Lecture-19-(11-23-10)

Lecture-19-(11-23-10) - APPH 4200 Physics of Fluids Fluid...

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Unformatted text preview: APPH 4200 Physics of Fluids Fluid Instabilities (Ch. 12) November 23, 2010 1.! ! Introduction 2.! Bénard Thermal Instability 1 ~ .. ~ v - J -- "' ~ lU 'l Q ~ ~ t- ~ ~ ~ i '0 't ~ .i .. Equilibria can be Stable or Unstable ,-..'* 'u .. '0 ~ ~ l"" '" '- .. t" " \U ~ ~ ~ l(. " \ ~ \. - ~~ .~ ~ , \I -i " ~~ ~ ,J t ~ ~ \. \i ~~ \. 2 N )- I'- .J .J l ui ~ ~ ~ ~ Vl ~ Linear Instability "' li cq ~ ~ J t ~ t, ~ tl ~ ~ (J 8 \U ¡ 'f VI -. ~ i,'" ~.. '\) C) ~ ~ " Q. " ~ '\ ~ ~ ). 'l &, \. " ). L I.. 'a tt ~ .. ~" It t J. ,. ~ Q ~~l t ~~ J l 'U .. ~" r o~ " .. ~ V1 ~ \L L -, J, ~ ~ í: '!\ ~ ~t ~ )( ~ ~' '." .. '3 1 ~~ - l tJ t. .. J Ii t ~ "'.. V i.. .. \J .. Yo .. "" ~ .. .. - ~ 'r tJ J ~~ ). tv ~ ~ '\ ~r i. .. 'w l~~ ~"~ Q~ , n ~~ ~ ~ \I \I ~~ (~ " I x'" "' J.." it': ~ ~ c: ~~ ..'- ~ ~ VI \ VJ 1 '\ a .. ~ n.Vl( Ç\ .~ t ~ \L ~~it~ v -: -Í t- 1 l ~S~~ ~~i~F \ t o .. - ~ ~ 'l ÇJ 't.. t~ ~i 3 Henri Bénard, (1874-1939) Henri Bénard was a French physicist who performed experiments on fluids for a Collège de France physics course given by Marcel Brillouin at the turn of the century. Bénard was among the first to study the behavior of a thin layer of liquid, about a millimeter in depth, when heated from below, the upper surface being in contact with air at a lower temperature. Experimenting with liquids of different viscosity, he observed in all cases the formation of a striking pattern of hexagonal cells. In his 1900 article, Bénard used a variety of means to visualize the structures he wanted to exhibit. They ranged from material substances he added to the liquid to optical contrivances such as lighting and the design of special photographic setups. His papers were abundantly illustrated w ith sketches and photographic clichés. In 1916, Lord Rayleigh provided a mathematical explanation for the onset of instability in such a convective system. 