[Ahluwaliya* et al., 5(7): July, 2016] ISSN: IC Value: 3.00 Impact Factor: 4.116

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1 [Ahluwalya et al., 5(7): July, 6] IN: IC Value: 3. Impat Fator: 4.6 IJERT INTERNATIONAL JOURNAL OF ENGINEERING CIENCE & REEARCH TECHNOLOGY TABILITY OF VICO-ELATIC( RIVLIN-ERICKEN) COMPREIBLE FLUI IN POROU MEIUM : EFFECT OF MAGNETIC FIEL Preet Ahluwalya, Aftab Alam, udhr Kumar epartment of Mathemats, H L M College,Ghazabad,U.P.(INIA) epartment of Mathemats, RM Unversty, NCR, Campus Ghazabad, U.P.(INIA) epartment of Mathemats,.(P.G.)College, Muzaffarnagar,U.P.-5(INIA) OI:.58/zenodo ABTRACT The hydro-magnet nstablty of Rvln-Erksen ompressble flud n porous medum s onsdered by applyng normal mode tehnque, the dsperson result s obtaned. The mportant result obtaned n ths present paper, by a number of theorems provdng ondtons for stablty or nstablty, and the ase of slghtly ompressble flud neutral or unstable modes are non-osllatory and osllatory mode s also stable. It s also found that the system are stable, unstable or neutral analyss for subson and superson dsturbanes wth weak applyng magnet feld (.e. ) and another bounds on the omplex wave veloty of unstable modes for N. KEYWOR: Hydromagnet nstablty, Rvln-Erksen, ompressble flud, porous medum,weak magnet feld. INTROUCTION In lassal hydrodynams, the majorty of the prevously known results related to the study of stablty of flud flows onerns ether ompressble flud n the absene of porous medum of nompressble flud through a porous medum. However, n many flows of engneerng nterest, flud speeds exeed the speed of sound and densty hanges an be qute large. These flows are alled ompressble flows. It seems that the effet of ary resstane n ombnaton wth ompressblty, to the best of our knowledge, s almost unnvestgated so far. However, sne the ompressblty s exhbted by all fluds n approprate rumstanes, t s neessary to nlude ts effet nto the stablty analyss of a system n the presene of a porous medum. Hydromagnet stablty of gravtatonally stratfed dusty flud rotated n a porous medum dsussed by Rahna and Jamala []. It s found that n ths paper nlude the equvalene of two and three dmensonal dsturbanes, suffent of stablty estmates on the growth rate of unstable modes and the exstene of varatonal prnple. Kumar and ngh [] have studed the stablty of the plane nterfae separatng two vso-elast (Rvln-Erksen) superposed fluds n he presene of suspended partles. The stablty analyss has been arred out, for mathematal smplty for two hghly vso-elast fluds of equal knemat vsostes and equal knemat vso-elasttes. The system s found to be stable for stable onfguraton and unstable for unstable onfguraton. harma and Kango [3] have studed the thermal onveton n Rvln-Erksen elasto-vsous flud n porous medum n the presene of unform magnet feld. Prakash and Kumar [4] studed the thermal nstablty n Rvln-Erksen elasto-vsous flud n the presene of larmar radus and varable gravty n porous medum. harma and Rana [5] studed the magnetogravtatonal nstablty of a thermally ondutng rotatng Rvln-Erksen flud through a porous medum wth fnte ondutng n the presene of Hall urrent. The wave propagaton has been onsdered for both parallel and perpendular axes of rotaton and the magnet feld s beng taken n the vertal dreton. Reently, Km et al. [6] nvestgated by the thermal nstablty of Vso-Elast flud n porous medum usng the modfed ary-oldroyd model. Kumar et al. [7] studed the thermal nstablty of Walter s B' Vso-Elast flud http: // Internatonal Journal of Engneerng enes & Researh Tehnology [5]

