Astrophysical Fluid Dynamics What is a Fluid?

Transcription

1 Astrophyscal Flud Dynamcs What s a Flud? 1

2 I. What s a flud? I.1 The Flud approxmaton: The flud s an dealzed concept n whch the matter s descrbed as a contnuous medum wth certan macroscopc propertes that vary as contnuous functon of poston (e.g., densty, pressure, velocty, entropy). That s, one assumes that the scales l over whch these quanttes are defned s much larger than the mean free path l of the ndvdual partcles that consttute the flud, 1 l ; n Where n s the number densty of partcles n the flud and s a typcal nteracton cross secton. 4 I. What s a flud? Furthermore, the concept of local flud quanttes s only useful f the scale l on whch they are defned s much smaller than the typcal macroscopc lengthscales L on whch flud propertes vary. Thus to use the equatons of flud dynamcs we requre Ll Astrophyscal crcumstances are often such that strctly speang not all crtera are fulflled. 4

3 I. What s a flud? Astrophyscal crcumstances are often such that strctly speang not all flud crtera are fulflled. Mean free path astrophyscal fluds (temperature T, densty n): 6 10 T / n cm 1) Sun (centre): ) Solar wnd: 3) Cluster: T 10 K, n10 cm 10 cm 10 R 710 cm flud approxmaton very good T 10 K, n10 cm 10 cm 13 AU cm flud approxmaton does not apply, plasma physcs 7 T 310 K, 3 3 n10 cm 4 10 cm 4 1 Mpc flud approxmaton margnal A B Sold vs. Flud By defnton, a flud cannot wthstand any tendency for appled forces to deform t, (whle volume remans unchanged). Such deformaton may be ressted, but not prevented. A B A B Sold C D C D C D Before applcaton of shear A B Shear force A B After shear force s removed A B Flud C D C D C D 5 3

4 Mathematcal Prelmnares Mathematcal prelmnares S F ds F dv V Gauss's Law C F dl F ds Stoe's Theorem S 6 4

5 Lagrangan vs. Euleran Vew There s a range of dfferent ways n whch we can follow the evoluton of a flud. The two most useful and best nown ones are: 1) Euleran vew Consder the system propertes Q densty, flow velocty, temperature, pressure at fxed locatons. The temporal changes of these quanttes s therefore followed by partal tme dervatve: ) Lagrangan vew Q t Follow the changng system propertes Q as you flow along wth a flud element. In a way, ths partcle approach s n the sprt of Newtonan dynamcs, where you follow the body under the acton of external force(s). The temporal change of the quanttes s followed by means of the convectve or Lagrangan dervatve DQ Dt Lagrangan vs. Euleran Vew Consder the change of a flud quantty 1) Euleran vew: change n quantty Q n nterval dt, at locaton r : Qrt (, ) ) Lagrangan vew: change n quantty Q n tme nterval dt, whle flud element moves from r to r r at a locaton r Q Qrt (, t) Qrt (, ) t t DQ Q( r r, t t) Q( r, t) Dt t Q v Q t D Dt v t Convectve/ Lagrangan Dervatve 5

6 Basc Flud Equatons Conservaton Equatons To descrbe a contnuous flud flow feld, the frst step s to evaluate the development of essental propertes of the mean flow feld. To ths end we evaluate the frst 3 moment of the phase space dstrbuton functon, correspondng to fve quanttes, f ( rv, ) For a gas or flud consstng of partcles wth mass m, these are 1) mass densty ) momentum densty 3) (netc) energy densty m u mv f r, v, tdv mvu / Note that we use u to denote the bul velocty at locaton r, and v for the partcle velocty. The velocty of a partcle s therefore the sum of the bul velocty and a random component w, v u w In prncple, to follow the evoluton of the (moment) quanttes, we have to follow the evoluton of the phase space densty f ( rv, ). The Boltzmann equaton descrbes ths evoluton. 6

