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12 Pages

### Lecture-14_(10-21-10)

Course: APPH 4200, Spring 2010
School: Columbia
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Word Count: 1730

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4200 Physics APPH of Fluids Stokes Flow (Ch. 9) October 21, 2010 1.! ! Viscous Decay of a Line Vortex 2.! Vortex sheet 3.! Stokes Solution for Viscous Flow around a Sphere 1 ~ ~ ~.. .. 'f - -i ~ -: ~ ~ y. ~~ &quot; Q ~.. ( .. -. l \A 0 :l .. t. .~ ci 1:1 J c: ~ ~~ &quot;) .i (! .. ~ .. ~ 'Q ~ y. ~. ~ I 1\ Viscous Decay of Line Vortex r&quot; ~ x .. 1 '\ '...

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4200 Physics APPH of Fluids Stokes Flow (Ch. 9) October 21, 2010 1.! ! Viscous Decay of a Line Vortex 2.! Vortex sheet 3.! Stokes Solution for Viscous Flow around a Sphere 1 ~ ~ ~.. .. 'f - -i ~ -: ~ ~ y. ~~ " Q ~.. ( .. -. l \A 0 :l .. t. .~ ci 1:1 J c: ~ ~~ ") .i (! .. ~ .. ~ 'Q ~ y. ~. ~ I 1\ Viscous Decay of Line Vortex r" ~ x .. 1 '\ ' \) 0 ~t\ ~ F ," ~ "0 ~ .. i \! .. ~ l.. ~ J. u t ~ ~ ~ ~ .. VI '\ " tI Q .. i ~r . '- '- v .\ . .. " ~ ,f ( L t ct "" l "" ~ ~ u 'U ': .. ~ l. ~ '- VI ~ ~ 'i Q,\ t Q \ ti t' ~\" l! .. '- i- ~ i ~\~ C1 C\ ., ~ (\ l \. ct ~ )l ef ": ( l; C' ~ 0 rl ": ci .,~ u ~l~ l~ "- .. -l ( N (\ .:40\ ( J ~ IcJc. ~~ ~(1. ~ . ~ o'\~ "i ~ "- tl~ ii ri fl l' \~ -,~ /" i (. ~ ~ f ~ l'j '- ~ ct ~ iI "" .. oJ ~ ~\ c. .. ~, + ow ~4 i~". ~ ~ . rl t . .. l:i tI ~ " ~ ~ 1 . r" .. ~ -l ~ ~ \ u ~ '3 ~ . ~ '" .. .. t " .. \~ ~ ~ , , f! ~ .. ~ ~ 1 .. 'u a ~ ~ \J t - f ~ 0. (" Q - "" t \~ .1' '" i-l tu ~ /' )~ ~ Vl 3 :i ... .i ~ "" F -. i- ~ II J \. : .. ,- c:i ~ a ~~ \b :i ~ 'I ) .. '0 \1 0 ~ () ~ U: 1 ct tJ 2 ~ l~ ~ ~ V\ L - -i ~ lI \) .. f, ) Similarity Solution f ~ \A ~ ~~ ~~ i "~ .. -0 V\ 'b l' ~ i ') ~ r: I( "".. J \J '1 '" ~ ' , va \I "l - i. , \I Gi ') L. r w -; ~ u: .. t: 'c i .. .. ui F '" i .. ~ 4 .. ~l '" \t 'C ~ () \A l ~t -~ ~ i) ~ ~ \ L t. N "' it .) .. tl ~ ~ ~ \J l \\ l 1 -l u. . l .. .~ ( "... ( ~ lt ~ - J "" \e \J ..' t' .. l 'l .J Q 0 ~ ~ ,. \ ~ . Ul i: - .l, 0 ~ Q. a l. L d t E 1: ~ ~ 0 i; ) .. ~ \I ~ .. l - .. .( .. (j ~ ~ l, ~ ~l G I" , 0 .~ ~ F J .. ~ l/ ! l.l I - Vl ~\~ t U 1:\ t -:\ ~ 1, l=\ t' ~ ,. :s) . U- ~ ("l" 0 ~\1J ~'(' .. u. \I c.i ~ N ~ '",. .. i ,. :t ~ ~I~~ )o Co J f:1 ( ~ (" '-" \i ?