BIOFLUIDODINAMICA Lorenzo Botti, Alessandro Colombo

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1 BIOFLUIDODINAMICA Lorenzo Botti, Alessandro Colombo Lezione #9 - Breve sommario dei principi di conservazione Versione 0.3 a.a. 2017/2018

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sha1_base64="c3tjnyxj/sdl/h0m1iqpjryhlxu=">aaacd3icbvc7tsnaedyhvwivagwaexesdgnkkabrakhdbj5selkns+bcmr5bn2ti0wwcz6br6cfig7r8gku/auo4wisptnrzk52d7xqcoo2/wuvvlbx1jekm6wt73dvfl+qcsekebq5iemdmdjbqrq0esbejqhbu7etre+gbmtyegjqjupu5d6ptspmrqciap5jyrvqnwujm4+ipnp/pqmsqtv+2anyeueycnvkj4a/e4oarz4o5ji03xsepsx0yi4hktuiwyeji/clopvcwh04+zjxj6hbmgaq1buyfpjslvij5xkx9l+30gt6yrw8m/ud1ixxe9mohwghb8dkifbkyryrkayddca0ill5ucfopxphghaum5kkpxku0d2fx+2xsoqs5ds25o6/wr/nkiqrcjsgjccgfqnb0ibnwskjesyv5nv6st6sd+tj3lqw8pld8gfw5w9yejz8</latexit> Relazioni integrali per il volume di controllo CV in possibile moto uniforme con velocità relativa dove Breve sommario dei principi di conservazione (1/3) ~U s è la velocità di trascinamento Conservazione della massa d dt d dt CV CV dv + ~ UdV + ~U r = ~ U ~ Us CS CS ( ~ U r ~n) ds =0 Conservazione della quantità di moto (equazione vettoriale) ~ U( ~ U r ~n) ds = X ~ F La sommatoria delle forze agenti sul sistema include tutte le forze di superficie originate dai tagli fluidi e solidi individuati dal CV e tutte le forze di volume, e.g., gravitazionale. Le forze di superficie sono originate dal fluido circostante e dovute alla pressione e agli sforzi viscosi oltre che all esposizione delle forze scambiate con corpi solidi che attraversano il volume di controllo. Il contributo di pressione alle forze di superficie è calcolato, convenientemente, come ~F pres = CS (p p atm )~nds!2

