Reading assignment: Chapter 4 of Aris

Transcription

1 Chaper 4 - The Knemacs of Flud Moon Parcle pahs and maeral dervaves Sreamlnes Sreaklnes Dlaaon Reynolds ranspor heorem Conservaon of mass and he equaon of connuy Deformaon and rae of sran Physcal nerpreaon of he deformaon ensor Prncpal axs of deformaon Vorcy, vorex lnes, and ubes Readng assgnmen: Chaper 4 of Ars Knemacs s he sudy of moon whou regard o he forces ha brng abou he moon. Already, we have descrbed how rgd body moon s descrbed by s ranslaon and roaon. Also, he dvergence and curl of he feld and values on boundares can descrbe a vecor feld. Here we wll consder he moon of a flud as mcroscopc or macroscopc bodes ha ranslae, roae, and deform wh me. We rea fluds as a connuum such ha he flud denfed o be a a specfc pon n space a one me wh neghborng flud wll be a anoher specfc pon n space a a laer me wh he same neghbors, wh he excepon of ceran bfurcaons. Ths denfcaon of he flud occupyng a pon n space requres ha he moon s deermnsc raher han sochasc,.e., random moons such as dffuson and urbulence are no descrbed. Cenral o he knemacs of flud moon s he concep of convecon or followng he moon of a parcle of flud. Parcle pahs and maeral dervaves Flud moon wll be descrbed as he moon of a parcle ha occupes a pon n space. A some me, say =0, a flud parcle s a a poson ξ = (ξ 1, ξ 2, ξ 3 ) and a a laer me he same parcle s a a poson x. The moon of he parcle ha occuped hs orgnal poson s descrbed as follows. x= x ( ξ, ) or x = x ( ξ ξ ξ, ) 1, 2, 3 The nal coordnaes ξ of a parcle wll be referred o as he maeral coordnaes of he parcles and, when convenen, he parcle self may be called he parcle ξ. The erms conveced and Lagrangan coordnaes are also used. The spaal coordnaes x of he parcle may be referred o as s poson or place. I wll be assumed ha he moon s connuous, sngle valued and he prevous equaon can be nvered o gve he nal poson or maeral coordnaes of he parcle whch s a any poson x a me ;.e., ξ = ξ( x, ) or ξ = ξ ( x, x, x, )

2 are also connuous and sngle valued. Physcally hs means ha a connuous arc of parcles does no break up durng he moon or ha he parcles n he neghborhood of a gven parcle connue n s neghborhood durng he moon. The sngle valuedness of he equaons mean ha a parcle canno spl up and occupy wo places nor can wo dsnc parcles occupy he same place. Excepons o hese requremens may be allowed on a fne number of sngular surfaces, lnes or pons, as for example a flud dvdes around an obsacle. I s shown n Appendx B ha a necessary and suffcen condon for he nverse funcons o exs s ha he Jacoban J ( x1, x2, x3) = ( ξ, ξ, ξ ) should no vansh. The ransformaon x=x(ξ,) may be looked a as he paramerc equaon of a curve n space wh as he parameer. The curve goes hrough he pon ξ, correspondng o he parameer =0, and hese curves are he parcle pahs. Any propery of he flud may be followed along he parcle pah. For example, we may be gven he densy n he neghborhood of a parcle as a funcon ρ(ξ,), meanng ha for any prescrbed parcle ξ we have he densy as a funcon of me, ha s, he densy ha an observer rdng on he parcle would see. (Poson self s a propery n hs general sense so ha he equaons of he parcle pah are of hs form.) Ths maeral descrpon of he change of some propery, say I(ξ,), can be changed o a spaal descrpon I(x,). [ ] I ( x, ) =I ξ( x, ), Physcally hs says ha he value of he propery a poson x a me s he value approprae o he parcle ha s a x a me. Conversely, he maeral descrpon can be derved from he spaal one. [ ξ ] I ( ξ, ) =I ( x, ), meanng ha he value as seen by he parcle a me s he value a he poson occupes a ha me. Assocaed wh hese wo descrpons are wo dervaves wh respec o me. We shall denoe hem by x = dervave wh respec o me keepng x consan. and D D = dervave wh respec o me keepng ξ consan ξ 4-2

