Squeezing Unsteady MHD Cu-water Nanofluid Flow Between Two Parallel Plates in Porous Medium with Suction/Injection

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1 Computational and Applied Mathematics Journal 018; 4(: ISSN: (Print; ISSN: (Online Squeezing Unsteady MHD Cu-water Nanoluid Flow Between Two Parallel Plates in Porous Medium with Suction/Injection Alok Kumar Pandey *, Manoj Kumar Department o Mathematics, Statistics and Computer Science, Govind Ballabh Pant University o Agriculture and Technology, Pantnagar, India address mr.alokpandey1@gmail.com (A. K. Pandey, mnj_kumar004@yahoo.com (M. Kumar * Corresponding author Citation Alok Kumar Pandey, Manoj Kumar. Squeezing Unsteady MHD Cu-water Nanoluid Flow Between Two Parallel Plates in Porous Medium with Suction/Injection. Computational and Applied Mathematics Journal. Vol. 4, No., 018, pp Received: January 11, 018; Accepted: January 9, 018; Published: March, 018 Abstract: In this article the iluence o suction/injection on low and heat transer in squeezing unsteady magnetohydrodynamics low between parallel plates in porous medium in the presence o thermal radiation or Cu-water nanoluid has been analyzed. The radiative heat lux is used to portray energy equation by using Rosseland approximation. The set o altered ODEs with appropriate boundary conditions have been solved numerically by applying shooting method along with Runge- Kutta-Fehlberg 4-5 th order o integration technique. The iluences o relatable parameters on dimensionless low ield and thermal ield have been shown in graphs and tabular orm. The results elucidate that heat transer coeicient decreases as increasing in thermal radiation parameter while the absolute values o coeicient o skin riction enhances with ampliy in magnetic ield parameter. The outcomes also declared that as enhance in the values o suction/injection parameter both the velocity and temperature proiles regularly decline. Keywords: MHD, Nanoluid, Porous Medium, Suction/Injection, Thermal Radiation 1. Introduction Analysis o heat and mass transer enhanced during last ew decade due to large application in several branches o science and engineering. Evaporation water rom tarn to the environment, blood sanitization inside the kidneys and liver are ew applications o mass transer, and heat transer entail in the ield o condensers and evaporators. The heat and mass transer rate entail in unsteady squeezing glutinous low ield has lot o use in lubrication method, chemical dispensation equipment, polymer dispensation, spoil o crops due to rosty, og ormation and dispersion. Azimin and Riazi [1] analyzed the impact o heat transer between two analogous disks or GO-water nanoluid. They ound that volume concentration o nanoparticle increases on increasing the values o Brownian number. Aziz et al. [] have discussed the eect o ree convection on nanoluid past a smooth plane plate implanted in porous medium and in the happening o gyrotactic microorganisms. They ound that on increasing the value o bio-convection parameters Nusselt number, rate o mass transer and motile density parameter enhanced while decreases on increasing the values o buoyancy parameter Nr. Das et al. [3] have examined the iluence o entropy exploration on MHD low during a vertical porous channel via convective heat source or nanoluid. Domairry and Hatami [4] have discussed numerical investigation o Squeezing lowthrough similar plates with Cu water nanoluid. They projected that on raising the values o volume raction o solid particle, there is no change in velocity boundary layer depth. Fakour et al. [5] deliberated the eect o magnetohydrodynamic and heat conduction o a nanoluids low through a channel with porous walls. Grosan et al. [6] considered the ree convection eect o heat transer within a square cavity packed through a porous medium in nanoluids. Gupta and Ray [7] have studied numerical analysis o the squeezing nanoluid low among two analogous plates. They ound that as increase in Prandtl number and Eckert number, temperature o the nanoparticle increases. Jha et al. [8] analyzed the eect o natural convection low inside an upright

