MHD Slip Flow and Heat Transfer on Stagnation Point of a Magnetite (Fe3O4 ) Ferrofluid towards a Stretching Sheet with Newtonian Heating

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1 11, Issue 1 (2019) CFD Letters Journal homepage: ISSN: MHD Slip Flow and Heat Transer on Stagnation Point o a Magnetite (Fe3O4 ) Ferroluid towards a Stretching Sheet with Newtonian Heating Open Access Muhammad Khairul Anuar Mohamed 1,, Nurul Ainn Ismail 1, Norhamizah Hashim 1, Norlianah Mohd Shah 1, Mohd Zuki Salleh 2 1 School o Foundation and Inter Disciplinary Studies, DRB-HICOM University o Automotive Malaysia, Peramu Jaya Industrial Area, Pekan, Pahang, Malaysia 2 Applied & Industrial Mathematics Research Group, Faculty o Industrial Sciences & Technology, Universiti Malaysia Pahang, Kuantan, Pahang, Malaysia ARTICLE INFO Article history: Received 2 October 2018 Received in revised orm 15 November 2018 Accepted 17 November 2018 Available online 10 January 2019 Keywords: Ferroluid, MHD, Newtonian heating, slip low, stagnation point, stretching sheet ABSTRACT Present paper investigates the low and heat transer o Magnetite (Fe3O4) water based nanoluid termed as erroluid on stagnation point past a stretching sheet with slip eect. The methodology starts with transorming the non-linear partial dierential equations that governed the model to ordinary dierential equations, then solved numerically by Runge-Kutta-Fehlberg (RKF45) method in Maple sotware. The inluenced and characteristics o pertinent parameters which are the stretching parameter, magnetic parameter, velocity slip parameter and solid volume raction or nanoluid are analyzed and discussed. It was ound that the magnetite (Fe3O4) erroluid provided higher wall temperature and heat transer capabilities compared to water. Meanwhile, the presence o slip eect had minimized the skin riction coeicient. Copyright 2019 PENERBIT AKADEMIA BARU - All rights reserved 1. Introduction Ferroluid termed as a nanoluid with nanoscale erromagnetic particles. It is strongly magnetized in the presence o magnetic ield. It was irst invented by NASA as a liquid rocket uel that can be injected toward pump inlet in a weightless or no gravity situation in aero space [1]. Further, erroluid is also employed in ield medicine as intelligent biomaterials or wound treatment and medicine drug targeting especially in cancer tumor treatment. Other applications involving this type o luid includes in electrical instruments such as computer hard disks, heat controlling agents in electric motor and hi-i speakers [2,3]. In considering the convective heat transer boundary layer low, Crane [4] pioneered this study in stretching surace. The study regarding this topic then etended with various type o luid include the erroluid. In modelling the mathematical model o erroluid, Tiwari and Das [5] introduced a Corresponding author. address: baa_khy@yahoo.com (Muhammad Khairul Anuar Mohamed) 17

2 single-phase model with a speciic case o nanoluid. This approaches is suitable to study the low and heat transer characteristic o speciic erroluid. The coeicient physical properties like thermal conductivity, density and speciic heat o nanoluid are taking account in computation. Recent study ocus on a stagnation point low past a stretching sheet studied by Zokri et al., [6], Kho et al., [7] and Mohamed et al., [8] whose considered the eects o MHD, thermal radiation, viscous dissipation and heat generation in viscous luid, Jerey luid and Williamson nanoluid. Present paper etends the study on stagnation point low past a stretching sheet in a magnetite (Fe3O4) erroluid with slip eects. Recent studies on erroluid includes the works rom Ramli et al., [9], Sheikholeslami [10], Jusoh et al., [11] and Hussanan et al., [12,13]. The Newtonian heating boundary conditions are more realistic to be taken account since it consider the proportional relation between the wall temperature and the heat transer rate Merkin [14]. Results published here are important or researchers in this area either in numerical or eperimental approach so it can be used as a comparison and reerence in uture. The act that the problem considered here has never been studied beore, thus the reported result in this paper is new. 2. Mathematical Formulation Consider a steady incompressible erroluid on a stagnation point past a stretching sheet with ambient temperature T. Assume that the ree stream velocity U and stretching velocity uw( ) are in the orms o linear uw( ) a and U b where a and b are positive constants [15]. Further, a uniorm magnetic ield o strength B0 is assumed to be applied in the positive y-direction normal to the stretching sheet. The magnetic Reynolds number is assumed to be small, and thus the induced magnetic ield is negligible. The physical model and coordinate system o this problem is shown in Figure 1. It is urther assumed that the plate is subjected to a Newtonian heating as proposed by Merkin [14]. The boundary layer equations are [16,17,9] u v 0, y (1) 2 2 u u du u Bo ( ) u v U n 2 ( u U ), y d y n (2) u T v T T n y y 2 2, (3) subject to the boundary conditions * u T u uw, v 0, hst y y at y 0, u U, T T as y, (4) where, u and v are the velocity components along the and y directions, respectively. Further, T is the erroluid temperature in the boundary layer, is the electrical conductivity, h s is the heat transer coeicient, n is the kinematic viscosity o erroluid, n is the erroluid density and n is 18

