MSC: 76U05, 76N20, 76A02

The inability of most uids to meet the in-

dustrial thermophysical requirements promp-

ted Maxwell [1] to consider suspending solid

particles, mostly millimetre-sized, in uids.

The development improved the thermal and

electrical characteristics of the uid but the

solid particles, being heavy, settle at the wall

and give rise to challenges such as clogging and

pipe erosion. The suggestion of Choi and East-

man [2] to replace the millimetre-sized particle

with a nanometre-sized particle revolutionised

the study of uid ow. The idea resolved the

clogging and pipe erosion problem of Maxwell's

approach [3, 4]. Since the innovation, authors

have considered the suspension of more than

one type of nanoparticle. Lee et al. [5] showed

that the thermal conductivity of the base uid

increased by more than 20% when two nano-

Coriolis Effect on the Flow of Water Carrying CNT, Graphene, and

Alumina Nanoparticles over a Heated Moveable Non-porous Surface

OLUWASEUN BIODUN ONUOHA1, FAWWAZ BATAYNEH2,

ABAYOMI SAMUEL OKE3,*,a, MARIO RASO4

1Department of Mathematical Sciences, Adekunle Ajasin University, NIGERIA

2Mathematics Division, College of Engineering, Kuwait College of Science and Technology, Doha, KUWAIT

3Department of Mathematical Sciences, Adekunle Ajasin University, NIGERIA

4Department of Computer Science, Sapienza University of Rome, ITALY

*Corresponding Author

aORCiD: 0000-0003-3903-4112

Abstract: Heat transfer fluids, heat exchangers, and coolants in electronics are typical industrial applications

where improved fluids are required for optimal performance. Stemming from the increasing demand, this study

examines the effects of suction, heat source and stretch- ing ratio on the flow and heat transfer in a

magnetohydrodynamic ternary hybrid nanofluid across a moveable rotating surface. Carbon nanotubes,

graphene and alumina are considered as the nanoparticles with water as the base fluid. The governing equations

are transformed from partial to ordinary differential equations. The equilibrium point of the system was ob-

tained, the conditions for the system stability were established and the emerging parameters were chosen within

the acceptable interval. The equations are numerically solved using the MATLAB bvp4c solver. The effects of

the flow parameters on the velocity and temperature distribution are graphically illustrated. The analysis shows

that the stretching ratio reduces flow temperature and velocity but increases skin friction. Coriolis force

enhances the heat transfer rate and increases the primary skin friction. Heat source increases flow temperature

and secondary skin friction.

Keywords: Ternary hybrid nanofluid; Coriolis force; Heat source; Suction; Stretching ratio

Received: February 25, 2024. Revised: September 7, 2024. Accepted: October 8, 2024. Published: November 21, 2024.

1. Background Information

International Journal of Applied Sciences & Development

DOI: 10.37394/232029.2024.3.26

Oluwaseun Biodun Onuoha, Fawwaz Batayneh,

Abayomi Samuel Oke, Mario Raso

E-ISSN: 2945-0454

284

Volume 3, 2024

particles were suspended in the base uid and

Baghbanzadeh et al. [6] showed that the heat

transfer rate increased by 14%. Suresh et al.

[7] considered the hybrid nanouid of copper-

alumina hybrid nanouid, Hayat and Nadeem

[8] studied silver-copper(II)oxide in water and

Oke et al. [9] considered the ow of copper-

alumina in water. Other recent studies on hy-

brid nanouid can be found in [10, 11, 12]. Re-

cently, it has been shown that ternary hybrid

nanouids enhance the thermal and electrical

properties of the uids better than the nano-

uids or hybrid nanouids [13]. Mousavi et al.

[14] examined the impact of volume fraction

and temperature on the thermophysical proper-

ties of a water-based ternary hybrid nanouid

of CuO, MgO, and TiO. The results showed

an improvement in the thermal conductivity

of the ternary hybrid nanouid with increasing

volume fraction. Furthermore, Elnaqeeb et al.

