ iO G l.ß y Osborn Reynolds (Reynolds Transport Theory) Q. ~ J .. '0 ~ j J \i ~ ~ ~!\ Ii l' ~ .) ') ~ ~ ~ .. I\\.j ~ ')- t ~ ~.\ ~ l. ~ 1 r\\, IV ~ 1842-1912 /" l\~ i I~ 1- l~ Q. ~ J ~ f ~ ~)( l~ ~~3 ~..) i t~ ~lL r " i~ \S t; o lJ (. "i ~\ Q + Qn ~ . l~ '- ~ '- ~ Çi c= lt ~ ~ + ~ I) /" W o~ IJ ~ ~~ ~ \~ ~1 ~æ \LJ - ~ \J l~ \ i 1; ~ I~ -r r \ l~ l~ ~~ . ~ /" ~ L ': Q + 1- ~;: 4- l~ /" \ ~ I": \1 ~ '\ ~~ t:~ ~ b\'C" l' l &. 1 ~ LJ r- ~ ~ ~~ ~~ tI l~ ~ ~ ~ ',- ~ ~ 'i ~ .J ÇJ\.8 li l~ \ .. Ci a ~ ~ ~ '- ,,1 ~.~ 7 Models for Stress 8 Navier & Stokes Claude-Lewis Henri Navier (1785-1836) Elasticity George Stokes (1819-1903) Viscous flow, Stokes’ Law 9 Pont des Invalides 10 Stokes Flow 11 Tay Bridge Disaster (1879) 12 Stokesian Fluid 13 Navier-Stokes Equation 14 Navier-Stokes & Euler 15 NS Properties ∂u 2 + u · ∇u = g − (1/ρ)∇p + ν ∇ u ∂t • Equation for rate of change of u(x,t) (not position) in lab frame • • Nonlinear. Complicated term: u · ∇u • NS is a starting point: Once u(x,t) is known, then other fluid properties can be determined. Viscous length scale: ν/U 16 Clay Prize: $1M 17 Simple Example: What is the steady flow between two parallel plates? Cartesian Coordinates No slip Boundary Conditions b ∂u + u · ∇u ∂t ∂ ux ux ∂x Solution: U(y) = −(1/ρ)∇p + ν ∇2 u 1 ∂p =− + ν ∇2 u x ρ ∂x y (b − y ) ux (y ) = 4U0 b2 ∂ 2 ux ν U0 2 ν∇ u = x ν ˆ = −x 8 2 ˆ ∂ y2 b 18 Not Simple Example: Steady, 1D Radial Flow, Cylindrical Coordinates ∂u + u · ∇u ∂t ∂ ur ur ∂r = = −(1/ρ)∇p + ν ∇2 u ￿2 ￿ 2 1 ∂p ∂ ur ∂ ur 1 ∂ ur ur − +ν + + −2 ρ ∂r ∂z2 ∂ r2 r ∂r r 1 ∂ (rUr ) = 0 → Ur (r, z ) = c1 (z )/r r ∂r 19 Energy Conservation 20 Mechanical Energy 21 Total Work Done by Stress 22 Mechanical Energy Density for a Stokes Fluid 23 Conservation of Total Energy Density 24 Equation for Internal Energy (Temperature) 25 Internal Energy Equation 26 Entropy (Disorder, Direction of Time) 27 Summary of Fluid Dynamical Equations 28 The Importance of Viscosity 29 Creation of Vorticity (Note: Flow at thin layer at surface of cylinder vanishes.) 30 Ch. 4, Exercise #1 31 Example 4.1 32 Integral Conservation Laws (Fixed Volume) 33 Example 4.1 (Solution) 34 Summary • Basic fluid dynamics involves “6” field variables: ρ, Ui, P, T • Conservation of Mass, Energy, Newton’s Law, and an equation of state provide a “closed” set of dynamical equations for a fluid • Viscous forces are represented by a stress tensor • Stokes’ model of viscosity gives us one of the most useful and well-known equations of applied science: Navier-Stokes equation. 35 ...
View Full Document

Page1 / 35

Lecture-6 - APPH 4200 Physics of Fluids Fluid Equations of...

This preview shows document pages 1 - 7. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online