4 1\ t ç: tU ~ ,~ ~ \- ~ :t ~ l J V' ~ ~ .. ~ '- .. Bénard Thermal Instability J- ~ I æ ~ J (\ f-Q ~ -i Q o _u ~ \ \ " ,~ ~ l-¡ ~ ~ t "'~ , ~,. ~ ~ ~ II. ~ r C! \, , "" \J ? 1 ). Q tQ .. ~ ~ (.i ~ lt t Q Il l~ . ~ ~~ ~ ~ i - ~ .~ ~ ø "" '0 ~ \d ~ .. J ~ VI \. ~ V' ~ () c! J ~ J \I V\ "- '\ '\ " '" \i ~ ~ " ~ ~ .. " ~ ~ .. (l I:. (-Q q: l- \l C- ~ ßl ~ 'e L .. .. \\ Q '" ~ , l ~ \1' " - .~~ "~\r' 0" lii 1\ C' r/'l (~ l (- '~ \-- ~ h ~ ~ t. ~ 11 v -l ~t 3~ u (.. .. I- ~ ~ ~i ll (I ~~ "" a oW ~ V(.J \. i Iu~~tt ,~ ~I -0 l~ ? q: ~ 1 ~ \a l) J~ .. \c oi \L 'l V\ ~) ) it È~ ~ o q: t ~. -) V\ " ~ '1 ~.J .. ~ .. "~t ~ t. 1- , ... cO :i ~ ~ ~ j~~ ., \t - ': ') .. ~ ll o0 '" t - .. 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C1 la-i '" Statics II 1- ~ I 'J r" ~ f- ~ 'i ~IÑ ~I~ +i h ." (' I l- r\ '- r- II 1-0 I ~l\\ ~ "~ .¡ ~IN r- lt ~ ~ ... ~ ~ .. u. ~ ~ 'l c; ") ~ $ ~ ~ \J ,: q: ~ .. .J i ~ '" \¡ ~ ( \J ~ ~ V\ t' ~ ~ 4 P .. ~ ~ ~ ~ ~ ~\Ñ ., ~ ~''1 I 1-.0 '- ~ - I~ 7 ~~ .. rf '- ~ l'ì ~ .l - -- L\r \Î I ,~ , ~~ n (Q. \ '\ ~0 £' '( 0. ii \ ." " c; ~ 1\ r\ 1: .. ~ 0 ., t ~l ~~ ., t. \J k i ~ \) ~ l~ \ l I ~( '" A , ~ 7 \Q 0 1 ~ ~ ~ ~ ~ \1 ~ ~ \.. t "l '" ~ ~ a: ~ \l .. ~ Q J" ~ l V "' Nonlinear Equations for Perturbations ~ .. l , 1 F: ~ 0: ~ Z . .. J~ ~ t c: ~ "' ~ .. ~ ~~ ~ .. q \r ') \1 ~ f 'I ~ ~ f) l j , F ~ l i, "" l- 0 '" 'l ~ l Q l) ~ \I ~ \, ~ Q Q )J i: (3" \) .. tl tt ~ ~ ~. Q ") \I ~rf l\ ~ ~ + ~i- ~ \T + )0.1 ~ f1 tl -1~ 1 II t1 ': r' l~ . (~ -l l' oj \ .¡ ~ C1 1( -: nI ~ ?1.i \ 'A C" rl ::). '" t) ~ i- l~ \ l- , (\ :: t\ )-\4J (" L , ~ ~ ,~ ~ ~ Cl t' -i~ -,~ l II l( :: ~ i~ . l"j "" + .( \ .i ~ () Cl Q Il ~l. C1 '\ -\ Sp. C1 t' T- ~( ~ C' Cl ~l- tvl) ~ II r~I- .¡ ~ 'l\-' "~ .. i~ \0 l ~ tI ~ ~ 1 ~ L ~ ~ ~ ~ '0 1 , tl lQ, "\ 'Jil i. \. ~ "- ~ .. 1 Q " I~ l ') u. l :: '- + i i ll- i ~ C" n ~ Q l .i ~ 8 ~ "" .t ~ ~ ~ - \U Q J i: " l7 Linear Equations '" -: ~ ~ r + tl~ a- + f ~'\\I C' r¡ -l~ i II ~ ~ '" -: k. (' ~ 'r )- ~ + 1'-\)- h \ -/~ C' t' T lQ" I il C) t\ -/~o , f\ ~ "i.u -j .