Boundary stratum exploration of unsteady 3D MHD stagnation point flow of Al–Cu water nanofluid

Accepted Manuscript

Boundary stratum exploration of unsteady 3D MHD stagnation point flow of Al-Cu water nanofluid N. Nithyadevi, P. Gayathri, N. Sandeep PII: DOI: Reference:

S0020-7403(17)31056-1 10.1016/j.ijmecsci.2017.08.003 MS 3813

To appear in:

International Journal of Mechanical Sciences

Received date: Revised date: Accepted date:

23 April 2017 7 August 2017 8 August 2017

Please cite this article as: N. Nithyadevi, P. Gayathri, N. Sandeep, Boundary stratum exploration of unsteady 3D MHD stagnation point flow of Al-Cu water nanofluid, International Journal of Mechanical Sciences (2017), doi: 10.1016/j.ijmecsci.2017.08.003

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Graphical Abstract:

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Consider a time dependent, incompressible, viscous, laminar, three-dimensional stagnation point flow of an electrically conducting nanofluid. The nanofluid is composed of aluminium rich copper nanoparticles homogeneously suspended in base fluid water. The Cartesian coordinate system (x, y, z) with the origin O at forward stagnation region, with x and y coordinates along the body surface and the flow is occupied by the domain z > 0 perpendicular to the body surface at origin as shown in Figure.

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Highlights • 3D magnetohydrodynamic stagnation point flow of AlCu-water nanofluid.

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• Electrical conductivity of AlCu alloy nanoparticals is introduced.

• Dual solutions obtained for water and AlCu-water nanofluid cases.

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• Increasing the mass proportion of Al enhances the heat transfer rate.

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Boundary stratum exploration of unsteady 3D MHD stagnation point flow of Al-Cu water nanofluid

1,2 Department 1

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N. Nithyadevi1,∗ , P. Gayathri2 , N. Sandeep3,∗ of Mathematics, Bharathiar University, Coimbatore 641046, India

Email:[email protected], 2 Email:[email protected]

3 Department

of Mathematics, VIT University, Vellore 632014, India Email:[email protected]

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Abstract

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A theoretical analysis of unsteady, three-dimensional stagnation-point flow of Al − Cu nanoparticle suspended water based nanofluid is performed. The thermophysical properties of alloy nanoparticles like density, specific heat capacity and thermal conductivity are computed using appropriate formula. A micro-convection based model framed by Patel et al. [1] is used for predicting the thermal conductivity of nanofluid. The governing physical model is mathematically modeled into a set of non-linear parabolic partial differential equations and solved by a shooting technique. A parametric study is performed for a varied range of nanoparticle volume fraction, ratio of the velocity gradient, unsteadiness parameter and the magnetic parameter. Additionally, the importance of the different compositional characteristics of the alloy nanoparticles on the flow and the thermal field is also examined. It is observed that the Nusselt number increases as the unsteadiness parameter λ decreases and the ratio of the velocity gradient c increases in the nodal point region. A novel result of the analysis reveals that the highest skin friction is obtained for Al50 Cu50 - water and the highest Nusselt number is obtained for Al90 Cu10 -water. Keywords: Stagnation point, three dimensional, magnetohydrodynamics, alloy nanoparticle, nanofluid.

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Introduction

In heat transfer technology, the emergence of nanofluids has become a potential replacement of the prevailing heat carrying fluids like water, ethylene glycol, engine oil, etc. Nano-sized solid particles are suspended stably and uniformly in ordinary fluids to yield the so-called ‘nanofluids’ that holds the enhanced thermal and transport properties relative to the base fluids. Numerous computational works on nanofluids have been confined 3

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to the nanoparticle suspensions of metals like Ag, Cu, Fe etc., and metal oxides like CuO, Al2 O3 , TiO2 , SiO2 , multi-walled carbon nanotubes(MWCNT), etc. Though heat transfer efficiency is the prime objective in any nanofluid usage, there are many other controlling factors like production cost, long term stability, corrosion resistivity, etc., in using a particular nanoparticle. This impediment can be overcome by amalgamating distinct solid particles or using alloys to obtain an alloy nanofluid. The experimental work done by Chopkar et al. [2] introduced the newly developed nanofluids that are formed by the suspension of alloy nanoparticles in ordinary base liquids. They fabricated Al2 Cu nanoparticles and dispersed those in low concentrations into the ethylene glycol base fluid. The newly formed nanofluid is found to possess the distinguished thermal conductivity ratio exceeding the other ethylene glycol based nanofluids which are found in the earlier available reports. They also disclosed that the composition, particle size, and thermophysical properties of the nanocomposite are the important controlling parameters in nanofluids.

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Subsequently, the above pioneering work was developed by Chopkar et al. [3] for two different aluminium rich alloy dispersoids in both water and ethylene glycol basefluids. While Cu, Ag suspended nanofluids are found to boost the thermal conductivity and hence heat transfer remarkably, the production cost hinders to incorporate these nanofluids in industrial applications. The authors hence utilized these materials in small fraction by coupling with aluminium to acquire a better thermal conductivity with less cost and high stability. By adopting high energy planetary milling, the elemental powder blends of Al-30 at.% Cu and Al-30 at.% Ag are mechanically alloyed and the size, distribution and morphology of the obtained powders are tested using bright field transmission electron microscopy(TEM). The obtained non-agglomerated nanoparticles(in size range of 20-80nm) of very low concentration 0.2-1.5 vol.% are dispersed in both the base fluids and the corresponding thermal conductivity is measured at room temperature by the thermal comparator device. Their experimental investigation revealed that the alloyed nanofluids are capable of increasing the thermal conductivity ratio(up to 2.4 times) than that of the base fluid(water) and the degree of enhancement is higher than other nanoparticles (Cu, CuO, Al2 O3 ) in an identic medium. They concluded that the nanoparticles of tiny sizes are found to be highly stable and also increase microconvection levels which inturn improves the heat transfer efficiency. In addition, their results convey that the type of basefluid influences more on heat transfer, by resulting that the nanoparticles dispersed in ethylene glycol are less thermally conducting than those dispersed in water. Recently, computational studies in utilizing the alloy nanofluids for heat transfer processes are done by very few researchers[3,4]. Having motivated by the promising applications of titanium alloy nanoparticles in surgical usages, Raju et al. [4] examined the fluid and heat flow of Ti-alloy (Ti90 Al6 V4 ) nanofluid over a vertical cone. They 4

