Jurnal Teknologi. g-jitter Induced Mixed Convection Flow of Heat and Mass Transfer Past an Inclined Stretching Sheet. Full paper

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1 Jurnal Teknologi Full paper g-jitter Induced Mied onvection Flo o Heat and Mass Transer Past an Inclined Stretching Sheet Noraihan Aiqah Rai a Abdul Rahman Mohd Kasim b Mukheta Isa a Sharidan Shaie a* a Department o Mathematical Sciences Faculty o Science Universiti Teknologi Malaysia UTM Johor Bahru Johor Malaysia b Mathematics Department Faculty o Industrial Science and Technology Universiti Malaysia Pahang Lebuhraya Tun Razak 6300 Kuantan Pahang Malaysia *orresponding author: sharidan@utm.my Article history Received : February 014 Received in revised orm : 3 August 014 Accepted :15 October 014 Graphical abstract Abstract This paper studies unsteady mied convection boundary layer lo o heat and mass transer past an inclined stretching sheet associated ith the eect o periodical gravity modulation or g-jitter. The temperature and concentration are assumed to vary linearly ith here is the distance along the plate. The governing partial dierential equations are transormed to a set o coupled ordinary dierential equations using non-similarity transormation and solved numerically by Keller-bo method. Numerical results or velocity temperature and concentration proiles as ell as skin riction Nusselt number and Sherood number are presented and analyzed or dierent values o inclination angle parameter. Keyords: g-jitter; mied convection; heat and mass transer; keller-bo; inclined stretching sheet Abstrak Kertas kerja ini mengkaji olakan campuran di dalam aliran lapisan sempadan bagi pemindahan haba dan jisim merentasi helaian regangan condong bersama kesan modulasi graviti berkala atau ketar-g. Suhu dan kepekatan permukaan diandaikan berubah secara linear terhadap dengan merupakan jarak di sepanjang permukaan. Persamaan perbezaan separa menakluk diubah ke sistem persamaan perbezaan biasa menggunakan persamaan tak serupa dan diselesaikan secara kaedah berangka menggunakan kaedah kotak-keller. Penyelesaian berangka untuk proil halaju suhu dan kepekatan termasuk geseran kulit nombor Nusselt dan nombor Sherood dipaparkan dan dianalisis untuk parameter sudut condongan yang berbeza. Kata kunci: Ketar-g; olakan campuran; pemindahan haba dan jisim; kotak-keller; helaian regangan condong 014 Penerbit UTM Press. All rights reserved. 1.0 INTRODUTION The production o sheeting material hich includes both metal and polymer sheets arises in a number o industrial manuacturing processes. The luid dynamics due to a stretching surace is important in many etrusion processes. For many practical applications the stretching suraces undergo cooling or heating that cause surace velocity and temperature variations. Problems involving the boundary layer lo due to a stretching surace in the vertical and inclined direction in a steady or unsteady viscous and incompressible luid hen the buoyancy orces are taken into account have been considered by many researchers such as Sharidan et al. 1 Ali et al. and Aurangzaib et al. 3. Meanhile various studies have been conducted involving g-jitter orces associated ith microgravity. g-jitter or periodical gravity modulation can be deined as the inertia eects due to quasisteady oscillatory or transient accelerations arising rom cre motions and machinery vibrations in parabolic aircrats space shuttles or other microgravity environment. The speciic amplitude and requency o the g-jitter accelerations depend on the dynamic behaviour o the spacecrat structure the location o the body and the type and location o the sources generating contributing orces 4. On the other hand Li 5 has ound that the requency and amplitude o the g-jitter play an important role in determining the convective lo behaviour o the system. Rees and Pop 6 have studied the eect o g-jitter on ree convection embedded in a porous medium near a stagnation point. A considerable amount o attention has been ocused in recent years by various researchers to study problems involving the eect o g-jitter on heat and mass transer These combined eects have many practical applications such as binary alloy solidiication systems since the quality o the inal products is strongly correlated to the concentration distribution in the melt during processing the migration o moisture through the air contained in ibrous insulations and grain storage insulations dispersion o chemical contaminants through ater-saturated soil 71:1 (014) 7 31.jurnalteknologi.utm.my eissn

