## ORIGINAL RESEARCH article

Front. Energy Res., 04 October 2022
Sec. Process and Energy Systems Engineering
Volume 10 - 2022 | https://doi.org/10.3389/fenrg.2022.986284

# Mathematical analysis of casson fluid flow with energy and mass transfer under the influence of activation energy from a non-coaxially spinning disc M. M. Alqarni1 Muhammad Bilal2* Reem Allogmany3 Elsayed Tag-Eldin4 Mohamed E. Ghoneim5 Mansour F. Yassen6,7
• 1Department of Mathematics, College of Sciences, King Khalid University, Abha, Saudi Arabia
• 2Department of Mathematics, City University of Science and Information Technology, Peshawar, Pakistan
• 3Department of Mathematics, Faculty of Science, Taibah University, Al-Madinah Al-Munawarah, Saudi Arabia
• 4Faculty of Engineering and Technology, Future University in Egypt, New Cairo, Egypt
• 5Department of Mathematical Sciences, Faculty of Applied Science, Umm Al-Qura University, Makkah, Saudi Arabia
• 6Department of Mathematics, College of Science and Humanities in Al-Aflaj, Prince Sattam Bin Abdulaziz University, Al-Kharj, Saudi Arabia
• 7Department of Mathematics, Faculty of Science, Damietta University, Damietta, Egypt

A Casson fluid is the most suitable rheological model for blood and other non-Newtonian fluids. Casson fluids hold yield-stress and have great significance in biomechanics and polymer industries. In this analysis, a numerical simulation of non-coaxial rotation of a Casson fluid over a circular disc was estimated. The influence of thermal radiation, second-order chemical reactions, buoyancy, and heat source on a Casson fluid above a rotating frame was studied. The time evolution of secondary and primary velocities, solute particles, and energy contours were also examined. A magnetic flux of varying intensity was applied to the fluid flow. A nonlinear sequence of partial differential equations was used to describe the phenomenon. The modeled equations were reduced to a non-dimensional set of ordinary differential equations (ODEs) using similarity replacement. The obtained sets of ODEs were further simulated using the parametric continuation method (PCM). The impact of physical constraints on energy, concentration, and velocity profiles are presented through figures and tables. It should be noted that the effect of the Casson fluid coefficient, the Grashof number, and the magnetic field reduces the fluid’s primary velocity contour. The mass transfer field decreases with the action of constructive chemical reactions, but is augmented by the effects of destructive chemical reactions. The accelerating trend in Schmidt number lowers the mass profile, while it is enhanced by increasing values of activation energy and Soret number.

## Introduction

Many natural and commercial processes use non-coaxial fluid motion. Non-coaxial and coaxial rotation of fluid and solids such as by discs, cylinders, and spheres are used in multiple industrial operations (Erdogan, 1977; Erdoḡan, 1997; Asghar et al., 2007). As a result of variations in temperature and chemical transmission, several types of non-coaxial and coaxial spinning of fluids in industrial systems arise. Swirling flows and spinning tubes while engaged in oil exploration and transit, and food manufacturing layouts are well-known instances of rotating mechanisms. In order to understand the underlying engineering mechanism, scientists and engineers have modeled the non-coaxial motion of fluids and calculated the results using a variety of approaches (Hayat et al., 2001; Hayat et al., 2004; Alharbi et al., 2022). Non-coaxial spinning of a disc has a wide range of uses, including food manufacturing and industrial production, jet turbines, squeeze bottles, hydrological flows, and chilling rotor blades, among others (Ullah et al., 2022a; Benhacine et al., 2022). Noranuar et al. (2021) used a computational approach to investigate the fluid flow and energy transport of MHD Casson ferrofluid affected by the coaxial rotation of a movable disc traveling through a permeable medium. Jabbar et al. (Jabbar et al., 2021) analyzed the influence of non-coaxial movement on the distribution of mass in a first-order biochemical reaction. It was discovered that the effect of buoyant force on secondary velocity was the polar opposite of its effect on primary velocity. Sharma et al. (Sharma et al., 2022) measured the thermal control of the flow of a hybrid nanoliquid over a stretchable rotating disc and discovered that a revolving disc with a static cone could attain the optimum condensation of disk-cone components while the exterior heat remained fixed. Das et al. (Das et al., 2018) determined the Hall impacts on MHD flow of an electrically charged particle induced by non-coaxial repetitions of a highly permeable disc and a fluid. The irregular flow of an oscillating disc in a Newtonian fluid about its own plane was reported by Ersoy (2017); when fluctuation occurred across the elliptical direction, the x- and y-terms exerted by the fluid on the disc varied in nearly opposite directions. The Darcy–Forchheimer hybrid nano-composite fluid flow across a permeable rotating disc was studied by Yaseen et al. (2022). Hassan et al. (2018) investigated the effects of a strong oscillating magnetic flux and a large concentration of nanoparticles on physical characteristics as well as energy and mass transport. Many researchers have recently reported on the study of non-coaxial motion of discs and fluids (Ali et al., 2020; Ali et al., 2022; Fei et al., 2022).