3 Thus I/ s he rae of change of I as observed a a fxed pon x, whereas DI/D s he rae of changed as observed when movng wh he parcle,.e., for a fxed value of ξ. The laer we call he maeral dervave. I s also called he conveced, convecve, or subsanal dervave and ofen denoed by D/D. In parcular he maeral dervave of he poson of a parcle s velocy. Thus pung I = x, we have Dx v = = x( ξ1, ξ2, ξ3, ) D or. D v = x D Ths allows us o esablsh a connecon beween he wo dervaves, for DI = I ( ξ, ) = I x(, ), D I I = + x x ξ I I = + v I = + ( v ) I [ ξ ] Sreamlnes We now have a formal defnon of he velocy feld as a maeral dervave of he poson of a parcle. Dx( ξ, ) ( ξ, ) vx, ( ) = = D The feld lnes of he velocy feld are called sreamlnes; hey are he soluons of he hree smulaneous equaons dx (, ) ds = vx where s s a parameer along he sreamlne. Ths parameer s s no o be confused wh he me, for n he above equaon s held fxed whle he equaons are negraed, and he resulng curves are he sreamlnes a he nsan. These may vary from nsan o nsan and n general wll no concde wh he parcle pahs. 4-3

4 To oban he parcle pahs from he velocy feld we have o follow he moon of each parcle. Ths means ha we have o solve he dfferenal equaons Dx (, ) D = vx subec o x=ξ a =0. Tme s he parameer along he parcle pah. Thus he parcle pah s he raecory aken by a parcle. The flow s called seady f he velocy componens are ndependen of me. For seady flows, he parameer s along he sreamlnes may be aken o be and he sreamlnes and parcle pahs wll concde. The converse does no follow as here are unseady flows for whch he sreamlnes and parcle pahs concde. If C s a closed curve n he regon of flow, he sreamlnes hrough every pon of C generae a surface known as a sream ube. Le S be a surface wh C as he boundng curve, hen v n ds S s known as he srengh of he sream ube a s cross-secon S. The acceleraon or he rae of change of velocy s defned as Dv a= = + ( v ) v D a = + v Noce ha n seady flow hs does no vansh bu reduces o a= ( v ) v for seady flow. Even n seady flow oher han a consan ranslaon, a flud parcle wll accelerae f changes drecon o go around an obsacle or f ncreases s speed o pass hrough a consrcon. Sreaklnes The name sreaklne s appled o he curve raced ou by a plume of smoke or dye, whch s connuously neced a a fxed pon bu does no dffuse. Thus a me he sreaklne hrough a fxed-pon y s a curve gong from y o x(y, ), he poson reached by he parcle whch was a y a me =0. A parcle s on he sreaklne f passed he fxed-pon y a some me beween 0 and. If hs me was, hen he maeral coordnaes of he parcle would be gven by ξ 4-4

5 = ξ(y, ). However, a me hs parcle s a x = x(ξ, ) so ha he equaon of he sreaklne a me s gven by [ ξ(, '), ] x= x y, where he parameer along les n he nerval 0. If we regard he moon as havng been proceedng for all me, hen he orgn of me s arbrary and can ake negave values -. The flow feld llusraed n 4.13 by Ars s assgned as an exercse. Dlaaon We noced earler ha f he coordnae sysem s changed from coordnaes ξ o coordnaes x, hen he elemen of volume changes by he formula ( x, x, x ) dv = dξ1dξ2dξ3 = J dv0 ( ξ1, ξ2, ξ3) If we hnk of ξ as he maeral coordnaes, hey are he Caresan coordnaes a = 0, so ha dξ 1 dξ 2 dξ 3 s he volume dv o of an elemenary recangular parallelepped. Consder hs elemenary parallelepped abou a gven pon ξ a he nal nsan. By he moon hs parallelepped s moved and dsored bu because he moon s connuous canno break up and so a some me s some neghborhood of he pon x = x(ξ, ). By he above equaon, s volume s dv = J dv o and hence J dv = = dv o rao of an elemenary maeral volume o s nal volume. I s called he dlaon or expanson. The assumpon ha x = x(ξ, ) can be nvered o gve ξ = ξ(x, ), and vce versa, s equvalen o requrng ha neher J nor J -1 vansh. Thus, 0 < J < We can now ask how he dlaon changes as we follow he moon. To answer hs we calculae he maeral dervave DJ/D. However, 4-5