2 3 Alok Kumar Pandey and Manoj Kumar: Squeezing Unsteady MHD Cu-water Nanoluid Flow Between Two Parallel Plates in Porous Medium with Suction/Injection similar plate. They ound that on increasing in the outcome o rareaction and luid wall interaction, volume low rate increases, while it reduces on enhancing the values o Hartmann number. Khalid et al. [9] purposed the ree convection eect o MHD low o past above an oscillating perpendicular plate entrenched in a porous medium or casson luid. Kuznetsov and Nield [10] investigated the impact o natural convective low inside a porous medium in nanoluids or Cheng-Minkowycz problem or: A revised model. They ound that decreased Nusselt number is ree o the values o Brownian number when on the boundary zero nanoparticle lux imposed. Mustaa et al. [11] have introduced the impact o heat and mass rate within squeezing unsteady nanoluid low along with corresponding plates. Pourmehran et al. [1] analyzed the squeezing unsteady low through two iinite parallel plates or nanoluid by analytical methods. They scrutinized that on increasing the values Nusselt number volume raction o nanoparticle enhances, while on enhancing the values o Eckert number, the values o squeeze number decreases. Sheikholeslami and Ganji [13] discussed the eect o nanoluids low and rate o heat transer between analogous plates in the presence o Brownian motion using dierential transormation method. They explained that Nusselt number is escalating with volume raction o nanoparticle while Hartmann number and heat source parameter are decreasing unction with squeeze number. Wubshet and Shankar [14] premeditated the iluence o MHD as well as heat transer over a stretching surace in the existence o thermal radiation and slip conditions with nanoluid. They ound that on enhancing the values o magnetic parameter velocity o nanoluid reduces and temperature o the surace decreases on enhancing the values o Prandtl number. Lately, Pandey and Kumar [15] have proposed the iluence o suction/injection and slip on MHD nanoluid low due to a porous wedge in continuation o viscous dissipation and convective boundary conditions. Once again, Pandey and Kumar [16] demonstrated the iluence o natural convection and thermal radiation on nanoluid low due to a stretching cylinder in the occurrence o porous medium, viscous dissipation and slip boundary conditions. Hatami et al. [17] have been applied the optimal collection method (OCM to obtained the results o orced convective Al O 3 -nanoluid low over a plate vertical plate by utilized variable magnetic ield. The control volume inite element method is taken by [18] to ind the outcomes to MHD orced convective Fe 3 O 4 -water nanoluid low in a semi annulus enclosure. Mahmoudi et al. [19] have used Lattice Boltzmann method (LBM is employed to ound the results to heat transer and nanoluid low in an open cavity. MHD low and heat transer analysis on nanoluid low between parallel plates with Duan-Rach approach (DRA was studied by [0]. The dierent approaches have been applied to acquire the results to nanoluid low over a stretching/shrinking surace [1-4]. The nanoluid low and heat transer in a microchannel is investigated by Karimipour et al. [5]. Behrangzade and Heyhat [6] examined the thermo and hydrodynamic eatures o Ag-water nanoluid by designing an experimental rig. Hayat et al. [7] have been employed homotopy analysis method (HAM to ind the solutions to nanoluid low due to convectively heated Riga plate. Ahmadi et al. [8] considered iluence o heat transer nanoluid low due to a stretching lat plate using Dierential Transormation Method (DTM. Recently, MHD micropolar nanoluid low between parallel plates was studied by Rashidi et al. [9]. Sheikholeslami and Ganji [30] have analyzed the heat transer iluence o Cuwater nanoluid low between two parallel plates utilizing homotopy perturbation method (HPM. Later on, the studied o heat transer analysis o steady nanoluid low between parallel plates is analyzed with the help o DTM is given by [31]. Ganga et al. [3] have utilized the homotopy analysis method to examine the iluence o heat generation/absorption and viscous-ohmic dissipation on nanoluid low over a vertical plate. Zin et al. [33] have investigated the iluence o MHD ree convective low o nanoluid over an oscillating vertical plate due to porous medium. They proposed the Laplace transorm technique to solve the principal equations. The implicit inite dierence method has been used to study the iluence o heat generation and natural convection in an inclined porous triangular enclosure illed with nanoluid by Mansour and Ahmed [34]. The aspire o the current analysis is to study the impact o thermal radiation and suction/injection on squeezing unsteady MHD Cu-water nanoluids low between analogous plates in the existence o porous medium. The equations that obtained have been cracked with the help o shooting method via 4-5 th Runge-Kutta-Fehlberg technique. The impact o dierent objective parameters on the velocity and temperature are depicted graphically and analyzed.. Mathematical Formulation Assumed a two dimensional unsteady squeezing MHD nanoluid low between two analogous plates in porous medium, in the existence o the thermal radiation and suction/injection. An oblique magnetic ield with variable strength B 0 acts normal to the plates. The space between two plates is z l ( 1 αt 0.5 h( t = ± = ±. For α > 0, the both plate 1 are squashed until they contact t = and or α < 0, the two α plate are take apart. The eect o viscous dissipation is also considered and it occurs at large values o Eckert number Ec >> 1. Copper (Cu nanoluid is based on regular luid (water with assertions that incompressible, no chemical reaction, solid particle (Cu and the regular luid (water are in equilibrium. Motion o the nanoluid considered as laminar and stable. The graphical mold o the physical model has been speciied along with low pattern and coordinate system in Figure 1.

