BIOFLUIDODINAMICA · Serge Lang, Algebra Lineare, Bollati Boringhieri !6. Riferimenti...

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

Transcript of BIOFLUIDODINAMICA · Serge Lang, Algebra Lineare, Bollati Boringhieri !6. Riferimenti...

BIOFLUIDODINAMICALorenzo Botti, Alessandro Colombo

Lezione #9 - Breve sommario dei principi di conservazione

a.a. 2017/2018

Versione 0.3

d

dt

Z

CV⇢~U dV +

Z

CS⇢~U( ~Ur · ~n) dS =

X~F

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d

dt

Z

CV⇢ dV +

Z

CS⇢( ~Ur · ~n) dS = 0

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!2

Relazioni integrali per il volume di controllo CV in possibile moto uniforme con velocità relativa

dove è la velocità di trascinamento

Conservazione della massa

Conservazione della quantità di moto (equazione vettoriale)

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

Breve sommario dei principi di conservazione (1/3)

~Us<latexit sha1_base64="crYqgABPotZU/dwACCjBGNEREx0=">AAAB+3icbVBNS8NAFNz4WetX1aOXxSJ4KokIepKCF48VTFtpQ9lsX+vS3U3YfSmUkF/hVU/exKs/xoP/xST2oK1zGmbe482bMJbCout+Oiura+sbm5Wt6vbO7t5+7eCwbaPEcPB5JCPTDZkFKTT4KFBCNzbAVCihE05uCr8zBWNFpO9xFkOg2FiLkeAMc+mhPwWe+gObDWp1t+GWoMvEm5M6maM1qH31hxFPFGjkklnb89wYg5QZFFxCVu0nFmLGJ2wMvZxqpsAGaRk4o6eJZRjRGAwVkpYi/N5ImbJ2psJ8UjF8tIteIf7n9RIcXQWp0HGCoHlxCIWE8pDlRuRNAB0KA4isSA5UaMqZYYhgBGWc52KSV1PN+/AWv18m7fOG5za8u4t683reTIUckxNyRjxySZrklrSITzhR5Ik8kxcnc16dN+f9Z3TFme8ckT9wPr4BzjGVDg==</latexit><latexit sha1_base64="crYqgABPotZU/dwACCjBGNEREx0=">AAAB+3icbVBNS8NAFNz4WetX1aOXxSJ4KokIepKCF48VTFtpQ9lsX+vS3U3YfSmUkF/hVU/exKs/xoP/xST2oK1zGmbe482bMJbCout+Oiura+sbm5Wt6vbO7t5+7eCwbaPEcPB5JCPTDZkFKTT4KFBCNzbAVCihE05uCr8zBWNFpO9xFkOg2FiLkeAMc+mhPwWe+gObDWp1t+GWoMvEm5M6maM1qH31hxFPFGjkklnb89wYg5QZFFxCVu0nFmLGJ2wMvZxqpsAGaRk4o6eJZRjRGAwVkpYi/N5ImbJ2psJ8UjF8tIteIf7n9RIcXQWp0HGCoHlxCIWE8pDlRuRNAB0KA4isSA5UaMqZYYhgBGWc52KSV1PN+/AWv18m7fOG5za8u4t683reTIUckxNyRjxySZrklrSITzhR5Ik8kxcnc16dN+f9Z3TFme8ckT9wPr4BzjGVDg==</latexit><latexit sha1_base64="crYqgABPotZU/dwACCjBGNEREx0=">AAAB+3icbVBNS8NAFNz4WetX1aOXxSJ4KokIepKCF48VTFtpQ9lsX+vS3U3YfSmUkF/hVU/exKs/xoP/xST2oK1zGmbe482bMJbCout+Oiura+sbm5Wt6vbO7t5+7eCwbaPEcPB5JCPTDZkFKTT4KFBCNzbAVCihE05uCr8zBWNFpO9xFkOg2FiLkeAMc+mhPwWe+gObDWp1t+GWoMvEm5M6maM1qH31hxFPFGjkklnb89wYg5QZFFxCVu0nFmLGJ2wMvZxqpsAGaRk4o6eJZRjRGAwVkpYi/N5ImbJ2psJ8UjF8tIteIf7n9RIcXQWp0HGCoHlxCIWE8pDlRuRNAB0KA4isSA5UaMqZYYhgBGWc52KSV1PN+/AWv18m7fOG5za8u4t683reTIUckxNyRjxySZrklrSITzhR5Ik8kxcnc16dN+f9Z3TFme8ckT9wPr4BzjGVDg==</latexit><latexit sha1_base64="crYqgABPotZU/dwACCjBGNEREx0=">AAAB+3icbVBNS8NAFNz4WetX1aOXxSJ4KokIepKCF48VTFtpQ9lsX+vS3U3YfSmUkF/hVU/exKs/xoP/xST2oK1zGmbe482bMJbCout+Oiura+sbm5Wt6vbO7t5+7eCwbaPEcPB5JCPTDZkFKTT4KFBCNzbAVCihE05uCr8zBWNFpO9xFkOg2FiLkeAMc+mhPwWe+gObDWp1t+GWoMvEm5M6maM1qH31hxFPFGjkklnb89wYg5QZFFxCVu0nFmLGJ2wMvZxqpsAGaRk4o6eJZRjRGAwVkpYi/N5ImbJ2psJ8UjF8tIteIf7n9RIcXQWp0HGCoHlxCIWE8pDlRuRNAB0KA4isSA5UaMqZYYhgBGWc52KSV1PN+/AWv18m7fOG5za8u4t683reTIUckxNyRjxySZrklrSITzhR5Ik8kxcnc16dN+f9Z3TFme8ckT9wPr4BzjGVDg==</latexit>