6 J ξ1 ξ2 ξ3 k = εk = ξ1 ξ2 ξ3 ξ1 ξ2 ξ ξ ξ ξ Now D Dx = = D ξ ξ D ξ for D/D s dfferenaon wh ξ consan so ha he order can be nerchanged. Now f we regard v as a funcon of x 1, x 2, x 3, m = ξ ξ m The above relaon can now be appled o dfferenaon of he Jacoban. DJ D k D x k D k = εk + εk + εk D D ξ1 ξ2 ξ3 ξ1 D ξ2 ξ3 ξ1 ξ2 D ξ 3 = ε + ε + ε m k m k k m k k k m ξ1 ξ2 ξ3 ξ1 m ξ2 ξ3 ξ1 ξ2 m ξ3 1 k 2 k 3 k = εk + εk + εk 1 ξ1 ξ2 ξ3 ξ1 2 ξ2 ξ3 ξ1 ξ2 3 ξ = J + J + J = v J ( ) where we made use of he propery of he deermnan ha he deermnan of a marx wh repeaed rows s zero. Thus, D(ln J) D = v We hus have an mporan physcal meanng for he dvergence of he velocy feld. I s he relave rae of dlaon followng a parcle pah. I s evden ha for an ncompressble flud moon, v =0 for ncompressble flud moon. 4-6

7 Reynolds ranspor heorem An mporan knemacal heorem can be derved from he expresson for he maeral dervave of he Jacoban. I s due o Reynolds and concerns he rae of change no of an nfnesmal elemen of volume bu any volume negral. Le I(x, ) be any funcon and be a closed volume movng wh he flud, ha s conssng of he same flud parcles. Then x dv F() = I(, ) s a funcon of ha can be calculaed. We are neresed n s maeral dervave DF/D. Now he negral s over he varyng volume so we canno ake he dfferenaon hrough he negral sgn. If, however, he negraon were wh respec o a volume n ξ -space would be possble o nerchange dfferenaon and negraon snce D/D s dfferenaon wh respec o keepng ξ consan. The ransformaon x = x(ξ, ), dv = J dv o allows us o do us hs, for has been defned as a movng maeral volume and so come from some fxed volume V o a = 0. Thus d d d I ( x, ) dv = I[ ( ξ, ), ] J dv d x V o DI = +I( v) JdV D V o DI = +I( v) dv D V o DI DJ = J +I dv D D Snce D/D = ( / ) + v we can express hs formula no a number of dfferen forms. Subsung for he maeral dervave and collecng he graden erms gves o o o d d I I ( x, ) dv = + v I+I( v) dv I = + ( Iv) dv Now applyng Green s heorem o he second negral we have 4-7