2 [Ahluwalya et al., 5(7): July, 6] IN: IC Value: 3. Impat Fator: 4.6 permeated wth suspended partles n hydromagnets n porous medum. Jamala and Agrawal [8] have dsussed the stablty of a densty stratfed flud wth horzontal streamng through a porous medum. harma and Kumar [9] have dsussed the thermal nstablty of a layer of Rvln-Erksen elasto-vsous flud ated on by a unform rotaton and found that rotaton has a stablzng effet and ntrodue osllatory modes n the system. Pundr and Pundr [] have dsussed the thermal nstablty of a ontnuously stratfed Vso-Elast (Oldroydan) flud n porous medum n the presene of a horzontal magnet feld. Pal [] dsussed the Raylegh-Taylor nstablty of Rvln-Erkson plasma n presene of a varable magnet feld and suspended partles n porous medum. Prakash and Chand [] have dsussed the thermal nstablty of Oldroydan vso-elast flud n the presene of fnte larmor radus, rotaton and varable gravty feld n porous medum. Agrawal et al. [3] have studed the shear flow nstablty of vso-elast flud n ansotrop porous medum. Kumar et al. [4] nvestgated by the thermal nstablty of a rotatng (Rvln-Erksen) vso-elast flud n the presene of unform vertal magnet feld. It s found that rotaton has a stablzng effet whereas the magnet feld has both stablzng and destablzng effets. Bahadur [5] has dsussed by the thermal nstablty of (Rvln- Erksen) elasto-vsous rotatng fluds n presene of suspended partles n porous medum. Wth the growng mportane of non-newtonan flud n modern tehnology and ndustres, the nvestgatons on suh fluds are desrable. There are many vso-elast fluds that an t be haraterzed by Maxwell s onsttutve relatons or Oldroyd s onsttutve relatons. One suh lass of Vso-Elast s the Rvln-Erksen flud. Ths and other lass of polymers, ropes, ushons, seats, foams, plast, engneerng equpments, et. Reently polymers are also used n agrulture, ommunaton applanes and n bomedal applatons. When flud permeates a porous materal the gross effet s represented by the ary s law. As a result of ths marosop law, the usual vsous s replaed by ' q, where and are the vsosty and vsoelastty of the thermosolutal nstablty of Rvln- k t Erksen flud, s the medum permeablty and q s the aran (flter) veloty. However, hydromagnet k ' nstablty of Rvln-Erksen ompressble flud n porous meda seems to the best of our knowledge unnvestgated so far. In ths hapter, therefore, we have made an attempt to rtally examne the hydromagnet nstablty of Rvln- Erksen ompressble flud n porous medum. It an be looked upon an extenson of hydromagnet stablty of a ompressble flud n a porous medum dsussed by Jamala [6]. EQUATION OF MOTION In the equatons of moton for the gas, the presene of partles adds an extra fore term proportonal to the veloty dfferene between partles and gas. Assumng that the usual vsous dsspaton along wth the dsspaton due to ary resstane s present, the governng equatons for the gas an be wrtten as q, (). q (. ) p q q v v' ( ) q H H g λ, () t k t, (3). q t P P [usng3] (4). q t t t H (5) ( H. ) q (. q) H t and. H, (6) where, H, s and are respetvely the magnet feld vetor, entropy per unt mass, magnet permeablty of the vauum. The operator s defned as t. t t U (7) and other parameters have ther usual meanng. The equatons from () to (6) represent respetvely, the non-lnearty http: // Internatonal Journal of Engneerng enes & Researh Tehnology [6]