7 Boltzmann Equaton In prncple, to follow the evoluton of these (moment) quanttes, we have to follow the evoluton of the phase space densty f ( rv, ). Ths means we should solve the Boltzmann equaton, f f t vf v f t c The rghthand collsonal term s gven by n whch f vv f v f v f v f v ddv t c v, v v, v s the angle W-dependent elastc collson cross secton. On the lefthand sde, we fnd the gravtatonal potental term, whch accordng to the Posson equaton 4 G( ext ) ext x, t s generated by selfgravty as well as the external mass dstrbuton. Boltzmann Equaton To follow the evoluton of a flud at a partcular locaton x, we follow the evoluton of a quantty c(x,v) as descrbed by the Boltzmann equaton. To ths end, we ntegrate over the full velocty range, f f f f v dv dv t x ( x x v t c, v) If the quantty s a conserved quantty n a collson, then the rghthand sde of the equaton equals zero. For elastc collsons, these are mass, momentum and (netc) energy of a partcle. Thus, for these quanttes we have, f t dv 0 The above result expresses mathematcally the smple noton that collsons can not contrbute to the tme rate change of any quantty whose total s conserved n the collsonal process. For elastc collsons nvolvng short-range forces n the nonrelatvstc regme, there exst exactly fve ndependent quanttes whch are conserved: mass, momentum (netc) energy of a partcle, m m; mv ; v c 7

8 Boltzmann Moment Equatons When we defne an average local quantty, for a quantty Q, then on the bass of the velocty ntegral of the Boltzmann equaton, we get the followng evoluton equatons for the conserved quanttes c, For the fve quanttes 1 Q n Q f dv t x x v n n v n 0 m m; mv; v the resultng conservaton equatons are nown as the 1) mass densty contnuty equaton ) momentum densty Euler equaton 3) energy densty energy equaton In the sequel we follow for reasons of nsght a slghtly more heurstc path towards nferrng the contnuty equaton and the Euler equaton. Contnuty equaton To nfer the contnuty equaton, we consder the conservaton of mass contaned n a volume V whch s fxed n space and enclosed by a surface S. S n n The mass M s The change of mass M n the volume V s equal to the flux of mass through the surface S, Where d dt n V M dv V dv s the outward pontng normal vector. S u n ds d V dv dv LHS: dt V V t RHS, usng the dvergence theorem (Green s formula): S V uds u dv 8

9 Contnuty equaton Snce ths holds for every volume, ths relaton s equvalent to t u 0 ( I.1) The contnuty equaton expresses - mass conservaton AND - flud flow occurrng n a contnuous fashon!!!!! S n n V One can also defne the mass flux densty as j u whch shows that eqn. I.1 s actually a contnuty equaton j 0 ( I.) t Contnuty Equaton & Compressblty From the contnuty equaton, t u 0 n we fnd drectly that, u u 0 t Of course, the frst two terms defne the Lagrangan dervatve, so that for a movng flud element we fnd that ts densty changes accordng to 1 D u Dt u 0 In other words, the densty of the flud element changes as the dvergence of the velocty flow. If the densty of the flud cannot change, we call t an ncompressble flud, for whch. 9

10 Momentum Conservaton mv t x x v When consderng the flud momentum, we obtan the equaton of momentum conservaton,, va the Boltzmann moment equaton, n n v n 0 t x x v vv 0 Decomposng the velocty v nto the bul velocty u and the random component w, we have vv uu ww By separatng out the trace of the symmetrc dyadc w w, we wrte ww p Momentum Conservaton By separatng out the trace of the symmetrc dyadc w w, we wrte where ww p P s the gas pressure p s the vscous stress tensor 1 p w 3 1 w 3 ww we obtan the momentum equaton, n ts conservaton form, u uu p t x x 10

11 Momentum Conservaton Momentum Equaton u uu p t x x Descrbes the change of the momentum densty u n the -drecton: The flux of the -th component of momentum n the -th drecton conssts of the sum of 1) a mean part: ) random part I, sotropc pressure part: 3) random part II, nonsotropc vscous part: uu p Force Equaton Momentum Equaton u uu p t x x By nvong the contnuty equaton, we may also manpulate the momentum equaton so that t becomes the force equaton Du p Dt 11

