
In an endeavor to encapsulate the dual aspects of volatility progression and periodicity inherent in autocorrelation frameworks demonstrated by various nonlinear time series, a novel conceptualization emerges—the periodic threshold autoregressive stochastic volatility (PTAR-SV) model. This model served as a viable alternative to the conventional periodic threshold generalized autoregressive conditional heteroskedasticity (TGARCH) process. The inherent probabilistic framework of the PTAR-SV model incorporated certain essential features, including strict periodic stationarity, enhancing its analytical robustness. Additionally, this study established the conditions for higher-order moments to exist within the PTAR-SV model. The autocovariance structure pertaining to the powers of the PTAR-SV process has been studied. The process of parameter estimation was scrutinized via the quasi-maximum likelihood technique. This estimation approach involved assessing likelihood using prediction error decomposition and Kalman filtering. Moreover, we extended our analysis to include a Bayesian Markov chain Monte Carlo (MCMC) method based on Griddy-Gibbs sampling, particularly suitable when the distribution of model innovations follows a standard Gaussian. Through a simulation study, we evaluated the performances of both the quasi-maximum likelihood (QML) and Bayesian Griddy Gibbs estimates, providing valuable insights into their respective strengths and weaknesses. Finally, we applied our newly developed methodology to model the spot rates of the euro against the Algerian dinar, demonstrating its applicability and efficacy in real-world financial modeling scenarios.
Citation: Ahmed Ghezal, Omar Alzeley. Probabilistic properties and estimation methods for periodic threshold autoregressive stochastic volatility[J]. AIMS Mathematics, 2024, 9(5): 11805-11832. doi: 10.3934/math.2024578
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In an endeavor to encapsulate the dual aspects of volatility progression and periodicity inherent in autocorrelation frameworks demonstrated by various nonlinear time series, a novel conceptualization emerges—the periodic threshold autoregressive stochastic volatility (PTAR-SV) model. This model served as a viable alternative to the conventional periodic threshold generalized autoregressive conditional heteroskedasticity (TGARCH) process. The inherent probabilistic framework of the PTAR-SV model incorporated certain essential features, including strict periodic stationarity, enhancing its analytical robustness. Additionally, this study established the conditions for higher-order moments to exist within the PTAR-SV model. The autocovariance structure pertaining to the powers of the PTAR-SV process has been studied. The process of parameter estimation was scrutinized via the quasi-maximum likelihood technique. This estimation approach involved assessing likelihood using prediction error decomposition and Kalman filtering. Moreover, we extended our analysis to include a Bayesian Markov chain Monte Carlo (MCMC) method based on Griddy-Gibbs sampling, particularly suitable when the distribution of model innovations follows a standard Gaussian. Through a simulation study, we evaluated the performances of both the quasi-maximum likelihood (QML) and Bayesian Griddy Gibbs estimates, providing valuable insights into their respective strengths and weaknesses. Finally, we applied our newly developed methodology to model the spot rates of the euro against the Algerian dinar, demonstrating its applicability and efficacy in real-world financial modeling scenarios.
Energy crises are among the most pressing issues the world faces today. Several researchers discussed various methods for producing energy at a lower cost. In the past years, common fluids such as water, engine oil, and ethylene glycol had poorer heat transfer rates due to their lower thermal conductivity. Due to the higher abilities of metals, which contain more thermal conductivity properties than ordinary fluids. The nanosize metals are added to the ordinary fluids which enhance the transfer rate due to an enhancement in thermal conductivity. In real-world applications, nanofluid is used in different procedures namely nano-technological and industrial developments such as nuclear reactors, vehicle cooling, heat exchanger, etc. Furthermore, magneto nanofluids are effective in wound treatments, cancer therapy, artery blockage removal, magnetic resonance imaging, and a variety of other applications. Hamad and Bashir [1] presented the influence of nanofluid under the power law model on a vertical stretching surface. They investigated Brownian motion and thermophoresis impacts on the vertical surface in their analysis. The influence of nanofluid flow on moving surfaces was studied by Bachok et al. [2]. They also developed the results by emphasizing the effects of the plate moving in the same or opposite direction in the free stream. The study of flow of nanofluid on an exponentially stretching sheet for nanomaterial fluid flow was presented by Nadeem and Lee [3]. Bég et al. [4] studied the steady flow of magnetic hydrodynamics mixed convection of nanofluid at the permeable nonlinear stretching sheet. Ramesh [5] highlighted the effects of nanofluid with Darcy-Forchheimer over stretching sheets. Khan et al. [6] discussed the Maxwell nanofluid model on a nonlinear stretching sheet. Khan and Nadeem. [7] analyzed the influence of chemically reactive nanomaterial Casson fluid with thermal slip over an exponentially stretching surface. Ramesh et al. [8] discussed the squeezing flow of micropolar Casson nanomaterial fluid with slip effects at stretching surfaces. Recently, a few authors developed results on the boundary layer flow of nanofluid at a stretching sheet (see [9,10]).
Many interesting works have been carried out on the hybrid nanofluid, which is an extended version of nanofluid. A hybrid nanofluid is a combination of two different nanosized particles and water as the base fluid. Devi and Devi [11] emphasized the impact of hybrid nanofluid on porous sheets numerically. Heat and mass transfer of hybrid nanofluid in a circular cylinder are discussed by Nadeem et al. [12]. They considered the magnetic hydrodynamic effects under the stagnation region. Nadeem et al. [13] worked on the fluid flow of hybrid nanomaterial at curved surfaces. Abbas et al. [14] inspected the impact of hybrid nanomaterial fluid flow with inclined magnetic hydrodynamics at a nonlinear stretching cylinder. Jyothi et al. [15] discussed the Casson hybrid nanomaterial fluid with squeezing flow with a sink or source. Several authors worked on the hybrid nanofluid for different flow assumptions and various physical aspects see [16,17].
The interest in magnetic hydrodynamics with hybrid nanofluid has been developed by the authors due to its many engineering applications. Because they can be used to control the rate of heat transfer by using an external magnetic field. Ali et al. [18] studied the influence of the laminar flow of the induced magnetic field on a stretching sheet. They implemented a numerical scheme to solve nonlinear differential equations. Thammanna et al. [19] utilized MHD to study the time-dependent flow of Casson nanomaterial fluid at an unsteadiness stretching sheet. Junoh et al. [20] emphasized the effects of the induced magnetic field with the stagnation point region. Moreover, the heat transfer rate at the stretching/shrinking sheet was analyzed by utilizing a two-phase model. Al-Hanaya et al. [21] discussed the micropolar hybrid nanomaterial fluid in the presence of a stagnation point region. They also highlighted the effects of the induced magnetic field on the curved surface. The hybrid nanomaterial fluid flow of the induced magnetic field transport mechanism has been studied by Alharbi [22]. Hafeez et al. [23] discussed the induced magnetic field of hybrid nanomaterial liquid for different aspects. Ali et al. [24] deliberated the influences of melting and MHD flow of nanomaterial liquid on the stretching surface. Some researchers have developed an interest in investigating the induced magnetic field for various flow assumptions, see [25,26,27,28].
In this study, we discuss the two-dimensional flow of Casson hybrid nanofluid over a vertical permeable exponential stretching sheet. We consider the induced magnetic field under the stagnation region. Furthermore, we discuss the effects of nonlinear radiation and heat generation. Besides, we present a study of three hybrid nanofluid models, namely: Xue, Yamada-Ota, and Tiwari Das. We also present a study on a single-wall carbon nanotube and multiwall carbon nanotube with base fluid water. We utilize boundary layer approximations to develop the governing equations under the assumptions of flow in the form of partial differential equations. Also, we use the Lie symmetry method to develop a suitable transformation. With the help of appropriate transformations, we convert partial differential equations into ordinary differential equations. Next, we use the fifth-order Runge-Kutta Fehlberg approach to analyze the ordinary differential equations. We investigate the effect of the concerning physical parameters by graphs and numerical values through tables. These findings are unique and may be helpful in the engineering and industrial fields.
The steady flow of incompressible Casson hybrid nanofluid over a permeable exponential stretching sheet is deliberated (see Figure 1).
