Numerical solution of MHD free convection of heat and mass transfer flow over an inclined hot plate with viscous dissipation and Hall current

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1 Proceedings o the International Conerence on Mechanical Engineering and Reneable Energ (ICMERE) 9 Noveber Chittagong Bangladesh ICMERE-PI- Nuerical solution o MHD ree convection o heat and ass transer lo over an inclined hot plate ith viscous dissipation and Hall current M. S. Ala * M. Ali M. M. Ala and S. M. Riat Itekher Departent o Matheatics Chittagong Universit o Engineering & Technolog Chittagong Bangladesh Matheatics Discipline Khulna Universit Khulna Bangladesh Departent o ME Chittagong Universit o Engineering & Technolog Chittagong Bangladesh * Shahalaaths@gail.co ali.ehidi9@gail.co aali@ath.buet.ac.bd riatitu@gail.co Abstract- The ai o the present ork is an investigation o stead MHD ree convection heat and ass transer lo o an incopressible electricall conducting luid over a sei-ininite inclined hot plate under the inluence o an applied unior agnetic ield ith viscous dissipation & Hall current. The non-linear governing boundar laer equations together ith the boundar conditions are transored to a sste o non-linear ordinar dierential equations b using the siilarit transoration. The sste o non-linear ordinar dierential equations are then solved b developing a suitable nuerical techniques such as shooting iteration technique along ith Runge- Kutta ourth order integration schee. The nuerical results concerned ith the priar velocit secondar velocit teperature and concentration proiles eects o various paraeters on the lo ields are investigated and presented graphicall. The results presented graphicall illustrate that priar velocit ield rises due to increase o Hall paraeter Eckert nuber and angle o inclination paraeter hile secondar velocit increases or agnetic paraeter and Hall paraeter. Again the priar velocit ield is decreased or agnetic paraeter. The teperature ield increases or the increasing values o agnetic paraeter and Eckert nuber but up to a certain values o eta it is decreased and then increased or angle o inclination hereas reverse result arises or Prandtl nuber. Also the negligible increasing eect on concentration proile are observed or increasing the values o agnetic paraeter. Again the concentration is decreased or Schidt nuber and increased or inclination paraeter and Eckert nuber. The nuerical results or the heat lux ro the stretching surace are copared ith the results reported b other authors hen the Grasho nuber odiied Grasho nuber Hall current and inclination o the angle are absent. The present results in this paper are in good agreeent ith the ork o the previous author. Keords: MHD Hall current Viscous Dissipation. INTRODUCTION The MHD lo probles has iportant in vie o its signiicant applications in industrial anuacturing processes such as poer generator plasa studies petroleu industries coolin Nuclear reactors boundar laer control in aerodnaics etc. Also MHD lainar boundar laer lo over an inclined stretching sheet has noticeable applications in glass bloing continuous casting paper production hot rolling ire draing drain plastic ils etal and poler extrusion etal spinning and spinnin ibbers. During its anuacturing process a stretched sheet interacts ith the abient luid therall and echanicall. Both the kineatics o stretching and the siultaneous heatinr cooling during such processes has a decisive inluence on the qualit o the inal products. In the extrusion o a poler sheet ro a die the sheet is soe tie stretched. B draing such a sheet in a viscous luid the rate o cooling can be controlled and the inal product o the desired characteristics can be achieved. In vie o its signiicant application various authors has been done a lot o orks related to this ield such as Venkatesulu and Rao [] has considered the eect o Hall Currents and Thero-diusion on convective heat and ass ttranser viscous lo through a porous ediu past a vvertical porous plate Sudha Mathe et al.[] studied the Hall eects on heat and ass transer through a porous ediu in a rotating channel ith radiation Kuar and Singh [] have studied Matheatical odelin Soret and Hall eects on oscillator MHD ree convective lo o radiating luid in a rotating vertical porous channel illed ith porous ediu Chauhan and Rastogi [] analzed the eect o Hall current on MHD slip lo and heat transer through a Porous ediu over an accelerated plate in a rotating sste and Nazul Isla & Ala [] studied ICMERE

