Communications on Applied Nonlinear Analysis 

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Thermal Radiation Effects of Convective Flow of a Hybrid Nanofluid 

with Different Base Fluids in Heat Transfer Over a Stretching Sheet 

 

Dr R . Lakshm𝒊𝟏 , A Subashin𝒊𝟐 , R Vijayakuma𝒓𝟑 

1Assistant Professor, Department of Mathematics, PSGR Krishnammal College for Women, Coimbatore – 641004 , 

Tamil Nadu, India. E-mail: lakshmir@psgrkcw.ac.in 

2Research scholar, Department of Mathematics, PSGR Krishnammal College for Women, Coimbatore – 641004 , Tamil 

Nadu, India. E-mail: subaarumugam11@gmail.com 

3Mathematics Section, FEAT, Annamalai University,Annamalainagar – 608002, India. 

Department of Mathematics, Periyar Arts College,Cuddalore, Tamil Nadu,India. 

E-mail: rathirath_viji@yahoo.co.in 

 

Article History: 

Received: 12-06-2024 

Revised: 10-07-2024 

Accepted: 01-08-2024 

Abstract:  

The study investigates a two-dimensional flow with uniform heat transfer processes in 

one direction, focusing on velocity profiles, temperature distribution, heat transfer 

characteristics, and potential effects of parameters on flow behavior of a hybrid nanofluid 

over a stretching sheet. Nanofluids are engineered colloidal suspensions of nanoparticles 

in a base fluid. In your case, TiO2 and Cu nanoparticles are dispersed in a base fluid 

consisting of water (H2O) and sodium citrate (NaC6H9O7). This mixture is known as a 

hybrid nanofluid. Additionally, consideration is given to the components of the magnetic 

field and nonlinear thermal radiation. The Runge-Kutta fourth-order method is a 

numerical technique used to solve ordinary differential equations, offering a balance 

between accuracy and computational efficiency, by simplification of partial differential 

equations. The results illustrate that significant parameter such as the magnetic parameter, 

nanoparticles of solid volume fractions, Radiation parameter significantly influence 

momentum and thermal profiles. These analyses protest that raising the Radiation 

parameter grounds an increase in the hybrid nanofluid temperature. Moreover, the Casson 

parameter has decreased the velocity profile in TiO2 - Cu/ sodium alginate hybrid 

nanofluids. Nonlinear thermal radiation effects become more pronounced at high 

temperatures. Studying their impact in hybrid nanofluids is essential for applications in 

high-temperature environments, such as in industrial processes or electronic devices. 

Keywords: Stretching sheet, Lorentz Force, Non-linear thermal radiation, Hybrid 

nanofluid. 

 

Nomenclature 

B magnetic field intensity, kgs-2a-1 

Cf skin friction coefficient 

Cp specific heat, jkg-1k-1 

R nonlinear radiation parameter 

k thermal conductivity, wm-1k-1 

M magnetic parameter 

Nux local Nusselt number 

Pr  prandtl number 

mailto:subaarumugam11@gmail.com


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𝑞𝑤 surface heat flux, wm-2 

Re local reynolds number 

F Dimensionless fluid velocity 

𝑇 temperature of fluid 

𝑇𝑤 ambient thermal 

𝑇∞ surface thermal 

U reference velocity, ms-1 

𝑈𝑤 constant velocity,  

𝑈∞ free stream momentum 

𝑢 momentum factor in the x way 

𝑥 , 𝑟 cylindrical directs 

Greek letters 

𝛼  angle 

β thermal expansion coefficient, k-1 

𝜙1 , 𝜙2 nanoparticles of volume fraction 

𝜇 dynamic viscosity, kgm-1s-1 

𝜐  kinematic viscosity, m2s-1 

𝜎 electrical conductivity, sm-1 

𝜌  density, kgm-3 

𝜃𝑤 Temperature ratio parameter 

𝜏𝑤 wall shear stress, nm-2 

𝑆𝑢𝑏𝑠𝑐𝑟𝑖𝑝𝑡𝑠 

hnf hybrid nanofluid 

nf nanofluid 

f fluid 

s solid 

 