[13] and Oke [15] considered the suspension of

carbon nanotubes, graphenes and alumina in

water. The study considered the eect of vari-

ous particle shapes and sizes on the thermal

properties of the uid. The results show that

temperature is enhanced at a small volume frac-

tion.

Power generation, cancer tumour treatment

and magnetic devices for cell separation are

all practical applications where the use of

magnetohydrodynamic (MHD) ows play vi-

tal roles. MHD ow becomes more applic-

able when the geometry of ow involves ro-

tation as seen in ([16, 17, 18, 19, 15]). Oke

et al. [20] explored the eect of rotating sur-

faces on the MHD ow of water and air. It

was discovered that increasing surface rota-

tion raises ow temperature and that increas-

ing magnetic eld strength counteracts the ef-

fects of increased Coriolis force on skin friction

and heat transfer rate. Oyem et al. [21] con-

sidered the eects of variable thermal conduct-

ivity on an MHD Blasius and Sakiadis ows.

The results show that velocity proles reduce

and the temperature increases with increasing

magnetic eld strength for both Sakiadis and

Blasius ows. The impact of magnetic eld on

the transport of micropolar uid was reported

by Fatunmbi et al. [22]. Ouru et al. [23] showed

that the magnetic eld strength also inhibits

the ow velocity and increases temperature pro-

les. The ow of a reactive tangent hyperbolic

uid over a non-linear stretching surface was

evaluated by Fatunmbi et al. [24]. The authors

pointed out that the momentum boundary layer

and uid velocity were reduced due to the drag-

like Lorentz force occasioned by the magnetic

eld. Insightful results on the MHD ow of

Eyring-Powell uid over a stretching sheet with

a convective boundary condition were shown in

Oke and Mutuku [25]. Williamson uid ow

over an inclined rotating surface is studied in

the presence of a magnetic eld by Juma et al.

[26].

Carbon is the fourth most abundant chemical

element in the universe, following the gases hy-

drogen, helium, and oxygen [27]. As the most

abundant solid element, carbon is an excel-

lent candidate for use in nanoparticles within

nanouids. Carbon exists in several allotropes

[28], including graphite, diamond, fullerenes,

graphene, and carbon nanotubes. These al-

lotropes have diverse industrial applications,

such as steel production, where graphite is used

for electrodes and lubricants, and diamonds for

cutting tools and abrasives [29, 30]. Given their

exceptional properties, this study focuses on

the use of carbon nanotubes and graphene as

nanoparticles in the nanouids. From the lit-

erature above, studies have been carried out to

consider the ow of ternary hybrid nanouid

over moveable surfaces. However, the studies

have been restricted to nanoparticles of dif-

ferent shapes, and non-rotating surfaces and

without considering heat source. These are the

research gaps lled with this study where the

ow of ternary hybrid nanouid. (made from

the suspension of carbon nanotube (CNT), alu-

mina and graphene nanoparticles in water) over

a rotating moveable surface under the inuence

of heat source. This study provides answers to

the following research questions;

1. How can the ow parameters be chosen

to sustain model stability?

2. what is the inuence of the stretching ra-

tio on the transport phenomenon of the

ow of ternary hybrid nanouid over a

moveable rotating surface?

3. to what extent does channel rotation af-

fect the heat transfer rate, skin friction

and other transport properties of the ow

of ternary-hybrid nanouid over a move-

able rotating surface?

International Journal of Applied Sciences & Development

DOI: 10.37394/232029.2024.3.26

Oluwaseun Biodun Onuoha, Fawwaz Batayneh,

Abayomi Samuel Oke, Mario Raso

E-ISSN: 2945-0454

285

Volume 3, 2024

4. how does the heat source aect the tem-

perature, velocity, skin friction and heat

transfer rate in the MHD ow of ternary

hybrid nanouid over a rotating surface?

This study models the transport processes of

a water-based ternary hybrid nanouid that

suspends three distinct nanoparticles of sim-

ilar shapes in a moveable rotating channel. The

ow is taken to be laminar and incompressible,

subject to a magnetic eld of constant mag-

netic eld strength

B0.

