2.10 Convected Coordinates
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1 Scton.0.0 onctd oordnats An ntroducton to curlnar coordnat was n n scton.6 whch srs as an ntroducton to ths scton. As ntond thr th orulaton o alost all chancs probls and thr nurcal plntaton and soluton can b achd usn a dscrpton o th probl n trs o artsan coordnats. Howr us o curlnar coordnats allows or a dpr nsht nto a nubr o portant concpts and aspcts o n partcular lar stran chancs probls. hs nclud th notons o th Push orward opraton L drats and obct rats. As wll bco clar not that all th tnsor rlatons prssd n sybolc notaton alrady dscussd such as N n l tc. ar ndpndnt o coordnat syst and hold also or th conctd coordnats dscussd hr..0. onctd oordnats In th artsan syst orthoonal coordnats wr usd. Hr ntroduc th curlnar coordnats. h atral coordnats can thn b wrttn as ( ) (.0.) so and d d d (.0.) ar th coarant bas ctors n th rrnc conuraton wth corrspondn whr contraarant bas ctors..0. wth (.0.) Sold Mchancs Part III 79
2 Scton.0 rrnc conuraton currnt conuraton ur.0.: urlnar oordnats h coordnat curs or a nt n th undord conuraton (or th suracs o constant ). On says that th curlnar coordnats ar conctd or bddd that s th coordnat curs ar attachd to atral partcls and dor wth th body so that ach atral partcl has th sa alus o th coordnats n both th rrnc and currnt conuratons. h coarant bas ctors ar tannt th coordnat curs. In th currnt conuraton th spatal coordnats can b prssd n trs o a nw currnt st o curlnar coordnats ( t) (.0.4) wth corrspondn coarant bas ctors and contraarant bas ctors wth d d d (.0.5) As th atral dors th coarant bas ctors dor wth th body bn attachd to th body. Howr not that th contraarant bas ctors ar not as such attachd; thy ha to b r-aluatd at ach stp o th doraton anw so as to nsur that th rlant rlatons.. ar always satsd. apl onsdr a pur shar doraton whr a squar dors nto a paralllora as llustratd n..0.. In ths scnaro a unt ctor n th squar ts appd to a ctor n th paralllora. h antud o s /sn. hs drs ro th apl workd throuh n scton.6; thr th ctor antand unt antud. Sold Mchancs Part III 80
3 Scton.0 ur.0.: A pur shar doraton onsdr now a paralllora (ntal condton) dorn nto a nw paralllora (th currnt conuraton) as shown n..0.. ur.0.: A pur shar doraton o on paralllora nto anothr Kpn n nd that th ctor wll b o antud /sn th transoraton quatons.0. or th conuratons shown n..0. ar tan tan tan tan (.0.6) onstants ha bn ottd ro ths prssons (whch rprsnt th translaton o th paralllora orn ro th artsan orn). Sold Mchancs Part III 8
4 Scton.0 ollown on ro.6 qns..6.9 th coarant bas ctors ar: tan tan (.0.7) and th nrs prssons tan tan (.0.8) Ln lnts n th conuratons can now b prssd as d dd d d d d d d d (.0.9) h scal actors.. th antuds o th coarant bas ctors ar (s qns..6.6) H H sn h h sn (.0.0) h contraarant bas ctors ar (s qn..6.) tan tan (.0.) and th nrs prssons Sold Mchancs Part III 8
5 Scton.0 tan tan (.0.) h antuds o th contraarant bas ctors ar H h sn H sn h (.0.) h trc cocnts ar (s qns..6.7) 0 0 tan sn tan 0 0 tan sn tan tan sn tan 0 0 tan sn tan (.0.4) h transoraton dtrnants ar (consstnt wth zro olu chan) ro qns dt dt J dt dt dt J dt (.0.5) Sold Mchancs Part III 8
6 Scton.0 apl onsdr a oton whrby a cub o atral wth sds o lnth L 0 s transord nto a cylndr o radus and hht H L 0 H L 0 ur.0.4: a cub dord nto a cylndr A plan w o on quartr o th cub and cylndr ar shown n P L 0 p ur.0.5: a cub dord nto a cylndr h oton and nrs oton ar n by χ() L 0 L 0 H L 0 (bass: ) (.0.6) and Sold Mchancs Part III 84
