Lecture-7 - APPH 4200 Physics of Fluids Fluid Equations of...

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Unformatted text preview: APPH 4200 Physics of Fluids Fluid Equations of Motion (Ch. 4) September 27, 2011 1.! ! Integral E.O.M.s Moving/Fixed Volumes 2.! Internal Energy 3.! Bernoulli’s Principle 4.! Equations in co-rotating frames 5.! Examples 1 Stokesian Fluid 2 NS Properties (incompressible) ∂u + u · ∇u = g − (1/ρ)∇p + ν ∇2 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 3 Fixed and Material Volumes 4 Identities For Conserved Quantities 5 ! f. '6 ~ .. Q ., :i u. r£ .. ~ Q "; ~ .. '" ~ :i -J 0 ( \i ~ "l. ~ l I) I:: l "' ': .) '\ ~ \ù ~l ~Q ¿j ~ -J v ~ ~ ( -- l.ù 4: 'f ~ .. ~ to 'A t 1 Example: Momentum Il~ r, .. lb .. I~ ~ ~ ~ ~ ~ II bl-+ ci q 'L ~ -i ~ '. \l~ (\) .. IQ) ~ II l-. , -t Ca \i IJ~ . \~ ~ ~ct 04 I~ Q- :i II ~ ~ Ci~ Q. ~ ~ ~~ ~ a\ 6 \t ( ~ .J 0 :i ~ \U .~ .. \. ll\V i i\~ . I~ + (~ + l~ ~ (~ ~ I' IJ I~ 1:: ~ '(~ .. (~ -l '3 ) \~ ~ ~ Q ~~ tt ~ L ': ': ~ ;-~ ~ ( Q .t \L l~. '¡ ~ ~ l "" + ~ ~~ (~ l\~. , ~,~ ~\~ ~ '- ~\~ , :: ~ . ~~ 4 Q. ~ r; ~ ~ L -\ :i + b\~ oa ~ 6 ~ \S 'i i :i .. ~ :i "" ~ .. ) o ~ \l ~ i\ (' ") 4t l~ l' l ~ L4 ~ \l -J \: \J ~ ~ \Ù (; l: '" 1 o~ l' ui t V\ "t . ") \U ~ r .. \l, l. ~ ': ~J Example: Mechanical Energy . c: 0- ., 1"3 . /0- l' 'l -I r(~ + to lb '- i IIN b~ ~ . i r~~ ~ ~ ií ~~ \~ ": ~~.. II ~ N ": Q. -(Ñ ': -t ~ ~ °lD ~I\V \ . I \T . l\ ;I~ ,, ': N /" ~ ( \j ~ ,i (,: \ ~ QQ c\ã c. ,At L~, ". i:i ~ i. ~ :) .J o ": ~ \U - )( lL ~ t; II "' (~. I\ L,"~ (-- ~ ,t b .. q" i~ ~ I~ ~ c:I ~N ~ ~ \Û L; '- \I\JC\~ l~ ~~ \-. ~ (\~ t UN I~ \~ tJ ,\~ . i~ I~ .. i\\ I ~ ~ (V t1\ ~ \(\~~ oJ ") \- )( ." t1 n\ .. :: -\(' l . '" ~.. t' + ~j' .'" \I .j .~ \i ( 0- ~ lJ " \ l r- . '\ "'.. . oJ '- ~ :) '\ N i. '. . l"- r- r" ~ , . ljv ~v ;;: ~ a ~'\ ~, ~ (; ~\X'\ .J + t: ~ l ~~ " N :: -t~ ": ~ ~~ ~ ~(~ I~ -i,. "- ~ ('~ /' + (t) . -,Q. '" ~ , ~ II Q, .. i l~ .. tõ N ~\ i. LJ ~ ': -\rJ .. Q () ~ ~("\~ 7 ! ) v \ø ~ -. \V s tL " ~ J Q :i J t lè ~ ~ '0 L r~ iJ l Fixed or Control Volume for Momentum-Force Calculations "' . ~ (~ l:i ~ Ct: ,l :J "0 llfJ b. ~~J ~ + ~ ~ ,~ Q. :- ~':i ~ " ": ~ Ij ~ ~\~ ( ~~ 1 co 3f ~( Il '\ \) ~ ~~ ~ 1i~ L;~ ~ l- ¿ í~ oQ ~ i-~ l ~" ~~ll r. (.... ". ~ l-¡- ~ ~ l ~~ :i il~~ ~t~ 5 Q l \J ~ ~ + ~ i . ~ l l. ~ .. ~ " Q ~ ~ l 1 '3 ! ~~ ~ ~ ~ \J )- 'J 1 ~ 1 .. '4 ~ \- j t In l l l. ~\~ ~ J. ~~ '- \l \l ~ L +- /; i- lJ (. t, Q~ ~ 'l '; '" )( ~ 'l :) ~ ~ t .. "' ~ l . lt ~ ,~ J u. t t l " \l \I ~i~ J ") t ~ t ~ V\ ~, l ~ ~ ,~ ~~lt ~ oJ '" ) 1~ J , '~ l tJ ~ ~ I \ \ \ \ -~~r -- - - 8 Example 4.1 9 Integral Conservation Laws (Fixed Volume) 10 Example 4.1 (Solution) 11 Energy