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discovered that the Ti-alloy nanofliuid is better in cooling processes than the fluids enriched with the pure Ti nanoparticles. Sandeep et al. [5] incorporated two different aluminium alloy nanoparticles AA 7072 and AA 7075 in nano-liquid film with extrinsic magnetic effects. They concluded that the amount of copper compositions in alloys plays a significant role in enhancing the thermal conductivity.

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A representative sample of the literatures concerning three-dimensional boundary layer flows is provided by references [6-10]. The stagnation point is a point of the flow field about a body where the fluid particles have zero velocity with respect to the body. This zero velocity entrains the region of highest pressure and highest heat transfer. Hiemenz [11] discovered that the boundary layer near a stagnation region can be analyzed by the Navier-Stokes equation. Malvandi et al. [12] numerically studied the Hiemenz flow of nanofluids over a stretching sheet. The exact solution of axisymmetric rotational stagnation point flow was analyzed by Weidman [13]. Dinarvand et al. [14] examined the axisymmetric stagnation-point flow of an incompressible nanofluid with combined convection. Howarth [14] laid the earlier ground to investigate the steady state stagnation point flow in a three dimensional plane. Eswara and Nath [16], and Xu et al. [17] fostered the above work for a time dependent flow near a three-dimensional stagnation body encompassing convective effects. Abbassi et al. [18] obtained the exact solutions for an unsteady three-dimensional stagnation point flow on a heated plate. Following this, Sabzevar et al. [19] extended the above work and found exact solutions in an oblique stagnation point flow domain. Rees et al. [20] made a detailed note on the mixed bioconvection flow of nanofluids in the stagnation region of a three-dimensional flat surface with anistropic slip effects. A theoretical analysis on the 3D magnetohydrodynamic mixed convection stagnation-point flow and heat transfer on a vertical flat plate was examined by Borelli et al. [21]. Nanofluid flow over three dimensional surfaces seems to be one of the active research area. Bachok et al. [22] investigated the nanoliquid flow in the neighborhood of three-dimensional stagnation point flow. Khan et al. [23] explored the applications of three-dimensional flow of nanofluids over a stretching plane and made a significant intuition for using nanofluids in solar energy. They concluded that the convective heat transfer performance of various liquids are enhanced by employing nanofluids. The three dimensional magneto-convective flow of Maxwell nanofluid in an inclined channel has been carried out by Jusoh et al. [24]. The study of magnetohydrodynamics has been discussed due to its widespread applications in the industrial processes, geothermal systems, power generation, etc., Shit and Haldar [25] investigated the hydromagnetic flow and heat transfer over a non-linearly shrinking porous sheet with radiation. Ishak et al. [26], Sajid et al. [27] analyzed mixed convective MHD flow near a two dimensional stagnation point. Sheikholeslami et al. [28] explored the impact of MHD effects on a free convective flow of Cu-water nanofluid. 5

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Nandy and Pop [29] intrigued the magnetic field effects on a nanofluid flow past a shrinking sheet. Pal and Mandal [30] studied the impact of magnetic field and mixed convection over nonlinearly stretching and shrinking sheets. Hayat et al. [31] explored the MHD second grade nanofluid flow over three-dimensional surface. Pal et al. [32] examined the magnetohydrodynamic boundary layer flow of a Casson nanofluid with non-linear stretching velocity and suction. Mahabaleshwar et al. [33] made an investigation on the flow of couple stress liquid over a stretching sheet with MHD effects.

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Many researchers are currently involved in nanofluid analysis both theoretically and experimentally by considering common metals and metal oxides like Cu, Ag, Al2 O3 , CuO, TiO2 , etc., as nanoparticles. Alloys are capable of enhancing the thermophysical properties and overcoming the inadequacies met by common base metals. As far as the authors knowledge is concerned, there is no numerical studies that investigated the compositional characteristics of the aluminium alloy nanofluids in the boundary layer flows. Motivated by the experimental investigation of Chopkar et al. [2], this paper provides the numerical perspective for predicting the fluid flow, heat transfer behavior with alloy based nanofluids. This is accomplished by computing the thermophysical properties of the aluminium alloys like density, heat capacity and thermal conductivity of the appropriate alloy composition used in the experimental work. Hence, this paper aims to solve the three dimensional stagnation point flow with the Al2 Cu - water nanofluids. In addition, an appropriate thermal conductivity model framed by Patel et al. [1] based on the size, particle volume fraction and the micro convection effects is used for predicting the thermal conductivity of the corresponding nanofluids.