The surface produces

the imaginary Coriolis force due to rotation

with an angular velocity of

Ω.

Assuming the

ow adheres to the boundary layer theory [31]

and the Sakiadis [32] theory on a moveable sur-

face, the physical conguration of the ow is

depicted in Figure (1). The uid layers adhered

to the surface obeyed the no-slip requirement,

and the surface is extended linearly in the

plane. The

- surface is assumed to move lin-

early while the uid layers glued to the surface

obey the no-slip condition. Hence, following

the work of Elnaqeeb et al. [13], Oke et al. [33]

and Oke [15], the ow can be modelled as the

following equations;

∂xu+∂yv+∂zw= 0,

(1)

(u∂x+v∂y+w∂z)u

=µtf

ρtf

∂zzu−2Ωu−σtf B2

ρtf

(2)

(u∂x+v∂y+w∂z)v

=µtf

ρtf

∂zzv+ 2Ωv−σtf B2

ρtf

(3)

(u∂x+v∂y+w∂z)T

=αtf ∂zzT+Q0

(ρCp)tf

(Tw−T∞) exp −zra

νbf .

(4)

At the base of the channel

(z= 0) ,

the uid

layers on the wall admit the same velocity as

that of the stretching wall and thus, the no-slip

condition is retained as

on the

-axis

u=ax, w =zw−(aνbf )1

2Zv

ay dη.

on the

-axis

v=ay, w =zw−(aνbf )1

2Zu

axdη,

on the

(x, y)

-plane

, T =Tw.

Furthermore, at the free stream

(z→ ∞),

the

free stream conditions are sustained as

u→0,

on the

-axis

v→0,

on the

-axis

T=T∞

on the

(x, y)

-plane

The characteristics of the ow that of practical

implication are the skin drags and heat transfer

rate in all directions which are dened as

Cfx =µtf ρbf a2x2−1∂u

∂z 



z=0

(5)

Cgy =µtf ρbf a2y2−1∂v

∂z 



z=0

(6)

Nux=−xκtf (κbf (Tw−T∞))−1∂T

∂z 



z=0

(7)

Nuy=−yκtf (κbf (Tw−T∞))−1∂T

∂z 



z=0

(8)

Since the properties of the resulting ternary

nanouid properties are inuenced by the prop-

erties of both the base uid and the nan-

oparticles, the following denitions extracted

from literature [33, 13] are adopted in this study

µtf =1

j=1

µjϕj, κtf =1

j=1

κjϕj,

(9)

ρtf = (1 −ϕ)ρbf +

j=1

ρjϕj⇒ρtf =ρbf A1,

(10)

(ρcp)tf = (1 −ϕ) (ρcp)bf +

j=1

(ρcp)jϕj

⇒(ρcp)tf = (ρcp)bf A2

where

A1= 1 −ϕ+1

ρbf

j=1

ρjϕj,

A2= 1 −ϕ+1

(ρcp)bf

j=1

(ρcp)jϕj.

The thermophysical properties of the nano-

particles and the base uid are provided in table

(3). Also, the ternary nanouid is made up of

three dierent nanouids. Considering platelet-

shaped nanoparticles, then the viscosity and

2. Research Methodology

2.1 Mathematical Formulation

International Journal of Applied Sciences & Development

DOI: 10.37394/232029.2024.3.26

Oluwaseun Biodun Onuoha, Fawwaz Batayneh,

Abayomi Samuel Oke, Mario Raso

E-ISSN: 2945-0454

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Volume 3, 2024

thermal conductivity of the nanouids are mod-

elled as

µj=1 + 37.1ϕj+ 612.6ϕ2

jµbf

κj

κbf

=κj+ 4.7κbf −4.7ϕ(κbf −κj)

κj+ 4.7κbf +ϕ(κbf −κj)

for

j= 1,2,3

and

ϕj>0.02.