1 Introduction: 

The study of nonlinear thermal radiation and viscous dissipation in hybrid nanofluids is a 

multidisciplinary field with applications in various thermal management systems. Ongoing research 

in this area is likely to contribute to the development of more efficient and advanced heat transfer 

technologies. In this research explained for developing advanced materials with improved thermal 

properties, which can have wide-ranging implications in various industries. The ability to enhance 

the thermal conductivity of fluids is crucial for optimizing heat transfer processes and improving the 

efficiency of thermal management systems [1]. This research may involve the development of 

mathematical models or equations that describe the viscoplastic flow over an inclined surface. This 

could include constitutive equations for viscoplastic materials, boundary conditions, and other 

relevant parameters [2]. The study investigates the impact of nanoparticles on natural convection in a 

cubic enclosure, examining heat transfer, fluid flow patterns, and thermal performance [3]. This 

research analyzes the effective thermal conductivity of a mixed convection flow of sodium alginate 



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and molybdenum disulphide nanoparticles using Maxwell Garnett and Brinkman models under 

specific conditions [4]-[5]. This research refers to a surface over which the fluid is flowing and 

stretching. Such sheets are often used as simplified models to represent industrial processes like 

extrusion or continuous casting [6]. The Casson model is a rheological model used to describe the 

flow behavior of certain fluids, particularly those with a yield stress. It's commonly employed in the 

study of non-Newtonian fluids as investigated in this research [7]. The study involves fluid flow, 

heat transfer, and electromagnetic effects, with sodium alginate as a base for nanofluids, known for 

their enhanced thermal properties due to nanoparticles [8]. This study explores the interaction of 

viscoelastic hybrid nanofluids in a porous space, aiming to enhance heat transfer efficiency and 

understand their behavior in specific applications [9]. This research to be focused on the free 

convection phenomenon, which occurs when a fluid is heated, becomes less dense, and rises due to 

buoyancy effects [10]. Fluid flow occurs through porous materials, indicating void spaces or pores, 

which are common in natural and engineered systems like soil and certain types of filters, as 

investigated in this research [11]. The paper investigates the natural convective flow of a nanofluid 

near a spinning cone, utilizing the Tiwari-Das nanofluid model, with specific findings and 

methodology available in the paper [12]. This research explores the influence of natural convection, 

thermal radiation, and nanoparticles on nanofluid flow over a non-linearly stretching surface, 

focusing on fluid dynamics and heat transfer properties [13]. This research explores mathematical 

modeling and computational analysis for solving complex fluid flow problems using magnetic fields, 

nanofluids, heat transfer, chemical reactions, and radiation [14]-[15]. This research investigates the 

Darcy-Forchheimer phenomenon, which refers to viscous and inertial effects in fluid flow through 

porous media [16]. The paper explores the impact of nanoparticles on forced convective heat transfer 

in a microchannel with asymmetric heating, emphasizing the importance of viscous dissipation in 

microscale fluid dynamics [17]. This research investigates magnetohydrodynamics (MHD) natural 

convection in a triangular cavity filled with a hybrid nanofluid of copper and aluminum oxide 

nanoparticles in water [18]. 

This research aims to understand how the combination of hybrid nanofluids, different base fluids, 

convective flow, thermal radiation, and Lorentz force affects heat transfer over a stretching sheet. 

Lorentz force is the force experienced by a charged particle in an electromagnetic field. Its inclusion 

implies that the study is considering the effects of an applied magnetic field on the nanofluid flow. 

This is often relevant in magnetohydrodynamics (MHD) studies. Thermal radiation refers to the 

electromagnetic radiation emitted by the fluid due to its temperature. Including thermal radiation in 

the study suggests an acknowledgment of the importance of radiative heat transfer mechanisms. The 

convective flow refers to the movement of fluid due to the temperature difference within the fluid. 

The stretching sheet implies that the fluid is flowing over a surface that is continuously stretching, 

which is a common scenario in boundary layer flows. This type of study is common in the field of 

fluid dynamics and heat transfer, particularly in applications related to nanotechnology, materials 

science, and energy systems. The outcomes of such research can have implications for improving 

heat transfer efficiency in various engineering applications.  

