Research on discrete differential solution methods for derivatives of chaotic systems

Xinyu Pan; Xinyu Pan

doi:10.3934/math.20241621

AIMS Mathematics

2024, Volume 9, Issue 12: 33995-34012. doi: 10.3934/math.20241621

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Research article Special Issues

Research on discrete differential solution methods for derivatives of chaotic systems

Xinyu Pan ^,

School of Electronics & Information Engineering, Suzhou University of Science and Technology, Suzhou, Jiangsu, 215009, China

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.

Keywords:

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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Abstract

1. Introduction

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.

2. Materials and methods

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.

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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 ${T}_{w}$ and the ambient fluid temperature is ${T}_{\infty }.$ ${U}_{e}$ is free stream velocity, and ${H}_{e}$ 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]}):

$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y} = 0 ,$

(1)

$\frac{\partial {H}_{1}}{\partial x}+\frac{\partial {H}_{2}}{\partial y} = 0 ,$

(2)

$u\frac{\partial u}{\partial x}+v\frac{\partial v}{\partial y}-\frac{{\mu }_{f}}{4\pi p}\left({H}_{1}\frac{\partial {H}_{1}}{\partial x}+{H}_{2}\frac{\partial {H}_{1}}{\partial y}\right)\\ = {U}_{e}\frac{d{U}_{e}}{dx}-\frac{\mu }{4\pi p}{H}_{e}\frac{d{H}_{e}}{dx}+{\mathrm{\nu }}_{f}\frac{{\rho }_{f}}{{\rho }_{hnf}}\left(\frac{1}{{\left(1-{\phi }_{1}\right)}^{2.5}(1-{{\phi }_{2})}^{2.5}}+\frac{1}{{\beta }_{1}}\right)\frac{{\partial }^{2}u}{\partial {y}^{2}}-\frac{{\rho }_{f}\mathrm{k}}{{\rho }_{hnf}}u ,$

(3)

$u\frac{\partial {H}_{1}}{\partial x}+v\frac{\partial {H}_{1}}{\partial y}-{H}_{1}\frac{\partial u}{\partial x}-{H}_{2}\frac{\partial u}{\partial y} = {\mu }_{e}\frac{{\partial }^{2}{H}_{1}}{\partial {y}^{2}} ,$

(4)

$u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y} = {\alpha }_{hnf}\frac{{\partial }^{2}T}{\partial {y}^{2}}+Q\frac{{\left(\rho {C}_{p}\right)}_{f}}{{\left(\rho {C}_{p}\right)}_{hnf}}\left(T-{T}_{\mathrm{\infty }}\right)-\frac{{\left(\rho {C}_{p}\right)}_{f}}{{\left(\rho {C}_{p}\right)}_{hnf}}\frac{\partial {q}_{r}}{\partial y} .$

(5)

With relevant boundary conditions are as follows:

$u = {\beta }^{*}{\mu }_{f}\left(\frac{1}{{\left(1-{\phi }_{1}\right)}^{2.5}(1-{{\phi }_{2})}^{2.5}}+\frac{1}{{\beta }_{1}}\right)\frac{\partial u}{\partial y} , v = {V}_{w} ,\\ {H}_{1} = 0 , {H}_{2} = 0 , T = {T}_{w}+{\gamma }^{*}\frac{\partial T}{\partial y} , \;{\rm{at}}\; y\to 0 , \\ u\to {U}_{e} , {H}_{1}\to {H}_{e} , T\to {T}_{\mathrm{\infty }} , \;{\rm{as }}\; y\to \infty .$

(6)

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

$u = \frac{\partial \psi }{\partial y} , v = -\frac{\partial \psi }{\partial x} , {H}_{1} = \frac{\partial {\psi }_{1}}{\partial y} , {H}_{2} = -\frac{\partial {\psi }_{1}}{\partial x} , T = {T}_{\mathrm{\infty }}+\left({T}_{w}-{T}_{\mathrm{\infty }}\right)\theta \left(\eta \right) .$

(7)

${\psi }_{1} = {H}_{o}{e}^{x/2L}\sqrt{\frac{\upsilon }{{U}_{o}}}$ g( $\eta)$ and $\psi = \sqrt{2{u}_{o}\upsilon L}{e}^{\frac{x}{2L}}f\left(\eta \right)$ are the stream functions of the magnetic field and velocity. By using these values, the suitable transformations can be written as