Mathematical formulation

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Consider a time dependent, incompressible, viscous, laminar, three-dimensional stagnation point flow of an electrically conducting nanofluid. The nanofluid is composed of aluminium rich copper nanoparticles homogeneously suspended in the base fluid water. Let us suppose a Cartesian coordinate system (x, y, z) with the origin O at forward stagnation region, with x and y coordinates along the body surface and the flow is occupied by the domain z > 0 perpendicular to the body surface at origin (see Fig. 1). The flow region in the vicinity of stagnation point can be segregated into two types, the potential flow region and the region of varying velocities inducing the boundary layer. When the free stream velocity exhibits an impulsive motion or varying arbitrarily with time, the unsteadiness in the flow field is imparted. The external free stream velocity components over the three dimensional body surface is given by ue =

ax , (1 − λt∗ )

ve = 6

by , (1 − λt∗ )

t∗ = at

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where λ is the dimensionless parameter characterizing the unsteadiness in the flow field. When λ > 0, the flow is accelerating, λ < 0, the flow corresponds to the decelerating case and when λ = 0, the steady state flow occurs. It is also assumed that near the stagnation point, the surface temperature Tw and the free stream temperature T∞ are constant.

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Suppose that a uniform, imposed magnetic field is exerted along the orthogonal body surface at x = 0, y = 0 and there is no electrostatic force imposed on the flow boundaries. Then the source term is the magnetic force F given by F = J × B, where B is the magnetic flux density(B = (0, 0, B0 ) and J is the induced current given by J = σnf u ¯ × B. Since the flow assumes low velocity(the velocity is slow in the vicinity of a stagnation point) and the applied magnetic field is assumed to be uniform, the magnetic Reynolds number takes small values. This small magnetic Reynolds number in turn facilitates the assumption of the negligible induced magnetic effects(refer Shercliff[34]). Substituting the expansion for J into F , we obtain x, y, z components of the source terms as Fx = −σnf uB02 , Fy = −σnf vB02 , Fz = 0, respectively. The time dependent three-dimensional boundary layers near the stagnation point for an ordinary fluid was studied by few investigators[35-37]. Using the fundamental boundary layer approximations, the unsteady conservation equations of mass, momentum, and thermal energy for the nanofluid flow can be determined as:

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∂u ∂v ∂w + + =0 ∂x ∂y ∂z ∂u ∂u ∂u ∂ue ∂ue µnf ∂ 2 u σnf B02 ∂u +u +v +w = + ue + − (u − ue ) ∂t ∂x ∂y ∂z ∂t ∂x ρnf ∂z 2 ρnf

(1) (2) (3)

∂T ∂T ∂T ∂2T ∂T +u +v +w = αnf 2 ∂t ∂x ∂y ∂z ∂z

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∂v ∂v ∂v ∂v ∂ve ∂ve µnf ∂ 2 v σnf B02 +u +v +w = + ve + − (v − ve ) ∂t ∂x ∂y ∂z ∂t ∂x ρnf ∂z 2 ρnf

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subject to the initial and boundary conditions u = v = w = 0;

T = Tw at z = 0,

x ≥ 0,

y≥0

u → ue , v → ve , w → W, T → T∞ as z → ∞, x ≥ 0, y ≥ 0

(5)

where (u, v, w) refers to the velocity components along the axes (x, y, z), T stands for the fluid temperature, where αnf is the thermal diffusivity of the nanofluid defined by αnf = knf /(ρcp )nf . The effective density ρnf and the heat capacitance (ρcp )nf of nanofluid are given by, ρnf = (1 − φ)ρf + φρp ,

(ρcp )nf = (1 − φ)(ρcp )f + φ(ρcp )p 7

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where φ is the nanoparticle volume fraction, ρf and (ρcp )f is the density and heat capacitance of the base fluid, respectively.

µnf =

µf (1 − φ)2.5

where µf is the dynamic viscosity of the base fluid.

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The effective dynamic viscosity is given by the Brinkman model,

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The electrical conductivity of the nanofluid can be expressed by     3 σσfs − 1 φ     σf σnf = 1 +  σs σs + 2 − − 1 φ σf σf

The effective thermal conductivity of nanofluid is obtained from Patel et al. [1] as follows:   knf kp df φ 2kb T =1+ 1 + c1 (6) kf kf dp (1 − φ) πµf dp αf

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where kp refers the thermal conductivity of the nanoparticles, kf is the thermal conduc˙ kb is the tivity of the base fluid, df is the molecular size of pure fluid(here, water 2A), Boltzmann constant, c1 is the constant to be determined experimentally(here, taken as 25000), T is the temperature(here, taken as 300K), and αf is the thermal diffusivity of the pure fluid medium. Equations(1)-(4) with boundary conditions(5) admits a similarity solution, by ax f 0 (η), v= g 0 (η), ∗ 1 − λt 1 − λt∗ r aνf w=− (f (η) + cg(η)) λt∗ < 1, 1 − λt∗ r a T − T∞ θ(η) = , η= z, Tw − T∞ νf (1 − λt∗ )

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u=

(7)

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If the flow pattern is bounded by both sides from one of the directions because of some physical limitations, a difference between the velocities in the x and y direction exists in the boundary layer region. Therefore, the parameter c occurring in Eq.(7) characterizes the ratio of velocity components x to y in the potential region which influences the boundary layer flow near the stagnation point. Here, a and b are the parameters that represent the curvatures of the body measured in the plane, y = 0, and x = 0 respectively. Moreover, 0 ≤ c ≤ 1 characterizes the flow in the nodal stagnation point of attachment region and −1 ≤ c ≤ 0 represents the saddle stagnation point of attachment. The special cases are as follows: c = 0 corresponds to the two dimensional flow (there is no velocity component in the x direction), and c = 1 represents the axisymmetric case (round a body of revolution placed symmetrically in a stream). 8

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The similarity equations hence derived are as follows: ρs η ){(f + cg)f 00 + λ(1 − f 0 − f 00 ) + (1 − f 02 )} ρf 2     3 σσfs − 1 φ     M (1 − f 0 ) = 0 +(1 − φ)2.5 1 +  σs σs σf + 2 − σf − 1 φ ρs η g 000 + (1 − φ)2.5 (1 − φ + φ ){(f + cg)g 00 + λ(1 − g 0 − g 00 ) + c(1 − g 02 )} ρf 2     σs 3 σf − 1 φ     M (1 − g 0 ) = 0 +(1 − φ)2.5 1 +  σs σs σf + 2 − σf − 1 φ      knf (ρcp )s 1 λη 0 00 0 θ + (1 − φ) + φ (f + cg)θ − θ =0 P r kf (ρcp )f 2

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f 000 + (1 − φ)2.5 (1 − φ + φ

(8)

(9)

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and the appropriate boundary conditions become f 0 (0) = g 0 (0) = 0,

f (0) = g(0) = 0, 0

0

f (η) → 1, νf αf

θ (η) → 0

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at

η=0 η → ∞.