ù ~ (l C"\ f\ ~ C\ ~\ ~ l' 'J ~ '" ..1 l" i\ i- i. "" ~ \ ¡. -\ t" (\ ~\.¡ l /ri () ~ It l- cQ(~ . (~I~5-¿ "' :J i(~\ ~ r6 C\ ê~~~~ '. 1 'I ,~ t .. \J \l oJ a ~ ~ i i ¿L. f ~ ~ "0 u .. .. Q '\ a. VI '" ;a l- l ") .. ( .. \I t) ~ .J \V ') lU . \I '0 Q -i ( - i .. t~ \l ~ "ó q: d ,. '" j .. "" ~ '" tv .. ,J 't ~ , '" fV' t ct .. ~, " \. r:i '" ~ .. ~ .. ~ .. Y ~ tI 1_ 0 ~(~ cl ~Q. ~ ~ ., c. ~ . ~ t ~ L ., \. ~ (. l ~ 9 ~ ( \I q" ~ ~ \J .. \\ - (:. "3 -l Simplification 0 '" ~.. t: ~.. ~ .. 4Q ~ ~ ~ , ll- Ii '" \ C\ ~ '" + ~ ~~\~ l lo. r' N(1 ,t lii -I r (V t' ') C1 ~ -1~ , " r- ":$l i\ nn l. )- :s \ ~ C' t\ + )( f1 ~ ~ \C" ~ ~\~ /" l~ . ~ + r If. 'N \I 0 \:s - , lQ . ~ " (~ . ii- \ .. t' C\ il ~ u ~Q. l' t) -\~ - ci \) u. l . " S \\ ct .. ~ \) , '" ~ \~ ~ t: . ~ ~ D '" (V~ ì- + li(V~ ~ '" + IV l ci b O\t ..1 ~ ) II ~'" -l ~ a\~ , rl :J ~ ~ IV ~ ., i ~ li- .. (1 (" tv \tN i tlcv ~ '- )l 0' fl ~f1 N ~ , la. n\c\ i: l! ~ .. ~ .. \U . 10 ~ ~ \A ~ w ~ ~ lo ~ ~ "- ~ '.t a. q. .. "' &. ~ .: l (Jj "0 '" '.. l ~ )o Two Coupled PDEs forms a Linear Eigensytem ¿l- .. ~ I l' N ,. \ IV (\ ~ "is tr f, IV :: N D '" ~ ~ ~ ': '" ~ ti\ tt ., Q~ tI .. c:r ,.. - ~ .. ~ .. ~ F t. Q ~ '1) \ù 6: ~ ~ t: ~l-i tu ~ '- t. i -r 1I ~l(V (: ~ 1 uól ~ ., '4f J " ~ ) ~ '. ~ . ~ ¿q. '\ ": 'I ~ ~ ~ ~, 1J rll J.. .. "''" '- '(1- Q \( ~ '1 4æ "" V rl .. " ), q 1-' '- ~ ~+l \ \tJ tt ~ .l i ~ ; .. ~¡~ ~ a \. ~ i: ~:, \\ ~ ~ i. tU 0 Q. .. 11 ~ 1' . "" I- ~ ~ \I u ~ 1- Q~ .. 'U ~, \) ~\ 0 ). ~ Q l. , ~ 0 " Q V) .. -t ~ ~i,y ci tf ~ ~ u l-: '" : l\-" '.. \a ~ ~ , 'J '- ~ \a " " ~ l \ ~ " ~ ~ "' '. " ~ ~ ., ~ , L ~ \4 1\ Q \1 ~ ~ . ~ , ') " . , . ~ 1 ) ~ t '- ~ ~ l: ~ i .. ~ 'ù ~ \I ~ ~ i. II ~ ., C1 ri ~\ J¡ ~~ I (1 \Ä 'tJ t '- ~ . ~ ~ \I ~ '" ( J ~ .. "' ~N .jr . .. Vi l ~ \J 1.' J~ ~~ ~ \. t ,; i, ~~ ~ g .. ~ ~ Vl \. '- ~'N +1 " rl 1l ~ 0 \..~ " :s ;J \¡ ,..~ í'", ' \.