Unsteady MHD Second Grade Fluids Flow in a Porous Medium with Ramped Wall Temperature

Transcription

1 Unsead MHD Second Grade Fluids Flow in a Porous Medium wih Ramped Wall Temperaure ZULKHIBRI ISMAIL Universii Malasia Pahang Facul of Indusrial Sciences & Technolog Lebuh Raa Tun Razak 6300 Kuanan Pahang zulkhibri@umpedum AHMAD QUSHAIRI MOHAMAD Universii Teknologi Malasia Deparmen of Mahemaical Sciences Facul of Science 830 Johor Bahru ahmadqushairi9@ahoocom ILYAS KHAN College of Engineering Majmaah Universi Basic Sciences Deparmen PO Box 66 Majmaah 95 SAUDI ARABIA ilaskhanqau@ahoocom SHARIDAN SHAFIE Universii Teknologi Malasia Deparmen of Mahemaical Sciences Facul of Science 830 Johor Bahru sharidan@umm Absrac: In his paper he unsead MHD flow of second grade fluids in a porous medium are analzed I is assumed ha he bounding infinie inclined plae has a ramped wall emperaure wih he presence of hea mass diffusion Closed-form soluions in a general form are obained b using he Laplace ransform echnique The obained resuls for veloci is found o saisf all he imposed iniial boundar condiions I can be reduced o known soluions from he lieraure as limiing cases Ke Words: Double diffusion MHD Porous medium Inclined plae Second grade fluids Laplace ransform Inroducion Free convecion flow pas an inclined plae is essenial due o is wide applicaions in modern echnolog [] invesigaed he effec of hermal radiaion on unsead MHD free convecion flow of an opicall hin gra gas pas an infinie inclined isohermal plae In recen ears ismail e al ismail e al [ 3] performed he effecs radiaion on MHD free convecion flow in a porous medium pas an infinie inclined plae wih ramped wall emperaure Non-Newonian fluids are imporan because of is significan applicaion such as perisalic ranspor [4] polmers processing biomechanics enhanced oil recover food producs [5] Samiulhaq e al [6] examined a free convecion flow of a second grade fluid wih ramped wall emperaure using Laplace ransform mehod Mos recenl samiulhaq e al [7] considered a porous medium wih he flow of free convecion second grade fluids However a lile work has been sudied regarding second grade fluids wih hea mass ransfer effecs over an inclined plae which occurs frequenl in naure hence he moivaion of his paper The objecive of his presen paper is o solve he double diffusions of second grade fluids on unsead MHD free convecion flow in a porous medium pas an infinie inclined plae wih ramped emperaure using Laplace ransform echnique Mahemaical Formulaion Consider he unsead MHD of second grade incompressible fluids wih combined hea mass ransfer b naural convecion flow near an infinie inclined plae embedded in a sauraed porous medium The x -axis is along o he plae wih he inclinaion angle φ o he verical he - axis is aken normal o he plae The plae is assumed o be elecricall conducing wih a uniform magneic field B of srengh B 0 applied in a direcion perpendicular o he plae The magneic Renolds number is assumed o be small in order o neglec he effec of applied magneic field The radiaion effec is also aken ino accoun Iniiall for ime 0 boh he fluid he plae are a res wih he consan emperaure T consan concenraion C A a ime > 0 he emperaure is raised or lowered o T Tw T 0 when 0 Thereafer i is mainained a a consan emperaure Tw when > 0 The concenraion is raised o a consan concenraion Cw The phsical vari- E-ISSN: 4-347X 76 Volume 06