~Ur = ~U � ~Us<latexit sha1_base64="c3TjnyxJ/SdL/H0m1iqpJryhLxU=">AAACD3icbVC7TsNAEDyHVwivAGWaExESDZGNkKABRaKhDBJ5SElknS+bcMr5bN2tI0WWCz6Br6CFig7R8gkU/AuO4wISptnRzK52d7xQCoO2/WUVVlbX1jeKm6Wt7Z3dvfL+QcsEkebQ5IEMdMdjBqRQ0ESBEjqhBuZ7Etre+GbmtyegjQjUPU5D6PtspMRQcIap5JYrvQnwuJm4+ipnp/PqmsQtV+2anYEuEycnVZKj4Za/e4OARz4o5JIZ03XsEPsx0yi4hKTUiwyEjI/ZCLopVcwH04+zJxJ6HBmGAQ1BUyFpJsLviZj5xkx9L+30GT6YRW8m/ud1Ixxe9mOhwghB8dkiFBKyRYZrkaYDdCA0ILLZ5UCFopxphghaUMZ5KkZpXKU0D2fx+2XSOqs5ds25O6/Wr/NkiqRCjsgJccgFqZNb0iBNwskjeSYv5NV6st6sd+tj3lqw8plD8gfW5w9YeJz8</latexit><latexit sha1_base64="c3TjnyxJ/SdL/H0m1iqpJryhLxU=">AAACD3icbVC7TsNAEDyHVwivAGWaExESDZGNkKABRaKhDBJ5SElknS+bcMr5bN2tI0WWCz6Br6CFig7R8gkU/AuO4wISptnRzK52d7xQCoO2/WUVVlbX1jeKm6Wt7Z3dvfL+QcsEkebQ5IEMdMdjBqRQ0ESBEjqhBuZ7Etre+GbmtyegjQjUPU5D6PtspMRQcIap5JYrvQnwuJm4+ipnp/PqmsQtV+2anYEuEycnVZKj4Za/e4OARz4o5JIZ03XsEPsx0yi4hKTUiwyEjI/ZCLopVcwH04+zJxJ6HBmGAQ1BUyFpJsLviZj5xkx9L+30GT6YRW8m/ud1Ixxe9mOhwghB8dkiFBKyRYZrkaYDdCA0ILLZ5UCFopxphghaUMZ5KkZpXKU0D2fx+2XSOqs5ds25O6/Wr/NkiqRCjsgJccgFqZNb0iBNwskjeSYv5NV6st6sd+tj3lqw8plD8gfW5w9YeJz8</latexit><latexit sha1_base64="c3TjnyxJ/SdL/H0m1iqpJryhLxU=">AAACD3icbVC7TsNAEDyHVwivAGWaExESDZGNkKABRaKhDBJ5SElknS+bcMr5bN2tI0WWCz6Br6CFig7R8gkU/AuO4wISptnRzK52d7xQCoO2/WUVVlbX1jeKm6Wt7Z3dvfL+QcsEkebQ5IEMdMdjBqRQ0ESBEjqhBuZ7Etre+GbmtyegjQjUPU5D6PtspMRQcIap5JYrvQnwuJm4+ipnp/PqmsQtV+2anYEuEycnVZKj4Za/e4OARz4o5JIZ03XsEPsx0yi4hKTUiwyEjI/ZCLopVcwH04+zJxJ6HBmGAQ1BUyFpJsLviZj5xkx9L+30GT6YRW8m/ud1Ixxe9mOhwghB8dkiFBKyRYZrkaYDdCA0ILLZ5UCFopxphghaUMZ5KkZpXKU0D2fx+2XSOqs5ds25O6/Wr/NkiqRCjsgJccgFqZNb0iBNwskjeSYv5NV6st6sd+tj3lqw8plD8gfW5w9YeJz8</latexit><latexit sha1_base64="c3TjnyxJ/SdL/H0m1iqpJryhLxU=">AAACD3icbVC7TsNAEDyHVwivAGWaExESDZGNkKABRaKhDBJ5SElknS+bcMr5bN2tI0WWCz6Br6CFig7R8gkU/AuO4wISptnRzK52d7xQCoO2/WUVVlbX1jeKm6Wt7Z3dvfL+QcsEkebQ5IEMdMdjBqRQ0ESBEjqhBuZ7Etre+GbmtyegjQjUPU5D6PtspMRQcIap5JYrvQnwuJm4+ipnp/PqmsQtV+2anYEuEycnVZKj4Za/e4OARz4o5JIZ03XsEPsx0yi4hKTUiwyEjI/ZCLopVcwH04+zJxJ6HBmGAQ1BUyFpJsLviZj5xkx9L+30GT6YRW8m/ud1Ixxe9mOhwghB8dkiFBKyRYZrkaYDdCA0ILLZ5UCFopxphghaUMZ5KkZpXKU0D2fx+2XSOqs5ds25O6/Wr/NkiqRCjsgJccgFqZNb0iBNwskjeSYv5NV6st6sd+tj3lqw8plD8gfW5w9YeJz8</latexit>