3 Computational and Applied Mathematics Journal 018; 4(: y l(1 αt 0.5 x Figure 1. Flow coiguration and coordinate system. The essential equations o mass, momentum and heat transer or nanoluid are expressed as [4]: D u v + = 0 x y u u u p u u νρ ρ + u + v = + µ + u σb 0u t x y x x y k1 v v v p v v νρ ρ + u + v = + µ + v σb 0v t x y y x y k1 (1 ( (3 T T T k T T u u v + u + v = t x y x y x µ 4 ( ρc p x y ( ρcp ( ρc Using Rosseland s approximation (see in Re. [14], The heat lux or radiation deined as: q r 4 = 3k σ 4 T y 1 r p q y (4 where ( σ, k locates the Stean Boltzmann constant and coeicient o mean absorption respectively. We suppose that the temperature deviation inside the low is to acilitate 4 T extended in a Taylor s series and growing 4 T about T and deserting terms o higher order. We acquire: By means o equations (5 and (6, equation (4 turns into: T T T T (6 (5 T T T k T k T T + u + v = + + t x y x k k y 3 16σ 1 3 ( ρcp ( ρcp ( ρc µ u u v x y x p (7 The coupled boundary conditions are: dh u = 0, v = Vw ( x, t =, T TH dt = at y h( t u αl = = 0, v = y 1 αt 0.5 Now, wstand or suction parameter i ( w> 0 and or injection parameter ( w < 0 velocity components are ( u, v correspondingly. Further w T, = 0 y at y = 0. (8. Again, along x and y direction µ be the eective dynamic viscosity, ρ be the eective density, ( ρc p be the eective heat capacity and k be the eective thermal conductivity o nanoluid and written as [4, 35]:

4 34 Alok Kumar Pandey and Manoj Kumar: Squeezing Unsteady MHD Cu-water Nanoluid Flow Between Two Parallel Plates in Porous Medium with Suction/Injection ( C ( C ( C µ k ks k ϕ ( k k s,.5 ( 1 ϕ k ks + k + ϕ ( k ks ρ = 1 ϕ ρ + ϕρs, ρ p = 1 ϕ ρ p + ϕ ρ p, s + µ = = (9 where φ be the solid volume raction urthermore the subscripts and s denote the regular luid (water and solid particle, respectively. The thermophysical properties o regular luid (water and solid particle (Cu are displayed in Table 1. ρ (kg/m 3 Table 1. Thermo physical properties o Cu-water nanoluid as in [4]. C p (j/kg K k (W/m K Pure water Copper (Cu Equations (, (3 and (7 can be distorted into non-linear ODEs by using the ollowing similarity variables (see also Re. [4]: η = l y 1 1 αt, αx u = 1 ( αt '( η, v = 1 αt 1 αl ( η T, θ = (10 T H The altered ODEs are: 1.5 ( ϕ ( η ( γ '''' SA 1 ''' + 3 '' + ' '' ''' + M '' = 0 (11 A PrEc 1 Nr θ + '' + Pr S θ ' ηθ ' + '' + 4 δ ' = 0.5 A 3 A3 ( 1 ϕ (1 The boundary conditions (8 become: w θ θ 0 =, '' 0 = 0, ' 0 = 1at η = 0 1 = 1, ' 1 = 0, 1 = 1as η = 1 (13 where number, αl S = be the Squeeze number, ν 3 σ ( ρc ( 1 αt p 16T Nr = be the thermal radiation parameter, k k 0 3 ( αt σ B l 1 M = the magnetic parameter, ρ µ reerence length and the constants A 1, A and A 3 are deined as: A s = ( ϕ +, A ( ϕ 1 1 ρ ϕ ρ ρ αx µ ρcp Ec = be the Eckert number, Pr = be the Prandtl ρ k w 1 1/ ( αt 1 l ν 1 γ = be the porous medium parameter, k µ (1 αt Vw x l = be the suction/injection parameter, δ = be the αl x ( ρc ( C p s = + ϕ, A ρ p 3 k k + k ϕ k k = = k k + k + ϕ k k s s s s For the practical interest, the coeicient o skin riction C and Nusselt number Nu are expressed as:. (14 C u µ y = ρ vw y= h t T lk, y y= h( t Nu = kt H (15 Thereore, the reduced skin riction coeicient C and reduced Nusselt number Nu are written as:

5 Computational and Applied Mathematics Journal 018; 4(: l.5 C = = A 1(1 φ ''(1, x (1 αtre C x 3 Nu = 1 αtnu = A 1 + Nr θ'(1 (16 3. Numerical Method The non-dimensional momentum equation (11 and energy equation (1 jointly with assisting boundary conditions (13 have been tackled numerically through shooting procedure with RKF 4-5 th order o integration ormula. For this plan, we irst changed the obtained primary dierential equations in the orm o irst order ODEs. Let us consider y 1 = η, y =, y3 = ', y4 = '', y5 = ''', y6 = θ, y7 = θ'. Now the succeeding irst order system obtained: 1 y1' y3 y' y 4 y3' y5.5 y4' = SA1 ( 1 ϕ ( y1y5 + 3y4 + y3y4 yy5 + ( γ + M y4 y5' y 7 y6' A PrEc y Pr ( ' S y y y y y + δ y A 3 A3 ( 1 ϕ 1+ Nr (17 and subsequent initial conditions. y1 0 y w y 3 p 1 y4 = 0 y 5 p y6 p3 y 1 7 (18 The system o irst order ODEs (17 via initial conditions (18 is solved using order o ourth-ith RKF-integration process and appropriate values o unknown initial conditions p 1, p and p3are preerred and then numerical integration is applied. Here we contrast the computed values o ( η, '( η and θ ( η as η = 1, through the speciied boundary condition (1 1 =, '( 1 = 0 and θ 1 = 0, and regulate the expected values o p 1, p and p 3 to achieve a enhanced approximation or result. The uamiliar p 1, pand p 3 have been approximated by Newton s scheme such a way that boundary conditions well-matched at highest numerical values o η = 1 with error less than Results and Discussion The transormed equations (11 and (1 related to the assisting boundary conditions (13 are solved numerically by the aid o ourth-ith order o Runge-Kutta-Fehlberg approach via shooting procedure. In this study, we analyzed the iluence o numerous pertinent parameter such as magnetic ield parameter M, porous medium parameter γ, suction/injection parameter w, thermal radiation parameter Nr, and nanoparticle volume raction φ on '( η and θ( η. To authenticate the numerical results obtained, we have compared our results o ( η and θ( η or divergent values o similarity variable η with Domairry and Hatami [4] as revealed in Table and are ound with admirable agreement. Table. Comparison o the values o ( η and θ( η or Cu-water when S = 0.1, Pr = 6., δ = 0.1, φ = 0.01, 0.05 Domairry and Hatami [4]. η Domairry and Hatami [4] ( η θ( η Present Result ( η θ( η Ec = and Nr = M = γ = w = 0 with

