Analytical investigation of unsteady CuO nanofluid flow, heat and mass transfer between two parallel disks

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1 Indian Journal o Cheical Technology Vol. 5, May 8, pp Analytical investigation o unsteady CuO nanoluid low, heat and ass transer between two parallel disks Azii M, Ganji DD, Azii A*,3 & Riazi R 4 Faculty o New Sciences and Technologies, University o Tehran, Tehran, Iran. Departent o Mechanical Engineering, Babol University o Technology, Babol University o Technology, Babol, Iran. 3 Departent o Cheical Engineering, College o Cheical Engineering, Mahshahr Branch, Islaic Azad University, Mahshahr, Iran. 4 Faculty o New Sciences and Technologies, University o Tehran, Tehran, Iran. E-ail:eysa.azii@gail.co Received 9 April 6; accepted 4 June 7 The heat transer in the unsteady CuO nanoluid low between two oving parallel disks has been investigated using analytical ethod called Galerkin Optial Hootopy Asyptotic Method (GOHAM). The eect o Brownian otion on heat transer enhanceent has been shown. The analytical investigation is carried out or various governing paraeters such as the squeeze paraeter, Hartan nuber, Brownian otion and therophoretic paraeters. The results show that concentration is an increasing unction o Brownian otion paraeter while it is a decreasing unction o the therophoretic paraeter.the coparison o obtained results with nuerical solutions assures us about the validity and accuracy o the current study. Keywords:Squeezing Flow, Nanoluid Flow, Heat Transer Enhanceent, GOHAM, CuO Nanoluid, a nae conceived by Choi, in Argonne National Laboratory to describe a luid in which nanoeter-sized particles are suspended. Nanoparticles have unique properties, such as large surace area to volue ratio, and lower kineatic energy which can be exploited in various applications. Nanoparticles are better stable when dispersed in base luids, due to their large surace area and they are ore stable when copared to icro luids which lead to any practical probles. In recent years, nanoluids have attracted ore and ore attention,. In paper, by Azii and Azii 3, DTM have successully applied to a non-linear MHD Jeery Hael proble with Graphene Oxide (GO) nanoparticle. The eects o graphene oxide solid volue raction, Reynolds nuber, Hartan nuber and the angle between parallel plates on velocity coponents were investigated. Their results showed that the velocity proile is strongly iluenced by solid volue raction o GO nanoparticles. The unsteady ixed convection squeezing low o an incopressible graphene oxide water nanoluid between two vertical parallel planes is discussed in paper by Azii and Riazi 4. The buoyancy orce due to theral and olecular diusion is taken as the source o the convective low. They concluded that when Graphene oxide solid volue raction increases, the rate o heat transer increases. Eckert nuber has signiicant eect on teperature proile and it can increase the rate o heat transer by increasing and their results showed that the teperature ield T decreases by increasing the ixed convection paraeter. The squeezing low between two parallel boundaries is an interesting topic o research due to its abundant applications. Exaples o such lows are quite prevalent in polyer processing, copression and injection odeling. The lubrication syste can be discussed through the squeezing low. The initial work on the squeezing low was investigated by Stean 5. Azii and Riazi 6 used analytical ethod called Reconstruction o Variational Iteration Method (RVIM) in order to ine approxiate solution or nanoluid squeezing low and heat transer between two oving parallel plates. They concluded that the

