Investigation of two phase unsteady nanofluid flow and heat transfer between moving parallel plates in the presence of the magnetic field using GM

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1 rans. Phenom. Nano Micro Scales, 4(): -8, Winter - Spring 06 DOI: 0.708/tpnms ORIGINAL RESEARCH PAPER Investigation o two phase unsteady nanoluid low and heat transer between moving parallel plates in the presence o the magnetic ield using GM N. Hedayati,*, A.Ramiar Babol University o echnology, Department o Mechanical Engineering, Babol, I. R.Iran Received 7 May 0; revised August 0; accepted 4 September 0; available online January 06 ABSRAC: In this paper, unsteady two phase simulation o nanoluid low and heat transer between moving parallel plates, in presence o the magnetic ield is studied. he signiicant eects o thermophoresis and Brownian motion have been contained in the model o nanoluid low. he three governing equations are solved simultaneously via Galerkin method. Comparison with other works indicates that this method is very applicable to solve these problems. he semi analytical analysis is accomplished or dierent governing parameters in the equations e.g. the squeeze number, Eckert number and Hartmann number. he results showed that skin riction coeicient value increases with increasing Hartmann number and squeeze number in a constant Reynolds number. Also, it is shown that the Nusselt number is an incrementing unction o Hartmann number while Eckert number is a reducing unction o squeeze number.his type o results can help the engineers to make better and researchers to investigate aster and easier. KEYWORDS: Brownian; Eckert number; Galerkin method (GM); Hartmann number; Nanoluid, hermophoresis INRODUCION One o most important engineering problems, peculiarly heat transer equations are nonlinear problems, so some them are solved using numerical methods and some are solved using dierent analytical methods. One o the known analytical simulation methods is the galerkin method, which is based on guessing the base unctions with polynomial equations. Sheikholeslami et al.[] Investigated rotating magnetic - hydrodynamic (MHD) viscous low and heat transer between stretching and porous suraces. he three dimensional analysis o steady deposition o luid on an inclined rotating disk is also investigated by Sheikholeslami et al. []. hey showed that by increasing normalized thickness, Nusselt number increase however this trend is more signiicant in bigger Prandtl numbers. In new years several researchers used dierent methods to solve these sort o problems [3 ]. Studying heat transer enhancement in various energy systems is essential due to the increment in energy prices. In the last decade, nanoluids technology was propounded and examined by various researchers numerically or experimentally to control heat transer in dierent applications. he nanoluid can be employed in dierent engineering applications, such as chemical processes, cooling o electronic equipment and heat exchangers. Most o the researchers supposed that nanoluids treat as standard pure luids. Khanaer et al. [6] *Corresponding Author nima.hedayati883@gmail.com el.: numerically investigated the heat transer increment due to adding nano-particles in a variously heated enclosure. Abu- Nada et al. [7] studied ree convection heat transer increment in a horizontal concentric annuli. hey showed that or some Rayleigh numbers, nanoparticles with higher thermal conductivities lead more increment in heat transer. Rashidi et al. [8] perormed the second law o thermodynamics analysis on an electrically conducting incompressible nanoluid lowing over a porous rotating disk. Heat transer and nanoluid low speciications between two parallel plates in a rotating system were studied by Sheikholeslami et al. [9]. hey demonstrated that Nusselt number increases with increment o nanoparticle Reynolds number and volume raction and decreases with increasing magnetic, rotation parameters and Eckert number. Sheikholeslami et al. [0] investigated the MHD ree convection in an eccentric semi annulus illed with nanoluid. hey ounded that the Nusselt number reduces with an increment in the position o internal cylinder at higher Rayleigh numbers. Recently, several researchers studied about heat transer and nanoluid low [ 4]. Whole o the above investigations supposed that there are not whatsoever slip velocities between nanoparticles and luid molecules and supposed that the concentration o nanoparticles is uniorm. It is believed that in natural convection o nanoluids, the nanoparticles could not attend luid molecules due to some slip mechanisms such as thermophoresis and Brownian motion, so the volume raction o nanoluids may not be uniorm and dierent concentration o nanoparticles will exist in a mixt-