The induced magnetic field is taken into account under the stagnation point flow. Heat generation and nonlinear radiation effects are discussed. The fluid wall temperature is Tw and the ambient fluid temperature is T∞. Ue is free stream velocity, and He is the free stream magnetic velocity function. A mathematical model in differential form is built for flow analysis, such as (see [29,30,31,32]):
∂u∂x+∂v∂y=0, | (1) |
∂H1∂x+∂H2∂y=0, | (2) |
u∂u∂x+v∂v∂y−μf4πp(H1∂H1∂x+H2∂H1∂y)=UedUedx−μ4πpHedHedx+νfρfρhnf(1(1−ϕ1)2.5(1−ϕ2)2.5+1β1)∂2u∂y2−ρfkρhnfu, | (3) |
u∂H1∂x+v∂H1∂y−H1∂u∂x−H2∂u∂y=μe∂2H1∂y2, | (4) |
u∂T∂x+v∂T∂y=αhnf∂2T∂y2+Q(ρCp)f(ρCp)hnf(T−T∞)−(ρCp)f(ρCp)hnf∂qr∂y. | (5) |
With relevant boundary conditions are as follows:
u=β∗μf(1(1−ϕ1)2.5(1−ϕ2)2.5+1β1)∂u∂y,v=Vw,H1=0,H2=0,T=Tw+γ∗∂T∂y,aty→0,u→Ue,H1→He,T→T∞,asy→∞. | (6) |
Introducing the stream functions are (see [29,30,31,32])
u=∂ψ∂y,v=−∂ψ∂x,H1=∂ψ1∂y,H2=−∂ψ1∂x,T=T∞+(Tw−T∞)θ(η). | (7) |
ψ1=Hoex/2L√υUog(η) and ψ=√2uoυLex2Lf(η) are the stream functions of the magnetic field and velocity. By using these values, the suitable transformations can be written as
u=∂ψ∂y=uoex/Lf'(η),v=−∂ψ∂x=−√uoυ2Lex/2L(f+ηf'),H1=∂ψ1∂y=Hoex/2L√2Lg'(η),H2=−∂ψ1∂x=−Hoex/2L2L√υUo(g+ηg'). | (8) |
Using the above suitable transformation, the partial differential equations are converted into ordinary differential equations as below:
ρfρhnf(1(1−ϕ1)2.5(1−ϕ2)2.5+1β1)f'''(η)+f''f+2(1−f')+γ1[2(g'2(η)−1)−gg'']−Mρfρhnf(f'(η)−1)+δθ=0, | (9) |
λg'''(η)−gf''+g''f=0, | (10) |
KhnfKf(ρcp)f(ρcp)hnf1Pr(1+43Rd)θ''+fθ'+(ρcp)f(ρcp)hnfKθ=0, | (11) |
with boundary conditions
f(0)=S,f'(0)=λ1(1(1−ϕ1)2.5(1−ϕ2)2.5+1β1)f''(0),f'(∞)=1,g(0)=0,g'(0)=0,g'(∞)=1,θ(0)=1+δ1KhnfKf(1+43Rd)θ'(0),θ(∞)=0. | (12) |
Some physical quantities of interest are melting rate at stretching sheet or Nusselt number Nux and skin friction Cf. The skin friction drag is presented as Cf=τwρfUw2, the ratio of conductive and convective heat transfer rate is defined as Nux=xqwk(T∞−Tw). The surface shear stress and heat flux are presented as below:
τw=μf(1(1−ϕ1)2.5(1−ϕ2)2.5+1β1)(∂u∂y)y=0,qw=−khnf(1+43)(∂T∂y)y=0. |
Using Eq (8), the above quantities become as
Re1/2xCf=(1(1−ϕ1)2.5(1−ϕ2)2.5+1β1)F''(0),Re1/2xNux=−khnf(1+43)θ'(0). |
The local Reynolds number is Rex=Uwxν. Some expressions are presented as:
μhnfμf=1(1−ϕ1)2.5(1−ϕ2)2.5,ρhnfρf=(1−ϕ1)(1−ϕ2)+ϕ1ρs1ρf+ϕ2ρs2ρf, |
(ρCp)hnf(ρCp)f=(1−ϕ1)(1−ϕ2)+ϕ1(ρCp)s1(ρCp)f+ϕ2(ρCp)s2(ρCp)f. |
The thermophysical characteristics of base fluid and nanoparticles in Table 1(see [21]).
Physical properties | Base fluid | Nanoparticles | |
Water | SWCNTs | MWCNTs | |
Cp(J/kgK) | 4179.0 | 425.00 | 796.0 |
ρ(kg/m3) | 997.10 | 2600.0 | 1600.0 |
K(W/mK) | 0.6130 | 6600.0 | 3000.0 |
The system of differential equations is a nonlinear boundary value problem. Several methods have been applied to solve the nonlinear boundary value problem arising in fluid dynamics. The system of nonlinear higher-order differential equations subject to boundary conditions is solved through the fifth-order Runge-Kutta-Fehlberg approach. The higher-order nonlinear differential equations are transformed into first-order differential equations. The procedure of the transformed equations is as follows (see [30]):
y(1)=f(η),y(2)=f'(η),y(3)=f''(η),yy1=f'''(η), | (13) |
yy1=[M{y(2)−1}−ρfγρhnf{2(y2(5)−1)−y(4).y(6)}−2(1−y(2)−y(3).y(1)+δy(7)]ρhnfρf.[1(1−ϕ1)2.5(1−ϕ2)2.5+1β1)−1], | (14) |
y(4)=g(η),y(5)=g'(η),y(6)=g''(η),yy2=g'''(η), | (15) |
yy2=[y(6).y(1)−y(4).y(3)]1λ, | (16) |
y(7)=θ(η),y(8)=θ'(η),yy3=θ''(η), | (17) |
yy3=−[y(1).y(8)−(ρcp)f(ρcp)hnfKy(7)]KfKhnf(ρcp)hnf(ρcp)fPr[1+43Rd]−1. | (18) |
With boundary conditions are
y0(1)−S;y0(2)−λ1(1(1−ϕ1)2.5(1−ϕ2)2.5+1β1)y0(3);yinf(2)−1;y0(4);y0(5);yinf(5)−1;y0(7)−1−δ1KhnfKf(1+43Rd)y0(8);yinf(7). | (19) |
The fifth-order Runge-Kutta-Fehlberg approach is used to solve the nonlinear higher-order differential system. The numerical outcomes will converge if the boundary residuals (R1(u∗1,u∗2,u∗3),R2(u∗1,u∗2,u∗3),R3(u∗1,u∗2,u∗3)) are not more than tolerance error i.e., 10−6. Newton's approach is used to change the initial approximations, and it is repeated until the required convergence basis is met. The residuals of boundary are offered as below:
R1(u∗1,u∗2,u∗3)=|y2(∞)−^y2(∞)|, |
R2(u∗1,u∗2,u∗3)=|y5(∞)−^y5(∞)|, |
R3(u∗1,u∗2,u∗3)=|y7(∞)−^y7(∞)|. |
Hence, ^y2(∞),^y5(∞), and ^y7(∞) are computed boundary values. Validation of the numerical scheme using the grid-independent test. The models of hybrid nanofluid are introduced. The Yamada-Ota model expression is defined as below (see [14]):
kbfkf=1+kfks1LRϕ0.21+(1−kfks1)ϕ1LRϕ0.21+2ϕ1(ks1ks1−kf)ln(ks1+kf2ks1)1−ϕ1+2ϕ1(kfks1−kf)ln(ks1+kf2kf), |
khnfkbf=1+kbfks2LRϕ0.22+(1−kbfks2)ϕ2LRϕ0.22+2ϕ2(ks2ks2−kbf)ln(ks2+kbf2ks2)1−ϕ2+2ϕ2(kbfks2−kbf)ln(ks2+kbf2kbf). |
Xue model expression is defined as below:
kbfkf=1−ϕ1+2ϕ1(ks1ks1−kf)ln(ks1+kf2kf)1−ϕ1+2ϕ1(kfks1−kf)ln(ks1+kf2kf), |
khnfkbf=1−ϕ2+2ϕ2(ks2ks2−kbf)ln(ks2+kbf2kbf)1−ϕ2+2ϕ2(kbfks2−kbf)ln(ks2+kbf2kbf). |
Tiwari-Das model expression is defined as below:
kbfkf=(n−1)kf−(kf−ks1)ϕ1(n−1)+ks1(kf−ks1)ϕ1+(n−1)kf+ks1, |
khnfkbf=(n−1)kbf−(kbf−ks2)ϕ2(n−1)+ks2(kbf−ks2)ϕ2+(n−1)kbf+ks2. |
A system of nonlinear ordinary differential Eqs (9)–(11) with boundary conditions (12) is solved through a numerical technique using the Matlab software packages. The involving physical parameters namely: M porosity parameter, β1 Casson fluid parameter, γ1 magnetic parameter, δ bouncy force parameter, λ reciprocal magnetic Prandtl number, Pr Prandtl number, K heat generation, Rd radiation parameter, S suction parameter, λ1 velocity slip parameter, and δ1 thermal slip parameter effects on the velocity profile, induced magnetic profile and temperature profile are revealed through Figures 2–13. In this analysis, we considered three models of hybrid nanofluid, namely: the Xue, Tiwari Das, and Yamada-Ota models. Two kinds of nanoparticles are discussed: single-wall and multi-wall carbon nanotubes with base fluid water. Thermo-physical characteristics of base fluid and nanoparticles are used which are revealed in Table 1. The variation of the velocity profile and β1 is shown in Figure 2. The curves of the velocity profile declined due to increasing the values of the Casson fluid parameter. The viscosity of fluid increased due to an increment in the Casson fluid parameter, which ultimately reduced the velocity of the fluid at the permeable vertical Riga sheet. The influence of γ1 on the velocity function is presented in Figure 3. The velocity profile curves decrease as the values of γ1 rise. As the electromagnetic forces enhanced due to increment of magnetic parameter which declined the velocity function.
