Nonlinear Asymmetric Kelvin Helmholtz Instability Of Cylindrical Flow

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The nonlinear asymmetric Kelvin Helmholtz stability of the cylindrical interface between the vapor and liquid phases of a °uid is studied when the phases are enclosed between two cylindri cal surfaces coaxial with the interface, and when there is mass and heat transfer across the inter face. The method of multiple time expansion is used for the investigation. The evolution of am plitude is shown to be governed by a nonlinear ¯rst order di®erential equation. The stability cri terion is discussed, and the region of stability is displayed graphically. Also investigated in this paper is the viscous linear potential °ow. DOO-SUNG LEE "Nonlinear Asymmetric Kelvin-Helmholtz Instability Of Cylindrical Flow With Mass And Heat Transfer And The Viscous Linear Analysis" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-2 | Issue-5 , August 2018, URL: https://www.ijtsrd.com/papers/ijtsrd17030.pdf Paper URL: http://www.ijtsrd.com/mathemetics/applied-mathematics/17030/nonlinear-asymmetric-kelvin-helmholtz-instability-of-cylindrical-flow-with-mass-and-heat-transfer-and-the-viscous-linear-analysis/doo-sung-lee

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International Journal of Trend in Scientiflc Research and DevelopmentIJTSRD International Open Access Journal ISSN:2456-6470 |www.ijtsrd.com|Volume -2|Issue-5 NONLINEAR ASYMMETRIC KELVIN-HELMHOLTZ INSTABILITY OF CYLINDRICAL FLOW WITH MASS AND HEAT TRANSFER AND THE VISCOUS LINEAR ANALYSIS DOO-SUNG LEE Department of Mathematics College of Education Konkuk University 120 Neungdong-Ro Kwangjin-Gu Seoul Korea e-mail address: dsleekonkuk.ac.kr Abstract The nonlinear asymmetric Kelvin-Helmholtz stability of the cylindrical interface between the vaporandliquidphasesofauidisstudiedwhen the phases are enclosed between two cylindri- cal surfaces coaxial with the interface and when there is mass and heat transfer across the inter- face. The method of multiple time expansion is used for the investigation. The evolution of am- plitude is shown to be governed by a nonlinear flrstorderdifierentialequation. Thestabilitycri- terion is discussed and the region of stability is displayed graphically. Also investigated in this paper is the viscous linear potential ow. Keywords Kelvin-Helmholtz stability Mass and heat Transfer Cylindrical ow. 0 1. Introduction In dealing with ow of two uids divided by aninterfacetheproblemofinterfacialstabilityis usuallystudiedwiththeneglectofheatandmass transfer across the interface. However there are situationswhentheefiectofmassandheattrans- fer across the interface should be taken into ac- count in stability discussions. For instance the phenomenon of boiling accompanies high heat and mass transfer rates which are signiflcant in determiningtheowfleldandthestabilityofthe system. Hsieh 1 presented a simplifled formulation of interfacial ow problem with mass and heat transfer and studied the problems of Rayleigh- Taylor and Kelvin-Helmholtz stability in plane geometry. The mechanism of heat and mass transfer