HAM Solutions on MHD Squeezing axisymmetric flow of Water Nanofluid through saturated Porous Medium between two parallel disks

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1 HAM Solutions on MHD Squeezing axisymmetric flow of Water Nanofluid through saturated Porous Medium between two parallel disks B. Siva Kumar Reddy 1, a), K.V. Surya Narayana Rao 2, b) and 3, c) R. Bhuvana Vijaya 1, 2 Department of Mathematics, RGM College of Engineering & Technology (Autonomous), Nandyal, Kurnool, Andhra Pradesh , India. 3 Department of Mathematics, JNTUA College of Engineering, Anantapuram, Andhra Pradesh , India. a) Corresponding author: sivajasmine707@gmail.com b) suryam_1968@yahoo.com c) bhuvanarachamalla@gmail.com Abstract: In this paper, we have considered the unsteady magnetohydrodynamic squeezing axi-symmetric flow of waternanofluid through saturated porous medium between two parallel disks. The equations for the governing flow are solved by Galerkin optimal Homotopy asymptotic method. The effects of non-dimensional parameters on velocity, temperature and concentration have been discussed with the help of graphs. Also we obtained local Nusselt number and computationally discussed with reference to flow parameters. Keywords: Nanofluid, squeezing flow, nanoparticle concentration, Heat transfer, porous medium, MHD flow, HAM solutions. INTRODUCTION Most common fluids such as water, ethylene, glycol, toluene or oil generally have poor heat transfer characteristics owing to their low thermal conductivity. A recent technique to improve the thermal conductivity of these fluids is to suspend nano-sized metallic particles such as aluminum, titanium, gold, copper, iron or their oxides in the fluid to enhance its thermal properties. The term nanofluid was envisioned to describe a fluid in which nanometer-sized particles were suspended in conventional heat transfer basic fluids. Nanotechnology aims to manipulate the structure of the matter at the molecular level with the goal for innovation in virtually every industry and public endeavor including biological sciences, physical sciences, electronics cooling, transportation, the environment and national security [2, 8]. There are four possible mechanisms in nanofluids contribute to thermal conduction: (a) ballistic nature of heat transport in nanoparticles, (b) Brownian motion of nanoparticles, (c) liquid layering at the liquid/particle interface, and (d) nanoparticle clustering in nanofluids. The Brownian motion of nanoparticles is too slow to directly transfer heat through nanofluid; however, it could have an indirect role to produce a convection like micro environment around the nanoparticles and particle clustering to increase the heat transfer. Recently, the problem of fluid flow and heat transfer in the presence of magnetic field has been the topic of many researches. Investigations on how this will affect the flow field and heat transfer parameters in a laminar boundary layer due to stretching sheet seems to be of crucial importance due to its wide application in a large number of industries. Unsteady MHD film flow over a rotating iinite disk was studied by Kumari and Nath [1].In a paper by Xu et al. [3] a series solution of unsteady three-dimensional MHD flow and heat transfer in the boundary