$u = \frac{\partial \psi }{\partial y} = {u}_{o}{e}^{x/L}{f}^{\mathrm{\text{'}}}\left(\eta \right) , v = -\frac{\partial \psi }{\partial x} = -\sqrt{\frac{{u}_{o}\upsilon }{2L}}{e}^{x/2L}\left(f+\eta {f}^{\mathrm{\text{'}}}\right) , \\ {H}_{1} = \frac{\partial {\psi }_{1}}{\partial y} = \frac{{H}_{o}{e}^{x/2L}}{\sqrt{2L}}{g}^{\mathrm{\text{'}}}\left(\eta \right) , \\ {H}_{2} = -\frac{\partial {\psi }_{1}}{\partial x} = -\frac{{H}_{o}{e}^{x/2L}}{2L}\sqrt{\frac{\upsilon }{{U}_{o}}}\left(g+\eta {g}^{\mathrm{\text{'}}}\right) .$

(8)

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

$\frac{{\rho }_{f}}{{\rho }_{hnf}}\left(\frac{1}{{\left(1-{\phi }_{1}\right)}^{2.5}(1-{{\phi }_{2})}^{2.5}}+\frac{1}{{\beta }_{1}}\right){f}^{\mathrm{\text{'}}\mathrm{\text{'}}\mathrm{\text{'}}}\left(\eta \right)+{f}^{\mathrm{\text{'}}\mathrm{\text{'}}}f+2\left(1-{f}^{\mathrm{\text{'}}}\right)\\+{\gamma }_{1}\left[2\left({{g}^{\mathrm{\text{'}}}}^{2}\left(\eta \right)-1\right)-g{g}^{\mathrm{\text{'}}\mathrm{\text{'}}}\right]-M\frac{{\rho }_{f}}{{\rho }_{hnf}}\left({f}^{\mathrm{\text{'}}}\left(\eta \right)-1\right)+\delta \theta = 0 ,$

(9)

$\lambda {g}^{\mathrm{\text{'}}\mathrm{\text{'}}\mathrm{\text{'}}}\left(\eta \right)-g{f}^{\mathrm{\text{'}}\mathrm{\text{'}}}+{g}^{\mathrm{\text{'}}}\text{'}f = 0 ,$

(10)

$\frac{{K}_{hnf}}{{K}_{f}}\frac{{\left({\rho }_{cp}\right)}_{f}}{{\left({\rho }_{cp}\right)}_{hnf}}\frac{1}{Pr}\left(1+\frac{4}{3}Rd\right){\theta }^{\mathrm{\text{'}}\mathrm{\text{'}}}+f{\theta }^{\mathrm{\text{'}}}+\frac{{\left({\rho }_{cp}\right)}_{f}}{{\left({\rho }_{cp}\right)}_{hnf}}K\theta = 0 ,$

(11)

with boundary conditions

$f\left(0\right) = S, {f}^{\text{'}}\left(0\right) = {\lambda }_{1}\left(\frac{1}{{\left(1-{\phi }_{1}\right)}^{2.5}(1-{{\phi }_{2})}^{2.5}}+\frac{1}{{\beta }_{1}}\right)f\text{'}\text{'}\left(0\right), {f}^{\text{'}}\left(\infty \right) = 1,\\ g\left(0\right) = 0, {g}^{\text{'}}\left(0\right) = 0, {g}^{\text{'}}\left(\infty \right) = 1, \theta \left(0\right) = 1+{\delta }_{1}\frac{{K}_{hnf}}{{K}_{f}}(1+\frac{4}{3}Rd)\theta \text{'}\left(0\right), \theta \left(\infty \right) = 0 .$

(12)

Some physical quantities of interest are melting rate at stretching sheet or Nusselt number ${Nu}_{x}$ and skin friction ${C}_{f}$ . The skin friction drag is presented as ${C}_{f} = \frac{{\tau }_{w}}{{\rho }_{f}{{U}_{w}}^{2}}$ , the ratio of conductive and convective heat transfer rate is defined as ${Nu}_{x} = \frac{x{q}_{w}}{k({T}_{\infty }-{T}_{w})}$ . The surface shear stress and heat flux are presented as below:

${\tau }_{w} = {\mu }_{f}\left(\frac{1}{{\left(1-{\phi }_{1}\right)}^{2.5}(1-{{\phi }_{2})}^{2.5}}+\frac{1}{{\beta }_{1}}\right){\left(\frac{\partial u}{\partial y}\right)}_{y = 0}, {q}_{w} = -{k}_{hnf}\left(1+\frac{4}{3}\right){\left(\frac{\partial T}{\partial y}\right)}_{y = 0}.$

Using Eq (8), the above quantities become as

${Re}_{x}^{1/2}{C}_{f} = \left(\frac{1}{{\left(1-{\phi }_{1}\right)}^{2.5}(1-{{\phi }_{2})}^{2.5}}+\frac{1}{{\beta }_{1}}\right){F}^{\text{'}\text{'}}\left(0\right), {Re}_{x}^{1/2}{Nu}_{x} = -{k}_{hnf}\left(1+\frac{4}{3}\right){\theta }^{\text{'}}\left(0\right).$

The local Reynolds number is ${Re}_{x} = \frac{{U}_{w}x}{\nu }$ . Some expressions are presented as:

$\frac{{\mu }_{hnf}}{{\mu }_{f}} = \frac{1}{{\left(1-{\phi }_{1}\right)}^{2.5}(1-{{\phi }_{2})}^{2.5}} , \frac{{\rho }_{hnf}}{{\rho }_{f}} = \left(1-{\phi }_{1}\right)\left(1-{\phi }_{2}\right)+{\phi }_{1}\frac{{\rho }_{{s}_{1}}}{{\rho }_{f}}+{\phi }_{2}\frac{{\rho }_{{s}_{2}}}{{\rho }_{f}} ,$

$\frac{{\left(\rho {C}_{p}\right)}_{hnf}}{{\left(\rho {C}_{p}\right)}_{f}} = \left(1-{\phi }_{1}\right)\left(1-{\phi }_{2}\right)+{\phi }_{1}\frac{{\left(\rho {C}_{p}\right)}_{{s}_{1}}}{{\left(\rho {C}_{p}\right)}_{f}}+{\phi }_{2}\frac{{\left(\rho {C}_{p}\right)}_{{s}_{2}}}{{\left(\rho {C}_{p}\right)}_{f}}.$

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
${C}_{p}\left(J/kg \; K \right)$	4179.0	425.00	796.0
$\rho (kg/{m}^{3})$	997.10	2600.0	1600.0
$K(W/mK)$	0.6130	6600.0	3000.0

| Show Table

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2.1. Numerical procedure

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]):

$y\left(1\right) = f\left(\eta \right), y\left(2\right) = {f}^{\mathrm{\text{'}}}\left(\eta \right), y\left(3\right) = {f}^{\mathrm{\text{'}}\mathrm{\text{'}}}\left(\eta \right), y{y}_{1} = {f}^{\mathrm{\text{'}}\mathrm{\text{'}}\mathrm{\text{'}}}\left(\eta \right),$

(13)

$y{y}_{1} = [\mathrm{M}\left\{\mathrm{y}\left(2\right)-1\right\}-\frac{{\rho }_{f}\gamma }{{\rho }_{hnf}}\left\{2\left({y}^{2}\left(5\right)-1\right)-\mathrm{y}\left(4\right).\mathrm{y}\left(6\right)\right\}-2\left(1-y\left(2\right)-y\left(3\right).\\y\left(1\right)+\delta y\left(7\right)\right]\frac{{\rho }_{hnf}}{{\rho }_{f}}.\left[{\frac{1}{{\left(1-{\phi }_{1}\right)}^{2.5}(1-{{\phi }_{2})}^{2.5}}+\frac{1}{{\beta }_{1}})}^{-1}\right] ,$

(14)

$y\left(4\right) = g\left(\eta \right), y\left(5\right) = {g}^{\mathrm{\text{'}}}\left(\eta \right), y\left(6\right) = {g}^{\mathrm{\text{'}}\mathrm{\text{'}}}\left(\eta \right), y{y}_{2} = {g}^{\mathrm{\text{'}}\mathrm{\text{'}}\mathrm{\text{'}}}\left(\eta \right) ,$