7 Scton.0 χ ( ) L0 L0 L0 H (bass: ) (.0.7) Introducn a st o conctd coordnats..0.6 th atral and spatal coordnats ar ( ) and (ths ar sply cylndrcal coordnats) ( ) L0 L0 L0 H cos sn tan (.0.8) (.0.9) A typcal atral partcl (dnotd by p) s shown n Not that th poston ctors or p ha th sa alus snc thy rprsnt th sa atral partcl. 4 p p ur.0.6: curlnar coordnat curs Sold Mchancs Part III 85
8 Scton.0.0. h Doraton radnt Wth conctd curlnar coordnats th doraton radnt s (.0.0) h doraton radnt oprats on a atral ctor (wth contraarant coponnts) V V rsultn n a spatal tnsor (wth th sa coponnts V ) or apl d d d d (.0.) o phass th pont ln lnts appd btwn th conuratons ha th sa coordnats : a ln lnt d d d ts appd to d d d d d d (.0.) hs shows also that ln lnts tannt to th coordnat curs ar appd to nw lnts tannt to th nw coordnat curs; th coarant bas ctors ar a ld o tannt ctors whch t appd to th nw ld o tannt ctors as llustratd n d d ur.0.7: Vctors tannt to coordnat curs Sold Mchancs Part III 86
9 Scton.0 h doraton radnt th transpos and th nrss ap th bas ctors n on conuraton onto th bas ctors n th othr conuraton (that th and n ths quaton ar ndd th nrss o and ollows ro.6.6): Doraton radnt (.0.) hus th tnsors and ap th coarant bas ctors nto ach othr whras th tnsors and ap th contraarant bas ctors nto ach othr as llustratd n contraarant bass coarant bass ur.0.8: th doraton radnt ts transpos and th nrss It was ntond abo how th doraton radnt aps bas ctors tanntal to th coordnat curs nto nw ctors tanntal to th coordnat curs n th currnt conuraton. In th sa way contraarant bas ctors whch ar noral to coordnat suracs t appd to noral ctors n th currnt conuraton. or apl th contraarant ctor s noral to th surac o constant and ts appd throuh to th nw ctor whch s noral to th surac o constant n th currnt conuraton. Sold Mchancs Part III 87
10 Scton.0 apl contnud arryn on apl ro abo n artsan coordnats 4 cornrs o an ntal paralllora (s..0.) t appd as ollows: / tan / tan / tan / tan (.0.4) hs corrsponds to a doraton radnt wth rspct to th artsan bass: whr 0 (.0.5) tan tan (.0.6) ro th arlr work wth apl th doraton radnt can b r-prssd n trs o drnt bas ctors: tan tan (.0.7) whch s qn In act can b prssd n a ulttud o drnt ways dpndn on whch bas ctors ar usd. or apl ro th abo can also b prssd as Sold Mchancs Part III 88
11 Scton.0 Sold Mchancs Part III 89 tan tan tan 0 tan 0 tan tan tan 0 0 (.0.8) (hs can b rd usn qn..0.0a blow.) oponnts o h arous coponnts o and ts nrss and th transposs wth rspct to th drnt bass ar: (.0.9) h coponnts o wth rspct to th rrnc bass ar
12 Scton.0 Sold Mchancs Part III 90 k k k k (.0.0) and slarly or th coponnts wth rspct to th currnt bass. oponnts o th Bas Vctors n drnt Bass h bas ctors thsls can b prssd altrnatly: (.0.) shown that so o th coponnts o th doraton radnt can b wd also as coponnts o th bas ctors. Slarly (.0.) or th contraarant bas ctors on has (.0.) and (.0.4).0. ducton to Matral and Spatal oordnats Matral oordnats Suppos that th atral coordnats wth artsan bass ar usd (rathr than th conctd coordnats wth curlnar bass ) hn
13 Scton.0 Sold Mchancs Part III 9 (.0.5) and rad rad (.0.6) whch ar qns hus rad s th notaton or and rad s th notaton or to b usd whn th atral coordnats ar usd to dscrb th doraton. ur.0.9: Matral coordnats and dord bass Spatal oordnats Slarly whn th spatal coordnats ar to b usd as ndpndnt arabls thn (.0.7) and currnt conuraton rrnc conuraton
