Effects of iron nanoparticles’ shape on convective flow of ferrofluid under highly oscillating magnetic field over stretchable rotating disk

Journal of Magnetism and Magnetic Materials 465 (2018) 531–539

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Research articles

Effects of iron nanoparticles’ shape on convective flow of ferrofluid under highly oscillating magnetic field over stretchable rotating disk

T

⁎

Mohsan Hassana, , C. Fetecaub, Aaqib Majeedc, Ahmad Zeeshanc a

Department of Mathematics, CUI, Lahore, Pakistan Academy of Romanian Scientists, Bucharest 050094, Romania c Department of Mathematics & Statistics, FBAS, IIUI, Islamabad, Pakistan b

A R T I C LE I N FO

A B S T R A C T

Keywords: Ferrofluid Highly oscillating magnetic field Particle shape Stretchable rotating disk

The persistence of the current article is to discuss the iron nanoparticles' shape in flows due to highly oscillating magnetic field over a stretchable rotating disk. For ferrofluid, water is considered as base fluid with suspension of iron nanoparticles having sphere, oblate ellipsoid and prolate ellipsoid shapes with different sizes. The impact of the nanoparticles' shape on velocity and temperature profiles, convective heat transfer coefficient, radial and transverse shear stress is deliberated through graphs and tables. The presence of highly oscillating magnetic field forces the particles to rotate faster than the fluid and, as a result, the total viscosity is certainly reduced. The governing equations, which are firstly modeled and thereafter converted into nonlinear ordinary differential equations in dimensionless form using similarity approach, are analytically solved using the Mathematica package BVPh 2.0 which is based on the homotopy analysis method (HAM).

1. Introduction Ferrofluids consist of a carrier fluid loaded with small (nanometer sized) magnetic particles. The behavior of these fluids varies due to the carrier fluid, temperature, particle size, shape and loading, magnetic characteristics of the particles and the applied magnetic field. Lately, many studies have been done to investigate the characteristic of ferrofluid as function of particle volume fraction and magnetic field strength, but these do not depend on particle size and shape effect [1–10]. Although the relationship amongst shape and magnetization is not as straight, the impact of different geometries on magnetic properties keeps on being assessed. De Vicente et al. [11] prepared magnetite rod-like particles with average diameter and length of 560 nm and 6.9 μm , respectively. These works showed that magneto-rheological performance is significantly improved for elongated magnetic particles under small-amplitude shear and simple steady shear flows, hence suggesting that particle shape strongly affects the structuration under an external field. In another study [12], magneto-rheological performances of magneto-rheological fluids was investigated by using iron particles of different shapes like spherical, plate-like, and rod-like and found better improvement by non-spherical particles. De Gans et al. [13] investigated the influence of particle size on the magneto rheological properties of an inverse ferrofluid. For small particles, a strong increase of magneto rheological properties was found. In view of heat

⁎

Corresponding author. E-mail address: [email protected] (M. Hassan).

https://doi.org/10.1016/j.jmmm.2018.06.019 Received 1 December 2017; Received in revised form 26 March 2018; Accepted 8 June 2018 Available online 18 June 2018 0304-8853/ © 2018 Elsevier B.V. All rights reserved.

transfer, Ellahi et al. [14] discussed the particle shape effects on heat transfer rate and nanofluid flow. Their fallouts show that heat transfer rate can be improved through taking different shapes of particles. In another study, R. Ellahi et al. [15] instigated theoretical study on ferrofluid by taking spherical nanoparticles. They found 7.86% heattransfer enhancement in the absence of a magnetic field and found 8.73% heat-transfer enhancement in present of a magnetic field. J. Fang et al. [16] used ellipsoids magnetic nanoparticles in their study, possessed remarkably enhanced thermal stability for maintaining tiny particle sizes and excellent dispersibility even under higher temperature. Effect of space dependent magnetic field on ferrofluid flow and heat transfer is investigated by M. Sheikholeslami and M. M. Rashidi [17]. They found that heat transfer coefficient is increased as increasing of magnetic number and nanoparticle volume fraction. A numerical study on the heat transfer of ferrofluids in microchannels was conducted by Xuan et al. [18]. They finally concluded that heat-transfer rate could increase if the directions of magnetic field gradient and fluid flow are the same. It is remarkable that there are still only relatively few such publications. To apply the ferrofluid to practical heat transfer processes, more studies on its flow and heat transfer feature are needed. To the best our knowledge, a theoretically study on the effect of particle shape on characteristics of ferrofluid flow, a study in which only shape changes while the rest of the parameters are kept practically constant, is missing in the literature. In present study, effects of particle

Journal of Magnetism and Magnetic Materials 465 (2018) 531–539

M. Hassan et al.

Nomenclature

V vr , vθ, vz k r , θ, z T M H υ Ho τs τB μ0 β kB m n Ω Mo ωo

ωp ξ I p α t ρ μ Cp χ′

Velocity Velocity components Thermal conductivity Cylindrical coordinates Temperature of fluid Magnetization of fluid Strength magnetic field Kinematic viscosity Amplitude of the field Relaxation time parameter Brownian relaxation time Permeability of free space, Thermal expansion coefficient Boltzmann constant particle Magnetic moment Number of particles Vorticity of the flow Equilibrium magnetization The angular frequency of the applied magnetic field

