MAGNETOHYDRODYNAMIC GO-WATER NANOFLUID FLOW AND HEAT TRANSFER BETWEEN TWO PARALLEL MOVING DISKS

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1 THERMAL SCIENCE: Year 8, Vol., No. B, pp MAGNETOHYDRODYNAMIC GO-WATER NANOFLUID FLOW AND HEAT TRANSFER BETWEEN TWO PARALLEL MOVING DISKS Introduction by Mohammadreza AZIMI and Rouzbeh RIAZI * Faculty o New Sciences and Technologies, University o Tehran, Tehran, Iran Original scientiic paper The unsteady MHD squeezing low o nanoluid with dierent type o nanoparticles between two parallel disks is discussed. The governing equations, continuity, momentum, energy, and concentration or this problem are reduced to coupled non-linear equations by using a similarity transormation. It has been ound that or contracting motion o upper disk combined with suction at lower disk, eects o increasing absolute values o squeeze parameter are quite opposite to the case o expanding motion. In this case, radial velocity near upper disk decreases while near the lower disk an accelerated radial low is observed. The comparison between analytical results and numerical ones achieved by orth order Runge-Kutta method, assures us about the validity and accuracy o problem. Key words: nanoparticle concentration, heat transer enhancement, nanoluid, Brownian motion, approximate analysis A nanoluid is achieved by dispersing nanoparticles in a base-luid. Nanoluids, i. e. luid suspensions o nanometer-sized solid particles and ibers, have been proposed as a route or surpassing the perormance o heat transer liquids currently available. Considerable attention has been recently given to nanoluids, nanoscale colloidal solutions, consisting o nanoparticles (with sizes o the order o to nm) dispersed in a base luid [, ]. Nanoluid describe a luid in which nanometer-sized particles are suspended [3]. Nanoparticles have unique properties, such as large surace area to volume ratio, and lower kinematic energy which can be exploited in various applications. Nanoparticles are more stable when dispersed in base luids, due to their large surace area and they are more stable when compared to micro-luids which lead to many practical problems. In recent years, nanoluids have attracted more and more attention. There are our possible mechanisms in nanoluids contribute to thermal conduction: ballistic nature o heat transport in nanoparticles, Brownian motion o nanoparticles, liquid layering at the liquid/particle interace, and nanoparticle clustering in nanoluids. The Brownian motion o nanoparticles is too slow to directly transer heat through nanoluid. However, it could have an indirect role to produce convection like micro-environment around the nanoparticles and particle clustering to increase the heat transer [4, 5]. Heat transer enhancement in various energy systems is vital because o the increase in energy prices. In recent years, nanoluids technology is proposed and studied by some researchers experimentally or numerically to control heat transer in a process. The nanoluid can * Corresponding author, ro_riazi@ut.ac.ir

2 384 THERMAL SCIENCE: Year 8, Vol., No. B, pp be applied to engineering problems, such as heat exchangers, cooling o electronic equipment, and chemical processes. Almost all o the researchers assumed that nanoluids treated as the common pure luid and conventional equations o mass, momentum, and energy are used and the only eect o nanoluid is its thermal conductivity and viscosity which are obtained rom the theoretical models or experimental data [6]. In various engineering problems, one may deal with mathematical relations which eventually reduce to equations in the orm o ODE or PDE. In most cases, scientiic problems are inherently o non-linearity that does not admit analytical solution, so these equations should be solved using special techniques. Some o these methods are reconstruction o variational iteration method [7], dierential transormation method [8, 9], homotopy perturbation method [], and optimal homotopy asymptotic method (OHAM) [], and others [-6]. The aim o this study is to investigate the MHD squeezing low o nanoluid between parallel disks and illustrate the eect o dierent parameters on the results. We will also compare the analytical solutions with numerical ones in order to show the eiciency o the method. z u r r = z = r n w (t) n w (t) Ht ( ) Figure. Physical model and co-ordinate system Mathematical ormulation Figure shows the geometry o the squeezing low o an incompressible viscous MHD nanoluid between two circular plates separated by a distance z = ±l( αt) / = = ±h(t). A uniorm magnetic ield o strength B(t) = B ( αt) / is applied perpendicular to the disks. The upper disk at z = h(t) approaching the stationary lower disk with the velocity dh/dt, whereas the low is assumed to be axisymmetric with respect to r =. The velocity components along the radial and axial directions are u(r, z, t), w(r, z, t), respectively. Considering unsteady axisymmetric low with negligible tangential velocity component, the conservation equations become: u u w + + = r r z u u u p u u u u ρ u w µ + + t r z = r r z r r r ρ w w w p w w w u w t r z z µ + + = r z r r T T T k T T T + u + w = t r z r z r z ( ρc p ) DT + τ DB m C T C T u u r r z z T x y () () (3) (4)