acrossaninterfaceisimportantinvariousindus- trialapplicationssuchasdesignofmanytypesof contacting equipment e.g. boilers condensers pipelines chemical reactors and nuclear reac- tors etc. In the nuclear reactor cooling of fuel rods by liquid coolants the geometry of the system in many cases is cylindrical. We have therefore considered the interfacial stability problem of a cylindrical ow with mass and heat transfer. Nayak and Chakraborty2 studied the Kelvin- Helmholtz stability of the cylindrical interface between the vapor and liquid phases of a uid when there is a mass and heat transfer across 0 IJTSRD| Available Onlinewww.ijtsrd.com|Volume-2|Issue-5|Jul-Aug 2018 Page:1405 1

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2 INTERNATIONAL JOURNAL OF TREND IN SCIENTIFIC RESEARCH AND DEVELOPMENT IJTSRDISSN:2456-6470 the interface while Elhefnawy3 studied the ef- fect of a periodic radial magnetic fleld on the Kelvin-Helmholtz stability of the cylindrical in- terface between two magnetic uids when there is mass and heat transfer across the interface. Theanalysisofthesestudieswasconflnedwithin theframeworkoflineartheory. Theybothfound that the dispersion relations are independent of the rate of interfacial mass and heat transfer. Hsieh4 found that from the linearized analysis when the vapor region is hotter than the liquid region as is usually so the efiect of mass and heat transfer tends to inhibit the growth of the instability. Thus for the problem of fllm boil- ing the instability would be reduced yet would persist according to linear analysis. It is clear that such a uniform model based on the linear theory is inadequate to answer the question of whether and how the efiect of heat andmasstransferwouldstabilizethesystembut the nonlinear analysis is needed to answer the question. Thepurposeofthispaperistoinvestigatethe Kelvin-Helmholtzasymmetricnonlinearstability of cylindrical interface between the vapor and liquid phases of a uid when there is a mass and heat transfer across the interface. The nonlinear problem of Rayleigh-Taylor in- stabilityof a system in a cylindrical geometry is howeverstudiedbythepresentauthorinLee5- 6. The multiple time scale method is used to ob- tain a flrst order nonlinear difierential equation from which conditions for the stability and in- stability are determined. In more recent yearsAwashi Asthana and Zuddin7 considered a problem in which a vis- cous potential ow theory is used to study the nonlinear Kelvin-Helmholtz instability of the in- terface between two viscous incompressible and thermally conducting uids. The basic equations with the accompanying boundary conditions are given in Sec.2. The flrst order theory and the linear dispersion re- lation are obtained in Sec.3. In Sec .4 we have derived second order solutions. In Sec.5 a flrst order nonlinear difierential equation is obtained andthesituations ofthe stabilityandinstability are summarized. In Sec.6 we investigate linear viscous potential ow. In Sec.7 some numerical examples are presented. 