2 layer for the case of impulsive stretching plate was given. As an example in the works by Azimi et al. [12] and Raftari et al. [9] the flow of two types of non-newtonian fluids namely viscoelastic fluid and Maxwellian fluid in channel flows and over stretching plates was addressed. Other works performed the MHD effects on the fluid flow and heat transfer through porous medium. Squeezing flows have many applications in food industry especially in chemical engineering. Some practical examples of squeezing flow include polymer processing, compression and injection molding. In addition the lubrication system can also be modeled by squeezing flows. The study of squeezing flows has its origins in the 19 th century and continues to receive considerable attention due to the practical applications in physical and biophysical areas. The theoretical and experimental studies of squeezing flows have been conducted by many researchers [4, 13]. Most scientific problems and phenomena are modeled by non-linear ordinary or partial differential equations. As an example boundary layer flows can be mentioned. Therefore the study on the various methods used for solving the non-linear differential equations is a very important topic for the analysis of engineering practical problems. There are a number of approaches for solving non-linear equations which range from completely analytical to completely numerical ones. Besides all advantages of using numerical methods closed form solutions appear more appealing because they reveal physical insights through the physics of the problem. Also parametric studies become more convenient with applying analytical methods. Some of these methods are homotopy perturbation method (HPM), differential transformation method (DTM) [5], adomian decomposition method (ADM) [6] and Galerkin optimal homotopy asymptotic method (GOHAM) [7]. The aim of this study is to study the MHD squeezing flow of nanofluid between parallel disks and illustrate the effect of different parameters on the results. We will also compare the analytical solutions with numerical ones in order to show the efficiency of the method. FORMULATION AND SOLUTION OF THE PROBLEM We consider the axisymmetric incompressible flow of MHD nanofluid between two parallel disks separated by a distance z l 1 t h t 1 2 for 0 as shown in figure 1. A uniform magnetic field of strength Bt B t 05 is applied perpendicular to the disks. The upper disk at z ht 0 1. moves towards or away from the stationary lower disk with the velocity dh / dt. 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. The governing equations for unsteady two dimensional (2D) axisymmetric flow of a nanofluid are as follows [10]. u u w 0 r r z u u u p u u 1 u u 2 u w B 2 0u u t r z r r z r r r k w w w p w w 1 w 2 u w B 0 w w t r z z r z r r k k 2 1 D T T T T T T C T C T T u u u w D 2 B t r z C r z r z r r z z T m x y C C C C 1 C C D B T 1 T T u w DB (5) t r z r r r z T m r r r z Where u and w are the velocities in the r and z directions respectively p is pressure, T is temperature, C is the nanoparticle concentration, DB is the Brownian motion coefficient, DT is the termophoretic diffusion coefficient, T m is the mean fluid temperature and k is the thermal conductivity. The last term in the energy equation is the total diffusion mass flux for nanoparticles, given as sum of two diffusion terms [10]. is the dimensionless parameter that gives the ratio of effective heat capacity of the nanoparticle material to heat capacity of the fluid. Effective (1) (2) (3) (4)

3 density, the effective dynamic viscosity, effective heat capacity conductivity k of the nanofluid are defined as [9] : and the effective thermal w z - axis u ɵ v z = h(t) = H(1-αt) 1/2 h(t) Porous medium O z = 0 B 0 1 f s s Cp Cp 1 Cp f s f k k 2k 2 k k ns k k 2k 2 k k v s f f s f s f f s FIGURE 1: physical model and coordinate system f (6) The relevant boundary conditions for the problem are: dh z ht u 0, w ww, T TH, C Ch dt w z 0 u 0, 0 w, T Tw, C Cw (7) 1t We can simplify above equations by introducing following parameters: r u f H z T T, w f 1 21 t 2,, H B0, B, 1 1 t 2 H 1t 1 2 TH TH 1 1 t 2 C Ch (8) C C w h

4 The above parameters are substituted into equations (2) and (3). Then the pressure gradient is eliminated from the resulting equations. We finally yield: IV f S f 3 f 2 ff M f M 0 D Using equations (8), (3) and (4) simplify to following equations: 1 2 PrS 2f PrNb PrNt 0 (10) Nt LeS 2f 0 (11) Nb With the following boundary conditions: f 0 A f 0 0,, 1 0 5, 0 0 1, f. f 1 0, (12) Where S is squeeze parameter, Pr is the prandtl number, A is the suction/blowing parameter, M is Hartman number, Nb is the Brownian motion parameter, D is the porosity parameter and Nt is the Lewis number which are defined as: 2 H v f w0 BH S, Pr, A, 0 v M, Le, D 2 v H v D, c DB Cw Ch s Nb, H c v 2 f c DT Tw Th s c T v f m e Nt (13) It is important to note that A 0 indicates the suction of fluid from the lower disk while A 0 represents injection flow. Set the page parameters for your manuscripts. The detailed page setup parameters are listed in Table 1. Galerkin optimal homotopy asymptotic method (GOHAM) solution: Following differential equation is considered: L u t N u t g t, Bu 0 (14) 0 Where L is a linear operator, is an independent variable, u is an unknown function, g is a known function, Nu is a nonlinear operator and B is a boundary operator. By means of OHAM one first constructs a set of equations: 1 pl, p g H pl, p g N, p B, p 0 H p denotes a nonzero auxiliary function for 0 Where p is an embedding parameter, is an unknown function. Obviously when p 0 and p 1, it holds that:, 0 u0,, 1 u Thus as p increases from 0 to 1, the solution,p varies from u to the solution obtained from Eq. (16) for p 0 f (9) (15) p and H 0 0,,p (16) Lu0 g 0, 0 0 We choose the auxiliary function H p in the form: 0 u where u0 is Bu (17) H p p1c 1 p2c2 (18) C,C, are constants which can be determined later. Expanding,p in a series with respect to p, one Where 1 2 has:, p,ci u0 u k,ci p k, i 12,, (19) k1