The impacts of λ1 on the velocity profile are revealed in Figure 4. The velocity profile increased due to growing values of λ1. This is due to the fact that when the slip condition arises, the velocity of the stretching sheet is different from the flow velocity nearby. The variation of the porosity parameter and the velocity profile is presented in Figure 5. It is prominent that the velocity function declined due to enhancing values of the porosity parameter. The velocity boundary layer thickness increases with the porosity value, reducing the fluid flow resistance.
Figure 6 highlights the influence of ϕ2 on the velocity profile. The curves of the velocity profile decline due to higher values of ϕ2. As a result, the magneto-Casson hybrid nanofluid's effective viscosity increases for positive values of the nanoparticle volume fraction and exhibits high resistance to liquid motion. The impact of solid nanoparticles concentration on the induced magnetic profile is presented in Figure 7. The induced magnetic profile declined due to increasing values of solid nanoparticle concentration. The magnetic field provides an impulse to the fluid that is being slowed down by viscous force as well as stabilizing the viscous impacts.
The variation of the magnetic Prandtl number with the induced magnetic profile is revealed in Figure 8. The induced magnetic profile increased due to increasing values of magnetic Prandtl number. The variation of the bouncy force parameter with temperature function is shown in Figure 9. The temperature profile increased due to rising values of the bouncy force parameter.
Figure 10 exposes the influence of thermal slip on the temperature profile. The curves of the temperature profile decline due to increasing values of thermal slip. The influence of K on the temperature profile is illustrated in Figure 11. As heat generation increased, the temperature function at permeable vertical sheet improved.
Figure 12 shows the impacts of Rd on the temperature profile. The curves of the temperature profile increased due to larger values of the radiation parameter. Radiation is a heat exchanger between two surfaces. As the temperature enhanced due to enhancing the values of radiation parameters. The variation of ϕ2 with the temperature profile is presented in Figure 13. The temperature profile shows an inciting nature towards higher concentration of solid nanoparticles. Because the thermal conductivity of the fluid was enhanced due to an increment in solid nanoparticles, which enhanced the temperature of the magneto-Casson hybrid nanofluid.
The involving physical parameters, namely: M porosity parameter, β1 Casson fluid parameter, γ1 magnetic parameter, δ bouncy force parameter, λ1 reciprocal magnetic Prandtl number, Pr Prandtl number, K heat generation, Rd radiation parameter, S suction parameter, λ velocity slip parameter, and δ1 thermal slip parameter effects on the skin friction and Nusselt number are presented in Tables 2 and 3. The variation of the thermal slip with skin friction is presented in Table 2. The values of thermal slip are enhanced which increases skin friction. The inspiration of Rd for skin friction is revealed in Table 2. The values of skin friction increased due to higher values of radiation parameters. As the radiation is enhanced it resists to enhance the skin friction.
Physical parameters | Re1/2xCf | ||||||||||||
δ1 | Rd | K | γ1 | M | δ | λ1 | β1 | λ | ϕ2 | S | Yamada-Ota Model | Xue-Model | Tiwari-Das Model |
0.2 | 04 | 0.4 | 0.1 | 0.3 | 0.5 | 0.4 | 0.3 | 0.5 | 0.04 | 0.4 | 3.9277 | 3.9265 | 3.8982 |
0.4 | - | - | - | - | - | - | - | - | - | - | 3.9331 | 3.9321 | 3.9217 |
0.6 | - | - | - | - | - | - | - | - | - | - | 3.9379 | 3.9371 | 3.9378 |
0.4 | 0.2 | - | - | - | - | - | - | - | - | - | 3.9293 | 3.9284 | 3.9259 |
- | 0.4 | - | - | - | - | - | - | - | - | - | 3.9331 | 3.9321 | 3.9217 |
- | 0.6 | - | - | - | - | - | - | - | - | - | 3.9364 | 3.9354 | 3.9191 |
- | 0.4 | 0.0 | - | - | - | - | - | - | - | - | 3.9284 | 3.9278 | 3.9321 |
- | - | 0.2 | - | - | - | - | - | - | - | - | 3.9302 | 3.9294 | 3.9276 |
- | - | 0.4 | - | - | - | - | - | - | - | - | 3.9331 | 3.9321 | 3.9217 |
-2 | - | 0.6 | - | - | - | - | - | - | - | - | 3.9380 | 3.9368 | 3.9130 |
- | - | 0.4 | 0.1 | - | - | - | - | - | - | - | 3.9331 | 3.9321 | 3.9217 |
- | - | - | 0.3 | - | - | - | - | - | - | - | 3.1784 | 3.1774 | 3.1647 |
- | - | - | 0.5 | - | - | - | - | - | - | - | 2.3019 | 2.3008 | 2.2825 |
- | - | - | 0.1 | 0.0 | - | - | - | - | - | - | 4.2479 | 4.2479 | 4.2479 |
- | - | - | - | 0.3 | - | - | - | - | - | - | 3.9331 | 3.9321 | 3.9217 |
- | - | - | - | 0.6 | - | - | - | - | - | - | 3.5766 | 3.5746 | 3.5555 |
- | - | - | - | 0.3 | 0.0 | - | - | - | - | - | 4.0232 | 4.0232 | 3.9422 |
- | - | - | - | - | 0.5 | - | - | - | - | - | 3.9331 | 3.9321 | 3.8047 |
- | - | - | - | - | 1.0 | - | - | - | - | - | 3.8436 | 3.8416 | 3.6662 |
- | - | - | - | - | 0.5 | 0.2 | - | - | - | - | 3.8979 | 3.8969 | 3.7675 |
- | - | - | - | - | - | 0.4 | - | - | - | - | 3.9331 | 3.9321 | 3.8047 |
- | - | - | - | - | - | 0.6 | - | - | - | - | 3.9540 | 3.9530 | 3.8266 |
- | - | - | - | - | - | 0.4 | 0.1 | - | - | - | 5.8775 | 5.8766 | 5.6973 |
- | - | - | - | - | - | - | 0.3 | - | - | - | 3.9331 | 3.9321 | 3.8047 |
- | - | - | - | - | - | - | 0.5 | - | - | - | 3.5737 | 3.5727 | 3.4533 |
- | - | - | - | - | - | - | 0.3 | 0.0 | - | - | 5.2763 | 5.2749 | 5.0492 |
- | - | - | - | - | - | - | - | 0.5 | - | - | 3.9331 | 3.9321 | 3.8047 |
- | - | - | - | - | - | - | - | 1.0 | - | - | 3.1172 | 3.1165 | 3.0330 |
- | - | - | - | - | - | - | - | 0.5 | 0.005 | - | 3.3073 | 3.3070 | 3.2240 |
- | - | - | - | - | - | - | - | - | 0.02 | - | 3.5631 | 3.5625 | 3.4621 |
- | - | - | - | - | - | - | - | - | 0.04 | - | 3.9331 | 3.9321 | 3.8047 |
- | - | - | - | - | - | - | - | - | 0.06 | - | 4.3394 | 4.3381 | 4.1788 |
- | - | - | - | - | - | - | - | - | 0.04 | 0.0 | 3.6943 | 3.6933 | 3.5161 |
- | - | - | - | - | - | - | - | - | - | 0.2 | 3.8118 | 3.8108 | 3.6568 |
- | - | - | - | - | - | - | - | - | - | 0.4 | 3.9331 | 3.9321 | 3.8047 |
- | - | - | - | - | - | - | - | - | - | 0.6 | 4.0575 | 4.0567 | 3.9526 |
Physical parameters | Re1/2xNux | ||||||||||||
δ1 | Rd | K | γ1 | M | δ | λ1 | β1 | λ | ϕ2 | S | Yamada-Ota Model | Xue-Model | Tiwari-Das Model |
0.2 | 04 | 0.4 | 0.1 | 0.3 | 0.5 | 0.4 | 0.3 | 0.5 | 0.04 | 0.4 | 3.5558 | 3.4796 | 1.8070 |
0.4 | - | - | - | - | - | - | - | - | - | - | 3.3573 | 3.2804 | 1.4694 |
0.6 | - | - | - | - | - | - | - | - | - | - | 3.1795 | 3.1024 | 1.2376 |
0.4 | 0.2 | - | - | - | - | - | - | - | - | - | 3.0678 | 3.0002 | 1.3549 |
- | 0.4 | - | - | - | - | - | - | - | - | - | 3.3573 | 3.2804 | 1.4694 |
- | 0.6 | - | - | - | - | - | - | - | - | - | 3.6307 | 3.5450 | 1.5672 |
- | 0.4 | 0.0 | - | - | - | - | - | - | - | - | 5.4398 | 5.3008 | 1.8099 |
- | - | 0.2 | - | - | - | - | - | - | - | - | 4.5319 | 4.4212 | 1.6671 |
- | - | 0.4 | - | - | - | - | - | - | - | - | 3.3573 | 3.2804 | 1.4694 |
- | - | 0.6 | - | - | - | - | - | - | - | - | 1.7182 | 1.6821 | 1.1640 |
- | - | 0.4 | 0.1 | - | - | - | - | - | - | - | 3.3573 | 3.2804 | 1.4694 |
- | - | - | 0.3 | - | - | - | - | - | - | - | 2.8416 | 2.7778 | 1.3849 |
- | - | - | 0.5 | - | - | - | - | - | - | - | 1.9747 | 1.9312 | 1.2451 |