2. Formulation of the problem and basic equations 0 We shall use a cylindrical system of coordi- nates rz so that in the equilibrium state z¡axis is the axis of asymmetry of the system. The central solid core has a radius a: In the equilibrium state the uid phase "1" of den- sity ‰ 1 occupies the region a r R and the uid phase "2" of density ‰ 2 occupies the region R r b: The inner and outer uids are streaming along the z axis with uniform ve- locities U 1 and U 2 respectively. The temper- atures at r ar R and r b are taken as T 1 T 0 and T 2 respectively. The bounding surfaces r a and r b are taken as rigid. The interface after a disturbance is given by the equation Frztr¡R¡·zt 0 2:1 where·istheperturbationinradiusoftheinter- face from its equilibrium value R and for which the outward normal vector is written as n rF jrFj ‰ 1+ 1 r · ¶ 2 + · z ¶ 2 ¡12 £ e r ¡ 1 r · e ¡ · z e z ¶ 2:2 we assume that uid velocity is irrotational in theregionsothatvelocitypotentialsare` 1 and ` 2 for uid phases 1 and 2. In each uid phase r 2 ` j 0: j 12 2:3 The solutions for ` j j 12 have to satisfy the boundary conditions. The relevant bound- ary conditions for our conflguration are i On the rigid boundaries r a and r b: 0 IJTSRD| Available Onlinewww.ijtsrd.com|Volume-2|Issue-5|Jul-Aug 2018 Page:1406

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INTERNATIONAL JOURNAL OF TREND IN SCIENTIFIC RESEARCH AND DEVELOPMENT IJTSRDISSN:2456-6470 3 Thenormalfleldvelocitiesvanishonbothcen- tral solid core and the outer bounding surface. ` 1 r 0 on r a 2:4 ` 2 r 0 on r b 2:5 ii On the interface r R+·zt : 1 The conservation of mass across the inter- face: •• ‰ F t +r`¢rF ¶‚‚ 0 or •• ‰ ` r ¡ · t ¡ 1 r · ` ¡ · z ` z ¶‚‚ 0 2:6 wherehrepresentsthedifierenceinaquantity as we cross the interfacei.e. h h 2 ¡h 1 where superscripts refer to upper and lower u- ids respectively. 2 The interfacial condition for energy is L‰ 1 F t +r` 1 ¢rF ¶ S· 2:7 whereListhelatentheatreleasedwhentheuid is transformed from phase 1 to phase 2. Phys- ically the left-hand side of 2.7 represents the latentheatreleasedduringthephasetransforma- tion while S· on the right-hand side of 2.7 represents the net heat ux so that the energy will be conserved. In the equilibrium state the heat uxes in the direction of r increasing in the uid phase 1 and 2 are ¡K 1 T 1 ¡ T 0 RlogaR and ¡K 2 T 0 ¡T 2 RlogRb where K 1 and K 2 are the heat conductivities of the two uids. As in Hsieh1978 we denote S· K 2 T 0 ¡T 2 R+·logb¡logR+· ¡ K 1 T 1 ¡T 0 R+·logR+·¡loga 2:8 and we expand it about r R by Taylor’s ex- pansion such as S·S0+·S 0 0+ 1 2 · 2 S 00 0+¢¢¢ 2:9 and we take S00 so that K 2 T 0 ¡T 2 RlogbR K 1 T 1 ¡T 0 RlogRa Gsay 2:10 indicating that in equilibrium state the heat uxes are equal across the interface in the two uids. From 2.1 2.7 and 2.9 we have ‰ 1 ` 1 r ¡ · t ¡ 1 r · ` 1 ¡ · z ` 1 z ¶ fi·+fi 2 · 2 +fi 3 · 3 2:11 where fi Glogba LRlogbRlogRa fi 2 1 R ¡ 3 2 + 1 logbR ¡ 1 logRa ¶ fi 3 1 R 2 • 11 6 ¡ 2logR 2 ab logbRlogRa + log 3 bR+log 3 Ra flogbRlogRag 2 logba ‚ : 3 The conservation of momentum balance by taking into account the mass transfer across the interface is ‰ 1 r` 1 ¢rF F t +r` 1 ¢rF ¶ ‰ 2 r` 2 ¢rF F t +r` 2 ¢rF ¶ +p 2 ¡p 1 + r¢njrFj 2 2:12 where p is the pressure and is the surface ten- sioncoe–cientrespectively. 0 Byeliminatingthe pressure by Bernoulli’s equation we can rewrite the above condition 2.12 as ||||||||||||||||||| 0 IJTSRD| Available Onlinewww.ijtsrd.com|Volume-2|Issue-5-Jul-Aug 2018 Page:1407