12 Vscous Stress A note on the vscous stress term : For Newtonan fluds: Hooe s Law states that the vscous stress s lnearly proportonal to the rate of stran /, u x where s the shear deformaton tensor, The parameters m and b are called the shear and bul coeffcents of vscosty. u 1u u 1 u x x 3 Euler equaton In the absence of vscous terms, we may easly derve the equaton for the conservaton of momentum on the bass of macroscopc consderatons. Ths yelds the Euler equaton. As n the case for mass conservaton, consder an arbtrary volume V, fxed n space, and bounded by a surface S, wth an outward normal. Insde V, the total momentum for a flud wth densty and flow velocty s V The momentum nsde V changes as a result of three factors: 1) External (volume) force, a well nown example s the gravtatonal force when V embedded n gravty feld. ) The pressure (surface) force over de surface S of the volume. (at ths stage we'll gnore other stress tensor terms that can ether be caused by vscosty, electromagnetc stress tensor, etc.): 3) The net transport of momentum by n- and outflow of flud nto and out of V n u dv u 1

13 Euler equaton 1) External (volume) forces,: f where s the force per unt mass, nown as the body force. An example s the gravtatonal force when the volume V s embeddded n a gravtatonal feld. ) The pressure (surface) force s the ntegral of the pressure (force per unt area) over the surface S, V f dv 3) The momentum transport over the surface area can be nferred by consderng at each surface pont the slanted cylnder of flud swept out by the area element ds n tme dt, where ds starts on the surface S and moves wth the flud, e. wth velocty u. The momentum transported through the slanted cylnder s so that the total transported momentum through the surface S s: S pn ds u u u n t S u u u n ds S Euler equaton Tang nto account all three factors, the total rate of change of momentum s gven by d dt u dv f dv pnds u un ds V V S S The most convenent way to evaluate ths ntegral s by restrctng oneself to the -component of the velocty feld, d dt u dv f dv pn ds u u n ds V V S S j j Note that we use the Ensten summaton conventon for repeated ndces. Volume V s fxed, so that d dt V V t Furthermore, V s arbtrary. Hence, p u u u f t x x j j 13

14 Euler equaton Reorderng some terms of the lefthand sde of the last equaton, p u u u f t x x leads to the followng equaton: j j u u p u u u f j j t x t x j j x From the contnuty equaton, we now that the second term on the LHS s zero. Subsequently, returnng to vector notaton, we fnd the usual exprsson for the Euler equaton, Returnng to vector notaton, and usng the we fnd the usual expresson for the Euler equaton: u t ( u ) u p f ( I.4) Euler equaton An slghtly alternatve expresson for the Euler equaton s u t u p u f ( I.5) In ths dscusson we gnored energy dsspaton processes whch may occur as a result of nternal frcton wthn the medum and heat exchange between ts parts (conducton). Ths type of fluds are called deal fluds. Gravty: For gravty the force per unt mass s gven by where the Posson equaton relates the gravtatonal potental j to the densty r: f 4 G 14

15 Euler equaton From eqn. (I.4) u t ( u ) u p f ( I.4) we see that the LHS nvolves the Lagrangan dervatve, so that the Euler equaton can be wrtten as Du Dt p f ( I.6) In ths form t can be recognzed as a statement of Newton s nd law for an nvscd (frctonless) flud. It says that, for an nfntesmal volume of flud, mass tmes acceleraton = total force on the same volume, namely force due to pressure gradent plus whatever body forces are beng exerted. Energy Conservaton In terms of bul velocty and random velocty the (netc) energy of a partcle s, The Boltzmann moment equaton for energy conservaton becomes u w m m ( ) mu mw v uw mwu t x x v n n v n 0 u w u w u w u 0 t x x Expandng the term nsde the spatal dvergence, we get u w u w u u u ww u w w w 15

16 Energy Conservaton Defnng the followng energy-related quanttes: 1) specfc nternal energy: w P 1 3 ) gas pressure 3) conducton heat flux 4) vscous stress tensor 1 P w 3 1 F w w 1 w ww 3 Energy Conservaton The total energy equaton for energy conservaton n ts conservaton form s u u u up u F u t x x Ths equaton states that the total flud energy densty s the sum of a part due to bul moton u and a part due to random motons w. The flux of flud energy n the -th drecton conssts of 1) the translaton of the bul netc energy at the -th component of the mean velocty, ) plus the enthalpy sum of nternal energy and pressure flux, Pu 3) plus the vscous contrbuton u u / u 4) plus the conductve flux F 16