- I, ~i ~ .. \i .. ~ '\ ). .. I"" 'l Q ~ ~ ~ - '- ~ Ii 1 .. h. J f- 1 4 i \:. \ .. \L .. l .. '0 v li \d ~ c: .. '.. f' t" ,, t, \\ ~ i n" ~I ( -r c. l .. r- '- ~\~ II ~ .. .. \U I . .. ~ ~ 'i .. t: \I l~ t V\ :rliJ ~I"t t1 t\ t . C" t :: \A -, ~ I r;(~ ~ ,. t' "' ~\lr u- .. .. .. ": ( ~\ iJ ~ fl N ~ c" l' .1 ~r~ r' II .. ~ c.1 ~ ~ l.. (~ .t '" . ri Ii " -. -:tl \l N .. c.1 J ll t: "" ,- ~ ~ 1 ~ ~ oJ ;:. -- \J ~ '" 0. ~ \1 V .. .. - ~ oJ ~ ~ ~ t- Similarity Solution '" ~~ ~~ L \A # q, ~ i. ~ .. Q ii .. "" .i -t IJ r" ". ~ l' 4i () l t! ~ '- ~ u. Y ~ i "" 'lt ~ s ., t- l J t .. ~ " ". ~ F- \L r '- '\L ~ .t ~ 'd .. i: J ~ ~Q .. ~ '" ~ , II 1 ~( ~ ~ ~ t-'" ~ '- ~ U t ow \l \l ~ c; a- ~ rJ , ~ ~ - - ~ /' C\ ~ ~ - r "t i t .. u 4 tt VI .. \e ~ r- ~ Q \f ~ ~ iJ Q "" 1c \J ~ V\ .. .. ) oJ Ie J .. ~ (. ~ J J ~ .. - , ) 4 .. ~ ~ \l. \L - 0 ( - "" 3 ~ \ \\ I' (; ,~ '9 'u. \l \1 - ~ v. Q) i: 1 ti '" \) : ~ ~ .. ~-- t: \L J: '" It ,, ~ .. \L l. - ~ .. \ 'l F- + t. - \ \L .." .. 11 , h -- .. '-\~ (i C" 4 (t i ~ 0 t' \- v - .. ~ Q - \L \L. ~ VI .. '= Vorticity Diffusion -- ~ .. ro~ ). ~ J~ 'r.\ ~ ii /" ~ l ~' ~if .. \ ', :s . .. ~ " .. .. "" "" o ~ ~.o .. ~ ~ .. ~ ~~ ~ ,. t" ~ I (\, -l )i ,. ~ .. .p rl ~ t' ,. l d r-\ ~ i1 i X ~ ~ , ~ ~ 0\ ~ "" t.\~ rl\~ i -l ~ " :i II C) .. . t" .. (- 'Q ~ ': t ~ 9 Ui .. Cl ~ '" ~ 3 :r .. -i n \& \4 J ct 't l . ~ G- o~ .. " ~ .. ~ \A tL ~ \. Q .. A t' 5 ~ f' . ~ l. ~ i: J ~ J' - ~ ~ l: '; i: V What is a Vortex Sheet? ~ .. ~ ~ ~ , :: .. "' 'f .. v .. ,. 'Y \6 5 l , -. ~ 'l ~ oJ ".. ~ ~ .. ~ .. .J ~ t ~ L. 1 1f .. tQ l9 ) -~- IE :: lt ~ cJ ~ ~~ l - ':- .. 0# t: A ~ - '" v~~ ie ~ . I~ ''t ~ ~ ~ i\ l": l~ II r. ~ I. ~ 4 ~ i . tI ~ .- (J :rll 1 ,~ ~ ~ ~ i -l :: tl ~ ~ II ~~ ~ '- - ~.. :r ,0 ~' ,-, i. '-~ ': .. u (~ .3 F .- ')\t" ~ 1- r" c1 Il ;: :s \ l' C" C\ .. VI ~ ll I': vi I ~ .. ) l 4C ~ 'Q i: '0 .. ~ \. ~ .. .. i .. .. ( - ~ f, tf i: .i ll ~ i. f. Ca ~ ,, ~ ). ~ u. It ~ i..' '- .. .. f\ i. \; Q " ~ ,. ~ l ~ ~ ~ '" .. ~ "" 0 IU ti i- .) l" . ~ c: \i ~ P- ~ -- ... - "1 i. 'l .. . ~ ~ ~ "" ~ 0 ~ - ~ I ~ ~ .. lt .) ~ 'V \I .. .. 'I i. .. "- li , /' ~ ~ i. '- i f '- ~ I .: ': ~ ~ .. "" h ~~ Q ~ i. '- -. ~ "\ \1 ~ "1 4l 6 ~ .. ~ .. ~ ~ 1 ': "" "" ~ l1 tl - Q w ,.. \l. i: V ') ~ c: ~ ,: ~ t, F: III .. ~~ ~ L- ).. "\ ~ .)~ "" ~ cJ ')1 ), t" M t" )li n 'l ': It) /' v ~' l. ,-, .. "' 11 ': "1:J .. U- ~(~ -l (I (i F( 'i t" C' \L .. " .. -lt II ~ U- .. ~ f:(~ -I~ \ cC \ ).. t" l\ \L .. II :ri il- ~ o l' ~\ l' , ~ ~ V\ ). '.. F: lL I (" \L - ti .. , () 'l , -\t ~ \L ~ h (0 ~ 0 ~ ~ (l ~"" ~ '" : ~ ~ C' \ U :: l. N d "I I ~ 'J \ ). t. " II .. ~ Fe , .. 1 "' ) "' ~ ~l ~~ i: ... ~~ ~ CJ F \ f .) ~ -. II '- L ~~ t: -~ N It ~i~ II " - )\J - 1- ~ .0 \- ~ .. ~ N\~ II 1 .. It \1 ~. ~ .f i :: ,~ '" ~ \l ~ ~ .. . t. ~ \L ~ ~ \~ i: .. .. l ~ tv \t ~ :c ~ . '" ). ,c \ ).. \L t" l\ ~ "" 1I C" ~ "J t" cv i)- ~ .. . x2 2 Out[3]= D x Erf , x In[3]:= 5 4 3 2 1 All ; "Erf x ", PlotRange x, 0, 5 , PlotLabel Plot Erf x , In[2]:= Vortex Sheet Diffusion 7 Error Function 1 Erf x 0.8 0.6 0.4 0.2 8 (g .. fl\ " I~ )( 4 I l(I ~ It: ~ 0 H \) ~~ (t Ie: . l~ ~ ~~ LI ~ - .. .. l ~ ~ .. .1 "" .. ~ \J l. .J ~ 10 (l ). 