3 THERMAL SCIENCE: Year 8, Vol., No. B, pp C C C C C C D B T T T + u + w = DB t r z r r r z Tm r r r z Here, T w and C w are the temperature and nanoparticles concentration at the lower disk while the temperature and concentration at the upper disk are T h and C h, respectively, u and w the velocities in the r, and z-directions, respectively, p the pressure, T the temperature, C the nanoparticle concentration, D B the Brownian motion coeicient, D T the termophoretic diusion coeicient, T m the mean luid temperature, and k the thermal conductivity. The last term in the energy equation is the total diusion mass lux or nanoparticles, given as sum o two diusion terms [3]. The dimensionless parameter, τ, gives the ratio o eective heat capacity o the nanoparticle material to heat capacity o the luid. Eective density (ρ ), the eective dynamic viscosity (µ ), eective heat capacity (ρ ) and the eective thermal conductivity k o the nanoluid are [7]: ρ = ρ ( ϕ) + ρϕ s s ( ρc ) = ( ρc ) ( ϕ) + ( ρc ) µ p p p s =.5 ( ϕ) k ks + k φ( k ks) ns = k k + k + φ( k k ) s s µ ν = ρ The relevant boundary conditions or the problem are: µ dh z= ht ( ) u=, w= ww =, T= TH, C= Ch dt w (7) z = u =, w=, T = Tw, C = Cw αt Previous equations can be simpliied by introducing ollowing parameters: αr αh z u = ( η), w= ( η), η = αt αt H αt (8) T TH B C Ch θ =, B =, ϕ = TW TH αt Cw Ch The previous parameters are substituted into eqs. () and (3). Then the pressure gradient is eliminated rom the resulting equations. We inally yield: ( IV ) ( ) S η + 3 Ha = (9) Using eq. (8), eqs. (3) and (4) are simpliy to the ollowing equations: θ + Pr S ( θ ηθ ) + Pr Nbθ ϕ + Pr Ntθ = () Nt ϕ + LeS( ϕ ηϕ ) + θ = Nb () (5) (6)

4 386 THERMAL SCIENCE: Year 8, Vol., No. B, pp With the ollowing boundary conditions: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = A, =, =.5, =, θ = ϕ =, θ = ϕ = () where S is the squeeze parameter, Pr the Prandtl number, A the suction/blowing parameter, Ha the Hartmann number, Nb the Brownian motion parameter, Nt the thermophoretic parameter, and Le the Lewis number which are deined: αh ν w σ BH ν S =, Pr =, A=, Ha =, Le = ν α αh ν D ( ρ ) ( ) s ( ) ( ρ ) ( ) s ( ) c D C C c D T T Nb =, Nt = ρc ν ρc T ν B w h T w h It is important to note that A > indicates the suction o luid rom the lower disk while A < represents injection low. Solution procedure The considered problem is a boundary value problem, and requires proper numerical solver. Most o the existing numerical solvers in MAPLE sotware are a combination o trapezoid or midpoint methods. Each o these basic schemes has its own characteristics. Methods implemented based on trapezoid method work eiciently or typical problems, however, midpoint based techniques are best suited or handling harmless end-point singularities. Fourth order Runge-Kutta method is a midpoint method which improves the Euler method by one order []. In this section, the normalized eqs. (9)-() that are coupled with the boundary conditions given in eq. () are solved numerically by Runge-Kutta method using MAPLE sotware. The most popular Runge-Kutta methods are ourth order. This is because, or the second order approaches, there are iinite numbers o versions. The ollowing ormulation is the most commonly used orm, i. e. the orth order Runge-Kutta method: where m e (3) y = i y + + i ( k k k3 k4) (4) k ( ) = E x y (5) i, k E = xi + h, yi + kh k3 E = xi + h, yi + kh (, ) 4 i i 3 i (6) (7) k = E x+ h y+ kh (8) The ourth order Range-Kutta numerical solver is a simple and eicient candidate technique or dierential equations and in most situations o interest this method represents an appropriate perormance. Results and discussion Eects o dierent low parameters on the velocity, temperature and concentration distributions are discussed in this section. The eects o increasing suction at lower disk on both axial and radial velocities are displayed in igs. and 3, respectively. It is evident that increasing value o