Thermal radiation is an electromagnetic wave-based heat transfer phenomenon. It is caused by a large temperature difference between two mediums. Many technological activities take place at extremely high temperatures. With regard to nuclear reactors, space technology, engineering and physics, glass production, power plants, furnace design, and other related areas, the contribution of thermal radiation to flow and thermal expansion may be observed. Satellites and other spacecraft, missiles, aircraft propulsion systems, atomic power plants, solar power plants, and combustion engines such as IC engines and furnaces are all dependent on radiation effects. Hassan et al. (2022) studied non-Newtonian fluid flows and determined their thermo-physical characteristics. In the presence of activation energy and a magnetic field, Rizwan and Hassan (2022), Raja et al. (2022), and Lin et al. (2014) documented the nanofluid flow of copper and cobalt nanocomposites under the influence of thermal radiation in different configurations. Hang et al. (2021) tested the ability of new smart materials to dynamically regulate thermal radiation and found that these materials could be employed in a variety of situations, resulting in better options and a significant increase in economic potential. Al and CuO nanoparticles were examined by Wakif et al. (2021) in the context of thermal radiation, and they concluded that the presence of convection cells was stabilized more by improving the strength of Lorentz forces, heat radiation, and surface roughness. The Jeffery hybrid nanofluid flow with thermal characteristics and magnetic effecst was studied by Ishtiaq et al. (2022), and their data showed that a $Cu−H2O$ nanoliquid was more stable than one of $TiO2−H2O$. Mabood et al. (2021) experimentally determined how hybrid nanoparticles influenced a variety of physical properties of hybrid nanoliquids across an overextended surface. The findings of their research were crucial in determining the influence of several key design elements on heat transmission and in improving industrial processes. The characteristics of fluid flow under the influence of thermal radiation and energy transmission over different geometries was reported by (Ahmed et al., 2018; Hassan et al., 2021a; Azam, 2022a; Azam, 2022b; Dadheech et al., 2022).

The minimum energy required to initiate an operation (such as a chemical reaction) is known as the activation energy, and the rate of a chemical reaction is directly proportional to its activation energy (Alsallami et al., 2022; Rehman et al., 2022). Punith et al. (2021) examined the effects of activation energy on the reactive species in a non-Newtonian nanofluid with energy and mass-exchange properties. The results showed that as the Marangoni number increased, the relative velocity improved, and the energy transmission decreased. Chemical processes, activation energy, and heat source/sink effects were used by Ramesh et al. (2022) to model the momentum and energy transport of a hybrid nanoliquid across two surfaces. The mass outlines were minimized by greater chemical reactions, whereas activation energy exhibited the reverse pattern. Hassan et al. (2021b) addressed the boundary layer behavior of the well-known non-Newtonian dilatant and pseudoplastic fluids over a moving belt. Ullah (Ullah et al., 2022b; Ullah et al., 2022c; Ullah, 2022) investigated the effects of activation energy and molten heat flux on an unstable Prandtl–Eyring model caused by a strained cylinder with varying thermal conductivity. It was discovered that increasing the reaction rate improved fluid temperature, while decreasing it had the opposite effect when activation energy and unsteadiness characteristics were considered. Many scholars have recently made significant contributions in this area (Azam and Abbas, 2021; Azam et al., 2021; Al-Mubaddel et al., 2022; Azam et al., 2022).

