On the Study of Magnetohydrodynamic Squeezing Flow of Nanofluid between Two Parallel Plates Embedded in a Porous Medium

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1 Journal o Computational Engineering and Physical Modeling -4 (8) -5 Contents lists available at CEPM Journal o Computational Engineering and Physical Modeling Journal homepage: On the Study o Magnetohydrodynamic Squeezing Flow o Nanoluid between Two Parallel Plates Embedded in a Porous Medium M.G. Sobamowo, L.O. Jayesimi, and M.A. Waheed 3. Department o Mechanical Engineering, University o Lagos, Akoka, Lagos, Nigeria. Works and Physical Planning Department, University o Lagos, Akoka, Lagos, Nigeria 3. Department o Mechanical Engineering, Federal University o Agriculture, Abeokuta, Nigeria Corresponding author: @gmail.com ARTICLE INFO Article history: Received: Revised: Accepted: Keywords: Squeezing low, Nanoluid, Magnetic ield, Parallel plates, Dierential Transormation Method. ABSTRACT A study o magnetohydrodynamic squeezing low o nanoluid between two parallel plates embedded in a porous medium is presented in this work. The ordinary dierential equation which is transormed rom the developed governing partial dierential equations is solved using dierential transormation method. The accuracy o the results o the approximate analytical method are established as they agree very well with the results numerical method using ourthith order Runge-Kutta-Fehlberg method. Using the developed analytical solutions, the parametric studies reveal that when the velocity o the low increases during the squeezing process, the Hartmann and squeezing numbers decrease while during the separation process, the velocity o the luid increases with increase in Hartmann and squeezing numbers. Also, the velocity o the nanoluids urther decreases as the Hartmann number increases when the plates move apart. However, it is revealed that increase in nanotube concentration leads to an increase in the velocity o the low during the squeezing low. The present study will be useul in various industrial, biological and engineering applications. CEPM is an open access journal under the CC BY license (

2 M. G. Sobamowo et al./ Journal o Computational Engineering and Physical Modeling -4 (8) -5. Introduction The various industrial, biological and engineering applications o low o squeezing low o luid between parallel plates have been the impetus or the continued interest and generation renewed interests on the subject. Also, the continuous technological developments have shown the various industrial, biological and engineering applications etc. These various applications coupled with the practical signiicance o the phenomena have in recent times made the process an increasing area o an active research ields in luid dynamics. Although, the pioneer work and the basic ormulations on the low phenomena was carried out Stean [], the analysis o the squeezing low process has continued to receive tremendous attention in the past ew decades. However, the applications o Reynolds equation or the squeezing low analysis in earlier studies [-3] led the studies to insuicient and inaccurate analyses as shown by Jackson [4] and Usha and Sridharan [5]. Consequently, urther studies have been presented in recent times to give a better insight into the low phenomena. Among these study, Usha and Sridharan [5] investigated the arbitrary squeezing low o a viscous luid between elliptic plates while in an earlier work, Yang [6] considered unsteady laminar boundary layers in an incompressible stagnation low. In a group o research studies, Kuzma [7], Tichy and Winer [8] and Grimm [9] examined the eects o luid inertial on squeezing low. Moreover, in order to gain better insight into the low process, urther works has been done [5-5]. Analyses o unsteady squeezing low o Casson and viscous luids between two plates have been carried out by Khan [5] and Rashidi et al. [6], respectively. Eects o heat transer on the squeezing low characteristics o viscous luid was investigated by Duwairi et al. [7] while Qayyum et al. [8] analyzed the squeezing low pattern o second grade and micropolar luids. The squeezing low behaviour o dusty luids was examined by Hamdam and Baron [9]. The low heat transer o viscous luid on a porous surace o a squeezing low problem was presented by Mahmood []. Using dierential transormation method, approximate analytical solutions were developed by Hatami and Jing [] to study the squeezing low o Newtonian and non-newtonian nanoluids. An extended work on heat and mass transer o a rotating squeezing low o nanoluid was submitted by Mohyud-Din et al. []. In another work [3], the authors scrutinized the squeezing low o Casson luid under the impacts o eects o thermal radiation. Qayyum and Khan [4] investigated the squeezing problem immersed in a porous medium while Qayyum et al. [5] studied the inluence o slip on the unsteady axisymmetric squeezing low o viscous luid through a porous medium channel. A study on heat and mass transer in the unsteady squeezing low between parallel plates by Mustaa et al. [6]. In dierent studies, the eects o magnetic ield on the steady and unsteady squeezing low o dierent Newtonian and Non-Newtonian luids have been examined by Siddiqui et al. [7], Domairry and Aziz [8], Acharya et al. [9], Ahmed et al. [3], Ahmed et al. [3], Khan et al. [3, 33], Hayat et al. [34], Khan et al. [35], Ullah et al. [36] etc. In a recent study, combined eects o slip and magnetic ield on the squeezing low problem was studied by Sobamowo and Jayesimi [37] using Chebychev spectral collocation method. In a previous study, methods such as regular and singular perturbation and dierential transormation has been used by Sobamowo and Akinshilo [38] and Sobamowo [39] to analyze such low problem under the magnetic ield. Eects o nanoparticle shapes and Ferro-magnetic magnetic ield on peristaltic low o Copper-water by