2 A.Ramiar & N. Hedayati Nomenclature B magnetic ield C p speciic heat at constant pressure D thermophoretic diusion coeicient Ha Hartmann number l distance o plate Nt thermophoretic parameter P Pressure S squeeze number luid temperature C nanoluid concentration D B Browniandiusion coeicient E C Eckert number k thermal conductivity Nb Brownian motion parameter Nu Nusselt numberr Pr Prandtl numberr Sc Lewis number Greek Symbols α thermal diusivity θ dimensionless temperature ρ density Φ Dimensionless concentration γ rate o squeezing μ dynamic viscosity o nanoluidd σ electrical conductivity o nanoluid -ure Nield and Kuznetsov [] investigated the natural convection in a horizontal layer o a porous medium. Khan and Pop [6] studied on boundary-layer low o a nanoluid past a stretching sheet. Free convection heat transer in an enclosure illed with nanoluid was studied by Sheikholeslami et al. [7]. hey studied two phase modeling o nanoluid in a rotating system with permeable sheet. hey showed that Nusselt number is a direct unction o injection parameter and Reynolds number but it is a reverse unction o Schmidt number, rotation parameter viscous, Brownian parameterr and thermophoretic parameter.he investigate o unsteady squeezing low between two parallel plates has vast spectrum o scientiic and engineering applications such as polymer processing,hydrodynamic machines, chemical processing equipment, lubrication system, ood processing and cooling towers ormation and dispersion o og. Mahmood et al. [8] studied the heat transer characteristics in the squeezed low over a porous surace. MHD squeezing low o a viscous luid between parallel disks was studied by Domairry and Aziz [9]. GOVERNING EQUAIONS Heat and mass transer analysis in the unsteady two- dimensional squeezing low o nanoluid between the ininite parallel plates is carried out (igure ). he two plates are placed at L(-γt) / =h(t). When γ> >0 the two plates are squeezed until they touch t=/γ and or γ<0 the two plates are separated. he viscouss dissipation eect, the generation o heat due to riction caused by shear in the low, is retained. Also, it is also assumed that the uniorm magnetic ield is applied, where is a unit vector in the Cartesian coordinate system. he electric current J and the electromagnetic orce F are deined by J=( ) and F= ( ), respectively. he governing equations or mass, momentum, energy and species transer in unsteady two dimensional low o nanoluid are [0]: u v 0 x y u u u p u u v ( t v y x x u y ) Fig.. Schematic geometry o problem Bu v v v p v u v ( t x y y x v ) y u v ( ) t x y x y u c P p (4( )) x cp C C DB.. x x y y ( D / c ) x y C C C u v t x y ( C C DB ) x y D c x y c p () () (3) (4) () 3

3 rans. Phenom. Nano Micro Scales, 4() -8, Winter - Spring 06 Here u and ν are the velocities in the x and y directions respectively, is the temperature, C is the Concentration, Pis the pressure, ρ is the base luid s density, μ is the dynamic viscosity, k is the thermal conductivity, C p is the speciic heat o nanoluid, D B is the diusion coeicient o the diusing species. he relevant boundary conditions are: C 0, v v dh/ dt,, C C at y h(), t v u / y / y C / y 0 at y 0. w H H he reduced governing equations will be obtained by introducing new dimensionless parameters: y t v t C. C / [ ( ) ] [( )] / [( ) ] h x, u ( ), t ( ),, Substituting the above parameters into equations and 3 and then eliminating the pressure gradient rom the resulting equations will give: iv S 3 Ha 0 Also substituting equation 7 into equations 4 and will lead to the ollowing dierential equations: Pr Pr S Ec Nb Nt 0 Nt SSc. 0 Nb With these boundary conditions:, 0,, 0 0, 0 0, 0 0, 0 0, H (6) (7) (8) (9) (0) () Where S is the squeeze number, Pr is the Prandtl number, Ec is the Eckert number, Sc is the Schmidt number, Ha is Hartman number o nanoluid, Nb is the Brownian motion parameter and Nt is the thermophoretic parameter, deined as: Sc, Ha B t, D Nb ( c) pdb( Ch)/ ( c), Nt ( c) D ( )/[( c) ]. p H c Nusselt number is deined as: k y Nu H yh t (3) In terms o dimensionless parameters o equation 7, it will change to: * Nu t Nu C l x t C '' * / ( )Re x () (4) he Galerkin method he method is used as a weighted residual method. It is an approximation technique or solving dierential equations.he method is perectly universal; it can be applied to elliptic, hyperbolic and parabolic equations, nonlinear problems, as well as complicated boundary conditions. Principles o the method In this method an approximate trial solution () containing a inite number o undetermined coeicients C,C,,C n can be constructed by the superposition o some basic unctions such as polynomial unctions written bellow t () ct ct +... () he method requires that the weighted averages o the residual Re (C,C,,C n,t) should vanish over the interval considered. he residual vanishes only with the exact solution or the problem. When (C,C,,C n ) are known, the trial solution () given by equation becomes a approximate solution or the problem.he weighting unctions w i (t) are taken as the same basis unctions used to construct the trial solution (). In this method weight unctions can be interpreted as: w i i,,..., n c i (6) l x S, Pr, Ec c p t () he weight unctions are used or the integration o the residual (Re) over the interval 0<t<. hus, the Galerkin method becomes 4