17 Wor Equaton Internal Energy Equaton For several purposes t s convenent to express energy conservaton n a form that nvolves only the nternal energy and a form that only nvolves the global PdV wor. The wor equaton follows from the full energy equaton by usng the Euler equaton, by multplyng t by and usng the contnuty equaton: u P u u u u u u t x x x x Subtractng the wor equaton from the full energy equaton, yelds the nternal energy equaton for the nternal energy u F u P t x x x where Y s the rate of vscous dsspaton evoed by the vscosty stress u x Internal energy equaton If we use the contnuty equaton, we may also wrte the nternal energy equaton n the form of the frst law of thermodynamcs, D PuFcond Dt n whch we recognze 1 D P u P Dt as the rate of dong PdV wor, and F cond as the tme rate of addng heat (through heat conducton and the vscous converson of ordered energy n dfferental flud motons to dsordered energy n random partcle motons). 17

18 Energy Equaton On the bass of the netc equaton for energy conservaton t u u u u P u F u g x we may understand that the tme rate of the change of the total flud energy n a volume V (wth surface area A),.e. the netc energy of flud moton plus nternal energy, should equal the sum of 1) mnus the surface ntegral of the energy flux (netc + nternal) ) plus surface ntegral of dong wor by the nternal stresses P 3) volume ntegral of the rate of dong wor by local body forces (e.g. gravtatonal) 4) mnus the heat loss by conducton across the surface A 5) plus volumetrc gan mnus volumetrc losses of energy due to local sources and sns (e.g. radaton) Energy Equaton The total expresson for the tme rate of total flud energy s therefore d dt V 1 1 u dv u u nda ˆ A upnda ugdv A V F nda ˆ dv A cond V æ P s the force per unt area exerted by the outsde on the nsde n the th drecton across a face whose normal s orented n the th drecton. For a dlute gas ths s P ww p æ G s the energy gan per volume, as a result of energy generatng processes. æ L s the energy loss per volume due to local sns (such as e.g. radaton) 18

19 Energy Equaton By applyng the dvergence theorem, we obtan the total energy equaton: 1 1 x u u u P F gu t Heat Equaton Implct to the flud formulaton, s the concept of local thermal equlbrum. Ths allows us to dentfy the trace of the stress tensor P wth the thermodynamc pressure p, P p Such that t s related to the nternal energy per unt mass of the flud,, and the specfc entropy s, by the fundamental law of thermodynamcs d Tds pdv Tds pd Applyng ths thermodynamc equaton and subtractng the wor equaton, we obtan the Heat Equaton, Ds T Fcond Dt 1 where Y equals the rate of vscous dsspaton, u x 19

20 Flud Flow Vsualzaton Flow Vsualzaton: Streamlnes, Pathlnes & Strealnes Flud flow s characterzed by a velocty vector feld n 3-D space. There are varous dstnct types of curves/lnes commonly used when vsualzng flud moton: streamlnes, pathlnes and strealnes. These only dffer when the flow changes n tme, e. when the flow s not steady! If the flow s not steady, streamlnes and strealnes wll change. 1) Streamlnes re a Famly of curves that are nstantaneously tangent to the velocty vector u any pont n tme.. They show the drecton a flud element wll travel at If we parameterze one partcular streamlne, wth l ( s 0 ) x, then streamlnes are defned as l S ( s) dls u ( l S ) 0 ds S 0 0

21 Flow Vsualzaton: Streamlnes Defnton Streamlnes: dls u ( l S ) 0 ds If the components of the streamlne can be wrtten as l ( x, y, z ) S Illustratons of streamlnes and then dl ( dx, dy, dz) u ( ux, uy, uz) re a dx dy dz u u u x y z ) Pathlnes Flow Vsualzaton: Pathlnes Pathlnes are the trajectores that ndvdual flud partcles follow. These can be thought of as a "recordng" of the path a flud element n the flow taes over a certan perod. The drecton the path taes wll be determned by the streamlnes of the flud at each moment n tme. Pathlnes l () P t dlp ul ( P, t) dt lp( t0) xp0 are defned by re a where the suffx P ndcates we are followng the path of partcle P. Note that at locaton the curve s parallel to velocty vector P, where the velocty vector s evaluated at locaton at tme t. l P l u 1