2 8 Sharidan Shaie et al. / Jurnal Teknologi (Sciences & Engineering) 71:1 (014) 7 31 etc. Li and Shu 15 have carried out a numerical study on double diusive convection driven by g-jitter in a microgravity environment. They ound that an increase o g-jitter orce (amplitude) may cause the nonlinear convective eects become much more obvious hich is drastically change the concentration ields. Ater that Shu et al. 16 etended the previous problem by describing a numerical study o g-jitter driven double-diusive convective los hich includes the thermal and concentration distributions in binary alloy melt systems subject to an eternal magnetic ield. On the other hand Li et al. 17 have presented a inite element model or the g-jitter induced double-diusive convection and solidiication phenomena ith and ithout the presence o magnetic ields in a Sn-doped Bi crystal groth system planned or space eperiments. Then Sharidan et al. 7 presented an eact analytical solution or the problem o laminar combined heat and mass transer by mied convection o a ully developed lo driven by a combination o g-jitter. They ound that the temperature and concentration ields are primarily controlled by diusion and the ree convection eect may be neglected. In spite o all these studies e have ound that the eect o double diusion has not been considered by other researchers in the case o inclined stretching sheet ith the eect o g-jitter. The aim o this paper is to study the eect o g-jitter on mied convection lo and mass transer past an inclined stretching sheet. The governing boundary layer equations are transormed into the non-dimensional partial dierential equations hich are solved numerically using Keller-bo method. The numerical results o the eect o requency o the oscillation amplitude o modulation Prandtl number Pr Schmidt number Sc mied convection parameter and the angle o inclination parameter on the skin riction Nusselt and Sherood number ill be presented graphically and in tabular orm. Folloing Rees and Pop 6 and Sharidan et al. 1 e consider a simple model problem in hich gravitational ields takes the orm ( ) ( ) 1 cos * g t g t k g0 t k here g0 is the time-averaged value o the gravitational acceleration * g t acting along the direction on the unit vector k hich is oriented in the upard direction is a scaling parameter hich gives the magnitude o the gravity modulation relative to g 0 t is the time and is the requency o oscillation o the g-jitter driven lo. I 1 then the orcing may be seen as a perturbation o the mean gravity. Since the governing equations o this problem are non-linear this kind o orcing leads to the phenomenon o streaming here a timeperiodic orcing ith zero means produces a periodic response consisting o a steady-state solution ith a non-zero mean and time-dependent luctuations involving higher harmonics 1..0 BASI EQUATION onsider an unsteady viscous incompressible double-diusive mied convection boundary layer lo over an inclined stretching sheet ith the presence o g-jitter eect. The coordinate system is such that measures the distance along the plate and y measures the distance normally into the luid. It is assumed that the plate has a linear velocity u ( ) moves in -direction o the lo. It is urther assumed that the temperature and concentration o the (1) plate varies linearly ith the distance along the plate here T ( ) T and ( ) ith T ( ) and T being the temperature o the plate and uniorm temperature o ambient luid hile ( ) and being the concentration o plate and uniorm concentration o the ambient luid. Meanhile the velocity temperature and concentration o the continuous stretching surace are assumed to be in the orm o u ( ) c T ( ) T a and () b here a b and c are constants and c 0. The lo conigurations and