In reviewing the existing literature, we found that no analysis of combined transit of energy and mass in the flow caused by non-coaxial motion of a disc and surrounding fluid with respect to second-order chemical reactions, activation energy and buoyancy force had been performed. Therefore, we numerically simulated the non-coaxial rotation of a Casson fluid and disc under the influence of thermal radiation, second-order chemical reactions, buoyancy, and heat source in a rotating frame. A nonlinear sequence of partial differential equations was used to describe the phenomenon. The modeled equations were reduced to a non-dimensional set of ODEs using similarity replacement. The obtained sets of ODEs were further simulated using PCM methodology. The effect of physical constraints on energy, concentration and velocity profiles are presented in figures and tables. In the next section, the model is formulated, and in succeeding sections, we present the results and discussion.

## Mathematical formulation

We considered the solute particles and thermal energy transmission in Casson fluid flow above a non-coaxial rotating disc with uniform angular velocity as displayed in Figure 1. The disk and ambient fluid both rotated on different axes under a fixed magnetic field, B0. The disc temperature was higher than the heat energy. Initially, the temperature and concentration of the disc and the ambient fluid were assumed to be the same. After a period of time, the temperature of the disc was changed. We observed that the motion of the fluid was created by the surface motion of the disc. The disc wall stretched and shrank with the positive and negative values of $Ω$. The gravitational and pressure effects are signified by g and p. In the proposed model, the opposing motions of the walls produced changes in vertical and horizontal velocities. The above-specified conditions produced results similar to (Erdogan, 1977; Benhacine et al., 2022)

$u=−(y−1)Ω, T=T∞, v=Ωx, C=C∞ at 0=tt>0, u=−Ωy, Tw=T, v=Ωx, Cw=C, z=0,z→∞, C=C∞, u=−Ωy+1, v=Ωx, T∞=T}(1)$

FIGURE 1

FIGURE 1. Physical sketch of fluid flow across a vertical spinning disc.

Based on the above conditions, the following field was considered:

$V=f−Ωy, g+Ωx+0, C=C(z,t), T=T(z,t).(2)$

The basic equations were designed according to (Erdogan, 1977; Benhacine et al., 2022)

$1ρ∂p∂x=2Ω2x+v(1β+1)∂2f∂z2−∂f∂t+Ωg−σB02ρ(f−Ωl)−μρkf+μΩρky,(3)$
$1ρ∂p∂y=2Ω2y+v(1β+1)∂2g∂z2−∂g∂t−Ωf−σB02ρg+βT(−T∞+T)g¯−μρkg+μΩρkx,(4)$
$∂T∂t=KρCp∂2T∂z2−1ρCp16σ1T∞33K*∂2T∂Z2+Q0ρCp(T−T∞),(5)$
$∂C∂t=D∂2C∂z2+DKTTm∂2T∂Z2−kr2(C−C0)(TT∞)n⁡exp(−EaκT),(6)$

Eliminating the pressure terms and employing the ambient conditions gives

$v(1β+1)∂2f∂z2−∂f∂t+Ωg−σB02ρ(f−Ωl)−μρkf(Ωl−f)=0,(7)$
$v(1β+1)∂2g∂z2−∂g∂t−Ωf−σB02ρg+Ω2l+βT(−T∞+T)g¯−μρkg=0,(8)$
$∂T∂t=KρCp∂2T∂Z2+1ρCp16σ1T∞33K∗∂2T∂Z2+Q0ρCp(T−T∞),(9)$
$∂C∂t=D∂2C∂z2+DKTT∞∂2T∂Z2−kr2(C−C0)(TT∞)n⁡exp(−EaκT),(10)$

Resulting in

$T=T, f=Ωl, g=0, C=C∞ t=0,t>0, T=T∞, Cw=C, g=0, f=0, z=0,C=0, f=Ωl, T=0, g=0 z→∞.}(11)$

The similarity variables are as (Benhacine et al., 2022)