2 8 INDIAN J. CHEM. TECHNOL., MAY 8 Nusselt nuber increases with increase o Eckert nuber and solid volue raction o graphene oxide nanoparticles in water. The eect o dierent types o nanoparticles (graphene oxide, aliiniu oxide, titaniu oxide, silver) on the Nusselt nuberin unsteady squeezing low between two oving parallel plates (which is illed with nanoluid) proble was investigated by Azii and Mirzaei 7. The results showed the nanoparticle type is an iportant actor in the cooling and heating processes and silver can cause ost heat transer enhanceent rate. Velocity proiles or various oving nuber have been also obtained in their study. In the heart o all the dierent engineering sciences, everything showed itsel in the atheatical relation that ost o these probles and phenoena are odeled by ordinary or partial dierential equations. In ost cases, scientiic probles are inherently o nonlinearity that does not adit exact solution, so these equations should be solved using special techniques. Soe o these ethods are Hootopy Perturbation Method (HPM) 8, Reconstruction o Variational Iteration Method (RVIM) 9, Glerkin Optial Hootopy Asyptotic Method (GOHAM) and others,. In this study, the Galerkin Optial Hootopy Asyptotic Method (GOHAM), is applied to ind the sei-analytical solutions o nonlinear dierential equations governing the proble o unsteady CuO nanoluid low, heat and ass transer between two oving disks. The eect o Brownian otion on nanoparticle concentration was also studied. Matheatical Forulation Figure shows the geoetry o the squeezing low o an incopressible viscous MHD nanoluid between two circular plates separated by a distance z l( - at) h( t). A unior agnetic ield o strength Bt ( ) B ( at ) is applied perpendicular to the disks. The upper disk at z h() t approaching the stationary lower disk with the velocity dh / dt. The low is axisyetric about r. The velocity coponents along the radial and axial directions are u( r, z, t ), wrzt, (,, ) respectively. Now speciy the basic equations or an unsteady axisyetric low and assue v é( u( r, z, t),, w( r, z, t) ) ù ê ë ú thus, the û unsteady ass and conservation Equations becoe: u u w + + r r z... () æ u u u ö r p u u u u u w æ ö ç è t r z ø r ç r z r r r è ø...() æ w w w ö r p w w w u w æ ö + + ç è t r zø z ç r z r r è ø...(3) T T T k æ T T Tö + u + w + + t r z ç r z r z è ø ( rc p ) é æ ù C T C Tö D éæ T uö æ uö ù + t D B ç è r r z z ø T çè x ø ç y ê ê è ø ë úû ú ë û C C C æ C C Cö + u + w D B + + t r z ç r r r z è ø æ ö ç ø DB T T T T ç è r r r z...(4)...(5) where u and w are the velocities in the r and z directions, respectively, pis pressure, Tis teperature, C is the nanoparticle concentration, D is the Brownian B otion coeicient, D T is the terophoretic diusion coeicient, T is the ean luid teperature and k is the theral conductivity. The last ter in the energy equation is the total diusion ass lux or nanoparticles, given as su o two diusion ters. is the diensionless paraeter that gives the ratio o Fig. Physical odel.

3 AZIMI et al.: UNSTEADY CuO NANOFLUID FLOW, HEAT AND MASS TRANSFER BETWEEN TWO PARALLEL DISKS 83 eective heat capacity o the nanoparticle aterial to heat capacity o the luid. Eective density ( ), the eective dynaic viscosity ( ), eective heat capacity ( C ) and the eective theral conductivity k o the nanoluid are deined as 8 : r r ( - j) + rsjs rc rc - j + rc ( p) ( p) ( ) ( p) s.5 ( - j) k ks + k -( k -ks) ns k ks + k + ( k - ks) v r...(6) The relevant boundary conditions or the proble are: dh z ht () u, w ww, T TH, C Ch dt w z u, w-, T Tw,, C Cw -at...(7) By introducing ollowing paraeters, the above Equation can be easily sipliied: ar ah z u ( h), w- ( h), h é ( at) ù é( at) ù H é( at) ù ê ë úû êë úû êë úû T-TH B C-Ch q, B, j TH -T é H ( -at) ù Cw-Ch êë úû...(8) The above paraeters are substituted into Equations. () and (3). Then the pressure gradient is eliinated ro the resulting Equations. We inally yield: ( IV ) ( ) - S h M 3... (9) Using equation (8), Equations (3) and (4) sipliy to ollowing equations: ( ) q + q - hq + q j + q Pr S Pr Nb Pr Nt Nt j + LeS ( j - hj ) + q Nb With the ollowing boundary conditions: ( ), ( ), ( ), ( ), q( ) j( ), q( ) j( )... ()... ()...