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is the Prandtl number, λ is the unsteadiness parameter, M =

M

where P r =

g (η) → 1,

θ(0) = 1

0

the magnetic field parameter (B1 =

σf B12 ρf a

is

√ B0 ). 1−λt∗

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The physical quantities of interest are the skin friction coefficient along the x and y axis, Cf x , Cf y and the local Nusselt number N ux , which are defined as τwx , ρf u2e

Cf y =

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Cf x =

τwy ρf u2e

N ux =

xqw , kf (Tw − T∞ )

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where the surface shear stress τwx , τwy and the surface heat flux qw are given by       ∂u ∂v ∂T τwx = µnf , τwy = µnf , qw = −knf ∂z z=0 ∂z z=0 ∂z z=0

(13)

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with µnf and knf being the dynamic viscosity and thermal conductivity of the nanofluids, respectively. Using the similarity variables(7), we obtain 1/2

Cf (Rex )

=

f 00 (0 , (1 − φ)2.5

1/2

Cf (Rex )

=c

y x

g 00 (0) , (1 − φ)2.5

where Rex = ue x/νf is the local Reynolds number.

Thermophysical properties of Al-Cu Alloys The density of the alloys can be computed by the formula, 1 m1 m1 = + ρalloy ρ1 ρ2 9

−1/2

N ux (Rex )

=−

knf 0 θ (0) kf

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where ρ1 , ρ2 and m1 , m2 corresponds to the density and mass compositions of first and second elements, respectively. The specific heat capacity of the alloys can be computed using the Neumann Kopp rule(described by Grimvall [38]) that expresses the molar heat capacity of an alloy as the weighted sum of the specific heat of the elements forming it. That is, (cp )alloy = a1 cp1 + a2 cp2

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where cp1 , cp2 and a1 , a2 corresponds to the specific heat capacity and atomic compositions of first and second elements, respectively. The thermal conductivity can be computed by kalloy = ke + kg , where ke is the electronic component(fundamental interaction processes which occur within metallic systems,) and kg is the lattice component(vibrations of the lattice ions in a solid). Based on this, Smith and Palmer [39] framed an empirical relation for finding the thermal conductivity of alloys given by kalloy =

CL0 T +D σe

(14)

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where C and D represent the constants which vary according to the materials used, L0 represents the Lorentz ratio, T is the temperature, and σe represents the electrical resistivity. For aluminium based alloy systems, the constants C and D in the Smith Palmer equation takes values 0.909 and 10.5, respectively, which holds for temperatures above 500K. Following this, Klemens and Williams [40] reformulated for the above values with Lorentz ratio 0.86L0 for temperatures around 300K and hence, the thermal conductivity of the Al-Cu alloy systems can be found by knowing the electrical resistivity σe . Meanwhile, the numerical values of thermal conductivity for particular compositions of Al-Cu alloys can also be obtained directly from Ho et al. [41]. The validation of these results with Klemens and Williams [40] showed only 5% deviation for the composition chosen in the present study. Hence, in this work we incorporate the values obtained from Ho et al. [41]. Similarly, the electrical conductivity of different compositions can be appropriately calculated by taking the inverse of the electrical resistivity values tabulated by Ho et al. [42]. Table 1 portrays the thermophysical properties of the Al-Cu alloy nanoparticles for various mass proportions. Table 2 displays the thermophysical properties of other nanoparticles Al2 O3 , CuO.

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Results and Discussion

Numerical solutions to the governing ordinary differential equations(8)-(10) with boundary conditions(11) are obtained using the shooting method in connection with the fifthorder Runge-Kutta-Fehlberg integration scheme. This method has been successfully used 10

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to solve various problems related to boundary layer flows and it is extremely accurate and reliable. In this algorithm, we firstly transform the governing partial differential equations into a system of non-linear ordinary differential equations. The boundary conditions for f 0 (η), g 0 (η) and θ(η) for η reaching infinity is transformed for a finite interval length(here, it is η = 15). The three missing slopes f 00 (0), g 00 (0) and θ0 (0) first assume an initial guess, and then iterate by shooting method coupled with the Runge-Kutta method, where the step size and convergence criteria are chosen to be 0.001 and 106 , respectively. When the two successive guesses are found to have a difference in the order less than 10−6 , the iteration is stopped. With the final iteration value for the missing slopes, the ordinary differential equations(8)-(10) are integrated using the Runge-Kutta method. In the absence of nanoparticles, the dimensionless surface skin friction coefficients are compared with the literature Kumari and Nath [35] and Ibrahim [36] in Table 3. These verify the validity of the results to possess a good accuracy.