$τ=Ωt, η=Ωvz, θ(Tw−T∞)=T−T∞, φ(Cw−C∞)=C−C∞, Γ=kΩv.(12)$

In Eqs 710, we get

$(1+1β)∂2F¯∂η2−(iF¯+1ΓF¯+pF¯)=−iGrθ¯,−1p=F¯(0,p), 0=F¯(∞,p),(13)$
$∂2θ∂η2−Prλ∂θ∂τ+Qθ=0, θ(0,τ)=1, θ(∞,τ)=0, θ(η,0)=0,(14)$
$∂2ϕ∂η2−Sc∂ϕ∂τ+ScSr∂2θ∂η2−Scγ(1+δθ)nφ exp(−E1+δθ)=0, ϕ(0,τ)=1, ϕ(∞,τ)=0, ϕ(η,0)=0.(15)$

The dimensionless parameters are

$F(η,τ)=(f+ig)Ωl−1, λ=1+43NR, NR=KK∗4σ1T∞3, Gr=βTg¯(Tw−T∞)Ω2l, Pr⁡=vpCpk,M2=σB02ρΩ, Sc=vD, Sr=DKT(Tw−T∞)vTm(Cw−C∞), γ=K1Ω, Γ=kΩv.}(16)$

## Numerical solution

Different researchers have used various numerical and computational techniques to solve nonlinear PDEs (Tie-Hong et al., 2019; Chu et al., 2021; Zhao et al., 2021; Chu et al., 2022a; Chu et al., 2022b; Iqbal et al., 2022; Jin et al., 2022; Nazeer et al., 2022; Rashid et al., 2022; Wang et al., 2022). The main steps in employing the PCM method were taken from (Shuaib et al., 2020a; Shuaib et al., 2020b; Elattar et al., 2022):

Step 1: Simplifying Eqs 1315 to first order:

$ƛ1(η)=f(η), ƛ2(η)=f′(η), ƛ3(η)=θ(η), ƛ4(η)=θ′(η), ƛ5(η)=ϕ(η), ƛ6(η)=ϕ′(η). }(17)$

By putting Eq. 17 in Eqs 1315, we get

$(1+1β)ƛ2′−(iƛ1+1Γƛ1+pƛ1)=−iGrƛ3,−1p=ƛ1(0,p), 0=ƛ1(∞,p),(18)$
$ƛ4′−Prλ*ƛ3+Qƛ3=0, ƛ3(0,τ)=1, ƛ3(∞,τ)=0, ƛ3(η,0)=0,(19)$
$ƛ6′−Scƛ5+ScSrƛ4′−Scσ(1+δƛ3)nφ exp(−E1+δƛ3)=0, ƛ5(0,τ)=1, ƛ5(∞,τ)=0, ƛ5(η,0)=0.(20)$

Step 2: Familiarizing the embedding term p in Eqs 1820:

$(1+1β)ƛ2′−(iƛ1+1Γƛ1+pƛ1)+ƛ2−(ƛ2−1)p=−iGrƛ3,−1p=ƛ1(0,p), 0=ƛ1(∞,p),(21)$
$ƛ4′−Prλ*ƛ3+ƛ4−(ƛ4−1)p+Qƛ3=0, ƛ3(0,τ)=1, ƛ3(∞,τ)=0, ƛ3(η,0)=0,(22)$
$ƛ6′−Scƛ5+ScSrƛ4′+ƛ6−(ƛ6−1)p−Scσ(1+δƛ3)nφ exp(−E1+δƛ3)=0, ƛ5(0,τ)=1,ƛ5(∞,τ)=0, ƛ5(η,0)=0.(23)$

Step 3: Apply the Cauchy principal and discretized Eqs 2123.

After discretization, the obtained set of equations was computed through Matlab, using the code PCM.