 

 



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2 Mathematical Formulation: 

● Consider two dissimilar base fluids (water (H2O), sodium alginate (NaC6H9O7)) with two 

distinct hybrid nanoparticles (titanium oxide (TiO2), copper (Cu)) and a uniform Lorentz force and 

nonlinear thermal radiation. The boundary layer flow should be stable, laminar, and convective along 

a stretching sheet. 

● The aim of the study is to examine the impact of hybrid nanoparticles on the flow 

characteristics and heat transfer of the two base fluids, in addition to the effects of magnetic fields 

and nonlinear thermal radiation. 

● A magnetic field would change the flow behavior by adding a Lorentz force term to the 

momentum equations and nonlinear thermal radiation to the energy equation. 

● Setting a boundary temperature at the stretching sheet's surface would govern the surface 

temperature and impact the system's heat transfer properties. 

● The physical sketch of present model is shown in Fig.I.  

● Beneath these assumptions, the governing equations in a stable state [6] 

 

 

 

 

 

 

 

 

 

 

 

Fig.I Geometry of the problem 

            

Continuity Equation:     

𝜕𝑢

𝜕𝑥
+

𝜕𝑣

𝜕𝑦
= 0                                                                                                                                            (1) 

Momentum Equation: 

𝑢
𝜕𝑢

𝜕𝑥
+ 𝑣

𝜕𝑢

𝜕𝑦
= 𝜈ℎ𝑛𝑓 (1 +

1

𝛽
) 𝜇ℎ𝑛𝑓

𝜕2𝑢

𝜕𝑦2
−

𝜎ℎ𝑛𝑓 

𝜌ℎ𝑛𝑓
𝐵0

2𝑢,                                                                     (2) 

Boundary Layer 

y, v 

x, u 

B0 



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Energy Equation: 

𝑢
𝜕𝑇

𝜕𝑥
+ 𝑣

𝜕𝑇

𝜕𝑦
=

𝑘ℎ𝑛𝑓

(𝜌𝐶𝑝)
ℎ𝑛𝑓

𝜕2𝑇

𝜕𝑦2
+  

16𝜎∗

3𝑘∗
 

𝜕

𝜕𝑦
(𝑇3

𝜕𝑇

𝜕𝑦
)

1

(𝜌𝐶𝑝)
ℎ𝑛𝑓

                                                   (3) 

2.1. Similarity Transformations and Boundary conditions: 

The suitable set of variables are 

𝑢 = 𝑐𝑥𝑓′(𝜂), 𝑣 = −(𝑐𝜈)
1
2 𝑓(𝜂),   𝜂 = (

𝑐

𝜈
)

1
2

𝑦, 𝑇 − 𝑇0 = (𝑇𝑤 − 𝑇∞)𝜃(𝜂),                                 (4) 

Relevant boundary circumstances are  

𝑢 = 𝑢𝑤 = 𝑐𝑥, 𝑣 = 0,        𝑇 = 𝑇𝑤     𝑎𝑡    𝑦 = 0,    

𝑢 = 0, 𝑇 = 𝑇∞    𝑎𝑡     𝑦 = ∞.                                                                                                      (5) 

2.2. Hybrid nanofluid Properties:  

𝜇ℎ𝑛𝑓 =
𝜇𝑓

(1 − 𝜙𝑛𝑝1)
2.5

(1 − 𝜙𝑛𝑝2)
2.5                                                                                                  (6) 

𝜌ℎ𝑛𝑓 = {(1 − 𝜙𝑛𝑝2)[(1 − 𝜙𝑛𝑝1)𝜌𝑓 + 𝜙𝑛𝑝1𝜌𝑛𝑝1]} + 𝜙2𝜌𝑛𝑝2                                                     (7) 

𝛼ℎ𝑛𝑓 =
𝑘ℎ𝑛𝑓

(𝜌𝐶𝑝)
ℎ𝑛𝑓

                                                                                                                                   (8) 

(𝜌𝐶𝑝)
ℎ𝑛𝑓

= {(1 − 𝜙𝑛𝑝2) [(1 − 𝜙𝑛𝑝1)(𝜌𝐶𝑝)
𝑓

+ 𝜙𝑛𝑝1(𝜌𝐶𝑝)
𝑛𝑝1

]} + 𝜙𝑛𝑝2(𝜌𝐶𝑝)
𝑛𝑝2

            (9) 

