Study of three dimensional stagnation point flow of hybrid nanofluid over an isotropic slip surface

Journal Pre-proof Study of three dimensional stagnation point flow of hybrid nanofluid over an isotropic slip surface Nadeem Abbas, M.Y. Malik, S. Nadeem

PII: DOI: Reference:

S0378-4371(19)32224-1 https://doi.org/10.1016/j.physa.2019.124020 PHYSA 124020

To appear in:

Physica A

Received date : 30 March 2019 Revised date : 11 September 2019 Please cite this article as: N. Abbas, M.Y. Malik and S. Nadeem, Study of three dimensional stagnation point flow of hybrid nanofluid over an isotropic slip surface, Physica A (2019), doi: https://doi.org/10.1016/j.physa.2019.124020. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

© 2019 Published by Elsevier B.V.

Journal Pre-proof

Study of three dimensional stagnation point flow of hybrid

Nadeem Abbas2 M.Y. Malik1*, and S. Nadeem2  Department of Mathematics, College of Sciences, PO Box 9004, King Khalid University, Abha 61413, Saudi Arabia. 2 Department of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000, Pakistan. Corresponding author [email protected], [email protected]

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1

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nanofluid over an isotropic slip surface

Abstract

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A three dimensional axisymmetric stagnation flow of hybrid nanofluid impinges on a moving plate with different slip coefficients in two orthogonal directions. On the solid boundary, no slip condition is replaced through the partial slip condition. Such anisotropic slip occurs in geometrically striated surface and super hydrophobic stripes. Using the experimental values, we have discussed the hybrid nanofluid theoretically for boundary layer flow due to stagnation point

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flow. Suitable similarity transformation, transform the Navier-Stokes equations to set of 12th order nonlinear ordinary differential equations. The problem is solved through asymptotic

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analysis for large slip and numerically integration. Effects of physical parameters are also discussed in detail.

Keywords: Hybrid nanofluid; asymptotic behavior; thermal axisymmetric; stagnation point

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flow; moving plate.

1. Introduction

Hybrid nanofluids are a composite mixture of metallic, polymeric or non-metallic nano-sized power with base fluid used to improve the heat transfer rate in different applications. The Hybrid

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nanofluid gain higher heat transfer rate as compared to pure fluid as it is proved experimentally and numerically by numerous researchers. Choi et al [1] was the first to claim about nanofluid which improves the heat transfer rate when compared with pure fluid. He presented his work in

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1995 at the Argonne National Laboratory. At that time, nanofluid was considered to be next generation of heat transfer rate fluids because they offer exciting new possibilities to improve

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heat transfer performance compared to pure fluid. After that, several researchers discussed the nanofluid experimentally and numerically [2-9]. This study opened new directions in engineering applications due to advantageous results after using nanofluids. Several experiments have been done and two types of the particles suspended in the base fluid, namely ‘‘Hybrid nanofluid’’.

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The purpose of combining two different particles in a fluid is to enhance its thermal conductivity which has been proved experimentally. Suresh et al. [10-11] initiated the idea of hybrid nanofluid which was proved through experiments and numerical solutions obtained. According to his views, the hybrid nanofluid boosts the heat transfer rate at the surface as compared to nanofluid and simple fluid. This idea gave a wide range of the researchers to do work in the field

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of hybrid nanofluid. Baghbanzadeh et al. [12] also discussed about the mixture of Multiwall/spherical silica nanotubes hybrid nanostructures and analysis of thermal conductivity

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of associated nanofluid. The analysis of 𝐴𝑙 𝑂

MWCNTs (Aluminium oxide-Multi Walled

Carbon Nanotubes) with base fluid water and their thermal properties are discussed by Nine et al. [13]. According to them spherical particles with hybrid nanofluid reveal a small increment in

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thermal conductivity as compare to cylindrical shape particle. Some recent studies on the hybrid nanofluids are cited [14-20]. On the moving solid boundary, the stagnation point flow is basis in several convective cooling procedures. For the first time, stagnation flow towards the moving plate have been discussed by

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Rott (1956), Wang [21], Libby [22] and was extended by Weidman and Mahalingam [23] while these researchers discussed no slip condition on solid boundary. The partial slip at the solid boundary occurs in the several fields like as rarefied gases, the slip regime exists where the

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Navier-Stokes equation is valid. The equivalence slip is presented on the solid surface which may be porous or rough was discussed by Wang [24]. Recently, a few results have been

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discussed about the stagnation point flow with different assumptions (see Refs. [25-27]). A few coated surfaces like as resist adhesion and Teflon. In all cases, no slip condition is replaced by Navier’s partial slip condition and the velocity slip is assumed to be proportional to the local shear stress. The basic purpose of our study is to gain heat flow rate improvement of Hybrid

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nanofluid flow in the presence of a Navier partial slip condition on the moving plate surface which has not been discussed so far. Our study is motivated due to heating/cooling issues

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appearing in industrial models.