3 the thermal diusivity o erroluid, which can be epressed in terms o the properties o base luid, erroparticles and solid volume raction as ollows [17,18] Fig. 1. Physical model and the coordinate system n kn n, n (1 ) s, n, n, 2.5 C (1 ) n n p n kn ks 2k 2 ( k ks ) (1 ),. k k 2 k ( k k ) C p C p C p n s s s (5) Note that kn, k and ks are the thermal conductivity o the erroluid, base luid and erroparticles, respectively while C p n is the heat capacity o erroluid. The non-linear partial dierential (Eq. 1-3) contains many dependent variables which in dimensional orms and diicult to solve. Thereore, the ollowing similarity variables are applied b T T y, b ( ), ( ), T (6) where, and are non-dimensional similarity variable, temperature and stream unction. The Eq. 1 satisied by deinition u and v, respectively. Substitute the Eq. 5 and 6 into (Eq. 1- y 3), then the ollowing ordinary dierential equations are obtained (1 ) 1 ( s)/( ) ( ) M (7) k n / k (1 ) ( C ) /( C ) p s p Pr 0, (8) 19

4 a B o2 ( ) where,( 0) is the stretching parameter, M b b is the magnetic parameter and n Cp Pr k is the Prandtl number. The transormed boundary conditions are (0) 0, (0) 1 (0), (0) 1 (0), ( ), ( ) 0 as y. (9) where, 1 2 * b v and 1 2 b h s v is the velocity slip parameter and conjugate parameter, respectively. The physical quantity interest are the wall temperature (0), (0) and the skin riction coeicient C which given by the heat transer rate C u w, 2 (10) u with surace shear stressw n y y0. Using the similarity variables in Eq. 6 gives C (0) Re, 2.5 (1 ) (11) where, Re U is the Reynolds number. 3. Results and Discussion The system o ordinary dierential Eq. 7 and Eq. 8 with boundary conditions Eq. 9 were solved numerically using the RKF45 technique in Maple. The boundary layer thickness between 3 and 6 was used in the computation, depending on the values o the parameters considered so that the boundary condition at ininity is achieved. The numerical results are obtained or wall temperature (0), the temperature gradient (0) and the reduced skin riction coeicient C Re or various values o pertinent parameter namely as the stretching parameter, magnetic parameter M, velocity slip parameter and solid volume raction or nanoluid. In order to validate the numerical results obtained, the comparison has been made. Table 1 shows the comparison values o (0) and (0) with previous published results by Salleh et al., [19] and Sari et al., [20] or ordinary viscous luid with various values o Pr. It is ound that the results are in a good agreement. Net, Table 2 shows data related to the thermophysical properties or water and magnetite (Fe3O4) applied by Khan et al., [17] and Ramli et al., [9]. Note that as water employed as a base luid in this study thereore, the numerical computation set Pr as