8 d d I I ( x, ) dv = dv + ( I v) ds n S() where S() s he boundng surface of. Ths adms of an mmedae physcal pcure for says ha he rae of change of he negral of I whn he movng volume s he negral of he rae of change of I a a pon plus he ne flow of I over he boundng surface. I can be any scalar or ensor componen, so ha hs s a knemacal resul of wde applcaon. I s gong o be he bass for he conservaon of mass, momenum, energy, and speces. Ths approach o he conservaon equaons dffers from he approach aken by Brd, Sewar, and Lghfoo. They perform a balance on a fxed volume of space and explcly accoun for he convecve flux across he boundares. Conservaon of mass and he equaon of connuy Alhough he dea of mass s no a knemacal one, s convenen o nroduce here and o oban he connuy equaon. Le ρ(x, ) be he mass per un volume of a homogeneous flud a poson x and me. Then he mass of any fne maeral volume s m = x ρ(, ) dv. If V s a maeral volume, ha s, f s composed of he same parcles, and here are no sources or snks n he medum we ake as a prncple ha he mass does no change. By nserng I = ρ n Reynolds ranspor heorem we have dm Dρ = ρ( ) dv d + v D ρ = + ( ρ v ) dv = 0 Ths s rue for an arbrary maeral volume and hence he negrand self mus vansh everywhere. I follows ha Dρ ρ + ρ( v) = + ( ρ v ) = 0 D whch s he equaon of connuy. A flud for whch he densy ρ s consan s called ncompressble. In hs case he equaon of connuy becomes v =0 ncompressble flow 4-8

9 and he moon s sochorc or he velocy feld solenodal. Combnng he equaon of connuy wh Reynolds ranspor heorem for a funcon I = ρ F we have d d D ρ FdV = ( ρf) + ρ F( v) dv D DF Dρ = ρ + F( + ρ v) dv D D ( sources) Thus DF ρ ( sources) dv = 0, for any V ( ) D DF hus, ρ ( sources) = 0 D = = DF ρ dv D Ths equaon s useful for dervng he conservaon equaon of a quany ha s expressed as specfc o a un of mass, e.g., specfc nernal energy and speces mass fracon. Deformaon and rae of sran The moon of fluds dffers from ha of rgd bodes n he deformaon or sran ha occurs wh moon. Maeral coordnaes gve a reference frame for hs deformaon or sran. Consder wo nearby pons P and Q wh maeral coordnaes ξ and ξ + dξ. A me hey are o be found a x(ξ, ) and x(ξ+dξ, ). Now 2 x( ξ + dξ, ) = x( ξ, ) + dξ + O( d ) ξ dv where O(d 2 ) represens erms of order dξ 2 and hgher whch wll be negleced from hs pon onward. Thus he small dsplacemen vecor dξ has now become where dx = x( ξ + dξ, ) x ( ξ, ) 4-9

10 dx =. ξ dξ I s clear from he quoen rule (snce dξ s arbrary) ha he nne quanes / ξ are he componens of a ensor. I may be called he dsplacemen graden ensor and s basc o he heores of elascy and rheology. For flud moon, s maeral dervave s of more drec applcaon and we wll concenrae on hs. If v = Dx/D s he velocy, he relave velocy of wo parcles ξ and ξ + dξ has componens D ξ ξ k dv = d k = d ξk D ξk However, by nverng he above relaon, we have ξ dv = dx = dx ξ k k x x expressng he relave velocy n erms of curren poson. Agan s evden ha he ( / ) are componens of a ensor, he velocy graden ensor, for whch we need o oban a sound physcal feelng. We frs observe ha f he moon s a rgd body ranslaon wh a consan velocy u, x = ξ + u and he velocy graden ensor vanshes dencally. Secondly, he velocy graden ensor can be wren as he sum of symmerc and ansymmerc pars, x 2 x x 2 = e +Ω or v = e+ω We have seen ha a relave velocy dv relaed o he relave poson dx by an ansymmerc ensor Ω,.e., dv = Ω dx, represens a rgd body roaon wh angular velocy ω = -vec Ω. In hs case 4-10