Fig.1: Stagnation flow of Hybrid nanofluid towards a moving plate. The X-axis aligns with the plate striations.

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2. Formulation The stagnation point flow of hybrid nanofluid for fixed Cartesian coordinates on the moving

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plate is considered which is seen in Fig. 1. Align the 𝑋 direction having the striations of the plate, 𝑌 direction normal to the striations and along the 𝑍 direction have the axis of the and 𝑍

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stagnation flow. 𝑈, 𝑉 and 𝑊 be the velocity components 𝑋 , 𝑌

directions

respectively of the moving plate. Navier [28] studied about the partial slip

𝑁𝜎 ,

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𝑈

(1)

Where, 𝑈, 𝜎 and 𝑁 are the tangential velocity, the tangential shear stress and the slip coefficient while the Navier’s partial conditions are satisfied on the plate. If the potential flow is far away from the plate

𝑎𝑋,

𝑉

𝑎𝑌,

𝑊

2𝑎𝑍,

𝜌

𝑝

2

𝑋

𝑌

𝑝 ,

(2)

al

𝑈

here, 𝑈, 𝑉 and 𝑊 are the velocity components aligns the Cartesian coordinates (𝑋, 𝑌, 𝑍)

nanofluid is 𝜌

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respectively. The strength of stagnation flow is 𝑎, the pressure is 𝑝, the density of Hybrid , the kinematic viscosity of Hybrid nanofluid is 𝑣

and the stagnation

pressure is 𝑝 . In the Newtonian fluid flow, the constant characteristics of the Navier Stokes

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equations are defined as follows (see Refs. [29-34] )

𝑈𝑈

𝑉𝑈

𝑈

𝑊𝑈

𝑉 𝑃 𝜌

𝑊

0,

𝑣

(3)

𝑈

𝑈

𝑈

,

(4)

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𝑈𝑉

𝑉𝑊

𝑊𝑉

𝑃 𝜌

𝑣

𝑉

𝑊𝑊

𝑃 𝜌

𝑣

𝑊

𝑉

𝑉

𝑊

,

𝑊

(5)

.

(6)

of

𝑈𝑊

𝑉𝑉

𝑈 ⎧ 𝑉 ⎪ ⎪ 𝑊 ⎨ ⎪ ⎪ ⎩

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By utilizing Eq. (2), the similarity transforms are chosen as follows

𝑎𝑋𝑓 𝜉 𝑢ℎ 𝜉 , 𝑎𝑌𝑔 𝜉 𝑣𝑘 𝜉 , 𝑎𝑣 𝑓 𝜉 𝑔 𝜉 , 𝑎 𝑍. 𝑣

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𝜉

(7)

With these transformations the continuity equation is identically satisfied. After applying the suitable similarity transformation the momentum and energy equations reduces to the system of

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non-linear ordinary differential equations, which are as follow

1

Φ

.

1

Φ

.

1

G F′′

Φ

.

1

Φ

.

F

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1

Φ

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1

FG

1

Φ

F F′′

1

GG

Φ

G

Φ

1

ρC ρC

Φ

ρC ρC

Φ 1

(8)

0,

1 1

.F

Φ 0,

ρC ρC

Φ

ρC ρC

.G (9)

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1 .

Φ

1

.

Φ

1

FH

1

Φ

GH

Φ

HF′

Φ

Φ

ρC

1

.

Φ

1

FK

Φ

GK

1

KF′

Φ

Φ 0.

𝑎𝑋, 𝑉

𝑎𝑌, 𝑊

2𝑎𝑍, 𝑝

𝑢

𝑁

𝜇 𝜇

→ → → →

𝜕𝑈 , 𝜕𝑍

𝑉

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𝑈

𝜉 𝜉 𝜉 𝜉

Φ

ρC

𝑌

𝑋

ρC

.K

ρC

𝑣

𝑁

𝜇 𝜇

𝜕𝑉 , 𝜕𝑍

Here, the dynamic viscosity of hybrid nanofluid is 𝜇 coefficients along the 𝑋 , and 𝑌

(11)

𝑝 , as 𝜉 → ∞

1 as 𝜉 → ∞, 1 as 𝜉 → ∞, 0 as 𝜉 → ∞, 0 as 𝜉 → ∞,

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F ⎧ G ⎨H ⎩K

ρC

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Subject to the potential flow boundary conditions

𝑈

(10)

p ro

.