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4 INTERNATIONAL JOURNAL OF TREND IN SCIENTIFIC RESEARCH AND DEVELOPMENT IJTSRDISSN:2456-6470 •• ‰ ‰ ` t + 1 2 ` r ¶ 2 + 1 2 1 r ` ¶ 2 + 1 2 ` z ¶ 2 ¡ ‰ 1+ 1 r · ¶ 2 + · z ¶ 2 ¡1 £ ` z · z + 1 r 2 ` · ¡ ` r ¶ · t + ` z · z + 1 r 2 ` · ¡ ` r ¶‚‚ R+·jrFj ‰ 1+ 1 r · ¶ 2 2 jrFj 2 ¡ jrFj 3 • 2 · z 2 ‰ 1+ 1 r · ¶ 2 ¡ 2 r 2 · 2 · z · z + 1 r 2 2 · 2 ‰ 1+ · z ¶ 2 ‚ : 2:13 |||||||||||||||||||||||||||| Whentheinterfaceisperturbedfromtheequi- librium · 0 to · Aexpikz+m ¡t the dispersion relation for the linearized problem is Dkma 0 2 +a 1 +ib 1 +a 2 +ib 2 0 2:14 where a 0 ‰ 1 E 1 m ¡‰ 2 E 2 m a 1 2kf‰ 2 E 2 m U 2 ¡‰ 1 E 1 m U 1 g b 1 fifE 1 m ¡E 2 m g a 2 k 2 f‰ 1 E 1 m U 2 1 ¡‰ 2 E 2 m U 2 2 g ¡ R 2 R 2 k 2 +m 2 ¡1 b 2 fikfE 2 m U 2 ¡E 1 m U 1 g where for the simplicity of notation we used E j m E j m kR j 12 where E j m kRj 12 are explained by 3.4-3.5. i When fi0 2.14 reduces to a 0 2 +a 1 +a 2 0: 2:15 Therefore the system is stable if a 2 1 ¡4a 0 a 2 0 2:16 or R 2 R 2 k 2 +m 2 ¡1 +k 2 ‰ 1 ‰ 2 E 1 m E 2 m U 2 ¡U 1 2 ‰ 1 E 1 m ¡‰ 2 E 2 m 0: 2:17 It is clear from the above inequality that the streaminghasadestabilizingefiectonthestabil- ity of a cylindrical interface because E 2 m is al- waysnegativefromthepropertiesofBesselfunc- tions. ii when fi 6 0 we flnd that necessary and su–cient stability conditions for 2.14 are 3 b 1 0 2:18 and a 0 b 2 2 ¡a 1 b 1 b 2 +a 2 b 2 1 0 2:19 since a 0 is always positive. 0 Putting the values of a 0 a 1 a 2 b 1 and b 2 from2.14 into2.18 and 2.19 we notice that the condition 2.18 is trivially satisfled since fi is always positive and from properties of Bessel functionsE 2 m isalwaysnegative. From2.19it can be shown that the condition for the stability of the system is R 2 R 2 k 2 +m 2 ¡1+k 2 ‰ 1 ‰ 2 E 1 m E 2 m U 2 ¡U 1 2 ‰ 1 E 1 m ¡‰ 2 E 2 m £ • 1¡ E 1 m E 2 m ‰ 1 ¡‰ 2 2 E 1 m ¡E 2 m 2 ‰ 1 ‰ 2 ‚ 0: 2:20 The stability condition 2.20 difiers from 2.17 by the additional last term: E 1 m E 2 m ‰ 1 ¡ ‰ 2 2 ‰ 1 ‰ 2 E 1 m ¡ E 2 m 2 : Thus the condition 2.20 is valid for inflnites- imal fi and when fi0 the last term is absent. Wenowemploymultiscaleexpansionnearthe critical wave number. The critical wave number 0 IJTSRD| Available Onlinewww.ijtsrd.com|Volume-2|Issue-5 |Jul-Aug 2018 Page:1408

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INTERNATIONAL JOURNAL OF TREND IN SCIENTIFIC RESEARCH AND DEVELOPMENT IJTSRDISSN:2456-6470 5 isattainedwhena 2 b 2 0. Thecorresponding critical frequency c is zero for this case. Introducing † as a small parameter we as- sume the following expansion of the variables: · 3 X n1 † n · n zt 0 t 1 t 2 +O† 4 2:21 ` j 3 X n0 † n ` j n rzt 0 t 1 t 2 +O† 4 j 12 2:22 where t n † n tn 012: 0 The quantities appearing in the fleld equations 2.3 and the boundaryconditions2.62.11and2.13can now be expressed in Maclaurin series expansion around r R: Then we use 2.21 and 2.22 and equate the coe–cients of equal power series in † to obtain the linear and the successive non- linearpartial difierentialequationsof variousor- ders. To solve these equations in the neighborhood of the linear critical wave number k c because of the nonlinear efiect we assume that the critical wave number is shifted to k k c +† 2 „: 3. First Order Solutions. We take ` j 0 U j z: j 12 The flrst order solutions will reproduce the lin- ear wave solutions for the critical case and the solutionsof2.3subjecttoboundaryconditions yield · 1 At 1 t 2 e i + „ At 1 t 2 e ¡i 3:1 ` 1 1 fi ‰ 1 +ikU 1 ¶ At 1 t 2 E 1 m kre i +c:c: 3:2 ` 2 1 fi ‰ 2 +ikU 2 ¶ At 1 t 2 E 2 m kre i +c:c: 3:3 where E 1 m kr I m krK 0 m ka¡I 0 m kaK m kr I 0 m kRK 0 m ka¡I 0 m kaK 0 m kR 3:4 E 2 m kr I m krK 0 m kb¡I 0 m kbK m kr I 0 m kRK 0 m kb¡I 0 m kbK 0 m kR 3:5 kz+m I 0 m ka r I m kr fl fl ra etc: with I m and K m are the modifled Bessel func- tions of the flrst and second kinds respectively. 4. Second order solutions. With the use of the flrst order solutions we obtainedtheequationsforthesecondorderprob- lem r 2 ` j 2 0 j 12 4:1 and the boundary conditions at r R: |||||||||||||||||||||||||||||| ‰ j ‰ ` j 2 r ¡ · 2 z U j ¡fi· 2 • ‰ j ‰ fi ‰ j +iU j ‰ 1 R ¡2 k 2 + m 2 R 2 ¶ E j m +fifi 2 ‚ £A 2 e 2i +‰ j A t 1 e i +c:c:+2fi 1 R +fi 2 ¶ jAj 2 j 12 4:2¡4:3 ‰ 2 U 2 ` 2 2 z ¡‰ 1 U 1 ` 1 2 z + 2 · 2 z 2 + 1 R 2 2 · 2 2 + · 2 R 2 ¶ ¡ 1 2 ‰•• ‰ fi ‰ +ikU ¶ 2 ‰ ¡1¡ m 2 R 2 +k 2 ¶ E 2 m +3fiUki¡2‰U 2 k 2 ‚‚ 0 IJTSRD| Available Onlinewww.ijtsrd.com|Volume-2|Issue5 |Jul-Aug 2018 Page:1409

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6 INTERNATIONAL JOURNAL OF TREND IN SCIENTIFIC RESEARCH AND DEVELOPMENT IJTSRDISSN:2456-6470 + R 3 R 2 k 2 +2¡7m 2 A 2 e 2i + •• ‰ k fi ‰ +ikU ¶ E m ‚‚ A t 1 e i +c:c: ¡ ‰•• ‰ fi 2 ‰ 2 +k 2 U 2 ¶‰ ¡1+E 2 m m 2 R 2 +k 2 ¶‚‚ + R 3 R 2 k 2 +m 2 ¡2 jAj 2 : 4:4 The non secularity condition for the existence of the uniformly valid solution is 0 A t 1 0: 4:5 Equations 4.1 to 4.4 furnish the second order solutions: · 2 ¡2 1 R +fi 2 ¶ jAj 2 +A 2 A 2 e 2i + „ A 2 „ A 2 e ¡2i 4:6 ` j 2 B j 2 A 2 e 2i E j 2m 2kr+c:c:+b j t 0 t 1 t 2 j 12 4:7 where A 2 1 D02k2m ‰•• ¡‰i2kUE 2m fl+ ‰ 2 ‰ E 2 m m 2 R 2 +k 2 ¶ +1 fi ‰ +ikU ¶ 2 +2‰kU 2 ¡i3fikU ‚‚ + 2R 3 2+R 2 k 2 ¡7m 2 4:8 B j 2 fl j + ‰ fi ‰ j +2ikU j A 2 4:9 fl j ‰ fi ‰ j +ikU j ‰ 1 R ¡2E j m m 2 R 2 +k 2 ¶ + fifi 2 ‰ j 4:10 ‰ 2 b 2 t 0 ¡‰ 1 b 1 t 0 ‰•• ‰ fi 2 ‰ 2 +k 2 U 2 ¶‰ 1¡E 2 m kR m 2 R 2 +k 2 ¶‚‚ ¡ R 3 k 2 R 2 +m 2 ¡4¡2Rfi 2 ¶ jAj 2 4:11 whereE j 2m E j 2m 2kR: 5. Third order solutions We examine now the third order problem: r 2 0 ` i 3 0: i12 5:1 On substituting the values of · 1 ` i 1 from 3.1-3.3 and · 2 ` i 2 from 4.6-4.7 into A.7 we obtain ` j 3 C j 3 E j 2m krA 2 „ Ae i +E j kr A t 2 e i +c:c: 5:2 where C j 3 ¡ •‰ E j 2m 2 m 2 R 2 +k 2 ¶ ¡ 1 R B j 2 ¡2 ‰ E j m m 2 R 2 +k 2 ¶ ¡ 1 R fi ‰ j +ikU j ¶ £ 1 R +fi 2 ¶ + 1 2 ‰ k 2 + 2+m 2 R 2 ¡ E j m R 3m 2 R 2 +k 2 ¶ 3fi ‰ j +ikU j ¶ 0 IJTSRD| Available Onlinewww.ijtsrd.com|Volume-2|Issue5 |Jul-Aug 2018 Page:1410