3 M. G. Sobamowo et al./ Journal o Computational Engineering and Physical Modeling -4 (8) -5 3 Akbar and Butt [4] while Sheikholesmi and Bhatti [4] examined the impacts o magnetic ield and shape o nanoparticles on the heat transer characteristic o nanoluid. Further studies on the eects o magnetic ields, thermal radiation, nanoparticles, chemical reactions etc. are presented in [4-5]. In the past and recent studies, dierent numerical and analytical approximate methods have been adopted to analyze the nonlinear problems o the low processes. In the present study, dierential transormation method is used to analyze the magnetohydrodynamic squeezing low o nanoluid between two parallel plates embedded in a porous medium. Parametric studies are carried out using the approximate analytical solutions.. Description o the problem and model development Fig. shows an unsteady two-dimensional squeezing low o nanoluid between two parallel plates placed at time-variant distance and under the inluence o magnetic ield. In such low problem as presented in the igure, it is assumed that the low o the nanoluid is laminar, stable, incompressible, isothermal, non-reacting chemically, the nanoparticles and base luid are in thermal equilibrium and the physical properties are constant. The luid conducts electrical energy as it lows unsteadily under magnetic orce ield. The luid structure is everywhere in thermodynamic equilibrium and the plate is maintained at constant temperature. Fig.. Model diagram o the low process. Following the assumptions, the governing equations or the low are given as [37-39] u x u y () u u u p u u n u n u v n B ou t x y x x y K p () n v v v v p v v n u v n t x y y x y K p (3) where n s (4-a)

4 4 M. G. Sobamowo et al./ Journal o Computational Engineering and Physical Modeling -4 (8) -5 n.5 (Brinkman model) (4-b) and the magnetic ield parameter is given as B Bt t (5) Under the assumption o no-slip condition, the prevailing boundary conditions are stated as y h( t) H at, u, v Vw, y, u, y v x, u (6) Table and present the thermal-physical properties o the base luid and the nanoparticles, respectively Table. Thermal-physical properties o the base luid [4-4, 5-5]. Base luid ρ (kg/m 3 ) c p (J/kgK) k (W/mK) Pure water Engine oil Kerosene Ethylene Glycol Table. Thermal-physical properties o nanoparticles [4-4, 5-5]. Nanoparticles ρ (kg/m 3 ) c p (J/kgK) k (W/mK) Copper (Cu) Silver (Ag) SWCNTs Aluminum oxide (Al O 3 ) Copper (II) Oxide (CuO) Titanium dioxide (TiO ) Introducing the ollowing dimensionless and similarity variables into Eq. () - (3). x x y u, t, v, t, t t H t HV H K Re SA A n w s n v H.5 p, S,,, (7) one arrives at iv Re 3 M (8)

