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Research on discrete differential solution methods for derivatives of chaotic systems

  • Received: 15 July 2024 Revised: 21 November 2024 Accepted: 22 November 2024 Published: 02 December 2024
  • MSC : 39A12, 39B12, 65P20

  • The pivotal differential parameters inherent in chaotic systems hold paramount significance across diverse disciplines. This study delves into the distinctive features of discrete differential parameters within three typical chaotic systems: the logistic map, the henon map, and the tent map. A pivotal discovery emerges: both the mean value of the first-order continuous and discrete derivatives in the logistic map coincide, mirroring a similar behavior observed in the henon map. Leveraging the insights gained from the first derivative formulations, we introduce the discrete n-order derivative formulas for both logistic and henon maps. This revelation underscores a discernible mathematical correlation linking the mean value of the derivative, the respective chaotic parameters, and the mean of the chaotic sequence. However, due to the discontinuous points in the tent map, its continuous differential parameter cannot characterize its derivative properties, but its discrete differential has a clear functional relationship with the parameter μ. This paper proposes the use of discrete differential derivatives as an alternative to traditional derivatives, and demonstrates that the mean value of discrete derivatives has a clear mathematical relationship with chaotic map parameters in a statistical sense, providing a new direction for subsequent in-depth research and applications.

    Citation: Xinyu Pan. Research on discrete differential solution methods for derivatives of chaotic systems[J]. AIMS Mathematics, 2024, 9(12): 33995-34012. doi: 10.3934/math.20241621

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  • The pivotal differential parameters inherent in chaotic systems hold paramount significance across diverse disciplines. This study delves into the distinctive features of discrete differential parameters within three typical chaotic systems: the logistic map, the henon map, and the tent map. A pivotal discovery emerges: both the mean value of the first-order continuous and discrete derivatives in the logistic map coincide, mirroring a similar behavior observed in the henon map. Leveraging the insights gained from the first derivative formulations, we introduce the discrete n-order derivative formulas for both logistic and henon maps. This revelation underscores a discernible mathematical correlation linking the mean value of the derivative, the respective chaotic parameters, and the mean of the chaotic sequence. However, due to the discontinuous points in the tent map, its continuous differential parameter cannot characterize its derivative properties, but its discrete differential has a clear functional relationship with the parameter μ. This paper proposes the use of discrete differential derivatives as an alternative to traditional derivatives, and demonstrates that the mean value of discrete derivatives has a clear mathematical relationship with chaotic map parameters in a statistical sense, providing a new direction for subsequent in-depth research and applications.



    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).

    Figure 1.  The flow pattern of Casson hybrid nanofluid.

    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]):

    ux+vy=0, (1)
    H1x+H2y=0, (2)
    uux+vvyμf4πp(H1H1x+H2H1y)=UedUedxμ4πpHedHedx+νfρfρhnf(1(1ϕ1)2.5(1ϕ2)2.5+1β1)2uy2ρfkρhnfu, (3)
    uH1x+vH1yH1uxH2uy=μe2H1y2, (4)
    uTx+vTy=αhnf2Ty2+Q(ρCp)f(ρCp)hnf(TT)(ρCp)f(ρCp)hnfqry. (5)

    With relevant boundary conditions are as follows:

    u=βμf(1(1ϕ1)2.5(1ϕ2)2.5+1β1)uy,v=Vw,H1=0,H2=0,T=Tw+γTy,aty0,uUe,H1He,TT,asy. (6)

    Introducing the stream functions are (see [29,30,31,32])

    u=ψy,v=ψx,H1=ψ1y,H2=ψ1x,T=T+(TwT)θ(η). (7)

    ψ1=Hoex/2LυUog(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=ψy=uoex/Lf'(η),v=ψx=uoυ2Lex/2L(f+ηf'),H1=ψ1y=Hoex/2L2Lg'(η),H2=ψ1x=Hoex/2L2LυUo(g+ηg'). (8)

    Using the above suitable transformation, the partial differential equations are converted into ordinary differential equations as below:

    ρfρhnf(1(1ϕ1)2.5(1ϕ2)2.5+1β1)f'''(η)+f''f+2(1f')+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(1ϕ1)2.5(1ϕ2)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(TTw). The surface shear stress and heat flux are presented as below:

    τw=μf(1(1ϕ1)2.5(1ϕ2)2.5+1β1)(uy)y=0,qw=khnf(1+43)(Ty)y=0.