3 [Ahluwalya et al., 5(7): July, 6] IN: IC Value: 3. Impat Fator: 4.6 equaton, the momentum equaton, the equaton of mass onservaton, the energy equaton, the nduton equaton and dvergene free ondton for magnet feld. In the equaton (), thrd term or rght hand sde represents the Lorenze fore. Fnally n energy equaton (4) s the square of the sound speed (.e., the veloty of propagaton of small z dsturbanes) n the medum, depends upon oordnate only and defned as p (8) s Followng the usual proedure and onsderng the perturbatons q (,, ), H [ H ( unform),, ], (9) ( z) and p p( z), We get and, we assume that the perturbatons an be expanded nto normal modes so that the dependene on general dsturbane f '( x, z, t) s taken as f '( x, z, t) f ( z) exp k x t () where k represents the wave number n -dreton and, n general, s omplex we get the fnal governng equaton k R k k k R k w w g k R k w g g k R k w N w () where, g N g s the square of the Väsälä Brunt frequeny, H s the magnet fore number and s a funton of beause of the presene of, U v d R s the nverse of the ary-reynolds number ku v' R ve d and k R k. The boundary ondtons now beome In the lmt x w at z and z. () R z x, z and t (.e., n the absene of porous medum) equaton () redues to the one obtaned by Agrawal and Rastog [7]. Here also, the harater of the problem s hanged due to the presene of porous medum and the system s stable, neutral or unstable aordng as, or. Beause of the omplated harater of the problem, t s extremely dffult to dsuss the problem n ts full generalty. Therefore, a number of physally mportant ases have been dealt wth n the subsequent setons. of a http: // Internatonal Journal of Engneerng enes & Researh Tehnology [7]

4 [Ahluwalya et al., 5(7): July, 6] IN: IC Value: 3. Impat Fator: 4.6 THE CAE OF A LIGHTLY COMPREIBLE FLUI The ompressblty of a flud s the nverse of ts bulk modulus of elastty. Therefore, the ompressblty of the flud s the rato of the relatve hange of the volume to the hange n appled pressure. Thus fluds whh requre a large pressure hange to densty, are alled slghtly ompressble fluds, whle fluds for whh even a small pressure hange makes an appreable hange n the densty are alled hghly ompressble or to say, smply ompressble fluds. It s well known that the gases are hghly ompressble whle the lquds are slghtly ompressble. Further, t s evdent that n hghly ompressble fluds, the veloty of sound wll be small whle n slghtly ompressble fluds, t wll be large. Therefore, t s very mportant, n the ontext of many physal stuatons to nvestgate extensvely, as we have done here, the ase of slghtly ompressble fluds. For slghtly ompressble fluds, g / w g / s assumed to be small and of the order of perturbatons so that the term beng of seond order n perturbatons an be gnored n omparson to w or ts dervatves. Under ths approxmaton, we get the redued stablty governng equaton as k R k k k R k w w N w. (3 ) In obtanng equaton (3), we have further assumed that g as k s small. w Multplyng equaton (3) by, the omplex onjugate of w, ntegratng over the range of and usng the boundary ondtons, we get after some arrangement of terms k R k w dz k k R k w dz N w dz. (4) The real and magnary part of ths equaton are gven by 4 ( )(. ( ) ) k R k k r ( k ) R k ( k ) (5) k R w dz w r ( k ) k R k w dz. (6) and We now prove the followng theorems. Theorem 3. : Neutral or unstable modes are non-osllatory. Proof : Equaton (6) shows that f, then s neessarly zero, showng thereby that modes, whether neutral or unstable must be non-osllatory. Hene, t s lear that osllatory modes are, stable. r It s to be noted that due to the presene of porous meda, we are not able to haraterze the stable modes n general. As t s lear from equatons (5) and (6), that, we an say nothng about r and hene about the harater of stable modes. Theorem 3. : If N everywhere n the flow doman, then the system s stable. z http: // Internatonal Journal of Engneerng enes & Researh Tehnology [8]