Conservation 12 Mechanical Energy 13 Total Work Done by Stress 14 Mechanical Energy Density for a Stokes Fluid 15 Conservation of Total Energy Density 16 Equation for Internal Energy (Temperature) 17 Internal Energy Equation 18 Entropy (Disorder, Direction of Time) 19 Summary of Fluid Dynamical Equations 20 : ..t ~~ Ql \~ -i rl J ui .. ~ ~ ;¿ .J ~v ~ uJ ~) oJ " 00 \U ø ~~ Bernoulli’s Equation ~ .. l! î ~ li ) F ~ ~~ D l 1\ J~ ~ ~ /~ ~ IJ (-: (tJ Q0 ~~ (Euler Equation) tt ~ , ~ ~ ~ i u .. II ~ ~ I:. \J ~, .. ,~ r) £ f. ~ ~~ Q. ~ l' - ~( \L CJ ~ Q.I~ J~ ~ (t ~\ ci ~~ (J \)\~ q. --~ rI ~ -,~ ~ l)~lo. ~ ~ '" " ). ~ ~ ~ , tu U lL -1~ Vl + Q '" t- \ (b ( :: f -\ l :: ~ t~ 1 \1 ~ ~ ~l ~ ~ + rJ ~ ~ 't J \( f. ! "" \J , ~ \1 t ) 'l ~ Vi ~~ ~ '\, ~ .. ~ ~ ..\!\ C' L \l ',. \ll ;: '1,. Q u- - rt tb J.i\,U '\ I~ " '- t. ~ \U l~ " L ~ -i ~ ~ '~ + (:1 \~ Cl "6 l.. J0~ ~ \f "" L ~\~ w ~ ,," ~ Q .£ ï.. t '- -£ L -. \h "\ ~\~ " ~ r" ~ :l ~ to ~ ~ri ~ ~ II \ /~ Il 'e) t:! y. ~ l': tV ~ -. + .. :: i~ -(N ll ~ l :i (~ \. ~ n (.) l ~ 'v +- 1:1 \ ~ n (\ ~ ~ (- ~ Q 21 ~ i , ~ .. i tU ~ il ~ l ~ ~ ~ ~ Vl 1 S0 ~~ ~ i; ") .J .. .~ .. '1 tu I: ~ Bernoulli’s Equation (Conservation of Energy) ). '" t: \ù ~ lu ~ , '3 (.. ~ ~ $ "J ~ ': \U l~ . D i ~ I~ " lllJ '. /~ .. i :J . l ri ~ ,) /' ,. :r l"l + ~ a\a ~ ~ v j; t f ~ "- \1 ,. I.. ~ a :i .. l. .. 'J ~ ~ 't tJ l ~ .. '" ~ It ~ - 0 ~ , ~ -l Q ~ t r ¡. 0# V\ 0 \o '" .. 0 ". i. ~ ~ ~ 3 'C 'l) ~ ~ u "" -f ~ ~ ~ - v ~ '" V\ ~ .. - j ~ ~ .. . "\ , -. lr I Q. 1 .. ~-J 'J If ,~ . b 0 Q ~ V\ \U l 0 .. ~ ~ ~ \t ~~~ t :i J -; ~ ~ l! ~ ~ ~ ~ /.. î t ~ ~ It ~ , . ~ I ~ ~ l-: . ( II .. ~ ~ " ~ Q. (~ 4- (L u l~ to '; D . b "- ~ r- ~ '~ ( :i ~ V ~~ ~ (~ ~ r" ( ~ * I~ 'Q. '- t~ ~ I~ .. ~ t I~ , A~ . ~ ~ .. :: O\~ ~, ~ \! '- i- -1~ id .. 1- Ir u ( h t. - r rl :J .. r& + J.. ~ ~ ~ is u. U ~ -C lii UJ ;. (l 1 a. l 0" + Ll L. . l\: (~ ~ I Hi CJ UJ L n-~ c; + .. ,. ~ -tÑ 'j II ~ ). VJ ! 'U ~i 1- t jl t: J o~ ~ t'1 -" i (. "" )- 0 (t VI rt ( ~ 22 Bernoulli’s Equation ∇B = u × Ω ￿ 12 dp B≡ u + + gz 2 ρ 12 p or u + + gz 2 ρ (Steady Flow) = constant, = constant 23 Lift 24 Pilot Tube 25 ..lv' - ~ .. ~ \) ~ ~ i - 3 ,,. ": .: ~ J -) ~s J ~ ~ 1: 11 iL O 1 '; 't " t~ -i \J IJ 'l tJ ~ ø '" t. ~ ~l ~ ,.. '" t 'l '" .. ~ 0 .. \i 'c ~ ~ -. Q' ÇL N~ T -- Q G- I. -i~ 4- II ~ 'l l ~ ~, --~ .. ~ -c f' '5 Ci l :r'; (o. .. ll J ~ Q. OJ n VJ ~ ~ ~. IU tf ï J ~ u. "' I.. .. "t t iC ~ \I 'C tJ "0 l ~t a: l: ~ .. :~ t (\ 't ,~ '\ \A 'C Vo 1\ ( t ¿ \I 1 '0 .J lL \t -i 'J .. i!, d \C t v i VI \:: l4 I! ~ ~ t ~ ~ ~ ~ ~ - l \t ~ t "" q: ~ ~ ~o I- a. q: ~J ("- ~ ~i ~ ~ \ '! l\ J~ ..' . . ~ /\t ~ ') "' Q ~ "" 1. ~ ci cl l Q.