(15)

$y{y}_{2} = \left[y\left(6\right).y\left(1\right)-y\left(4\right).y\left(3\right)\right]\frac{1}{\lambda } ,$

(16)

$y\left(7\right) = \theta \left(\eta \right), y\left(8\right) = {\theta }^{\mathrm{\text{'}}}\left(\eta \right), y{y}_{3} = {\theta }^{\mathrm{\text{'}}\mathrm{\text{'}}}\left(\eta \right) ,$

(17)

$y{y}_{3} = -[\mathrm{y}\left(1\right).\mathrm{y}\left(8\right)-\frac{{\left({\rho }_{cp}\right)}_{f}}{{\left({\rho }_{cp}\right)}_{hnf}}Ky\left(7\right)]\frac{{K}_{f}}{{K}_{hnf}}\frac{{\left({\rho }_{cp}\right)}_{hnf}}{{\left({\rho }_{cp}\right)}_{f}}Pr[{1+\frac{4}{3}Rd]}^{-1} .$

(18)

With boundary conditions are

$y0\left(1\right)-S;y0\left(2\right)-{\lambda }_{1}\left(\frac{1}{{\left(1-{\phi }_{1}\right)}^{2.5}(1-{{\phi }_{2})}^{2.5}}+\frac{1}{{\beta }_{1}}\right)y0\left(3\right);yinf\left(2\right)-1;y0\left(4\right);\\y0\left(5\right);yinf\left(5\right)-1;y0\left(7\right)-1-{\delta }_{1}\frac{{K}_{hnf}}{{K}_{f}}(1+\frac{4}{3}Rd)y0\left(8\right);yinf\left(7\right) .$

(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 ( ${\mathrm{R}}_{1}\left({{u}^{*}}_{1}, {{u}^{*}}_{2}, {{u}^{*}}_{3}\right), {\mathrm{R}}_{2}\left({{u}^{*}}_{1}, {{u}^{*}}_{2}, {{u}^{*}}_{3}\right), {\mathrm{R}}_{3}\left({{u}^{*}}_{1}, {{u}^{*}}_{2}, {{u}^{*}}_{3}\right))$ 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:

${\mathrm{R}}_{1}\left({{u}^{*}}_{1}, {{u}^{*}}_{2}, {{u}^{*}}_{3}\right) = \left|{\mathrm{y}}_{2}\left(\mathrm{\infty }\right)-\widehat{{\mathrm{y}}_{2}}\left(\mathrm{\infty }\right)\right| ,$

${\mathrm{R}}_{2}\left({{u}^{*}}_{1}, {{u}^{*}}_{2}, {{u}^{*}}_{3}\right) = \left|{\mathrm{y}}_{5}\left(\mathrm{\infty }\right)-\widehat{{\mathrm{y}}_{5}}\left(\mathrm{\infty }\right)\right| ,$

${\mathrm{R}}_{3}\left({{u}^{*}}_{1}, {{u}^{*}}_{2}, {{u}^{*}}_{3}\right) = \left|{\mathrm{y}}_{7}\left(\mathrm{\infty }\right)-\widehat{{\mathrm{y}}_{7}}\left(\mathrm{\infty }\right)\right| .$

Hence, $\widehat{{y}_{2}}\left(\infty \right), \widehat{{y}_{5}}\left(\infty \right)$ , and $\widehat{{y}_{7}}\left(\infty \right)$ 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]):

$\frac{{k}_{bf}}{{k}_{f}} = \frac{1+\frac{{k}_{f}}{{k}_{{s}_{1}}}\frac{L}{R}{\phi }_{1}^{0.2}+\left(1-\frac{{k}_{f}}{{k}_{{s}_{1}}}\right){\phi }_{1}\frac{L}{R}{\phi }_{1}^{0.2}+2{\phi }_{1}\left(\frac{{k}_{{s}_{1}}}{{k}_{{s}_{1}}-{k}_{f}}\right)ln\left(\frac{{k}_{{s}_{1}}+{k}_{f}}{{2k}_{{s}_{1}}}\right)}{1-{\phi }_{1}+2{\phi }_{1}\left(\frac{{k}_{f}}{{k}_{{s}_{1}}-{k}_{f}}\right)ln\left(\frac{{k}_{{s}_{1}}+{k}_{f}}{2{k}_{f}}\right)} ,$