3 Breve sommario dei principi di conservazione (2/3) Relazioni integrali per il volume di controllo CV in possibile moto uniforme con velocità relativa ~r = U ~ U dove U~s è la velocità di trascinamento <latexit sha1_base64="c3tjnyxj/sdl/h0m1iqpjryhlxu=">aaacd3icbvc7tsnaedyhvwivagwaexesdgnkkabrakhdbj5selkns+bcmr5bn2ti0wwcz6br6cfig7r8gku/auo4wisptnrzk52d7xqcoo2/wuvvlbx1jekm6wt73dvfl+qcsekebq5iemdmdjbqrq0esbejqhbu7etre+gbmtyegjqjupu5d6ptspmrqciap5jyrvqnwujm4+ipnp/pqmsqtv+2anyeueycnvkj4a/e4oarz4o5ji03xsepsx0yi4hktuiwyeji/clopvcwh04+zjxj6hbmgaq1buyfpjslvij5xkx9l+30gt6yrw8m/ud1ixxe9mohwghb8dkifbkyryrkayddca0ill5ucfopxphghaum5kkpxku0d2fx+2xsoqs5ds25o6/wr/nkiqrcjsgjccgfqnb0ibnwskjesyv5nv6st6sd+tj3lqw8pld8gfw5w9yejz8</latexit> U~s <latexit sha1_base64="cryqgabpotu/dwaccjbgnerex0=">aaab+3icbvbns8nafnz4wetx1aoxxsj4kokiepkcf48vtftpq9lsx+vs3u3yfsmukf/hvu/exks/xop/xst2ok1zgmbe482bmjbcout+oiura+sbm5wt6vbo7t5+7ecwbapecpb5jcptdkfktt4kfbcnzbavcihe05ucr8zbwnfpo9xfkog2filkeamc+mhpwwe+gobdwp1t+gwomvem5m6mam1qh31hxfpfgjkklnb89wyg5qffxcvu0nfmlgj2wmvxqpsagark4o6ejrjrgawvkpyi/n5imbj2psj8ujf8titeif7n9ricxqwp0hgcohlxciwe8pdlrurnab0ka4issa5uamqyyhgbgwc52ksv1pn+/awv18m7fog5za8u4t683retiuckxnyrjxysrklrsitzhr5ik8kxcnc16dn+f93tfme8ckt9wpr4bzjgvdg==</latexit> Conservazione dell energia totale d dt <latexit sha1_base64="folgmgospy065pwjfta/sxibpq=">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</latexit> CV ~2 U ei + + gz 2! ~2 U dv + h+ + gz 2 CS! (U~r ~n) ds = Q + W S + W F V con: <latexit sha1_base64="xqfyynoo6h5pqncokualbm6qhi=">aaace3icbvdlssnafj3uv62vqesxdhbeeoigm6ugiauw7qpaeutg/r0mmdmruhhcz9bl/cra7civs/wix/yhkdaovzogce7hzjxniodgypozc3pzc4ljxubsyura+yw5unbufkg4n7ktftr2mqqopgihqqjtqwfxhqssx6r+6w6ufr53g5maei4bewioomne6pu792bj1e9pszevty//mhrtpum2wrymwgs8tosnkqpxnzytoqxc85jjp3bgtahsruyi4hljudtueji/cdoj9gluhdlh8r0p9qmfrqaoklstitfiyi5wk9cj5l0gd7qas8v//m6iq5pe5hwghdb4+kifbkyrorktqedcauill050cfrzltdbguoizzraytykpjh/b09bokevsxrypdpy5xz/nmimsh7jedypmtuivxpeyahjn78kieylpxylwyr8bb92jbydpb5a+m9y8qpj5q</latexit> entalpia h = ei + p/ Q derivata nel tempo del calore fornito al sistema da una macchina, e.g., una pompa W S derivata nel tempo del lavoro fornito al sistema tramite macchine W F V derivata nel tempo del lavoro degli sforzi viscosi <latexit sha1_base64="gcscwlx2ftgl1ur1576d7rfh1qy=">aaab/nicbvdlssnafj3uv62vqks3g0uqhjqiobul4mlbfuajptj9lydokmgmruhhijf4vx7sstv+lcfzgjwwjrwr3oud77vgvfat+9mqls2vrk6v1ysbm1vbo9xdvbajys2hxsm6a7pdegrqgsfsugqdszwjxt8yu3mdx5agxgf9zhv4avsfiqh4axtyr1fqv+cqfnxj6n+twbx7rx0ktgfqeczx71yx1epa4grc6mt3hvugltkpgemyvnzaggj+wefrsgriajjfkmwf0kdymi6pauyfplslvjyqfxkwdp50mgi7nvjej/3m9giexxijcfsoepdueqkj+yhat0jkadoqgrjylbypcyplmikafynypy2u0n7coa/xytts7pj152781rjumimta7iitkmdrkgdxjlmqrfofhkitytf+vrerxerpef0jv7oytp7a+vgeu5w0</latexit> <latexit