5 [Ahluwalya et al., 5(7): July, 6] IN: IC Value: 3. Impat Fator: 4.6 Proof : Let f possble, the system be unstable when N. Then and Equaton (5) now beomes 3 4 ( )( ( ) ) k k R k ( k ) 3 R k ( k ) k R w dz k ( k R ) R k w dz N w dz (7) ve r n vew of the theorem (). whh s mathematally nonsstent. Hene, t follows that the system must be stable under the ondton everywhere n the flow doman. Corollary : If N and N k everywhere n the flow doman, then the system s stable. Theorem 3.3 : An upper bound on N k max mn. kr assoated wth an arbtrary unstable mode f exsts when Proof : Multplyng equaton (4) by and separatng ts magnary part, we have 4 {( ) } k k R ( k ). ( k R )( ) ve r k R w dz ( ) R k w dz k k k R w dz k w dz N w dz. (8) For N, rearrangement of terms n equaton (8) yelds ( ) k k R ( k ). 4 ( k R )( ) ve r k R w dz k ( k ) w dz R k w dz R k ( k N ) w dz. (9) N N s gven by Let the system be unstable. Then equaton (9) n vew of theorem () s onsstent f R k ( k N ) somewhere n the flow doman Comparng the above bounds for unstable modes wth those obtaned by Agrawal and Rastog [7], we see that porous medum has a stablzng effet n vew of the observaton that the bounds on redue as the ary resstane nreases. THE CAE OF A WEAK APPLIE MAGNETIC FIEL (.e., ) It s well known [Chandrasekhar [9]] that a sutable magnet feld tends to suppress the nstablty of varous flows and n many ases t makes an otherwse unstable flow ompletely stable to small perturbatons. For ths reason, many http: // Internatonal Journal of Engneerng enes & Researh Tehnology [9]

6 [Ahluwalya et al., 5(7): July, 6] IN: IC Value: 3. Impat Fator: 4.6 authors have nvestgated the effet of a magnet feld on varous unstable flows. Our purpose s also to see the effet of magnet feld on the stablty of ompressble flud flowng through a porous medum by restrtng our analyss, due to the omplex nature of the problem under onsderaton, to weak appled magnet feld. [Agrawal and Agrawal []], When and w are small enough and therefore an be negleted n omparson to (sne w s a perturbaton and ther dervatves are small. It s n fat true when s large). The fnal governng equaton () redues to k R k g k k R k w w g g k R k w N w. () Let us now apply the transformaton. () q we k w w where, g E exp dz. () Equaton () now redues to Rq k R k k R k R k q R( k N ) q. (3) where, R E together wth the boundary ondtons q at z and q z. (5) Multplyng equaton (3) by and usng the boundary ondtons, we get k R k R q dz k R k R k q dz R ( N ) dz. (6) Real and magnary parts of equaton (6) are respetvely gven by 4 ( k )( k R ( k ) ) R r ( k ) R ( ) k k k R q dz, the omplex onjugate of q, ntegratng over the range of z k R ( )( k R ) k R q dz r ve R ( k N ) q dz (7) and q r ( k ) k R k q dz. (8) We now prove the followng theorems : Theorem 4. : Neutral or unstable modes are non-osllatory. Proof : Proof mmedately from equaton (8). By mplaton, t follows that the osllatory modes are stable. Theorem 4. : If N everywhere n the flow doman, the system s stable. k (4) http: // Internatonal Journal of Engneerng enes & Researh Tehnology []

7 [Ahluwalya et al., 5(7): July, 6] IN: IC Value: 3. Impat Fator: 4.6 Proof : For, let the system be unstable,.e.,, then n vew of theorem (), should be zero. But N equaton (7) beomes mathematally nonsstent on substtutng greater than zero. The smlar remark s germane f we start wth doman, r neessarly mplyng thereby the stablty of the system. when. Thus for N N r. Therefore an not be everywhere n the flow ). In fat, we are more We observe that ths result holds true even n the absene of a magnet feld(.e., nterested n the ase when, n order to demonstrate the stablzng role of magnet feld and porous medum. s the ase when ether (statally unstable arrangement) or f, then g havng a possblty of the unstable modes to our n the absene of magnet feld. Therefore, n the subsequent theorems, we wll dsuss the ase when. N N N Theorem 4.3 : If and k N / everywhere n the flow doman, the system s stable. Proof : For, equaton (7) an be wrtten as 4 ( k )( k R ( k ) ) R r ( k ) N N R ( ) k k k R q dz k R ( )( k R ) k R q dz r ve R ( k N ) q dz. (9) Theorem () shows that f, equaton (9) beomes nonsstent f the ondton N k (3) holds everywhere n the flow doman. It follows that the system s stable for the wave numbers gven by (3) f N. It s lear from the above theorem that the range of stable wave numbers nreases as magnet feld nreases. Ths establshes the stablzng harater of magnet feld n the present ontext. It follows from the above dsusson that nstablty mght our for small wave numbers n the range N. k In the next theorem, therefore, we have made an attempt to fnd the bounds on modes, f exst n the wave number range of N k. assoated wth arbtrary unstable ANALYI FOR UBONIC ITURBANCE Ths seton deals wth the hydromagnet stablty of subson dsturbanes n the presene of an arbtrary magnet feld. The stablty governng equaton () for subson dsturbanes redues to k R k k ( k R k ) w w g ( k R k ) w http: // Internatonal Journal of Engneerng enes & Researh Tehnology []