5 Table. 1 Reduced Nusselt number for case S = 1, A = 2, N t = 0.1, L e = 1, Pr = 6.2 with various Brownian motion parameter N b Nu GOHAM Nu HAM [31] Numerical Substituting Eq. (9) into Eq. (5) collecting the same powers of p, and equating each coefficient of p to zero we obtain set of differential equation with boundary conditions. Solving differential equations by boundary conditions u,u,c,u,c are obtained. Generally speaking the solution of Eq. (4) can be determined approximately in the form:, p,ci u0 u k,ci p k, i 12,, (20) k1 0 m u m u u,c (21) k1 Note that the last coefficient residual: k m i C m can be function of. Substituting Eq. (10) into Eq. (4) there results the following m i i i R,C L u,c g N u,c (22) If R,C i 0 then m u,c i happens to be the exact solution. Generally such a case will not arise for nonlinear problems, but we can minimize the functional by Galerkin method: R,C 1,C 2,,Cm w i, i 12,,,m Ci (23) C i 12,,,m can be identified from the conditions: The unknown constants b 1 2 i 1 2 m 0 a i J C,C w R,C,C,,C d (24) Where a and b are two values depending on the given problem. With these constants the approximate solution (of order m ) Eq. (23) is well determined. It can be observed that the method proposed in this work generalizes these two methods using the special (more general) auxiliary function H p. RESULTS AND DISCUSSION The main objective of this paper is to discuss the iluence of pertinent parameters on the velocity, temperature and Nano-particle concentration fields. The computational results for f and f represent radial and axial velocities respectively. Figures (2) represent velocity profiles and Figure (3) represent temperature distribution and Figure (4) depicts the nano-particle concentration profiles. Figure (5) depicts the iluence of suction or blowing parameter on temperature and nano-particle concentration. We noticed that, from the figure (2) the effect of squeeze parameter S on axial and radial velocities here S 0 corresponds to the movement of upper disk towards lower one, while 0 S S describes away the movement of the same disk. When we increase the squeeze parameter then the radial velocity and axial velocity reduces through at the fluid medium being the other parameters fixed D 1,N b 2,N t 0. 1,L e 1,P r 0. 71,M 1,A From the same Figure the radial velocity reduces for t 5 and then it is improved for t 5 at the upper disk, where as the axial velocity enhances with increasing the permeability parameter D. We also noticed that, lower the permeability of the porous medium lesser the fluid speed is observed in the entire region between the upper and lower disks ( N b 2,N t 0. 1, L e 1, Pr 0. 71,M 1,S 1,A ). The opposite trend is noticed in case of increasing the intensity of the magnetic field. i.e. both the velocity components f and f gradually reduces with

6 increasing Hartman number M. This is because of the reason that effects of a transverse magnetic field on an electrically conducting fluid gives rise to a resistive type force (called Lorentz force) similar to drag force and upon increasing the values of M increases the drag force which has tendency to slow down the motion of the fluid. ( D 1,N 2,N 0. 1,L 1, Pr 0. 71,A 0. 05,S 1). b t e FIGURE 2: The Radial and Axial velocity profiles against S, D and M The iluence of Lewis parameter N t, squeeze parameter S, Prandtl number Pr and Brownian motion parameter N on temperature distribution is displayed from the Figure (3). We noticed that, the temperature b reduces with increasing Lewis number N t and Prandtl number Pr. Also the simultaneous effects of Brownian motion parameter N and squeeze parameter S on the temperature field, we observed that an increasing the b Brownian motion parameter N b results in the uniform increasing in the temperature. Similar behavior is observed for increasing squeeze parameter S. Figure (4) shows the effects of Lewis parameter N t, squeeze parameter S, Schmidt number Le and Brownian motion parameter N b. An increasing squeeze parameter S or Brownian motion parameter N b increase the Nano-particle concentration, where as it reduces with increasing Lewis parameter N and Schmidt number Le. t