11 1 1 ω = ε Ω = ε 2 2 k k k or 1 ω = v 2 k Thus he ansymmerc par of he velocy graden ensor corresponds o rgd body roaon, and, f he moon s a rgd one (composed of a ranslaon plus a roaon), he symmerc par of he velocy graden ensor wll vansh. For hs reason he ensor e s called he deformaon or rae of sran ensor and s vanshng s necessary and suffcen for he moon o be whou deformaon, ha s, rgd. Physcal nerpreaon of he (rae of) deformaon ensor The (rae of) deformaon ensor s wha dsngushes flud moon from rgd body moon. Recall ha a rgd body s one n whch he relave dsance beween wo pons n he body does no change. We show here ha he (rae of) deformaon ensor descrbes he rae of change of he relave dsance beween wo parcles n a flud. Also, descrbes he rae of change of he angle beween hree parcles n he flud. Frs we wll see how he dsance beween wo maeral pons change durng he moon. The lengh of an nfnesmal lne segmen from P o Q s ds, where ds = dx dx = dξ dξ k. 2 ξ ξk now P and Q are he maeral parcles ξ and ξ + dξ so ha dξ and dξ k do no change durng he moon. Also, recall ha Dx /D = v. Thus D 2 ( ds ) = + d ξ d ξk = 2 d ξ d ξ k D ξ ξk ξ ξ k ξ ξk by symmery. However, Thus dξ = dv = dx and dξk = d ξ ξ k [ ξ ] sn ce v= v x( ) and x= x( ξ) x 4-11

12 1 D D ds ds ds dx dx e dx dx e dx dx 2 D D x 2 ( ) = ( ) ( ) = = ( +Ω ) = by symmery, or 1 D dx dx ( ds ) = e. ( ds) D ds ds Now dx /ds s he h componen of a un vecor n he drecon of he segmen PQ, so ha hs equaon says ha he rae of change of he lengh of he segmen as a fracon of s lengh s relaed o s drecon hrough he deformaon ensor. In parcular, f PQ s parallel o he coordnae axs 01 we have dx/ds = e (1) and 1 D ( dx 1) = e 11 n drecon of 01 dx D 1 Thus e 11 s he rae of longudnal sran of an elemen parallel o he 01 axs. Smlar nerpreaons apply o e 22 and e 33. Now les examne he angle beween wo lne segmens durng he moon. Consder he segmen, PQ and PR where PQ s he segmen ξ + dξ as before and PR s he segmen ξ + dξ. If θ s he angle beween hem and ds s he lengh of PR, we have from he scalar produc, ds ds 'cos θ = dx dx ' Dfferenang wh respec o me we have D [ ds ds 'cos θ ] = dv dx ' + dx dv ' D = dx dx ' + dx dx ' snce dv ' = ( / ) dx '. The and are dummy suffxes so we may nerchange hem n he frs erm on he rgh, hen performng dfferenaon we have afer dvdng by ds ds 4-12

13 1 D 1 D Dθ cos θ ds + ds ' snθ ds D ds ' D D dx dx' = + x x ds ds ' dx ' dx = 2e ds ' ds Now suppose ha dx s parallel o he axs 01 and dx o he axs 02, so ha ( dx ' / ds ') = δ 1 and ( dx / ds) = δ 2 and θ12 = π / 2. Then dθ d 12 = 2e12. Thus e 12 s o be nerpreed as one-half he rae of decrease of he angle beween wo segmens orgnally parallel o he 01 and 02 axes respecvely. Smlar nerpreaons are approprae o e 23 and e 31. The fac ha he rae of deformaon ensor s lnear n he velocy feld has an mporan consequence. Snce we may supermpose wo velocy felds o form a hrd, follows ha he deformaon ensor of hs s he sum, of he deformaon ensors of he felds from whch was supermposed. If v = λ() x (no summaon on ), we have a deformaon whch s he superposon of hree srechng parallel o he hree axs. However, f v 1 =f(x 2 ), v 2 = 0, v 3 = 0 so ha only nonzero componen of he deformaon ensor s e 12 = ½ f (x 2 ), he moon s one of pure shear n whch elemens parallel o he coordnae axs s no sreched a all. Noe however ha n pure srechng an elemen no parallel or perpendcular o he drecon of srechng wll suffer roaon. Lkewse n pure shear an elemen no normal o or n he plane of shear wll suffer srechng. Prncpal axs of deformaon The rae of deformaon ensor s a symmerc ensor and he prncpal axs of deformaon can be found. They correspond o he egenvalues of he marx and he egenvalues are he prncpal raes of sran. A se of parcles ha s orgnally on he surface of a sphere wll be deformed o an ellpsod whose axes are concden wh he prncpal axs. Vorcy, vorex lnes, and ubes We have frequenly remnders of roang bodes of flud such as ropcal sorms, hurrcanes, ornadoes, dus devls, whrlpools, eddes n he flow behnd obecs, urbulence, and he vorex n dranng bahubs. The knemacs of hese flud moons s descrbed by he vorcy. The ansymmerc par of he rae of sran ensor Ω represens he local roaon, ω k. Recall 4-13