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The computational analysis is carried out for Al2 Cu(Al wt%. 50, Cu wt.% 50) alloy nanofluid for the effect of the nanoparticles volume fraction φ, unsteadiness parameter λ, ratio of the velocity gradient c and the magnetic parameter M . Finally, the computational analysis is performed on the comparison of the three different compositions of Al-Cu alloys. The results of the velocity profiles and the temperature distributions are presented as graphs and the local skin-friction coefficient, local Nusselt number values are displayed as graphs and tables. Fig. 2 depicts the impact of nanoparticle size on the thermal conductivity ratio of nanofluids for varying Al2 Cu nanoparticle volume fraction. The experimental work of Chopkar et al. [2] encompasses a wide range of nanometer size enclosed by 20–80nm and they stated that the particle size has a significant impact in the variation of thermal conductivity. This figure displays the thermal conductivity ratio obtained from Eq.(6) and we can observe that the thermal conductivity ratio increases with a reduction in the crystallite size. The smaller size attributes to the increased collision and interaction between particles and thereby, increasing the micro-convection of the fluid which accounts to the greater energy transport. For further analysis, an appropriate size of 40 nm is chosen. Table 4 shows that the local Nusselt number value gets elevated as we increase the nanoparticles volume fraction when λ = 1, c = 0.5, M = 0 remain fixed. There is an excellent enhancement of 97.56% for a nanoparticle volume fraction of 5% which motivates our initiative for numerically analyzing the flow behavior of the Al2 Cu-water nanofluid. Fig. 3 portrays the comparison of the local Nusselt number values of water and nanofluid(φ = 0.05) for different values of the velocity gradient ratio. An appreciable increase in the local Nusselt number is seen for nanofluids as compared to the pure fluid and the heat is transferred at its maximum for the axisymmetric case(c = 1). Fig. 4 exhibits a tremendous improvement in the heat transmission rate for the chosen alloy 11

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nanoparticles compared with ordinary nanoparticles Al2 O3 , CuO.

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The eminence of the nanoparticle volume fraction (0 ≤ φ ≤ 0.05) on the surface shear stress and the surface heat transfer rate is displayed in Table 5. We can observe that f 00 (0), g 00 (0) has a slight increase as φ increases and −θ0 (0) decreases sharply. Table 6 reveals the effect of the velocity gradient ratio for c = 0(the two-dimensional case), c = 0.5(the nodal region case) and c = 1(the axisymmetric case) in the absence of magnetic field. The comparison on different values of c is exhibited for both water(φ = 0) and alloy nanofluid(Al2 Cu-water). It demonstrates that the surface shear stress values along both x and y axis are larger for a nanofluid whereas the surface heat transfer rate is lesser for a nanofluid for all values of c. Further, as the velocity gradient ratio is raised, the surface stress and heat transmission intensifies and it is found to be maximum for the axisymmetric case.

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The influence of unsteadiness parameter λ and the velocity gradient ratio c for water and nanofluid on the local skin-friction coefficients along the x and y directions and the local Nusselt number can be seen in the Figs. 5 and 6, respectively. From Fig. 5, we observe a similar fluid and thermal flow behavior for both nanofluid and pure fluid cases. A substantial increase in the skin-resistance along both x and y axes and the Nusselt number values are seen for nanofluid compared to the base fluid. Moreover, as λ increases, the skin-friction values also increase, however, the Nusselt number decreases. This is because as λ increases, the momentum boundary layer gets broadened, and as φ increases, the viscosity of the fluid is raised, which as a wholesome increases the skinfriction values. Fig. 6 yields the interesting result of the influence of c on the flow parameters and thermal field. The skin-friction coefficient along the x axis shows a linear increase as c increases. On the other hand, the skin-friction coefficient along the y direction assumes a parabolic curve-like pattern with an overturn at c = 0 for both water and nanofluid. The reason is that in the nodal point zone (0 ≤ c ≤ 1), the velocity vectors along both the x and y coordinates are directed away from the stagnation point, whereas, in the saddle point zone(−1 ≤ c ≤ 0), there will be outward flow in one direction and fully or at least partial flow along another coordinate. This flow pattern also strongly influences the thermal flow behavior which can be seen from the Nusselt number curve. In addition, the heat transfer rate is higher for nanofluids for all values of c.

The boundary layers of the flow and heat satisfying the far field boundary conditions in an asymptotic fashion are illustrated in Figs. 7-14. Fig. 7 and 8 manifests the effect of the nanoparticle volume fraction (0 ≤ φ ≤ 0.05) for the velocity components f 0 and g 0 along the x and y axes, respectively, that shows only small variation. The effect of φ in Fig. 9 shows that the temperature raises by increasing φ. The influence of the magnetic field M on the flow and thermal boundary layer can be observed from Figs. 10-12. In contrast to Figs. 7 and 8, there is a noticeable variation in the momentum boundary layer 12

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between the pure fluid and the nanofluid because of the highly electrically conducting nanoparticles influenced by the magnetic field. The momentum boundary layer f 0 , g 0 augments with the increase in the magnetic field and the velocity of the pure fluid is higher than the nanofluid. Although, the thermal boundary layer seems to reduce for raising values of M , the temperature is found to be higher for the nanofluid than the pure fluid.

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A qualitative analysis on the thermal boundary layer with unsteadiness parameter λ and the velocity gradient ratio c are depicted in Figs. 13 and 14, respectively. For increasing λ, the temperature rises and with increase in c, the temperature distribution falls. These figures convey the fact that the heat transfer rate decreases with increase in temperature distribution and vice versa. This phenomena is a replication of the result observed from Figs. 5 and 6, respectively. However, for both λ and c, the temperature boundary layer is higher for Al2 Cu-water nanofluid relative to the pure fluid.

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Figs. 15-17 display the different compositional variation of the Al-Cu alloys in terms of the local skin-friction coefficients along the x, y directions and the local Nusselt number, respectively. The skin-friction values are found to be dominant with the mass composition of 50:50 which reveals that the skin-friction coefficient increases as the copper composition is at its highest. But to the contradictory, the Nusselt number rate is found to be high for 90:10 composition, that is, with the lowest copper composition. This is a very interesting result that specifies the importance of the alloy compositional characteristics. The reason behind this is copper solubility in aluminium decrease with increase in copper proportion after a certain mass proportion which, in turn, reduces the thermal conductivity of the alloy formed. Hence, this results in a decrease in the Nusselt number for the proportional increase in copper.