Hence, with this

model, we have

µtf =1

j=1

µjϕj

=µbf

j=1 1 + 37.1ϕj+ 612.6ϕ2

jϕj

=µbf

ϕA3

(11)

κtf =1

j=1

κjϕj

=κbf

j=1

κj+ 4.7κbf −4.7ϕ(κbf −κj)

κj+ 4.7κbf +ϕ(κbf −κj)ϕj

=κbf

ϕA4,

(12)

where

A3=

j=1 1 + 37.1ϕj+ 612.6ϕ2

jϕj,

A4=

j=1

κj+ 4.7κbf −4.7ϕ(κbf −κj)

κj+ 4.7κbf +ϕ(κbf −κj)ϕj.

Employing the similarity variables

u=axf′, v =acyg′, w =−(aνbf )1

2(f+cg)

(13)

T=T∞+ (Tw−T∞)θ, η =za

νbf 1

(14)

the equations are rendered dimensionless and

the number of parameters is reduced to become

ϕA1

f′′′ + (f+cg)f′′ −f′2−K+M

A1f′

= 0,

(15)

ϕA1

g′′′ + (f+cg)g′′ −cg′2+K−M

A1g′

= 0,

(16)

ϕA2

Θ′′ +P rQ

e−η+P r (f+cg) Θ′

(17)

with the boundary and initial conditions

η= 0; f=fw, f′= 1, g =fw

c, g′= 1,Θ = 1,

(18)

η→ ∞;f′→0, g′→0,Θ→0.

(19)

A1= 1 −ϕ+1

ρbf

j=1

ρjϕj,

A2= 1 −ϕ+1

(ρcp)bf

j=1

(ρcp)jϕj,

A3=

j=1 1 + 37.1ϕj+ 612.6ϕ2

jϕj,

A4=

j=1

κj+ 4.7κbf −4.7ϕ(κbf −κj)

κj+ 4.7κbf +ϕ(κbf −κj)ϕj.

K=2Ω

a, M =σB2

aρbf

, Q =Q0

a(ρcp)bf

P r =νbf

αbf

, αbf =κbf

(ρcp)bf

The ow characteristics that are of practical in-

terest are also reduced to

xCfx =ϕ−1A1f′′ (0) , Re

yCgy =ϕ−1A1g′′ (0)

(20)

Re−1

xNux=Re−1

yNuy=−ϕ−1A3Θ′(0) .

(21)

The system of equations (15) - (17) has bound-

ary conditions (18) and (19) which makes it

dicult to adopt a direct numerical solution.

2.2 Similarity Transformation

2.3 Method of Solution

International Journal of Applied Sciences & Development

DOI: 10.37394/232029.2024.3.26

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E-ISSN: 2945-0454

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Hence, the shooting technique is required to

convert the boundary conditions to initial con-

ditions. To start with, equations (15) - (17) are

rewritten as a system of rst-order dierential

equations by setting

h1=f, h2=f′, h3=f′′, h4=g, h5=g′,

h6=g′′, h7= Θ, h8= Θ′

to have

h′

1=h2,

(22)

h′

2=h3,

(23)

h′

3=−ϕA1

A3(h1+ch4)h3−h2

2−K+M

A1h2,

(24)

h′

4=h5,

(25)

h′

5=h6,

(26)

h′

6=−ϕA1

A3(h1+ch4)h6−ch2

5+K−M

A1h5,

(27)

h′

7=h8,

(28)

h′

8=−ϕA2

P r (h1+ch4)h8+Q

exp (−η).

(29)

However, we have a combination of the initial

and boundary conditions (18) and (19). The

shooting technique requires that all these condi-

tions be rewritten as initial conditions, by pla-

cing an arbitrary initial guess for the variables

whose conditions are not known. In this case,

we have the conditions as

h1=fw, h2= 1, h3=ε1, h4=fw

(30)

h5= 1, h6=ε2, h7= 1, h8=ε3.

(31)

where

ε1, ε2, ε3

are arbitrary guesses that will

be updated after every iteration until the

boundary conditions

h2(∞)=0, h5(∞)=0, h7(∞)=0

are satised up to a specied tolerance level.

