Research Article, J Nanomater Mol Nanotechnol Vol: 12 Issue: 1

# Influence of Arrhenius Activation Energy and Radiation Over the Flow of a Chemically Reacting Third Grade Hybrid Nanofluid on a Moving Flat Surface

**Ogunsola AW, Ajala OA and Ajayi Tunde M ^{*}**

Department of Mathematics, Ladoke Akintola University of Technology, Ogbomoso, Nigeria

***Corresponding Author:** **Ajayi Tunde M, Department of Mathematics**

Ladoke Akintola University of Technology, Ogbomoso, Nigeria

**E-mail:** stillmetunde@gmail.com

**Received date**: 08 August, 2022, Manuscript No. JNMN-22-71853; **Editor assigned date**: 11 August, 2022, PreQC No. JNMN-22-71853 (PQ); **Reviewed date**: 25 August, 2022, QC No. JNMN-22-71853; **Revised date**: 06 January, 2023, Manuscript No. JNMN-22-71853 (R); **Published date**: 16 January, 2023, DOI: 10.4172/2324-8777.1000352

**Citation: ***Ogunsola AW, Ajala OA, Ajayi Tunde M (2023) Influence of Arrhenius Activation Energy and Radiation Over the Flow of a Chemically Reacting
Third Grade Hybrid Nanofluid on a Moving Flat Surface. J Nanomater Mol Nanotechnol 12:1*.

## Abstract

In this era of global quest for more energy, the hybrid nanofluid is a better choice compared to the conventional nanofluid. The intention here is to present the influence of both radiation and Arrhenius activation energy over a moving fluid. Similarity transformation is being employed to convert the governing equations of the problem to coupled ordinary differential equations and numerical solutions are obtained using MATLAB bvp5c. Effects of radiation, Arrhenius parameter, fluid parameter, Reynold number were examined, presented graphically and a discussion was made on same. It was noted that the hybrid nanofluid shows better result compared to nanofluid.

### Keywords: Hybrid nanofluid, Moving plat, Third grade fluid, Radiation, Arrhenius activation energy

## Keywords

Hybrid nanofluid; Moving plate; Third grade fluid; Radiation; Arrhenius activation energy

## Introduction

Fluid flow has numerous applications in many fields not limited to
applications in oil extraction, engine cooling systems, blood flow in
the body, drug targeting, cleaning of soil from containment, hydraulic
machines, refrigerators and air conditioners, hydroelectric plants. The
quest for advancement in almost every sphere of life has necessitated
researchers and scientists to develop new ideas and inject these in
modern equipment and devices used in the mechanical, electrical, day
to day life and industries, such as air conditioners, refrigerators, heat
exchangers, electronic cooling, car radiators, solar thermal, energy
storage, hydraulics system and heat pipes [1]. Simple liquids like
water, oil, etc. were earlier being used for heat conduction and transfer
of heat but it has been realized that they have fallen below the recent
world’s demand for energy due to their poor thermal conductivity. A
major achievement in this quest to solve the energy problem was
recorded in 1995 which can be linked to Choi [2]. His research and
subsequent works on what is today known as nanofluid indicated that
nanofluids comparatively shows more thermal capacity and efficiency for heat transfer rate than simple or convectional fluids [3].
Deliberated on enhancing thermal conductivity of fluids with
nanoparticles. The potential benefits of nanofluid with copper
nanophase materials were estimated and one of such benefits is the
dramatic reductions in the heat exchanger pumping power. Thermal
transport in nanofluids was explicitly analyzed by Eastman, et al.
made a review of nanofluid thermal conductivity and their heat
transfer enhancements based on available experimental results [4-6].
Examined the energy destruction of forced convective nanofluid
through a duct with constant wall temperature. Thermal energy
storage behavior of Cu/paraffin nanofluids PCMs was numerically
simulated [7]. Entropy analysis of third order nanofluid flow on a
porous sheet was considered [8]. The role of slip on a two phase
flow of Newtonian nanofluid was investigated [9]. In this era of
emerging technology, a new version of nanofluid called hybrid
nanofluid has been developed. Unlike the nanofluid, the hybrid
nanofluid is made up of more than one metallic nanoparticle.
Research shows that the hybrid nanofluid exhibits a higher heat
transfer rate and more efficiency than the conventional fluids and
nanofluids [10]. Analyzed the properties of hybrid nanofluid and its
heat transfer phenomenon. The importance of hybrid nanofluid over
simple nanofluids was discussed by [11]. The impacts of magnetic
dipole on hybrid nanofluid flow were examined by [12]. The flow
of Cu-Al_{2}O_{3} hybrid nanofluid over a moving permeable surface was
described by [13]. The energy transference of a hybrid nano suspension
within a heated chamber was simulated [14]. The word ‘activation
energy’ was introduced by Svante Arrhenius in 1889 and this refers to
the minimum energy required to start a chemical reaction. Activation
energy is a non-negotiable idea in oil emulsions, food processing,
geothermal reservoirs and chemical engineering. The use and limitations
of the activation energy as a means of evaluating thresholds, excitation
functions and tunneling processes were discussed [15,16]. Addressed the
characterization of the stagnation point flow of Carreau fluid induced due
to the stretching of a chemically reacting surface. The influence of
Arrhenius activation energy on the heat and mass transfer of second
grade nanofluid flow was analyzed [17]. Reported that the Arrhenius
activation energy influences a rise in fluid concentration. The effects of
activation energy and dual stratification on the MHD flow of Maxwell
nanofluid was discussed [18]. Analyzed the influence of activation
energy on MHD buoyancy induced nanofluid flow past a vertical surface
in the presence of radiation and chemical reaction. The knowledge of
radiative heat transfer mechanism is essential in the field of power
generation, fossil fuel combustion, solar power technology, nuclear
reactor cooling, etc. [19]. Investigated the effects of radiation heat
transfer on the convective flow of nanofluid. The subject was
investigated [20] for a boundary layer flow over a convective flat surface
[21]. A micropolar fluid to inspect the concept [22]. Examined the
radiation effects of a nanofluid flow in the company of chemical reaction
and activation energy.