Internal angular momentum of particles The ratio of magnetization energy Moments of inertia of the particles Pressure Phase angle Time Density Viscosity Specific heat Magnetic susceptibility

Subscripts

c int nf s nc a f

Backbone particles Cluster Composition of particles and fluid Solid particle Dead ends particles Aggregation Base fluid

d

shapes, particle size and high oscillating magnetic field on fluid flow and heat transfer over rotating stretchable disk are demonstrated. To achieve this goal, the present work is organized in the following way. In mathematical formulation section, a controllable force is introduced into the fundamental hydrodynamic equations which produced as negative viscosity effects. These equations are transformed into ordinary differential equations by applying appropriate transformations. In addition, the correlation models of physical properties for spherical and non-spherical particles are also discussed in this section. The solution for problem and the accuracy of method is discussed in solution of the problem section. The impact of pertinent flow quantities on velocity and temperature profiles as well as on convective heat transfer coefficient are demonstrated and discussed in results and discussion section. In last section, achievements of study are concluded and a way to enhance the convective heat transfer in fluid flow is given.

∇ × H = 0, ∇ ·B = 0,

2. Mathematical modeling of the problem

Consider the axially symmetric laminar and non-conducting flow of an incompressible nano-Ferrofluid past a stretchable rotating disk that has an angular velocity varying with time Ω v /(1−βv t ) . The coordinate system and geometry of the problem are shown in Fig. 1. We consider that the disk rotation speed has a form of Ω v r /(1−βv t ) and the disk stretching velocity is α v Ω v r /(1−βv t ) , which is proportional to the radius r. The basic governing equations containing continuity, motion, temperature, magnetization and rotational motion equations in vector form are [8,19]

I

dt

= (M × H )−

I (ωp−Ω), τS

(2)

(3)

dM 1 = ωp × M − (M −M0), dt τB

(4)

dT = knf ∇2 T . dt

(5)

(ρCp)nf

s

(1)

dV I = −∇p + μnf ∇2 V + μ0 (M ·∇) H + ∇ × (ωp−Ω), dt 2τs

dωp

H mH (t ) 1 ,ξ= , L (ξ ) = cothξ − . H kB Ta ξ

(7)

In above, magnetic moment of the particles and number of particles are denoted by m and n respectively. Langevin function is L (ξ ) in which ξ is the ratio of magnetization energy. Boltzmann constant and absolute temperature are denoted with kB and Ta symbols. Here, the inertial term is as small as compared to relaxation term dω p ωp I dt ≪ I τ . So, Eq. (3) can be rewritten as

2.1. Flow modeling

∇ ·V = 0,

(6)

where B = μ0 (H + M ) . Instantaneous equilibrium magnetization Mo at τB = 0 in terms of the Langevin function is defined as

Mo = nmL (ξ )

ρnf

∂

In above, dt = ∂t + V·∇, V = (vr , vθ, vz ) is velocity, T is temperature, M is magnetization of the fluid, H is strength magnetic field, τs is Relaxation time parameter, τB is Brownian relaxation time, μ0 is permeability of free space, I is sum of moments of inertia of the particles per unit volume, ωp is internal angular momentum due to the self-rotation of particles and Ω is the vorticity of the flow. Mean angular velocity of the particle The complete set of equations also includes the Maxwell’s equations

Fig. 1. Geometry of the problem. 532

Journal of Magnetism and Magnetic Materials 465 (2018) 531–539

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ωp = Ω +

τs (M × H ). I

Mθ+ = χ ′ Hocosα+sin(ωo t −α+) =

(8)

Using Eq. (8), Eqs. (2) and (4) can be written as

dM 1 τ = Ω × M − (M −M0)− s M × (M × H ). dt τB I

Mθ− = −χ ′ Hocosα+sin(ωo t −α−) = −

(9)

If the field is linearly polarized along the radial direction, the tangential component of magnetization is:

(11)

Mθ =

Here ωo is the angular frequency and Ho is the amplitude of the applied magnetic field. We assume that the field amplitude is small in the sense mHo ⩽ kB Ta . This supposition allows us to avoid the difficulties that are related with the nonlinearity of the Langevin function in Eq. (7) as a replacement with

(M × H ) = −ΩτB χ ′ Ho2cos2 α cosωo t cos(ωo t −2α )

(12)

(23) Now averaging Eq. (22) over the period of field variation 1

(24)

Using (24) and (9), we get the equation of motion as

dV 1 ρnf = −∇p + μo (M . ∇) H + ⎛μnf + μf ϕ ξ 2cos2 α cos2α ⎞ ∇2 V. dt 4 ⎝ ⎠ (25) Let ∼ p be reduced pressure due to magnetic force as ∼ −∇p = ∇p + μ (M . ∇) H .

