Flow pattern induced by the plane piston moving in a non-ideal gas with weak gravitational field

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1 Ain Shm Engineering Journl (), 5 3 Ain Shm Univerity Ain Shm Engineering Journl ENGINEERING PHYSICS AND MATHEMATICS Flow pttern induced by the plne piton moving in non-idel g with wek grvittionl field L.P. Singh, S.D. Rm *, D.B. Singh Deprtment of Applied Mthemtic, Intitute of Technoly, Bnr Hindu Univerity, Vrni 5, Indi Received Februry revied 8 June ccepted 3 July Avilble online Augut KEYWORDS Wek grvittionl field Non-idel g Perturbtion method Abtrct A ytemtic perturbtion cheme i ued to nlyze the flow pttern induced by the motion of plne piton mov ing with contnt velocity in non-idel g with wek grvittionl field. The flow vrible re epnded erie of mll prmeter e, which i the rtio of the typicl ecpe velocity to the plm velocity. The zeroth order olution give the uniform flow which i influenced by the prmeter of non-idelne without grvity. The firt order reltionhip how the influence of non-idelne of the g well the pplied grvity on the flow field. Ditribution of the flow vrible h been preented grphiclly for vriou vlue of prmeter of non-idelne. The tructure of the hock front in ce of mll perturbtion theory i lo dicued. Ó Ain Shm Univerity. Production nd hoting by Elevier B.V. All right reerved.. Introduction In the tudy of mthemticl theory of compreible fluid dynmic, the piton problem i bic prototype problem. Generlly, if the piton i pulled bck, then rrefction wve will be formed, nd otherwie, if the piton i puhed forwrd, then the puh will cue compreed wve moving * Correponding uthor. Mobile: E-mil ddre: rm.r.pm@itbhu.c.in (S.D. Rm) Ó Ain Shm Univerity. Production nd hoting by Elevier B.V. All right reerved. Peer review under reponibility of Ain Shm Univerity. doi:.6/j.ej..7. Production nd hoting by Elevier into the g. In [ 4,6] it i decribed by tking long tube cloed by piton t one end nd open t the other end nd uming tht the g in the tube i ttic with uniform preure p nd denity q then ny motion of the piton will cue correponding motion of the g. Cheter [5] invetigted theoreticlly the gdynmic diturbnce produced in g-filled tube by vibrting piton uing mll-mchnumber compreible flow model. Wen-rui [7] ued the perturbtion method to tudy the wek hock nd trong hock problem generted by the piton motion in wek grvittionl field. Ro nd Purohit [8] hve nlyzed the elf imilr flow of non-idel g driven by n epnding g. Wu nd Robert [9] nd Robert nd Wu [] conidered the flow of g driven by n epnding piton nd dicued the hock wve theory of onoluminecence by tking imilr eqution of tte of the medium. Skuri [] invetigted the propgtion of phericl hock wve though elf grvitting polytropic g phere uch tr by uing the method of power erie epnion.

2 6 L.P. Singh et l. In the preent pper, untedy gdynmic concept re ued to model the piton-driven compreion of confined g. Perturbtion method, bed on the limit of low Mch number, i ued to contruct olution. We nlyze the vrition of hock trength nd it propgtion in non-idel fluid with wek grvittionl field. The effect of vn der Wl ecluded volume nd wve front geometry on the evolutionry behviour of hock wve re dicued. Thi type of problem in non-idel g h importnce t high temperture becue the vlidity of umption of the idel g t high temperture i not vlid. The eqution of tte for non-idel ge uch low denity ge i tken