## Materials and Methods

Motivated by all the above works, here we have attempted a mathematical investigation on the influence of Arrhenius activation energy and radiation on the forced convection flow of a chemically reacting third grade hybrid nanofluid with temperature dependent fluid properties. To the best of our knowledge, these combined effects on a third grade hybrid nanofluid were not carried out in the past. Using similarity transformations, the governing equations are transformed into a system of ordinary differential equations. These obtained equations are solved numerically using matlab bvp5c. The results obtained are presented in tables and graphs and discussions thereafter made.

**Nomenclature**

B: Magnetic field strength

*C*: Concentration

*C _{p}*: Heat capacity at constant Pressure

*C _{w}*: Concentration at the wall

*C _{∞}*: Concentration at the free stream

*D _{B}*: Brownian diffusion coefficient

*D _{T}*: Thermophoretic diffusion coefficient

*E _{a}*: Activation energy

*k*: Fluid thermal conductivity

*k _{0}*: Universal gas constant

*k _{r}*: Chemical reaction rate

*n*: Rate constant

*P*: Permeability of porous medium

*T*: Fluid Temperature

*T _{w}, T_{∞}*: Temperature at the wall, ambient temperature

U_{0}: Velocity stretching rate

*u*: Component of velocity in x direction

*u _{e}*: Free stream velocity

*u _{w}*: Velocity of the moving plate

*v*: Component of velocity in y direction

*v _{w}*: Velocity normal to the plate (

*v*>0 signifies injection)

_{w}*β _{3}*: Material fluid parameters

*δ*: Velocity parameter

ϑ: kinematic viscosity

*ρ _{nf}*: Nanofluid density

*ρ _{s}*: Density of nanosolid

*μ*: Fluid viscosity

*μ _{nf}*: Nanofluid viscosity

*σ*: Electrical conductivity of the fluid

ψ: Stream function

*τ=(ρC _{ρ})_{hnf}/(ρC_{ρ})_{f}*: Heat capacity ratio of nanoparticles to base fluid

ω: Nanoparticle volume fraction

**Mathematical formulation**

Consider a two dimensional, steady, boundary layer flow of a
hybrid nanofluid made up of Ag/Au nanoparticles with third grade
base fluid past a half infinite plate moving in a uniformly free flow, U
as shown in **Figure 2**. The system is such that the x-axis is aligned to
the plate surface while the y-axis is the coordinate measured normal to
the plate. The hybrid nanofluid is taken to be in thermal equilibrium with no slip condition. In addition, the ambient fluid velocity of the
plate is considered to be uw= δUx where δ is the parameter for plate
velocity [23]. The hybrid nanofluids thermo physical characteristics are
written in **Figure 1**.

Based on the above assumptions together with ideas from previous works [24,25], the governing equations for the flow takes the form:

Where the component of velocity for and x axes y are u and v
respectively, u_{e} stands for the free stream velocity δ is the parameter
for plate velocity. Furthermore, the density of hybrid nanofluid, *ρ _{hnf}*,
viscosity of hybrid nanofluid,

*μ*heat capacity of hybrid nanofluid (

_{hnf},*ρCρ*)

*thermal conductivity of hybrid nanofluid*

_{hnf}*K*, are as follow [26,27].