1 ~ ! Analogous to Magnetic Diffusion ~ .. ) .. ~ ") ~ Q \) ~ .. "" tJ ~ v I'" ~ \U Il \0 lh \! If , \! (" n Q1\ .u II UJ )( ~ i- -i 0 ~ ld 't ib \. )( l~ i "l~ If rh 'A "" , c: ~ (I ~ \(~ .. I " b \ \J o f\ 'c ~ la ( Il ~ t J ". .. lL .. ~ 0 V .. va J \I : 't ~ C ~ .. ~ ).. ,. ~ .. " l- ~ l r ( ( ~\~ (h 0 \1 ~ ~ \) 1 'Q ~ ~ ~ .. 1 ~ ~ ) .. '" 'l 1 ~ .. Q 'V " It ~ \) ~ ~ ~ 'V 1 .. ~ .. ": .. u. '\. a ~ " "' ~ V' v J . 9 Stokes Flow 10 ~ ~ ~ I\ ~ ~ ~ c: l 3 ~ ". l: ll ~~ l\ (:r tV l) ., ~ ~; ~ t: . ~ l~ II , t) \) ~ (l ... J .. "" ~ \: ~ ). tL .. , ~ rl ~ 1 ~~ r'" ~ ~ .. .3 . ~ c: l. l- l' . .. .. - ~ 'C' i :i .j t .I '0 't 'C 1: ~ ~~ '- .. . . 3: 0. ~ (l JI lu '\ ~ \L ~ (~ ~ i~ ~ t: ' ~ ~ '- l ). 'l "' .. 1 ~ '\ ~ ;( ~ t" ?0 .~ t: ~ 1 V) "0 i '\ I J ~ \) "\ c: 1 ~ '" ~ J~ l .. o t "" b~ (I t' II I (L i~ /~ .. tJ l\ ~ .. ~ .. " '" ~ ~ ~ o ~~~ \) ~ 1I ~ () ~ ) lt ~ .. 4 ~ 1 .. lL ~ .. 1: '& b V\ I~ Q. ~ (I j l -( f. ~ .. (~ J tv~ ~ , ~ tL .. J ~ ,i ~ ~ ~" tL .. ~ ~ ~ r ~ r ~ ') .. ~ "' ~ o 'j .\ ~ ,~ tV' VI L/ -. Q (: "i L l ": .. Vl ~ \J i -. tL 1 \) ~ ~ i: t~ ~ v Stokes Solution for Viscous Flow Around a Sphere (1851) . .. \) ,. ). ! ) ~ !~~ ~~i b f' I': i~ ~i ) ~ ~ ~ \~ \l "'" tl. (( ~ ~ ~ VI ~ i,. i .. l-- ~ txJ i:" u. il ~., t b ~~lI (O ~ - ~ -\ri ~ ': ~~ ~ \A () Q .. 1 i"" ~ '3 j tV 't ~ ~ 1; ..,. ~ li q, II " , ~3 :i l.. ~..~ ~ ~ '- :) ~ \"~ ~ ~( :j t ~ I~ t II ~ ~ I) ( 1 ~~ I\: .l .. ~ .. 0 , ~ ti \A , I -. ~~ ~ I~ ~ r: '. ~ t \A '* :t .. ~ 4: ~ ~ Vl ~ ot l. '" ~ ".. .. ).. " .. J \J c: '.0 \u ~ .: -i 'l ~ .. Steady, Creeping, Flow Around an Object 11 ' . ~~ \I ~ .. " ~" i ~l ~~ i: VI '- t J _. Il ~ "t ~~ :r \o.. \l -~ F~ t ~ ~ ,. '1\ " ~ '- x ~ \ ~ ~ ~ I ~ '4l /" J~ 3 l~ " \.) ~ ~t" (\ \) I~ II i~ ) ~ h I ': l"tJ )! () ~ ). .. \l - ., l - '" t "' ~ .. ~ ~ "3 ~ ~ v .. ~ ~~ l. \. \) ~ '" t( I~ ':= /' ,. J VI ~ .. to t ;; .. 'l! to ( r ~ ~b ~i lJ , 1: ~ V\ ~ .. 12 Stokes Flow (1-3) 13 Stokes Flow (4-5) 14 ~ ~ 1 ~ \I .i r; :) "3 .) ~ Vl ~ '0 ~ :J ~ 'i ~ Stokes Solution (some algebra...) r~' i-\ b ~" ~ -I ~ ~ N\~ -\~~ .. ., " 'C ~\J --~ .. )( ~ l c; L -. -l .t l U l:~ ~'\ ~ ~ ~ \ l\ i r- _\ l" ~ t'\ ~ -I! ~ cJ\~ ~ , ~~ ~~ ~ ri\~ ~~ -; \ i ,~"~ l.. rJ l- ( 1 ~ ~\~ b ci ,~ -l ! "'.. ~ 1 \ .l ~~ + T .. -i ~ \ I\~ l \ II -\( ~ l- ci ") '\ ~ .. Q) \I ~ ? ~ )( -1 /' Ii ~\~ Lr ~ '-lt rl\~ ~ ~ -\ ~ \t. li l~ ~ ~ rr- 1 ~ ~ \\ t V. ~ tl\~ C) t\ (" \ \\ l I.. \; 1\ ~ ,. ~ ~ r N t" ~ -i l "" ~ "" .. , l i ~ " ~ ~\~ \I .. ~ ~ rl\~ ~ l -i "- ~ ~ 'i ~ /'l ~ , '" -. \b .. i _. i~ ,. --, ~ti . -' ~ \\ ~ f\ ~ :) -i cl\~ i -l~ ~ r(\~ iI \I '\ ~iJ VI T .. , \~ N .. L II I.. l3 \: U\ ~ ~ Vi ~ ~ ~ X ~ \\ r ~ "" \ -H ~~ ~ cL 15 Stokes Flow (6-7) 16 ~ .. \l ~ ll t. "" \I ~ -l ') r: ~ () b ~ t' \\ ':t ob \,i rI . (l LJ T ~rt .: cJ \ ~ -\ V\ ~ '- cJl~ r) ~ r' Stokes Solution (some algebra...) rJ C) q ~ ". ~.. .. -) I\ t'l~\ ( r C) u ~ 'h t' '" ~ ':t \ :: ~ + (' \ tI" ~ '- tl \,. .. ~ I .. NI~ .. , l' )..