(14)

o

Here α is the phase angle between magnetization and magnetic field at rest in the fluid. Substituting Eqs. (12)–(14) into Eqs. (3) and (4), we get

τ M = χ ′ Hocosα, ωp = ⎛ s ⎞ MHosinα, tanα = (ωo−ωp) τB. ⎝I⎠

ρ

dV p + (μnf + μf ϕξ 2cos2 α cos2α ) ∇2 V. = −∇∼ dt

∂vz ∂vr v + r + = 0, ∂r r ∂z

χ ′ Ho

−

(16)

3μf V

kB Ta mHo . kB Ta 2

(28)

∂∼ p ∂ 2v ∂ vr ∂2vr ⎤ 1 ⎛ ⎞+ + ⎛μnf + μf ϕξ 2cos2 α cos2α ⎞ ⎡ 2r + ⎢ ∂r ⎥ ∂r ∂r ⎝ r ⎠ ∂z 2 ⎦ 4 ⎝ ⎠⎣ ∂v ∂v ∂v v 2 = ρnf ⎡ r + vr r + vz r − θ ⎤, ⎢ ∂t ∂r ∂z r ⎥ ⎦ ⎣

is the root means square value of Hocosωo t and ϕ = nV is

the volume concentration of the particles, τβ =

(27)

Now, the equations of continuity, motion and energy under boundary layer approximations can be written in form

(15)

ξ 2/3 ξ 2/3 ⎞ , tanα = ωo τB ⎜⎛1− , ωp = ωo ⎟. 2 2 2 2 1 1 + + ω τ ωo2 τB2 ⎠ 1 + ωo τB o B ⎝

(26)

In the extant of reduced pressure, Eq. (25) is expressed as

= 6μf ϕ , we eliminate the angle α from Eq.

is Brownian re-

laxation time, V is volume of particle and ξ = According to Eq. (8), the rotation rates in the left- and right-polarized fields prove to be different. Under the two rotating fields, Eq. (15) can be written in the presence of hydrodynamical vortex Ω = (0, 0, Ω) as:

τ = ⎛ s ⎞ M+Hosinα+ + Ω, tanα+ = (ωo−ω p+) τB, ⎝I⎠ (17)

τ M− = χ ′ Hocosα−, ω p− = ⎛ s ⎞ M−Hosinα− + Ω, tanα− = (ωo−ωp −) τB. ⎝I⎠ (18) In above, it is shown that magnetics particles rotations are faster when ωo and Ω have the same sign. So, last terms in Eqs. (17) and (18) are

tanα+ = (ωo−Ω) τB, tanα− = (ωo + Ω) τB.

the first

(M × H ) = −μf ϕΩξ 2cos2 α cos2α.

(13)

M± = (M cos(ωo t −α ), ± M sin(ωo t −α ), 0), ω p± = (0, 0, ± ωp).

I τs

2π , ωo

term yields 2 cos2α and the second term vanishes. Thus, we find

For each of these fields, Eqs. (2)–(4) admit solution with the fluid is quiescent (Ω = 0) and the magnetization rotates when an angular frequency of the field, lagging behind the field at some angle α . Then

M+ = χ ′ Hocosα+, ω p+

(22)

= −2μf Ωϕξ 2cos2 α (cos2 ωo t cos2α + sinωo t × cosωo t sin2α ).

nm2 . 3kB Ta

1 H± = (Hocosωo t , ± sinωo t , 0), H = (H+ + H−). 2

Ho 2

1 (Mθ+ + Mθ−) = ΩτB χ ′ Hocos2 α cos(ωo t −2α ). 2

The magnetization component in Eq. (22) is responsible for the magnetic torque on the particle in a field polarized in θ -direction and it obtains:

In above χ ′ is called the magnetic susceptibility. The magnetic field is considered as a superposition of two rotating fields known as left-hand (subscript +) and the right-hand (subscript−) polarized fields

Here

(21)

−(ωo + Ω) τB cosωo t ).

Hr = Hocosωo t , Hθ = Hz = 0.

M=

χ ′ Ho (sinωo t 1 + (ωo−Ω)2τB2

(10)

Magnetic field is functional in the radial direction as

Using the expression (15)

(20)

−(ωo−Ω) τB cosωo t ),

dV 1 ρnf = −∇p + μo (M . ∇) H + μnf ∇2 V + ∇ × (M × H ), dt 2

Mo = χ ′ H , χ ′ =

χ ′ Ho (sinωo t 1 + (ωo−Ω)2τB2

(29)

∂ 2v ∂ vθ ∂2vθ ⎤ 1 ⎛μnf + μf ϕξ 2cos2 α cos2α ⎞ ⎡ θ + ⎛ ⎞+ 2 ⎢ ∂r ⎝ r ⎠ ∂z 2 ⎥ 4 ⎝ ⎠ ⎣ ∂r ⎦ ∂ ∂ ∂ v v v v v θ θ θ r θ ⎤, = ρnf ⎡ + vr + vz + ∂r ∂z r ⎦ ⎣ ∂t

(30)