14 Scton.0 rad rad (.0.8) h dscrptons ar llustratd n Not that th bas ctors sa n ach o ths cass (curlnar atral and spatal). ar not th rad rad ur.0.0: doraton dscrbd usn drnt ndpndnt arabls Sold Mchancs Part III 9
15 Scton Stran nsors h auchy-rn tnsors h rht auchy-rn tnsor and th lt auchy-rn tnsor b ar dnd by qns b b b b (.0.9) hus th coarant coponnts o th rht auchy-rn tnsor ar th trc cocnts. hs hhlhts th portanc o : th a clar asur o th doraton occurrn. (It s possbl to aluat othr coponnts o.. and also ts coponnts wth rspct to th currnt bass but only th coponnts wth rspct to th rrnc bass ar (norally) usd n th analyss.) h Strtch Now analoous to..9.. ds ds d d dd d d db d (.0.40) so that th strtchs ar analoous to..7 ds ds ds ds d d d d d b d d d d d db d d d d b d (.0.4) h rn-laran and ulr-alans nsors h rn-laran stran tnsor and th ulr-alans stran tnsor ar dnd throuh....4 Sold Mchancs Part III 9
16 Scton.0 ds ds ds ds d d I d dd I b d dd (.0.4) h coponnts o and can b aluatd throuh (wrtn I th dntty tnsor prssd n trs o th bas ctors n th rrnc conuraton and I th dntty tnsor prssd n trs o th bas ctors n th currnt conuraton) b (.0.4) Not that th coponnts o and wth rspct to thr bass ar qual ths s not tru rardn thr othr coponnts.. ). (althouh apl contnud arryn on apl ro abo consdr now an apl ctor V V V (.0.44) y h contraarant and coarant coponnts ar V V V y V tan V (.0.45) V V V y y tan h antud o th ctor can b calculatd throuh (s qn..6.5 and.6.49) V VV V V y V V VV VV V V V V tan tan y y y y V V y y VV VV V V V V tan tan (.0.46) Sold Mchancs Part III 94
17 Scton.0 h nw ctor s obtand ro th doraton radnt: V V Vy V 0 V y V y Vy 0 V V Vy V tan tan 0 V V y y (.0.47) In trs o th contraarant ctors: V Vy V V (.0.48) y tan tan Not that th contraarant coponnts do not chan wth th doraton but th coarant coponnts do n nral chan wth th doraton. h antuds o th ctors bor and atr doraton ar n by th auchy-rn stran tnsors whos cocnts ar thos o th trc tnsors (th rst o ths s th sa as.0.46) VV b k l k l VV V V VV V V VV k l k l (.0.49) ro ths th antud o th ctor atr doraton s VV V V V V V (.0.50) y y y.0.5 Intrdat onuratons Strtch and otaton nsors h polar dcopostons ha bn dscrbd n..5. h dcopostons ar llustratd n..0.. In th atral dcoposton th atral s rst strtchd by and thn rotatd by. Lt th bas ctors n th assocatd ntrdat conuraton b ĝ. Slarly n th spatal dcoposton th atral s rst rotatd by and thn strtchd by. Lt th bas ctors n th assocatd ntrdat conuraton n ths cas b. hn analoous to qn..0. { Probl } Sold Mchancs Part III 95
18 Scton.0 Sold Mchancs Part III 96 (.0.5) (.0.5) ur.0.: th atral and spatal polar dcopostons Not that and sytrc so (.0.5) (.0.54) Slarly or th rotaton tnsor wth orthoonal (.0.55) (.0.56) ĝ Ĝ
19 Scton.0 Sold Mchancs Part III 97 h abo rlatons can b chckd usn qns..0. and tc. Varous rlatons btwn th bas ctors can b drd or apl (.0.57) Doraton radnt latonshp btwn Bass h arous bas ctors ar rlatd abo throuh th strtch and rotaton tnsors. h ntrdat bass ar rlatd drctly throuh th doraton radnt. or apl ro.0.5a.0.55b (.0.58) In th sa way (.0.59) nsor oponnts h strtch and rotaton tnsors can b dcoposd alon any o th bass. or th ost natural bass would b and or apl (.0.60) wth. On also has
20 Scton.0 Sold Mchancs Part III 98 (.0.6) wth slar sytry. Also (.0.6) and (.0.6) wth slar sytry. Not that coparn.0.60a.0.6a.0.6a.0.6a and usn.0.57 (.0.64) Now not that rotatons prsr ctors lnths and n partcular prsr th trc.. (.0.65) hus aan usn.0.57 and th contraarant coponnts o th abo tnsors ar also qual.