- | - | - | 0.1 | 0.0 | - | - | - | - | - | - | 3.5143 | 3.4338 | 1.4973 |
- | - | - | - | 0.3 | - | - | - | - | - | - | 3.3573 | 3.2804 | 1.4694 |
- | - | - | - | 0.6 | - | - | - | - | - | - | 3.1609 | 3.0883 | 1.4347 |
- | - | - | - | 0.3 | 0.0 | - | - | - | - | - | 3.4099 | 3.3318 | 1.4704 |
- | - | - | - | - | 0.5 | - | - | - | - | - | 3.3573 | 3.2804 | 1.4587 |
- | - | - | - | - | 1.0 | - | - | - | - | - | 3.3036 | 3.2278 | 1.4465 |
- | - | - | - | - | 0.5 | 0.2 | - | - | - | - | 3.3233 | 3.2475 | 1.4544 |
- | - | - | - | - | - | 0.4 | - | - | - | - | 3.3573 | 3.2804 | 1.4587 |
- | - | - | - | - | - | 0.6 | - | - | - | - | 3.3762 | 3.2986 | 1.4612 |
- | - | - | - | - | - | 0.4 | 0.0 | - | - | - | 2.5806 | 2.5223 | 1.3223 |
- | - | - | - | - | - | - | 0.3 | - | - | - | 3.3573 | 3.2804 | 1.4587 |
- | - | - | - | - | - | - | 0.6 | - | - | - | 3.6019 | 3.5190 | 1.5034 |
- | - | - | - | - | - | - | 0.3 | 0.0 | - | - | 2.5647 | 2.4973 | 1.1972 |
- | - | - | - | - | - | - | - | 0.5 | - | - | 3.3573 | 3.2804 | 1.4587 |
- | - | - | - | - | - | - | - | 1.0 | - | - | 3.7602 | 3.6767 | 1.5657 |
- | - | - | - | - | - | - | - | 0.5 | .005 | - | 2.5001 | 2.4666 | 1.4006 |
- | - | - | - | - | - | - | - | - | 0.02 | - | 2.8923 | 2.8389 | 1.4266 |
- | - | - | - | - | - | - | - | - | 0.04 | - | 3.3573 | 3.2804 | 1.4587 |
- | - | - | - | - | - | - | - | - | 0.06 | - | 3.7799 | 3.6816 | 1.4876 |
- | - | - | - | - | - | - | - | - | 0.04 | 0.0 | 1.6248 | 1.5507 | 0.196851 |
- | - | - | - | - | - | - | - | - | - | 0.2 | 2.5003 | 2.4258 | 0.85367 |
- | - | - | - | - | - | - | - | - | - | 0.4 | 3.3573 | 3.2804 | 1.4587 |
- | - | - | - | - | - | - | - | - | - | 0.6 | 4.1876 | 4.1061 | 1.8678 |
The influence of K on skin friction is highlighted in Table 2. The values of skin friction increased due to higher values of the heat generation parameter. Table 2 reveals the variation of porosity parameters and skin friction. The values of skin friction decline due to higher values of the porosity parameter. The porous body's pore volume to total nominal volume ratio resists decreasing skin friction. The impact of the bouncy force parameter on the skin friction reveals in Table 2. The values of skin friction are reduced due to higher values of the bouncy force parameter. A body submerged partially or completely in a fluid appears to drop weight or to be lighter due to the buoyant force.
The impact of velocity slip on skin friction is revealed in Table 2. Values of skin friction increased due to higher values of λ1. This is due to the fact that when the slip condition arises, the velocity of the stretching sheet is different from the flow velocity nearby as well as skin friction is enhanced. Table 2 reveals the influence of the Casson fluid parameter on skin friction. The skin friction declines due to larger values of β1. The variation of the magnetic Prandtl number and skin friction is presented in Table 2. The skin friction is found to be decreasing behavior due to higher values of magnetic Prandtl number. The influence of ϕ2 on skin friction is revealed in Table 2. The values of skin friction are found to be increasing due to higher values of ϕ2 because of solid particles in the fluid are enhanced, which increases the skin friction phenomena.
The variation of suction parameters with skin friction is revealed in Table 2. The values of skin friction increased due to larger values of S because the section parameter enhanced, which resisted the flow of fluid as well as enhanced skin friction. The influence of thermal slip on the Nusselt number is revealed in Table 3. The values of the Nusselt number increased due to enhancing values of δ1. The variation of radiation parameter and Nusselt number is revealed in Table 3. The values of the Nusselt number are enhanced due to boosting values of the radiation parameter. As a result of the addition of radiation, the heat transfer phenomenon was enhanced. The impacts of heat generation on the Nusselt number are presented in Table 3. The heat generation parameter is enhanced, which declines the values of the Nusselt number. The impacts of magnetic parameters on the Nusselt number are revealed in Table 3. The values of the Nusselt number are found to be declining due to higher values of the magnetic field parameter.
The variation of the porosity parameter and Nusselt number is revealed in Table 3. The values of the Nusselt number are reduced due to porosity parameter enhancement. The variation of δ bouncy force parameter and Nusselt number are presented in Table 3. The values of the Nusselt number decay due to greater values of δ bouncy force parameter. The influence of velocity slip on the Nusselt number is shown in Table 3. The values of the Nusselt number are enhanced due to higher values of λ1. The variation of β1 and Nusselt number reveals in Table 3. As the value of the β1 grew, the Nusselt number moved up. The impact of the magnetic Prandtl number on the Nusselt number is revealed in Table 3. The Nusselt number enhances for higher values of λ.
The influence of solid nanoparticle concentration on the Nusselt number is shown in Table 3. The heat transfer rate enhances due to higher values of ϕ2. The thermal conductivity of the fluid increased due to the increment in solid nanoparticles in the base fluid, which boosted the values of the heat transfer rate. The inspiration of the suction parameter on the Nusselt number is highlighted in Table 3. The Nusselt number rises as the value of the suction parameter rises. Table 4 offers the comparative results of Zainal et al. [33] and Bachok et al. [34] for various values ϕ2 on skin friction when the rest of values are zero such as M=γ1=δ=λ1=K=Rd=S=λ=δ1=ϕ1=0 and β1→∞. It should be noted that our results were found to be in good agreement with existing results.
ϕ2 | Zainal et al. [33] | Bachok et al. [34] | Present results |
0.0 | 1.687218 | 1.687200 | 1.687221 |
0.1 | 2.579342 | 2.579400 | 2.579317 |
0.2 | 3.590122 | 3.590100 | 3.590111 |
In this analysis, the steady flow of hybrid Casson nanofluid over a vertical permeable exponential stretching sheet is considered. The impacts of the induced magnetic field are studied in this analysis. The influence of heat generation and radiation with a slip effect is studied. Three models of hybrid nanofluid, namely: Yamada-Ota, Xue, and Tiwari Das are debated. The key points are presented as follows:
● The skin friction achieved higher values in the Yamada-Ota model of hybrid nanofluid as compared to the Xue model and the Tiwari Das model.
● The Nusselt number achieved higher values in the Yamada-Ota model of hybrid nanofluid as compared to the Xue model and Tiwari Das model.
● Temperature profile increased due to larger values of the radiation parameter. Radiation is a heat exchanger between two surfaces.
● The temperature profile shows in inciting nature towards higher concentration of nanoparticles. Because the thermal conductivity of the fluid was enhanced due to an increment in solid nanoparticles, which enhanced the temperature of magneto-Casson hybrid nanofluid.
● The velocity profile increased due to increasing the values of λ1. This is due to the fact that when the slip condition arises, the velocity of the stretching sheet is different from the flow velocity nearby.
