Magnetohydrodynamic Flow and Heat Transfer of Two Immiscible Fluids through a Horizontal Channel

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1 Research Article Internional Jornal of Crrent Engineering Technolog ISSN INPRESSCO. All Rights Reserved. Available Magnetohdrodnamic Flow He Transfer of Two Immiscible Flids throgh a Horizontal Channel Abdl Meen Ȧ* Ȧ Department of UG & PG Stdies Research in Mhemics Government College Glbarga Karnaka India Accepted 5 November 3 Available online December 3 Vol.3 No.5 (December 3) Abstract In this paper a theoretical analsis of stead laminar fll developed flow he transfer of two immiscible Magnetohdrodnamic viscos flids throgh a horizontal channel taking into an accont the effect of viscos Jole dissipion is presented. The flow below the interface is assmed to be electricall condcting while the other flid the channel walls are electricall insling. The channel walls are maintained two different constant temperres. Partial differential eqions governing the flow he transfer are transformed to ordinar differential eqions are solved exactl for both the region. The variion of velocit temperre profiles with respect to varios parameters sch as Magnetic field parameter Prtl nmber Eckert nmber rios of viscosit thermal condctivit are depicted graphicall. Kewords: He transfer Magnetohdrodnamics immiscible flids viscos dissipion.. Introdction The flow he transfer of electricall condcting flids in channels is a classical problem nder the effect of a transverse magnetic field which occrs in Magnetohdrodnamic (MHD) power generors pmps accelerors aerodnamic heing electrostic precipition polmer technolog petrolem indstr prificion of crde oil. This problem has received considerable tention in the case of horizontal parallel ples than vertical parallel ples. Hartmann (Hartmann 937) carried ot the pioneer work in the std of stead MHD channel flow of a condcting flid nder the niform magnetic field transverse to an electricall insled channel wall. The MHD flow of electricall condcting flid in the presence of an external magnetic field de to the implsive motions of an infinite fl ple has been discssed b Rossow (Rossow 957) he has obtained an exact soltion of the modified Navier-Stokes eqions Maxwell s eqion for the particlar case when the rio of the viscos magnetic Renolds nmber is nit. Then the problem was extended in nmeros was (Alpher 96 Alireza Sahai 99 Attia Kotb 996 Ha Abbass 8 T. Ha. et. al. Stamenković M. Ž. et. al. Simon Shagaia Daniel 3). All the mentioned stdies pertain to a single-flid model. Most of the problems reling to the petrolem *Corresponding athor: Abdl Meen indstr plasma phsics magneto-flid dnamics etc. involve mlti-flid flow sitions. The interest in these tpes of problems stems from the possibilit of redcing the power reqired to pmp oil in a pipeline b sitable addition of wer. The first investigions were associed with the LM-MFM generor project the Argone Nional Laboror. Packham (Packham Shail 97) analzed strified laminar flow of two immiscible liqids in a horizontal pipe. Hartmann flow of condcting flid in a channel with a laer of non-condcting flid between pper channel wall the condcting flid has been stdied b Shail (Shail 973). He fond th increase of order 3% cold be achieved in the flow re for sitable rios of depths viscosities of two flids. Loharsbi (Loharsbi Sahai 988) dealt with two-phase MHD flow he transfer in a parallel-ple channel. Both phases are incompressible the flow was assmed to be stead one-dimensional fll developed. There have been some experimental analtical stdies on hdrodnamic aspects of two-flid flow reported in the recent literre (Malasahett Leela 99 Chamkha Malashett et al. ) Umavhi (Umavhi et al. 4) investiged fll developed flow he transfer in horizontal channel consisting of an electricall condcting flid laer swiched between two flids laers. Recentl Umavhi (Umavhi et. al ) have presented analtical soltions for nstead/oscillor two-flid flow he transfer in a horizontal channel. Stamenković (StamenkovićM. Ž. et. al. ) investiges the Magnetohdrodnamic flow of 95