~Fpres = �Z

CS(p� patm)~ndS

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Conservazione dell’energia totale

!3

Breve sommario dei principi di conservazione (2/3)Relazioni integrali per il volume di controllo CV in possibile moto uniforme con velocità relativa

dove è la velocità di trascinamento~Ur = ~U � ~Us

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d

dt

Z

CV⇢

ei +

~U2

2+ gz

!dV +

Z

CS⇢

h+

~U2

2+ gz

!( ~Ur · ~n) dS

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= Q+ WS + WFV<latexit sha1_base64="XqfyyNoO6h5PQNcokuZALbm6QHI=">AAACE3icbVDLSsNAFJ3UV62vqEsXDhZBEEoigm6UgiAuW7QPaEuZTG/r0MmDmRuhhCz9BL/Cra7ciVs/wIX/YhKDaOvZzOGce7hzjxNIodGyPozC3PzC4lJxubSyura+YW5uNbUfKg4N7ktftR2mQQoPGihQQjtQwFxHQssZX6R+6w6UFr53g5MAei4beWIoOMNE6pu7Z92Bj1E9PszeVty//mHRZTPum2WrYmWgs8TOSZnkqPXNzyTOQxc85JJp3bGtAHsRUyi4hLjUDTUEjI/ZCDoJ9ZgLuhdlh8R0P9QMfRqAokLSTITfiYi5Wk9cJ5l0Gd7qaS8V//M6IQ5Pe5HwghDB4+kiFBKyRZorkTQEdCAUILL050CFRzlTDBGUoIzzRAyTykpJH/b09bOkeVSxrYpdPy5Xz/NmimSH7JEDYpMTUiVXpEYahJN78kieyLPxYLwYr8bb92jByDPb5A+M9y8QPJ5q</latexit><latexit sha1_base64="XqfyyNoO6h5PQNcokuZALbm6QHI=">AAACE3icbVDLSsNAFJ3UV62vqEsXDhZBEEoigm6UgiAuW7QPaEuZTG/r0MmDmRuhhCz9BL/Cra7ciVs/wIX/YhKDaOvZzOGce7hzjxNIodGyPozC3PzC4lJxubSyura+YW5uNbUfKg4N7ktftR2mQQoPGihQQjtQwFxHQssZX6R+6w6UFr53g5MAei4beWIoOMNE6pu7Z92Bj1E9PszeVty//mHRZTPum2WrYmWgs8TOSZnkqPXNzyTOQxc85JJp3bGtAHsRUyi4hLjUDTUEjI/ZCDoJ9ZgLuhdlh8R0P9QMfRqAokLSTITfiYi5Wk9cJ5l0Gd7qaS8V//M6IQ5Pe5HwghDB4+kiFBKyRZorkTQEdCAUILL050CFRzlTDBGUoIzzRAyTykpJH/b09bOkeVSxrYpdPy5Xz/NmimSH7JEDYpMTUiVXpEYahJN78kieyLPxYLwYr8bb92jByDPb5A+M9y8QPJ5q</latexit><latexit sha1_base64="XqfyyNoO6h5PQNcokuZALbm6QHI=">AAACE3icbVDLSsNAFJ3UV62vqEsXDhZBEEoigm6UgiAuW7QPaEuZTG/r0MmDmRuhhCz9BL/Cra7ciVs/wIX/YhKDaOvZzOGce7hzjxNIodGyPozC3PzC4lJxubSyura+YW5uNbUfKg4N7ktftR2mQQoPGihQQjtQwFxHQssZX6R+6w6UFr53g5MAei4beWIoOMNE6pu7Z92Bj1E9PszeVty//mHRZTPum2WrYmWgs8TOSZnkqPXNzyTOQxc85JJp3bGtAHsRUyi4hLjUDTUEjI/ZCDoJ9ZgLuhdlh8R0P9QMfRqAokLSTITfiYi5Wk9cJ5l0Gd7qaS8V//M6IQ5Pe5HwghDB4+kiFBKyRZorkTQEdCAUILL050CFRzlTDBGUoIzzRAyTykpJH/b09bOkeVSxrYpdPy5Xz/NmimSH7JEDYpMTUiVXpEYahJN78kieyLPxYLwYr8bb92jByDPb5A+M9y8QPJ5q</latexit><latexit