from the ttiticl phyic in the implified form Aniimov nd Spiner [].. Bic eqution For the motion of trnient g in locl region of tellr tmophere, we dopt coordinte ytem in which the origin i t the center of the tr, nd the -i in the direction of the tellr rdiu. oq ot þ u oq o þ q ou o ¼ ð:þ ou ot þ u ou o þ op q o ¼ Gm ð:þ o ot p ðm bþ c þ u o o p ðm bþ c ¼ ð:3þ where the time t nd the pce co-ordinte re the two independent vrible, q i the denity, p i the preure, u i the velocity of the prticle, m =/q i the pecific volume, b i the Vn der Wl ecluded volume which lie in the rnge b 6. 3, G i the grvittionl contnt, m i the tellr m nd c the pecific het rtio. Eq. (.), (.), (.3) my be reduced to pure one dimenionl non-idel gdynmic eqution if we ignore the grvity. In the dimenionl nlyi, velocity dimenion my be contructed n ¼ t : ð:4þ However, Eq. (.), (.), (.3) contin the grvittionl contnt G which introduce nother quntity with velocity dimenion, tht i rffiffiffiffiffiffiffi Gm u g ¼ : ð:5þ The ytem of Eq. (.), (.), (.3) i upplemented with n eqution of tte [4] pð bqþ ¼qRT ð:6þ where R i the g contnt nd T i the temperture. In the ce of wek grvity, the typicl grvittionl velocity (.5) i mller thn both the onic velocity nd the plm velocity nd bic flow i the Riemnn flow. We deire to find the influence of the non-idelne on the flow field in the preence of grvity. Let u introduce the non-dimenionl prmeter ~u ¼ u ~ ¼ ~t ¼ t u u =t ~ ¼ ð:7þ nd the mll non-dimenionl prmeter e ¼ u g ð:8þ u p where ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cp=qð bqþ i the ound peed in non-idel medium, u *, * nd t * repreent repectively the typicl velocity, length nd time. In ce of ientropic flow, the ytem of Eq. (.), (.), (.3) tke the following form o ot þ u o o ðc þ bqþ þ ou ð bqþ o ¼ ð:9þ ou ot þ u ou o þ ð bqþ ðc þ bqþ o o ¼ e : ð:þ The olution of Eq. (.9), (.) cn be contructed in the power erie of e [5] 9 ¼ þ e þ e þ >= u ¼ u þ eu þ e u þ > : ð:þ q ¼ q þ eq þ e q þ With the help of Eq. (.) the bic eqution of zero order tke the form o ot þ u o ou ot þ u ou o þ o þ ðc þ bþ ð bþ ð bþ ðc þ bþ ou o ¼ o o ¼ ð:þ ð:3þ where b ¼ bq. Auming = (n), u = u (n), Eq. (.) nd (.3) trnform to ðu nþ d dn þ ðc þ bþ du ð bþ dn ¼ ð:4þ ðu nþ du dn þ ð bþ ðc þ bþ d dn ¼ : ð:5þ The olution of the Eq. (.4) nd (.5) yield u ð bþ ¼ cont: ð:6þ c which i the Riemnn invrint nd thu Eq. (.4) nd (.5) repreent the Riemnn flow in non-idel gdynmic. A pecil olution of the piton problem my be found in the following mnner. If the piton move with contnt velocity, the flow prmeter hed of the piton i tken u ¼ cont: ¼ cont: ð:7þ Similrly, the firt-order eqution my be written o ot þ u o ou ot þ u ou o þ o þ ðc þ bþ ð bþ ð bþ ðc þ bþ ou o ¼ o o ¼ ð:8þ ð:9þ nd o on for eqution of higher order. In thi pproch, the zeroth order eqution nd ll eqution of higher order, ecept the firt order eqution, re non-liner. Thu the influence of the pplied grvity in non-idel g i non-liner. Now we dicu bout the olution of the firt order eqution which re the function of n nd t. Let u ume