5 M. G. Sobamowo et al./ Journal o Computational Engineering and Physical Modeling -4 (8) -5 5 Alternatively, Eq. (8) can also be expressed as.5 iv SA 3 M (9) and the boundary conditions are,, () The boundary conditions depict the condition o no-slip on the disk. 3. Analysis o the dierential equation using dierential transorm method The developed nonlinear equation in Eq. (9) cannot be solved exactly and analytically. However, we apply an approximated analytical method such as dierential transormation method (DTM) as introduced by Ζhou [53] to solve the equation. The basic deinitions, the properties and some applications o the method can be ound in [39, 54-58] Using the DTM operational properties as stated in [39, 53-58], the dierential transorm o Eq. (8) is m m m 3 m 4) F m 4 3m m F m m l m l m l m l 3 F m l 3 l Re m M m m F m F m l l l l 3 F l 3 l m m l F m l l l F l l where l l l The boundary conditions are l F, m m F m m, F() m m m F m, m, F 3 k (6) l F m, m, F k, (5)

6 6 M. G. Sobamowo et al./ Journal o Computational Engineering and Physical Modeling -4 (8) -5 Thereore, we have the ollowing boundary conditions in DTM domain, (),, 3 F F F k F k (7) where k and k are the constants which will be determined through the boundary conditions M m m F m m Re l m l m l m l 3 F m l 3 l F m 4 3Re m m F m m m m 3 m 4) m Re F m l l l l 3 F l 3 l m Re m l F m l l l F l l Using m=,,, 3 in the above recursive relations, one arrives at F 4 F 5 M k 3Rek F 6 F M M k Rek Re k Rek Re k k 84 F(8) M M k 3Rek M 3 F 9 9Re k Rek 6Re kk Re k 8Re k 54Re kk 48kk Re 48Rekk 4Re kk F() (8) (9) Following rom above and according to the deinition o DTM, one can write the solution o Eq. (8) as 3 5 M M k 3Rek 7 3 k k M k Rek 84 9Re k Rek 6Re kk 3 M M M k 3Rek 9 k Rek 6kkR e Re k 8Re k 54Re kk 48kkRe 48Rekk 4Re kk... ()

7 M. G. Sobamowo et al./ Journal o Computational Engineering and Physical Modeling -4 (8) -5 7 The unknown values o constants k and k can be ound using Eq. (5) which states that at,. Thereore, according to the boundary conditions, we have M M k 3Rek k k M k 3Rek 84 9Re k Rek 6Re kk M M M k 3Rek 9 k Rek 6k k Re Re k 8Re k 54Re kk 48kk Re 48Rekk 4Re kk... 3 M M k 3Rek ' k 3k M k 3Rek 4 9Re k Rek 6Re kk 3 M M M k 3Rek 9 k Rek 6kk Re Re k 8Re k 54Re kk 48kk Re 48Rekk 4Re kk... () It should be noted that when Eq. () and () are solved, dierent values or k and k are gotten or respective values o α and Re. In order to ind the skin riction, the second-order derivatives o () at the wall is developed as 3 M M k 3Rek '' 6k M k 3Rek 9Re k Rek 6Re kk M M M k 3Rek 9 k Rek 6k k Re 84 7Re 3 k 3 8Re k 54Re kk 48kkRe 48Rekk 4Re kk... 3 It should be noted that Re SA.5 Judging rom physical point o view, the skin riction is an important physical quantity o interest in the low analysis o luid. The skin riction can be expressed as () (3)

8 8 M. G. Sobamowo et al./ Journal o Computational Engineering and Physical Modeling -4 (8) -5 C u n y V n yh() t w Using the dimensionless variables in Eq. (7), we developed a non-dimensional orm o Eq. (4) as H C A x t Re C which gives, C A.5 x.5 M M k 3Rek 6k M k 3Rek 9Re k Rek 6Re kk M M M k 3Rek 9 k Rek 6k k Re Re k 8Re k 54Re kk 48kk Re 48Rekk 4Re kk 4. Results and Discussion 3... In order to establish the accuracy o the results o the applied approximate analytical method, the results are compared with the results o numerical method (NM) using ourth-ith order Runge- Kutta-Fehlberg method as presented in Tables 3-5. The table shows the comparisons o results o DTM and NM or dierent values o permeation Reynolds and Hartmann numbers. Also, the Tables presented the various impacts o controlling low parameters on the squeezing low process. It is shown that during the separation low, the velocity o the low increases while the skin riction coeicient decreases. However, in the squeezing low process, increase in the squeezing and Hartmann numbers cause the skin riction coeicient to decrease. Table 3. Results o NM and DTM or large squeezing number in the absence o magnetic ield. Squeezing S= M=, /= η NM DTM (4) (5) (6)