Φ

ρC

0, 1

1

.H

ρC

of

1

ρC

(12)

𝑎𝑠 𝜉 → 0.

𝜌

𝑣

(13)

. 𝑁 and 𝑁 are the slip

directions respectively. By applying the similarities

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variables, the above equations are reduced into the following forms

𝐹 0

𝛾

𝐹

0 , 𝐺 0

𝛾

𝐺

0 , 𝐻 0

1

𝛾

𝐻′ 0 , (14)

𝐾 0

1

𝛾

𝜇 𝜇

𝐾 0 ,

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𝛾

𝑁

are the dimensionless slip factors while it is also noted that normal velocity on

plate is zero which is defined as 𝐹 0

𝐺 0

0 without loss of the generality to the velocity

The no slip effects ( 𝛾

0,

𝐺 0

0.

(15)

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𝐹 0

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which can set as

0 ) on stagnation point flow on plate was analyzed by Heimenz (1911).

The numerical integration [35] of the Eqs. [8-9] by applying the bvp4c algorithms to execute the 0

1.31033 and 𝐺

result is the potential flow 𝐹

𝐺

𝜉, 𝐻

𝐹

𝜉

𝐺

𝜉

0 𝐾

0. Suppose 𝜖 ≡ ≪ 1 and expand

𝜖𝐹

𝜖 𝐹

⋯,

(16)

𝜖𝐺

𝜖 𝐺

⋯,

(17)

𝜖𝐻

𝜖 𝐻

⋯,

(18)

𝐾

𝜖𝐾

𝜖 𝐾

⋯,

(19)

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𝐻

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Eqs (8

1.31033. For the full slip 𝛾 → ∞ mean that

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initial values of 𝐹

11) are the set of nonlinear coupled ordinary differential equations with boundary

conditions (12 , 14 , and 15 for the case of isotropic slip surface.

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3. Asymptotic behavior for large slip Consider that 𝛾 and 𝛾 are the slip factors which are very large when compared to unity. Eq. (12) then suggest the expansions of the following form

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𝐹

𝜒 𝜉 𝛾

𝐴

⋯, 𝐺

𝜉

𝜓 𝜉 𝛾

𝐴

(20)

1, 2, 3, …) are constants and 𝜒, 𝜓 ≪ 1. Eqs. (8-9) linearize to .𝜒

𝜉

.𝜓

𝜉

.

2𝜉𝜒 ′ 𝜉

2𝜒 𝜉 ,

2𝜉𝜓

2𝜓 𝜉 .

𝜉

p ro

.

.

⋯,

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Here, the 𝐴 (𝑖

𝜉

.

(21)

(22)

The corresponding boundary conditions Eqs. (12, 14) becomes 0,

𝜓 0 and Eqs. (16

𝜒′′ 0

0,

1,

𝜓′′ 0

𝜒′ 𝜉 → 0 𝑎𝑠 𝜉 → ∞,

1,

19) have been integrated with the results of 𝜒 0

0.526126. Subsequently, Eqs. (16 𝐹 0

1

.

0.526126 and 𝜓 0

19) are identical. Therefore for large slip

⋯,

𝐺 0

1

.

(24)

⋯

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The decline rate for large 𝜉 will be

(23)

𝜓′ 𝜉 → 0 𝑎𝑠 𝜉 → ∞,

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𝜒 0

𝜒 𝜉

𝑐 𝜉.

(25)

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Applying the variation parameters the solutions that declines at infinite are 𝜒 𝜉

𝑑𝜉,

𝜉

where 𝐴

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.

(26)

⁄

𝑒

𝜉

. .

The asymptotic behavior for large 𝜉, applying the integration by parts 𝜒~ 𝐴

⁄

𝑒 𝜉

.

(27)

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This fast decline allows numerical integration to be very efficient. For lateral movement at large slip, we consider

Eqs (11

𝜙 𝜉 𝛾

⋯,

𝜅 𝜉 𝛾

𝐾

12) are reduced by using the Eqs (18

19), and are defined as follow

1 .

Φ

1

.

Φ

1

2𝜉𝜙 𝜉

φ

1

𝜙 𝜉

Φ

Φ

0,

1

Φ

.

1

Φ

.

1

2𝜉𝜅 𝜉

Φ

1

𝜅 𝜉

𝜅 0

1,

ρC

Φ

ρC

𝜅 𝜉 → 0,

ρC

Φ

𝜉

𝜅

𝜉

(29)

ρC

ρC

(30)

ρC

𝜅 𝜉 → 0, 𝑎𝑠 𝜉 → ∞.