6 36 Alok Kumar Pandey and Manoj Kumar: Squeezing Unsteady MHD Cu-water Nanoluid Flow Between Two Parallel Plates in Porous Medium with Suction/Injection η Domairry and Hatami [4] ( η θ( η Present Result ( η θ( η Furthermore, Table 3 depicts that the assessment o Nusselt number or the diverse values o Prandtl number Pr and Eckert number Ec in the non-existence o Nr = 0 = M = w = γ, which shows an terriic agreement. Table 3. Comparison o the values o Nusselt number θ' ( 1 or Copper (Cu when S = 0.5, 0.1, δ = Nr = M = γ = w = 0 and φ = 0. Pr Ec Mustaa et al. [11] Gupta and Ray [7] Pourmehran et al. [1] Present result The scenarios o leading physical parameters on the velocity and temperature proiles are depicted in Figures Figures and 3 exhibit that the impact o magnetic ield parameter M on velocity and temperature sketches or Nr = γ =, = 0.5, Pr = 6. and φ = 0.0. The variation w in velocity proiles due to several values o M is depicted in Figure, the graph cleared that velocity near the wall o range [ 0,0.5 reduces with increase in M. However, Table 4. Values o skin riction coeicient ''( 1 and Nusselt number θ' ( 1 Nr γ M w velocity accelerates in the region o 0.5 < η 1. Figure 3 clears that thermal ield decelerates close to the vicinity o the surace o plate, while ar away rom the wall o the plate it enhances. The values o low rate and Nusselt number are shown in Table 4 or special values o Nr, γ, M, w and φ. It is obvious rom Table 4 that as enlarge in the values o volume raction o nanoparticle, the absolute value o shear stress rate and coeicient o heat transer are reduced. or Copper (Cu when Ec = 0.01, Pr = 6., S = 1 and δ = ϕ ''( 1 θ' (

7 Computational and Applied Mathematics Journal 018; 4(: Figure. Variation in velocity (η due to several values o M. Figure 3. Variation in temperature θ(η due to several values o M. The low ield and thermal ield are aected by the porous medium parameter γ as looking at Figures 4 and 5, respectively. In Figure 4, initially momentum boundary layer width raise along with γ, ater then its behavior changed due to away rom the wall o plate. Moreover, in Figure 5 temperature curves o Cu-water nanoluid continuously depreciates with ampliied in the values o γ, in the region o 0 η 1 also we observed the behavior o skin riction and Nusselt number rom the Table 4, in other words we can say that the heat transer rate and magnitude o skin riction coeicient are ampliied with γ. The temperature proiles o nanoluid are iluenced by thermal radiation parameter Nr or the ixed values o governing parameters as such M = = γ, Pr = 6., w = 0.5, Ec = 0.01 = δ, S = 1 and φ = 0.0, is revealed in Figure 6. It is evident rom this curves that thermal ield requently decelerates with augmentation in Nr or every values o η. Moreover, thermal boundary layer width also diminishes as raised the radiation parameter. Furthermore, Table 4 indicates that the heat transer coeicient continuously decreases.

8 38 Alok Kumar Pandey and Manoj Kumar: Squeezing Unsteady MHD Cu-water Nanoluid Flow Between Two Parallel Plates in Porous Medium with Suction/Injection Figure 4. Variation in velocity (η due to several values o γ. Figure 5. Variation in temperature θ(η due to several values o γ.

9 Computational and Applied Mathematics Journal 018; 4(: Figure 6. Variation in temperature θ(η due to several values o Nr. The change in velocity and temperature graphs due to dimensionless suction/injection parameter ware exhibited in Figures 7 and 8, correspondingly. On ocusing Figure 7, we observed that low ield o nanoluid decreases as accelerate in the values o w, in the speciied domain [0,1]. The change in absolute value o skin riction ''( 0 and Nusselt number θ' ( 0 corresponding to suction/injection parameter is shown in Table 4. On looking at this table we noticed that the values o ''( 0 and θ' ( 0 are reduced with enhance in w. Similarly, Figure 8 shows that thermal ield reduces as increase in suction/blowing parameter in the range o 0 η 1. Figure 7. Variation in velocity (η due to several values o w.