~,V \ " \. - .6 \l (\, "- "" \. '\ " \\ "- \. n. 1\ , ~,r' f.' - ~.. '" " _ ~ ,../ \. "- --,~ .. ~ C1 1\ .. ~ l c: .. '" "' ~ ( ~ I~ 1 "0 . "" Boundary Conditions 12 Gj '" (AI ~ va Q ~ 0 ~ ~ ~ ~ ~ lc -. C: ~ ~ 3 "* Q. Q r 0\ a Normal Modes: Rewrite PDE as ODE .) q, ~ ~ 0 .. ~ ,- " III "4 ~ ''0 ~ "- () \L ~ 0 ~ -. - rJ ~ Q ~ t ~ V\ , 0 \L .. ~ .... '4 ~ , l .o ~ ~ \ ~3 .I J.. oJ ~ v "' "' () .. ~ ~ \1 ~ ~ " 0 1\ /' U. ~ A -. .. ~~~ '" ~ .. ~' + l it rJ ~.. ~ 0 ~ " !i ., , ~ ~ . .. io II ~ dw N i. + .a ~ .. J. 9 ~i. ~ -l 'X ~'I \. .", .l vi ~ ~ v tl ',I- \":tV " I' "" ici .. '- na' i-'" .. )- (i- , .... .. "'"" ~ I- \Û ~ ,. V\ i M\ ~ ~ ~ ~ .J~ l r't- \Ñ ~ T (\ ~ " ~ "'l 0 \J t ,. ~ , rt' i.. \a\ ~ '0 a. 13 0) 1 .. t t E- o () ,~ ~ ~ ;; Mode Equations ?i,J ~ 'l " l l, ~ -: ri ~ ~~ I 1: N \'A '" C'( ~r4 i- V\ .. . .. . f (e 1 .. .. ~ ') \I ~ '0 ~ ~ ~ \(tv~ t ~rl ~(- ~l- N t' \ (\ C' (\ '- N~ + "'1l! ~ " . ~\~ t.l ~ (I , ;: i\ fl f\ I o ~ ~ \I \ )i , f/ \ ,. C" "' J tJ~ '( "' Ñl ~ ~ ~ f ~ ~ ~ . .. \. )- t ~ ~~ ") ~ ¿ \L u: -l , (\ ç cÎ \Ü J l~ ~ '" ~~ ~ :; t. . , tu ~ \/ ~ oJ ~ ~~ r" ~ ~ :" t 'l~-.. ;~l ,:Vl ( i ~ ~\~ '- ~ .. ~ .. ~ ~ I\ ~ ~ ~ '5 ') .. . ~ \U ~ cr ~ ~ it ~ t. ~ ~r1 ~l! Q (\- , x ~~\~ "- 2)- " I ~(-~ \1 ~ ,, "; 1 /' '( "3 rl I fV~ .l -(:s \~ .. vi (Vt' ra ~ ~ ~rt ~ ~ .. ~ i: ~ ~~~ 0. :) ") 14 ~ ~ ~ \J ~ V) ", ~ ~ I:: l "i .. "' ( t( " ~ ~ ~ ,l V "" ~ i " '# lc t ~ '0 ~ ~ t ~ "\ ¿ ,i ~ .. Q 'c V\ t-l t \c ~ ~ ~ J l ~ ~ ~ Q .. ,. . ~ Dimensionless Form . 1- ~ t t) ~ 'J ~ l' ~ ~ ~ "t r" ~ '\ * i- V" \ ,. ( li i l " l( :: 4J .. 1 i 1 W 1 " . '3 "" ~ 'ii f ~\.. 0 J '( ~tl ~ ~f\ ~\x , (.\ ~ X"\~ "' W\ 'í l 4 .. ~ .. . L l .. \ ~ f- \J l Q ~ ::~ ). ~ ~ 1I i-tiv a "" "" .. f .. .l -, c. r '" + '- N~ +- v\~ "(l/' '1 'c i1 ~~\~lJ ~ 't l- \ i~~ t .. \. ~ ~ tl- \ NI c. J ~ fl ~' 'iì t- 15 ~ \' "" " 0 ~ - ,~ Jl ') - l j ld -h ') "t t Dimensionless Equations ~I,J~ 0' \~Ñ . C: II '- ~~ .. V\ ,lr' fj ~-:"" r'~ ~ , ., rl \'" l' ~ l.iL~ :ril; ( (" (1 (V/' ,J~ I ,. ~r"\~ ~ LJ l .. fI v 0 '"S v ~ 1 .. " l~ "l ~ .. tL " 'l ( ¡ :i ll 0 .. (l 1 ~ " ~ ( i. \I ~ i, ~ 0 \J '4 ~ .. . ~ ~ ') t i: ~ J ~ .. ~ 11 ~ ~ V\ ~ ~ ~ ~ ~ ~ " t\ .. ~ \0 ~ ~ " ~ \) , 10' t( ~ .. ~ "l ~ V' ( .. ~ (; ~ II 'ó S ~ ~ "' - '4 .. 