~ - -l l" i .. '" rJ ,. "'I~ I .. Oo~ 1 /' ~ \ri ,-\ ~ " I ~ ~ (~ 3" \ ., \ C l- L ~ Ii o '\ I .. (~ .. + "\ l" ~\ '\ - .. .. .. \. , ~ \,J ~ ,, ~ ~ ~ iJ ~i t. -,'" ~ ;. '" "" .. r -i J W(~ '" .. l/\ t M'\ t\ Q) I ~ l , ~ " tv ~ ..1,J II -. ~ . . ~ '- .. l: '" tI "3 "" " tlN .. ~ l- C' 't ~ ., ~; i- r" IJ 1\ ~ ( ~ ~~ ff \ ~~ ,. ,. " ~.. -i~ l\ r- j- , ~ ~ .. .. Q ~ 17 Stokes Flow (8-9) 18 Stokes Law Stokes, G. G. "On the Effect of the Internal Friction of Fluids on the Motion of Pendulums." Cambridge Philos. Trans. 9, 8-106, 1851. CGS unit for is poise (Pl) and for / is stokes (St) 19 Problem 9.1 20 # "" .. ~ ~ o ~ l:i 'b l' .. I (l" ~ Ii .. ~ l:s C1': '~ - i ~ ;it l"~ rc ~ "l 4) ~ .. ~ .. '" \( i. ,. . ~ \). t\N I ,) l. .. '\ 1. 'J Q 1; 'l , ti " ~\). '0 '" ~ t ., t. - c l '" \ ~ \I f~ ~\l II ~tj ~ J.1.c i ~~t ~). ~ L ~.. \ ,. ~ ~ .. Ii -c \J /' GJ ~ '- ~ tf r\.. r n fi i I' II -i ~)-\ .. ~ i-' .. ~\~ *~~ ~ I Irl ~ , 6'\). '- '" ~ \\ va ~ " . , . ~ i: ~ l/ 3 ~ t: Vi ~~ ii (I ~ --\,i ~ i, II t.. 1,J II .. _oJ .; (\$\ ). ~ \)", '- t , C'" J. ii ~ ~ \ ~\rIC1 t1 II .. . ~ i. .\ )~ Q .s .. ~"' ~ ,. . . .. \A ~ 1 '4 C) ~.. ~ \\ i. ~ VI It h :: Q t~ -: tJ .), '- li ~ ~ .i () .. ) i , t o i tl t('C \I ~ v~ 9 F .. L ~ tI ,.. l 1 Y Q Viscous decay of a line vortex Stream lines and vortex lines Leap-frog vortex rings Viscous relaxation towards steady ow Problem 9.1 21 HW #3 22 Summary Viscosity causes vorticity diffusion and decreases velocity gradients. For strong viscosity, viscous forces balance pressure forces. Drag coefcient scales inversely with Reynolds number as expected. 23
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Punctuation and MechanicsWriting Paragraphs/Thesis/Outline1The Living End: The Period (.) The period is the red light at the end of asentence. When you reach the period, it's all over. Whatever thought you were trying to conveyhas been delivered.
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General-specific TextsTask One: Selling Cities: Promoting New Images for Meetings Tourism1 Meetings tourism, which we define as travel associated with attendance at corporateor association meetings, conferences, conventions, or congresses or public or