2 Abdl Meen Internional Jornal of Crrent Engineering Technolog Vol.3 No.5 (December 3) two immiscible electricall condcting flids between isothermal insled moving ples in the presence of an applied electric inclined magnetic field with the effects of indced magnetic field. Keeping in view the wide area of practical importance of mlti-flows he transfer nder the effect of a transverse magnetic field as mentioned the objective of this std is to investige MHD flow of two immiscible flids throgh a horizontal channel. The flow above the interface is assmed to be electricall condcting while the other flid the channel walls are assmed to be electricall insling.. Mhemical Formlion Y B -h Figre. Phsical configrion Fig. Phsical Configrions h The phsical configrion is shown in Fig. consisting of two infinitel long parallel ples maintained different constant temperres T w T w. The region-i ( h ) is occpied b a viscos flid having densit dnamic viscosit specific he constant C p thermal condctivit K. The region-ii ( h ) is occpied b a different viscos flid having densit dnamic viscosit specific he constant C p thermal condctivit K. A constant magnetic field of strength B is applied transverse to the flow direction the indced magnetic field is neglected. Also it is assmed th no electric field exists th the Hall effect of MHD is negligible. However the effect of Jolean he dissipion is inclded. The flow is assmed to be laminar stead fll developed is driven b temperre gradients Tw Tw p also b pressre gradient. x With these assmptions taking Cp Cp Cp the governing eqions of the flow field is given b: p () x T K () T w X T w p B (3) x T K B (4) where is the x-component of flid velocit T is the flid temperre. The sbscripts correspond to region-i region-ii respectivel. The bondar conditions on velocit are the no-slip bondar conditions which reqired th the x-component of velocit mst vanish the wall. The bondar conditions on temperre are isothermal conditions. We also assme the continit of velocit shear stress temperre he flx the interface between the two flid laers =. The bondar interference condition for this problem is written as T T h T T K K T T w T T w h B the se of following non-dimensional qantities: * U i Ti T i T T w C Pr K i p w w * h M B h p h P U w U Ec C ( T T w ) P x for simplicit dropping the asterisks eqions () to (5) becomes d P d d d Ec Pr d d d mm m P d d kec Pr d Pr kec M () d m d K where m is the rio of viscosities k is K the rio of thermal condctivities. The bondar interference condition in the nondimensional form becomes (5) (6) (7) (8) (9) 953

3 Abdl Meen Internional Jornal of Crrent Engineering Technolog Vol.3 No.5 (December 3) m k h h () on temperre the velocit profiles are illstred in Figs. 9. Ths one can conclde th flow can be controlled b varing the phsical parameters governing the flow sch a Magnetic field parameter Prtl nmber Eckert nmber rios of viscosit thermal condctivit of different flids.. 3. Soltion The governing eqions (7) to () are solved exactl for sing the conditions (). The soltions of velocit temperre for both the regions are discssed in the next section. 4. Reslts Discssion M=. The problem of two immiscible flid flow he transfer in a horizontal channel taking into an accont the effect of viscos Jole dissipion is stdied for the flow below the interface is assmed to be electricall condcting while the other flid the channel walls are electricall insling. The velocit temperre profile are obtained analticall. To std the effect of magnetic field Prtl nmber Eckert nmber rios of viscosit rios of thermal condctivit on flow flied nmerical vales are compted for exact soltion. The following parameter vales except the varing one are fixed as P 5 M h m k Pr. 7 Ec.. Figre illstred th the presence of transverse magnetic field prodces a resistive force called the Lorentz force which acts in the direction opposite to th of the flow which slow down the motion of the flid. It is obvios th increase in the magnetic field parameter decrease the velocit the temperre profiles is also decreases as shown in Fig. 3. Figres 4 5 depict the effect of viscosit rio m on the velocit temperre distribtions for both the regions respectivel. It is seen form these figres th the vale of m reslt in enhancement of the velocit fields also enhance the temperre distribtion de to the viscos dissipion effect. Fig. 6 displas the inflence of the thermal condctivit rio k on the temperre profiles. Increases in the thermal condctivit rio have the tendenc to cool down the thermal ste in the channel. This is depicted in the redction in flid temperres as k increases. As the Prtl nmber increases correspondingl increase the strength of the he sorces in the temperre eqion hence this strength in trn increases the overall flid temperre as shown in Fig. 7. De to the implicit relionship between the Prtl nmber the velocit field via the magnetic field terms it is noted th increases in the Prtl nmber have limited effects on the flid velocities as illstred in Fig. 8. The effects of the Eckert nmber are similar to those for the Prtl nmber Fig. Velocit profile for different vales of M Fig. Velocit profile for different vales of M M=. Fig. 3 Temperre profile for of M Fig.3 Temperre profile for different vales of M. -.. m = Fig. 4 Velocit profile for different of m Fig.4 Velocit profile for different vales of m 954