sha1_base64="XqfyyNoO6h5PQNcokuZALbm6QHI=">AAACE3icbVDLSsNAFJ3UV62vqEsXDhZBEEoigm6UgiAuW7QPaEuZTG/r0MmDmRuhhCz9BL/Cra7ciVs/wIX/YhKDaOvZzOGce7hzjxNIodGyPozC3PzC4lJxubSyura+YW5uNbUfKg4N7ktftR2mQQoPGihQQjtQwFxHQssZX6R+6w6UFr53g5MAei4beWIoOMNE6pu7Z92Bj1E9PszeVty//mHRZTPum2WrYmWgs8TOSZnkqPXNzyTOQxc85JJp3bGtAHsRUyi4hLjUDTUEjI/ZCDoJ9ZgLuhdlh8R0P9QMfRqAokLSTITfiYi5Wk9cJ5l0Gd7qaS8V//M6IQ5Pe5HwghDB4+kiFBKyRZorkTQEdCAUILL050CFRzlTDBGUoIzzRAyTykpJH/b09bOkeVSxrYpdPy5Xz/NmimSH7JEDYpMTUiVXpEYahJN78kieyLPxYLwYr8bb92jByDPb5A+M9y8QPJ5q</latexit>con:

• entalpia

• derivata nel tempo del calore fornito al sistema da una macchina, e.g., una pompa

• derivata nel tempo del lavoro fornito al sistema tramite macchine

• derivata nel tempo del lavoro degli sforzi viscosi

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Relativamente a

• nullo su pareti solide

• trascurabile su superfici d’ingresso/uscita

• il contributo relativo alle superfici in moto delle macchine viene inglobato in

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h1 +U21

2+ gz1 = h2 +

U22

2+ gz2 � q � wS � wFV

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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

Conservazione del trinomio di Bernoulli

Le ipotesi sotto le quali è valida la conservazione del trinomio sono

Breve sommario dei principi di conservazione (3/3)Caso stazionario con ingressi ed uscite “monodimansionali”(1: ingresso, 2: uscita)

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)

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U2

2+

p

⇢+ gz = const

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✓p

⇢g+

U2

2g+ z

in

=

✓p

⇢g+

U2

2g+ z

out

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Riferimenti bibliografici 1/4

Çengel-Cimbala, Meccanica dei Fluidi, McGraw-Hill

Quartapelle-Auteri, Fluidodinamica Incomprimibile, Casa Editrice Ambrosiana

Frank M. White, Fluid Mechanics, McGraw-Hill

!5

Riferimenti bibliografici 2/4

Stephen B. Pope, Turbulent Flows, Cambridge University Press

Serge Lang, Algebra Lineare, Bollati Boringhieri

!6

Riferimenti bibliografici 3/4

S. Volker, B. Weigand, H. Gomaa, Dimensional Analysis for Engineers, Springer

!7

G. P. Galdi, R. Rannacher, A. M. Robertson, S. Turek Hemodynamical Flows Modeling, Analysis and Simulation

A. J. Chorin, J. E. Marsden A Mathematical Introduction to Fluid Mechanics, Springer

Riferimenti bibliografici 4/4

!8