coordinate system are shon in Figure 1. Figure 1 Physical model and coordinate system Under the usual boundary layer approimation along ith Boussinesq approimations the governing equations are given by: u v 0 y u u u u u v t y y T u T v T T t y y g t T T T c cos u v D t y y subject to the olloing initial and boundary conditions t 0 : u v 0 T T or any y t 0 : u( ) c v 0 T( )=T + a ( )= + b on y 0 u 0 T T as y here u and v are the velocity components along and y aes a and c are constants T is the luid temperature is the concentration o the luid is the coeicient o thermal diusivity and is the thermal and concentration T epansion coeicient respectively D is mass diusivity is the angle o inclination and is kinematic viscosity respectively. Folloing Sharidan et al. 1 the compleity o the problem is reduced by introducing the olloing non-dimensional variables () (3) (4) (5) (6)

3 9 Sharidan Shaie et al. / Jurnal Teknologi (Sciences & Engineering) 71:1 (014) 7 31 t c y c 1 1 ( ) ( ) ( ) ( T T) ( ) ( ) ( ) ( T T ) ( ) gt () g( ) g here is the stream unction hich is deined as u and y v. It is noted that equation () is satisied ith the nondimensional variables introduced. By using Equation (7) Equations (3) (4) and (5) become cos( ) ( N)cos 1 Pr 1 Sc. The boundary conditions (6) become on 0 (11) as here Pr is the Prandtl number is the non-dimensional requency N is the buoyancy ratio and is the mied convection parameter hich are deined as ( ) Gr c Pr N c T( T T ) Re 3 here ( ) Gr g0t T T being the local Grasho number and Re u ( ) is the local Reynolds number respectively. We notice that 0 corresponds to aiding lo and 0 to opposing lo respectively. The physical quantities o interest include the skin riction coeicient the local Nusselt number Nu and the Sherood number Shr hich are deined as ( ) q( ) Nu u k( T T ) m ( ) Shr D( ) here the ( ) q ( ) and m ( ) are given by 0 (7) (8) (9) (10) (1) (13) u T q k y y y0 y0 m D y y0 Here k is the thermal conductibility is the dynamic viscosity D is the mass diusivity is the shear stress at the all q is the average convective heat transer coeicient and m is the average mass transer coeicient. Using variables (6) e obtain 1 Re ( 0) ( 0) 1 Re Sh Re r 1 Nu ( 0). 3.0 RESULTS AND DISUSSION. The system o the unsteady governing equations (8) (9) and (10) together ith the boundary conditions (11) is nonlinear dierential equations depending on the various values o the parameters such as requency o the oscillation amplitude o modulation Prandtl number Pr Schmidt number Sc mied convection parameter and the angle o inclination parameter. These equations are solved numerically by using inite dierence method hich is knon as Keller-Bo method. This method has been ound to be very suitable in dealing ith nonlinear parabolic problems. Table 1 omparison o the mean skin riction transer rate 0 or and Pr=0.7 c F 0 F 0 and mean heat Present Sharidan et al Present Sharidan et al (14) (15) Table 1 represents the comparison o the result beteen Sharidan et al. 1 or vertical stretching sheet ith the present result by neglecting the buoyancy ratio (N=0) inclination angle ( 0) and Schmidt number (Sc=0). The mean skin riction and heat transer rate are the averages o the set o values o 0 and 0 respectively in the range o 0 and obtained by using trapezoidal rule. It can be seen rom the Table 1 that very good agreement beteen the results eists. This agreeable comparison lends conidence in the numerical results obtained in this paper. Folloing Sharidan et al. 1 in all result e vary rom 0 to 1 since values o above 1 is equivalent to having the perceived gravity reverse its direction over part o the g-jitter cycle. The computation or the appropriate steady solution ere alays started ith 0 and convergence to a steady periodic state as demanded to have taken place hen