$\frac{{k}_{hnf}}{{k}_{bf}} = \frac{1+\frac{{k}_{bf}}{{k}_{{s}_{2}}}\frac{L}{R}{\phi }_{2}^{0.2}+\left(1-\frac{{k}_{bf}}{{k}_{{s}_{2}}}\right){\phi }_{2}\frac{L}{R}{\phi }_{2}^{0.2}+2{\phi }_{2}\left(\frac{{k}_{{s}_{2}}}{{k}_{{s}_{2}}-{k}_{bf}}\right)ln\left(\frac{{k}_{{s}_{2}}+{k}_{bf}}{2{k}_{{s}_{2}}}\right)}{1-{\phi }_{2}+2{\phi }_{2}\left(\frac{{k}_{bf}}{{k}_{{s}_{2}}-{k}_{bf}}\right)ln\left(\frac{{k}_{{s}_{2}}+{k}_{bf}}{2{k}_{bf}}\right)} .$

Xue model expression is defined as below:

$\frac{{k}_{bf}}{{k}_{f}} = \frac{1-{\phi }_{1}+2{\phi }_{1}\left(\frac{{k}_{{s}_{1}}}{{k}_{{s}_{1}}-{k}_{f}}\right)ln\left(\frac{{k}_{{s}_{1}}+{k}_{f}}{2{k}_{f}}\right)}{1-{\phi }_{1}+2{\phi }_{1}\left(\frac{{k}_{f}}{{k}_{{s}_{1}}-{k}_{f}}\right)ln\left(\frac{{k}_{{s}_{1}}+{k}_{f}}{2{k}_{f}}\right)} ,$

$\frac{{k}_{hnf}}{{k}_{bf}} = \frac{1-{\phi }_{2}+2{\phi }_{2}\left(\frac{{k}_{{s}_{2}}}{{k}_{{s}_{2}}-{k}_{bf}}\right)ln\left(\frac{{k}_{{s}_{2}}+{k}_{bf}}{2{k}_{bf}}\right)}{1-{\phi }_{2}+2{\phi }_{2}\left(\frac{{k}_{bf}}{{k}_{{s}_{2}}-{k}_{bf}}\right)ln\left(\frac{{k}_{{s}_{2}}+{k}_{bf}}{2{k}_{bf}}\right)} .$

Tiwari-Das model expression is defined as below:

$\frac{{k}_{bf}}{{k}_{f}} = \frac{\left(n-1\right){k}_{f}-\left({k}_{f}-{k}_{{s}_{1}}\right){\phi }_{1}\left(n-1\right)+{k}_{{s}_{1}}}{\left({k}_{f}-{k}_{{s}_{1}}\right){\phi }_{1}+\left(n-1\right){k}_{f}+{k}_{{s}_{1}}} ,$

$\frac{{k}_{hnf}}{{k}_{bf}} = \frac{\left(n-1\right){k}_{bf}-\left({k}_{bf}-{k}_{{s}_{2}}\right){\phi }_{2}\left(n-1\right)+{k}_{{s}_{2}}}{\left({k}_{bf}-{k}_{{s}_{2}}\right){\phi }_{2}+\left(n-1\right){k}_{bf}+{k}_{{s}_{2}}} .$

3. Results and discussion

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, ${\beta }_{1}$ Casson fluid parameter, ${\gamma }_{1}$ magnetic parameter, $\delta$ bouncy force parameter, $\lambda$ reciprocal magnetic Prandtl number, $Pr$ Prandtl number, $K$ heat generation, $Rd$ radiation parameter, $S$ suction parameter, ${\lambda }_{1}$ velocity slip parameter, and ${\delta }_{1}$ thermal slip parameter effects on the velocity profile, induced magnetic profile and temperature profile are revealed through Figures 2–. 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 . The variation of the velocity profile and ${\beta }_{1}$ is shown in . 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 ${\gamma }_{1}$ on the velocity function is presented in . The velocity profile curves decrease as the values of ${\gamma }_{1}$ rise. As the electromagnetic forces enhanced due to increment of magnetic parameter which declined the velocity function.