● The velocity profile declined due to increasing the values of the Casson fluid parameter. Higher values of the Casson fluid parameter cause a reduction in viscosity of the fluid as a result velocity reduces.
● The values of skin friction are found to be increasing due to higher values of ϕ2 because solid particles in the fluid are enhanced, which increases the skin friction coefficient.
Dr. Taqi A.M. Shatnawi wishes to thank Hashemite University for paying the Article Publication Fee. Dr. Nadeem Abbas and Prof. Dr. Wasfi Shatanawi wish to express their gratitude to Prince Sultan University for facilitating the publication of this article through the research lab Theoretical and Applied Sciences Lab (TAS).
The authors declare no conflict of interest.
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20. | Umair Rashid, Ali Akgül, Dianchen Lu, Impact of nanosized particles on hybrid nanofluid flow in porous medium with thermal slip condition, 2023, 1040-7790, 1, 10.1080/10407790.2023.2279085 | |
21. | Muhammad Usman Ghani, Muhammad Imran, S. Sampathkumar, Fairouz Tchier, K. Pattabiraman, Ahmad Zubair Jan, A paradigmatic approach to the molecular descriptor computation for some antiviral drugs, 2023, 9, 24058440, e21401, 10.1016/j.heliyon.2023.e21401 | |
22. | Muhammad Azam, Waqar Azeem Khan, Manoj Kumar Nayak, Abdul Majeed, Three dimensional convective flow of Sutterby nanofluid with activation energy, 2023, 50, 2214157X, 103446, 10.1016/j.csite.2023.103446 | |
23. | Azzh Saad Alshehry, Humaira Yasmin, Abdul Hamid Ganie, Rasool Shah, Heat transfer performance of magnetohydrodynamic multiphase nanofluid flow of Cu–Al2O3/H2O over a stretching cylinder, 2023, 21, 2391-5471, 10.1515/phys-2023-0142 | |
24. | K. Sudarmozhi, D. Iranian, Fahima Hajjej, Ilyas Khan, Abdoalrahman S.A. Omer, M. Ijaz Khan, Exploring thermal diffusion and diffusion-thermal energy in MHD Maxwell fluid flow around a stretching cylinder, 2023, 52, 2214157X, 103663, 10.1016/j.csite.2023.103663 | |
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26. | Kiran Batool, Fazal Haq, Faria Fatima, Kashif Ali, Significance of Interfacial Nanolayer and Mixed Convection in Radiative Casson Hybrid Nanofluid Flow by Permeable Rotating Cone, 2023, 13, 2191-1630, 1741, 10.1007/s12668-023-01191-1 | |
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31. | Ashish Paul, Bhagyashri Patgiri, Neelav Sarma, Transformer oil‐based Casson ternary hybrid nanofluid flow configured by a porous rotating disk with hall current, 2024, 104, 0044-2267, 10.1002/zamm.202300704 | |
32. | Shafiq Ahmad, Muhammad Naveed Khan, Rifaqat Ali, Showkat Ahmad Lone, Analysis of free bioconvective flow of hybrid nanofluid induced by convectively heated cone with entropy generation, 2024, 38, 0217-9849, 10.1142/S0217984924500155 | |
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34. | Lokendra Kumar, Magnetic Dipole and Mixed Convective Effect on Boundary Layer Flow of Ferromagnetic Micropolar Hybrid Nanofluid, 2024, 13, 2169-432X, 435, 10.1166/jon.2024.2123 | |
35. | M.R. Zangooee, Kh. Hosseinzadeh, D.D. Ganji, Hydrothermal analysis of Ag and CuO hybrid NPs suspended in mixture of water 20%+EG 80% between two concentric cylinders, 2023, 50, 2214157X, 103398, 10.1016/j.csite.2023.103398 | |
36. | Waseem Abbas, Nejla Mahjoub Said, Nidhish Kumar Mishra, Zafar Mahmood, Muhammad Bilal, Significance of coupled effects of resistive heating and perpendicular magnetic field on heat transfer process of mixed convective flow of ternary nanofluid, 2024, 149, 1388-6150, 879, 10.1007/s10973-023-12723-y | |
37. | Munawar Abbas, Nargis Khan, Ali Saleh Alshomrani, M.S. Hashmi, Mustafa Inc, Performance-based comparison of Xue and Yamada–Ota models of ternary hybrid nanofluid flow over a slendering stretching sheet with activation energy and melting phenomena, 2023, 50, 2214157X, 103427, 10.1016/j.csite.2023.103427 | |
38. | Wei Fan, High-temperature superconductivity mechanism and an alternative theoretical model of Maxwell’s classical electromagnetism theory, 2024, 38, 0217-9849, 10.1142/S0217984924501513 | |
39. | Mohsin Ali, Nadeem Abbas, Wasfi Shatanawi, Soret and Dufour effects on Sutterby nanofluid flow over a Riga stretching surface with variable thermal conductivity, 2024, 0217-9849, 10.1142/S0217984924504554 | |
40. | S. Nadeem, Bushra Ishtiaq, Jehad Alzabut, Hassan A. Ghazwani, Ahmad M. Hassan, Unsteady magnetized flow of micropolar fluid with prescribed thermal conditions subject to different geometries, 2023, 53, 22113797, 106946, 10.1016/j.rinp.2023.106946 | |
41. | Ramesh Reddy, S. Abdul Gaffar, Chemical Reaction and Viscous Dissipative Effects on Buongiorno’s Nanofluid Model Past an Inclined Plane: A Numerical Investigation, 2024, 10, 2349-5103, 10.1007/s40819-024-01723-7 | |
42. | Dharmendar Reddy Yanala, Shankar Goud Bejawada, Kottakkaran Sooppy Nisar, Influence of Chemical reaction and heat generation/absorption on Unsteady magneto Casson Nanofluid flow past a non-linear stretching Riga plate with radiation, 2023, 50, 2214157X, 103494, 10.1016/j.csite.2023.103494 | |
43. | Sobia Nisar, Ali Alsalme, Rimsha Zulfiqar, Muhammad Rizwan, Deok-Kee Kim, Ghulam Dastgeer, Zafar Muhammad Shazad, Laser-induced breakdown spectroscopy for rapid and accurate qualitative analysis of elemental composition in fertilizers, 2024, 38, 0217-9849, 10.1142/S0217984924501240 | |
44. | Hanumesh Vaidya, D. Tripathi, Fateh Mebarek-Oudina, C. Rajashekhar, Haci Mehmet Baskonus, K.V. Prasad, , Scrutiny of MHD impact on Carreau Yasuda (CY) fluid flow over a heated wall of the uniform micro-channel, 2024, 87, 05779073, 766, 10.1016/j.cjph.2023.12.015 | |
45. | Nilanchala Sethy, S. R. Mishra, Illustration of thermal buoyancy on the flow of MHD Casson CNT–water nanofluids over a vertical stretching surface, 2024, 38, 0217-9849, 10.1142/S0217984924502312 | |
46. | Aaqib Majeed, Taoufik Saidani, Nouman Ijaz, Ahmed Osman Ibrahim, Hela Gnaba, Sadia Samar Ali, Boost heat transfer efficiency through thermal radiation and electrical conductivity in nanofluids, 2024, 0044-2267, 10.1002/zamm.202400005 | |