2 ables become funcions of he ime he space onl since plae is infinie in x z -direcions The geomer of he problem is shown in Figure Figure : Geomer diagram coordinae ssem From he above assumpions he unsead MHD free convecion viscous fluids flow pas an infinie inclined plae in a porous medium is governed b u = v u α ρ K v α θ 3 u σb 0 ρ u u gβ T cos φ T T gβ C cos φ C C T = k ρc p T ρc p q r C = D C 3 wih he following iniial boundar condiions: u = 0 T = T C = C a 0 0 u = 0 C = C w a = 0 > 0 T = T T w T 0 4 a = 0 0 < 0 T = T w a = 0 > 0 u C T 0 as > 0 u T C denoe he veloci emperaure concenraion respecivel υ is he kinemaic viscosi α represens he maerial parameer of second grade fluid σ is he elecrical conducivi of he fluid ρ is he fluid densi ϕ is a porosi of porous medium K is he permeabili of he porous medium g is he acceleraion due o he gravi β T β C are he hermal expansion concenraion expansion k is he fluid hermal diffusivi c p is he specific hea q r is he radiaive hea flux D is he mass diffusion I is assumed ha he radiaive hea flux erm is produced b plaes emperaure T T w simplified b using Rossel approximaion q r = 4α 0σ T 4 T 4 5 α 0 is he mean radiaion absorpion coefficien σ is he Sefan-Bolzmann consan We assume ha he emperaure differences wihin he flow are sufficienl small such ha T 4 ma be expressed as a linear funcion of he emperaure Using Talor series b exping T 4 abou T neglecing higherorder erms hus T 4 = 4T 3 T 3T 4 6 Inroducing he following dimensionless variables = L = vg / 3 u = u L vg / 3 T = T T T T C = C C C 7 w w C L = v/ 3 g / 3 K = ϕl K M = σ BoL µ Subsiuing equaion 6 equaion 7 ino equaions o 3 b eliminae he noaion he dimensionless momenum equaion can be wrien as c u = u α 3 u bu GrT cos φ GmC cos φ 8 T = T RT 9 C = C 0 Sc wih he dimensional iniial boundar condiions are u C T = 0 a 0 0 u = 0 C = a = 0 > 0 T = a = 0 0 < T = a = 0 > u C T 0 as > 0 Here u T C K M R Gr Gm α Sc are nondimensional fluid veloci emperaure concenraion permeabili of he porous medium magneic parameer known as Harmann number radiaion parameer hermal Grashof number he mass E-ISSN: 4-347X 77 Volume 06

3 Grashof number l number maerial module of second grade fluids Schmid number respecivel The consans appearing in he above equaions are defined as α = α L b = M K c = α ϕl K Sc = υ D = µc p k R = 6ασ T L k M = σb 0 L ρυ Gr = g/ 3β T Tw T L υ / 3 K = K L Gm = g/ 3β m Cw C L υ / 3 3 oblem Soluion The analical soluions for he ssem of equaions 8 0 wih he iniial boundar condiions will be deermined b he Laplace ransform mehod The following ransformed equaions is s-domain are obained b cs ū Gr T cos φ Gm C cos φ = αs ū 3 T s R T = 0 4 C Scs C = 0 5 The equaions 4 4 are uncoupled from equaion 3 heir soluions are e s T s = e Φs 6 s C s = s e Scs 7 Φ = R In order o ge T C b using he inverse Laplace ransform Sc C = erfc 8 T r = T T H 9 T = 4 4 Φ Φ e Φ erfc e Φ erfc Φ Φ In order o find he soluion of equaion 3 le ū s = ū s = ū ū 0 s Φαs bcs e s s e bcs αs e Φs ū s = s[scsαs bcs] e bcs αs e Scs To find he soluion for u s we solve ū ū separael Le ū = ū = s s α [sm m ] e bcs αs e Φs ū = αsc[sm 3 m 4] ū = s e bcs αs e Scs m = m = m 3 = Sc c m 4 = α Φ c α α Φ c 4α Φ b α αsc Sc c 4αbSc αsc 3 The inverse Laplace ransforms for ū ū are given b u = α [ e m m m m sinh m m m cosh m u = ] 4 αm 4 Sc e m 3 sinh m 4 5 In order o deermine he inverse Laplace ransforms of ū ū we consider he following funcion: F s = s e bcs αs 6 E-ISSN: 4-347X 78 Volume 06