10 40 Alok Kumar Pandey and Manoj Kumar: Squeezing Unsteady MHD Cu-water Nanoluid Flow Between Two Parallel Plates in Porous Medium with Suction/Injection Figure 8. Variation in temperature θ(η due to several values o w. The variation in momentum boundary layer and thermal boundary layer proiles versus similarity variable η are represented in Figures 9 and 10 or three dierent value o nanoparticle volume raction φ. It is noticeable in Figure 9 that on increasing in the values o volume o solid particle velocity boundary layer is decreased near the vicinity o the plates, while ar rom the surace o plates it increased. Figure 10 depicts that temperature o nanoparticle regularly declines with raise in volume o the solid particle, and due to this cause thickness o thermal boundary layer depreciates. According to Table 4 as increase in solid volume raction, the '' 0 is reduced, while heat transer rate is values o enhanced with elevate in volume raction φ. Figure 9. Variation in velocity (η due to several values o Φ.

11 Computational and Applied Mathematics Journal 018; 4(: Figure 10. Variation in temperature θ(η due to several values o Φ. 5. Conclusion The impact o various governing parameters like thermal radiation parameter Nr, magnetic parameter M, porous medium parameter γ, suction/injection parameter wand the nanoparticle volume action φ on low and heat transer o a squeezing unsteady nanoluid MHD low between analogous plates in porous medium in the occurrence o thermal radiation and suction/injection taking water like regular luid and Cu alike nanoluid particle was analyzed. The dimensionless velocity and temperature outlines have studied. It was concluded that on growing the value o suction/injection parameter w, both the velocity and temperature o nanoparticle signiicantly decreased, while temperature o nanoparticle decreases on mounting the values o the thermal radiation parameter. As increasing in magnetic parameter, the absolute value o skin riction '' 0 θ' 0 are augmented. The and Nusselt number coeicient o heat transer reduces on elevating the values o radiation parameter and volume concentration o nanoparticle. Reerences [1] M. Azimi, R. Riazi, Heat transer analysis o GO-water nanoluid low between two parallel disks, Prop. Power Res. 4 (1 ( [] A. Aziz, W. A. Khan, I. Pop, Free convection boundary layer low past a horizontal lat plate embedded in porous medium illed by nanoluid containing gyrotactic microorganisms, Int. J. Therm. Sci. 56 ( [3] S. Das, A. S. Banu, R. N. Jana, O. D. Makinde, Entropy analysis on MHD pseudo-plastic nanoluid low through a vertical porous channel with convective heating, Alexandria Eng. J. 54 (3 ( [4] G. Domairry, M. Hatami, Squeezing Cu water nanoluid low analysis between parallel plates by DTM-Padé Method, J. Mol. Liq. 193 ( [5] M. Fakour, D. D. Ganji, M. Abbasi, Scrutiny o underdeveloped nanoluid MHD low and heat conduction in a channel with porous walls, Cas. Stud. Therm. Eng. 4 ( [6] T. Groşan, C. Revnic, I. Pop, D. B. Ingham, Free convection heat transer in a square cavity illed with a porous medium saturated by a nanoluid, Int. J. Heat Mass Trans. 87 ( [7] A. K. Gupta, S. S. Ray, Numerical treatment or investigation o squeezing unsteady nanoluid low between two parallel plates, Powd. Technol. 79 ( [8] B. K. Jha, B. Aina, A. T. Ajiya, MHD natural convection low in a vertical parallel plate microchannel, Ain Shams Eng. J. 6 (1 ( [9] A. Khalid, I. Khan, A. Khan, S. Shaie, Unsteady MHD ree convection low o Casson luid past over an oscillating vertical plate embedded in a porous medium, Eng. Sci. Technol. Int. J. 18 (3 ( [10] A. V. Kuznetsov, D. A. Nield, The Cheng Minkowycz problem or natural convective boundary layer low in a porous medium saturated by a nanoluid: a revised model, Int. J. Heat Mass Trans. 65 ( [11] M. Mustaa, T. Hayat, S. Obaidat, On heat and mass transer in the unsteady squeezing low between parallel plates, Meccan. 47 (7 (

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