2 Duour and Soret eects on stead MHD ree convection and ass transer luid lo through a porous ediu in a rotating sste Raptiset et al. [] have studied the viscous lo over a non-linearl stretching sheet in the presence o a cheical reaction and agnetic ield. Tan et al. [] studied various aspects o this proble such as the heat ass and oentu transer in viscous los ith or ithout suction or bloing. Abel and Mahesh [8] presented an analtical and nuerical solution or heat transer in a stead lainar lo o an incopressible viscoelastic luid over a stretching sheet ith poer-la surace teperature including the eects o variable theral conductivit and non-unior heat source and radiation. So the present paper is ocused on stead MHD ree convection heat and ass transer lo o an incopressible electricall conducting luid over a sei-ininite inclined hot plate under the inluence o an applied unior agnetic ield ith viscous dissipation & Hall current.. FORMULATION OF THE PROBLEM AND SIMILARITY ANALYSIS Let us consider stead to diensional MHD ree convection heat and ass transer lo in an incopressible electricall conducting luid over a sei-ininite inclined hot plate under the inluence o an applied unior agnetic ield ith viscous dissipation and Hall current. The lo is subjected to a transverse agnetic ield o strength B hich is assued to be applied in the positive direction noral to the surace. The pressure gradient bod orce viscous dissipation and joule heating eects are neglected. Under the above assuptions and usual boundar laer approxiation the diensional governing equations o continuit oentu energ and concentration under the inluence o externall iposed agnetic ield ith the presence o Hall current are: u v Equation o continuit: () x Moentu equation: u u u u v ν gβt T x gβ * C C σb cos ρ u W cos W W W σb u v ν u - W () x ρ Energ Equation: T T k T μ u W u v x ρcp ρc p () σb ρc p u W Concentration Equation: C C C u v D () x here u and v are the velocit coponents along x and directions and W is the secondar velocit coponent along () T z-axis T T and are the luid teperature the stretching sheet teperature and the ree strea teperature respectivel hile C and are the C C corresponding concentrations is the hall paraeter k is the theral conductivit o the luid c p is the speciic heat ith constant pressure α is the angle o inclination μ is the coeicient o viscosit ν is the kineatic viscosit σ is the electrical conductivit ρ is the luid densit β is the theral expansion coeicient β* is the concentration expansion coeicient B is the agnetic ield intensit U is the stea velocit g is the acceleration due to gravit D is the coeicient o ass diusivit respectivel. The above equations are subject to the olloing boundar conditions: u U v W T T C C at and u v T T C C as To convert the governing equations into a set o siilarit equations e introduce the olloing siilarit transoration: W U g η η U ν x ψ ν xu η T T C C ψ ψ θη η u v T T C C x Fro the above transorations the non-diensional nonlinear and coupled ordinar dierential equations are obtained M Grθcos G cos g () M g g - g () θ ηpr θ PrEc g (8) PrEcM g Sc (9) The transor boundar conditions: and Where g g g θ θ at η as η θ and are the diensionless priar velocit secondar velocit teperature and concentration respectivel η is the siilarit variable is the value o η at hich boundar conditions is achieved the prie denotes dierentiation ith respect to η. Also xσ B M ρu μc Pr k p Gr Sc gβ ν D * T T x xgβ C C U and Ec c p G T U T are the agnetic paraeter Grasho nuber odiied Grasho nuber Prandtl nuber Schidt nuber and Eckert nuber respectivel. U η ICMERE