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INTERNATIONAL JOURNAL OF TREND IN SCIENTIFIC RESEARCH AND DEVELOPMENT IJTSRDISSN:2456-6470 7 + fi ‰ j ¡ikU j ¶ 2m 2 R 3 E j m ¡ m 2 R 2 ¡k 2 ¶ + fi ‰ j ‰ 4fi 2 1 R +fi 2 ¶ ¡3fi 3 ¡ ‰ fi ‰ j ¡ikU j ¶‰ E j m m 2 R 2 +k 2 ¶ + 1 R + 2fifi 2 ‰ j A 2 ‚ : j 12 5:3 We substitute the flrst- and second-order solutions into the third order equation. In order to avoid nonuniformity of the expansion we again impose the condition that secular terms vanish. Then from A.8 we flnd 0 i D0km A t 2 + ‰ 2k c „+ •• ‰U fi ‰ +ikU ¶ E m ‚‚ k c i„ A+qA 2 „ A0 5:4 where q •• ‰ ikUC 3 E m +A 2 i fi ‰ 3kU¡k 2 U 2 ¶ +B 2 fi ‰ ¡iUk ¶‰ 2E m E 2m m 2 R 2 +k 2 ¶ ¡1 ¡i 5 2R +2fi 2 ¶ kU fi ‰ +ikU ¶ + 3 R fi 2 ‰ 2 ¡E m m 2 R 2 +k 2 ¶ kU 2 i 7 fi ‰ +5kUi ¶ + m 2 R 2 E 2 m fi 2 ‰ 2 +3k 2 U 2 ¡i2 fi ‰ kU ¶‚‚ ¡ R 4 2A 2 R¡4¡4Rfi 2 1¡m 2 ¡2A 2 Rm 2 +k 2 R 2 ¡ 3 2 m 2 +k 2 R 2 2 + 1 2 9m 2 +k 2 R 2 ¡6 “ : 5:5 |||||||||||||||||||||||||||||| We rewrite 5.4 as A t 2 + a 1 + a 2 jAj 2 A0 5:6 which can be easily integrated as jAt 2 j 2 a 1r jA 0 j 2 exp¡2a 1r t £a 1r +a 2r jA 0 j 2 ¡a 2r jA 0 j 2 exp¡2a 1r t ¡1 5:7 where A 0 is the initial amplitude and a jr a j j 12 . With a flnite initial value jA 0 j jAj may be- comeinflnitewhenthedenominatorin5.7van- ishes. Otherwise jAj will be asymptotically bounded. The situation can be summarized as follows: 1 a 2r 0 stable. i a 1r 0 jAj 2 0 as t 2 1 ii a 1r 0 jAj 2 ¡a 1r a 2r as t 2 1 2 If a 2r 0 i a 1r 0 unstable. iia 1r 0andjA 0 j 2 ¡a 1r a 2r :unstable. iii a 1r 0 and jA 0 j 2 ¡a 1r a 2r : stable andjAj 2 0 as t 2 1:Thusasu–cientcon- ditionforstabilityisa 2r 0whichisduetothe flnite amplitude efiect. The cylindrical system is nonlinearly stabile if a 1r 0 and the initial amplitude is su–ciently small. 6.Viscous asymmetric linear cylindrical ow In this section we consider the viscous poten- tial ow. For the viscous uid 2.12 is now replaced by ‰ 1 r’ 1 ¢rF F t +r’ 1 ¢rF ¶ ‰ 2 r’ 2 ¢rF F t +r’ 2 ¢rF ¶ +p 2 ¡p 1 ¡2„ 2 n¢r›r’ 2 ¢n +2„ 1 n¢r›r’ 1 ¢n+ r¢njrFj 2 6:1 0 IJTSRD| Available Onlinewww.ijtsrd.com|Volume-2|Issue5 |Jul-Aug 2018 Page:1411

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8 INTERNATIONAL JOURNAL OF TREND IN SCIENTIFIC RESEARCH AND DEVELOPMENT IJTSRDISSN:2456-6470 where „ 1 „ 2 are viscosities of uid ’1’ and ’2’ respectively and we modify 2.13 accordingly. The nonlinear analysis for the viscous uid is too onerous when the perturbation is asymmet- ric wearecontenthere with thelinearanalysis. Then linearizing 2.6 2.11 and 6.1 we have •• ‰ ` r ¡ · t ¶ ¡ · 1 z U ¶‚‚ 0 6:2 0 ‰ 1 ` 1 r ¡ · t ¡ · z U ¶ fi· 6:3 •• ‰ ` t + ` z U ¶ +2„ 2 ` r 2 ‚‚ ¡ 2 · z 2 + · R 2 + 1 r 2 2 · 2 ¶ : 6:4 When the interface is perturbed to · Aexpikz+m ¡t we recover the flrst order solutions 3.1-3.3 and the dispersion relation for the viscous uid is same as 2.14 however a 0 ‰ 1 E 1 m ¡‰ 2 E 2 m a 1 2kf‰ 2 E 2 m U 2 ¡‰ 1 E 1 m U 1 g b 1 fifE 1 m ¡E 2 m g+2„ 1 E 1 t ¡„ 2 E 2 t a 2 k 2 f‰ 1 E 1 m U 2 1 ¡‰ 2 E 2 m U 2 2 g ¡ R 2 R 2 k 2 +m 2 ¡1 ¡2fi „ 1 ‰ 1 E 1 t ¡ „ 2 ‰ 2 E 2 t ¶ b 2 fikfE 2 m U 2 ¡E 1 m U 1 g ¡2k„ 1 U 1 E 1 t ¡„ 2 U 2 E 2 t with E i t E i m k 2 + m 2 k 2 ¶ ¡ 1 R and necessary and su–cient stability conditions are b 1 0 6:5 and a 0 b 2 2 ¡a 1 b 1 b 2 +a 2 b 2 1 0 6:6 since a 0 is always positive. 7. Numerical examples Inthissectionwedonumericalworksusingtheexpressionspresentedinprevioussectionsforthe fllm boiling conditions. The vapor and liquid are identifled with phase 1 and phase 2 respectively FIGURE 1. The critical wave number for m1. 0 IJTSRD| Available Onlinewww.ijtsrd.com|Volume-2|Issue5 |Jul-Aug 2018 Page:1412