∂2vz ∂2vz ⎤ ∂∼ p 1 1 ∂vz + ⎛μnf + μf ϕ ξ 2cos2 α cos2α ⎞ ⎡ 2 + + − ∂z ∂z 2 ⎥ r ∂r 4 ⎝ ⎠⎢ ⎣ ∂r ⎦ ∂vz ∂vz ∂vz ⎤ = ρnf ⎡ + vr + vz , ∂r ∂z ⎦ ⎣ ∂t

(31)

∂T ∂T ∂T ⎤ 1 ∂T ∂ 2T ∂ 2T ⎤ + vr + vz (ρCp )nf ⎡ = knf ⎡ 2 + + ⎢ ∂ ∂ ∂ t r z r r r ∂ ∂ ∂z 2 ⎥ ⎣ ⎦ ⎦ ⎣

(32)

along the following boundaries conditions

at z = 0; vr (r , z , t ) =

αv Ωv r , 1 − βv t

vθ (r , z , t ) =

Ωv r , 1 − βv t

vz = 0,

⎫ ⎪ . T (r , z , t ) = Tw ⎬ at z → ∞; vr (r , z , t ) → 0, vθ (r , z , t ) → 0, T (r , z , t ) → T∞⎪ ⎭

(19)

The θ -components of magnetization due to rotating magnetic field become 533

(33)

Journal of Magnetism and Magnetic Materials 465 (2018) 531–539

M. Hassan et al.

Now, we introduce the following similarity transformation Ωv νf

η=

z 1 − βv t

vz (r , z , t ) =

, vr (r , z , t ) = Ωv ν f

1 − βv t

Ωv r F (η), 1 − βv t

E (η) , θ (η) =

T − T∞ , Tw − T∞

vθ (r , z , t ) =

Ωv r G (η), 1 − βv t

Ωv μ f ∼ p = − 1 − β t P(η) v

Table 1 Comparison of the obtained results with results from the existing literature when ϕ = 0% and α v = 2 .

⎫ ⎪ . ⎬ ⎪ ⎭ (34)

By using Eq. (34) into Eqs. (28)–(33), we obtain to dimensionless nonlinear system of ordinary differential equations is

2F (η) + E′ (η) = 0,

G′ (0)

F ′ (0)

S

Present

Rashidi

Present

Rashidi

−0.1 −0.5 −1

−3.1187 −2.9632 −2.7621

−3.1178 −2.9601 −2.7622

−2.0532 −1.9907 −1.9204

−2.0530 −1.9901 −1.9111

(35)

ρn f

⎡F 2−G 2 + EF ′ + S ⎛F + η F ′⎞ ⎤ 2 ⎠⎦ ρf ⎣ ⎝ ξ2 / 3 ⎛ ⎞ ⎞ 1−ωo2 τB2 ⎛1− 1 + ωo2 τB2 1 ⎜ μnf ⎝ ⎠ ⎟ F ″, 2 = + ϕξ 2⎟ ⎜ μf 4 ξ2 / 3 2 2 ⎛1 + ωo τB (1− )⎞ ⎟ ⎜ 1 + ωo2 τB2 ⎝ ⎠ ⎠ ⎝

(36)

ρn f

⎡EG′ + 2FG + S ⎛G + η G′⎞ ⎤ 2 ⎠⎦ ρf ⎣ ⎝ ξ2 / 3 ⎛ ⎞ ⎞ 1−ωo2 τB2 ⎛1− μ + ωo2 τB2 1 1 ⎜ nf ⎝ ⎠ ⎟ G″, 2 = + ϕξ 2⎟ ⎜ μf 4 ξ2 / 3 2 2 ⎛ 1 + ωo τB (1− )⎞ ⎟ ⎜ 2 2 1 + ωo τB ⎝ ⎠ ⎠ ⎝

(37)

ρn f

⎡EE′ + S (E + ηE′) ⎤ = − ∂P 2 ∂η ρf ⎣ ⎦ ξ2 / 3 ⎛ ⎞ ⎞ 1−ωo2 τB2 ⎛1− 1 + ωo2 τB2 1 2 ⎜ μnf ⎝ ⎠ ⎟ E″, + + ϕξ 2⎟ ⎜ μf 4 ξ2 / 3 2 2 ⎛1 + ωo τB (1− )⎞ ⎟ ⎜ 1 + ωo2 τB2 ⎝ ⎠ ⎠ ⎝ (38)

Fig. 2. The effect of particle concentration on radial velocity profile.