3 Flow pttern induced by the plne piton moving in non-idel g with wek grvittionl field 7 u ðn tþ ¼t m fðnþ ðn tþ ¼t n rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gðnþ: ð:þ cp ¼ We infer from Eq. (.8) nd (.9), m = n = +. If the q ð bq Þ dimenionl typicl velocity u * i umed the piton velocity u p which i contnt, the bic velocity i u =. ¼ p d q The boundry condition t the piton become p q ð3:6þ ð3:7þ fðþ ¼ ð:þ with d =( bq )/( bq ),which i contnt. With the help of reltion (3.5), Eq. (3.7) trnlte Introducing the non-dimenionl quntity g ¼ =u t, Eq. (.8) nd (.9) my be written in the following form ðþ ¼ 4ð bq Þq U G ðf gþ ¼ð gþ ðgþ þ ðc þ bþ ðc þ Þp þ q ð bq ÞðU ofðgþ Þ ð bþ gðgþ 4dð bq Þ U ðc þ bq ÞU þ ð bq ð3:8þ Þ U ¼ ð:þ with G ðf gþ ¼ð gþ ofðgþ þ ð bþ ðc þ bþ ðgþ U fðgþ ¼ ðu Þ u U ðu þ Þ : u ð3:9þ ¼ g : ð:3þ On uing the bic tte condition (3.6), the ytem of Eq. (3.8), (3.9) yield the condition which relte the function f(g ) nd g(g ) 3. Jump condition t hock fðg Þ¼gðg Þ ð3:þ where the contnt i given The jump condition t the hock re given by the principle of conervtion of m, momentum nd energy cro the hock front nd re written [,3] ¼ ð bq Þq U u ðc þ Þp þ q ð bq ÞðU ðc þ bq q ¼ q Þ þ ð bq Þ Þ c þ c þ U ð bq d Þ U ðc þ bq ÞU þ ð bq Þ U þ : ð3:þ u ¼ ð bq Þ ðu c þ Uing the boundry condition (.) nd (3.), Eq. (.), U ð3:þ (.3) my be olved in the region 6 g 6 U =u p. The bic tte olution re computed from the bic p ¼ p þ ð bq reltion (3.4) for three vlue of Þ /U =.5,.5,.75 ðu c þ ð3:3þ nd preented in the tble () (3). where ubcript denote the flow vrible evluted hed of the hock front nd U i hock peed. Epnding the vrible u, p, q nd U erie of mll prmeter e, the zero order jump condition become " ðc þ bq q ¼ q Þ þ ð bq # 9 Þ c þ c þ U >= u ¼ ð bq Þ ðu c þ Þ U p ¼ p þ ð bq Þ ðu c þ Þq > nd the firt order jump condition re q ðþ ¼ u ðþ 4ð bq Þ ð cþ U Þ 9 q U ðc þ bq Þ þ ð bq Þ ð U Þ c þ c þ U >= ¼ ð bq Þ ðu c þ þ Þ U U ¼ 4ð bq Þ q c þ U U p ðþ > ð3:4þ ð3:5þ where upercript () denote the vlue t the hock g = g. Uing (.), the effective ound peed for zero order nd firt order tke the form 4. Three emple The differentil eqution (.) nd (.3) nd the boundry condition (.) nd (3.8) re liner. The olution of the plne piton problem my be reduced to the following two elementry olution G ðf g Þ¼ G ðf g Þ¼ g f ðþ ¼ g ðþ ¼ ð4:þ G ðf g Þ¼ G ðf g Þ¼ f ðþ ¼ g ðþ ¼: ð4:þ Therefore, the olution of the plne piton problem i the liner combintion of the two elementry olution fðgþ ¼f ðgþþxf ðgþ gðgþ ¼g ðgþþxg ðgþ ð4:3þ where X i contnt nd defined X ¼ f ðg Þ g ðgþ g ðgþ f ðg Þ : The effect of the pplied grvity on the ditribution of function f(g) nd g(g) in non-idel flow field i preented in Fig. 3 for the vlue of Tble 3. In wek grvittionl field the internl nd, ometime, kinetic energy of the g will

4 8 L.P. Singh et l. Tble Bic tte olution for /U =.5. /U =.5 b ¼ : b ¼ : b ¼ :4 U /u p /u p /u p Tble Bic tte olution for /U =.5. /U =.5 b ¼ : b ¼ : b ¼ :4 Figure Ditribution of the function f(g) nd g(g) for different vlue of b with hock wve condition for /U =.5 nd c =.67. U /u p /u p /u p Tble 3 Bic tte olution for /U =.75. /U =.75 b ¼ : b ¼ : b ¼ :4 U /u p o /u p /u p Figure Ditribution of the function f(g) nd g(g) for different vlue of b with hock wve condition for /U =.5 nd c =.67. ehut to overcome the pplied grvity nd the proce i lowed down due to incree in the vlue of prmeter of non-idelne b compred to wht it would in n idel g. The propgtion velocity of the hock wve i U þ eu. The trength of the hock wve i increed if U i poitive. Further, the epreion (3.9) how tht U h the me ign u nd therefore f(g). Fig. 3 how tht f(g ) for the ll ce. Alo, the monotonic decreing vlue of g(g) from the piton g = to hock wve g = g decree more rpidly with n incree in the vlue of the prmeter