## Results and discussion

In this section, we present the trends and physical mechanism behind each figure. Table 1 compares our calculations to those in the published literature. It can be concluded that the proposed technique and results are reliable. The following explanations have been noticed from velocity, energy and concentration profiles:

Velocity Profile $\left({f}^{\prime }\left(\eta ,\text{\hspace{0.17em}}\tau \right),\text{\hspace{0.17em}}g\left(\eta ,\text{\hspace{0.17em}}\tau \right)\right)$:

TABLE 1

TABLE 1. Comparison of the present outcomes with published work for skin friction, Nusselt and Sherwood numbers, where $β=Gr=0.$

Figures 25 display the trend of velocity outlines $f′(η, τ)$ versus the Casson fluid factor, $β$, Grashof number, Gr, magnetic term, M, and parameter,$τ$, respectively. The primary velocity of the fluid significantly declines with the impact of Casson fluid factor, Grashof number, magnetic term and parameter $τ$. It can be seen that the effect of Casson fluid factor lessens the fluid velocity $f′(η, τ)$as presented in Figure 2. The gravitational force enhances, while the angular rotation of the spinning disc decreases the effect of Grashof number, which is why the increasing effect of Gr reduces the velocity field$f′(η, τ)$as shown in Figure 3. The improving tendency of the magnetic field causes the generation of Lorentz forces, which resist the flow field and decrease its velocity (Figure 4). Figure 5 illustrated that the influence of parameter$τ$declines the primary velocity contour.

FIGURE 2

FIGURE 2. Velocity outlines $f′(η, τ)$ versus Casson fluid factor $β$.

FIGURE 3

FIGURE 3. Velocity outlines $f′(η, τ)$ versus Grashof number Gr.

FIGURE 4

FIGURE 4. Velocity outlines $f′(η, τ)$ versus magnetic term M.

FIGURE 5

FIGURE 5. Velocity outlines $f′(η, τ)$ versus parameter$τ$.

Figures 68 illustrate the velocity outlines $g(η, τ)$ versus Casson fluid factor $β$, Grashof number Gr, magnetic term M and parameter $τ$ , respectively. It can be seen in Figure 6, that the Casson fluid factor lessens the fluid velocity $g(η, τ)$. The gravitational effect augments, while the angular revolution of revolving disc moderates, the value of the Grashof number, because the increasing effect of Gr boosts the secondary velocity field$g(η, τ)$as shown in Figure 7. Figures 8 and 9 illustrate that the secondary velocity frameworks decrease with the influence of magnetic field, while they increase with the increase in $τ$. Physically, the repellant force of the magnetic flux resists the fluid motion, decreasing the velocity field, $g(η, τ)$.

Energy Profile $\left(\theta \left(\eta ,\text{\hspace{0.17em}}\tau \right)\right)$:

FIGURE 6

FIGURE 6. Velocity outlines $g(η, τ)$ versus Casson fluid factor $β$.

FIGURE 7

FIGURE 7. Velocity outlines $g(η, τ)$ versus Grashof number Gr.

FIGURE 8

FIGURE 8. Velocity outlines $g(η, τ)$ versus magnetic term M.

FIGURE 9

FIGURE 9. Velocity outlines $g(η, τ)$ versus parameter $τ.$

Figures 1012 illustrate the mechanism behind the energy outlines $θ(η, τ)$ versus the thermal radiation, NR, the Prandtl number, Pr, and the parameter, $τ$, respectively. Figures 10, 11 revealed that the energy contours decline with the rising values of thermal radiation and Prandtl number. Physically, radiation from the surface of the fluid transfers thermal energy to the surrounding system, which lowers the fluid temperature and results in the lessening of the energy outline $θ(η, τ)$ as shown in Figure 10. Fluids with higher Prandtl numbers always have lower thermal diffusivity; therefore, an increase in Prandtl number decreases the energy field. Figure 12 shows that the action of parameter $τ$ diminishes the thermal distribution, $θ(η, τ)$.

Concentration Profile $\left(\varphi \left(\eta ,\text{\hspace{0.17em}}\tau \right)\right)$:

FIGURE 10

FIGURE 10. Energy outlines $θ(η, τ)$ versus thermal radiation NR.

FIGURE 11

FIGURE 11. Energy outlines $θ(η, τ)$ versus Prandtl number Pr.