4 The inverse Laplace ransform for he equaion 6 is given b f = b uc e α erfc u α 0 0 I dsdu 0 α us c αb uc c e α erfc u α 0 I du 0 α u c αb 7 Thus u = f e Φ erfc e Φ erfc u = f erfc Φ Φ 8 Sc 9 I 0 is modified Bessel funcion of he firs kind of order zero erfc is complimenar error funcion Consequenl he expression for veloci in he domain can be wrien in simple form as u = U H U H U H 30 Figure : comparison of veloci u in equaion 30 wih equaion from Samiulhaq e al [7] momenum soluion was reduced o published resul As expeced he resul are found idenical which show he validi of he obainable soluion From Figure b neglecing he effec of mass ransfer inclinaion angle he momenum equaion from 30 is idenical o he equaion b Samiulhaq e al [7] I is worh o menion ha equaions saisf all he imposed iniial boundar condiions Hence his also provides a useful mahemaical check o our calculi The obained soluions are also sudied numericall in order o deermine he effecs of several involved parameers such as second grade parameer α Harmann number M Figure 3 shows he effec of second grade parameers α on veloci profiles I is noiced ha he fluid veloci decreases hen increases on decreasing second grade parameers U = u u u = u s u s ds 0 U = u u u = u s u s ds 0 The smbol of u u u u denoes convoluion produc of wo funcions 4 Graphical Resuls Discussion Analical soluions of unsead MHD free convecion flow in a porous medium are invesigaed In his case we consider second grade fluids pas an infinie inclined plae wih ramped wall emperaure The problem soluions are solved b using Laplace ransform echnique Limiing case show ha he presen Figure 3: veloci profiles a various second grade parameers α The variaion of veloci for differen values of Harmann numbers M is ploed in Figure 4 We can see ha he applicaion of ransverse magneic field will resul a resisive pe force namel he Lorenz force This force ends o resis he fluid flow hus reducing he veloci E-ISSN: 4-347X 79 Volume 06

5 [7] Samiulhaq S Ahmad D Vieru I Khan S Shafie Unsead magneohdrodnamic Free Convecion Flow of a Second Grade Fluid in a Porous Medium wih Ramped Wall Temperaure Plos One 9 04 pp e88766 Figure 4: veloci profiles a various Harmann numbers M Acknowledgemens: The auhors graefull acknowledge financial suppor from Minisr of Educaion Malasia Universii Malasia Pahang hrough a gran RDU3405 References: [] M Narahari An Exac Soluion of Unsead MHD Free Convecion Flow of a Radiaing Gas pas an Infinie Inclined Isohermal Plae Appl Mech Maer pp 8 33 [] Z Ismail A Hussanan I Khan S Shafie MHD Radiaion Effecs on Naural Convecion Flow in a Porous Medium Pas an Infinie Inclined Plae Wih Ramped Wall Temperaure: An Exac Analsis In J Appl Mah Sa pp [3] Z Ismail I Khan A Imran A Hussanan S Shafie MHD Double Diffusion Flow b Free Convecion pas an Infinie Inclined Plae wih Ramped Wall Temperaure in a Porous Medium Malasian J Fundam Appl Sci 0 04 pp 37 4 [4] T Haa S Hina A A Hendi Influence of Wall operies on Perisalic ranspor of Second Grade Fluid wih Hea Mass Transfer Hea Transfer Asean Res 40 0 pp [5] T Haa M Nawaz M Sajid S Asghar The Effec of Thermal Radiaion on he Flow of a Second Grade Fluid Compu Mah wih Appl pp [6] Samiulhaq I Khan F Ali S Shafie Free Convecion Flow of a Second-Grade Fluid wih Ramped Wall Temperaure Hea Transfer Res pp E-ISSN: 4-347X 80 Volume 06