9 M. G. Sobamowo et al./ Journal o Computational Engineering and Physical Modeling -4 (8) -5 9 Table 4. Results o NM and DTM o skin riction parameter or large separation number under the inluence o magnetic ield. S M NM DTM Table 5. Results o NM and DTM or small squeezing number in the absence o magnetic ield. Squeezing S =.5, M=, /= Squeezing S =.5, M=, /= η NM DTM NM DTM Using nanoparticle parameter value o.5 i.e. ϕ =.5, Figs. and 3 show the variation o velocities o low o the luid over the length. As depicted in the igure, when the axial velocity o luid low near the wall region decreases, there is an increase in velocity gradient at the wall region. This behaviour occurs because o the conservativeness o the mass low rate ().5 l () Fig. a. Variation o (η) with the low length Fig. b. Variation o (η) with the low length Impact o magnetic ield parameter on the low behaviour o the luid is shown in Fig. 3. As the Hartmann number increases, the low velocity decreases in the range o η.5 and then increases in the range.5 < η. This is due to the act that the magnetic ield created a retarding orce, Lorentz orce created which decreases the motion o the luid at boundary layer

10 M. G. Sobamowo et al./ Journal o Computational Engineering and Physical Modeling -4 (8) -5 during the squeezing low process. During the separation o the plates, as the magnetic ield parameter increases, the low velocity o the luid urther decreases. Such a behaviour is caused by the act that the luid low with high velocity to ill a vacant space that occurs such that the law o conservation o mass is not violated. Fig. 4 displays the inluence o rcy number on the squeezing low pattern o the nanoluid. As it is shown in the igure, there is an opposite trend to that o the squeezing number eects on the low process. The igure shows that increase in the rcy number causes the low velocity to increase in the range o η.5 and a decrease is witness in the range.5 < η..5 M = M = 3 M = 5 M = 7 l () Fig. 3. Eects o magnetic number on the low velocity o the luid..5 = 8. = 6. = 4. =. l () Fig. 4. Eects o rcy number on the low velocity o the luid. As it is illustrated in Fig. 5, the eects o squeezing number on the low velocity is shown. As the squeezing number increases, there is a decrease in low velocity in the range o η.5 and then increase in the low velocity in the range.5 < η. Fig. 6 shows that impacts o nanoparticle raction on the luid velocity. It is shown that as the nanoparticle raction increases, the velocity decreases in the range o η.5 and then increases in the range.5 < η. This

11 M. G. Sobamowo et al./ Journal o Computational Engineering and Physical Modeling -4 (8) -5 is because increase in the nanoparticle raction leads to an increased more collisions between nanoparticle and particles at the boundary surace o the plates. Consequently, a retardation in the low process occurs which decreases the low velocity near the boundary layer..5 S = S = 4 S = 7 S = l () Fig. 5. Eects o Squeezing number on the low velocity o the luid..5 =. =.5 =. =.5 l () Fig. 6. Eects o nanoparticle raction on the low velocity o the luid. Table 6. Skin riction or dierent squeezing, Hartmann and rcy numbers. C S M

12 M. G. Sobamowo et al./ Journal o Computational Engineering and Physical Modeling -4 (8) -5 Numerical values or skin riction coeicient are presented in Table 6. The Table also presented the eects o squeezing (S), magnetic parameters (M) and rcy () on the skin coeicient. It is shown that the numerical value o the skin riction coeicient increases as the squeezing (S) and Hartmann (magnetic ield, M) numbers increase, the skin-riction coeicient also increases while the skin-riction coeicient decreases as the rcy number increases. 5. Conclusion In this work, analysis o magnetohydrodynamic squeezing low o nanoluid between two parallel plates embedded in a porous medium have been presented using dierential transormation method. The accuracy o the results o the approximate analytical method was established numerically using ourth-ith order Runge-Kutta-Fehlberg method. Parametric studies were carried out which established the impacts o the controlling low parameter on the low process. The present study will be useul in various industrial, biological and engineering applications. Nomenclature B(t) H Kp M P py Re U V Vw x y Greek Symbol τ τo μ Φ Magnetic ield strength rcy number Squeezing low height Permeability Hartmann parameter Pressure yield stress o the luid. Reynold number Squeezing low Parameter velocity in x direction velocity in y Direction Dimensionless velocity in y direction injection/suction velocity horizontal axis o low Perpendicular axis to the low Eective thermal conductivity Eective dynamic viscosity Eective density Dimensionless similarity variable shear stress Casson yield stress dynamic viscosity shear rate raction o nanoparticle in nanoluid