4 30 Sharidan Shaie et al. / Jurnal Teknologi (Sciences & Engineering) 71:1 (014) 7 31 ma ( 0) ( 0) 10 over the hole period. Figure to 5 illustrate the eect o amplitude o modulation requency o oscillation and inclination angle parameter or ied values c Pr=0.7 Sc=1.5 N=1 0. and 5. The results obtained sho the variation o skin riction ( 0) rate o heat and mass transer ( 0) and ( 0). From these igures e can see that the eect o increasing give an almost proportional increase or decrease in the skin riction and rate o heat and mass transer. We can also observe that skin riction heat and mass transer coeicients increase as the inclination angle parameter increase. This is toards the act that as the plate is inclined rom the vertical the buoyancy orce eect due to the thermal and mass diusions decrease as cos decreases. 6 (16) Hoever hen the values increase the corresponding curves sho the dierent trends as can be seen rom Figure to 5 particularly or the variation o rate o heat and mass transer. Also it is observed rom the Figure 3 and 5 as the values o requency o oscillation is larger the rate o heat and mass transer are changed airly small. Figure 6 depicts the eect o amplitude o modulation on the velocity temperature and concentration proile. Three dierent values o amplitude ( 00.51) are chosen. It is observed that the temperature and concentration proile are decrease ith the increase o amplitude o modulation hereas the velocity proile increases. These graphical behaviour o proiles can be veriied rom the boundary conditions shon in Equation (11). 0 Figure Variation o skin riction and rate o heat and mass transer 0 and 0 c Pr 0.7 Sc and dierent values o or N=1 ith or 0 Figure 3 Variation o skin riction and rate o heat and mass transer 0 and 0 c Pr 0.7 Sc and dierent values o or N=1 ith or 0 Figure 4 Variation o skin riction and rate o heat and mass transer 0 and 0 c Pr 0.7 Sc and dierent values o or N=1 ith or

5 31 Sharidan Shaie et al. / Jurnal Teknologi (Sciences & Engineering) 71:1 (014) Figure 5 Variation o skin riction and rate o heat and mass transer 0 and 0 c Pr 0.7 Sc and dierent values o or N=1 ith or Figure 6 Velocity (a) temperature (b) and concentration (c) proiles or dierent values o hen Pr 0.7 Sc and ONLUSION In this paper the eect o g-jitter induced mied convection lo past an inclined stretching sheet has been numerically investigated. The governing boundary layer equations are transormed into a non-dimensional orm and the resulting nonlinear system o partial dierential equations is solved numerically using the Keller-bo method. The eects o the requency o the oscillation amplitude o modulation mied convection parameter and the angle o inclination parameter or Prandtl number Pr = 0.7 and Schmidt number Sc = 1.5 on the skin riction coeicient and rate o heat and mass transer have been eamined in detail. Furthermore the numerical results or velocity temperature and concentration ields also graphically displayed. omparison o the results shos that the obtained results agree very ell ith the previous results or vertical stretching plate ( 0) reported by Sharidan et al. 1 Acknoledgement The authors ould like to acknoledge MOE and Research Management entre UTM or the inancial support through vote numbers 09j71 and 06h67 or this research. The irst author also ould like to epress many thanks to UTM or Zamalah scholarship. Reerences [1] S. Shaie N. Amin I. Pop Microgravity Sci. Technol.:XVIII [] F. M. Ali R. Nazar N. M. Ariin I. Pop Meccanica. 46: [3] Aurangzaib A. R. M. Kasim N. F. Mohammad S. Shaie. Heat Trans. Asian Research. 4: 89. [4] B. N. Antar V. S. Nuotio-Antar Fundamentals o Lo Gravity Fluid Dynamics and Heat Transer. Boca Raton: R Press. [5] B. Q. Li Int. J. Heat Mass Trans. 39: 853. [6] D. A. S. Rees I. Pop Heat Mass Trans. 37: 403. [7] S. Shaie N. Amin I. Pop Int. ommun. Heat Mass Trans. 3: 657. [8] S. Shaie N. Amin I. Pop Int. J. Appl. Mech. Eng. 10: 311. [9] S. Shaie N. Amin I. Pop Int. J. Heat Mass Trans. 48: 456. [10] S. Shaie N. Amin I. Pop Int. J. Fluid Mech. 3: 91. [11] S. Shaie N. Amin I. Pop Int. J. Energ. Heat Mass Trans. 8: 1. [1] S. Shaie N. Amin I. Pop Mech. Res. ommun. 34: 115. [13] D. A. S. Rees I. Pop Int. ommun. Heat Mass Trans. 7: 415. [14] N. Naser A. R. M. Kasim Aurangzaib S. Shaie Recent Advances in Mathematics [15] Y. Shu B. Q. Li An Int. J. omp. Metho : 345. [16] Y. Shu B. Q. Li H.. De Groh III Numer. Heat Trans. A- Appl : 345. [17] K. Li B. Q. Li H.. De Groh J. Thermophys. Heat Trans. 17: 199.