5 THERMAL SCIENCE: Year 8, Vol., No. B, pp suction strength results in higher absolute values o both the velocities. As increasing suction allows more luid to low near the lower disk thereore a decrease in boundary-layer thickness is expected. 4 ( h) ( h) h A = A = A = 3 A = 4 Figure. Eect o suction/blowing number on axial velocity in case o, Ha =, S = h A = A = A = 3 A = 4 Figure 3. Eect o suction/blowing number on radial velocity in case o, Ha =, S = ( h) ( h) h S = 4 S = S = S = S = h S = 4 S = S = S = S = 4 Figure 4. Eect o squeeze number on axial velocity in case o, Ha =, A = Figure 5. Eect o squeeze number on radial velocity in case o, Ha =, A = Iluences o squeeze parameter on axial and radial velocities are displayed in igs. 4 and 5, respectively. Here S > denotes the movement o upper disk away rom the lower disk. It can be seen rom ig. 4 that or squeezing motion o upper disk combined with suction axial velocity near the center is increased while or dilating motion a decrease in axial velocity is observed. From ig. 5 one can see the behavior o radial velocity or same variations in S. It is evident or expanding motion; an accelerated radial low is observed near the upper disk, however, this trend changes gradually as we move away rom it. Somewhere near the center this trend gets converted into an opposite one, that is, rom that point to lower disk a delayed motion

6 388 THERMAL SCIENCE: Year 8, Vol., No. B, pp is observed. For contracting motion o upper disk, combined with suction at lower disk, eects o increasing values o squeeze parameter, S, are opposite to the case o expanding motion. q.8 q h S = S = S = 3 S = 4 Figure 6. Eect o squeeze number on temperature in case o, Nb =.3, Nt =., Le = Ha =, Pr = 6., A =.5 q h A =.5 A =.5 A =.75 A =. Figure 7. Eect o suction/blowing parameter on temperature in case o Nb =., Nt =., Le = Ha = S =, Pr = 6. q h Nb =.5 Nb =.75 Nb =. Nb =.5 Figure 8. Eect o Brownian motion parameter on temperature in case o Nt =., Le = A = Ha = S =, Pr = h Nt =. Nt =. Nt =.3 Nt =.4 Figure 9. Eect o thermophoretic parameter on temperature in case o Nb =., Ha =, Le = A =, S =, Pr = 6. The eects o interesting physical parameters, namely squeeze parameter, suction/ blowing parameter, Brownian motion parameter, and thermoporetic parameter on the dimensionless temperature proile are sketched in the igs The iluence o squeeze number on temperature is illustrated in ig. 6. It can be seen rom ig. 6 that the absolute o non-dimensional temperature increases or increasing S in the range o.5 < η <.

7 THERMAL SCIENCE: Year 8, Vol., No. B, pp Consequence o increasing suction parameter is presented in ig. 7 whereas the temperature reduces with increasing A. In another word, the temperature and thermal boundary-layer thickness are enhanced when we decrease the value o suction/blowing number. Table illustrates the eects o Hartmann number on the squeezing low o nanoluid. The comparison o approximate (analytical) results [7] with numerical solutions obtained by ourth order Runge-Kutta, in the current study, have been also presented in case A =, S =, Nb =, Nt =., Le =, Pr = 6.. Table. Eect o Hartmann number on Ha h num GOHAM[7] Conclusions This article investigates MHD squeezing low o graphene oxide water nanoluid between parallel disks. At irst the mathematical ormulation is presented. Governing PDE are converted via similarity transormations. The numerical solution o governing equations were compared with analytical results previously obtained by using Galerkin OHAM method [7]. The ollowing concluding remarks are achieved. y The suction parameter decreases the thermal boundary-layer thickness hence at the disks we have higher rate o heat transer. y The eect o squeeze number on the axial velocity proiles is minimal. y For contracting motion o upper disk combined with suction at lower disk, eects o increasing absolute values o squeeze number are quite opposite to the case o expanding motion. In this case, radial velocity near upper disk decreases while near the lower disk an accelerated radial low is observed. y For both the cases o suction and injection, the temperature and concentration unction values increases monotonically as the similarity variable, A, increases. y The axial component o velocity increases near the central axis o the channel but deceases near the walls. y The transverse magnetic ield decreases the luid motion. y The temperature signiicantly rise and proiles move closer to upper disk as Brownian motion parameter increase in both suction and blowing cases. y Increasing Brownian motion parameter enhances nanoparticle concentration value. A numerical solution using well known orth order Runge-Kutta method has also been obtained or the sake o comparison. It is ound that the numerical and analytical results are in good agreement. Reerences [] Saidur, R., et al., A Review on Applications and Challenges o Nanoluids, Renewable and Sustainable Energy Reviews, 5 (), 3, pp [] Bahiraei, M., A Comprehensive Review on Dierent Numerical Approaches or Simulation in Nanoluids: Traditional and Novel Techniques, Journal o Dispersion Science and Technology, 35 (4), 7, pp [3] Azimi, M., Riazi, R., Heat Transer Analysis o Magnetohydrodynamics Graphene Oxide-Water Nanoluid Flow through Convergent-Divergent Channels, Journal o Computational and Theoretical Nanoscience, 3 (6),, pp