The MATLAB bvp4c solver is used to solve

the system (See Oke [34] for some other semi-

analytical methods of solution). The bvp4c

solver [35] executes the nite dierence code

that implements the three-stage Lobatto IIIa

formula of the fourth order. The system is

solved with an absolute tolerance of

10−4

and

a relative tolerance of

10−4.

The stability analysis of the equations (22 - 29)

is carried out in this section. This becomes ne-

cessary to identify the range of values accept-

able for the parameter values. By setting right

hand side of the equations to

we have the

equilibrium point as

(h1, h2, h3, h4, h5, h6, h7, h8)

=a1,0,0, a4,0,0, a7,Qexp (−η)

A2(a1+ca4).

The characteristic equation obtained by substi-

tuting the equilibrium point into the Jacobian

of the system is



−λ1 0 0 0 0 0 0

0−λ1 0 0 0 0 0

0a32 a33 −λ0 0 0 0 0

0 0 0 −λ1000

0 0 0 0 −λ100

0 0 0 0 0 a66 −λ0 0

0 0 0 0 0 0 −λ1

γ0 0 −γc 000a88 −λ



= 0,

where

γ=−ϕA2

P rh8, a32 =ϕA1

A3K+M

A1,

a33 =−ϕA1

(a1+ca4), a66 =−ϕA1

(a1+ca4),

a88 =−ϕA2

P r (a1+ca4).

The characteristic equation becomes

λ4(a88 −λ) (a66 −λ)λ2−a33λ−a32= 0, λ1,2,3,4= 0,

λ5=a88 <0, λ6=a66 <0

provided

a1+ca4>0,

λ7,8<0,

provided

a33 <0, a32 <0.

Hence, the system is stable provided

a1+ca4>

The values of the parameters will therefore

be chosen so that

a1+ca4>0

and since

ϕA1

A3K+M

A1>0,

then the values of

and

can be chosen arbitrarily.

The results obtained in this study are validated

with the results obtained using the MATLAB

bvp5c solver when

M→0, K →0.

The res-

ults are shown in Table 4. There is an excel-

lent agreement between the results. Hence, the

bvp4c is used to numerically study the problem

under consideration.

2.4 Qualitative analysis

2.5 Validation of Results

International Journal of Applied Sciences & Development

DOI: 10.37394/232029.2024.3.26

Oluwaseun Biodun Onuoha, Fawwaz Batayneh,

Abayomi Samuel Oke, Mario Raso

E-ISSN: 2945-0454

288

Volume 3, 2024

In the ow under consideration, the ternary hy-

brid nanouid is made from the suspension of

carbon nanotubes, graphene nanoparticles and

Al2O3

nanoparticles in water base uid. By

rendering the governing equations dimension-

less, the ow becomes equipped with some di-

mensionless parameters. This section discusses

the dynamics of the ow as the dimensionless

parameters (stretching parameter

suction ve-

locity

fw,

Coriolis force

magnetic strength

and heat source

) vary during the ow.

The ow properties are visualised graphically

to illustrate the behaviour of the ow proper-

ties (the primary velocity, secondary velocity

and temperature) as the ow parameters are

varied. Figures (2a) and (2b) visually illustrate

the behaviour of the velocity components in the

- and

-directions. It is observed that the ve-

locity decreases in all directions as the stretch-

ing ratio increases. The stretching ratio is var-

ied over the interval

[0.1,1.0]

where

c= 0.1

indicates that stretching in the

-direction is

faster than stretching in the

-direction and

c= 1.0

indicates that stretching is happen-

ing at the same rate in both directions. Sim-

ilarly, temperature is shown in Figure (2c) to

reduce as the stretching ratio increases. The

inuence of suction velocity

(chosen from

the interval

[0.1,0.7]

) on the ow velocity and

temperature is shown in Figures (3a), (3b) and

(3c). As observed in Figures (3a) and (3b), in-

creasing suction reduces velocity in all direc-

tions and Figure (3c) shows that temperature

reduces with increasing suction. The eects of

Coriolis force, measured by the dimensionless

parameter

are studied in Figures (4a) and

(4b). By varying the Coriolis force over the

interval

K∈[0.01,1.00] ,

where

K= 0.01

in-

dicates the presence of nearly no rotation while

K= 1.00

represents a moderately rotating sur-

face. Coriolis force reduces the primary velo-

city decreases (as shown in Figure (4a)) but

increases the secondary velocity (as shown in

Figure (4b)). The ow is equipped with some

heat source and it is observed in Figure (5) that

the temperature of the ow increases as heat

source increases.