~ i- Hole in Bucket ~ ,&) ~ ~ '" ~ ~ '3 ( . i: d;~. , I ~ ~4 26 "" l ~ 1\ t. ~ ~ VI \ '; f:: -t ) I' j 'c ~ l. ~ ~ t ~ ~ l .. l- i ~ ~~ cl t ,C J l \~ i ~ ,( What are E.O.M. in Co-Rotating Frame of Reference? 3 ~ \1l F .. q, "" l, \J ~ (: " N ~ t .. ~ t" 0 - .) '" " ~ () .. ~ \J '" ~ V .. tJ ~'" II \. "l ~ \I .. ~ ~ tr '- o :: ôo . ~ l l .. 'l \0 "' ~( 3~ ~ I~ II ~ .. ~ ~ ~ , \ i. t- ~ \U J'" C:\l 27 Gaspard-Gustave Coriolis 1792-1843 “On the equations of relative motion of a system of bodies”, published 1835 28 ~ ~ le ~ .g .i t ~ Ll J ~ 1-: ~ u. 1) æ ( ~ .. V\ 'l Small Rotation of Frame in δt "t ~ ~ u. -i ~ t 0 .0 "" 'f 3 ~ ~ '\ ~ tt ~ Ir C ~ Q ~ i ~ ~ ii fa. ~ \ i ,. .G\ - ~ 'l Ai .... .. .... ) .: J.. ~~ n. o l ~ l1 ~ J. ~~ ~ l (~ -~ \ij L q, il tu ~~ \1 ~ \i ~ ~~ "- 10- .. ). ).. ~ " lL 0- ~ 1- .. ~ 1 .. III V W 'I ~ ~ ~ I- ct \ 0- C: , ,. ~ .~ ... a. "' ~ I iJ l: ~ ~ .:. ~ ~ ~.~ 0. . .. "-. I~ '\ i '" - .. -i t. Ii ~ ". ~ lL, (lJ .. ,- ~ of .l .. .¡ "" I, - -l --~ \L " \o 0- 1 ~ tt ~ ~ ;l~ ". 'i :- Q! . "" 0- ,. ~".. .., V I' ,. ì"' u. 'oJ Q. " " 'U ~ L, ~ ~ i--., 0 f)~ l III r! ~ ~ C: ~ .. ~ ~ 'i ~ l . "" '.. ~ ~ , .) ~ " of - (J 'c ~ 1: ~ (; ~ 3 1 () ii /' ~ Q U ! "l ~ ~ \\ ~ '" VI , cD Q "" .. 0 .o 1 ~ \I 1\1 ~ \) -. ~ 'J ~~ " "' (I ~.. oJ ~ \ê \A ( \) ,~ t .. ~ i: J 29 ~ ~ ~ \il '0 1 \ t) t¡ j C: () \ .. h= l L \J \L ~ 1I \. ~ ~ ') Co-Rotating Frame ~ a. ~ "" , oJ ~ -l 0. J .. J ~ J 0- ~ '3 + ~ \l CL "" ~ ~"J L ~ Oi VJ .- t ~ ~ ~ ~ ~ fQ l~ I.. ~ i. ~ i.c '- (?' \ ,, ~ ~ x. (~ x i i= ;. ~ I~ X- l) .. ~ l~ \~ ~ +- ~ l,\ x. '( '~ 1- ~ &. ~ .. f ~~ ~~ " ~t J 'f ~( \i J tt. ,~ II ~i~ \L \l II lls\ ~ ~ ~\~ ~~ ~ ~~lL/~~'t '~ ~ ~ J. 'f 1-ic) t, J) ~ \~ z 1- l~ ') t~ ~~ ~~ 3~ 't .. ~ \L' I~ ~ ~~ I :J \1. h ~ \Ù I- I(L u IL .. .. \U ~ h ~.J \ ~ -l \W ~ \l~ () t if .: ~ Ii ;~ \l L '- vS , .. t 'J ~ . "' ~ ~~ - "¥ ~ r\ \ ~ a.~ ~ I ~ ~ t :) "' (continued) 30 ..' ~ .. '4 \i ~ "\ .. ~ /" i~ 'J I~ v '" /~ 1- ~ J~ 'L (, ~\~ \L 'i\t -r ~ J~ ~ ~~ ~ u ~i ~ Ê 0 à l ~ .. ~ "" at t: \J 1 ~ a .. ~ ~ -.. Centrifugal and Coriolis Forces .J ," ,v~ ,.( ~ 1"\ . "; (l. .. '- 1"\ ~ (~ '- ~ l~ h ,, I~ I~ ~ (u t1 .. J - Q () (: c: rJ 0. ~ JL l '- I' I~ l h, 1/ f~ -l \\ ~ .. (ià \l \~ ~ Ii) It \) l i.. lù ~ Q " ( '0 "" r .. ~ -i ) u .. q .f ~~ c: :z \\ I) II I C:.. i ~ -( )( I~ ~ ~ 0 (! 'l Ie¿ I -l ~ (~ i. I :J ~ r-~ - IV (\ ~l~ \i ~\~ Ç) \. ~ ~ ~': oQ~ ~J. 'I ~ ~ ~ 31 ~ ~ ~ " ,~ 1l ~ ~ - a l \. 'l ~ CV - .J l. 