5 Table 1 Comparison results with previously published or Pr when M 0 and 1. Pr Salleh et al., [19] Sari et al., [20] Present (0) (0) (0) (0) (0) (0) Table 2 Thermophysical properties o water base luid and magnetite (Fe 3 O 4 ) nanoparticles Physical Properties Water Base Fluid Magnetite (Fe3O4) Nanoparticles (kg/m 3 ) C (J/kg K) p k(w/m K) Figures 2-5 show the temperature proiles or various values o,,m and. It is worth mentioning here that in Newtonian heating, the eect o heat transer coeicient (0) is similar since this quantity is directly proportional to the wall temperature (0). In Figure 2, it is ound that the increase o reduced the temperature as well as its thermal boundary layer thicknesses. The increase o stretching velocity over stream velocity has thinning the boundary layer thicknesses. Contrary with Figures 3-5, the increase o, M and gives rise on temperature. It is realistic since the increase volume o magnetite nanoparticles in the luid had increase the erroluid capabilities in thermal conductivity hence increase the temperature as well as the heat transer rate. Further, the increase in magnetic ield had attract the erroluid particles stick to the surace which enhanced the eiciency o the heat transer rate. Notice that in Figure 5, the boundary layer thicknesses increases more signiicantly with the increase in compared to the changes in M and rom Figures 3 and 4. Fig. 2. Eect on and 0.1. when Pr 6.2, 0.5, M 1 21

6 Fig. 3. Eect M on and 0.1. when Pr 6.2, 0.5, 1 Fig. 4. Eect on when Pr 6.2, 0.5, M 1and 0.1. Net, Figure 6 illustrates the velocity proiles or various values o. It is ound that the presence o stretching eects or 1 has increase the luid velocity while reduced the thickness o velocity boundary layer. Thinning in velocity boundary layer thicknesses physically denoted to the increase in skin riction coeicient. For 1, which imply to the case o stretching velocity is equal to the luid eternal velocity outside the boundary layer, the velocity is constant which cause the zero skin riction coeicient, C Re 0. The velocity boundary layer thickness is increases as the stretching velocity is less than stream velocity. This is shown clearly at 1. Further, the presents o slip velocity had reduced the luid velocity and the velocity boundary layer thickness. Eect o on velocity proiles are shown clearly in Figure 7. 22

7 Fig. 5. Eect on and M 1. when Pr 6.2, 0.5 Fig. 6. Eect on 0.5, M 1and 0.1. when Pr 6.2, 23

8 Fig. 7. Eect on 0.5, M 1and 0.1. when Pr 6.2, 1/ Figures 8 and 9 illustrate the eect o M and on variation o (0) and C Re 2, respectively. It is ound that the increase in M and had increase the wall temperature (0) as discussed in Figures 4 and 5. Notice that 0 imply to the case that no magnetite nanoparticle or only water base luid. From Figure 8, it is suggested that the erroluid ( 0) provided higher (0) compared to the water. It is realistic due to the magnetite nanoparticles is very good in thermal conductivity as stated in Table 2. Blending this nanoparticle with water was enhanced the based luid characteristic. It also had increase the luid density and viscosity which relects to a reducing in luid velocity as well as the skin riction coeicient as shown in Figure 9. Thus, erroluid can be concluded had a low skin riction coeicient compared to its based luid. Meanwhile, the value o C Re is decreases as M increases in Figure 9. The increase o M on a surace had attracted the magnetite particles in nanoluid which then reduced the luid velocity. This relects to the low in skin riction coeicient and the rise o temperature. Fig. 8. Eect M and on variation o 0.5 and 1. 0 when Pr 6.2, 24

9 Fig. 9. Eect M and on variation o Pr 6.2, 0.5 and 1. C Re when Lastly, the variation o C Re with various values o and are presented in Figure 10. Generally, the increase o results to the increase in skin riction coeicient. For 1, it is suggested that the value o C Re is negative due to the opposing direction as 1 (reer to Eq. 9). The presence o slip eect 0.5 has dominant the eects o this small stretching value and enhanced the value o C Re 1/2. As 1, the skin riction becomes zero and the eect o slip is negligible at this stage as the relations as described in boundary conditions (9). Further, or 1, it is ound that the presence o slip eect has opposed the stretching eects which then reduced the skin riction coeicient. Physically, the increase o stretching velocity over ambient velocity had rise the luid low velocity. Surprisingly, slip eect then minimize the speed dierence between luid low and surace which reduced the value o the C Re 1/2. From this igure, it is suggested that suicient strength o slip eect may eliminated the stretching eects. Fig. 10. Eect and on variation o Pr 6.2, M 1, 0.1 and 1. C Re when 25