4 Abdl Meen Internional Jornal of Crrent Engineering Technolog Vol.3 No.5 (December 3).... m = Ec= Fig. 5 Temperre profile for different vales of m Fig.5 Temperre profile for different vales of m Fig. 9 Temperre profile for of Ec Fig.9 Temperre profile for different vales of Pr.... k = - - Ec= Fig. 6 Temperre profile for of k Fig.6 Velocit profile for different vales of k Fig. Velocit profile for vales of Ec Fig. Velocit profile for different vales of Bc. Acknowledgment 5. The athor wold like to thank UGC-New Delhi for the financial spport nder UGC-Minor Research Project. [MRP(S)-75/-/KAGUG9/UGC-SWRO] Nomenclre -.7 Pr= Fig. 7 Temperre profile for vales of Pr Fig.7 Temperre profile for different vales of k. C P Ec k M P Pr T T w t U specific he constant pressre Eckert nmber thermal condctivit Hartmann nmber pressre Prtl nmber temperre wall temperre time average velocit - Pr= Greek letters flid densit viscosit of flid non-dimensional temperre Fig. 8 Velocit profile for of Pr Fig.8 Velocit profile for different vales of Pr Sbscripts qantities for region-i region-ii respectivel. 955

5 Abdl Meen Internional Jornal of Crrent Engineering Technolog Vol.3 No.5 (December 3) Reference Alireza Setaesh Sahai V. (99) He transfer in developing MHD poiseille flow variable transport properties Int. J. He Mass Transfer 33(8) 7-7. Alpher R.A. (96) He transfer in magnetohdrodnamics flow between parallel ples Int. J. He Mass Transfer Attia H.A. Kotb N.A. (996) MHD flow between two parallel ples with he transfer Acta Mechanica Chamkha A. J. () Unstead laminar hdromagnetic flidparticle flow he transfer in channels circlar pipes Int. J. He Flid Flow Hartmann J. (937) Theor of the laminar flow of an electricall condctive liqid in a homogeneos magnetic field Mh. Fs. Med Lohrasbi J. Sahai V. (988) Magnetohdrodnamic he transfer in two phase flow between parallel ples Appl. Sci Res Malashett M. S. Leela V. (99) Magnetohdrodnamic he transfer in two phase flow Int. J. Engg. Sci Packham B. A. Shail. R. (97) Strified laminar flow of two immiscible flids Proc. Camb. Phil. Soc Rossow V. J. (957) Implsive motion of an infinite fl ple in a hdromagnetic viscos incompressible flid NACA-TN Report Shail R. (973) On laminar tow-phase flow in magnetohdrodnamic Int. J Engg. Sci Simon Daniel Shagaia Y. Daniel (3) Convective flow two immiscible flids he transfer with poros along an inclined channel with pressre gradient Research Invent: Internional Jornal of Engineering Science (4) -8. Stamenković M. Ž. et. al. () MHD flow he transfer of two immiscible flids between moving ples Transactions of the Canadian Societ for Mechanical Engineering 34(3-4) StamenkovićM. Ž. et. al. () Magnetohdrodnamic flow he transfer of two immiscible flids with indced magnetic fields effects Thermal Science 6() T. Ha R. Sajjad S. Asghar () Series soltion for MHD channel flow of a Jeffer flid Commnicion in Nonlinear Science Nmerical Simlion 5(9) T. Ha Z. Abbas (8) He transfer analsis on the MHD flow of a second grade flid in a channel with poros medim Chaos Solitons & Fracrals 38() Umavhi J. C. Malashett. M.S. Abdl Meen (4) Fll developed flow he transfer in a horizontal channel containing electricall condcting flid swiched between two flid laer Int. J. Applied Mechanical Engineering 9(4) Umavhi J.C. Abdl Meen Chamkha A.J. Mdhaf A.A. (6) Oscillor Hartmann two flid flow he transfer in a horizontal channel Internional Jornal of Applied Mechanics Engineering () Umavhi J.C. Chamkha A.J. Abdl Meen Mdhaf A.A. (5) Unstead two-flid flow he transfer in a horizontal channel He Mass Transfer Umavhi J.C. Chamkha A.J. Meen A. Mdhaf A.A. (9) Unstead oscillor flow he transfer in a horizontal composite poros medim channel Nonlinear Analsis: Modelling Control 4(3)