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Conclusion

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Laminar boundary layers are obtained for a time dependent three dimensional stagnation point flow of Al2 Cu dispersed water-based nanofluid flow subject to magnetic effect. The thermophysical properties of alloy nanoparticles like the density, specific heat capacity and the thermal conductivity are computed using appropriate formulae. The mathematical model is similarity transformed and the numerical output of the resultant non-dimensional equations are acquired by shooting technique. Key findings of the computational analysis are summarized below: • A favorable enhancement in Nusselt number is attained for Al2 Cu nanoparticles compared to the commonly used nanoparticles such as Al2 O3 , and CuO and the enhancement rate of Al2 Cu- water is 97.60% (for nanoparticle concentration of 5%) 13

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relative to water.

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• Increase in magnetic effect and nanoparticle volume fraction enlarges the momentum boundary layer for both pure fluid(water) and Al2 Cu- water nanofluid. However, the thermal boundary layer condenses with increase in magnetic effect and enlarges with nanoparticle volume fraction. • The skin-friction coefficients and the Nusselt number values increase with increase in the unsteadiness parameter but an opposite trend can be remarked for the velocity gradient ratio.

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• The highest skin friction is obtained for Al50 Cu50 - water and the highest Nusselt number is obtained for Al90 Cu10 -water(in mass composition). This is due to the solubility of copper and the changes in thermophysical properties with mass proportion variation.

References

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Acknowledgments The author P. Gayathri would like to thank UGC-BSR fellowship for their financial support.

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[1] H.E. Patel, T. Pradeep, T. Sundararajan, A. Dasgupta, S.K. Das, A microconvection model for thermal conductivity of nanofluid, Pramana J. Phys 65 (2005) 863–869.

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[3] M. Chopkar, S. Kumar, D.R. Bhandari, P.K. Das, I. Manna, Development and characterization of Al2 Cu and Ag2 Al nanoparticle dispersed water and ethylene glycol based nanofluid, Mater. Sci. Eng. B 139 (2007) 141-148. [4] C.S.K. Raju, N. Sandeep, V. Sugunamma, Unsteady magnetonanofluid flow caused by a rotating cone with temperature dependent viscosity: A surgical implant application, J. Mol. Liq. 222 (2016) 1183–1191. [5] N. Sandeep, R.P. Sharma, M. Ferdows, Enhanced heat transfer in unsteady magnetohydrodynamic nanofluid flow embedded with aluminum alloy nanoparticles, J. Mol. Liq. 234 (2017) 437–443. [6] A.J. Chamkha, Hydromagnetic plane and axisymmetric flow near a stagnation point with heat generation, Int. Comm. Heat Mass Transfer 25 (1998) 269–278. 14

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[19] M.H.H. Sabzevar, A.B. Rahimi, H. Mozayeni, Three-dimensional unsteady stagnation point flow and heat transfer impinging obliquely on a flat plate with transpiration, J. Appl. Fluid Mech. 9 (2016) 925–934.

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[24] R. Jusoh, R. Nazar, I. Pop, Flow and heat transfer of magnetohydrodynamic three dimensional Maxwell nanofluid over a permeable stretching/shrinking surface with convective boundary conditions, Int. J. Mech. Sci, 124 (2017) 166–173.

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[25] G.C. Shit, R. Haldar, Effects of thermal radiation on MHD viscous fluid flow and heat transfer over nonlinear shrinking porous sheet, Appl. Math. Mech. (Eng Ed.) 32 (2011) 677-688.

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[26] A. Ishak, R. Nazar, N. Bachok, I. Pop, MHD mixed convection flow near the stagnation-point on a vertical permeable surface, Physica A 389 (2010) 40-46.

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[27] M. Sajid, B. Ahmed, Z. Abbas, Steady mixed convection stagnation point flow of MHD Oldroyd-B fluid over a stretching sheet, J. Egypt. Math. Society 23 (2015) 440-444. [28] M. Sheikholeslami, M.G. Bandpy, R. Ellahi, M. Hassan, S. Soleimani, Effects of MHD on Cu-water nanofluid flow and heat transfer by means of CVFEM, J. Magn. Magn. Mater. 349 (2014) 1882-00. [29] S.K. Nandy, I. Pop, Effects of magnetic field and thermal radiation on stagnation flow and heat transfer of nanofluid over a shrinking surface, Int. Comm. Heat Mass Transf. 53 (2014) 50-55.

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[30] D. Pal, G. Mandal, Hydromagnetic convectiveradiative boundary layer flow of nanofluids induced by a non-linear vertical stretching/shrinking sheet with viscousOhmic dissipation, Powder Technol. 279 (2015) 61-74.

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[31] T. Hayat, I. Ullah, T. Muhammad, A. Alsaedi, Magnetohydrodynamic (MHD) three-dimensional flow of second grade nanofluid by a convectively heated exponentially stretching surface, J. Mol. Liq. 220 (2016) 1004-1012. [32] D. Pal, N. Roy, K Vajravelu, Effects of thermal radiation and Ohmic dissipation on MHD Casson nanofluid flow over a vertical non-linear stretching surface using scaling group transformation, Int. J. Mech. Sci. 114 (2016) 257–267.

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[33] U.S. Mahabaleshwar, I.E. Sarris, A.A. Hill, G. Lorenzini, I. Pop, An MHD couple stress fluid due to a perforated sheet undergoing linear stretching with heat transfer, Int. J. Heat Mass Transfer 105 (2017) 157–167. [34] J. A. Shercliff, A textbook of magnetohydrodynamics, Pergamon Press, 1965, p. 45.