7 FIGURE 3: Temperature Profiles against N t, S, Pr and N b FIGURE 4: Concentration Profiles against N t, S, Le and N b

8 FIGURES 5: Temperature & Concentration Profiles against A (Suction / Blowing parameter) From the Figure (5) it is evident that by increasing in positive value for a number, the temperature value is decreased but in glowing an opposite treatment is observed as increasing suction allows more fluid to the flow near the lower disk. Therefore a decrease in boundary layer thickness is expected. And an increasing value of suction or blowing parameter the results in higher absolute value of the nano-particle concentration. A comparison between GOHAM and HAM solutions [11] for reduced nusselt number as shown in table 2. The rate of heat transfer reduces with increasing Lewis parameter N t and Brownian motion parameter N b and enhances with increasing squeeze parameter S and Prandtl number Pr. TABLE 2: Reduced Nusselt number for the case A=2, Le=1 N t S Pr N b Nu HAM [10] Present Nu [GOHAM] Present Numerical CONCLUSIONS We have considered the unsteady MHD queezing axisymmetric flow of water-nanofluid through saturated porous medium between two parallel disks. The equations for the governing flow are solved by Galerkin optimal Homotopy asymptotic method. The conclusions are made as the following. 1. The effect of squeeze number on the axial velocity profiles is minimal. 2. For contracting motion of upper disk combined with suction at lower disk, effects of increasing absolute values of S are quite opposite to the case of expanding motion. However, radial velocity near upper disk decreases while near the lower disk an accelerated radial flow is observed. 3. For both the cases of suction and injection, the temperature and concentration distributions increase monotonically as the suction /blowing parameter A increases. 4. The axial velocity increases near the central axis of the channel but decreases near the walls.

9 5. The application of transverse magnetic field reduces the fluid motion and fluid velocity enhances with increase in the permeability of the porous medium. 6. The temperature drastically augment and profiles move closer to upper disk as Brownian motion parameter increase in both suction and blowing cases. 7. An inncreasing Brownian motion parameter enhances nano-particle concentration. REFERENCES 1. Kumara, M. and Nath, G. unsteady MHD film flow over a rotating iinite disk, International Journal of Engineering Science. 42, pp , Yu, W., France, D.M., Choi, S.U.S., and Routbort, J.L., Review and assessment of nanofluid technology for transportation and other applications, ANL/ESD/07-9, Xu, H., Liao, S.J., and Pop, I., Series solution of unsteady three dimensional MHD flow and heat transfer in the boundary layer over an impulsively stretching plate, European Journal of Mechanics B/Fluids. 26, pp , Ran, X.J., Zhu, Q.Y., Li, Y., An explicit series solution of the squeezing flow between two iinite plates by means of the homotopy analysis method, Communications in Nonlinear Science and Numerical Simulation, 14, pp , Rashidi, M.M., The modified differential transform method for solving MHD boundary-layer equations, Computer Physics Communications, 180 (11), pp , Hayat, T., Hussain, Q., Javed, T., The modified decomposition method and pade approximants for the MHD flow over a non-linear stretching sheet, Nonlinear Analysis: Real World Applications, 10, pp , Idrees, M., Islam, S., Haq, S., Islam, S., Application of the optimal homotopy asymptotic method to squeezing flow, Computers and Mathematics with Applications, 59, pp , Saidur, R., Leong, K.Y., Mohammad, H.A., A review on applications and challenges of nanofluids, Renewable and Sustainable Energy Reviews, 15 (3), pp , Raftari, B., Vajravelu, K., Homotopy analysis method for MHD viscoelastic fluid flow and heat transfer in a channel with a stretching wall, Communications in nonlinear science and numerical simulation 17, pp , Hayat, T., Yousaf, A., Mustafa, M., Obaidat, S., MHD squeezing flow of second grade fluid between two parallel disk, International Journal of Numerical Methods Fluids 69 (2), pp , Hashmi, M.M., Hayat, T., Alsaedi, A., On the analytic solutions for squeezing flow of nanofluid between parallel disks, Nonlinear Analysis : Modelling and control 17 (4), pp , Azimi, A., Azimi, M., Analytical Investigation on 2-D unsteady MHD viscoelastic flow between moving parallel plates using RVIM and HPM, 11 (11), pp , Azimi, M., Azimi, A., Mirzaei, M., Investigation of the unsteady grapheme oxide nanofluid flow between two moving plates, Journal of Computational and Theoritical Nanoscience.11 (10), pp , Siva Kumar Reddy, B., Veera Krishna, M., Surya Narayana Rao, K.V., Bhuvana Vijaya, R., HAM Solutions on MHD flow of Nanofluid through saturated porous medium with Hall effects, Materials Today proceedings. xx, pp.xx-xx, 2016 (In press).