(31)

30) subject to the boundary conditions (31) are solved numerically by applying the

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Eqs (29

1,

Φ

Φ

𝜙

0,

the corresponding conditions are 𝜙 0

ρC

Pr e-

1

ρC

p ro

1

(28)

⋯,

of

𝐻

bvp4c technique, and obtain the values of 𝜙 0

urn

values we can write

𝐻 0

0.630343 𝛾

⋯,

𝐾 0

0.630343 and 𝜅 0

0.630343 𝛾

0.630343 with these

⋯

(32)

For the general 𝛾 the original equations can be solved easily. Eqs (8, 9, 12, 14 and 15) are

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integrated numerically. More accurate results of the initial values are given in Tables [1-2]. Table-1: The initial and final values of 𝛾 for different values of slip factor 𝛾 and 𝛾

for the

hybrid nanofluid. The values in the each cell from top: 𝐹′ 0 , 𝐺′ 0 , 𝐾 0 and 𝐻 0 . The results from approximate formulas Eq. (24) and Eq. (32) are given in parenthesis.

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𝛄𝟐 𝛄𝟏

𝛄

𝟎

𝛄𝟏

𝛄𝟏

𝟎. 𝟏

𝛄𝟏

𝟎. 𝟐

𝛄𝟏

𝟎. 𝟓

𝛄𝟏

𝟏 

𝛄𝟏

𝟐 

0

0.179677

0.311965

0.545037

0.713014

0.835916

0

0

0

0

0

0

0

0

1

0.860584

0.745032

0.517791

0.337205

0.19700

0

1

1 

1 

1 

1 

1 

0.25

0

0.180515

0.314201

0.55019

0.719429

0.841726

0.25 

0

0.0511468

0.099742

0.22626

0.380542

0.562789

0.25 

1

0.859442

0.74182

0.509764

0.32671

0.187206

0.25 

1

0.96197

0.923622

0.816185

0.673372

0.490662

0.5

0

0.181281

0.316061

0.553456

0.722243

0.843261

0.5 

0

0.0984738

0.184695

0.377134

0.560483

0.726423

0.5 

1

0.858401

0.739163

0.504744

0.322204

0.18469

0.5 

1

0.92527

0.853791

0.678204

0.494107

0.315608

0.75

0

0.181981

0.317614

0.555644

0.723762

0.843945

0.75 

0

0.142213

0.257054

0.481598

0.661105

0.801479

0.75 

1

0.857452

0.736954

0.501407

0.319793

0.183578

0.75 

1

0.890105

0.791038

0.576098

0.387707

0.231774

1.0

0

0.182621

0.318921

0.557194

0.724704

0.844331 

1.0 

0

0.182621

0.318921

0.557194

0.724704

0.844331 

1.0 

1

0.856585

0.735103

0.499058

0.318307 

0.182954 

1.0 

1

0.735103

0.499058

0.318307 

0.182954 

p ro

Pr e-

al

urn

Eqs. (16

0.856585

19) have been integrated with the results of 𝜒 0

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0.564871. Subsequently, equations (19 𝐹 0

1

.

⋯,

𝟓 

0.928402 (0.8948) 0 0.0872909 (0.1261) 1  0.931973 (0.8948) 0.770963 0.0811592 (0.1261) 0.265617 0.932395 (0.8948) 0.872654 (0.7895) 0.0804606 (0.1261) 0.15001 (0.2521) 0.932555 (0.8948) 0.911877 (0.8597) 0.0801968 (0.1261) 0.104415 (0.1681) 0.932639  (0.8948)  0.932639  (0.8948)  0.0800584  (0.1261)  0.0800584  (0.1261) 

of

0

𝛄𝟏

0.564871 and 𝜓 0

20) are identical, therefore, for large slip 𝐺 0

1

.

⋯

(33)

𝛄𝟏

𝟏𝟎 

0.963106 (0.9474) 0 0.0452321 (0.06301) 1  0.965201 (0.9474) 0.872895 0.0416155 (0.06301) 0.149621 0.965328 (0.9474) 0.93277 (0.8948) 0.0414033 (0.06301) 0.0798425 (0.1261) 0.965374 (0.9474) 0.954309 (0.9298) 0.0413282 (0.06301) 0.0544332 (0.0840) 0.965397 (0.9474) 0.965397 (0.9474) 0.0412894  (0.06301)  0.0412894  (0.0630) 