(ρCp)n f

η k Pr[Eθ′ + S θ′] = nf θ″ (ρCp)f 2 kf

(39)

and corresponding boundary condition are

F (0) = α v, F (∞) = 0,

G (0) = 1, E (0) = 0, θ (0) = 1,⎫ . G (∞) = 0, θ (∞) = 0 ⎬ ⎭

In above relations, Pr =

μf Cp f kf

(40)

is modified Prandtl number and

βv

S = Ω is the unsteadiness parameter. In the total viscosity, second term v illustrates the rotational viscosity due to an alternating magnetic field. In the presence of a stationary field, the rotational viscosity always remains positive. However, for ωo τB = 1 with ωo as the frequency of an alternating magnetic field and τB as Brownian relaxation time, reaches a state where the field frequency matches with the relaxation of magnetization and thus, the field effect on the viscosity will vanish. For ωo τB > 1, rotational viscosity produced the negative resulting in a decrease in the viscosity of the ferrofluid. In this paper, the results are calculated by considering the negative viscosity effects on ferrofluid. 2.2. Physical properties In Eqs. (36)–(39), effective density ρnf and heat capacitance (Cp)nf are defined as

ρnf = (1−ϕ) ρf + ϕρs ,

(Cp)nf =

(1−ϕ) (ρCp)f + ϕ (ρCp)s ρeff

Fig. 3. The effect of particle concentration on tangential velocity profile.

(41)

n∗ϕe A∗ ⎞ knf = ⎜⎛1 + ⎟ kf , 1−ϕe A∗ ⎠ ⎝

. (42)

In order to include the shape effects of nanoparticles, the thermal conductivity model can be expressed as [20]

where the parameter A∗ is defined by 534

(43)

Journal of Magnetism and Magnetic Materials 465 (2018) 531–539

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Fig. 6. The effect of highly field frequency on radial velocity profile.

Fig. 4. The effect of particle concentration on axial velocity profile.

Fig. 5. The effect of particle concentration on temperature profile. Fig. 7. The effect of highly field frequency on tangential velocity profile.

A∗

1 = 3

∑ j = a, b, c

kpj−kf kpj + (n∗−1) kf

. (44)

r∗ =

In Eq. (44), n∗ = 3/ ψ1.55 is the empirical shape factor (ψ is the sphericity defined as ratio between the surface area of the sphere and the surface area of the real particle with equal volumes), kf is thermal conductive of base fluid and kpj are thermal conductivities along the axes of the particle which are defined by

(46)

(a2 + v∗)(b2 + v∗)(c 2 + v∗) 2

d (j, v∗) = ×

∫0

kp−kl

⎤k . kpj = ⎡1 + s ⎢ kp {r ∗d (j, 0)−d (j, δ )}−kl {r ∗d (j, 0)−d (j, δ )−r ∗} ⎥ ⎣ ⎦

(a2 + δ )(b2 + δ )(c 2 + δ ) , abc

∞

dw

(45)

(j 2 + v∗ + w∗) (a2 + v∗ + w∗)(b2 + v∗ + w∗)(c 2 + v∗ + w∗) (47)

Here j (=a, band c ) is along the semi-axes directions of the particle kp and kl are the thermal conductivities of the solid particle and its surrounding layer, r ∗ is the volume ratio and d (j, v ) is depolarization factor defined by

with v = 0 for outside of solid ellipsoid and v∗ = δ for outside surface of its surrounding layer. The viscosity model for different shapes of particle is defined as [21] 535

Journal of Magnetism and Magnetic Materials 465 (2018) 531–539

M. Hassan et al.

ω=

102 + 2904p−1855p1.5 + 1604p2 + 80.44p3 1497p + p2

(49)

In above, p is the aspect ratio of nanoparticles. So that aspect ratio is p > 1 for prolate ellipsoids, p = 1 for spheres and p < 1 for oblate ellipsoids.

2.3. Convection heat transfer To understand the convection boundary layers, it is necessary to understand convective heat transfer between a surface and a fluid flowing past it. A thermal boundary layer develops if the fluid free stream and the surface temperatures are different. As a result of temperature difference, energy is exchanged and profiled temperature is produced. The convective heat transfer can be calculated as

Qz = hA (Tw−T∞),

(50)

where A is area of disk and h is the convection heat transfer coefficient of the flow. The heat transfer at the surface by conduction is

Qz = −knf A

Fig. 8. The effect of highly field frequency on axial velocity profile.

∂ (T −T∞) ∂y

. (51)

z=0

These two terms have to be equal; thus

−knf h=

∂ (T − T∞) ∂y z=0

(Tw−T∞)

(52)

2.4. Solution of the problem Due to nonlinear nature of Eqs. (35)–(39), an exact solution is not possible. Now, we opted to go for analytic solution. To this end, we use the Mathematica package BVPh 2.0 which is based on the homotopy analysis method employed for solving nonlinear ordinary differential equations using computational software Mathematica 9. It is found that BVPh 2.0 package provided most accurate solution [22,23]. In this package, it is needed to put appropriate initial guess of solutions and auxiliary linear operators to find the desired solution and they are given as

£E (E ) = Fig. 9. The effect of highly field frequency on temperature profile.

