Amin Haliloic Mah Eciss E-mail : 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
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
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
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
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
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 6 6 8 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:
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 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 : Incompssibl conini qaion 0 q a Nai-Soks qaions in lindical coodinas: -componn: μ q b -componn: μ q c -componn:
Amin Haliloic 8 Mah Eciss μ q d
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 000 0 Answ: ad di o 000 0 Δ 0 0 Answ: Th ncion ln saisis h Laplac qaion Answ: Δ di cl ad Solion: Δ 6
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
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 0 8 0 6 8 S onsqnl
Amin Haliloic Mah Eciss S 8 8 9 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
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
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:
Amin Haliloic Mah Eciss a 8 8 7 7 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 6 6 8 di U di Γad di cl U 6 6 8 sinc cl U 0 w ha di cl U 0
Amin Haliloic 6 Mah Eciss di w di Γ Γ Γ 6 6 8 w Γ Γ Γ 6 6 8 q W sbsi U and in h qaion q and 8 8 φ Γ Γ No ha Γ is a consan 6 6 6 8 6Γ 6 6 8 0 6Γ 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
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
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
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
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 0 000 0 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
Amin Haliloic Mah Eciss Sp W ind h loci componn w W sol q -c wih bondais c and c6: w 0 q -c 0 000 w 0 c w 0 0 c6 dw Rmak: Tchnicall w can wi insad d aiabl om q -c w ha 0 0 000 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