Journal Pre-proof

Eqs (29

30) subject to the boundary conditions (33) are solved numerically by applying the

bvp4c technique, which gives 𝜙 0

0.676772 and 𝜅 0

0.676772 𝛾

⋯,

𝐾 0

0.676772 𝛾

⋯

(34)

of

𝐻 0

0.676772. Thus

Table-2: The initial and final values for different slip factor 𝛾 and 𝛾

for the Nanofluid. The

p ro

values in the each cell from top: 𝐹′ 0 , 𝐺′ 0 , 𝐾 0 and 𝐻 0 . The results from approximate formulas Eq. (33) and Eq. (34) are given in parenthesis. 𝛄𝟐 𝛄𝟏

𝛄

𝛄𝟏

𝟎

𝛄𝟏

𝟎. 𝟏

𝛄𝟏

𝟎. 𝟐

0

0.151154

0.2687

0

0

0

0

0

1

0.884114

0

1

0.25

𝟎. 𝟓

0.492349

𝛄𝟏

𝟏 

0.668505

Pr e-

0

𝛄𝟏

𝛄𝟏

𝟐 

0.805788

0

0

0.783957

0.571635

0.386335

0.231972

1 

1 

1 

1 

1 

0

0.151761

0.27043

0.496878

0.674746

0.811938

0.25 

0

0.0420657

0.082527

0.191217

0.331629

0.510154

0.25 

1

0.883296

0.781512

0.564699

0.376249

0.221675

0.25 

1

0.968914

0.937421

0.846909

0.719907

0.544876

0.5

0

0.152326

0.5 

0

0.0815616

0.5 

1

0.5 

1

0.75

0

0.75 

0

0.75  1.0

0.499981

0.677835

0.813841

0.154975

0.328153

0.507446

0.682912

urn

0.271919

0.882537

0.779418

0.56000

0.371348

0.218573

0.93867

0.878998

0.724764

0.548862

0.363421

0.15285

0.273203

0.502184

0.679611

0.814732

0.118608

0.21851

0.428328

0.612125

0.766385

1

0.881834

0.777619

0.55669

0.368558

0.217133

1

0.909398

0.825205

0.629345

0.440404

0.271412

0

0.153338

0.274315

0.503808

0.680752

0.815246

Jo

0.75 

al

0

𝛄𝟏

𝟓 

0.913751 (0.8870) 0 0.104903 (0.1354) 1  0.917825 (0.8870) 0.732357 0.0979268 (0.1354) 0.308583 0.918402 (0.8870) 0.848158 (0.7741) 0.0969707 (0.1354) 0.178262 (0.2707) 0.918627 (0.8870) 0.894124 (0.8494) 0.0966009 (0.1354) 0.125161 (0.1805) 0.918747 (0.8870)

𝛄𝟏

𝟏𝟎 

0.95527 (0.9435) 0 0.0547702 (0.0677) 1  0.957736 (0.9435) 0.848496 0.0505178 (0.0677) 0.177719 0.957918 (0.9435) 0.918935 (0.8870) 0.0502149 (0.0677) 0.0960951 (0.1354) 0.957984 (0.9435) 0.944681 (0.9247) 0.0501058 (0.0677) 0.0658224 (0.0902) 0.958018 (0.9435)

Journal Pre-proof

1.0

0

0.153338

0.274315

0.503808

0.680752

0.815246

1.0

1

0.881181

0.776066

0.554261

0.366777

0.216306

1.0

1

0.881181

0.776066

0.554261

0.366777

0.216306

of

4. Thermal axisymmetric stagnation flow

0.918747 (0.8870) 0.0964046 (0.1354) 0.0964046 (0.1354)

p ro

As discussed in [35], the temperature far away from the plate is considered to be 𝑇

0.958018 (0.9435) 0.0500496 (0.0677) 0.0500496 (0.0677)

And

temperature on the surface is 𝑇 . The energy equation takes the form 𝑈𝑇

𝑉𝑇

𝑊𝑇

𝛼

𝑇

𝛩 𝜉

𝑇 𝑇

𝑇 . 𝑇

𝑇

Pr e-

Introducing

𝑇

.

(35)

(36)

After applying the above transformation, the dimensionless energy equation becomes κ κ Θ 1

Φ

1

Φ

Φ

ρC

Φ

ρC

𝐺

𝐹 𝛩

0,

(37)

ρC

al

Pr

ρC

while Pr is Prandtl number. The temperature slip may exist on the plate surface which was

urn

presented by White (2006) and Shidlovskiy (1967). The temperature slip condition similar to Navier’s Stocks condition is 𝑘

𝜕𝑇 𝜕𝑧

ℎ

𝑇

𝑇 ,

𝑎𝑠 𝑧 → 0.