of non-idelne ( b) which implie tht the internl energy of the g between the piton nd hock wve ehut more rpidly in nonidel g. Further, the effect of non-idelne of the g i to decree the vlue of f(g) from piton to the hock which how tht the proce of ccelertion/decelertion i lowed down due to non-idelne of the g. 5. Strong hock wve pproimtion For the ce of trong hock wve, the flow region become nrrow, nd g 6 g. Conequently the ytem of Eq. (.) nd (.3) tke the following form Figure 3 Ditribution of the function f(g) nd g(g) for different vlue of b with hock wve condition for /U =.75 nd c =.67. ðc þ bþ ð bþ ofðgþ gðgþ ¼ ð5:þ ð bþ ðc þ bþ ðgþ fðgþ ¼ g : ð5:þ

5 Flow pttern induced by the plne piton moving in non-idel g with wek grvittionl field 9 Eliminting o g(g)/ o g from the bove eqution we find the differentil eqution in term of f(g) follow o fðgþ þ ofðgþ g fðgþ ¼ g : ð5:3þ g4 The generl olution of the bove differentil eqution i determined eily fðgþ ¼c g b þ c g b þ b ðb 4Þ g : ð5:4þ Uing the bove reltion in Eq. (5.) we determine the vlue of g(g) follow gðgþ ¼ ðc þ bþ ð bþg c g b þ c g b b ð5:6þ ðb 4Þ g where b =/, contnt c nd c re obtined by the boundry condition (.) nd (3.) c ¼ b 4 b h b c þ b þð i bþg g b ðc þ bþ þð bþbg =g h c þb ð bþg i h g b þ c þ b þð bþg i g b ð5:7þ nd c ¼ b 4 b ð bþg Šg b ½ðc þ bþ þð bþbg Š=g ½ c þ b ð bþg Šg b þ½ c þ b þð : bþg Šg b ð5:8þ With the help of the originl reltion (.), the profile of the firt order olution re given u ð tþ ¼ c þb b b t þ c þ ð5:9þ t t ðb 4Þ b½c þ b ð tþ ¼ ðc þ bþ ð bþ b b b t c c : t t ðb 4Þ ð5:þ We now dicu the ce of trong hock wve, i.e., /U fi, which give g =(c + )/. For c = 5/3, we hve fðgþ ¼c g :38889 þ c g :38889 :9345 ð5:þ g gðgþ ¼ ðc þ bþ ð bþ c g :38889 þ c g :38889 þ :348 ð5:þ g 3 of the hock wve decree. Alo, the monotonic decreing function g(g) how further decreing trend with n incree in the vlue of b, therefore, the g internl energy ehut which gree with the erlier reult dicued in [7]. Since (g ) i much mller thn one. The nlyticl reult preented here give the pproimte olution nd decribe qulittively the bic feture of the influence of pplied grvity in non-idel g. Likewie, we derive the nlyticl olution ner the piton, which re epected to be more ccurte. Eq. (5.) nd (5.) re reduced to f ðþ ¼ ð bþ ðc þ bþ gðþ g ðþ ¼ ðc þ bþ ð bþ ½ fðþš: The g motion will be ccelerted ner the piton if g() >, on the other hnd decelerted if g() <. From the bove reltion it i cler tht the internl energy lwy decree in the region ner the piton. Alo, the non-idelne of the flow field contribute to decree the internl energy further. 6. Reult nd dicuion The hyperbolic ytem of Eq. (.), (.), (.3) poee three fmilie of chrcteritic d/dt = u, the trjectory of fluid prticle, nd d/dt = u ±, the outgoing nd incoming wvelet. With the help of epnion (.) the chrcteritic reltion hve been chnged follow. For outgoing wvelet d dt ¼ðu þ Þþeðu þ Þ. For incoming wvelet d dt ¼ðu Þþeðu Þ 3. For the trjectory of the fluid prticle, d dt ¼ u þ eu : where the contnt c nd c re given numericlly c ¼ :664ð:335 þ bþ :446 :437ð:335 þ bþ :5748 c ¼ :6397ð:335 þ bþþ:9595 :437ð:335 þ bþ :5748 : The vlue of c nd c ppering in the Eq. (5.) nd (5.) incree with n incree in the vlue of b conequently the negtive vlue of f(g) incree, nd conequently the trength Figure 4 Ditribution of the function f(g) nd g(g) for different vlue of b with hock wve condition for /u p = nd c =.67.