    Using Eq (8), the above quantities become as

    Re1/2xCf=(1(1ϕ1)2.5(1ϕ2)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(1ϕ1)2.5(1ϕ2)2.5,ρhnfρf=(1ϕ1)(1ϕ2)+ϕ1ρs1ρf+ϕ2ρs2ρf,
    (ρCp)hnf(ρCp)f=(1ϕ1)(1ϕ2)+ϕ1(ρCp)s1(ρCp)f+ϕ2(ρCp)s2(ρCp)f.

    The thermophysical characteristics of base fluid and nanoparticles in Table 1(see [21]).

    Table 1.  Thermophysical characteristics of base fluid and nanoparticles.
    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

     | Show Table
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    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(1y(2)y(3).y(1)+δy(7)]ρhnfρf.[1(1ϕ1)2.5(1ϕ2)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(1ϕ1)2.5(1ϕ2)2.5+1β1)y0(3);yinf(2)1;y0(4);y0(5);yinf(5)1;y0(7)1δ1KhnfKf(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(u1,u2,u3),R2(u1,u2,u3),R3(u1,u2,u3)) are not more than tolerance error i.e., 106. 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(u1,u2,u3)=|y2()^y2()|,
    R2(u1,u2,u3)=|y5()^y5()|,
    R3(u1,u2,u3)=|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+(1kfks1)ϕ1LRϕ0.21+2ϕ1(ks1ks1kf)ln(ks1+kf2ks1)1ϕ1+2ϕ1(kfks1kf)ln(ks1+kf2kf),
    khnfkbf=1+kbfks2LRϕ0.22+(1kbfks2)ϕ2LRϕ0.22+2ϕ2(ks2ks2kbf)ln(ks2+kbf2ks2)1ϕ2+2ϕ2(kbfks2kbf)ln(ks2+kbf2kbf).

    Xue model expression is defined as below:

    kbfkf=1ϕ1+2ϕ1(ks1ks1kf)ln(ks1+kf2kf)1ϕ1+2ϕ1(kfks1kf)ln(ks1+kf2kf),
    khnfkbf=1ϕ2+2ϕ2(ks2ks2kbf)ln(ks2+kbf2kbf)1ϕ2+2ϕ2(kbfks2kbf)ln(ks2+kbf2kbf).

    Tiwari-Das model expression is defined as below:

    kbfkf=(n1)kf(kfks1)ϕ1(n1)+ks1(kfks1)ϕ1+(n1)kf+ks1,
    khnfkbf=(n1)kbf(kbfks2)ϕ2(n1)+ks2(kbfks2)ϕ2+(n1)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 213. 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.

    Figure 2.  Effect of β1 on the F'(η).
    Figure 3.  Effect of γ1 on the F'(η).
    Figure 4.  Effect of λ1 on the F'(η).
    Figure 5.  Effect of M on the F'(η).
    Figure 6.  Effect of ϕ2 on the F'(η).
    Figure 7.  Effect of ϕ2 on the g'(η).
    Figure 8.  Effect of λ1 on the g'(η).
    Figure 9.  Effect of δ on the θ(η).
    Figure 10.  Effect of δ1 on the θ(η).
    Figure 11.  Effect of K on the θ(η).
    Figure 12.  Effect of Rd on the θ(η).
    Figure 13.  Effect of ϕ2 on the θ(η).

    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.

    Table 2.  Comparative numerical analysis of the Yamada-Ota model, Tiwari-Das model, and Xue model for skin friction with physical parameters.
    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

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    Table 3.  Comparative numerical analysis of Yamada-Ota model, Tiwari-Das model, and Xue model for Nusselt number with physical parameters.
    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

     | Show Table
    DownLoad: CSV

    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.

    Table 4.  Comparison with Zainal et al. [33] and Bachok et al. [34] for different values ϕ2 on the skin friction when the rest of values are zero such as M=γ1=δ=λ1=K=Rd=S=λ=δ1=ϕ1=0 and β1.
    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

     | Show Table
    DownLoad: CSV

    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.