(38)

Jo

The dimensionless form of the above equation is defined as follows Θ 0

γ

k k

Θ′ 0

1.

(39)

For away from the surface of plate, the boundary condition is written as 𝑇 → 𝑇 , 𝑎𝑠 𝑧 → ∞,

(40)

Journal Pre-proof

and the dimensionless form is written as Θ ∞

0,

(41)

The velocity slip factor 𝛾 that enter in the function 𝐹 and 𝐺 while the thermal slip effect introduce the two additional parameters γ

. For the large parameter 𝜖 ≡ ≪ 1

of

and α

yield the dominant equation 𝜖 Θ

𝜖Θ

⋯

p ro

Θ

(42)

Such that 𝛾 is very large, after using the Eqs (16, 17, 42) in equations (37, 39, 41), we obtain the following equations

Pr

1

φ

1

Φ

Φ

Θ 0

Pr e-

κ κ θ

ρC

Φ

ρC

S

k k

Θ ′ 0

ρC

2𝜉Θ

0,

(43)

ρC

1.

(44)

al

The solution of above initial values problem is directly defined as 𝐶𝑒𝑟𝑓𝑐

urn

Θ

where 𝑃

Jo γ

(45)

and 𝐶 is found from conditions at zero 1

𝐶

where 𝛽

𝑃 , 𝜉

1

𝛽

2𝑃 𝜋

,

. For large 𝛾, temperature is defined as

(46)

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2𝑃 𝜋

Θ 0 1

𝛽

.

𝑂 𝛾

2𝑃 𝜋

(47)

Θ 𝜉

𝐹

1,

0

𝐺

𝐹 𝜉 𝐺′′ 𝜉 𝛽 𝛾 𝐹 0

0

and 𝐺 0 𝐹 0 𝐹 0 𝛾 𝐺 0

𝛾

0

1

,

(48)

𝛽

1

(49)

,

Pr e-

Θ′ 0

𝛾 𝐺

p ro

9). Thus for 𝑃

Eqs (8

of

We noted from [35] that 𝛩 is proportional to 𝐹′′ and 𝐺 . Eq (43) is identical to the differential

for the arbitrary 𝑃 and large values of 𝛾

𝛾

𝛾 , 𝐹 and 𝐺 ~ 𝜉 and solution of Eq (43) is

proportional to erfc( 𝑃 𝜉).the initial values found to be 2 𝑃 ⁄𝜋

Θ 0

1

2𝛽

𝑃 ⁄𝜋

𝑜 𝛾

,

(50)

for the general 𝛾, Eqs (41, 43, 44)are integrated numerically. The numerical values are calculated

al

in the Tables [3-4] for both Hybrid nanofluid and Nanofluid.

urn

Table-3: Thermal initial values of Hybrid nanofluid for axisymmetric flow. The values in the bracket are calculated from Eq. (50). 𝐏𝐫

𝛄

𝟎. 𝟎

𝛄

𝟎. 𝟐

𝛄

𝟎. 𝟓

𝛄

𝟏 

𝛄

𝟐 

1

-2.51242

-1.42303

-0.862236

-0.520419

-0.290273

6.2  2 6.2  3 6.2  4

-2.64461 -2.69537 -2.72215

-1.4645 -1.47993 -1.48797

-0.877286 -0.882800 -0.885654

-0.525864 -0.52784 -0.528859

-1.45819

-1.00961

-0.69083

-0.452635

-1.51921 -1.54268 -1.55508 -0.74996

-1.03849 -1.0494 -1.05513 -0.610461

-0.70423 -0.709233 -0.711843 -0.477291

-0.458349 -0.460463 -0.461562 -0.350029

-0.291959 -0.292567 -0.292880 -0.267896 (-0.2609) -0.269888 -0.270619 -0.270998 -0.228289

2.0  1 2.0  2.0  2.0  0.5 

2 3 4 1

Jo

6.2

𝛄𝟏

𝛄

𝟓 

-0.124758 (-0.1240) -0.125068 -0.125179 -0.125237 -0.120434 (-0.1190) -0.120835 -0.120981 -0.121057 -0.11172

𝛄

𝟏𝟎 

-0.0639669 (-0.0638) -0.0640485 -0.0640777 -0.0640927 -0.0628108 (-0.0624) -0.0629197 -0.0629593 -0.0629798 -0.0603557