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[35] M. Kumari, G. Nath, Unsteady flow and heat transfer of a viscous fluid in the stagnation region of a three-dimensional body with a magnetic field, Int. J. Eng. Sci. 40 (4) (2002) 411–432.

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[36] F.S. Ibrahim, Unsteady Mixed Convection Flow in the Stagnation Region of a Three Dimensional Body Embedded in a Porous Medium, Nonlinear Anal. Model. Control, 13 (1) (2008) 31-46.

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[37] A. J. Chamkha and S.E. Ahmed, Similarity solution for unsteady MHD flow near a stagnation point of a three-dimensional porous body with heat and mass transfer, heat generation/absorption and chemical reaction, Journal of Applied Fluid Mechanics, 4 (2) (2011) 87–94.

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[38] G. Grimvall, Thermophysical properties of materials, Enlarged and revised version, Elsevier Science BV, First edition (1999). [39] C.S. Smith, E.W. Palmer, Thermal and electrical conductivities of copper alloys, Trans AIME Papers (1935). [40] P.G. Klemens, R.K. Williams, Thermal conductivity of metals and alloys, Int. metals reviews, 31 (1986) 197–215. [41] C.Y. Ho, M.W. Ackerman, K.Y. Wu, S.G. Oh, T.N. Hanvill, Thermal conductivity of ten selected binary alloy systems, J. Phys. Chem. Ref. Data 7 (1978) 959–1177. [42] C.Y. Ho, M.W. Ackerman, K.Y. Wu, S.G. Oh, T.N. Hanvill, Electrical resistivity of ten selected binary alloy systems, J. Phys. Chem. Ref. Data 12 (1983) 183–322. 17

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[43] S.P. Jang, S.U.S. Choi, Effects of various parameters on nanofluid thermal conductivity, J. Heat Transf. 129 (2007) 617–623.

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prescribed constants externally imposed z component of B ratio of velocity gradient along y and x axes specific heat at constant pressure skin friction along x and y direction, respectively diameter of the nanoparticle Thermal conductivity Boltzmann number magnetic parameter Nusselt number Prandtl number wall heat flux local Reynolds number temperature of the fluid temperature of the fluid at wall temperature of the fluid in free stream velocity components along x, y and z axes, respectively velocity component in the free stream along x and y, respectively

PT

a, b B0 c cp Cf x , Cf y dp k kb M Nu Pr qw Rex T Tw T∞ u, v, w ue , ve

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Nomenclature

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Greek Symbols α thermal diffusivity φ nanoparticle volume fraction ρ density µ dynamic viscosity ν kinematic viscosity

18

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electrical conductivity unsteadiness parameter similarity varaiable shear stress dimensionless temperature

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σ λ η τ θ

AC

CE

PT

ED

M

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Subscripts f base fluid nf nanofluid p nanoparticle

19

AC

CE

PT

ED

M

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Fig. 1. Schematic flow model

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6 25 40 55 70

5

nm nm nm nm

3

2

0.01

0.02

0.03

0.04

0.05

AN US

1 0.00

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knf/kf

4



0.06

0.07

Fig. 2. Variation of the thermal conductivity ratio knf/kf for different nanoparticle size.

1.6

1.2 1.0

PT

NuxRex-1/2

1.4

water Al2Cu-water

ED

1.8

M

2.0

0.8

CE

0.6 0.4

AC

0.2 0.0

0

0.5

c

1

Fig. 3.Variation of local Nusselt number with c for water and alloy nanofluids when λ = 1,  = 0.05, M = 0.

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2.1 Al2Cu Al2O3 CuO

1.9

1.5 1.3 1.1 0.9

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

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0.7 0.00

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NuxRex-1/2

1.7

0.09

0.10



Fig. 4.Variation of local Nusselt number with for different nanoparticles Al2Cu, Al2O3, CuO when= 1, c = 0.5, M = 0.

M ED

C fxRex1/2

PT

3

Solid line --> water Dashed line --Al2Cu - water

2

(x/y)C fyRex1/2

AC

CE

CfxRex1/2, (x/y) C fyRex1/2, NuxRex-1/2

4

1

NuxRex-1/2

0 -2

-1



0

1

2

Fig. 5. Variation of local skin friction coefficients along x, y directions and local Nusselt number with  for water and alloy nanofluids when c = 0.5, M = 0.

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1.6 1.4 1.2

CfxRex1/2

1.0

NuxRex-1/2

0.8

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0.6 0.4 0.2

Solid line --> water Dashed line --Al2Cu - water

0.0

(x/y)CfyRex1/2

-0.2 -0.4 -1.0

-0.8

-0.6

-0.4

-0.2

-0.0

0.2

0.4

0.6

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CfxRex1/2, (x/y)C fyRex1/2, NuxRex-1/2

1.8

c

0.8

1.0

M

Fig. 6. Variation of local skin friction coefficients x, y directions and local Nusselt number with c for water and alloy nanofluids when=1, M = 0.

ED

1.0

 = 0.0, 0.01, 0.03, 0.05

0.9

PT

0.8 0.7 0.6

f

CE

0.5

0.692

AC

0.4 0.3 0.2

0.684 0.70

0.1 0.0

0

1

0.71



2

3

Fig. 7. Variation of velocity profiles ffor water and alloy nanofluids when= 1, c = 0.5, M = 0.

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1.0 0.9

 = 0.0, 0.01, 0.03, 0.05

0.8 0.7

0.5 0.4

0.693

0.3 0.2 0.1 0

1

0.83

2

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0.0

0.683 0.81

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g

0.6

3



M

Fig. 8. Variation of velocity profiles g for water and alloy nanofluids when= 1, c = 0.5, M = 0.