£G (G ) =

Table 2 The values of convection heat transfer coefficient and thermal conductivity corresponding to different values of particle concentration, highly field frequency and particle size for ferrofluid containing spherical particles. Re1/2Nu ϕ

knf

h

Size

knf

h

ωo τB

h

0.613 0.680 0.753

308.406 319.097 328.011

a=b=c=4 a=b=c=8 a = b = c = 12

0.753 0.728 0.720

328.011 323.892 322.670

2 4 6

328.011 328.044 328.057

μnf = (1 + ωϕ) μf ,

£F (F ) =

dG dη

d2G

+

dη2

dF dη

+

d2F , dη2 dθ

, £ θ (θ) = 2 dη

Eo = 0, Fo (η) = α v e−η, ⎫ . Go (η) = e−η, θo (η) = e−2η ⎬ ⎭

⎫ ⎪ + 2⎬ dη ⎪ ⎭ d2θ

(53)

(54)

To check the accuracy of this method, we compare values of F ′ (0) and G′ (0) with the results obtained by Rashidi et al. [24] in Table 1. We get the results for velocity, temperature distribution up to 30th iterations of package. Further, the results for velocity components and temperature up to first iteration are as follow

a=b=c 0.0% 2.5% 5.0%

dE , dη

E=− (48)

where ω is defined as follows

536

2499α v 2499α v −η e , + 1250 1250

(55)

Journal of Magnetism and Magnetic Materials 465 (2018) 531–539

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Table 3 The values of convection heat transfer coefficient and thermal conductivity corresponding to different values of particle concentration, highly field frequency and particle size for ferrofluid containing prolate particles. Size

h

knf

Re1/2Nu ϕ

knf

h

ωo τB

h

0.823 0.774 0.761

338.883 331.430 329.365

2 4 6

338.883 338.921 338.935

a≫b=c 0.0% 2.5% 5.0%

0.613 0.714 0.823

308.406 325.322 338.883

a = 7, b = c = 2 a = 14, b = c = 4 a = 21, b = c = 6

Table 4 The values of convection heat transfer coefficient and thermal conductivity corresponding to different values of particle concentration, highly field frequency and particle size for ferrofluid containing oblate particles. Size

h

knf

Re1/2Nu ϕ

knf

h

ωo τB

h

0.777 0.752 0.745

331.894 327.866 326.737

2 4 6

331.894 331.929 331.943

a=b≫c 0.0% 2.5% 5.0%

0.613 0.692 0.777

308.406 321.358 331.894

a = b = 5, c = 2 a = b = 10, c = 4 a = b = 15, c = 6

Table 5 The values of radial shear and transversal shear stress at the wall corresponding to different values of particle concentration and highly field frequency for ferrofluid containing spherical particles. Re1/2Nu ϕ

τzr |z = 0× 10−5

τzθ |z = 0× 10−4

ωo τB

τzr |z = 0× 10−5

τzθ |z = 0× 10−4

0.0% 2.5% 5.0%

−2.08983 −2.3306 −2.56704

−1.10922 −1.23723 −1.36277

2 4 6

−2.56704 −2.56803 −2.56842

−1.36277 −1.36329 −1.36350

ξ2 / 3 ⎛ ⎞ 1−ωo2 τB2 ⎛1− 1 + ωo2 τB2 257α v 1 2 ⎜ 433α v ρnf ⎝ ⎠ η + 433 μnf η G = 1− + ϕξ 2 2 ⎜ 2000 ρf 2000 4 2000 μf ⎛1 + ωo2 τB2 (1− ξ /23 ) ⎞ ⎜ 1 + ωo τB2 ⎝ ⎠ ⎝

−

⎞ 433S ρnf 433S ρnf 2 433a v ⎟ −η ⎛ 433α v ρnf ⎞ −2η η+ η ++ η e +⎜ e , 4000 ρf 8000 ρf 2000 ⎟ 2000 ρf ⎟ ⎝ ⎠ ⎟ ⎠ (57)

Table 6 The values of radial shear and transversal shear stress at the wall corresponding to different values of particle concentration and highly field frequency for ferrofluid containing prolate particles. Re1/2Nu ϕ

τzr |z = 0× 10−5

τzθ |z = 0× 10−4

ωo τB

τzr |z = 0× 10−5

τzθ |z = 0× 10−4

0.0% 2.5% 5.0%

−2.08983 −2.33071 −2.56705

−1.10922 −1.23722 −1.36276

2 4 6

−2.56705 −2.56804 −2.56843

−1.36276 −1.36329 −1.36349

141 knf 141 (ρCp)nf 141 (ρCp)nf T = ⎜⎛1 + η+ PrSη + PrSη2⎟⎞ e−2η. 250 2000 ( ) 2000 (ρCp)f k ρC f p f ⎝ ⎠ (58)