21 Scton.0 As ntond th tnsors can b dcoposd alon othr bass or apl (.0.66).0.6 nctors and nalus Analoous to..5 th nalus o ar dtrnd ro th nalu probl dt ladn to th charactrstc quaton..5 wth prncpal scalar narants..6-7 I 0 (.0.67) I II III 0 (.0.68) I II III tr A (tr) tr( ) dt k k (.0.69) h nctors ar th prncpal atral drctons h spctral dcoposton s thn IN 0 N wth (.0.70) N N (.0.7) whr and th ar th strtchs. h rann spctral dcopostons n..7 hold also. Not also that th rotaton tnsor n trs o prncpal drctons s (s..5) n N n N (.0.7) whr n ar th spatal prncpal drctons. Sold Mchancs Part III 99
22 Scton Dsplacnt and Dsplacnt radnts onsdr th dsplacnt u o a atral partcl. hs can b wrttn n trs o coarant coponnts and u : u u. (.0.7) h coarant drat o u can b prssd as u u (.0.74) h snl ln rrs to coarant drntaton wth rspct to th undord bass.. th hrstol sybols to us ar unctons o th. h doubl ln rrs to coarant drntaton wth rspct to th dord bass.. th hrstol sybols to us ar unctons o th. Altrnatly th coarant drat can b prssd as u (.0.75) and so u u (.0.76) h last qualts ollown ro.0.-. h coponnts o th rn-laran and ulr-alans stran tnsors.0.4 can b wrttn n trs o dsplacnts usn rlatons.0.76 { Probl }: n n n u u un u (.0.77) In trs o spatal coordnats / coponnts o th ulr-laran stran tnsor ar / th Sold Mchancs Part III 00
23 Scton.0 (.0.78) n k k n whch s h Doraton o Ara and Volu lnts Drntal Volu lnt onsdr a drntal olu lnt ord by th lnts conuraton qn..6.4: d n th undord dv d d d (.0.79) whr qn dt (.0.80) h sa olu lnt n th dord conuraton s dtrnd by th lnts d : d d d d (.0.8) whr dt (.8.8) ro.6.5 t sq..0. k k dt k k (.0.8) whr k s th artsan prutaton sybol and so th Jacoban dtrnant s (s..5) d J dt (.0.84) dv Sold Mchancs Part III 0
24 Scton.0 and dt s th dtrnant o th atr wth coponnts. Drntal Ara lnt onsdr a drntal surac (paralllora) lnt n th undord conuraton () () boundd by two ctor lnts d and d and wth unt noral N. hn th ctor noral to th surac lnt and wth antud qual to th ara o th surac s usn.6.54 n by N ds d d d d () () k () () () () k (.0.85) d d whr k s th prutaton sybol assocatd wth th bass.. k k k k. (.0.86) sn k k on has N ds () () k k d d (.0.87) Slarly th surac ctor n th dord conuraton wth unt noral n s n ds () () () () () () k d d d d k d d (.0.88) whr k s th prutaton sybol assocatd wth th bass.. k k k k. (.0.89) oparn th two prssons or th aras n th undord and dord conuratons on nds that n ds NdS dt NdS (.0.90) whch s Nanson s rlaton qn hs s consstnt wth was sad arlr n rlaton to..0.8 and th contraarant bass: aps ctors noral to th coordnat curs n th ntal conuraton nto corrspondn ctors noral to th coordnat curs n th currnt conuraton. Sold Mchancs Part III 0
25 Scton Probls. Dr th rlatons s rlatons.0.76 wth and to dr.0.77 n n n u u un u Sold Mchancs Part III 0
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