1.0

ED

0.9 0.8

PT

0.7



0.6 0.5

  

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0.4

AC

0.3 0.2 0.1 0.0

0

1



2

3

Fig. 9. Variation of temperature profiles with  for different when= 1, c = 0.5, M = 0.

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1.0 0.9 Solid line - - water Dashed line - - Al2Cu-water

0.8 0.7

0.5

M = 0, 1, 2

0.4 0.3 0.2 0.1 0.5

1.0

1.5

2.0

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0.0 0.0

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f

0.6



2.5

M

Fig. 10. Variation of velocity profile along x direction for different M with water and alloy nanofluids when = 1, c = 0.5.

0.9 0.8

PT

0.7

ED

1.0

Solid line - - water Dashed line - - Al2Cu-water

0.5

CE

g

0.6

M = 0, 1, 2

0.4

AC

0.3 0.2 0.1

0.0 0.0

0.5

1.0



1.5

2.0

2.5

Fig. 11. Variation of velocity profile along y direction for different M with water and alloy nanofluids when = 1, c = 0.5.

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1.0 0.9

Solid line - - water Dashed line - - Al2Cu-water

0.8 0.7

0.5 M = 0, 1, 2

0.4 0.3 0.2 0.1 0

1

2

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0.0

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

0.6



3

M

Fig. 12. Variation of temperature profile with M for water and alloy nanofluids when= 1, c = 0.5.

ED

1.0 0.9 0.8

PT

0.7

Solid line - - water Dashed line - - Al2Cu-water

0.6



CE

0.5

 = -1, 0, 1

AC

0.4 0.3 0.2 0.1 0.0

0

1



2

3

Fig. 13. Variation of temperature profiles with  for different when c = 0.5, M = 0.

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1.0 Solid line - - water Dashed line - - Al2Cu-water

0.9 0.8 0.7

0.5

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

0.6 c = 0, 0.5, 1

0.4 0.3 0.2

0.0

0

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0.1 1

2

3



4

Fig. 14. Variation of temperature profiles with  for different c when= 1, M = 0.

ED

PT

1.65

1.60

CE

CfxRex1/2

1.70

50-50 70-30 90-10

M

1.75

AC

1.55

1.50 0.00

0.01

0.02



0.03

0.04

0.05

Fig. 15. Variation of local skin friction coefficient along x-direction with  for different Al-Cu mass proportions (50:50, 70:30, and 90:10) when= 1, c = 0.5, M = 0.

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0.75 0.74

0.72 0.71 0.70 0.69 0.68 0.67 0.66 0.01

0.02

0.03

0.04

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0.65 0.00

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(x/y)C fyRex1/2

0.73

50-50 70-30 90-10



0.05

1.7

1.4

PT

1.3 1.2 1.1

CE

NuxRex-1/2

1.5

50-50 70-30 90-10

ED

1.6

M

Fig. 16. Variation of local skin friction coefficient along x-direction with for different Al-Cu mass proportions (50:50, 70:30, and 90:10) when= 1, c = 0.5, M = 0.

1.0

AC

0.9 0.8

0.7 0.00

0.01

0.02



0.03

0.04

0.05

Fig. 17. Variation of local Nusselt number with  for different Al-Cu mass proportions (50:50, 70:30, and 90:10)when= 1, c = 0.5, M = 0.

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Table 1 Thermophysical properties of alloy nanoparticles at 300K Al50Cu50

Al70Cu30

Al90Cu10

Cp(J/Kg K)

747.90

822.40

878.71

(Kg/m3)

4133.68

3402.48

2891.08

K(W/m K)

112

128

161

(/m)-1

1.4993107

1.8484107

2.7248107

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Physical properties

Table 2 Thermophysical properties of fluid and nanoparticles at 300K(Jang and Choi [43])

Cp(J/Kg K)

4170

(Kg/m3)

997

K(W/m K)

0.613

Al2O3

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Fluid phase(water)

CuO

729

540

3880

6510

42.34

18

M

Physical properties

ED

Table 3 Comparison of numerical results for the values of f(0) and g(0) for a pure fluid(water) for the steady state flow = 0. f(0)

c

1.311857

1.31194

0.5

1.266725

1.26687

1.2654

0.997901

0.99813

1.0142

1.235348

1.23538

1.2268

0.570326

0.57049

0.5848

1.230095

1.23019

1.2304

-0.111459

-0.11150

-0.1113

1.271499

1.27154

1.2732

-0.794162

-0.79449

-0.8110

CE

AC

-0.5 -1

Ibrahim [34]

PT

1.0

Kumari and Nath [35] 1.3153

0

Present results

g(0) Present results

Ibrahim [34]

1.311857

1.31194

Kumari and Nath [35] 1.3153

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Table 4 Values of the local Nusselt number for different nanoparticle volume fraction  when λ = 1, c = 0.5, M = 0. 0.0

0.0125

0.025

0.0375

0.05

NuxRex-1/2

0.7101

0.8962

1.0724

1.2407

1.4029

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

Table 5 Values of the surface shear stress and surface heat transfer rate with  when λ = 1, c = 0.5, M=0.



f″(0)

g″(0)

0.0

1.5242

1.2991

0.01

1.5287

0.03

1.5350

0.05

1.5379

-θ′(0)

AN US

0.7101 0.6879

1.3083

0.6460

1.3108

0.6098

M

1.3029

c



0

0.0

AC

1

g″(0)

-θ′(0)

1.4985

0.9845

0.4004

0.05

1.5120

0.9934

0.3890

0.0

1.5242

1.2991

0.7101

0.05

1.5379

1.3108

0.6098

0.0

1.5570

1.5570

0.9787

0.05

1.7859

1.7859

0.7968

PT

f′′(0)

CE

0.5

ED

Table 6 Values of the surface shear stress and surface heat transfer rate for different nanoparticle volume fraction  when λ = 1,  = 0.05, M = 0.