3. Results and discussion In this section, to understand the behavior of different particle shapes, particle concentration and highly oscillating magnetic field frequency effects on velocity and temperature fields as well as on convective heat transfer coefficient after back substitution of similarity transformations, the results are displayed through graphical and tabular form. To perceive the influences of these parameters, some assumptions are taken to account. Consider that nano-ferrofluid is produced by water and iron nanoparticles of sphere, oblate ellipsoid and prolate ellipsoid shapes. We take sphere particles of 4 nm radius, oblate ellipsoid particle (approximating platelets) having radius 5 nm and 2 nm thickness and prolate ellipsoids (approximating fibers) particles have 7 nm length and 2 nm radius. Moreover, nanolayer thinness around minor axes of each particle is taken 1 nm. The explorations of certain parameters influence on velocity and temperature profiles are represented in Figs. 2–9. The results of radial, tangential and axial velocities profiles with particles concentrations effects are shown in Figs. 2–4 as consider S = 0.5 rotation and stretching speed of disk is 0.057 m / s and 0.028 m / s at time t = 5sec . It is perceived that several velocity lines have been exposed in resultant of different nanoparticles concentrations in these Figures. By different concentration, divergent collisions between neighboring particles in a fluid are happened to produce diverse velocity lines. It is noticed that when the nanoparticle concentration is enhanced, resistance between adjacent layers of moving fluid is enhanced which leads to fall down in velocity profiles. In many studies [8,9,14,15,17], same trend is seen for velocity profiles against particles concentration. It is also seen that the

Table 7 The values of radial shear and transversal shear stress at the wall corresponding to different values of particle concentration and highly field frequency for ferrofluid containing oblate particles. Re1/2Nu ϕ

τzr |z = 0× 10−5

τzθ |z = 0× 10−4

ωo τB

τzr |z = 0× 10−5

τzθ |z = 0× 10−4

0.0% 2.5% 5.0%

−2.08983 −2.33064 −2.56705

−1.10922 −1.23722 −1.36277

2 4 6

−2.56705 −2.56804 −2.56842

−1.36277 −1.36329 −1.36350

ξ2 / 3 ⎛ ⎞ 1−ωo2 τB2 ⎛1− 1 + ωo2 τB2 257 ρnf 257α v 1 2 ⎜ ⎝ ⎠ η F = αv + + ϕξ 2 ⎜ 4000 ρf 2000 4 ξ2 / 3 2 2 ⎛1 + ωo τB (1− )⎞ ⎜ 1 + ωo2 τB2 ⎝ ⎠ ⎝

⎞ 257α v μnf 257Sα v ρnf 257Sα v ρnf 2 257α v2 ρnf ⎟ −η η− η+ η− e + 2000 μf 4000 ρf 8000 ρf 4000 ρf ⎟ ⎟ ⎠ 257α v2 ρnf ⎞ −2η ⎛ 257 ρnf e , + ⎜− + 4000 ρf 4000 ρf ⎟ ⎝ ⎠

(56)

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influence of concentration is dominant on axial velocity as compare to other velocity components. In additions, shapes with fixed sizes of nanoparticles also produced effects on velocity profiles due its impact on physical properties of fluid. In velocities profiles, it is noticed that the maximum velocities are decreased caused by particles of prolate shape, followed by oblate and spherical shape shapes respectively. Fig. 5 shows the effect of particle concentrations on temperature profile under the assumption of surface temperature 300 K and 298 K for free stream region. It is seen that the temperature of ferrofluid is enhanced by increasing nanoparticles concentration of all types of shapes. But, diverse temperature profiles are noted by different shapes of particles contained ferrofluid due to their different impact on thermal conduction and heat capacity of fluid. It seen that maximum temperature of fluid is gain by prolate shape of particles as compare to others and noticed also in [25] that cylindrical particles are more effective as compared to others. Moreover, it is also noted that thermal boundary layer occupied 0.015 m region along z-axis which is minimum 0.005 m than velocity boundary layer region. So, heat slowly diffuses relative to momentum but much improved by particle concentration as compare to base fluid. The effects of field frequency on velocities profiles are illustrated by Figs. 6–8. It is seen that velocity in all directions is improved when field frequency is increased. It is due to oscillating magnetic field which forces particles to rotate faster than fluid. In this consequence, some part of energy of the oscillating magnetic field transforms into kinetic energy of the fluid. This transformation of a part of the alternating field energy into kinetic energy of the fluid just manifests itself in a certain reduction of the total viscosity. This certain reduction of viscosity became as caused to increase the velocity profiles. Similar behavior of results under impact of field frequency for velocity profile is noted as seen in this study [19]. The impact of field frequency on temperature is shown in Fig. 9. In this, it is observed that temperature is increased due to decreasing of field frequency. The second set of results not only display the effects of particle shape and its concentration but also shows the effect of particle size on convective heat transfer coefficient, thermal conductivity and shear stresses under highly oscillating magnetic field. Tables 2–4 illustrate results of convective heat transfer coefficient and thermal conductivity under influence of particle shape, particle size, particle concentration and magnetic field frequency. It is known that convection heat transfer is depending on heat transfer coefficient, thermal conductivity of fluid and temperature difference. When nanoparticles are dispersed in the base fluid, thermal conductivity of fluid is much improved. In these tables, it is noted that convective heat transfer coefficient along with thermal conductivity is improved by increasing particle concentrations which leads to improvement of convection heat transfer. But this development also depends on nanomaterial, particle shape and particle size etc. Chopkar et al. [26] were the first to show experimentally that the effective thermal conductivity of Al70Cu30 nanofluids strongly depends on the nanoparticle size. This is a significant feature of nanofluids. More recently, Kim et al. [27] showed that the thermal conductivity of nanofluids increases linearly with decreasing particle size. In the present study, it is noted that when the size of particle is increased, the values of thermal conductivity and convection heat transfer coefficient are decreased. So, small size particles are more effective for heat transfer. In addition, maximum convection heat transfer is found by particles of prolate ellipsoid shape followed by oblate ellipsoid and spherical shapes respectively. The minimum convective heat transfer is found by spherical particles as compare to other particles. Similar behavior for spherical and prolate particles is noticed in [25,28]. The impact of magnetic field frequency on convective heat transfer coefficient of ferrofluids that contained different shapes of nanoparticles is also shown in these tables. It is noticed in these tables that under all these parameters, the heat transfer coefficient is increased. Results indicate that heat transfer is more improved under magnetization effect but not boost as particle concentration play a rule in enhancement.

