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1 Amin Haliloic Mah Eciss amin@shkhs wbpa : wwwshkhs/amin MATH EXERISES GRADIENT DIVERGENE URL DEL NABLA OERATOR LALAIAN OERATOR ONTINUITY AND NAVIER-STOKES EQUATIONS VETOR RODUTS I and hn scala o do podc k j i co o coss podc In som books is also considd o podc dind b GRADIENT DIVERGENE URL DEL NABLA OERATOR LALAIAN OERATOR GRADIENT L b a scala ild Th adin is h co ild dind b ad DIVERGENE L R Q b a co ild coninosl diniabl wih spc o and Thn h dinc o is h scala ild dind b R Q di URL Th cl o is h co ild dind b k Q j R i Q R R Q k j i cl Q R Q R DEL NABLA OERATOR Th co dinial opao k j i is calld dl o nabla

2 Amin Haliloic Mah Eciss Usin w can dno ad di and cl as blow: ad di cl No ha is no h sam as R Q LALAIAN OERATOR Th Laplacian opao Δ is dind o a scala ild U b U U U U U Δ and o a co ild R Q b R Q Δ Δ Δ Δ lindical coodinas : ansomaion: cos sin olm lmn: d d d dv local basis: k j i j i cos sin sin cos Vco componns laionship: sin cos ϑ cos sin scala ild: adin: ad laplacian: Δ co ild: dinc: di o cl: cl k

3 Amin Haliloic Mah Eciss EXERISES ind a di b ad di i and c cl ind ad di cl i Which on o h ollowin ncions a b ln c p saisis h Laplac qaion Δ 0? ind Δ i Wi h nal anspo qaion U Γ S φ wiho sin opaos di Δ cl o ad H U w ncions Γ S w a al ncions o and 6 Which on i an o h ollowin ncions a b c saisis h qaion U Γ S? H Γ U and S 7 ind which on i an o h ollowin ncions a b c saisis h qaion di U di Γad S wh Γ U and S 8

4 Amin Haliloic Mah Eciss 8 am 008 A Wi h nal anspo qaion U Γ S q wiho sin opaos di Δ cl o ad H U w ncions Γ S w a al ncions o and B L Γ U ind S in h qaion q i w now ha h ncion saisis h qaion 9 Q6 am 008 onsid h ollowin qaion U Γ U 6 L Γ consan U ind h consan Γ in h qaion q i w now ha h ncion saisis h qaion q 0 I possibl ind o h in paial diais and a and b and c and d and Hin: Ncssa condiion: I has conins diais hn h mid diais o shold b qal Ths * is h ncssa condiion o h isnc o a ncion ha has h in diais

5 Amin Haliloic Mah Eciss I possibl ind o h in paial diais and a and b and c and d and Hin: Ncssa condiion: I has coninos diais hn h mid diais o shold b qal Ths on : on : on : a h ncssa condiion o h isnc o a ncion ha has h in diais W consid an incompssibl dnsi cons sad sa aiabls do no dpnd on im isohmal Nwonian low wih a in loci ild w V Us h ollowin qaions conini and Nai Soks qaions o ind n pssion o pss as a ncion o and wh consan μ consan 00 i 0 and / 98 wh s m Incompssibl conini qaion: 0 w q Nai Soks qaions: componn: w μ q componn: w μ q

6 Amin Haliloic 6 Mah Eciss componn: w w w w w w w w μ q a V 0 b V c V am 009 A onsid h ollowin qaion U Γ U L Γ consan U ind h consan Γ in h qaion q i w now ha h ncion saisis h qaion q B W consid an incompssibl dnsi cons sad sa aiabls do no dpnd on im isohmal Nwonian low wih a in loci ild V w Us h ollowin qaions conini and Nai Soks qaions o ind n pssion o pss as a ncion o and wh consan μ consan 00 i 0 and wh 98m / s and V 6 am 009 W consid an incompssibl dnsi cons sad sa aiabls do no dpnd on im isohmal Nwonian low wih a in loci ild V w Us h ollowin qaions conini and Nai Soks qaions o ind is i paam a and hn ii n pssion o pss as a ncion o and wh consan μ consan 00 i 0 and wh 98m / s and V a onsid sad incompssibl isohmal lamina saiona Nwonian low in a lon ond pip in h -dicion wih consan cicla coss-scion o adis R m Us h conini and h Nai-Soks qaions in clindical coodinas o ind h loci ild V and h pss ild i h lid low saisis h ollowin condiions:

7 Amin Haliloic 7 Mah Eciss c0 All paial diais wih spc o im a 0 Sad low c μ000 k/m s and 000 k/m c A onsan pss adin / /0 a/m is applid in h hoional ais -ais in o noaion: / /0 c Th low is paalll o h ais ha is 0 and 0 c W assm ha h low is aismmic Th loci dos no dpnd on ha is 0 c Bonda cond No-slip bonda condiion V lid V wall : I hn 0 c6 Bonda condiion : has maimm a 0 ha is Th conini and h Nai-Soks qaions o an incompssibl isohmal Nwonian low dnsi cons iscosi μ cons wih a loci ild V in lindical coodinas : Incompssibl conini qaion 0 q a Nai-Soks qaions in lindical coodinas: -componn: μ q b -componn: μ q c -componn:

8 Amin Haliloic 8 Mah Eciss μ q d

9 Amin Haliloic 9 Mah Eciss ANSWERS AND SOLUTIONS: Solion: Q R a Sinc di w ha di 0 0 Answ a di b Sinc ad w ha o di ad di 00 Answ b ad di 00 i j k i d c cl Q R i j k Answ c cl j k Solion: i j cl k i j k 0 Ths di cl 0 and ho ad di cl Answ: ad di o Δ 0 0 Answ: Th ncion ln saisis h Laplac qaion Answ: Δ di cl ad Solion: Δ 6

10 Amin Haliloic 0 Mah Eciss Γ φ S ad di U di Γ Γ Γ φ S di w di φ S w Γ Γ Γ 6 Which on i an o h ollowin ncions a b c saisis h qaion U S Γ? H Γ U and S Solion : Th qaion U S Γ can b win as q Γ di di S ad di U di a L Vi calcla h diais o and sbsi in h l hand sid LHS and ih hand sid o h qaion q LHS: RHS 60 Whnc RHS LHS Ths h ncion is no a solion o h qaion b

11 Amin Haliloic Mah Eciss LHS RHS Whnc LHS RHS and h ncion is no a solion o h qaion c L Thn LHS 6 RHS 7 Ths LHS RHS and h ncion is no a solion o h qaion Answ: Non o h ncions saisis h qaion 7 Answ: ncion saisis h qaion 8 am 98 A Wi h nal anspo qaion U Γ S q wiho sin opaos di Δ cl o ad H U w ncions Γ S w a al ncions o and B L Γ U ind S in h qaion q i w now ha h ncion saisis h qaion Solion: A U Γ S di U di Γad S di w di Γ Γ Γ S w Γ Γ Γ S q B W sbsi Γ U and in h qaion q and 8φ 0 S S onsqnl

12 Amin Haliloic Mah Eciss S Q6 am 008 onsid h ollowin qaion U Γ U 6 q L Γ consan U ind h consan Γ in h qaion q i w now ha h ncion saisis h qaion Solion: U Γ U 6 di U di Γad di cl U 6 sinc cl U 0 w ha di cl U 0 di w di Γ Γ Γ 0 6 w Γ Γ Γ 0 6 q W sbsi U and in h qaion q and φ Γ No ha Γ is a consan 6 0 Γ Γ 6 8 Γ Γ Γ 6 Γ Answ: Γ 0 I possibl ind o h in paial diais and a and b and c and d and Hin: Ncssa condiion: I has coninos diais hn

13 Amin Haliloic Mah Eciss h mid diais o shold b qal i * is h ncssa condiion o h isnc o a ncion ha has h in diais Answ: a b c d No solion sinc h condiion * is no lilld Solion a Sinc and h diais a coninos h condiion * is lilld and w can ind o h in diais In od o ind w ina wih spc o h is o h qaions q q and d Ths i W ha inad wih spc o ho h consan sill dpnd on Now o ind w dinia and sbsi i in q and : inall sbsiin in i w ha wh is a consan Answ: a b c d No solion sinc h condiion on is no lilld Solion a

14 Amin Haliloic Mah Eciss a and Sinc h condiions on a lilld and w can ind o h in diais In od o ind w ina wih spc o h is o h qaions q q q and d Ths i W ha inad wih spc o ho h consan sill dpnd on and Now o ind w dinia and sbsi i in q and : W ha inad wih spc o ho h consan sill dpnd on and Ths ii Now sbsiin ii in q w ha inall sbsiin in ii w ha wh is a consan alclaion o h pss ild o a known loci ild o an incompssibl sad sa isohmal Nwonian low Answ:

15 Amin Haliloic Mah Eciss a b c Solion a W sbsi w 0 in q and no ha al diais wih spc o a 0: onini qaion: 0 0 qi idnicall lilld Nai Soks qaions: componn: 6 qi componn: 6 qi componn: 0 qi Now qi is 8 * Sbsiion in qi implis 6 8 Hnc om * w ha 8 8 ** Now w sbsi ** in qi and 0 0 wh is a consan inall sbsiin in ** w ha 8 8 wh is a consan Solion A: U Γ U di U di Γad di cl U sinc cl U 0 w ha di cl U 0

16 Amin Haliloic 6 Mah Eciss di w di Γ Γ Γ w Γ Γ Γ q W sbsi U and in h qaion q and 8 8 φ Γ Γ No ha Γ is a consan Γ Γ Γ Answ A: Γ Solion B: W sbsi 6 w in q and no ha al diais wih spc o a 0: onini qaion: 0 0 qi idnicall lilld Nai Soks qaions: componn: 6 qi componn: 8 qi componn: qi Now qi is 8 * Sbsiion in qi implis 8 8 Hnc om * w ha 8 8 ** Now w sbsi ** in qi and wh is a consan inall sbsiin in ** w ha Γ 6 6 8

17 Amin Haliloic 7 Mah Eciss 8 8 Answ B: 8 8 wh is a consan Solion a V is w sbsi a w in q and no ha al diais wih spc o a 0: onini qaion: 0 a a No w ha V Usin h Nai Soks qaions w : componn: 6 9 qi componn: qi componn: qi Now qi is 6 9 * Sbsiion in qi implis Hnc om * w ha 6 9 ** W sbsi ** in qi and wh is a consan inall sbsiin in ** w ha 6 9 Answ : 6 9 wh is a consan

18 Amin Haliloic 8 Mah Eciss Q onsid sad incompssibl isohmal lamina saiona Nwonian low in a lon ond pip in h -dicion wih consan cicla coss-scion o adis R m Us h conini and h Nai-Soks qaions in clindical coodinas o ind h loci ild V and h pss ild i h lid low saisis h ollowin condiions: c0 All paial diais wih spc o im a 0 Sad low c μ000 k/m s and 000 k/m c A onsan pss adin / /0 a/m is applid in h hoional ais -ais in o noaion: / /0 c Th low is paalll o h ais ha is 0 and 0 c W assm ha h low is aismmic Th loci dos no dpnd on ha is 0 c Bonda cond No-slip bonda condiion V lid V wall : I hn 0 c6 Bonda condiion : has maimm a 0 ha is 0 0 Th conini and h Nai-Soks qaions o an incompssibl isohmal Nwonian low dnsi cons iscosi μ cons wih a loci ild V in lindical coodinas : SOLUTION Incompssibl conini qaion 0 q a Nai-Soks qaions in lindical coodinas: -componn: μ q b -componn: μ q c

19 Amin Haliloic 9 Mah Eciss -componn: μ q d W choos as a ical ais an a in a hoional plan and h low is paalll wih h -ais W dno loci co V wh and a -componn - componn and -componn in clindical coodinas Accodin o h assmpions w ha 0 0 and dos no dpnd on Sinc is h ical ais w ha ha co - 00 wh 98 m/s which in clindical coodinas is cos sin and 0 Now w sbsi / /0 a/m μ000k /ms in h conini and Nai- Soks qaions: Sinc 0 and 0 accodin o c conini qaion in clindical coodinas 0 is 0

20 Amin Haliloic 0 Mah Eciss This lls s ha is no a ncion o hmo sinc loci dos no dpnd on assmpion c w concld ha dpnds onl on To simpli noaion w dno w * Now w sbsi cos sin and 0 / /0 a/m μ000k /ms in h Nai-Soks qaions: Th -componn o h Nai-Soks qaion is: 0 cos q -c Th -componn o h Nai-Soks qaion: 0 sin q -c Th Z-componn o h Nai-Soks qaion wh w and w 0 q -c is: Sp W ind h pss In od o ind h pss w sol q -c q -c and h qaion is 0 ha cos sin 0 om hos qaions w cos 0

21 Amin Haliloic Mah Eciss Sp W ind h loci componn w W sol q -c wih bondais c and c6: w 0 q -c w 0 c w 0 0 c6 dw Rmak: Tchnicall w can wi insad d aiabl om q -c w ha w w w sinc w is now a ncion o onl on w sbsiion 0 and c6 0 w w w sbsiion and c w Ths w and V 0 0 Answ : cos 0 V 0 0

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The Simply Typed Lambda Calculus Type Inference Instead of writing type annotations, can we use an algorithm to infer what the type annotations should be? That depends on the type system. For simple type systems the answer is yes, and

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