'S ~ , & ( .. :; ~ ~ , ~ ~ ~ Incompressible Navier-Stokes in Co-Rotating Frame I~ rv ~ ~ ~(~ 'l- ~\~ i ( (ï (l I~ I ~~ c: d: -J .. r ;. \: \\) ~ \ 0 li ~ I t: "'~ b l t: I~ :i 'o "" \J oT .. ~ 'Q ~ .. il :~ i' " . \~ ( "" l~ ¡ ~ \l ~ i b(~ L Ii l~ + ,:4 l (" c; '~(~ 32 ~ ~ ~ 1 '- o \ ~ I"\ l(\ ~ " ~ i ~ t. VI ~ l" Co-Rotating MagnetoSphere -- (! l\n ii I (, ~ ~ I t. '- J l~ ~ 1 f ~ ~ ~ ~ I.. ~ ~ \ ~ ~ c. () "' \L '\. IU \r ~ ).. \t( c: ~ l" II ~ ~ iij \\ I" C: r' J \o 'l ~ \n .L ~ ~ .. V) ~ ~, Ii ~ l Q .. ~ ~~ \c \i f v t ~ \. ~ t \J lø o It ~ .. (l ~ I; ,I L ~ 'å r ~ ~C: '\Y \r I \L ~ ~ .. ') ~~ ~ ~ \~ ~ 1: J~ 1 J l- ~ Ib J ltJ -i \A 1J) Ii tr ~Itv ~~ J ~ ). i: , ) t rv .- . I: ,. '\ ') l: ~ .. \I \d .. \L \L \L G ') \~ f ~-l ~~ ri \ ~ ~~ ~\ '- C; ~ ~ 1- t~ ~ "- ,. ~ (¡ 1.. t -,. '"~ f J ~ il l~ 33 (.. J \i \J ~ 3 I~ "' ~ " ~ ie. J o ~ l f- ~ · ~ .l l I~ ~ V) ~ .. .r ! l \ti Q \= Q Q) ~ o ~ ~ l "" æ. ~ Coriolis Force \~ ~ -i .. "' '" \ l~ L~ "- "J .J ~ i: ~ ~ l)' , 'j t" ~ t: .. . t~ ~ '\ ~~ VI ~~ ~ a: i \QO % oJ ~ Q o ~G ~l j .. "' aÙ .. ~~ .. ~ '- ~ ~ .. V\ 1: " ~ l. 1: ~ ~ ~ Q L ø (\ v \"" .. ~ ( oJ "' J '0 - ~ 'J ~ -- V\ \J ~ d 3 ~ 1- - l l ~3 1 .. o .~ ': v ~~ t: ~ i ( .. ~~ "" () (" ~ t" "' t~ X~ .. J \ ~ \~ '- L, ~ vt ~ ~ \~ 'l J \) ( :i -t lU I~ n \) 34 ~ t ~ .. \t -i l ~ S. ! \1 ~ Angular Momentum , ~ r: ~ ~ 'L \\ .0 .l .. " :) 1: \J '\ ~ '- " ~ \i V ~ ¿ ~ l 1 ~ il 1) ( .. '" J~ 'i I -l \ l e: ll) Ii t ~ I-J u ~ ! ) \r i \" ") ~ ~ . ~ (l \ \- \~ l\ I~I =( "" .. )i ~ ,~ t :J ~ (ù Q VJ \:i (\ '-1 .. I~ ~ ~ n\J l. ~ (~ ~ J~~ + .J ~ ~ ~ ~ ( ¡~ ~ '- t~ '\ ~\.. ~ ~ ~ i J 0. ") '" ~ :- ~ "Á Q. "- l :: ~ ~ :J l. ~~ ~ ~ ~ q: 'c; C: ~ ~ ~ "c ~ r- () ~ ll- \l ~l~ , ~~ \t ~ q, ~ ~ -l ( \\ I C: \ \- II L- ~ ~~~~ ~ Ie: l) tJ ~ ~ J J ~ \l ~ ~ 'J \L 35 Example 4.2 ~ r6 -i .. c: ( ~ W "Q ~ ~ ') l~ "" * ''i ) ~i " .. ~ (l .) ~ ~ ~~t t ~tJ ~~~ o IU .( 1 ~ t le ~ ~ ~! Q. if ~ I~ . d ~ :J ~ ~ ò' I:) '- "" ~ l ") ~ I~ v ~~ 'C~ i' \_ nJ VI Ç) ct (" ~ ~ '\ ~ (~ Ç\ \! - J ~ ;i c: V) 1.. .k ~ '1! ~ .. G: l" . ¡. V i ~ .. e: 'l: -: \I ~ \0 ~ ~ ~ ~ 1: ~ .J -J ~ 1S Q \A \J ~ l" ~ ~ N " 36 Ch. 4 Exercises • • • • • 4.3/Angular momentum & stress tensor 4.4/Differential form of E.O.M. 4.5/Stokes’ expression for stress tensor 4.8/Rocket equation 4.9/Trust generated from a “propeller” 37 2. In Section 3 we derived the continuity equation (4.8) by staring from the inte- Exercise 4.3 gral form of the law of conservation of mass for afied region. Derive equation (4.8) by starng from an integral form for a material volume. (Hint: Formulate the principle for a material volume and then use equation (4.5).) 