The effects of particle shape and its concentration on radial and transversal shear stresses at wall are shown in Tables 5–7. In these, it is seen that the radial and transversal shear stresses at wall are increased in absolute sense by increasing particle concentrations. The main reason in this fact is increment of friction force and this friction is improved through assistant of increasing concentrations of particles. The negative value of stresses which is presented in these tables means the solid surface exerts a drag force on the ambient fluid. It is also noticed that surface exerts a maximum drag force in radial direction as compare to transversal direction. The effect of shape of particles on these wall shear stresses has no significance in influence, but drag force is reduced as taking spherical particles as compare to others. The effect of field frequency on radial and transversal shear stresses is also shown in these tables. It is seen that, transversal shear stress more dominant as compare to radial shear stress in all cases of particle’s shapes and total shear stress is little bit increased by increasing the field frequency. 4. Concluding remarks In this paper, effects of particle shape and size along with highly oscillating magnetic field on fluid flow and heat transfer over rotating stretchable disk are discussed. The physical properties of fluid are affected by different shapes of particles, different particle concentration and highly field frequency which generate an impact on velocity, temperature as well as convection heat transfer. It is perceived that velocity is decreased by increasing of particle concentrations and slower down by taking sphere particles as compare to other. In addition, temperature profile is increased by increasing particle concentration and more improved by taking prolate particles than other shapes. On the other hand, it is found that highly oscillating magnetic field forces the particles to rotate faster than the fluid and as a result total viscosity is certain reduced. In this consequence, an improved in flow speed is observed. In view of convection heat transfer, maximum heat transfer is gained by prolate particles and seen similar behavior in another study [25]. But this improvement can be boosted by small size particles and this trend is also noted in [26,27]. Moreover, the heat transfer is enhancement by magnetic field but not so rapid as the particles' concentration does this. Thus, in heat transfer fluids, nanoparticles can provide new innovative technologies with potential to tailor the heat transfer fluid’s thermal properties through control over particle size, shape, and others. In addition, these results can help design a chemical process, chemical engineers can predict the behavior of fluid in terms of heat transfer, temperature and velocity profiles to design a more efficient and environmental friendly process. References [1] A. Satoh, Rheological properties and orientational distributions of dilute ferromagnetic spherocylinder particle dispersions, J. Colloid Interface Sci. 234 (2001) 425–433. [2] Y. Qi, W. Wen, Influences of geometry of particles on electrorheological fluids, J. Phys. D Appl. Phys. 35 (2002) 2231–2235. [3] B.J. Park, I.B. Jang, H.J. Choi, A. Pich, S. Bhattacharya, H.J. Adler, Magnetorheological characteristics of nanoparticle-added carbonyl iron system, J. Magn. Magn. Mater. 303 (2006) 290–293. [4] P.C. Fannin, L. Vekas, C.N. Marin, I. Malaescu, On the determination of the dynamic properties of transformer oil based ferrofluid in the frequency range 0.1–20 GHz, J. Magn. Magn. Mater. 423 (2007) 61–65. [5] S.B. Trisnanto, Y. Kitamoto, Nonlinearity of dynamic magnetization in a superparamagnetic clustered-particle suspension with regard to particle rotatability under oscillatory field, J. Magn. Magn. Mater. 400 (2016) 361–364. [6] N.S. Gibanov, M.A. Sheremet, H.F. Oztop, Khaled Al-Salem, MHD natural convection and entropy generation in an open cavity having different horizontal porous blocks saturated with a ferrofluid, J. Magn. Magn. Mater. 452 (2018) 193–204. [7] F. Selimefendigil, H.F. Öztop, Natural convection in a flexible sided triangular cavity with internal heat generation under the effect of inclined magnetic field, J. Magn. Magn. Mater. 417 (2016) 327–337. [8] M. Hassan, A. Zeeshan, A. Majeed, R. Ellahi, Particle shape effects on ferrofuids flow and heat transfer under influence of low oscillating magnetic field, J. Magn. Magn. Mater. 443 (2017) 36–44. [9] M. Sheikholeslami, M. Gorji-Bandpy, D.D. Ganji, S. Soleimani, Natural convection

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