The skin friction in both

- and

-directions are

displayed in Figures (6a) and (6b) for varying

Coriolis force, stretching ratio and heat source.

Figure (6a) shows that skin friction in the

direction increases as either Coriolis force or

stretching ratio increases. More so, increas-

ing heat source only slightly decreases the skin

friction in the

-direction. Figure (6b) shows

that skin friction in the

-direction increases as

stretching ratio increases but decreases with in-

creasing Coriolis force. More so, increasing the

heat source only slightly increases the skin fric-

tion in the

-direction. Figure (7) shows the

behaviour of heat transfer rate. Increasing the

Coriolis force only contributes very slowly to an

increase in the heat transfer rate, a higher heat

source leads to a reduction in the heat transfer

rate and the stretching ratio decreases the heat

transfer rate.

A summary of the results is provided in Table

(1). The results are discussed in this section

and our ndings are compared with existing lit-

erature.

The stretching ratio described by Makhdoum

et al. [36] is the ratio of wall stretching in

one direction to wall stretching in another dir-

ection. In this case, the boundary condition

u(η= 0) = axf′(η)

and

v(η= 0) = acyg′(η)

gives a stretching ratio

Establishing that

the primary direction is the

-direction, then

the ratio stretching ratio describes the ratio of

stretching in the

-direction to the stretching

in the

-direction. The values of

were restric-

ted to the interval

[0.1,1.0]

so that the stretch-

ing in the

-direction is less than or equal to

the stretching in the

-direction (where equal-

ity only holds when

c= 1

). According to the

study by Oke [15], the stretching ratio decreases

the primary velocity but increases the second-

ary velocity. However, our results show that

both the primary and the secondary velocities

are reduced by increasing the stretching ratio.

The conict in the eect of the stretching ratio

on the secondary velocity could be due to re-

strictions placed on the secondary wall so that

the secondary wall cannot move faster than the

primary wall. The stretching ratio increases the

skin friction in all directions but decreases the

heat transfer rate. This outcome agrees with

the study of Makhdoum et al. [36]. Suction

3. Analysis and Discussion of Results

3.1 Analysis of Results

3.2 Discussion of Results

International Journal of Applied Sciences & Development

DOI: 10.37394/232029.2024.3.26

Oluwaseun Biodun Onuoha, Fawwaz Batayneh,

Abayomi Samuel Oke, Mario Raso

E-ISSN: 2945-0454

289

Volume 3, 2024

measures the amount of partial vacuum cre-

ated when the uid is drawn towards a region

of lower pressure [37]. Makhdoum et al. [37]

showed that suction decreases the temperat-

ure and this is in agreement with our ndings.

The partial pressure created by suction leads

to a reduction in the thermal energy distribu-

tion in the ow and hence, increasing the suc-

tion leads to a reduction in the temperature.

Coriolis force as described by [12] is the iner-

tia force generated in a rotating system and is

responsible for the ctitious deection of the

path of moving objects in the rotating frame.

Coriolis force increases as angular velocity in-

creases. As angular velocity increases, the ow

experiences sideways lurching to the secondary

direction. The dynamic equilibrium is main-

tained by the reduction in ow in the primary

direction [12]. The heat source, measured by

the heat source parameter

represents the ex-

ternal heat source introduced to the ow. In-

creasing the heat source enhances the inherent

thermal energy in the ow and thereby raises

the temperature. This nding agrees with that

found in the ref [4].

u v T Cfx Cf y N u

c↓ ↓ ↓ ↑ ↑ ↓

fw↓ ↓ ↓

- - -

K↓ ↑

↑ ↓ ↑

- -

↑ ↓ ↑ ↓

Table 1: Summary of analysis

This work analyses the ow of ternary hy-

brid nanouid obtained by the suspension of

graphene and alumina nanoparticles in water.