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General-Specific TextsAudience1AudienceBackground knowledge presumedBackgroundText A targets an educated, but not highly specialized,audience.audience.It provides a lot of background material and is carefulto avoid too much technical vocabulary.
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A.1 People have been pulling freshwater out of the oceans for centuries usingtechnologies that involve evaporation, which leaves the salts and other unwantedconstituents behind. 2 Salty source water is treated to speed evaporation, and theevaporated w
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Numbers
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Technical WritingSeptember 16, 20111 September 16 Orientation September 23 Paragraph and essay writing;Punctuation and mechanics September 30 Features of Academic/Technicalwriting October 7 Writing General-Specific texts;Audience October 14 Writ
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Features of Academic WritingAnd What is Technical Writing1Why write a research report?A research report is a paper written by aninvestigator to describe a research studythat he or she has completed.thatThe purpose of the report is to explain toot
National Taipei University - EECS - 101
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National Taipei University - EECS - 101
Use the circuit in Figure 1 to supply a variable voltage to a load of R=5. The source is 120VRMS, 60 Hz. The load power has a maximum value of 2100 W and minimum of 500 W. Find:a) the required range of b) the SCR average current requirementc) the curr
National Taipei University - EECS - 101
Tsogbayar OnhProblem_1A buck chopper like that in Figure 7.1 has the following data: E=40 V, T=50 s, L=250H, D=0.4,C=60 F, and R=10 .Find:(a) the value of L necessary for the continuous current mode(b) VC(c) Imax and Imin(d) CVa) TR 0.00005 *
National Taipei University - EECS - 101
O.TsogbayarHomeworkProblemA dc shunt motor drives a centrifugal pump at a speed of 1000 rpm when the terminal voltageand line currents are 200 V and 50 A, respectively. The armature and field resistances are 0.1 and 100 , respectively.(a) Designe a
National Taipei University - EECS - 101
SUBJECT1.2.3.4.5.Diode Single-Phase RectifierSingle-Phase Bridge Phase-Controller RectifierSingle-Phase Bridge Phase-Controller RectifierSingle-Phase Bridge Phase-Controller Rectifier and DC Load VoltageThree-Phase Half RectifierSINGLE - PHASE
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DC LOAD VOLTAGESome circuit arrangement effectively place a DC voltage in the circuit as part of the load. This mightresult from capacitor voltage, another source may be the back electromotive force (EMF) of DC motor.As shown, if the inductor is ideal