6 3 L.P. Singh et l. the vlue of b, in the preence of wek grvittionl field, which i lo illutrted through Fig Since the medium in n trophyicl environment i not umed to follow the idel g umption, it i the necery to nlyze the gdynmic procee in non-idel medium including the pplied grvity. The pper hed light on n elementry proce, nd preent the effect of the pplied grvity on the flow field in non-idel medium. Acknowledgement Figure 5 Ditribution of the function f(g) nd g(g) for different vlue of b with hock wve condition for /u p = 4 nd c =.67. Figure 6 Ditribution of the function f(g) nd g(g) for different vlue of b with hock wve condition for /u p = nd c =.67. From the bove reult we dopt tht the chrcteritic line re curved. It my be noted here tht the effect of non-idelne of the medium i to incree the vlue of. Therefore the lope of the out going wvelet incree ner the piton in comprion with the ce of idel medium with pplied grvity nd chrcteritic line () h oppoite behvior tht of out going wvelet. An incree in lope of incoming wvelet i due to the pplied grvity in non-idel medium. The poition of the hock wve in perturbed region i t ¼ U þ eu ð6:þ We now dicu bout the plne piton problem with boundry condition written t the piton fðþ ¼ gðþ ¼: ð6:þ The olution of Eq. (.), (.3), with the boundry condition (6.), repreent tht the function f(g) nd g(g) re monotoniclly decreing function of g. The Fig. 4 6 contitute the ditribution of f(g) nd g(g) with different piton boundry condition for /u p =, 4,, repectively. In Fig. 4 6, b ¼ : repreent the ce of idel g nd i me given in [7]. From Eq. (5.) nd (5.) one my conclude tht the function f(g) remin unffected with repect to different vlue of b, where the function g(g) i gretly ffected by The econd nd third uthor cknowledge the finncil upport from the UGC nd CSIR, New Delhi, Indi, under the SRF cheme. Reference [] Cournt R, Friedrich KO. Superonic flow nd hock wve. New York: Wiley-Intercience 948. [] Whithm GB. Liner nd nonliner wve. New York: Wiley- Intercience 974. [3] Dfermo CM. Hyperbolic conervtion lw in continuum phyic. New York: Springer-Berlin Heidelberg. [4] Li TT. Globl clicl olution for quiliner hyperbolic ytem. New York: Wiley-Mon Pri 994. [5] Cheter W. Reonnt ocilltion in cloed tube. J Fluid Mech 9648: [6] Wng M, Koy DR. Dynmic compreion nd wek hock formtion in n inert g due to ft piton ccelertion. J Fluid Mech 99:67 9. [7] Wen-rui H. The pne piton problem in wek grvittionl field. Appl Mth Mech 9856: [8] Rng Ro MP, Purohit NK. Self imilr piton problem in nonidel g. Int J Eng Sci 9764:9 7. [9] Wu CC, Robert PH. Shock-wve propgtion in onoluminecing g bubble. Phy Rev Lett 9937: [] Robert PH, Wu CC. Structure nd tbility of phericl imploion. Phy Lett A 9963: [] Skuri Akir. Propgtion of phericl hock wve in tr. J Fluid Mech 956(4): [] Aniimov SI, Spiner OM. Motion of n lmot idel g in the preence of trong point eploion. J Appl Mth Mech 975: [3] C Wu C, Robert PH. Structure nd tbility of phericl hock wve in Vn der Wl g. Qurt J Mech Appl Mth 99649:5 43. [4] Singh LP, Huin Akml, Singh M. Evolution of wek dicontinuitie in non idel rditing g. Commun Nonliner Sci Numer Simul 6:69 7. [5] Singh LP, Rm SD, Singh DB. Propgtion of wek hock wve in non-uniform rditive Mgnetdynmic. Act Atronut 67:96 3. L.P. Singh, Profeor of Applied Mthemtic, Intitute of Technoly, Bnr Hindu Univerity. Hi re of interet i non-liner wve in gdynmic nd computtionl fluid dynmic.

7 Flow pttern induced by the plne piton moving in non-idel g with wek grvittionl field 3 S.D. Rm, M.Sc., from Bnr Hindu Univerity 6. He i currently prepring to Ph.D. degree t Intitute of Technoly, Bnr Hindu Univerity. Hi re of interet include non-liner wve in gdynmic. D.B. Singh, M.Sc., from Bnr Hindu Univerity 5. He i currently prepring to Ph.D. degree t Intitute of Technoly, Bnr Hindu Univerity. Hi re of interet include non-liner wve in gdynmic.