_{hnf}Where *ρ _{s1},ω_{1}, (ρC_{ρ})_{s1}, k_{s1}* the thermophysical properties for gold nanoparticle are,

*ρ*are the thermophysical properties for silver nanoparticle,

_{s2},ω_{1}, (ρC_{ρ})_{s2}, k_{s2}*ρ*are the thermophysical properties for the base fluid (

_{f}, μ_{f}, (ρC_{p})_{f}, k_{f}**Table 1**).

Material | Density (kg/m^{^3}) |
Specific Heat Capacity C_{p} (J/KgK) |
Electrical Conductivity σ×10^{^(-5)} (S/m) |
Thermal Conductivity K(W/mk) |
---|---|---|---|---|

Aluminium Oxide (Al_{2}O_{3}) |
3970 | 765 | 0.85 | 40 |

Blood | 1050 | 3617 | 0.18 | 0.52 |

Copper (Cu) | 8933 | 385 | 1.67 | 401 |

Gold (Au) | 19300 | 129 | 4.1 | 318 |

Silver (Ag) | 10500 | 235 | 18.9 | 429 |

**Table 1:** Thermo physical properties of some nanofluids.

The Roseland approximation for radiation [28]:

Using Taylor’s series and neglecting higher order terms for the expansion of *T ^{4}* about

*T*we finally have the radiation term in equation (3) to be

_{∞}Introducing the established similarity variables and stream function *Ψ* used by Shehzard, Mohammed as

In this study, the fluid viscosity and thermal conductivity are assumed to be linear function of temperature in the form

Using equations (1-5) and variables (6-10), the continuity equation (1) is automatically satisfied and the governing equations reduced to the following ordinary differential equations [29,30]:

Where the parameters for third grade fluid *β _{3}=α_{3}U0^{2}/pv*, chemical reaction parameter

*C*, activation energy parameter

_{r}=k_{r}^{2}/U_{0}*E*, Local magnetic parameter

_{1}=E_{a}/ k_{0}T_{∞}*H*, Local Reynold number

_{α}=σ_{hnf}B2/ρ_{hnf}U_{0}*Φ=U*, Temperature dependent thermal conductivity parameter

_{0}x^{2}/v*∈=γ(T*, Temperature dependent viscosity parameter

_{w}-T_{∞})*ξ=b(T*, Lewis number

_{w}-T_{∞})*L*, Brownian motion parameter

_{e}=v/D_{R}*N*, Thermophoretic parameter

_{b}=D_{B}/vT_{∞}(C_{w}-C_{∞})*N*, Prandtl number

_{t}=D_{T}/vT_{∞}(T_{w}-T_{∞})*P*, Radiation parameter

_{r}=(ρC_{p})_{f}θ/k_{f}*R*, temperature difference parameter

_{a}=k_{f}k_{1}/4σ^{2}T_{∞}^{3}*T*, The physical qualities of engineering interest in this study are the skin friction coefficient

_{d}=(T_{w}-T_{∞})/T_{∞}*C*, local Nusselt number

_{f}*Nu*and Sherwood number

_{f}*S*which are defined as

_{h}**Numerical solution**

The coupled ordinary differential equations (11-13) and their corresponding boundary conditions (14) are first reduced to a system of first order equations using the method of superimposition [31]. This system of equations is thereafter solved numerically using the Matlab bvp5c solver.

## Results and Discussion

In order to analyze our results, numerical computation has been
carried out for various values of Brownian motion parameter (*N _{b}*),
Darcy number (

*D*) Forchheimer number (

_{a}*F*), Hartmann number (

_{s}*H*), Fluid material parameter (

_{a}*β*), Lewis number (

_{3}*L*), Prandtl number (

_{e}*P*), Thermal conductivity parameter (∈ ), Thermophoretic parameter (

_{r}*N*) and Velocity parameter (ε), using the numerical scheme discussed in the previous section. The numerical values obtained from the result are plotted in

_{t}**Figures 2-12**and

**Table 2**. The influence of fluid parameter on fluid velocity was depicted in

**Figure 2**.

It was observed that the fluid parameter reduces fluid motion. The
Physics behind this is that increasing the fluid parameter decreases the
stress that increases the value of the dynamic viscosity thereby
producing fluid retardation. The variation in velocity with respect to
local Reynold number was graphically illustration in **Figure 3**.