3 . METHODOLOGY The governing undaental equations o oentu theral and concentration in Netonian luids are essentiall nonlinear coupled ordinar or partial dierential equations. Generall the analtical solution o these nonlinear dierential equations is alost diicult so a nuerical approach ust be ade. Hoever no single nuerical ethod is applicable to ever nonlinear dierential equation. The various tpes o ethods that are available to solve these nonlinear dierential equations are inite dierence ethod shooting ethods quasi-linearization local siilarit and non-siilarit ethods inite eleent ethods etc. Aong these the shooting ethod is an eicient and popular nuerical schee or the ordinar dierential equations. This ethod has several desirable eatures that ake it appropriate or the solution o all parabolic dierential equations. Hence the sste o reduced nonlinear ordinar dierential equations together ith the boundar conditions have been solved nuericall using ourth-order Runge-Kutta schee ith a shooting technique. Thus adopting this tpe o nuerical technique described above a coputer progra ill be setup or the solution o the basic nonlinear dierential equations o our proble here the integration technique ill be adopted as the ourth order Runge-Kutta ethod along ith shooting iterations technique. First o all higher order non linear dierential equations are converted into siultaneous linear dierential equations o irst order and the are urther transored into initial value proble appling the shooting technique. Once the proble is reduced to initial value proble then it is solved using Runge -Kutta ourth order technique. The eects o the lo paraeters on the velocit teperature and species concentration are coputed discussed and have been graphicall represented in igures and also the values o skin riction rate o teperature and rate o concentration shon in Table or various values o dierent paraeters. In this regard deining ne variables b the equations g g θ θ 8 9 The higher order dierential equations () () (8) and (9) a be transored to nine equivalent irst order dierential equations and boundar conditions respectivel are given belo: M d d d M Grcosα Gcosα 8d M M d d d ηpr PrEc PrEc PrEcM PrEcM d8 9 d9 Sc 9 The boundar conditions are as ollos: at η and 8 8 as η. RESULTS AND DISCUSSION Nuerical calculation or the distribution o priar velocit secondar velocit teperature and concentration proiles across the boundar laer or dierent values o the paraeters are carried out. For the purpose o our siulation e have chosen M =. =. Pr =. Gr = -. G = -. Ec =. Sc =. and α= o hile the paraeters are varied over range as shon in the igures. The eects o various paraeters on priar velocit proile are shon in Fig. - Fig.. Fro Fig. it is observed that the priar velocit proile starts ro axiu value at the surace and then decreasing until it reaches to the iniu value at the end o the boundar laer or all the values M. It is interesting to note that the eect o agnetic ield is ore proinent at the point o peak value because the presence o M in an electricall conducting luid introduces a orce like Lorentz orce hich acts against the lo i the agnetic ield is applied in the noral direction as in the present proble. As a result velocit proile is decreased. Again ro Fig. and Fig. it is observed that the velocit is increased or the increasing values o Hall paraeter and angle o inclination. Besides Fig. shos the negligible increasing eect o Eckert nuber on priar velocit proile. Fig. - Fig.8 sho the secondar velocit proile or various values o M α and Ec. Fro these igures it is observed that the secondar velocit proile is increased or the increasing values o M and but decreasing eect are observed in case o Eckert nuber hereas up to a certain values o eta the secondar velocit proile is decreased and then increased or increasing values o inclination paraeter. Again Fig. 9 Fig. sho the teperature proile obtained b the nuerical siulation or various values o entering paraeters. Fro Fig.9 and Fig. it is clearl deonstrates that the theral boundar laer thickness increases or the increasing values o agnetic paraeter and Eckert nuber because heat energ is stored in the liquid due to the rictional heating. It is obvious that the increasing eect o Ec enhanced the teperature at an point o lo region. Fro this igure it is concluded that greater viscous dissipative heat be the cause o increasing teperature proile. Fro Fig. it is observed that in the certain interval o η the teperature proile is decreased and then increased or increasing values o inclination paraeter but reverse trend arises or the increasing values o Pr hich shon in Fig.. That is the theral boundar laer thickness increases as the Pr increases ipling loer heat transer. It is due to the act that grater values o Pr eans increasing theral conductivit and thereore it is able to diuse heat ro the plate ore slol than saller values o Pr hence the rate o heat transer is reduced as a result the heat o the luid in the boundar laer increases. Again Fig. - Fig. shos the concentration proiles obtained b the nuerical siulation or various values o entering non-diensional paraeters. Fro Fig. it is seen that the concentration proile is increased or the eect o Ec due to the act that heat energ is stored in the liquid due to the rictional heating but reverse results arises or the increasing values o agnetic paraeter hich is shon in Fig.. It is obvious that the increasing eect o Ec enhanced the concentration ICMERE

4 at an point o lo region or the case o present ork. Fro this igure it is concluded that greater viscous dissipative heat be the cause o increasing concentration proile. Again the concentration proile is increased or the increasing values o inclination paraeter hereas reverse results arises in case o Schidt nuber are shon in Fig. and Fig.. Again a negligible increasing eect are observed in case o agnetic paraeter hich are shon in Fig..Further the nuerical solutions or the skin riction and the Nuselt nuber have been copared ith those o Ishak et al. [9].These results are given in Table. and it is observed that the present results and those o Ishak et al. [9] are in good agreeent.. CONCLUSIONS In the present paper is an investigation o stead MHD ree convection heat and ass transer lo o an incopressible electricall conducting luid over a stretching sheet in a rotating sste under the inluence o an applied unior agnetic ield ith Hall current. The leading equations are solved nuericall b the shooting ethod along ith Runge- Kutta ourth-ith order integration schee. The results are presented to displa the lo characteristic like velocit teperature and concentration. Folloing are the conclusions ade ro above analsis: The agnitude o priar velocit proile decreases ith increasing agnetic paraeter. The agnetic paraeter is ound to retard the velocit at all points o the lo ield. It is because that the application o transverse agnetic ield ill result in a resistive tpe orce (Lorentz orce) siilar to drag orce hich tends to resist the luid lo and thus reducing its velocit but there is no eect on priar velocit proile or rotational and reaction paraeter. The secondar velocit starts ro iniu value at the plate and increases until it attains the axiu value ithin the boundar laer and then starts decreasing until it reaches the ree strea area satising the ar ield boundar condition. Thereore the noticeable increasing eect are observed or agnetic and Hall paraeter on secondar velocit proile but reverse result arises or Eckert nuber. It is interesting to note that up to a certain values o eta the secondar velocit proile is decreased and then increased or increasing values o inclination paraeter. The teperature proile is starting at the initial point o the plate surace and increases until it attains the axiu value ithin the boundar laer and then starts decreasing until it reaches to zero ar aa ro the plate satising the boundar condition. So noticeable increasing eect are observed on teperature proile or agnetic paraeter and Eckert nuber. Again up.8... to a certain interval o η the teperature proile is decreased and then increased or increasing values o inclination paraeter but reverse trend arises or the increasing values o Pr. That is the theral boundar laer thickness increases as the Pr increases ipling loer heat transer. It is due to the act that grater values o Pr eans increasing theral conductivit and thereore it is able to diuse heat ro the plate ore slol than saller values o Pr hence the rate o heat transer is reduced as a result the heat o the luid in the boundar laer increases In this case teperature asptoticall approaches to zero in ree strea region. The eect o Sc is ound to decrease the concentration proile because increasing in Sc decreases olecular diusivit hich result a decrease o the boundar laer. Hence the concentration o the species in loer or large values o Sc. Again negligible increasing eect are observed or the reaining paraeters. Table.Coparison o skin riction [ ] and rate o heat transer [ ] or dierent values o M hen Pr =. Gr = G = Ec =. Sc =. α = =. M Ishak et al. [9] Present results M=. M=. M= Fig.: Priar velocit proile or various values o M ICMERE