(𝜌𝛽𝑇)ℎ𝑛𝑓 = {(1 − 𝜙𝑛𝑝2)[(1 − 𝜙𝑛𝑝1)(𝜌𝛽𝑇)𝑓 + 𝜙1(𝜌𝛽𝑇)𝑛𝑝1]} + 𝜙2(𝜌𝛽𝑇)𝑛𝑝2                   (10) 

𝑘ℎ𝑛𝑓

𝑘𝑛𝑓
=

𝑘𝑛𝑝2 + 2𝑘𝑛𝑓 − 2𝜙𝑛𝑝2(𝑘𝑛𝑓 − 𝑘𝑛𝑝2)

𝑘𝑛𝑝2 + 2𝑘𝑛𝑓 + 𝜙𝑛𝑝2(𝑘𝑛𝑓 − 𝑘𝑛𝑝2)
                                                                                (11) 

where 

𝑘𝑛𝑓

𝑘𝑓
=

𝑘𝑛𝑝1 + 2𝑘𝑓 − 2𝜙𝑛𝑝1(𝑘𝑓 − 𝑘𝑛𝑝1)

𝑘𝑛𝑝1 + 2𝑘𝑓 + 𝜙𝑛𝑝1(𝑘𝑓 − 𝑘𝑛𝑝1)
                                                                                       (12) 

𝜎ℎ𝑛𝑓

𝜎𝑛𝑓
=

𝜎𝑛𝑝2 + 2𝜎𝑛𝑓 − 2𝜙𝑛𝑝2(𝜎𝑛𝑓 − 𝜎𝑛𝑝2)

𝜎𝑛𝑝2 + 2𝜎𝑛𝑓 + 𝜙𝑛𝑝2(𝜎𝑛𝑓 − 𝜎𝑛𝑝2)
                                                                                 (13) 

where 

𝜎𝑛𝑓

𝜎𝑓
=

𝜎𝑛𝑝1 + 2𝜎𝑓 − 2𝜙𝑛𝑝1(𝜎𝑓 − 𝜎𝑛𝑝1)

𝜎𝑛𝑝1 + 2𝜎𝑓 + 𝜙𝑛𝑝1(𝜎𝑓 − 𝜎𝑛𝑝1)
                                                                                        (14) 

 

 



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2.3. Explanation for Momentum: 

Using above equations (4) and (5), eqns. (1) – (2) can be transmuted as, 

(1 +
1
𝛽

)

(1 − 𝜙𝑛𝑝1)
2.5

(1 − 𝜙𝑛𝑝2)
2.5 𝑓′′′  −

𝜎ℎ𝑛𝑓

𝜎𝑓
𝑀𝑓′         

+ [(1 − 𝜙𝑛𝑝2) [1 − 𝜙𝑛𝑝1 + 𝜙𝑛𝑝1

𝜌𝑛𝑝1

𝜌𝑓
] + 𝜙𝑛𝑝2

𝜌𝑛𝑝2

𝜌𝑓
] [𝑓𝑓′′ − 𝑓′2

] = 0                           (15) 

2.4. Temperature Distribution: 

Using above equations (4) and (5), eqn (3) can be transmuted as, 

𝜃′′ [
𝑘ℎ𝑛𝑓

𝑘𝑓
+ 𝑅((𝜃𝑤 − 1)𝜃 + 1)

3
]

+ 𝑃𝑟𝑓𝜃′ [(1 − 𝜙𝑛𝑝2) [1 − 𝜙𝑛𝑝1 + 𝜙𝑛𝑝1

(𝜌𝐶𝑝)
𝑠1

(𝜌𝐶𝑝)
𝑓

] + 𝜙𝑛𝑝2 (
(𝜌𝐶𝑝)

𝑠2

(𝜌𝐶𝑝)
𝑓

)]

= 0                                                                                                                                                          (16) 