Journal Pre-proof

0.5  2 0.5  3 0.5  4

-0.77117 -0.77934 -0.78365

-0.624444 -0.629784 -0.632596

-0.485796 -0.489022 -0.490716

( -0.2165) -0.230217 -0.230939 -0.231316

-0.354581 -0.356297 -0.357195

(-0.1088) -0.11218 -0.112351 -0.11244

(-0.0595) -0.0604896 -0.0605393 -0.0605652

are calculated from Eq (50). 𝛄𝟏

𝟎. 𝟎

𝛄

𝟎. 𝟐

𝛄

6.2

1 -2.63778

-1.55646

-0.963806

-0.589623

6.2  6.2  6.2  2.0 

2 -2.80721 3 -2.87409 4 -2.90983 1 -1.5358

-1.61393 -1.63582 -1.64733 -1.09349

-0.98554 -0.993658 -0.997895 -0.763608

-0.597686 -0.600662 -0.602208 -0.508125

2.0  2.0  2.0  0.5 

2 3 4 1

-1.61538 -1.64689 -1.66373 -0.79413

-1.13324 -1.14866 -1.15682 -0.65676

-0.782782 -0.790108 -0.793963 -0.52146

-0.516545 -0.519724 -0.52139 -0.388178

-0.331907 (-0.3281) -0.334447 -0.335377 -0.335858 -0.304422 (-0.2974) -0.307424 -0.308548 -0.309134 -0.256869

0.5  2 -0.82252 0.5  3 -0.83375 0.5  4 -0.83976

-0.67605 -0.68363 -0.68767

-0.533548 -0.538258 -0.540757

-0.394837 -0.39741 -0.398771

-0.259768 -0.26088 -0.261465

𝛄

𝟏 

𝛄

𝟐 

𝛄

𝟓 

-0.143604 (-0.1429) -0.144078 -0.14425 -0.144339 -0.138206 (-0.1367) -0.138821 -0.13905 -0.139169 -0.127491 (-0.1243) -0.128201 -0.128471 -0.128613

𝛄

𝟏𝟎 

-0.0738114 (-0.0736) -0.0739363 -0.0739816 -0.074005 -0.0723586 (-0.0720) -0.0725269 -0.0725893 -0.0726217 -0.0693088 (-0.0684) -0.0695181 -0.0695975 -0.0696391

al

5. Results and Discussions

𝟎. 𝟓

p ro

𝐏𝐫

Pr e-

𝛄

of

Table-4: Thermal initial valued of Nanofluid for axisymmetric flow. The values in the bracket

By applying the similarity transforms, three dimensional stagnation flow of Hybrid nanofluid on

urn

the moving plate having an isotropic slip is governed by a set of nonlinear ordinary differential equations which solved numerically. The results of the exact similarity solutions of the NavierStokes equations are discussed in this study. An isotropic slip occurs on the rough striated surface and also on super hydrophobic nano surface. The slip factor can also be considered as the

Jo

ratio of slip length into the plate and is extended velocity profile to a nominal length. The solution is obtained by applying the bvp4c technique. First of all, we noted that the functions 𝜓 and 𝜒 are partially non-coupled from 𝐹 and 𝐺 respectively. The initial value problem start having the equation (16) for the given slip factors 𝛾

and 𝛾

respectively, while a guessed

Journal Pre-proof

𝐹

0 and 𝐺

0 . Now we check the values of 𝐹 𝜉

1

𝐺 𝜉

1

→ 0 for the large

values of 𝜉. The quantity of 𝐹 0 and 𝐺 0 denoted as a slip velocities due to the stagnation point flow. The quantities 𝐾 0 and 𝐻 0 are the fluid slip velocities due to moving plate. Suppose that 𝛾

of

be the ratio of slip factors. The range of 𝛾 is taken to be 0.1

having ‘‘1’’ being the case of isotropic. In the case of isotropic, we noted that 𝐹

1 and

𝐺 and 𝐾

Our large slip asymptotic solution agrees well with the numerical results when 𝛾

p ro

𝛾

𝐻.

10. It is

noted that as the slip factor increases from zero, the normalized slip velocities 𝐹 0 and 𝐺 0 are increased in both cases Nanofluid and Hybrid nanofluid as shown in Tables [1-2]. It is also

Pr e-

noted that velocity profiles 𝐹 𝜉 and 𝐺 𝜉 increases for large values of slip parameter of Hybrid nanofluid and Nanofluid respectively as shown in Figs. (3) and (4). The universal function 𝐹 𝜉 and 𝐺 𝜉 are governed by the stagnation flow only, and are not influenced by the lateral motion of the plate. Some typical similarity velocities as shown in Tables [1-2]. The slip velocities due to lateral are described by the universal functions 𝐻 0

1 across the striations. For no slip, 𝐻 0 , 𝐾 0 are unity and decline to zero as slip

al

and 𝐾 0

1 aligns the striations

parameter is increased. Note that the velocities due to lateral motion do depend on the stagnation

urn

flow. Typically universal profiles are revealed in Tables [1-2] for both Nanofluid and Hybrid nanofluid.