3. Consider conservation of angular momentum derived from the angular momentum principle by the word statement: Rate of increase of angular momentum in volume V = net influx of angular momentum across the bounding surface A of V + torques due to surface forces + torques due to body forces. Here, the only torques are due to the same forces that appear in (linear) momentum conservation. The possibilities for body torques and couple stresses have been neglected. The torques due to the surface forces are manipulated as follows. The torque about a point 0 due to the element of surace force imkdAm is J EijkXjimkdAm, where x is the position vector from 0 to the element dA. Using Gauss' theorem, we write this as a volume integral, ¡Bijk-(XjimÜdV = Bijk . 8imk)dV 8 ¡ (8Xj -imk +Xj- V 8xm .v' 8xm 8xm Exercise.ç = Bijk r ijk +8imk) dV, lv ( x j 8xm 1 where we have used 8xjl8xm = 8jm. The second term is Ivx x V. 'tdV and combines with the remaining terms in the conservation of angular momentum to give the left-hand Iv x x (Linear Momentum: equation (4.17)) dV = Iv Eijkijk dV. Since side = 0 for any volume V, we conclude that Bijkikj = 0, which leads to iij = iji. 4. Near the end of Section 7 we derived the equation of motion (4.15) by staring from an integral form for a material volume. Derive equation (4.15) by staring from the integral statement for afied region, given by equation (4.22). 38 -0 I~ ~ -C U . :r :x Y. \J oJ VI v " .. ~ ~ t, \c F ~ ", l~ U . 'IJ '- ~ do ~~ ii ~ "" 'C: ~r ~~ t~ ~t ~ ~ l~ ~. ~( ~ li~ " "' -: \\~ '- ~ ~'" "" OJ . .. . .. ~ 0'; \L ( '- \L ~~ ~'t ~ . oJ \~ \1. ~ \) \è 1: ~ ~ '" \\ 1- ~~ CJ \.c \-; vi ': £~ ~ '1 ~ oJ t\ ~ lO \. ~ .0 VI Vl i. v, ~ f l,y ~ Exercise 4.3 .: ~ r( ('. ~ :J ~ \) ~ iJ r .) ~ C1 . -- ') + ~' (r '- (v.. L; ~ ~\~~ ~ (E(~( J -\I oJ ., ~ .~ __ \U-., ~ ('-.,"" :i ~ -- 1 '¿ i.. l- " 1: Il .t L. ~ I~ 't. 'I ott ~" j .: ~ ''' ~~ ~ ..:') J (-:-' u o C lì ~ .~ N-S i '~ " "" ~ r~ '- a ç: \0 ~ ~ ,j (V~ II () ii \1 ~ '-'J ~0 '- l' A\ ~ f i .. .' ~ )( ~\ t ~ c2 ~ l 39 where we have used 8xjl8xm = 8jm. The second term is Ivx x V. 'tdV and combines with the remaining terms in the conservation of angular momentum to give the left-hand Iv x x (Linear Momentum: equation (4.17)) dV = Iv Eijkijk dV. Since Exercise 4.4 side = 0 for any volume V, we conclude that Bijkikj = 0, which leads to iij = iji. 4. Near the end of Section 7 we derived the equation of motion (4.15) by staring from an integral form for a material volume. Derive equation (4.15) by staring from the integral statement for afied region, given by equation (4.22). 5. Verify the validity of the second form of the viscous dissipation given in equation (4.60). (Hint: Complete the square and use 8ij8ij = 8ii = 3.) 