22 3) Strealnes Flow Vsualzaton: Strealnes Strealnes are are the locus of ponts of all the flud partcles that have passed contnuously through a partcular spatal pont n the past. Dye steadly njected nto the flud at a fxed pont extends along a strealne. In other words, t s le the plume from a chmney. l T dlt ul ( T, t) dt lt( T) xt0 ul (, t) Strealnes can be expressed as where T s the velocty at locaton at tme t. The parameter T parameterzes the strealne lt t, T and 0 wth t 0 tme of nterest. T t 0 l re a T Flow Vsualzaton: Streamlnes, Pathlnes, Strealnes The followng example llustrates the dfferent concepts of streamlnes, pathlnes and strealnes: æ red: pathlne æ blue: strealne æ short-dashed: evolvng streamlnes re a

23 Steady flow Steady flow s a flow n whch the velocty, densty and the other felds do not depend explctly on tme, namely / t 0 In steady flow streamlnes and strealnes do not vary wth tme and concde wth the pathlnes. Knematcs of Flud Flow 3

24 Stoes Flow Theorem Stoes flow theorem: The most general dfferental moton of a flud element corresponds to a 1) unform translaton ) unform expanson/contracton dvergence term 3) unform rotaton vortcty term 4) dstorton (wthout change volume) shear term The flud velocty u ( Q ) at a pont Q dsplaced by a small amount R from a pont P wll dffer by a small amount, and ncludes the components lsted above: uq ( ) up ( ) HRSR unform translaton Dvergence unform expanson/contracton Shear term dstorton Vortcty unform rotaton Stoes Flow Theorem Stoes flow theorem: the terms of the relatve moton wrt. pont P are: ) Dvergence term: unform expanson/contracton H 1 u 3 3) Shear term: unform dstorton S : shear deformaton scalar : shear tensor 4) Vortcty Term: unform rotaton 1 S RR 1u u 1 u x x u u 4

25 Stoes Flow Theorem Stoes flow theorem: One may easly understand the components of the flud flow around a pont P by a smple Taylor expanson of the velocty feld u ( x ) around the pont P: u u u ( x R, t) u ( x, t) R x Subsequently, t s nsghtful to wrte the rate-of-stran tensor u / x terms of ts symmetrc and antsymmetrc parts: u 1 u u 1 u u x x x x x The symmetrc part of ths tensor s the deformaton tensor, and t s convenent -and nsghtful to wrte t n terms of a dagonal trace part and the traceless shear tensor, u x 1 u 3 n Stoes Flow Theorem where 1) the symmetrc (and traceless) shear tensor s defned as 1 u u 1 u x x 3 ) the antsymmetrc tensor as 1 u x u x 3) the trace of the rate-of-stran tensor s proportonal to the velocty dvergence term, u 1 u u 3 u x x x 1 3 5

26 Stoes Flow Theorem Dvergence Term u 1 u u 3 u x x x 1 3 We now from the Lagrangan contnuty equaton, 1 D Dt that the term represents the unform expanson or contracton of the flud element. u Stoes Flow Theorem Shear Term The traceless symmetrc shear term, 1 u u 1 u x x 3 represents the ansotropc deformaton of the flud element. As t concerns a traceless deformaton, t preserves the volume of the flud element (the volume-changng deformaton s represented va the dvergence term). (ntenton of llustraton s that the volume of the sphere and the ellpsod to be equal) 6

27 Stoes Flow Theorem Shear Term Note that we can assocate a quadratc form e. an ellpsod wth the shear tensor, the shear deformaton scalar S, 1 S such that the correspondng shear velocty contrbuton s gven by S u, R R We may also defne a related quadratc form by ncorporatng the dvergence term, 1 v u u R R x x Evdently, ths represents the rrotatonal part of the velocty feld. For ths reason, we call the velocty potental: v RR D RR u RR 3 v m m m m m u u 0 v Stoes Flow Theorem Vortcty Term The antsymmetrc term, represents the rotatonal component of the flud element s moton, the vortcty. Wth the antsymmetrc we can assocate a (pseudo)vector, the vortcty vector where the coordnates of the vortcty vector, 1 3 the vortcty tensor va u u u x x x where 1 u x u x s the Lev-Cevta tensor, whch fulfls the useful dentty u (,, ), are related to m m m m m m m ps p s s p 7