    [1] S. Zhou, X. Wang, Simple estimation method for the second-largest Lyapunov exponent of chaotic differential equations, Chaos Soliton. Fract., 139 (2020), 109981. https://doi.org/10.1016/j.chaos.2020.109981 doi: 10.1016/j.chaos.2020.109981
    [2] Z. Z. Ma, Q. C. Yang, R. P. Zhou, Lyapunov exponent algorithm based on perturbation theory for discontinuous systems, Acta Phys. Sin., 70 (2021), 240501. https://doi.org/10.7498/aps.70.20210492 doi: 10.7498/aps.70.20210492
    [3] F. Nazarimehr, S. Panahi, M. Jalili, M. Perc, S. Jafari, B. Fercec, Multivariable coupling and synchronization in complex networks, Appl. Math. Comput., 372 (2020), 124996. https://doi.org/10.1016/j.amc.2019.124996 doi: 10.1016/j.amc.2019.124996
    [4] N. Zandi-Mehran, S. Jafari, S. M. R. H. Golpayegani, Signal separation in an aggregation of chaotic signals, Chaos Soliton. Fract., 138 (2020), 109851. https://doi.org/10.1016/j.chaos.2020.109851 doi: 10.1016/j.chaos.2020.109851
    [5] S. J. Cang, L. Wang, Y. P. Zhang, Z. Wang, Z. Chen, Bifurcation and chaos in a smooth 3D dynamical system extended from Nosé-Hoover oscillator, Chaos Soliton. Fract., 158 (2022), 112016. https://doi.org/10.1016/j.chaos.2022.112016 doi: 10.1016/j.chaos.2022.112016
    [6] V. V. Klinshov, V. A. Kovalchuk, I. Franović, M. Perc, M. Svetec, Rate chaos and memory lifetime in spiking neural networks, Chaos Soliton. Fract., 158 (2022), 112011. https://doi.org/10.1016/j.chaos.2022.112011 doi: 10.1016/j.chaos.2022.112011
    [7] K. D. S. Andrade, M. R. Jeffrey, R. M. Martins, M. A. Teixeira, Homoclinic boundary-saddle bifurcations in planar nonsmooth vector fields, Int. J. Bifurcat. Chaos, 32 (2022), 22300099. https://doi.org/10.1142/S0218127422300099 doi: 10.1142/S0218127422300099
    [8] N. Yadav, S. Shah, Topological weak specification and distributional chaos on noncompact spaces. Int. J. Bifurcat. Chaos, 32 (2022), 2250048. https://doi.org/10.1142/S0218127422500481 doi: 10.1142/S0218127422500481
    [9] X. Y. Pan, H. M. Zhao, Research on the entropy of logistic chaos, Acta Phys. Sin., 61 (2012), 200504. https://doi.org/10.7498/aps.61.200504 doi: 10.7498/aps.61.200504
    [10] H. P. Wen, S. M. Yu, J. H. Lü, Encryption algorithm based on Hadoop and non-degenerate high-dimensional discrete hyperchaotic system, Acta Phys. Sin., 66 (2017), 230503. https://doi.org/10.7498/aps.66.230503 doi: 10.7498/aps.66.230503
    [11] X. Y. Wan, J. M. Zhang, A novel image authentication and recovery algorithm based on dither and chaos, Acta Phys. Sin., 63 (2014), 210701. https://doi.org/10.7498/aps.63.210701 doi: 10.7498/aps.63.210701
    [12] B. Yang, X. Liao, Some properties of the Logistic map over the finite field and its application, Signal process., 153 (2018), 231–242. https://doi.org/10.1016/j.sigpro.2018.07.011 doi: 10.1016/j.sigpro.2018.07.011
    [13] M. Lazaros, V. Christos, J. Sajad, J. M. Munoz-Pacheco, J. Kengne, K. Rajagopal, et al., Modification of the logistic map using fuzzy numbers with application to pseudorandom number generation and image encryption, Entropy, 22 (2020), 474. https://doi.org/10.3390/e22040474 doi: 10.3390/e22040474
    [14] M. Wang, X. Wang, T. Zhao, C. Zhang, Z. Xia, N. Yao, Spatiotemporal chaos in improved cross coupled map lattice and its application in a bit-level image encryption scheme, Inform. Sciences, 554 (2021), 1–24. https://doi.org/10.1016/j.ins.2020.07.051 doi: 10.1016/j.ins.2020.07.051
    [15] X. Y. Wang, S. Gao, X. L. Ye, S. Zhou, M. X. Wang, A new image encryption algorithm with cantor diagonal scrambling based on the PUMCML system, Int. J. Bifurcat. Chaos, 31 (2021), 2150003. https://doi.org/10.1142/S0218127421500036 doi: 10.1142/S0218127421500036