47. | Esraa N. Thabet, Zeeshan Khan, A. M. Abd-Alla, F. S. Bayones, Thermal enhancement, thermophoretic diffusion, and Brownian motion impacts on MHD micropolar nanofluid over an inclined surface: Numerical simulation, 2023, 1040-7782, 1, 10.1080/10407782.2023.2276319 | |
48. | Azhar Mustafa Soomro, Mustafa Abbas Fadhel, Liaquat Ali Lund, Zahir Shah, Mansoor H. Alshehri, Narcisa Vrinceanu, Dual solutions of magnetized radiative flow of Casson Nanofluid over a stretching/shrinking cylinder: Stability analysis, 2024, 10, 24058440, e29696, 10.1016/j.heliyon.2024.e29696 | |
49. | B.S. Bhadauria, Anish Kumar, Sawan Kumar Rawat, Moh Yaseen, Thermal instability of Tri-hybrid Casson nanofluid with thermal radiation saturated porous medium in different enclosures, 2024, 87, 05779073, 710, 10.1016/j.cjph.2023.12.032 | |
50. | Mohammed A. Albedah, Zhixiong Li, Iskander Tlili, A tripe diffusion bioconvective model for thixotropic nanofluid with applications of induced magnetic field, 2024, 14, 2045-2322, 10.1038/s41598-024-58195-4 | |
51. | Syed Muhammad Raza Shah Naqvi, Umair Manzoor, Hassan Waqas, Dong Liu, Hamzah Naeem, Sayed M. Eldin, Taseer Muhammad, Numerical investigation of thermal radiation with entropy generation effects in hybrid nanofluid flow over a shrinking/stretching sheet, 2024, 13, 2191-9097, 10.1515/ntrev-2023-0171 | |
52. | P.M. Patil, Bharath Goudar, Mrinalgouda Patil, E. Momoniat, Unsteady magneto bioconvective Sutterby nanofluid flow: Influence of g-Jitter effect, 2024, 89, 05779073, 565, 10.1016/j.cjph.2023.10.043 | |
53. | N. Gomathi, De Poulomi, Entropy optimization on EMHD Casson Williamson penta-hybrid nanofluid over porous exponentially vertical cone, 2024, 108, 11100168, 590, 10.1016/j.aej.2024.07.092 | |
54. | J.C. Umavathi, M. Sankar, O.Anwar Bég, Ali J. Chamkha, Computation of couple stress electroconductive polymer from an exponentially stretching sheet, 2023, 86, 05779073, 75, 10.1016/j.cjph.2023.10.002 | |
55. | Muhammad Amjad, Nabeela Ramzan, Shahzad Ahmad, Haider Ali, Mansoor Alshehri, Nehad Ali Shah, Retracted: Numerical study of heat and mass transfer for micropolar fluid flow due to two symmetrical stretchable disks, 2023, 98, 0031-8949, 105227, 10.1088/1402-4896/acf813 | |
56. | Xue Wang, Saeid Razmjooy, Improved Giza pyramids construction algorithm for Modify the deep neural network-based method for energy demand forecasting, 2023, 9, 24058440, e20527, 10.1016/j.heliyon.2023.e20527 | |
57. | M. Shanmugapriya, R. Sundareswaran, S. Gopi Krishna, U. Fernandez-Gamiz, S. Narasimman, Magnetized Casson hybrid nanofluid flow under the influence of surface-catalyzed reactions over a porous moving wedge, 2024, 14, 2158-3226, 10.1063/5.0216570 | |
58. | Nadeem Abbas, Noor Ul Huda, Wasfi Shatanawi, Zead Mustafa, Melting heat transfer of Maxwell–Sutterby fluid over a stretching sheet with stagnation region and induced magnetic field, 2024, 38, 0217-9849, 10.1142/S0217984924500854 | |
59. | Muhammad Asad Iqbal, Muhammad Usman, F.M. Allehiany, Muzamil Hussain, Khalid Ali Khan, Rotating MHD Williamson nanofluid flow in 3D over exponentially stretching sheet with variable thermal conductivity and diffusivity, 2023, 9, 24058440, e22294, 10.1016/j.heliyon.2023.e22294 | |
60. | R. Sindhu, S. Eswaramoorthi, K. Loganathan, Reema Jain, Comparative approach of Darcy–Forchheimer flow on water based hybrid nanofluid (Cu-Al2O3) and mono nanofluid (Cu) over a stretched surface with injection/suction, 2024, 11, 26668181, 100786, 10.1016/j.padiff.2024.100786 | |
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62. | Viacheslav Kovtun, Torki Altameem, Mohammed Al-Maitah, Wojciech Kempa, Entropy-metric estimation of the small data models with stochastic parameters, 2024, 10, 24058440, e24708, 10.1016/j.heliyon.2024.e24708 | |
63. | Nadeem Abbas, Wasfi Shatanawi, Fady Hasan, Zead Mustafa, Thermodynamic flow of radiative induced magneto modified Maxwell Sutterby fluid model at stretching sheet/cylinder, 2023, 13, 2045-2322, 10.1038/s41598-023-40843-w | |
64. | M. Shanmugapriya, R. Sundareswaran, S. Gopi Krishna, Abdu Alameri, Saranya Shekar, Investigation of Magnetized Casson Nanofluid Flow along Wedge: Gaussian Process Regression, 2024, 2024, 1687-0425, 1, 10.1155/2024/2880748 | |
65. | Nur Syahirah Wahid, Mohd Shafie Mustafa, Norihan Md Arifin, Ioan Pop, Nur Syazana Anuar, Najiyah Safwa Khashi'ie, Numerical and statistical analyses of three-dimensional non-axisymmetric Homann's stagnation-point flow of nanofluids over a shrinking surface, 2024, 89, 05779073, 1555, 10.1016/j.cjph.2023.11.034 | |
66. | Santosh Chaudhary, Ajay Singh, Devendra Kumar, Dumitru Baleanu, Numerical analysis for MHD blood-nanofluid flow through a non-linearly stretched sheet interpolated in a permeable medium along heat generation, 2023, 52, 2214157X, 103786, 10.1016/j.csite.2023.103786 | |
67. | Zahoor Shah, Muhammad Asif Zahoor Raja, Waqar Azeem Khan, Muhammad Shoaib, Vineet Tirth, Ali Algahtani, Kashif Irshad, Tawfiq Al-Mughanam, Computational intelligence paradigm with Levenberg-Marquardt networks for dynamics of Reynolds nanofluid model for Casson fluid flow, 2024, 191, 0301679X, 109180, 10.1016/j.triboint.2023.109180 | |
68. | D.O. Soumya, P. Venkatesh, Pudhari Srilatha, Jasgurpreet Singh Chohan, B.C. Prasannakumara, Mansoor Alshehri, Nehad Ali Shah, Significance of TiO2- water nanofluid, buoyant strength and ohmic heating in the enhancement of microchannel efficiency, 2024, 60, 2214157X, 104605, 10.1016/j.csite.2024.104605 | |
69. | Muhammad Azhar Iqbal, Nargis Khan, A.H. Alzahrani, Y. Khan, Thermophoretic particle deposition in bioconvection flow of nanofluid with microorganisms and heat source: Applications of nanoparticle and thermal radiation, 2025, 18, 16878507, 101305, 10.1016/j.jrras.2025.101305 | |
70. | S. Baskaran, R. Sowrirajan, S. Divya, S. Eswaramoorthi, K. Loganathan, Analysis of water based Casson hybrid nanofluid (NiZnFe2O4+MnZnFe2O4) flow over an electromagnetic actuator with Cattaneo–Christov heat-mass flux: A modified Buongiorno model, 2025, 13, 26668181, 101079, 10.1016/j.padiff.2025.101079 | |