4 A.Ramiar & N. Hedayati t 0 w i Re( c, c,..., c, t ) dt 0 Equation 7 provide several algebraic equations or the determination o the unknown coeicients (C,C,,C n ). Applying the GM to the problem For this problem, we wish to obtain an approximate solution in the speciic interval 0<η<. o construct a trial solution (η),θ(η),φ(η), the basis unctions are chosen to be polynomials in η and create a trial solution o the orm: ( ) c0c c + c c + c c ( ) c c c + c ( ) c c c + c c c c c c c c c c c n c + c c + c ( 7) ( 8) For the speciic problem considered here, η,η,η 3,,η 9,ηη 0 are the weight unctions to be used or the integration o the residuals Re( C,C,,C 0,η) over the interval 0<η<. So the Galerkin method or irst w yields: RESULS AND DISCUSSION In this paper, nanoluid low and heat transer among movingg parallel plates is studied. he eects o important parameters on heat and mass speciications are investigated. he present GM code is validated by comparing the obtained results with Mustaa et al. [0]. As it is shown in able, the results are in a very good agreement. Eect o the Hartmann number and squeeze number on the velocity proiles and skin riction coeicient is shown in Figure. R0 Re( c, c, c, c, c, c, c, dr ) (9a) R0 c, c, c, c 4 6 Re( c7, c8, c9, c0, c, c, c3, R0 Re( c c, c, c, c 4 6 7, ) dr 0, c, c, c , ) dr 0, c, c, (9b) (9c) Including another equation rom boundary condition equation we have thirty algebraic equations that provide the unknown coeicients C 0,C, C 9. By introducing these coeicients into equation (8,9,0), temperature,concentration and base luid distribution equations will be determined. For (Ha=, Sc=0., Ec=0., S=0., Pr= 0, Nt=0., Nb= =0.) he base luid, temperature and concentration unctions are respectively: 3 4 ( ) ( ) ( ) (0a) (0b) 0c)) (c) Fig.. Eect o the squeeze number and Hartmann number on the velocity proiles and skin riction coeicient in a constant Reynolds Numberand Ha =, S=0. It is important to mention that the squeeze number S characterizes the motion o the plates. S > 0 is related to the plates moving separate, and S < 0 is related to the plates movingg together (thatt called squeezing low). In this paper positivee values o squeeze number are attended.

5 rans. Phenom. Nano Micro Scales, 4() -8, Winter - Spring 06 able Comparison between the present work and Mustaa et al [0]. Mustaa et al. [0] Present Mustaa Present Work et al. [0] Work S As squeeze slightly. () () () () number increases, horizontal velocity reduces It is valuable noting that the eect o magnetic ield is reducing the value o the velocity all over the enclosure becausee the presence o magnetic ield deines a orce that is called the Lorentz orce and acts against the low i the magnetic ield is applied in the normal direction. his is a resisting orce that slows down the luid velocity and consequently reducess the skin riction coeicient. At a constant Reynolds number, by ncreasing the Hartmann numberr, the luid momentum orce is collated with a resisting Force ineective on velocity value so skin riction coeicient value increases. Figure 3 shows the eect o the squeeze number and Hartmann number on the temperature proile and Nusselt number. An increment in the squeeze number ψ can be relevant to the reduction in the kinematic viscosity, an increment in the distance o plates or an increment in the speed at which the plates move.emperature grows as the squeezee number and Hartmann number increase. hus Nusseltt number increases with an increment in Hartmann numberr and squeeze number. Eect o the squeeze number and Hartmann number on the concentration proile is shown in Figure 4. Fig. 4. Eect o the squeeze number and Hartmann number on the concentration proile when Nt = Nb = 0., Ha =, S = 0., Sc = 0. and Pr = 0 (c) Fig. 3. Eect o the squeeze number and Hartmann number on the temperature proile and Nusselt number or Ec = 0., Nt = 0., Nb = 0., Sc = 0. and Pr = 0 he concentration proile increases with an increment in squeezee number and Hartmann number when η<0. while it conversely decreasess with increasing o these parameters or η>0.. Eect o Eckert number on the Nusselt number 6