28 Stoes Flow Theorem Vortcty Term The contrbuton of the antsymmetrc part of the dfferental velocty therefore reads, u 1 u u R 1 R R, m m m m x x The last expresson n the eqn. above equals the -th component of the rotatonal velocty v rot R 1 u of the flud element wrt to ts center of mass, so that the vortcty vector can be dentfed wth one-half the angular velocty of the flud element, Lnear Momentum Flud Element The lnear momentum p of a flud element equal the flud velocty u ( Q ) ntegrated over the mass of the element, Substtutng ths nto the equaton for the flud flow around P, we obtan: p u ( Q ) dm uq ( ) up ( ) HRSR p u ( P ) dm R dm H R dm S dm If P s the center of mass of the flud element, then the nd and 3 rd terms on the RHS vansh as Rdm 0 Moreover, for the 4 th term we can also use ths fact to arrve at, Sdm R dm R dm0 8

29 Lnear Momentum Flud Element Hence, for a flud element, the lnear momentum equals the mass tmes the center-of-mass velocty, p u ( Q ) dm m u ( P ) Angular Momentum Flud Element Wth respect to the center-of-mass P, the nstantaneous angular momentum of a flud element equals We rotate the coordnate axes to the egenvector coordnate system of the deformaton tensor D m (or, equvalently, the shear tensor m), n whch the symmetrc deformaton tensor s dagonal and all strans Then J R u ( Q ) dm 1 1 v D mr m R D R D R D R D m are extensonal, u u u D ; D ; D x 1 x x 3 J R u ( Q ) R u ( Q ) dm

30 Angular Momentum Flud Element In the egenvalue coordnate system, the angular momentum n the 1-drecton s where wth u /and D m evaluated at the center-of-mass P. After some algebra we obtan where J R u ( Q ) R u ( Q ) dm u 3 ( Q ) u 3 ( P ) 1R R 1 D 33R 3 u ( Q ) u ( P ) R R D R J I I I I D D I jl s the moment of nerta tensor jl jl j l Notce that I jl s not dagonal n the prmed frame unless the prncpal axes of I happen to concde wth those of. I R R R dm D m jl Angular Momentum Flud Element Usng the smple observaton that the dfference D snce the sotropc part of I does not enter n the dfference, jl we fnd for all 3 angular momentum components wth a summaton over the repeated l s. Note that for a sold body we would have D j jl l For a flud an extra contrbuton arses from the extensonal stran f the prncpal axes of the moment-of-nerta tensor do not concde wth those of D. Notce, n partcular, that a flud element can have angular momentum wrt. ts center of mass wthout possesng spnnng moton, e. even f u / 0! J I I 1 1l l 3 33 J I I l l J I I 3 3l l 1 11 J I 30

31 Invscd Barotropc Flow Invscd Barotropc Flow In ths chapter we are gong to study the flow of fluds n whch we gnore the effects of vscosty. In addton, we suppose that the energetcs of the flow processes are such that we have a barotropc equaton of state P P (, S ) P ( ) Such a replacement consderably smplfes many dynamcal dscussons, and ts formal justfcaton can arse n many ways. One specfc example s when heat transport can be gnored, so that we have adabatc flow, Ds Dt s t v s 0 wth s the specfc entropy per mass unt. Such a flow s called an sentropc flow. However, barotropc flow s more general than sentropc flow. There are also varous other thermodynamc crcumstances where the barotropc hypothess s vald. 31

32 Invscd Barotropc Flow For a barotropc flow, the specfc enthalpy h dh T ds Vdp becomes smply dh Vdp dp and h dp Kelvn Crculaton Theorem Assume a flud embedded n a unform gravtatonal feld,.e. wth an external force so that gnorng the nfluence of vscous stresses and radatve forces - the flow proceeds accordng to the Euler equaton, u t u u g To proceed, we use a relevant vector dentty f whch you can most easly chec by worng out the expressons for each of the 3 components. The resultng expresson for the Euler equaton s then g p 1 u u u u u u 1 u u u g t p 3