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INTERNATIONAL JOURNAL OF TREND IN SCIENTIFIC RESEARCH AND DEVELOPMENT IJTSRDISSN:2456-6470 9 so that T 1 T 0 T 2 . In the fllm boiling the liquid-vapor interface is of saturation condition and the temperature T 0 is set equal to the saturation temperature. The properties of both phases are determined from this condition. First in flgure 1 we display critical wave number k c i.e. the value for which 0 in 2.14 Here we chose ‰ 1 0:001gmcm 3 ‰ 2 1gmcm 3 72:3dynecmb2cma1cmR 1:2cmfi0:1gmcm 3 s 0 FIGURE 2. The stability diagram for the ow when m1. The system is stable in the region between the two upper and lower curves. Fig.3.Viscous cylindrical ow for m0.The region above the curve is stable region. Fromthisflgurewecannoticethatcriticalwavenumberincreasesasthevelocityofuidincreases theincrementrateoftheinvisciduidbeingsharperathigheruidvelocities. Inflgure2wedisplay 0 IJTSRD| Available Onlinewww.ijtsrd.com|Volume-2|Issue5 |Jul-Aug 2018 Page:1413

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10 INTERNATIONAL JOURNAL OF TREND IN SCIENTIFIC RESEARCH AND DEVELOPMENT IJTSRDISSN:2456-6470 the region of stability of uid in the nonlinear analysis as the velocity of one uid increases while that of the other uid remains unchanged. In these flgures u 1 remains constant as 1 cm/sec while u 2 varies from 1 cm/sec to 10cm/sec. The region between the two curves is the region of stability while in the region above the upper curve the uid is unstable. 0 In Fig.3 and Fig.4 we present the results for viscous cylindrical linear ow.Here we chose ‰ 1 0:0001gmcm 3 ‰ 2 1gmcm 3 72:3dynecmb 2cma 1cmR 1:2cmfi :1gmcm 3 s„ 1 0:00001poise„ 2 0:01poise Fig.4.Viscous cylindrical ow for m1.The region above the curve is stable region. 8. Conclusions. The stability of liquids in a cylindrical ow when there is mass and heat transfer across the interface which depicts the fllm boiling is studied. Using the method of multiple time scales a flrst order nonlinear difierential equation describing the evolution of nonlinear waves is obtained.With the linear theory the region of stability is the whole plane above a curve like in Fig.34 however with the nonlinear theory it is in the form of a band as shown in Fig.2. Unlike linear theory with nonlinear theory it is evident that the mass and heat transfer plays an important role in the stability of uid in a situation like fllm boiling. Appendix The interfacial conditions are given on r R as Order O† •• ‰ ` 1 r ¡ · 1 T 0 ¶ ¡ · 1 z ` 0 z ¶‚‚ 0 A:1 ‰ 1 ` 1 1 r ¡ · 1 T 0 ¡ · 1 z ` 0 z ¶ fi· 1 A:2 0 IJTSRD| Available Onlinewww.ijtsrd.com|Volume-2|Issue-5|Jul-Aug 2018 Page:1414