                                                                                                                                                      

and the related boundary restrictions are 

𝑓(0) = 0, 𝑓′(0) = 1,   𝜃(0) = 1,        

𝑓′(∞) = 0,     𝜃(∞) = 0                                                                                                                  (17) 

where, 

𝑀 =
𝜎𝑓𝐵0

2

𝑐𝜌𝑓
, 𝑃𝑟 =

𝜈𝑓

𝛼𝑓
,   𝑅𝑒 =

𝑥𝑢𝑤

𝜈𝑓
 , 𝑞𝑟 =

−16𝜎∗

3𝑘∗

𝜕

𝜕𝑦
(𝑇3

𝜕𝑇

𝜕𝑦
)                                              

 

2.5. Nusselt number and Skin friction Analysis: 

The Heat transfer rate and Surface drag force, which is determined by, 

The dimensionless form of Cf  and  Nu, are 

𝐶𝑓 =
𝜏𝑤

𝜌𝑓𝑢𝑤
2

,    𝑁𝑢 =
𝑥𝑞𝑤

𝑘𝑓(𝑇𝑤 − 𝑇∞)
                                                                                                 (18) 

 

𝜏𝑤 = 𝜇ℎ𝑛𝑓 (1 +
1

𝛽
) (

𝜕𝑢

𝜕𝑦
)

𝑦=0

,   𝑞𝑤 = 𝑞𝑟−𝑘ℎ𝑛𝑓 (
𝜕𝑇

𝜕𝑦
)

𝑦=0

                                                         (19) 

  𝑁𝑢𝑅𝑒
−1
2 = − (

𝑘ℎ𝑛𝑓

𝑘𝑓
) + 𝑅((𝜃𝑤 − 1)𝜃 + 1)

3
∗ 𝜃′.                                                              (20) 



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𝐶𝑓𝑅𝑒
1
2 = (1 +

1

𝛽
) (

𝜇ℎ𝑛𝑓

𝜇𝑓
) 𝑓′′(0)                                                                                            (21) 

 

3.Physical Explanation: 

The resulting ordinary differential equations expressed in Eqns. (15) & (16) confined to the boundary 

constraints (17) have been resolved by IVth order Runge-Kutta form with shooting manner with the 

support of BVP4C code in MATLAB software. Fig.II is mentioned the numerical solution for flow 

chart, we indicate 𝐹 = 𝑦1, 𝜃 = 𝑦4 for our present model and render a prime step of the system are  

𝑦1
′ = 𝑦2 

𝑦2
′ = 𝑦3 

𝑦3
′ =  (1 +

1

𝛽
)

−1

((1 − 𝜙𝑛𝑝1)
2.5

(1 − 𝜙𝑛𝑝2)
2.5

) {
𝜎ℎ𝑛𝑓

𝜎𝑓
𝑀𝑦2

− [(1 − 𝜙𝑛𝑝2) [1 − 𝜙𝑛𝑝1 + 𝜙𝑛𝑝1

𝜌𝑛𝑝1

𝜌𝑓
] + 𝜙𝑛𝑝2

𝜌𝑛𝑝2

𝜌𝑓
] [𝑦1𝑦3 − 𝑦2

2]} 

𝑦4
′ = 𝑦5 

𝑦5
′ = −𝑃𝑟 (

𝑘ℎ𝑛𝑓

𝑘𝑓
+

1

𝑅((𝜃𝑤 − 1)𝑦4 + 1)
3) [(1 − 𝜙𝑛𝑝2) [1 − 𝜙𝑛𝑝1 + 𝜙𝑛𝑝1

(𝜌𝐶𝑝)
𝑠1

(𝜌𝐶𝑝)
𝑓

]

+ 𝜙𝑛𝑝2 (
(𝜌𝐶𝑝)

𝑠2

(𝜌𝐶𝑝)
𝑓

)] [𝑦1𝑦5] 

and  

𝑦1(0) = 0, 𝑦2(0) = 1𝑦4(0) = 1   

 

 



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4. Graphical and tabulated outcomes with discussion: 

Figure I depict the items are placed of the conundrum. In regard to physical behaviour and attitude 

are explanations in the Figures. 1-8 have been drawn for the hybrid nanofluid flow, which consists of 

TiO2 and Cu nanoparticles, (water (H2O), sodium alginate (NaC6H9O7)) are considered as base fluids 

to demonstrate the impression of numerous parameters on acceleration, thermal profiles. The base 

fluids' and the nanoparticles' thermophysical characteristics were described in Table 1. The values 

for surface drag force and nusselt number are given in Table 2. 

