On Similarity Transformations A Classical Approach

Transcription

1 O Similari rasformaios A Classial Aroah Edard M. Rose EMR eholo Gro Irodio Similari rasformaios are ofe ilized o oer arial differeial eqaios o a se of ordiar differeial eqaios []. he ordiar differeial eqaios ma he e soled meriall [,] or a losed form solio ma ossil e fod. Sramaia [] meios ha sdes ofe hae diffilies i arri o he rasformaio. he reaso for his ma e ha he deriaio of he rasform is lo ad edios. I ma also e ha he roedres for he deriaio are o ell ko or ma o e easil rerieed from he lierare. He sess he se of Male [5] o aid he sde. I he rose of his aer o deail he se of oe sh rasformaio ad sho ho he deelome of aoher is arried o iho he se of a omer alera ssem. A Simle rasformaio Cosider he rasi hea-odio rolem i a sla. he aliale eqaios ad odar odiios are:,0 0 0, ad, 0 Assme folloi rasformaio ariale is ko: o imleme he rasformaio deermie he folloi: / /

2 - - Also: 5 Ssii Eqaios ad 5 io Eqaio - Simlifi: 0 Sie is ol a fio of ad has ee elimiaed, a e relaed U: 0 d du d U d 6 B ssii he odar odiios of Eqaios io he e odar

3 odiios eome: U0 U 0 7 A solio o Eqaio 6 sje o Eqaios 7 is ko o e [] U erf I erms of he oriial ariales:, erf Fidi a rasformaio he eqaios for he free oeio from a erial fla lae hae ee eesiel sdied [,,6]. he aliale eqaios are [6] Coii 0 8 Hdrodami υ 9 Eer 0 I erms of he dimesioless ariale Coii 0 Hdrodami υ

4 Eer Bodar odiios: 0: 0, 0. : 0, 0, Phsial roeries are assmed osa I order o deelo a rasformaio roeed as follos [6] : Irode a sream fio i ad - ad ssie io Eqaios -: 0 υ 5 6 No assme ha:, 7 ad 8 he 9 ad 0

5 5 Callae he folloi deriaies oi ha is a fio of : A. B. C. D. { E. Ssie A, B, C, D, E io Eqaio 5, simlif ad diide : ν If is o disaear from Eqaio he folloi ms hold: 0

6 6 0 Sole for ad : Ssie he ales of ad io Eqaio ad mlil hroh : ν Sie ad are arirar, le ad ν Sole for ad : υ ad ν Eqaio eomes 0 Proeed i a similar maer for hih is assmed o e ol a fio of : F: G:

7 7 H: Ssie Eqaios A, D ad F, G, H ad io Eqaio 6 ad simlif: Ssie for, ad ad simlif: Pr Bodar odiios are: 0 0 ad a 0 ad 0, 0 a he deede ariales ad are he: / ν ν / he merial solio o Eqaios ad is disssed i []. Colsios he se of similari rasformaios o oer arial differeial eqaios o ordiar differeial eqaios a e lo ad edios. Hoeer, he roedre is sraihforard ad a e arried o i a lassial fashio.

8 8 Referees. Cho, Y ad Aessoos, A., Similari rasformaio mehods i he aalsis of he o dimesioal sead omressile lamiar odar laer, erm aer:.6 Comressile Flid Damis, Sri 00, Mehaial Eieeri, MI. h://o.mi.ed/nr/rdolres/mehaial-eieeri/-6sri00/9a55f7-c6ed- BA-878C-97FADC9C7C/0/a_ho_aessoos.df. Rose, E. Femla.: he Verial Heaed Plae, CACHE Nes, No 60 Sri 005. Goldsei, A. S., A Comaioal Model for eahi Free Coeio, Chemial Eieeri Edaio, 8, o, Fall Sramaia, V. R., Comer-Failiaed Mahemaial Mehods i CHE Similari Solio, Chem. E Edaio, 0, o, Fall 006, < h://.malesof.om> 6. Boeler, L. M. K, Cherr, V. H., Johso, H. A. ad Marielli, R. C., Hea rasfer Noes, Uiersi of Califoria Press, Berkele ad Los Aeles, 98. Nomelare Free Coeio rasformaio Elish seifi hea, J/k K raiaioal osa,m/s k hermal odii, J/s m K Pr Pradl Nmer µ /k emerare, K isohermal emerare of he lae, K emerare far from he lae, K

9 9 eloi i direio, m/s eloi i direio, m/s disae measred from oom of lae, ard, m disae measred from loer ede of lae o rih, m Greek hermal diffsii, k / ρ - m /s hermal easio oeffiie, / for ideal as ideede ariale dimesioless emerare - }/ - µ dami isosi k/m s fio i similari rasformaio ρ desi, k/m υ kiemai isosi, m /s µ / ρ sream fio