8 [Ahluwalya et al., 5(7): July, 6] IN: IC Value: 3. Impat Fator: 4.6 g g ( k R k ) w N w. (3) w Multplyng equaton (3) by ntegratng over the range of k R k w dz k ( k R k ) w dz g k R k w dz z and usng the boundary ondtons, we get N w dz. (3) The real and magnary parts of equaton (3) are respetvely gven by k R ( k ) ( r )( k ) k R k R w dz k R ( k ) g ( r )( k ) k R k R w dz g k R ( k ) ( )( r k ) k R w k R dz k ( )( k R ) k R ) w dz N w dz r ve and w g r ve ( ( k R ) k R ) dz w dz g w k w dz dz (33). (34) We now prove some theorems gven below Theorem 5. : Neutral or unstable modes are non-osllatory under the ondton ( / ). Theorem 5. : If N and ( / ), everywhere n the flow doman, the system s stable. Theorem 5.3 : If N, ( / ) and k N /, the system s stable. Theorem 5.4 : If N and ( / ) everywhere n the flow doman, then an upper bound on the omplex wave veloty of arbtrary unstable mode, f exsts, s gven by N max k mn. kr Proof : The magnary part of equaton (3) after multplyng t by s gven by ( k ) k R k R ( k ) k R w http: // Internatonal Journal of Engneerng enes & Researh Tehnology []

9 [Ahluwalya et al., 5(7): July, 6] IN: IC Value: 3. Impat Fator: 4.6 g g w dz w dz k ( k ) k R ( ) w dz N w dz. (35 ) Or after some rearrangement of terms, t beomes ( k ) k R k R ( k ) k R w g g w dz w dz k ( k ) w dz kr ( k N ) w dz. (36) Now, let the modes be unstable under the ondtons and ( / ) everywhere n the flow doman, then for the onssteny of equaton (36), we must neessarly have kr ( k N ) somewhere n the flow doman. N In vew of doman of theorem (), t follows that an estmate on assoated wth an arbtrary unstable mode s gven by N k max kr mn. ANALYI FOR UPERONIC ITURBANCE A dsturbane s sad to be superson [Lees and Ln [], Lesson, Fox and Zen [] and Blumen [3], et.] ff ts wave speed relatve to the flow veloty n the dreton of wave propagaton s greater than the loal son veloty. Therefore, for superson dsturbanes, we must have ( U ) /, (37 ) where, U s the flow veloty. In the absene of any flow veloty, ondton (37) redues to /. ( 38) In the present ontext, for superson dsturbanes, we have k R k, so that k R k R. (39) ve Thus, under ths approxmaton, the stablty governng equaton () redues to ( ) w k ( k R ) k R w. (4 ) ve Multplyng equaton (4) by ondtons, we get w, the omplex onjugate of w, ntegratng over the range of z and usng boundary ( ) w dz k k R k R w dz. ve The real and magnary parts of equaton (4) are gven by ( ) w dz k ( ) k R k R w dz (4 ) r ve http: // Internatonal Journal of Engneerng enes & Researh Tehnology [3]