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72. | Khaleeq ur Rahman, Syed Zulfiqar Ali Zaidi, Refka Ghodhbani, Dana Mohammad Khidhir, Muhammad Asad Iqbal, Iskander Tlili, Thermomechanics of radiated hybrid nanofluid interacting with MHD and heating source: Significance of nanoparticles shapes, 2025, 18, 16878507, 101544, 10.1016/j.jrras.2025.101544 |
Physical properties | Base fluid | Nanoparticles | |
Water | SWCNTs | MWCNTs | |
Cp(J/kgK) | 4179.0 | 425.00 | 796.0 |
ρ(kg/m3) | 997.10 | 2600.0 | 1600.0 |
K(W/mK) | 0.6130 | 6600.0 | 3000.0 |
Physical parameters | Re1/2xCf | ||||||||||||
δ1 | Rd | K | γ1 | M | δ | λ1 | β1 | λ | ϕ2 | S | Yamada-Ota Model | Xue-Model | Tiwari-Das Model |
0.2 | 04 | 0.4 | 0.1 | 0.3 | 0.5 | 0.4 | 0.3 | 0.5 | 0.04 | 0.4 | 3.9277 | 3.9265 | 3.8982 |
0.4 | - | - | - | - | - | - | - | - | - | - | 3.9331 | 3.9321 | 3.9217 |
0.6 | - | - | - | - | - | - | - | - | - | - | 3.9379 | 3.9371 | 3.9378 |
0.4 | 0.2 | - | - | - | - | - | - | - | - | - | 3.9293 | 3.9284 | 3.9259 |
- | 0.4 | - | - | - | - | - | - | - | - | - | 3.9331 | 3.9321 | 3.9217 |
- | 0.6 | - | - | - | - | - | - | - | - | - | 3.9364 | 3.9354 | 3.9191 |
- | 0.4 | 0.0 | - | - | - | - | - | - | - | - | 3.9284 | 3.9278 | 3.9321 |
- | - | 0.2 | - | - | - | - | - | - | - | - | 3.9302 | 3.9294 | 3.9276 |
- | - | 0.4 | - | - | - | - | - | - | - | - | 3.9331 | 3.9321 | 3.9217 |
-2 | - | 0.6 | - | - | - | - | - | - | - | - | 3.9380 | 3.9368 | 3.9130 |
- | - | 0.4 | 0.1 | - | - | - | - | - | - | - | 3.9331 | 3.9321 | 3.9217 |
- | - | - | 0.3 | - | - | - | - | - | - | - | 3.1784 | 3.1774 | 3.1647 |
- | - | - | 0.5 | - | - | - | - | - | - | - | 2.3019 | 2.3008 | 2.2825 |
- | - | - | 0.1 | 0.0 | - | - | - | - | - | - | 4.2479 | 4.2479 | 4.2479 |
- | - | - | - | 0.3 | - | - | - | - | - | - | 3.9331 | 3.9321 | 3.9217 |
- | - | - | - | 0.6 | - | - | - | - | - | - | 3.5766 | 3.5746 | 3.5555 |
- | - | - | - | 0.3 | 0.0 | - | - | - | - | - | 4.0232 | 4.0232 | 3.9422 |
- | - | - | - | - | 0.5 | - | - | - | - | - | 3.9331 | 3.9321 | 3.8047 |
- | - | - | - | - | 1.0 | - | - | - | - | - | 3.8436 | 3.8416 | 3.6662 |
- | - | - | - | - | 0.5 | 0.2 | - | - | - | - | 3.8979 | 3.8969 | 3.7675 |
- | - | - | - | - | - | 0.4 | - | - | - | - | 3.9331 | 3.9321 | 3.8047 |
- | - | - | - | - | - | 0.6 | - | - | - | - | 3.9540 | 3.9530 | 3.8266 |
- | - | - | - | - | - | 0.4 | 0.1 | - | - | - | 5.8775 | 5.8766 | 5.6973 |
- | - | - | - | - | - | - | 0.3 | - | - | - | 3.9331 | 3.9321 | 3.8047 |
- | - | - | - | - | - | - | 0.5 | - | - | - | 3.5737 | 3.5727 | 3.4533 |
- | - | - | - | - | - | - | 0.3 | 0.0 | - | - | 5.2763 | 5.2749 | 5.0492 |
- | - | - | - | - | - | - | - | 0.5 | - | - | 3.9331 | 3.9321 | 3.8047 |
- | - | - | - | - | - | - | - | 1.0 | - | - | 3.1172 | 3.1165 | 3.0330 |
- | - | - | - | - | - | - | - | 0.5 | 0.005 | - | 3.3073 | 3.3070 | 3.2240 |
- | - | - | - | - | - | - | - | - | 0.02 | - | 3.5631 | 3.5625 | 3.4621 |
- | - | - | - | - | - | - | - | - | 0.04 | - | 3.9331 | 3.9321 | 3.8047 |
- | - | - | - | - | - | - | - | - | 0.06 | - | 4.3394 | 4.3381 | 4.1788 |
- | - | - | - | - | - | - | - | - | 0.04 | 0.0 | 3.6943 | 3.6933 | 3.5161 |
- | - | - | - | - | - | - | - | - | - | 0.2 | 3.8118 | 3.8108 | 3.6568 |
- | - | - | - | - | - | - | - | - | - | 0.4 | 3.9331 | 3.9321 | 3.8047 |
- | - | - | - | - | - | - | - | - | - | 0.6 | 4.0575 | 4.0567 | 3.9526 |
Physical parameters | Re1/2xNux | ||||||||||||
δ1 | Rd | K | γ1 | M | δ | λ1 | β1 | λ | ϕ2 | S | Yamada-Ota Model | Xue-Model | Tiwari-Das Model |
0.2 | 04 | 0.4 | 0.1 | 0.3 | 0.5 | 0.4 | 0.3 | 0.5 | 0.04 | 0.4 | 3.5558 | 3.4796 | 1.8070 |
0.4 | - | - | - | - | - | - | - | - | - | - | 3.3573 | 3.2804 | 1.4694 |
0.6 | - | - | - | - | - | - | - | - | - | - | 3.1795 | 3.1024 | 1.2376 |
0.4 | 0.2 | - | - | - | - | - | - | - | - | - | 3.0678 | 3.0002 | 1.3549 |
- | 0.4 | - | - | - | - | - | - | - | - | - | 3.3573 | 3.2804 | 1.4694 |
- | 0.6 | - | - | - | - | - | - | - | - | - | 3.6307 | 3.5450 | 1.5672 |
- | 0.4 | 0.0 | - | - | - | - | - | - | - | - | 5.4398 | 5.3008 | 1.8099 |
- | - | 0.2 | - | - | - | - | - | - | - | - | 4.5319 | 4.4212 | 1.6671 |
- | - | 0.4 | - | - | - | - | - | - | - | - | 3.3573 | 3.2804 | 1.4694 |
- | - | 0.6 | - | - | - | - | - | - | - | - | 1.7182 | 1.6821 | 1.1640 |
- | - | 0.4 | 0.1 | - | - | - | - | - | - | - | 3.3573 | 3.2804 | 1.4694 |
- | - | - | 0.3 | - | - | - | - | - | - | - | 2.8416 | 2.7778 | 1.3849 |
- | - | - | 0.5 | - | - | - | - | - | - | - | 1.9747 | 1.9312 | 1.2451 |
- | - | - | 0.1 | 0.0 | - | - | - | - | - | - | 3.5143 | 3.4338 | 1.4973 |
- | - | - | - | 0.3 | - | - | - | - | - | - | 3.3573 | 3.2804 | 1.4694 |
- | - | - | - | 0.6 | - | - | - | - | - | - | 3.1609 | 3.0883 | 1.4347 |
- | - | - | - | 0.3 | 0.0 | - | - | - | - | - | 3.4099 | 3.3318 | 1.4704 |
- | - | - | - | - | 0.5 | - | - | - | - | - | 3.3573 | 3.2804 | 1.4587 |
- | - | - | - | - | 1.0 | - | - | - | - | - | 3.3036 | 3.2278 | 1.4465 |
- | - | - | - | - | 0.5 | 0.2 | - | - | - | - | 3.3233 | 3.2475 | 1.4544 |
- | - | - | - | - | - | 0.4 | - | - | - | - | 3.3573 | 3.2804 | 1.4587 |
- | - | - | - | - | - | 0.6 | - | - | - | - | 3.3762 | 3.2986 | 1.4612 |
- | - | - | - | - | - | 0.4 | 0.0 | - | - | - | 2.5806 | 2.5223 | 1.3223 |
- | - | - | - | - | - | - | 0.3 | - | - | - | 3.3573 | 3.2804 | 1.4587 |
- | - | - | - | - | - | - | 0.6 | - | - | - | 3.6019 | 3.5190 | 1.5034 |
- | - | - | - | - | - | - | 0.3 | 0.0 | - | - | 2.5647 | 2.4973 | 1.1972 |
- | - | - | - | - | - | - | - | 0.5 | - | - | 3.3573 | 3.2804 | 1.4587 |
- | - | - | - | - | - | - | - | 1.0 | - | - | 3.7602 | 3.6767 | 1.5657 |
- | - | - | - | - | - | - | - | 0.5 | .005 | - | 2.5001 | 2.4666 | 1.4006 |
- | - | - | - | - | - | - | - | - | 0.02 | - | 2.8923 | 2.8389 | 1.4266 |
- | - | - | - | - | - | - | - | - | 0.04 | - | 3.3573 | 3.2804 | 1.4587 |
- | - | - | - | - | - | - | - | - | 0.06 | - | 3.7799 | 3.6816 | 1.4876 |
- | - | - | - | - | - | - | - | - | 0.04 | 0.0 | 1.6248 | 1.5507 | 0.196851 |
- | - | - | - | - | - | - | - | - | - | 0.2 | 2.5003 | 2.4258 | 0.85367 |
- | - | - | - | - | - | - | - | - | - | 0.4 | 3.3573 | 3.2804 | 1.4587 |