sha1_base64="pffhajppich2f4zqhn+sroybbau=">aaab+xicbvdlsgnbeoynrxhfuy9ebopgkeykoccjepgyghlaeslspbohzd6y6rxcko/wqidv4twv8ec/olvuqrprvfr10d3lx0oact1pp7s2vrg5vd6u7ozu7r9ud486jkq0wlaivkr7pjeoihtkqswf2vkga+w689um7/7inriklyneyzdge9dogck5w6g3feawsxqtbcupudrrkviduo0bxvv2xsjaggjbq3pu+5mq1trkkkhyvkideyczhju+xbgviaztdnz12ws8rwilimmknfchf/j1iegdmpfdschowy14m/uf1e5pcd1mxglhkljfjbxmi4zq0vaabcw1evhscmqyijrtorami6efrnbtmx24s1/v0o6f3xprxuty1rjpmimdcdwcufgwru04a6a0aybm3ic3hxuufvexpef0ltpe5hj9wpr4bpp+ulq==</latexit> <latexit sha1_base64="rikiybhoshf09jcg5s1m27fk2nw=">aaab+3icbvc7tsnaedzzdoevokq5esfrrtcggpfoqemgjxqykxnyyaccmdbd2ukypjx0ejfh2j5gar+hbnxaqltjw2tlstxfiydn1p2l5xvtvbjr3dza3tmt7e13tjrodm0eyuj3amaihdakfbcl9bavcchg0yvcr/7cnqiklzdwqy+ypnqjavnakx7wsjctjsnb4e1uttwc9bf4pwktkq0hruvm+wjghc5mb0ptdgp2uabeqvqejgjxkta39kqktb+whyc0epemixodjoksqsrfidspoyqcbokoypt7lxf+8foljcz8vywghdxfhejcschwlwwtqedcaylllwcqqsqoigbwwcwzgx1vrth97894ukc9rw3i3c1vxpbnvmghosinxcpnpemusyu0csekpjfn8ujkzqvz5rz/jc45eaa/ihz8q2tt5t5</latexit> <latexit sha1_base64="lx9jo1tiosv219fjoif0od0tgys=">aaab/nicbvdlssnafj34rpvvdelmsaiusikcrqqgimsk9gftkjppbr06syag6gegf/hvlfuxk2/4sj/crkz0nazopxzd/feeygpdlrup7o0vlk6tl7qg5ube/s1vb2oyonic2j2wsewezieuebrqooac0sdcq0a2mv7nffqbtrbzd4uybh7jjjmacm7tsydckme1mw/s6kw1rdbfhfqclxctjnrodwtfns6teclkkhnt91yffso0ci4hqw4sa4rxkta39kihwd8tlg5o8ejyrhtboksqsrfidsfhozcwm7gtk8n/nelv7n9rmcx/ipifscepf8eqojxsldtbblab0jdygsvxyoichnmigcfprxbsxetlo1fxjz3y+szmndcxve7vm9evk2uygh5iicei+ckya5is3sjpwo8kseyyvz6lw6b877z+isu2yoyb84h98dfjy</latexit> Relativamente a W F V <latexit sha1_base64="lx9jo1tiosv219fjoif0od0tgys=">aaab/nicbvdlssnafj34rpvvdelmsaiusikcrqqgimsk9gftkjppbr06syag6gegf/hvlfuxk2/4sj/crkz0nazopxzd/feeygpdlrup7o0vlk6tl7qg5ube/s1vb2oyonic2j2wsewezieuebrqooac0sdcq0a2mv7nffqbtrbzd4uybh7jjjmacm7tsydckme1mw/s6kw1rdbfhfqclxctjnrodwtfns6teclkkhnt91yffso0ci4hqw4sa4rxkta39kihwd8tlg5o8ejyrhtboksqsrfidsfhozcwm7gtk8n/nelv7n9rmcx/ipifscepf8eqojxsldtbblab0jdygsvxyoichnmigcfprxbsxetlo1fxjz3y+szmndcxve7vm9evk2uygh5iicei+ckya5is3sjpwo8kseyyvz6lw6b877z+isu2yoyb84h98dfjy</latexit> nullo su pareti solide trascurabile su superfici d ingresso/uscita il contributo relativo alle superfici in moto delle macchine viene inglobato in W S <latexit sha1_base64="0dvvk4lu18jbzh5w2rrxokdzt8=">aaab/xicbvdlssnafj34rpvvdelmsaiusikcrqtgxmvf+4a2lmn0tg6dtmlmjvbc8cvc6sqdupvbxpgvtmiw2npwh3pu4d57glgkg6776swtr6yurvc2qptb2zu7tb39jokszahnixnpxsamskggjqil9ginlawkdippve53h0abeak7nmxgh2yixfhwhlbqd0yrpt1smn5mw1rdbbgf6clxslinjvrd2pdn8yqehvwyy/qeg6ofmo2cs8iqg8razpiutabvqwihgd8tts7ocwiyrjqgtywkhqi/eykljmfg0mgd6bes8x//p6cy4v/fsooefqpf+eqkkxyhatbbdar0idissvbyou5uwzrnccms6tmnhyqrypb/77rdi5bxhuw7s5qzcvy2yq5jackrpikxpsjnekrdqek4g8kwfy4jw6r86b8/4zuusumqpyb87hn4jilgu=</latexit>!3