33 Kelvn Crculaton Theorem If we tae the curl of equaton u 1 we obtan where p u u u g t u g p t s the vortcty vector, u and we have used the fact that the curl of the gradent of any functon equals zero, 1 0; 0 u p Also, a classcal gravtatonal feld g satsfes ths property, so that gravtatonal felds cannot contrbute to the generaton or destructon of vortcty. g 0 Vortcty Equaton In the case of barotropc flow, e. f p p p ( ) p so that also the nd term on the RHS of the vortcty equaton dsappears, 1 1 p p 0 The resultng expresson for the vortcty equaton for barotropc flow n a conservatve gravtatonal feld s therefore, t whch we now as the Vortcty Equaton. u 0 33

34 Kelvn Crculaton Theorem Interpretaton of the vortcty equaton: t u 0 Compare to magnetostatcs, where we may assocate the value of wth a certan number of magnetc feld lnes per unt area. B Wth such a pcture, we may gve the followng geometrc nterpretaton of the vortcty equaton, whch wll be the physcal essence of the Kelvn Crculaton Theorem The number of vortex lnes that thread any element of area, that moves wth the flud, remans unchanged n tme for nvscd barotropc flow. Kelvn Crculaton Theorem To prove Kelvn s crculaton theorem, we defne the crculaton G around a crcut C by the lne ntegral, C u dl Transformng the lne ntegral to a surface ntegral over the enclosed area A by Stoes theorem, we obtan u n da A A n da Ths equaton states that the crculaton G of the crcut C can be calculated as the number of vortex lnes that thread the enclosed area A. 34

35 Kelvn Crculaton Theorem Tme rate of change of G Subsequently, we nvestgate the tme rate of change of G f every pont on C moves at the local flud velocty u. Tae the tme dervatve of the surface ntegral n the last equaton. It has contrbutons: d dt n da tme rate of change of area A t where ˆn s the unt normal vector to the surface area. Kelvn Crculaton Theorem d dt Tme rate of change of G n da tme rate of change of area A t The tme rate of change of area can be expressed mathematcally wth the help of the fgure llustratng the change of an area A movng locally wth flud velocty. On the bass of ths, we may wrte, d dt nda ˆ u dl A t C We then nterchange the cross and dot n the trple scalar product u dl u dl 35

36 Kelvn Crculaton Theorem Tme rate of change of G Usng Stoes theorem to convert the resultng lne ntegral d dt to a surface ntegral, we obtan: d dt nda ˆ u dl A t C u nˆ da A t The vortcty equaton tells us that the ntegrand on the rght-hand sde equals zero, so that we have the geometrc nterpretaton of Kelvn s crculaton theorem, d dt 0 Kelvn Crculaton Theorem Tme rate of change of G Usng Stoes theorem to convert the resultng lne ntegral d dt to a surface ntegral, we obtan: d dt nda ˆ u dl A t C u nˆ da A t The vortcty equaton tells us that the ntegrand on the rght-hand sde equals zero, so that we have the geometrc nterpretaton of Kelvn s crculaton theorem, d dt 0 36

37 the Bernoull Theorem Closely related to Kelvn s crculaton theorem we fnd Bernoull s theorem. It concerns a flow whch s steady and barotropc,.e. u 0 t and p p( ) Agan, usng the vector dentty, 1 u u u u u we may wrte the Euler equaton for a steady flow n a gravtatonal feld f u p u u u u t 1 p u u u the Bernoull Theorem The Euler equaton thus mples that where h s the specfc enthalpy, equal to for whch 1 u u h h h dp p We thus fnd that the Euler equaton mples that 1 u u h 37

38 the Bernoull Theorem Defnng the Bernoull functon B 1 B u h whch has dmensons of energy per unt mass. The Euler equaton thus becomes u B Now we consder two stuatons, the scalar product of the equaton wth and and, u ( u ) B 0 1) B s constant along streamlnes ths s Bernoull s streamlne theorem ( ) B 0 ) B s constant along vortex lnes e. along ntegral curves ( x ) * vortex lnes are curves tangent to the vector feld ( x ) 0 38