- | - | - | - | - | - | - | - | - | - | 0.6 | 4.1876 | 4.1061 | 1.8678 |
Physical properties | Base fluid | Nanoparticles | |
Water | SWCNTs | MWCNTs | |
Cp(J/kgK) | 4179.0 | 425.00 | 796.0 |
ρ(kg/m3) | 997.10 | 2600.0 | 1600.0 |
K(W/mK) | 0.6130 | 6600.0 | 3000.0 |
Physical parameters | Re1/2xCf | ||||||||||||
δ1 | Rd | K | γ1 | M | δ | λ1 | β1 | λ | ϕ2 | S | Yamada-Ota Model | Xue-Model | Tiwari-Das Model |
0.2 | 04 | 0.4 | 0.1 | 0.3 | 0.5 | 0.4 | 0.3 | 0.5 | 0.04 | 0.4 | 3.9277 | 3.9265 | 3.8982 |
0.4 | - | - | - | - | - | - | - | - | - | - | 3.9331 | 3.9321 | 3.9217 |
0.6 | - | - | - | - | - | - | - | - | - | - | 3.9379 | 3.9371 | 3.9378 |
0.4 | 0.2 | - | - | - | - | - | - | - | - | - | 3.9293 | 3.9284 | 3.9259 |
- | 0.4 | - | - | - | - | - | - | - | - | - | 3.9331 | 3.9321 | 3.9217 |
- | 0.6 | - | - | - | - | - | - | - | - | - | 3.9364 | 3.9354 | 3.9191 |
- | 0.4 | 0.0 | - | - | - | - | - | - | - | - | 3.9284 | 3.9278 | 3.9321 |
- | - | 0.2 | - | - | - | - | - | - | - | - | 3.9302 | 3.9294 | 3.9276 |
- | - | 0.4 | - | - | - | - | - | - | - | - | 3.9331 | 3.9321 | 3.9217 |
-2 | - | 0.6 | - | - | - | - | - | - | - | - | 3.9380 | 3.9368 | 3.9130 |
- | - | 0.4 | 0.1 | - | - | - | - | - | - | - | 3.9331 | 3.9321 | 3.9217 |
- | - | - | 0.3 | - | - | - | - | - | - | - | 3.1784 | 3.1774 | 3.1647 |
- | - | - | 0.5 | - | - | - | - | - | - | - | 2.3019 | 2.3008 | 2.2825 |
- | - | - | 0.1 | 0.0 | - | - | - | - | - | - | 4.2479 | 4.2479 | 4.2479 |
- | - | - | - | 0.3 | - | - | - | - | - | - | 3.9331 | 3.9321 | 3.9217 |
- | - | - | - | 0.6 | - | - | - | - | - | - | 3.5766 | 3.5746 | 3.5555 |
- | - | - | - | 0.3 | 0.0 | - | - | - | - | - | 4.0232 | 4.0232 | 3.9422 |
- | - | - | - | - | 0.5 | - | - | - | - | - | 3.9331 | 3.9321 | 3.8047 |
- | - | - | - | - | 1.0 | - | - | - | - | - | 3.8436 | 3.8416 | 3.6662 |
- | - | - | - | - | 0.5 | 0.2 | - | - | - | - | 3.8979 | 3.8969 | 3.7675 |
- | - | - | - | - | - | 0.4 | - | - | - | - | 3.9331 | 3.9321 | 3.8047 |
- | - | - | - | - | - | 0.6 | - | - | - | - | 3.9540 | 3.9530 | 3.8266 |
- | - | - | - | - | - | 0.4 | 0.1 | - | - | - | 5.8775 | 5.8766 | 5.6973 |
- | - | - | - | - | - | - | 0.3 | - | - | - | 3.9331 | 3.9321 | 3.8047 |
- | - | - | - | - | - | - | 0.5 | - | - | - | 3.5737 | 3.5727 | 3.4533 |
- | - | - | - | - | - | - | 0.3 | 0.0 | - | - | 5.2763 | 5.2749 | 5.0492 |
- | - | - | - | - | - | - | - | 0.5 | - | - | 3.9331 | 3.9321 | 3.8047 |
- | - | - | - | - | - | - | - | 1.0 | - | - | 3.1172 | 3.1165 | 3.0330 |
- | - | - | - | - | - | - | - | 0.5 | 0.005 | - | 3.3073 | 3.3070 | 3.2240 |
- | - | - | - | - | - | - | - | - | 0.02 | - | 3.5631 | 3.5625 | 3.4621 |
- | - | - | - | - | - | - | - | - | 0.04 | - | 3.9331 | 3.9321 | 3.8047 |
- | - | - | - | - | - | - | - | - | 0.06 | - | 4.3394 | 4.3381 | 4.1788 |
- | - | - | - | - | - | - | - | - | 0.04 | 0.0 | 3.6943 | 3.6933 | 3.5161 |
- | - | - | - | - | - | - | - | - | - | 0.2 | 3.8118 | 3.8108 | 3.6568 |
- | - | - | - | - | - | - | - | - | - | 0.4 | 3.9331 | 3.9321 | 3.8047 |
- | - | - | - | - | - | - | - | - | - | 0.6 | 4.0575 | 4.0567 | 3.9526 |
Physical parameters | Re1/2xNux | ||||||||||||
δ1 | Rd | K | γ1 | M | δ | λ1 | β1 | λ | ϕ2 | S | Yamada-Ota Model | Xue-Model | Tiwari-Das Model |
0.2 | 04 | 0.4 | 0.1 | 0.3 | 0.5 | 0.4 | 0.3 | 0.5 | 0.04 | 0.4 | 3.5558 | 3.4796 | 1.8070 |
0.4 | - | - | - | - | - | - | - | - | - | - | 3.3573 | 3.2804 | 1.4694 |
0.6 | - | - | - | - | - | - | - | - | - | - | 3.1795 | 3.1024 | 1.2376 |
0.4 | 0.2 | - | - | - | - | - | - | - | - | - | 3.0678 | 3.0002 | 1.3549 |
- | 0.4 | - | - | - | - | - | - | - | - | - | 3.3573 | 3.2804 | 1.4694 |
- | 0.6 | - | - | - | - | - | - | - | - | - | 3.6307 | 3.5450 | 1.5672 |
- | 0.4 | 0.0 | - | - | - | - | - | - | - | - | 5.4398 | 5.3008 | 1.8099 |
- | - | 0.2 | - | - | - | - | - | - | - | - | 4.5319 | 4.4212 | 1.6671 |
- | - | 0.4 | - | - | - | - | - | - | - | - | 3.3573 | 3.2804 | 1.4694 |
- | - | 0.6 | - | - | - | - | - | - | - | - | 1.7182 | 1.6821 | 1.1640 |
- | - | 0.4 | 0.1 | - | - | - | - | - | - | - | 3.3573 | 3.2804 | 1.4694 |
- | - | - | 0.3 | - | - | - | - | - | - | - | 2.8416 | 2.7778 | 1.3849 |
- | - | - | 0.5 | - | - | - | - | - | - | - | 1.9747 | 1.9312 | 1.2451 |
- | - | - | 0.1 | 0.0 | - | - | - | - | - | - | 3.5143 | 3.4338 | 1.4973 |
- | - | - | - | 0.3 | - | - | - | - | - | - | 3.3573 | 3.2804 | 1.4694 |
- | - | - | - | 0.6 | - | - | - | - | - | - | 3.1609 | 3.0883 | 1.4347 |
- | - | - | - | 0.3 | 0.0 | - | - | - | - | - | 3.4099 | 3.3318 | 1.4704 |
- | - | - | - | - | 0.5 | - | - | - | - | - | 3.3573 | 3.2804 | 1.4587 |
- | - | - | - | - | 1.0 | - | - | - | - | - | 3.3036 | 3.2278 | 1.4465 |
- | - | - | - | - | 0.5 | 0.2 | - | - | - | - | 3.3233 | 3.2475 | 1.4544 |
- | - | - | - | - | - | 0.4 | - | - | - | - | 3.3573 | 3.2804 | 1.4587 |
- | - | - | - | - | - | 0.6 | - | - | - | - | 3.3762 | 3.2986 | 1.4612 |
- | - | - | - | - | - | 0.4 | 0.0 | - | - | - | 2.5806 | 2.5223 | 1.3223 |
- | - | - | - | - | - | - | 0.3 | - | - | - | 3.3573 | 3.2804 | 1.4587 |
- | - | - | - | - | - | - | 0.6 | - | - | - | 3.6019 | 3.5190 | 1.5034 |
- | - | - | - | - | - | - | 0.3 | 0.0 | - | - | 2.5647 | 2.4973 | 1.1972 |
- | - | - | - | - | - | - | - | 0.5 | - | - | 3.3573 | 3.2804 | 1.4587 |
- | - | - | - | - | - | - | - | 1.0 | - | - | 3.7602 | 3.6767 | 1.5657 |
- | - | - | - | - | - | - | - | 0.5 | .005 | - | 2.5001 | 2.4666 | 1.4006 |
- | - | - | - | - | - | - | - | - | 0.02 | - | 2.8923 | 2.8389 | 1.4266 |
- | - | - | - | - | - | - | - | - | 0.04 | - | 3.3573 | 3.2804 | 1.4587 |
- | - | - | - | - | - | - | - | - | 0.06 | - | 3.7799 | 3.6816 | 1.4876 |
- | - | - | - | - | - | - | - | - | 0.04 | 0.0 | 1.6248 | 1.5507 | 0.196851 |
- | - | - | - | - | - | - | - | - | - | 0.2 | 2.5003 | 2.4258 | 0.85367 |
- | - | - | - | - | - | - | - | - | - | 0.4 | 3.3573 | 3.2804 | 1.4587 |
- | - | - | - | - | - | - | - | - | - | 0.6 | 4.1876 | 4.1061 | 1.8678 |
ϕ2 | Zainal et al. [33] | Bachok et al. [34] | Present results |
0.0 | 1.687218 | 1.687200 | 1.687221 |
0.1 | 2.579342 | 2.579400 | 2.579317 |
0.2 | 3.590122 | 3.590100 | 3.590111 |