6. A rectangular tank is placed on wheels and is given a constant horizontal acceleration a. Show that, at steady state, the angle made by the free surface with the horizontal is given by tan e = a I g. 7. A jet of water with a diameter of 8 cm and a speed of25 ml s impinges normally on a large stationar flat plate. Find the force required to hold the plate stationar. Compare the average pressure on the plate with the stagnation pressure if the plate is 20 times the area of the jet. 8. Show that the thrst developed by a stationar rocket motor is F = pAU2 + A(p - Paim), where Paim is the atmospheric pressure, and p, p, A, and U are, respectively, the pressure, density, area, and velocity of the fluid at the nozzle exit. 9. Consider the propeller of an airplane moving with a velocity UI. Take a reference frame in which the air is moving and the propeller (disk) is stationary. Then the effect of the propeller is to accelerate the fluid from the upstream value 1J1 to 40 (ê .: :t X :: U -è il ~ \J .. - ~ Exercise 4.4 !b J Q f~ l~ :J -L t:~ l-i '(~ ~ l ~ ~ ~ Io l" It :i ~t l '" .. l( + ul 0 ~ ~(~ " ~l~ II l~ ~ J -t .. + :: h . ii \J ~ ~ (i L~ -~ ~-t,:;~ l"\ ~ :: ~ ", ~ ~ I :: lJ r "' ' ~ 'u ~ ~ -l 1: ,~ Q. l'- "" \A ~ ~ ~ V\ ~ "- ~ '~ ~~ ~ ~'" 'l tP .. 12 \ ~ ~~ ~ ~ (\~ ~ &-~~ ~ ( ~ :" / t. ,. ~ '- )( l4 "" ~ " ~ t i ~ C\ ~ l-A ~ l '\ ~ tL ~ g 'u ~ 0 ~ \e u i. l( l: ~ \i l ~ r~ ~ tL ~ (0 ~) .. ") ~~ ul ") \eI - ") :s D . \ i . 'It "" t "J ") ~ .. i ~ ( ) 'i: 1': I t- Q. + t~ ~ 1l(\I ii ~. ~ tb 1:r1;J + L :: ri (\ ~~ \) 4- l:) h . ~ i~ ~ .l Q; Ç) íD . F- d \~ i"" -: ~ Q) i. ~ Q . \, l\\~ (b + (\n ~ Il l~ \ ": ~ I~ '- ~ "' n (' I:\U ~ . .. 41 side = 0 for any volume V, we conclude that Bijkikj = 0, which leads to iij = iji. Exercise 4.5 4. Near the end of Section 7 we derived the equation of motion (4.15) by staring from an integral form for a material volume. Derive equation (4.15) by staring from the integral statement for afied region, given by equation (4.22). 5. Verify the validity of the second form of the viscous dissipation given in equation (4.60). (Hint: Complete the square and use 8ij8ij = 8ii = 3.) 6. A rectangular tank is placed on wheels and is given a constant horizontal acceleration a. Show that, at steady state, the angle made by the free surface with the horizontal is given by tan e = a I g. 7. A jet of water with a diameter of 8 cm and a speed of25 ml s impinges normally on a large stationar flat plate. Find the force required to hold the plate stationar. Compare the average pressure on the plate with the stagnation pressure if the plate is 20 times the area of the jet. 8. Show that the thrst developed by a stationar rocket motor is F = pAU2 + A(p - Paim), where Paim is the atmospheric pressure, and p, p, A, and U are, respectively, the pressure, density, area, and velocity of the fluid at the nozzle exit. 9. Consider the propeller of an airplane moving with a velocity UI. Take a reference