4.1. Deviation of the Velocity Profile: 

               Fig.1. shows, with the use of water and sodium alginate mixing hybrid nanofluid, the 

velocity profile of the stretching sheet is observed to decrease. The decrease in the velocity profile 

could be attributed to the magnetic field's impact (𝑀 = 1,2,3)on the fluid, potentially altering its 

viscosity or flow pattern. The sodium alginate in the hybrid nanofluid may also contribute to the 

changes in the velocity profile due to its rheological properties.                   

                As the radiation parameter (𝑅 = 1,1.5,2) increases, the velocity profile of the stretching 

sheet decreases. This suggests that the influence of radiation on the motion or flow of the water 

mixing hybrid nanofluid causes a reduction in the velocity profile of the stretching sheet. Also, the 

velocity profile of the stretching sheet increases with the radiation parameter when sodium alginate is 

used as the base fluid in the hybrid nanofluid. This implies that higher radiation levels enhance the 

velocity profile in the case of sodium alginate-based hybrid nanofluids, as explain Fig.2. 

                Fig.3.displays, when the base fluid is a water mixing hybrid nanofluid, the velocity profile 

of the stretching sheet decreases as the temperature ratio parameter (𝜃𝑤 = 0.1,0.2,0.3) increases. 

This could be due to changes in viscosity, thermal conductivity, or other fluid properties influenced 

by the temperature ratio. In contrast, when using a sodium alginate mixing hybrid nanofluid as the 

base fluid, the velocity profile of the stretching sheet increases. This suggests that the presence of 

sodium alginate, a natural polymer, in the nanofluid may lead to different thermal and flow 

characteristics compared to water, resulting in an increased velocity profile as the temperature ratio 

parameter rises. 

             An increasing Casson parameter (𝛽 = 0.1,0.2,0.3) generally leads to a decrease in the 

velocity of the fluid. Higher yield stress reduces fluid movement, causing slower velocity. Increased 

Casson parameter indicates shear-thinning behavior, decreasing viscosity. Shear forces influence 

fluid velocity in stretching sheets, with higher Casson parameters indicating greater fluid sensitivity, 

as appeared in Fig .4. 

4.2. Deviation of the Thermal Profile: 

                Fig.5. explains, As the magnetic field parameter (𝑀 = 1,2,3) increases, there is an 

increase in the temperature profile of the stretching sheet. In this case, it's a hybrid nanofluid, 

meaning it's a combination of water and sodium alginate mixed with nanoparticles. The increase in 

the magnetic field parameter suggests that there's a stronger magnetic field applied to the system. 

Magnetic fields can affect the behavior of fluids, especially when they contain nanoparticles. The 

magnetic field may induce changes in the alignment and movement of nanoparticles within the 



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nanofluid. 

                  As the radiation parameter (𝑅 = 1,1.5,2)  increases, it becomes a more dominant mode of 

heat transfer, indicating its significant influence on heat transfer behavior. The presence of hybrid 

nanofluids, whether water-based or sodium alginate-based, further modifies the heat transfer 

characteristics of the system. The use of nanoparticles in these hybrid nanofluids contributes to 

changes in thermal conductivity, and the overall temperature increases on the stretching sheet. as 

exposed in Fig.6. 

             Fig.7. explains, as the temperature ratio parameter (𝜃𝑤 = 0.1,0.2,0.3)  increases, the 

temperature profile of the stretching sheet in the context of water mixing hybrid nanofluid and 

sodium alginate mixing hybrid nanofluid is subject to changes in heat transfer, fluid flow, and 

thermal boundary layer characteristics. The specific effects will depend on the interplay of factors 

such as nanofluid properties, additive effects, and the overall thermal conditions imposed on the 

system.so finally as the temperature ratio parameter increases, it suggests that the temperature 

distribution on the sheet is increasing.             

                   The increase in the Casson parameter (𝛽 = 0.1,0.2,0.3) could lead to a higher resistance 

to flow, affecting the boundary layer near the stretching sheet. This change in flow behavior, 

combined with the enhanced thermal properties of the sodium alginate mixing hybrid nanofluid. 

hybrid nanofluid, may contribute to an increase in the temperature profile along the stretching sheet. 