10 [Ahluwalya et al., 5(7): July, 6] IN: IC Value: 3. Impat Fator: 4.6 (4) and k r ( k R ) w dz. (43) We now prove the followng theorem : Theorem 6. : The system s stable. Proof : Let the system be unstable or neutral,.e.,, then n vew of theorem (), r. But for r, equaton (4) s mathematally nonsstent. Therefore, there exsts not even a sngle mode for whh the system s unstable or neutral or n other words the system s stable. THE GENERAL CAE An attempt has been made n seton, to dsuss the problem n ts full generalty and followng results are obtaned. Theorem 7. : For, and N, unstable modes, f exst, are osllatory and the non-osllatory N k modes, f exst, are stable. Proof : After some smplfaton, fnal stablty governng equaton () an be wrtten as ( k R ) k R ve k ( k R k ) w w g ( k ) k R w g ( k ) k R w Multplyng equaton (44) by boundary ondtons, we get ( k ) k R w dz w N w. (44), the omplex onjugate of w, ntegratng over the range of z and usng the ( k k ) k R w dz g k R k ( ve ) g k R k R N w dz. (45) Under the ondtons stated n the theorem, equaton (45) beomes mathematally nonsstent n the followng two stuatons : [] When. Therefore, for arbtrary unstable or neutral modes, an not be zero or n other words, arbtrary unstable or neutral modes are osllatory. [] If for r, we put. Therefore, f non-osllatory modes exst, should neessary be less than zero or the non-osllatory modes are stable. ACKNOWLEGEMENT The author frst wshes to thanks Assoate Professor udhr Kr. Pundr for ths paper work. REFERENCE [] Rahana and Jamala :Ata Cena Inda, XIXM, 3, 6 (993). [] Kumar, P. and ngh, G. J. : Rom. Journ. Phys., 5 (9), 97 (6). [3] harma, R. C. and Kango,. K. : Czeh. J. Phys., 49, 97 (999). [4] Prakash, K. and Kumar, N. : J. Phys. o. Jpn., 68, 68 (). r and http: // Internatonal Journal of Engneerng enes & Researh Tehnology [4]

11 [Ahluwalya et al., 5(7): July, 6] IN: IC Value: 3. Impat Fator: 4.6 [5] harma, V. and Rana, G. G. : Indan J. Pure Appl. Maths., 3, 559 (). [6] Km, W. C., Lee,. B. and Km,. : Int. J. Heat Transfer, 46, 5 (3). [7] Kumar, P., ngh. G. J. and Lal, R. : Thermal ene, 8(), 5 (4). [8] Jamala and Agrawal,. C. : nd. J. Pure Appl. Maths., (7), 6 (99). [9] harma, R. C. and Kumar, P. : Z. Natur. for h., 3a, 8 (998). [] Pundr,. K. and Pundr, R. : Ata Cena Inda, XXIXM, 3, 487 (3). [] Pal, R. : Ph.. Thess, C.C.. Unversty, Meerut (). [] Prakash, K. and Chand, R.: Pro. Nat. Aad. Inda, IV, 7(A) (). [3] Agrawal, A., Jamala and Agrawal,. C. : Ganta,, 55, 6 (4). [4] Kumar, P., Mohan, H. and Lal, R. : Int. Journal of Maths and Mathematal enes, (6). [5] Bahadur, R. : Ph.. Thess, C.C.. Unversty, Meerut (8). [6] Jamala : Ph.. Thess, C.C.. Unversty, Meerut (99). [7] Agrawal,. C. and Rastog, V. P. : The Maths. tudent, 48(4), 356 (956). [8] attnger,. H. : am. J. Appl. Maths., 5, 45 (967). [9] Chandrasekhar,. : Oxford Unversty Press, London, (99). [] Agrawal,. C. and Agrawal, G.. : J. Phys. o. Japan, 7, 8 (989). [] Lees, L. and Ln, C. C. : N.A.C.A. Teh., Note, 5 (946). [] Lesson, M., Fox, J. A. and Zen, H. M. : J. Flud Meh.,, 9 (965a) [3] Blumen, W. : J. Flud Meh., 4, 769 (97). http: // Internatonal Journal of Engneerng enes & Researh Tehnology [5]