frame in which the air is moving and the propeller (disk) is stationary. Then the effect of the propeller is to accelerate the fluid from the upstream value 1J1 to the downstream value U2 ;: U I. Assuming incompressibilty, show that the thrst developed by the propeller is given by pA 2 2 42 ~ L, :l (0 ~ ~ -C :i li \) (j) X t Exercise 4.5 '0 "" (' r '" " '- ~ l" :: ~ '-1-, I . "" ".1, "' \ "oJ ~J i, '", n , .~ 'oJ \ó 'Ú . '" NI ,. ~ , . ~ ~ N , i oJ t ~ ~ "v " t ~ l~ L 7 l. .. V vI~ ~ " )t \J .. '" \d 'l Ë r ~ ~ -t ~ 0 VI \, 1: ~ :£ I~ .. ~ ). -~ 4 ( ~ \L ì l: S~ ~ 0- . 1 "i . .J . "' V\ . .. v- l" tv '). ~ --l~ + r\ ': . .~ ~ ' ,J "' .'" , \(,J I \b ~ ' oJ ''' \J ~ .J . '" N \ t" L ~ \\ l\ V\ i: ~ . t~ ~ 1/ '" .~ \L ~'V . '\ "oJ V\ ... l-J Q 1 t f" ii · oJ ' .J V\ ~'J l' . ') , .. V' .~ . .. V\ \- ~ 0 r -i ~ rJ ~ ~ -l 'i l' .. tv "" :: . b '- \ Ç\ IC ~ ~ r(' ~ . ~ -t~ . '" l. .J '" ..1 .( N \Ó ~ \\ ~ Ci 't V\ " ~ -. \Í ~ .. ., . oJ \! ' "" N \~ ( () ~ 43 horizontal is given by tan e = a I g. 7. A jet of water with a diameter of 8 cm and a speed of25 ml s impinges normally on a large stationar flat plate. Find the force required to hold the plate stationar. Compare the average pressure on the plate with the stagnation pressure if the plate is 20 times the area of the jet. Exercise 4.8 8. Show that the thrst developed by a stationar rocket motor is F = pAU2 + A(p - Paim), where Paim is the atmospheric pressure, and p, p, A, and U are, respectively, the pressure, density, area, and velocity of the fluid at the nozzle exit. 9. Consider the propeller of an airplane moving with a velocity UI. Take a reference frame in which the air is moving and the propeller (disk) is stationary. Then the effect of the propeller is to accelerate the fluid from the upstream value 1J1 to the downstream value U2 ;: U I. Assuming incompressibilty, show that the thrst developed by the propeller is given by pA 2 Ui), 2 F = T(U2 where A is the projected area of the propeller and p is the density (assumed constant). Show also that the velocity of the fluid at the plane of the propeller is the average value U = (Ui + U2)/2. (Hint: The flow can be idealized by a jump, of magnitude pressure !lp = F I A right at the location of the propeller. Also apply Bernoull's equation between a section far upstream and a section immediately upstream of the propeller. Also apply the Bernoull equation between a section immediately downstream of the propeller and a section far downstream. This wil show that !lp = p(Ul - Ul)/2.) 44 Exercise 4.8 ~ ~ 'l l~ ~ '" v j .: .. ~ ~i ~i I J ~ .. ') lL ~ ~ ~ ~ 1/ ~ ~ Q: l '0 ~ II "" . "' ".. q: , ~ ~~ 1 rJ. '; c: ~ '") :i ~ (L "- r, ci l _. :r ~ -l ~ n. ~ , ~ ~ II .. ~ ~ ~ ~ .. " F ~ "..