The ow is subjected to some heat source

while the wall of the channel is moveable and

rotating. The ow equations are developed,

rendered dimensionless and simulated to illus-

trate the eects of various parameters. The

main outcomes of the study include;



The stretching ratio reduces ow temper-

ature, velocity in all directions and heat

transfer rate but increases the skin fric-

tion in all directions.



Heat transfer rate is enhanced with in-

creasing Coriolis force.



Increasing rotation leads to an increase in

both the skin friction in the primary ow

direction and the heat transfer rate but a

reduction in the skin friction in the sec-

ondary ow direction.



More heat source increases ow temper-

ature and secondary skin friction but de-

creases primary skin friction and heat

transfer rate.

This study focuses on ows subject to a low to

moderately high rotation, excluding extremely

high rotation. In instances of high rotation, the

uid dynamics becomes turbulent. Examining

the behaviour of hybrid ternary uid ow over

a rapidly rotating surface could be of scientic

interest, oering insight into a domain char-

acterized by turbulent uid dynamics. This

study has emphasised on using similar nano-

particle shapes, and some other works of lit-

erature (see [38, 39, 13, 33]) have emphasised

dissimilar nanoparticles. It will be of benet

to the industry if comparisons between ows of

ternary hybrid nanouid with similar and dis-

similar nanoparticle shapes can be made.

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Appendix

Table 2:

Nomenclature

Dimensional quantities

x, y, z

Distance in three-dimensional space

(

)

electrical conductivity

(

M−1L−3T3A2

)

u, v, w

Velocity component in the

x, y, z

-directions (

LT −1

)

thermal diusivity (

L2T−1

)

Dimensional uid temperature (

)

magnetic eld strength (

L−1A

)

wall temperature (

)

thermal conductivity (

K−1

)

T∞

free stream temperature (

)

density (

ML−3

)

Specic heat capacity (

L2T−2K−1

)

Ω

angular velocity of the surface

(

T−1

)

kinematic viscosity (

L2T−1

)

dynamic viscosity (

ML−1T−1

)

Dimensionless quantities

f′

primary velocity

rotation parameter

g′

secondary velocity

P r

Prandtl number

temperature

suction

Magnetic eld parameter

stretching ratio

Cfx

local skin friction in

-direction

volume fraction

Heat source parameter

Nux

Nusselt number in

-direction

Cfy

local skin friction in

-direction

Nuy

Nusselt number in

-direction

Subscripts

ternary hybrid nanouid

base uid

1,2,3

CNT, graphene,

Al2O3

nanoparticle

u=ax, w =zw−√aνbf Rv

ay dη

v=acy, w =zw−(aνbf )1

2Ru

ax dη

B¯

wΩ

Figure 1: Flow conguration

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Table 3: thermal and physical properties [20]

ρkgm−3kW m−1K−1cpJkg−1K−1

carbon nanotubes 2100 3007.4 410

graphene 2200 5000 790

Al2O3

3970 40 765

water 997.1 0.613 4179

Table 4: Results Validation for

M=K= 0, fw= 0.3, P r = 6.2, η∞= 70

c Q f′′(0) g′′ (0) Θ′(0)

bvp4c bvp5c bvp4c bvp5c bvp4c bvp5c

5.0 0.1 -0.6751 -0.6727 -0.9173 -0.9138 2.7848 -2.7685

5.5 0.1 -0.6943 -0.6918 -0.9568 -0.9531 2.8759 -2.858

6.0 0.1 -0.713 -0.7102 -0.9946 -0.9907 2.9636 -2.9441

0.3 0.5 -0.443 -0.4422 -0.3553 -0.3549 1.09 -1.0905

0.3 1.0 -0.443 -0.4422 -0.3553 -0.3549 0.4711 -0.4717

0.3 1.5 -0.443 -0.4422 -0.3553 -0.3549 -0.1478 0.1472

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012345678910

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

f'( )

c = 0.1000

c = 0.3250

c = 0.5500

c = 0.7750

c = 1.0000

Horizontal velocity reduces

with increasing stretching

ratio c

c = 0.5; K = 0.2;