10 4. Conclusions As a conclusion, the increase o magnetic parameter, velocity slip parameter and nanoparticles volume raction results to a decrease o temperature proiles as well as the thermal boundary layer thicknesses while stretching parameter does contrary. The presence o slip eect has reduced the velocity boundary layer thickness. Net, it is ound that the magnetite (Fe3O4) erroluid provided higher wall temperature and heat transer capabilities compared to water. The magnetite (Fe3O4) erroluid also results the low skin riction coeicient which physically reduce chances o erosion thus etending lie-time o the surace components. Lastly, it is worth to conclude that the presence o slip eect had opposed the stretching eect capabilities thus minimize the skin riction coeicient. Suicient strength o slip eect may eliminated the stretching eects. Acknowledgement Authors thanks DRB-HICOM University o Automotive Malaysia and Universiti Malaysia Pahang or the inancial and acilities supports. Reerences [1] Stephen, Papell Solomon. "Low viscosity magnetic luid obtained by the colloidal suspension o magnetic particles." U.S. Patent 3,215,572, issued November 2, [2] Zeeshan, A., A. Majeed, and R. Ellahi. "Eect o magnetic dipole on viscous erro-luid past a stretching surace with thermal radiation." Journal o Molecular liquids 215 (2016): [3] Rashad, A. M. "Impact o anisotropic slip on transient three dimensional MHD low o erroluid over an inclined radiate stretching surace." Journal o the Egyptian Mathematical Society 25, no. 2 (2017): [4] Crane, Lawrence J. "Flow past a stretching plate." Zeitschrit ür angewandte Mathematik und Physik ZAMP 21, no. 4 (1970): [5] Tiwari, Raj Kamal, and Manab Kumar Das. "Heat transer augmentation in a two-sided lid-driven dierentially heated square cavity utilizing nanoluids." International Journal o Heat and Mass Transer 50, no (2007): [6] Zokri, Syazwani Mohd, Nur Syamilah Ariin, Muhammad Khairul Anuar Mohamed, Mohd Zuki Salleh, Abdul Rahman Mohd Kasim, and Nurul Farahain Mohammad. "Inluence o radiation and viscous dissipation on magnetohydrodynamic Jerey luid over a stretching sheet with convective boundary conditions." Malaysian Journal o Fundamental and Applied Sciences 13, no. 3 (2017). [7] Kho, Y. B., A. Hussanan, M. K. A. Mohamed, N. M. Sari, Z. Ismail, and M. Z. Salleh. "Thermal radiation eect on MHD Flow and heat transer analysis o Williamson nanoluid past over a stretching sheet with constant wall temperature." In Journal o Physics: Conerence Series, vol. 890, no. 1, p IOP Publishing, [8] Mohamed, Muhammad Khairul Anuar, Nor Aida Zuraimi Md Noar, Zulkhibri Ismail, Abdul Rahman Mohd Kasim, Norhaizah Md Sari, Mohd Zuki Salleh, and Anuar Ishak. "Slip eect on stagnation point low past a stretching surace with the presence o heat generation/absorption and Newtonian heating." In AIP Conerence Proceedings, vol. 1867, no. 1, p AIP Publishing, [9] Ramli, Norshaira, Syakila Ahmad, and Ioan Pop. "Slip eects on MHD low and heat transer o erroluids over a moving lat plate." In AIP Conerence Proceedings, vol. 1870, no. 1, p AIP Publishing, [10] Sheikholeslami, Mohsen. "CuO-water nanoluid low due to magnetic ield inside a porous media considering Brownian motion." Journal o Molecular Liquids 249 (2018): [11] Jusoh, Rahimah, Roslinda Nazar, and Ioan Pop. "Magnetohydrodynamic rotating low and heat transer o erroluid due to an eponentially permeable stretching/shrinking sheet." Journal o Magnetism and Magnetic Materials 465 (2018): [12] Hussanan, Abid, Mohd Zuki Salleh, and Ilyas Khan. "Microstructure and inertial characteristics o a magnetite erroluid over a stretching/shrinking sheet using eective thermal conductivity model." Journal o Molecular Liquids 255 (2018): [13] Hussanan, Abid, Iran Ahmed, and Mohd Zuki Salleh. "Mathematical analysis o erroparticles suspended Casson blood low in vessels under eternal magnetic ield." Biomath Communications Supplement 5, no. 1 (2018). 26

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