ε | Pr | φ_{1} |
β_{3} |
f"(0) |
-θ′(0) | -φ′(0) |
---|---|---|---|---|---|---|

0.3 | 0.7 | 1 | 0.1 | -0.919171632869 | 3.371615 | 0.372866 |

0.5 | -0.919171632869 | 3.384494 | 0.372866 | |||

0.7 | -0.919171643828 | 3.393485 | 0.372866 | |||

0.7 | 0.6 | 1 | 0.1 | -0.919171643828 | 0.293713 | 3.397298 |

0.8 | -0.919171643828 | 0.336425 | 3.389693 | |||

1 | -0.919171643828 | 0.373302 | 3.3822 | |||

0.7 | 0.7 | 1 | 0.1 | -0.469620427220 | 0.320053 | 0.154215 |

-1.237028774901 | 0.293558 | 3.10958 | ||||

- 0.919171643828 | 0.315945 | 3.393485 | ||||

0.7 | 0.7 | 3 | 0.1 | -0.791743710779 | 0.31021 | 3.379875 |

5 | -0.725581578956 | 0.305244 | 3.372605 | |||

7 | -0.681583078104 | 0.300736 | 3.367966 | |||

0.7 | 0.7 | 1 | 0 | -1.079721786205 | 0.319372 | 3.407922 |

0.2 | -0.956916721305 | 0.317026 | 3.397209 | |||

0.4 | -0.888882527640 | 0.314909 | 3.390374 |

**Table 2:** Numerical results for different values of the controlling parameters and corresponding local skin friction, nusselt number and the local sherwood number.

The figure showed that velocity profile reduces for an increase in
local Reynold number. The reason for this boils from the fact that the
local Reynold number given as ϕ_{1}=U0X^{2}/*v* is a function of the free
stream velocity and plastic velocity. Hence, the two parameters acts to
repel fluid flow which led to the decrease observed in fluid velocity as
local Reynold increases. The effect of radiation parameter on the
velocity profile is illustrated in **Figure 4**.

The figure revealed that radiation slightly increases fluid flow. This
is in good agreement with **Figure 5** reported Makinde and Olanrewaju
[32]. On the other hand, a decrease in temperature was observed for an
increase in the value of the radiation parameter as demonstrated in **Figure 6**.

The reason for this decrease in temperature is because large values
of radiation parameter correspond to an increase in dominance of
conduction over radiation, thus decreasing the thickness of the
thermal boundary layer and increasing the heat loss to the ambient.
Similar decrease in temperature due to increase in radiation
parameter was reported in **Figure 7** by Makinde and Olanrewaju [33]
and **Figure 8** by Zaib, et al. [34].

**Figure 9** illuminates the variation in temperature caused by a
change in thermal conductivity parameter. The figure revealed that
temperature decreases as thermal conductivity parameter increases.

This observation is in harmony with reported by Hazarika and
Konch Jadav. The influence of chemical reaction parameter on
fluid concentration is graphically represented **Figure 10**.

The Figure revealed that chemical reaction parameter suppresses
fluid concentration. The physical significance of this, is that the
number of solute molecules undergoing chemical reaction gets
increased as chemical reaction parameter increases whereas other
parameters (conditions) are constant, this therefore result in decrease
in the concentration profile. This is in good agreement with reported
by Makinde and Olanrewaju (**Figure 11**).

The variation of thermosphere tic parameter with concentration is elucidated. The figure shows that the concentration field is an increasing function of thermophoresis parameter. This observation is in good harmony with reported in Shehzad, et al. The influence of activation energy parameter on temperature is elucidated. The figure showed that activation energy increases fluid temperature. Showed the impact of activation energy on fluid concentration.

The figure elucidated that activation energy enhances the
concentration profile. The Physics behind this is that a higher
energy activation and weaker temperature leads to lower rate of
reaction, which results in a slowdown in the chemical reaction and
thus fluid concentration increases. This is in good agreement.
This portrayed the influence of temperature difference
parameter on fluid temperature. The figures showed that
temperature difference reduce fluid temperature. The variation of
temperature difference parameter with the fluid concentration is
demonstrated in **Figure 12**. The figure revealed that temperature
difference deflates the fluid concentration profile. This observation is
in harmony with **Figure 12** reported by Kiran Kumar, et al.

The effect of various parameters on the local skin friction, Nusselt
number and Sherwood number is depicted in **Table 2**. The thermal
conductivity parameter (ε) causes a negligible decrease in skin friction
but increases the Nusselt number. Prandtl number has no effect on skin
friction. Viscoelastic third grade parameter (β_{3}) reduces the Nusselt
number. The Nusselt number is an increasing function of the Prandtl
number. However, viscoelastic third grade parameter (β_{3}) reduces the
Sherwood number.

## Conclusion

The analysis of various parameters on steady laminar flow of an incompressible, electrically conducting non Newtonian hybrid nanofluid over a moving plate has been carried out. The following were detected: The fluid motion decreases on increasing the fluid parameter and Reynold number while the radiation parameter increases fluid motion.

• Fluid temperature decreases on increasing thermal conductivity, temperature difference and radiation parameters but Arrhenius energy parameter raises fluid temperature.

• The arrhenius energy and thermophoretic parameters increase the concentration of the fluid while the chemical reaction and temperature difference parameters reduce the fluid concentration profile.

• Thermal conductivity parameter and the Prandtl number because a rise in the Nusselt number but the material fluid parameter reduces it.

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