    [16] Z. P. Zhao, S. Zhou, X. Y. Wang, A new chaotic signal based on deep learning and its application in image encryption, Acta Phys. Sin., 70 (2021), 230502. https://doi.org/10.7498/aps.70.20210561 doi: 10.7498/aps.70.20210561
    [17] B. X. Mao, Two methods contrast of sliding mode synchronization of fractional-order multy-chaotic systems, Acta Electronica Sin., 48 (2020), 2215–2219. https://doi.org/10.3969/j.issn.0372-2112.2020.11.017 doi: 10.3969/j.issn.0372-2112.2020.11.017
    [18] B. X. Mao, D. X. Wang. Self-adaptive sliding mode synchronization of uncertain fractional-order high-dimension chaotic systems, Acta Electronica Sin., 49 (2021), 775–780. https://doi.org/10.12263/DZXB.20200316 doi: 10.12263/DZXB.20200316
    [19] Z. C. Zhu, Q. X. Zhu, Adaptive neural prescribed performance control for non-triangular structural stochastic highly nonlinear systems under hybrid attacks, IEEE T. Automat. Sci. Eng., 2024. https://doi.org/10.1109/TASE.2024.3447045 doi: 10.1109/TASE.2024.3447045
    [20] Q. X. Zhu, Event-triggered sampling problem for exponential stability of stochastic nonlinear delay systems driven by Lexvy processes, IEEE T. Automat. Control, 2024. https://doi.org/10.1109/TAC.2024.3448128 doi: 10.1109/TAC.2024.3448128
    [21] Y. Xue, J. Han, Z. Tu, X. Y. Chen, Stability analysis and design of cooperative control for linear delta operator system, AIMS Math., 8 (2023), 12671–12693. https://doi.org/10.3934/math.2023637 doi: 10.3934/math.2023637
    [22] H. Bi, G. Qi, J. Hu, P. Faradja, G. Chen, Hidden and transient chaotic attractors in the attitude system of quadrotor unmanned aerial vehicle, Chaos Soliton. Fract., 138 (2020), 109815. https://doi.org/10.1016/j.chaos.2020.109815 doi: 10.1016/j.chaos.2020.109815
    [23] L. X. Fu, S. B. He, H. H. Wang, K. H. Sun, Simulink modeling and dynamic characteristics of discrete memristor chaotic system, Acta Phys. Sin., 71 (2022), 030501. https://doi.org/10.7498/aps.71.20211549 doi: 10.7498/aps.71.20211549
    [24] J. Y. Ruan, K. H. Sun, J. Mou. Memristor-based Lorenz hyper-chaotic system and its circuit implementation, Acta Phys. Sin., 65 (2016), 190502. https://doi.org/10.7498/aps.65.190502 doi: 10.7498/aps.65.190502
    [25] J. V. N. Tegnitsap, H. B. Fotsin, Multistability, transient chaos and hyperchaos, synchronization, and chimera states in wireless magnetically coupled VDPCL oscillators, Chaos Soliton. Fract., 158 (2022), 112056. https://doi.org/10.1016/j.chaos.2022.112056 doi: 10.1016/j.chaos.2022.112056
    [26] H. Xiao, Z. Li, H. Lin, Y. Zhao, A sual rumor spreading model with consideration of fans versus ordinary people, Mathematics, 11 (2023), 2958. https://doi.org/10.3390/math11132958 doi: 10.3390/math11132958
    [27] Q. Yang, X. Wang, X. Cheng, B. Du, Y. Zhao, Positive periodic solution for neutral-type integral differential equation arising in epidemic model, Mathematics, 11 (2023), 2701. https://doi.org/10.3390/math11122701 doi: 10.3390/math11122701
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    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
    61. Ri Zhang, Mostafa Zaydan, Mansoor Alshehri, C.S.K. Raju, Abderrahim Wakif, Nehad Ali Shah, Further insights into mixed convective boundary layer flows of internally heated Jeffery nanofluids: Stefan's blowing case study with convective heating and thermal radiation impressions, 2024, 55, 2214157X, 104121, 10.1016/j.csite.2024.104121
    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
    71. Mohammed Alrehili, Stagnation flow of a nanofluid with rough surface and energy dissipation: A tangent hyperbolic model with practical engineering uses, 2025, 26, 25901230, 104825, 10.1016/j.rineng.2025.104825
    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
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