The nanoparticles in the hybrid nanofluid can potentially enhance heat transfer, resulting in a higher 

temperature gradient, explained in Fig.8. 

Velocity Distributions: 

 

Fig. 1. The effects of  𝑀 = 1,2,3 on 𝑓′ . 

 

 

Fig. 2. The impacts of  𝑅 = 1,1.5,2 on 𝑓′  



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Fig. 3. The effects of  𝜃𝑤 = 0.1,0.2,0.3 on 𝑓′ . 

 

Fig. 4. The impacts of  𝛽 = 0.1,0.2,0.3 on 𝑓′ 

.

 

Fig. 5. The effects of  𝑀 = 1,2,3 on 𝜃. 

 

Fig. 6. The impacts of  𝑅 = 1,1.5,2 on 𝜃. 

 

Fig. 7. The effects of  𝜃𝑤 = 0.1,0.2,0.3 on 𝜃. 

 

Fig. 8. The impacts of  𝛽 = 0.1,0.2,0.3 on 𝜃. 

 

 

 



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Table 1: Thermophysical characterstics of the base fluids and the nanoparticles [10],[12] 

Base fluid and nanoparticle Cp/(J/(kg.K)) ρ/(kg/m3) k/(W/(m.K)) β/10-5(1/K) σ(sm-1) 

H2O 4179 997 0.613 21 0.05 

NaC6H9O7 4175 989 0.6376 99 2.6x10-4 

TiO2 686.2 4250 8.954 0.9 2.38x106 

Cu 385 8933 400 1.67 5.96x107 

 

Table 2: The values of 𝐶𝑓 coefficient and Nu for M, 𝛽 with surface temperature is listed below 

 

M 

 

R 

 

𝜃𝑤 

 

β 
𝐶𝑓𝑅𝑒

1
2 𝑁𝑢𝑅𝑒

−1
2  

TiO2 - Cu TiO2 - Cu 

H2O NaC6H9O7 H2O NaC6H9O7 

1  

1 

 

0.1 

 

1 

-1.5309 -2.2377 7.2386 7.4772 

2 -1.8479 -2.6540 7.1926 7.4472 

3 -2.1220 -3.0246 7.1528 7.4204 

 

1 

1  

0.1 

 

1 

-1.5309 -2.2377 7.2386 7.4772 

1.5 -1.5309 -2.2377 5.9887 6.1986 

2 -1.5309 -2.2377 5.2565 5.4500 

 

1 

 

1 

0.1  

1 

-1.5309 -2.2377 7.2386 7.4772 

0.2 -1.5309 -2.2377 5.1425 5.3424 

0.3 -1.5309 -2.2377 4.3040 4.4915 

 

1 

 

1 

 

0.1 

0.4 - -3.1450 - 7.5083 

0.7 - -2.5075 - 7.4896 

1 - -2.2377 - 7.4772 

Conclusion: 

          A summary of their findings and a discussion of the implications of taking into account the 

Lorentz force and nonlinear thermal radiation were covered in the study's conclusion section on the 

convective flow of a hybrid nanofluid with different base fluids in heat transfer over a stretching 

sheet. It has been discussed that the following results and ramifications: 

⮚ This refers to the motion of a fluid driven by temperature differences within the fluid itself. 

Convection plays a significant role in heat transfer processes. 

⮚ This refers to a surface over which a fluid is flowing and the sheet is simultaneously 

stretched. The stretching sheet problem is a classical scenario in fluid mechanics and heat transfer. 

⮚ Thermal radiation is the emission of electromagnetic waves (radiation) due to the temperature 

of an object. In the context of heat transfer, thermal radiation plays a role alongside conduction and 

convection. 

⮚ The Lorentz force is the force experienced by a charged particle moving through an 

electromagnetic field. This force is perpendicular to both the velocity of the particle and the magnetic 

field.           



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⮚ This type of study is common in the field of fluid dynamics and heat transfer, particularly in 

applications related to nanotechnology, materials science, and energy systems. The outcomes of such 

research can have implications for improving heat transfer efficiency in various engineering 

applications. 

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