8 39 THERMAL SCIENCE: Year 8, Vol., No. B, pp [4] Azimi, M., Azimi, A., Flow Simulation o GO Nanoluid in Semi Porous Channel, Turkish Journal o Science and Technology, 9 (4),, pp [5] Azimi, M., Ommi, F., Using Nanoluid or Heat Transer Enhancement in Engine Cooling Process, Nano Energy and Power Research, (3),, pp. -3 [6] Sheikholeslami, M., et al., Numerical Simulation o Two Phase Unsteady Nanoluid Flow and Heat Transer between Parallel Plates in Presence o Time Dependent Magnetic Field, Journal o the Taiwan Institute o Chemical Engineers, 46 (5), Jan., pp [7] Azimi, A., Azimi, M., Analytical Investigation on -D Unsteady MHD Viscoelastic Flow between Moving Parallel Plates Using RVIM and HPM, Walailak Journal o Science and Technology, (4),, pp [8] Ganji, D. D., Azimi, M., Application o DTM on MHD Jeery Hamel Problem with Nanoparticles, U.P.B. Scientiic Bulletin Series A, 75 (3),, pp. 3-3 [9] Sheikholeslami, M., et al., Application o Dierential Transormation Method or Nanoluid Flow in a Semi-Permeable Channel Considering Magnetic Feild Eect, International Journal o Computational in Engineering Science and Mechanics, 6 (5), 4, pp [] Azimi, M., Azimi, A., Investigation on the Film Flow o a Third Grade Fluid on an Inclined Plane Using HPM, Mechanics and Mechanical Engineering, 8 (4),, pp [] Azimi, M., et al., Investigation o the Unsteady Graphene Oxide Nanoluid Flow between Two Moving Plates, Journal o Computational and Theoretical Nanoscience, (4),, pp. 4-8 [] Azimi, M., Ganji, D. D., Application o Max Min Approach and Amplitude Frequency Formulation to the Nonlinear Oscillation Systems, The Scientiic Bulletin Series A, 74 (), 3, pp. 3-4 [3] Ganji, D. D., et al., Energy Balance Method and Amplitude Frequency Formulation Based o Strongly Non-Linear Oscillator, Indian Journal o Pure & Applied Physics, 5 (), 9, pp [4] Azimi, M., Riazi, R., Analytical Simulation o Mixed Convection between Two Parallel Plates in Presence o Time Dependent Magnetic Field, Indian Journal o Pure & Applied Physics, 54 (6), 5, pp [5] Azimi, M., et al., Analytical Investigation o MHD Jeery Hamel Problem with Graphene Oxide Nanoparticles Using GOHAM, Journal o Computational and Theoretical Nanoscience, (5), 6, pp [6] Azimi, M., Riazi, R., Flow and Heat Transer o MHD Graphene Oxide Water Nanoluid between Two Non-Parallel Walls, Thermal Science, (7), 5, pp [7] Azimi, M., Riazi, R., Heat Transer Analysis o GO-Water Nanoluid between Two Parallel Disks, Propulsion and Power Research, 4 (5),, pp. 3-3 Paper submitted: July 3, 5 Paper revised: May, 6 Paper accepted: June 3, 6 7 Society o Thermal Engineers o Serbia Published by the Vinča Institute o Nuclear Sciences, Belgrade, Serbia. This is an open access article distributed under the CC BY-NC-ND 4. terms and conditions