13 M. G. Sobamowo et al./ Journal o Computational Engineering and Physical Modeling -4 (8) -5 3 Reerences [] M. J. Stean. Versuch Uber die scheinbare adhesion, Sitzungsberichte der Akademie der Wissenschaten in Wien. Mathematik-Naturwissen 69, 73 7, 874. [] O. Reynolds. On the theory o lubrication and its application to Mr Beauchamp Tower s experiments, including an experimental determination o the viscosity o olive oil. Philos. Trans. Royal Soc. London 77, 57 34, 886. [3] F. R. Archibald, F.R., 956. Load capacity and time relations or squeeze ilms. J. Lubr. Technol. 78, A3 A45. [4] J. D. Jackson. A study o squeezing low. Appl. Sci. Res. A, 48 5, 96. [5] R. Usha and R. Sridharan, R. Arbitrary squeezing o a viscous luid between elliptic plates. Fluid Dyn. Res. 8, 35 5, 996. [6] Wole, W.A., 965. Squeeze ilm pressures. Appl. Sci. Res. 4, Yang, K.T., 958. Unsteady laminar boundary layers in an incom- pressible stagnation low. J. Appl. Math. Trans. ASME 8, [7] D. C. Kuzma. Fluid inertia eects in squeeze ilms. Appl. Sci. Res. 8, 5, 968. [8] J. A. Tichy, W. O. Winer. Inertial considerations in parallel circular squeeze ilm bearings. J. Lubr. Technol. 9, , 97. [9] R. J. Grimm. Squeezing lows o Newtonian liquid ilms: an analysis include the luid inertia. Appl. Sci. Res. 3 (), 49 66, 976. [] G. Birkho. Hydrodynamics, a Study in Logic, Fact and Similitude, Revised ed. Princeton University Press, 37, 96. [] C. Y. Wang. The squeezing o luid between two plates. J. Appl. Mech. 43 (4), , 976. [] C. Y. Wang, L. T. Watson. Squeezing o a viscous luid between elliptic plates. Appl. Sci. Res. 35, 95 7, 979. [3] M. H. Hamdan and R. M. Baron. Analysis o the squeezing low o dusty luids. Appl. Sci. Res. 49, , 99. [4] P. T. Nhan. Squeeze low o a viscoelastic solid. J. Non-Newtonian Fluid Mech. 95, ,. [5] U. Khan, N. Ahmed, S. I. U. Khan, B. Saima, S. T. Mohyud-din. Unsteady Squeezing low o Casson luid between parallel plates. World J. Model. Simul. (4), 38 39, 4. [6] M.M. Rashidi, H. Shahmohamadi and S. Dinarvand, Analytic approximate solutions or unsteady two dimensional and axisymmetric squeezing lows between parallel plates, Mathematical Problems in Engineering, Vol. (8), pp. -3, 8. [7] H.M. Duwairi, B. Tashtoush and R.A. Domesh, On heat transer eects o a viscous luid squeezed and extruded between parallel plates, Heat Mass Transer, vol. (4), pp.-7, 4. [8] A. Qayyum, M. Awais, A. Alsaedi and T.Hayat, Squeezing low o non-newtonian second grade luids and micro polar models, Chinese Physics Letters, vol. (9), 347, [9] M.H Hamdam and R.M. Baron, Analysis o squeezing low o dusty luids, Applied Science Research, vol. (49), pp , 99. [] M.Mahmood,S.Assghar and M.A. Hossain, Squeezed low and heat transer over a porous surace or viscous luid, Heat and mass Transer, vol.(44), 65-73, 7. [] M. Hatami and D.Jing, Dierential Transormation Method or Newtonian and non-newtonian nanoluids low analysis: Compared to numerical solution, Alexandria Engineering Journal, vol. (55), [] S. T. Mohyud-Din, Z. A. Zaidi, U. Khan, N. Ahmed. On heat and mass transer analysis or the low o a nanoluid between rotating parallel plates, Aerospace Science and Technology, 46, 54-5, 4.

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