14 1 ω = ε Ω 2 ω = vecω k k 1 = v 2 The curl of velocy s known as he vorcy, w= v Thus he vorcy and he ansymmerc par of he rae of sran ensor s a measure of he roaon of he velocy feld. An rroaonal flow feld s one n whch he vorcy vanshes everywhere. The feld lnes of he vorcy feld are called vorex lnes and he surface generaed by he vorex lnes hrough a closed curve C s a vorex ube. The srengh of a vorex ube s defned as he surface negral of he normal componen. I s equal o he crculaon around he closed curve C ha bounds he cross-secon S by Sokes heorem. S w nds = ( v) nds S = v ds =Γ We observe ha he srengh of a vorex ube a any cross-secon s he same, as w s a solenodal vecor. The surface negral of he normal componen of a solenodal vecor vanshes over any closed surface. The surface negral on he surface of a vorex ube s zero because he sdes are angen o he vorcy vecor feld. Thus he surface negral across any cross-secon mus be equal n magnude. The magnude of he vorcy feld can be vsualzed from he relave wdh of he vorex ubes n he same manner ha he magnude of he velocy feld can be vsualzed by he wdh of he sream ubes. Because he srengh of he ube does no vary wh poson along he ube, follows ha he vorex ubes are eher closed, go o nfny or end on sold boundares of roang obecs. In a real flud sasfyng he no-slp boundary condon, vorex lnes mus be angenal o he surface of a body a res, excep a solaed pons of aachmen and separaon, because he normal componen of vorcy vanshes on he saonary sold. When he vorex ube s mmedaely surrounded by rroaonal flud, wll be referred o as a vorex flamen. A vorex flamen s ofen us called a vorex, bu we shell use hs erm o denoe any fne volume of vorcy mmersed n rroaonal flud. Of course, he vorex flamen and he vorex requre he flud o be deal (zero vscosy) o make src sense, because vscosy dffuses vorcy, bu hey are useful approxmaons for real fluds of small vscosy. 4-14

15 Helmholz gave hree laws of vorex moon n For he moon of an deal (zero vscosy) baroropc (densy s a sngle valued funcon of pressure) flud under he acon of conservave exernal body forces (graden of a scalar), hey can be expressed as follows: I. Flud parcles orgnally free of vorcy reman free of vorcy. II. Flud parcles on a vorex lne a any nsan wll be on a vorex lne a subsequen mes. Alernavely, can be sad ha vorex lnes and ubes move wh he flud. III. The srengh of a vorex ube does no vary wh me durng he moon of he flud. The equaons for he dynamcs of vorcy wll be developed laer. P. G. Saffman, Vorex Dynamcs, Cambrdge Unversy Press, C. Truesdell, The Knemacs of Vorcy, Indana Unversy Press, Assgnmen 4.1 Plo he sreamlnes, parcle pahs, and sreaklnes of he flow feld descrbed n Sec Fnd he CHBE 501 web page. Download he fles n CENG501/Problems/lnes. Execue lnes wh MATLAB. Assgnmen 4.2 Execue he program deform n CHBE 501/Problems/deform and prn he fgures. I compues he parcle pahs for a pach of parcles deformng n Couee flow, sagnaon flow, and ha of Sec Look a he respecve subroune o deermne he equaons of he flow feld. For each of hese flow felds calculae: a) dvergence b) curl c) rae of deformaon ensor d) ansymmerc ensor e) Whch felds are solenodal or rroaonal? 4-15