4: ). I' Cd )f ~" " \f \\ l- x: 4 ). ,. t t (I S ~ ~ -i ~ J l0 ~ ,, ë It ).\ ~ ul \\~ \C ~ \è 1. .J ~ .. l. ~ ç ") () d ~ ~ l, -\~ 0 -\Ñ -+\ II " II T\ -i~ (\ '- r' /' +1 ,. "- r+ " ~ (\ (\ '\- '\"'iV t't ~ 16 ~ t ~ VI - ~ ~ ~ ~ "" ~ " .. 1 "' .. .. f ~ j Marginal Instability/Stability l. ., J lì ~ I, ~ .. 1: ~ ~ ~ ~ t. 1 "' '\ ~ ~ l~ 't (~ (~ ~ l- ~ :t ~~ ~ .. L. l ~ '/' Ct '- ( \U f- V\ ci \A t\ -= t u. -- ~ \t ~ , R" ~ .. 'l ~ ~~ ~ n ~ .. iL " () II V\ Iu1 -i ~ ~l- ~ ~~ II \:slv rJ~ I ~ r\ ~ \., N "( :: ~ ,. ~ c:-l i , ( ~t\ II u \L- ~ CO (\ ~ (~ "- I M /' "'~ \~,. I (\ \~ t" rv - ~ 0 ~ ~ '- t; Q .. . .l ~ \L "0 \) -i q ~~ ~ l i \A .., '4 t~~ uJ · ~ ~ Q t. ~ - , " ,,\. Q~ 9~ ~~ \~ ~ ~ '\ :i - '" ~ t rlQ v '0 l" "' l 0 v ~ ~ '" ~ -: ~ ~ '" \ , ~ -. .. ~I ~ , ~. , " , ,,' " '\ \ ui ~ )0 l. ~ . ii , t:: \" l 't , " " 17 ~J ~ 'l ~ ~ 0 .. 0 '" lc .. 't, Vl t .l (~ .. '" \' :: ~ t ~ ~ '" ~ '- ~ tQ .. \. " L \I l\ I ~ l 4. N tv ~ rv ~ \ t" - co I .1 ~ r(" ~ "" .i ,. ~ 0 \o ~ 'i .. .. "I ~\l " ~ ~ I- ~ ~ II .. N(t ~ t' -\ M r- f/ cJ ~ - ~ \J ~ .. í \~ q ~ ( ~ IL J .w \i - ~ rJ ~ ~ '" Q J "- ~ A. Solution for Marginal Instability/Stability . , ~ lt ~ u. ~ ~ ~ ~ .. - l- '- + " J ~~ I rI~ ~\\I .. U ~~ o -. l '- ~~ .. , '= -,~ .. crl - t, Il " _.l " l' C' . oJ '" .. .s ': - o~ 'V , () u ~ -"' .. r- .. 'd x. ~ .. 18 Bénard Photos 19 Fun 2D Simulation Shows Linear to Nonlinear Evolution 20 Bénard Photos Benard-Maragoni 21 Mathematica Notebook Bernard-Marginal-Mode.nb 22 Summary • Instabilities are (initially) small perturbations that grow in time. • The method used to find the mode structure and growth time for a linear instability is similar to the method used to find wave structure and dispersion. • Bénard-Rayleigh thermal instability occurs when the temperature gradient exceeds a threshold. • As the ΔT increases, a single mode becomes unstable first. 23 ...
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This document was uploaded on 10/18/2011.

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