() where S is squeeze paraeter, Pr is the Prandtl nuber, M is Hartan nuber, Nb is the Brownian otion paraeter Nt is therophoretic paraeter and Le is the Lewis nuber which are deined as: ah v sbh v S, Pr, M, Le v a v De ( rc) DB ( Cw-Ch) ( rc) DT( Tw-Th) s s Nb, Nt ( rc) v ( rc) T v...(3) It is iportant to note that A indicates the suction o luid ro the lower disk while A represents injection low. Solution Procedure Following dierential Equation is considered: Lut ( ()) + Nut ( ()) + gt (), Bu ( )...(4) where L is a linear operator, is an independent variable, ut () is an unknown unction, g t is a known unction, N ( u( t)) is a nonlinear operator and B is a boundary operator. By eans o OHAM, one irst constructs a set o Equations: ( - p) él( vt (, p) g( t) ) ù ê + -H( p ë úû ) él( v( t, p) ) g( t) N( v( t, p) ) ù ê + + ú B( v( t, p) ) ë û (5) where p [,] is an ebedding paraeter, H ( p ) denotes a nonzero auxiliary unction or p and H (), is an unknown unction. Obviously, when p and p, it holds that: (,) u ( ), (,) u( ) vt t vt t...(6) Thus, as p increases ro to, the solution vt, p varies ro u () to the solution u ( ), where ( ) u () is obtained ro Eq. (6) or p : ( ( t) ) ( t) ( ) Lu + g, Bu... (7) We choose the auxiliary unction H ( p ) in the or: H( p) pc + pc+... (8)

4 84 INDIAN J. CHEM. TECHNOL., MAY 8 where C, C, are constants which can be deterined later. Expanding (, p ) in a series with respect to p, one has: å ( ) ( ) ( ) vt, pc, u t+ u t, C p, i,, i k i k k>...(9) Substituting Equation. into Equation.6, collecting the sae powers o p, and equating each coeicient o p to zero, we obtain set o dierential equation with boundary conditions. Solving dierential Equations by boundary conditions u( ), u(, C), u(, C),... are obtained. Generally speaking, the solution o Equation.5 can be deterined approxiately in the or: å vt (, pc, ) u( t) + u ( t, C) p, i,, i k i k k> +å k ( ) ( t) ( t, ) k i... () u u u C...() Note, that the last coeicient C can be unction o. Substituting Equation. into Equation.4, there results the ollowing residual: ( ( ) ( )) ( ) ( ( ) ( )) Copper Oxide R( t, Ci) L u t, Ci + g t + N u t, C... () i () ( I R(, C i ) then u ) (, C i ) happens to be the exact solution. Generally, such a case will not arise or nonlinear probles, but we can iniize the unctional by Galerkin ethod: ( t,,,, ) R C C C wi, i,,..., Ci... (3) The unknown constants Ci ( i,,..., ) can be identiied ro the conditions: b ò ( ) ( t ) J C, C wr, C, C,..., C dt i a... (4) where a and b are two values, depending on the given proble. With these constants, the approxiate solution (o order ) (Eq. (4)) is well deterined. It can be observed that the ethod proposed in this work generalizes these two ethods using the special (ore general) auxiliary unction H ( p ). Results and Discussion In this section, we will discuss about the obtained results o squeezing CuO-Water nanoluid low between parallel disks proble or various solid volue raction and oving paraeter. The physical properties o Copper Oxide- Water nanoluid can be ound in Table.. Figure shows the eect o the squeeze nuber on the teperature proile in the case o H,Pr 7, Ec.5, Nt.5, Nb.5, Sc 3.As it can be seen in Fig. the non-diensional teperature is direct unction o squeezing paraeter. In the other words, an increase in the squeeze nuber can be related with the decrease in the kineatic viscosity, an increase in the distance between the plates and an increase in the speed at which the plates ove. Theral boundary layer thickness increases as the squeeze nuber increases. It is iportant to note that paraeters N and b N t characterize the strengths o Brownian otion and therophoresis eects. Table Thero physical properties o water and CuO nanoparticle 3 (kg / ) C p ( j / kgk ) k(w /.k ) Pure water Fig. Eect o squeeze paraeter on teperature.