6 A.Ramiar & N. Hedayati and temperature proile is depicted in Figure. considering viscous dissipation eects signiicantly increases the temperature. Nusselt number increases with an increment in the Eckert number becausee o the reduction in thermal boundary layer thickness near the upper plate. Fig.. Eect o Eckert number on the temperaturee proile and Nusselt number when Nt = Nb = 0., Ha =, S = 0., Sc = 0. and Pr = 0. CONCLUSION Unsteady two phase heat transer and low o nanoluid in the presence o MHD is analytically investigated in this paper.he signiicant eect o thermophoresis and Brownian motion have been contained in the model. GM is used to solve the governingg equations. he eects o the Hartmann number, squeezee number, Brownian motion, Schmidt number, thermophoretic parameter and Eckert number on the temperature and concentration proiles are studied. he results indicated that skin riction coeicient reduces with increasing squeeze number and Hartmann number. Also it can be ound that the Nusselt number enhances with an increment in the Schmidt number Eckert number and Hartmann number, but it reduces with increasing squeeze number. REFERENCES [] M.Sheikholeslami, H..R.Ashorynejad, D.D. Ganji, A. Kolahdooz: Investigation o Rotating MHD Viscous Flow and Heat ranser between Stretching and Porous Suraces Using Analytical Method. Hindawi Publishing Corporation Mathematical Problems in Engineering (0), (Article ID 8734, 7 pages). [] M. Sheikholeslami, H.R. Ashorynejad, D.D. Ganji, A. Yıldırım: Homotopy perturbation method or three-dimensional problem o condensation ilm on inclined rotating disk, Scientia Iranica B 9 (0) [3] D.D.Ganji, H.B.Rokni, M.G.Sahani, S.S.Ganji. Approximate traveling wave solutions or coupled shallow water, Advances in Engineeringg Sotware 4 (00) [4] M. Keimanesh, M.M.Rashidi, A.J. Chamkha, R.Jaari: Study o a third grade non- Newtonian luid low between two parallel plates using the multi-step dier-ential transorm method, Computers and Mathematics with Applications 6 (0) [] M.Hatami, Kh.Hosseinzadeh, G. Domairry, M.. Behnamar: Numerical study o MHD two-phase Couette low analysis or luid-particle suspension between moving parallel plates, Journal o the aiwan Institute o Chemical Engineers 4 (04) 38-4 [6] K.Khanaer, K.Vaai, M.Lightstone: Buoyancy- driven heat transer enhance-ment in a twonanoluids, dimensional enclosure utilizing International Journal o Heat and Mass ranser 46 (003) [7] E.Abu-Nada,Z. Masoud, A.Hijazi: Natural convection heat transer enhance-ment in horizontal concentric annuli using nanoluids, International Communications in Heat and Mass ranser 3 (008) [8] M.M. Rashidi, S.Abelman, N. Freidooni Mehr: Entropy generation in steady MHD low due to a rotating porous disk in a nanoluid, International Journal o Heat and Mass ranser 6 (03). [9] M.Sheikholeslami,Sh. Abelman, D.D.Ganji: Numerical simulation o MHD nanoluid low and heat transerr considering viscous dissipation, International Journal o Heat and Mass ranser 79 (04). [0] M.Sheikholeslami, M.Gorji-Bandpy,D.DD Ganji: MHD ree convection in an eccentric semi-annulus illed with nanoluid, Journal o the aiwan Institute o Chemical Engineers 4(04)04 6. [] M.Sheikholeslami, M.Gorji-Bandpy, D.D.Ganji, S.Soleimani: MHD natural convection in a nanoluid illed inclined enclosure with sinusoidal wall using CVFEM, Neural Comput & Applic 4 (04) [] A.Malvandi,D.D. Ganji: Brownian motion and thermophoresis eects on slip low o alumina/water nanoluid inside a circular microchannel in the 7

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