~. ~~ \~\~ , \~t I \ ~ J9 ~ \~!~ - .. 'I .. .. )''- , l1 45 20 times the area of the jet. 8. Show that the thrst developed by a stationar rocket motor is F = pAU2 + A(p - Paim), where Paim is the atmospheric pressure, and p, p, A, and U are, Exercise 4.9 respectively, the pressure, density, area, and velocity of the fluid at the nozzle exit. 9. Consider the propeller of an airplane moving with a velocity UI. Take a reference frame in which the air is moving and the propeller (disk) is stationary. Then the effect of the propeller is to accelerate the fluid from the upstream value 1J1 to the downstream value U2 ;: U I. Assuming incompressibilty, show that the thrst developed by the propeller is given by pA 2 2 F = T(U2 - Ui), where A is the projected area of the propeller and p is the density (assumed constant). Show also that the velocity of the fluid at the plane of the propeller is the average value U = (Ui + U2)/2. (Hint: The flow can be idealized by a jump, of magnitude pressure !lp = F I A right at the location of the propeller. Also apply Bernoull's equation between a section far upstream and a section immediately upstream of the propeller. Also apply the Bernoull equation between a section immediately downstream of the propeller and a section far downstream. This wil show that !lp = p(Ul - Ul)/2.) 10. A hemispherical vessel of radius R has a small rounded orifice of area A at the bottom. Show that the time required to lower the level from h I to h2 is given by A.. 3 I 2 5 I 2 t = ~ (~R (h3/2 _ h3/2) _ ~ (h5/2 - h5/2)). 46 () f\ Ie: ~ - i ~ c: 'Q ~ () " '= \t ~ - \J ~ u. .. ~ " l. -i o . ~ .. ~ ·í - -- - 3 1'_ '. "' .i V) '" :r .ø .. " .. .. .. c: - .. ~ lL ~ ~ nl Exercise 4.9 ~ - :: ~ ~ tt t "' '" -i 'l ~ ~ 'tt ~ ld~O~ ~ _ ci 'II l\ \L N Q. ~ ~~ ". L (l ~ II ~ d ~ ~ c: ~ ..-: '( --~) - ~t.-- ~ ~ (L l- .. ( ;¿ ~ ': ~ f~ .J 1' I ~ ~- I ,"", ~- ~ ~ ll~ 1/ i ~ i \L ~ .: ~ ~,~ .: f' , ii :i '- + ~ \ 0- .. ri ~ ~ I, ~\Ñ ': "- I NN ::- ~ N ~ J ~ 'L il 0. I' ~ \J ~ \ê ~ ': .. .J ::Q. \ ~ 3 ~ ~ ~ Q. \ ct \A3~~ ~ l-: + :: \1' N~~i.. rJ \L D \L '0 ~\ ii Ii \ ~ ~\~I ~ w " ~\~ " L \l (L ,. ~-\ f' ~ ~\ cJ 1\ .. .. ~ ~ 11 '- ": 4- 4- \\ \ i c( ~ ~ c: .J o.l~ ~\~ ") ': ~ ~ CJ 47 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 Integral or differential formulation of equations of motion (E.O.M.) are equivalent • Integral form of momentum-force equation can be combined with Bernoulli’s Principle for a powerful way to compute flow/force parameters. • Apparent forces, centrifugal and Coriolis, appear in a corotating frame of reference 48 ...
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