5 ..8. =. =. =....8 =. =. = Fig.: Priar velocit proile or various values o.. 8 Fig.: Secondar velocit proile or various values o.8.8. = = =.. = o = o = o Fig.: Priar velocit proile or various values o α.8. Ec=. Ec=. Ec=. 8 Fig.: Secondar velocit proile or various values o α.8.. Ec=. Ec=. Ec= Fig.: Priar velocit proile or various values o Ec M=. M=. M=. 8 Fig.: Secondar velocit proile or various values o M 8 Fig.8: Secondar velocit proile or various values o Ec M=. M=. M=. Fig. 9: Teperature proile or various values o M ICMERE

6 .. = o = o = o.8. = o = o = o... Fig. : Teperature proile or various values o α Fig. : Concentration proile or various values o α Ec=. Ec=. Ec=..8. Ec=. Ec=. Ec=.9 Fig. : Teperature proile or various values o Ec.. Fig. : Concentration proile or various values o Ec Pr=. Pr=. Pr=..8.. Sc=. Sc=. Sc=.9 Fig. : Teperature proile or various values o Pr.8... M=. M=. M=.9 Fig. : Concentration proile or various values o M. Fig. : Concentration proile or various values o Sc. ACKOWLEDGEMENT The authors are thankul to UGC (HEQEP World Bank) to get enough support ro Siulation lab (CP-8) under Matheatics Departent CUET hich helped us copletinur ork successull. Also e ould like to extend our sincere regards to VC and Pro-VC o CUET or their tiel support to establish the Siulation lab.. REFERENCES [] D.Venkatesulu and U. R. Rao Eect o Hall Currents and Thero-diusion on convective Heat and Mass Transer lo o a Viscous Cheicall reacting rotating luid through a porous ediu past a Vertical porous plate International Journal o Scientiic & ICMERE

7 Engineering Research vol. no.. [] S. Mathe P. R. Nath and R.D. Prasad Hall Eects on Heat and Mass Transer Through a Porous Mediu in a Rotating Channel ith Radiation Advances in Applied Science Research vol. no. pp. -9. [] R. Kuar and K. D. Singh Matheatical odelin Soret and Hall eects on oscillator MHD ree convective lo o radiating luid in a rotating vertical porous channel illed ith porous ediu International Journal o Applied Matheatics and Mechanics vol. 8 no. pp [] D. S. Chauhan and P. Rastogi Hall Eects on MHD Slip Flo and Heat Transer Through a Porous Mediu over an Accelerated Plate in a Rotating Sste International Journal o Nonlinear Science vol. no. pp. 8-. [] N. Isla and M. M. Ala Duour and Soret eects on stead MHD ree convection and ass transer luid lo through a porous ediu in a rotating sste Journal o Naval Architecture and Marine Engineering. [] A. Raptis and C. Perdikis Viscous lo over a non-linearl stretching sheet in the presence o a cheical reaction and agnetic ield International Journal o Nonlinear Mechanics vol. pp. -9. [] Y. Tan X. You Xu Hang and SJ. Liao A ne branch o the teperature distribution o boundar laer los over an ipereable stretching plate Heat Mass Transer vol. pp [8] M. S. Abel and N. Mahesha Heat transer in MHD viscoelastic luid lo over a stretching sheet ith variable theral conductivit non-unior heat source and radiation Applied Matheatics and Modelling vol. pp [9] A. Ishak R. Nazar and I. Pop Hdro agnetic Flo and Heat Transer Adjacent to a Stretching Vertical Sheet Heat Mass Transer vol. pp ICMERE