5 AZIMI et al.: UNSTEADY CuO NANOFLUID FLOW, HEAT AND MASS TRANSFER BETWEEN TWO PARALLEL DISKS 85 Fig.3 Eect o Brownian otion paraeter on teperature. Figure 3 shows the eect o squeezing paraeter on non-diensional concentration proile in the case o H 4,Pr 7, Ec., Nt.5, Sq,5, Sc 3. As it can be seen in Fig.3, an increase in N b eectively increases the nanoparticles concentration. This increase is due to the eective oveent o nanoparticles ro the upper disk to the luid. Figure 4 shows the iluence o therophoretic paraeter on concentration unction in the case o H 3,Pr 7, Ec., Nb.5, Sq.5, Sc 3.The non-diensional concentration unction decreases by increasing the therophoretic paraeter. Fro the physical point o view, an increase in the therophoretic eect generates the larger ass lux due to teperature gradient which decreases the concentration. Figure 5 shows the eect o Hartan nuber on velocity proile when Sq. It is iportant to note that the iluence o external agnetic ield is to decrease the value o the velocity agnitude throughout the enclosure because the presence o agnetic ield introduces a orce called the Lorentz orce, which acts against the low, i the agnetic ield is applied in the noral direction. The igure also gives ioration about the accuracy o our solution by presenting a coparison between analytical solutions obtained by GOHAM and nuerical ones achieved by orth order Runge Kutta Fig.4 Eect o therophoretic paraeter on concentration. Fig.5 Iluence o H on velocity. ethod. As it can be illustrated in Fig.5, analytical solutions have good agreeent with nuerical ones. This igure assures us about the accuracy and validity o our approxiate analytical solution. Conclusion In this study, unsteady MHD nanoluid low and heat transer between parallel disks are investigated.

6 86 INDIAN J. CHEM. TECHNOL., MAY 8 GOHAM is used to solve the governing equations. The eect o the squeeze nuber on heat and ass transer are investigated. The results show that the higher values o heat transer enhanceent are obtained when Brownian otion increases. Also, it can be ound that concentration is an increasing unction o Brownian otion paraeter while it is a decreasing unction o the therophoretic paraeter. Velocity is decreasing unction o agnetic eect. The coparison between GOHAM and Runge Kutta ethod assures us about the validity and accuracy o our solution. Reerences Choi S U S, ASME Fluids Eng Div, 3 (995) 99. Azii M & Oi F, J Nano Energy Power Res, (3) 4. 3 Azii M & Azii A, Coput & Theoret Nanosci, (5) Azii M & Riazi R, Indian J Pure Appl Phys, 54 (6) Stean M J, Sitzungsberichteder Akadeie der Wissenschaten in Wien Matheatik-Naturwissen, 69 (874) 7. 6 Azii M & Riazi R, Indian J Che Technol, 3 (6) Azii A & Mirzaei M, Coput & Theoret Nanosci, (5) Azii M & Riazi R, Ther Sci, (5). Doi:.98/ TSCI553A 9 Azii A & Azii M, Walailak J Sci & Tech, () (4) 955. Ganji D D & Azii M, Sci Bull Ser A, 75 () (3) 3. Ganji D D & Azii M,Sci Bull Ser A, 3 () 3. Ganji D D, Azii M & Mostoi M, Indian J Pure Appl Phys, 5 () 67.