FIGURE 12

FIGURE 12. Energy outlines $θ(η, τ)$ versus parameter$τ.$

Figures 1317 reveal the mass profile outlines $ϕ(η, τ)$ versus the constructive chemical reaction, $+γ,$ the destructive chemical reaction, $−γ,$ the Schmidt number, Sc, the activation energy, E, and the Soret number, Sr, respectively. Figures 13, 14 depict that the mass transfer field is reduced by the action of positive chemical reaction, $+γ$, but augmented by negative chemical reaction, $−γ$. The chemical reaction is inversely related to the angular rotation of the circular disk. The increase in chemical reaction regulates the angular motion of disc, as shown in Figures 13, 14. The kinetic viscosity of the fluid is increased, while the molecular diffusion of particles is reduced by the intensifying influence of Sc, which is why an increase in Sc reduces the mass field as presented in Figure 15. Figures 16, 17 show that the concentration contour increases with increasing activation energy and Soret number. The increased activation energy boosts the kinetic energy of fluid particles, which results in acceleration of mass transfer as depicted in Figure 16. Molecular diffusion is enhanced, while the kinetic viscosity increases at high Soret numbers, and, as a consequence, mass transition is enhanced (Figure 17).

FIGURE 13

FIGURE 13. Mass outlines $ϕ(η, τ)$ versus constructive chemical reaction$γ.$

FIGURE 14

FIGURE 14. Mass outlines $ϕ(η, τ)$ versus destructive chemical reaction$γ.$

FIGURE 15

FIGURE 15. Mass outlines $ϕ(η, τ)$ versus Schmidt number Sc.

FIGURE 16

FIGURE 16. Mass outlines $ϕ(η, τ)$ versus activation energy E.

FIGURE 17

FIGURE 17. Mass outlines $ϕ(η, τ)$ versus Soret number Sr.

## Conclusion

We have examined the numerical simulation of non-coaxial rotation of a Casson fluid and disc. The influence of thermal radiation, second-order chemical reaction, buoyancy, and heat source in a Casson fluid over a rotating frame is also studied. A nonlinear sequence of partial differential equations was used to describe the phenomenon. The modeled equations were reduced to a non-dimensional set of ODEs using similarity replacement. The obtained sets of ODEs were simulated using PCM. The key findings are:

• The Casson fluid coefficient, Grashof number, and magnetic field reduce the fluid’s primary velocity contour${f}^{\prime }\left(\eta ,\text{\hspace{0.17em}}\tau \right)$.

• The secondary velocity $g\left(\eta ,\text{\hspace{0.17em}}\tau \right)$outline decreases with the increasing value of magnetic field and Casson fluid parameter, but is increased from the effects of parameter $\tau$and the Grashof number.

• The energy field $\theta \left(\eta ,\text{\hspace{0.17em}}\tau \right)$ decays with the rising values of thermal radiation, parameter $\tau$, and Prandtl number.

• The mass transfer field is decreased by the action of the positive chemical reaction, $+\gamma$, but increases f tomhe effects of negative chemical reactions.

• An increase in the Schmidt number results in a decrease in the mass profile, while the mass profile is enhanced by increasing values of activation energy and Soret number.

• Our mathematical model may be modified for other types of non-Newtonian fluid models and may be solved through fractional and analytical techniques.

## Data availability statement

The original contributions presented in the study are included in the article/Supplementary Material, and further inquiries can be directed to the corresponding author.

## Author contributions

All authors listed have made a substantial, direct, and intellectual contribution to the work and approved it for publication.

## Acknowledgments

The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through Large Groups [Project under grant number (RGP.2/116/43)]. The authors would like to thank the Deanship of Scientific Research at Umm Al-Qura University for supporting this work by Grant Code: 22UQU4350057DSR004.

## Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

## Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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Keywords: non-coaxial rotation, activation energy, pcm, heat source, chemical reaction, thermal radiation

Citation: Alqarni MM, Bilal M, Allogmany R, Tag-Eldin E, Ghoneim ME and Yassen MF (2022) Mathematical analysis of casson fluid flow with energy and mass transfer under the influence of activation energy from a non-coaxially spinning disc. Front. Energy Res. 10:986284. doi: 10.3389/fenrg.2022.986284

Received: 04 July 2022; Accepted: 09 August 2022;
Published: 04 October 2022.

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