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INTERNATIONAL JOURNAL OF TREND IN SCIENTIFIC RESEARCH AND DEVELOPMENT IJTSRDISSN:2456-6470 11 •• ‰ ` 1 T 0 + ` 1 z ` 0 z ¶‚‚ ¡ 2 · 1 z 2 + · 1 R 2 + 1 r 2 2 · 1 2 ¶ : A:3 Order O† 2 0 •• ‰ ` 2 r + 2 ` 1 r 2 · 1 ¡ · 2 T 0 ¡ · 1 T 1 ¡ · 1 z ` 1 z ¡ 1 r 2 · 1 ` 1 ¡ · 2 z ` 0 z ¶‚‚ 0 A:4 ‰ 1 ` 1 2 r + 2 ` 1 1 r 2 · 1 ¡ · 2 T 0 ¡ · 1 T 1 ¡ · 1 z ` 1 1 z ¡ 1 r 2 · 1 ` 1 ¡ · 2 z ` 0 z ¶ fi· 2 +fi 2 · 2 1 A:5 •• ‰ ‰ ` 2 T 0 + ` 1 T 1 + 2 ` 1 T 0 r · 1 + 1 2 • ` 1 r ¶ 2 + 1 r 2 ` 1 ¶ 2 + ` 1 z ¶ 2 ‚ + ` 2 z ` 0 z + ` 1 r · 1 T 0 ¡ ` 1 r ¶ + ` 0 z ¡ · 1 z · 1 t 0 +2 · 1 z ` 1 r ¡ · 1 z ¶ 2 ` 0 z + 2 ` 1 zr · 1 ¶‚‚ ¡ ‰ 2 · 2 z 2 + 1 2R · 1 z ¶ 2 + · 2 R 2 ¡ · 2 1 R 3 ¡ 3 2 1 R 3 · 1 ¶ 2 + 1 r 2 2 · 2 2 ¡ 2 r 3 · 1 2 · 1 2 : A:6 Order O† 3 ‰ j ‰ ` j 3 r + 2 ` j 2 r 2 · 1 + 2 ` j 1 r 2 · 2 + 1 2 3 ` j 1 r 3 · 2 1 ¡ · 3 T 0 ¡ · 2 T 1 ¡ · 1 T 2 ¡ · 1 z ` j 2 z + 2 ` j 1 zr · 1 ¶ ¡ · 2 z ` j 1 z ¡ · 3 z ` j 0 z ¡ 1 R 2 · 2 ` j 1 ¡ 1 R 2 · 1 ` j 2 ¡ 1 R 2 · 1 · 1 2 ` j 1 r + 2 R 3 · 1 · 1 ` j 1 fi· 3 +2fi 2 · 1 · 2 +fi 3 · 3 1 j 12 A:7 •• ‰ ‰ ` 3 T 0 + ` 2 T 1 + ` 1 T 2 + 2 ` 1 T 0 r · 2 + 2 ` 1 T 1 r + 2 ` 2 T 0 r ¶ · 1 + 1 2 3 ` 1 T 0 r 2 · 2 1 + ` 1 r ` 2 r + 2 ` 1 r 2 · 1 ¶ + ` 1 z ` 2 z + 2 ` 1 rz · 1 ¶ + 1 R 2 ` 1 2 ` 1 r ¡ 1 R ` 1 ¶ · 1 + 1 R 2 ` 1 ` 2 ¡ ` 1 r ¡ · 1 t 0 ¶ ` 2 r ¡ ` 1 z · 1 z ¡ 1 R 2 ` 1 · 1 ¶ + ` 1 r · 2 t 0 + · 1 t 1 ¡ ` 2 r + ` 1 z · 1 z + 1 R 2 ` 1 · 1 ¶ +· 1 2 ` 1 r 2 · 1 T 0 ¡2 ` 1 r ¶ + ` 0 z ` 3 z + 2 ` 1 rz · 2 + 2 ` 2 rz · 1 + 3 ` 1 r 2 z · 2 1 2 +2 ` 2 r · 1 z +2 ` 1 r · 2 z +2 2 ` 1 r 2 · 1 z · 1 ¡ · 2 t 0 · 1 z ¡ · 1 t 0 · 2 z ¡2 · 1 z · 2 z ` 0 z ¡2 · 1 z ¶ 2 ` 1 z ¡ 2 R 2 · 1 z · 1 ` 1 ¶‚‚ ¡ • 2 · 3 z 2 ¡ 3 2 2 · 1 z 2 · 1 z ¶ 2 0 IJTSRD| Available Onlinewww.ijtsrd.com|Volume-2|Issue-5|Jul-Aug 2018 Page:1415

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12 INTERNATIONAL JOURNAL OF TREND IN SCIENTIFIC RESEARCH AND DEVELOPMENT IJTSRDISSN:2456-6470 ¡ 1 2 · 1 R 2 · 1 z ¶ 2 + 1 R · 1 z · 2 z + · 3 R 2 ¡ 2· 1 · 2 R 3 + · 3 1 R 4 ¡ 3 R 3 · 1 · 2 ¡ 1 2R 2 2 · 1 z 2 · 1 ¶ 2 + 9 2R 4 · 1 · 1 ¶ 2 + 1 R 2 2 · 3 2 + 1 R 2 2 · 1 2 ‰ ¡ 1 2 · 1 z ¶ 2 ¡ 2· 2 R + 3· 2 1 R 2 ¡ 3 2R 2 · 1 ¶ 2 ¡ 2 R 3 · 1 2 · 2 2 ¡ 2 R 2 · 1 2 · 1 z · 1 z ‚ : A:8 REFERENCES 0 1. Hsieh D.Y. Interfacial stability with mass and heat transfer Phys. Fluids 1978 215: 745-748 2. Nayak A.R. and Chakraborty B.B. Kelvin- Helmholtz stability with mass and heat transfer Phys. Fluids 1984 278: 1937-1941 3. Elhefnawy A. R. F. Stability properties of a cylindrical ow in magnetic uids:efiect of mass and heat transfer and periodic radial fleld. Int. J. Engng Sci. 1994325: 805-815 4. Hsieh D.Y. Efiect of heat and mass transfer on Rayleigh-Taylor instability. Trans ASME1972 94D: 156-162. 5. Lee D.-S. Nonlinear stability of a cylindrical interface with mass and heat transfer Z. natur- forsch. 2000 55a: 837-842 6. Lee D.-S. Nonlinear instability of cylindrical interface with mass and heat transfer in mag- netic uids. Z. Angew. Math. Mech. 2002 82 8: 567-575 7. Awashi M.K. Asthana R. and Uddin Z. Nonlinear study of Kelvin-Helmholtz instability of cylindrical ow with mass and heat transfer Inter.Comm. Heat and Mass Trans. 201671: 216-224 0 IJTSRD| Available Onlinewww.ijtsrd.com|Volume-2|Issue-5|Jul-Aug 2018 Page:1416

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