M = 1; Pr = 6.2;

fw = 0.3; Q = 2;

(a) primary velocity with stretching ratio

0 2 4 6 8 10 12

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

g'( )

c = 0.1000

c = 0.3250

c = 0.5500

c = 0.7750

c = 1.0000

vertical velocity reduces

with increasing stretching

ratio c

c = 0.5; K = 0.2;

M = 1; Pr = 6.2;

fw = 0.3; Q = 2;

(b) secondary velocity with stretching ratio

0123456

0.2

0.4

0.6

0.8

1.2

( )

c = 0.1000

c = 0.3250

c = 0.5500

c = 0.7750

c = 1.0000

c = 0.5; K = 0.2;

M = 1; Pr = 6.2;

fw = 0.3; Q = 2;

Temperature reduces

with increasing

stretching ratio c

Figure 2: stretching ratio eects

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012345678910

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

f'( )

fw = 0.1000

fw = 0.2500

fw = 0.4000

fw = 0.5500

fw = 0.7000

c = 0.5; K = 0.2;

M = 1; Pr = 6.2;

fw = 0.3; Q = 2;

Horizontal velocity reduces

with increasing fw

(a) primary velocity with suction

012345678910

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

g'( )

fw = 0.1000

fw = 0.2500

fw = 0.4000

fw = 0.5500

fw = 0.7000

c = 0.5; K = 0.2;

M = 1; Pr = 6.2;

fw = 0.3; Q = 2;

vertical velocity reduces

with increasing fw

(b) secondary velocity with suction

0123456

0.2

0.4

0.6

0.8

1.2

( )

fw = 0.1000

fw = 0.2500

fw = 0.4000

fw = 0.5500

fw = 0.7000

c = 0.5; K = 0.2;

M = 1; Pr = 6.2;

fw = 0.3; Q = 2;

Temperature reduces

with increasing fw

Figure 3: suction eects

012345678910

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

f'( )

K = 0.0100

K = 0.2575

K = 0.5050

K = 0.7525

K = 1.0000

c = 0.5; K = 0.2;

M = 1; Pr = 6.2;

fw = 0.3; Q = 2;

horizontal velocity reduces

with increasing Coriolis force

(a) primary velocity with Coriolis force

0 5 10 15

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

g'( )

K = 0.0100

K = 0.2500

K = 0.5000

K = 0.7500

K = 1.0000

c = 0.5; K = 0.2;

M = 1; Pr = 6.2;

fw = 0.3; Q = 2;

vertical velocity increases

with increasing Coriolis force

(b) secondary velocity with Coriolis force

Figure 4: Coriolis force eect

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012345678

0.2

0.4

0.6

0.8

1.2

1.4

1.6

1.8

( )

Q = 0.1000

Q = 1.3250

Q = 2.5500

Q = 3.7750

Q = 5.0000

Temperature increases

with increasing heat

source

c = 0.5; K = 0.2;

M = 1; Pr = 6.2;

fw = 0.3; Q = 2;

Figure 5: temperature with heat source

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Parameters

3.1

3.2

3.3

3.4

3.5

3.6

3.7

3.8

Cfx

(a) skin friction in

-direction

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Parameters

1.2

1.4

1.6

1.8

2.2

2.4

2.6

2.8

Cfy

(b) skin friction in

-direction

Figure 6: skin friction

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Parameters

Cfy

Figure 7: heat transfer rate

International Journal of Applied Sciences & Development

DOI: 10.37394/232029.2024.3.26

Oluwaseun Biodun Onuoha, Fawwaz Batayneh,

Abayomi Samuel Oke, Mario Raso

E-ISSN: 2945-0454

298

Volume 3, 2024