4 <latexit sha1_base64="jl39exmglwv3zd8lv9y3iyxj+w=">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</latexit> <latexit 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sha1_base64="4vcwq4nqa5s7xaios6pom2g2tda=">aaachxicbvdlsgnbejz1gemr6thlybauiewgqs9kwivhccyrkhh6j504odu7zpqkcdlv8bp8cq968iexyp/4ibuwvedqqu66e7yiyutue67mzu9mzs3x1goli4tr6yw1tabnoynwiyivwguflcopmygsvj4ermewffy8q9pxn7rbo2vot6nuytdaiadqqayqreabfthxgqseoymibvdc8voztpmksw3rve8imuqm2pvyq7fxcc/pd4osmzhpve6aptd0ucocahwnq250butccqfartyie2gig4hig2m6ohqntnji+lfdu2qcgp0hcp+ete7xmjbnaoaj/rdicu7g9vlp7ntwmahhytqaoyuivxipikj4usmdllcnlfgisc8exipeycdbchkryeymq4c6+y5eh9/v4vavyrnlvxzvbltem8mqlbftsh3nsgnxykauzbhpsjj2wr/bk3dvpzovz+tu65eqzg+whnldpp1iiq==</latexit> Breve sommario dei principi di conservazione (3/3) Caso stazionario con ingressi ed uscite monodimansionali (1: ingresso, 2: uscita) w S w FV h 1 + U gz 1 = h 2 + U gz 2 q w S w FV dove, e sono calore e lavoro trasferito per unità di massa. Ipotizzando il lavoro delle forze viscose nullo e dividendo per per l accelerazione di gravità si ottiene una relazione che lega delle altezze (head) p g + U 2 2g + z in = Conservazione del trinomio di Bernoulli p g + U 2 2g + z U p + gz = const Le ipotesi sotto le quali è valida la conservazione del trinomio sono out + h attriti h pompa + h turbina 1. Flusso stazionario (ipotesi comune) 2. Fluido incomprimibile 3. Flusso non viscoso, non sono presenti attriti (ipotesi restrittiva) 4. Valido lungo una linea di flusso, linee di flusso differenti hanno costanti differenti!4

5 Riferimenti bibliografici 1/4 Frank M. White, Fluid Mechanics, McGraw-Hill Çengel-Cimbala, Meccanica dei Fluidi, McGraw-Hill Quartapelle-Auteri, Fluidodinamica Incomprimibile, Casa Editrice Ambrosiana!5

6 Riferimenti bibliografici 2/4 Serge Lang, Algebra Lineare, Bollati Boringhieri Stephen B. Pope, Turbulent Flows, Cambridge University Press!6

7 Riferimenti bibliografici 3/4 S. Volker, B. Weigand, H. Gomaa, Dimensional Analysis for Engineers, Springer G. P. Galdi, R. Rannacher, A. M. Robertson, S. Turek Hemodynamical Flows Modeling, Analysis and Simulation!7

8 Riferimenti bibliografici 4/4 A. J. Chorin, J. E. Marsden A Mathematical Introduction to Fluid Mechanics, Springer!8