Present study is focused on heat transfer of Hybrid nanofluid and nanofluid for stagnation slip flow on moving plate. The exact similarity solution in closed form is obtained for energy

Jo

equation. The flow due to literal motion of the plate depends heavily upon the velocity slip parameter 𝛾

𝛾

𝛾

of Hybrid nanofluid and nanofluid both on the boundary and in

governing equations through the stagnation flow. The following concludes that asymptotic formulas resulting from Eqs (47) and (50) agrees for large 𝛾 and 𝛽 as revealed in Tables [3-4] for

Journal Pre-proof

both hybrid nanofluid and simple nanofluid respectively. The heat transfer increases with increase in 𝛾 , decreases with increase in 𝛾 and increases for Prandtl number (Pr). It is seen that Θ 0 is high for axisymmetric stagnation flow of Hybrid nanofluid and Nanofluid. But for three

of

dimensional stagnation flow the heat transfer rate for the axisymmetric case is actually less. Due to the fact, the normal velocities at infinity are equal and one should divide Eq. (50) through √2. 0. Temperature profile increases for the increment of

p ro

However, the denominator is larger for 𝛾

Φ and thermal slip parameter 𝛽 as revealed in Figs. (7-8) both for Hybrid nanofluid and Nanofluid.

Pr e-

6. Final results

Three dimensional Hybrid nanofluid stagnation flow and heat transfer on the moving plate is considered in this study. The stagnation flow with an isotropic slip and asymptotic behavior for large slip in Hybrid nanofluid is also discussed here. The following key factors are concluded: The asymptotic formulas and the exact solution are accurate for large slip 𝛾 and 𝛽.



With the increment of physical parameters 𝛽 and Pr temperature profile for both cases

al



Hybrid nanofluid and Nanofluid increases. 𝐻 0 and 𝐾 0 are unity for no slip condition and decreases to zero as slip is increased.



Slip factors increases with the increase in normalized slip velocities 𝐹′ 0 and 𝐺′ 0 .



An isotropic is reflected in the various slip velocities 𝐹′ 0 and 𝐺 0 . It is found that

urn





Jo

relative velocities are higher at small slip. Slip velocities increases with increase in slip parameter in both cases Hybrid nanofluid and Nanofluid.

Pr e-

p ro

of

Journal Pre-proof

Jo

urn

al

Fig. 2: Impact of Φ on the 𝐹′ 𝜉 and 𝐺′ 𝜉 .

Fig. 3: Impact of γ on the 𝐹′ 𝜉 .

Pr e-

p ro

of

Journal Pre-proof

Jo

urn

al

Fig. 4: Impact of γ on the 𝐺′ 𝜉 .

Fig. 5: Impact of Φ on the 𝐻 𝜉 .

Pr e-

p ro

of

Journal Pre-proof

Jo

urn

al

Fig. 6: Impact of Φ on the 𝐾 𝜉 .

Fig. 7: Impact of Φ on the 𝜃 𝜉 .

Pr e-

p ro

of

Journal Pre-proof

Fig. 8: Impact of β on the 𝜃 𝜉 .

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35. Wang, C. Y. (2006), Stagnation slip flow and heat transfer on a moving plate, Chemical Engineering Science, 61(23), 7668-7672. Acknowledgement: The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University, Abha 61413, Saudi Arabia for funding this work through research groups program under grant number R.G.P-2/29/40.

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Nomenclatures Prandtl number

Φ

Nanoparticle of Al 𝑂

Φ

Nanoparticle of 𝐶𝑢

𝛩

Temperature profile Velocities profiles

ρ

Fluid Density

p ro

𝐹 and G

of

Pr

𝛾 and 𝛾   T

Wall temperature

T

Pr e-

Velocity Slip Factors

Ambient temperature

ν

Fluid kinematic Viscosity

νnf

Nanofluid kinematic Viscosity

νhnf

Heat capacity of hybrid nanofluid

hnf

al

ρCp

Hybrid nanofluid kinematic Viscosity

κ

Thermal conductivity of fluid

κnf

urn

Thermal conductivity of nanofluid

κhnf

Thermal conductivity of hybrid nanofluid

μhnf ρCp nf αhnf αnf

Jo

μnf

𝛾 and 𝛾  

Viscosity of hybrid nanofluid Viscosity of nanofluid Heat capacity of nanofluid

Thermal diffusivity of hybrid nanofluid Thermal diffusivity of nanofluid Thermal slip factors