#### Abstract

In this numerical investigation, three-dimensional tangent hyperbolic nanofluid flow past a stretching surface has been analyzed under the Smoluchowski thermal and Maxwell velocity slip conditions at the boundary. For the modification of the flow fields, the magnetic effect has also been incorporated. Nonlinear thermal radiation and Joule heating have also been included to improve heat transmission. The modelled PDEs have been turned into nonlinear ODEs after the application of the appropriate transformation. MATLAB bvp4c algorithm is utilized to achieve the solution of the problem. The code’s legitimacy is accomplished by repeating previously published results in the literature. Efforts are made to discuss the effects of the important parameters on velocities, energy profile, and concentration distribution in detail via graphs and tables. Based on our simulations, an enhancement in the slip parameter leads both velocities to decline. The ratio parameter causes a decline in velocity in the horizontal direction whereas it increases in the vertical direction. The heat transfer rate is boosted as the temperature ratio parameter is increased.

#### 1. Introduction

Magnetohydrodynamic gained the consideration of researchers because of natural phenomena such as astrophysics and geophysics to various significant engineering applications, that is, liquid metal, plasma confinement, and electromagnetic casting. Magnetohydrodynamic flow plays a vital role in the medical fields such as reducing bleeding in various injuries, magnetic resonance treatment, and treatment of cancer tumors. Well-known applications of magnetohydrodynamic flows involve combustion modeling, cooling of the nuclear reactor, plasma, and geophysics studies [1–4]. The famous application of magnetohydrodynamic flows is the use of the magnetic field in semiconductor industries. Laouer et al. [5] reported the analysis of the stability of MHD fluid flow across a flat plate. They determined that the critical Reynolds number is substantially influenced by the geometry of the velocity profile. Dynamics of multiple slip boundaries effect on MHD Casson-Williamson double-diffusive nanofluid flow past an inclined magnetic stretching sheet was ascertained by Humane et al. [6]. Adem [7] reported the analytical treatment of the EMHD non-Newtonian flow of the fluid past a stretching surface in a porous medium. Their vital remark was that the implementation of the magnetic field is counterproductive in the enhancement of the concentration and the velocity, whereas in the presence of moderate values of the magnetic field effect, both chemical reaction and porosity parameter results in the temperature and concentration distribution decline in the whole flow domain. Muhammad Atif et al. [8] ascertain the EMHD effects on micropolar nanofluid. Irfan et al. [9] discussed the numerical solution of the nanofluid flow of magnetohydrodynamic free stream over a radiating stretching sheet. It was noted that the rate of heat transfer is an increasing function of the Prandtl number. Darcy-Forchheimer MHD Jeffery fluid with nanoparticles past a permeable cone was studied by Gupta et al. [10]. One of the main observations was that the velocity is hiked as the porosity parameter is boosted. Patil et al. [11] reported the analysis on double-diffusive time-dependent MHD Prandtl nanofluid flow due to linear stretching sheet with convective boundary conditions. Gopal et al. [12] ascertained the EMHD nanofluid with higher-order chemical reaction and viscous dissipation effect. Khan and Alzahrani [13] studied the entropy-optimized dissipative flow of Carreau–Yasuda fluid with radiative heat flux and chemical reaction and concluded that the concentration reduces for both reaction parameters and Lewis number. Farooq et al. [14] investigated the nonsimilar analysis for Darcy–Forchheimer–Brinkman model of Casson fluid in a porous media and found that increasing numerical values of Prandtl number enhances the rate of heat transfer. Shafiq et al. [15] studied the designing of the artificial neural network of nanoparticle diameter and solid-fluid interfacial layer on single-walled carbon nanotubes/ethylene glycol nanofluid flow on thin slendering needles.

For non-Newtonian fluids, there are many models in the literature which describe the different rheological fluid properties. The tangent hyperbolic is one of the non-Newtonian four constant fluid models which can describe the shear-thinning behaviour; that is, its viscosity drops by an enhancement in the shear rate. Few of the examples may include blood, paints, ketchup, lava, and whipped cream. The advantages of this model over the other models are due to its simplicity, computational ease, and physical robustness. Furthermore, the kinetic theory of liquids is used to derive its constitutive relation rather than the empirical relation. Laboratory experiments show that shear-thinning behaviour can be described very precisely by this model, and the description of the blood is a very good example of this model. Recently, tangent hyperbola nanofluid models constitute a recent development in heat and mass transfer fluid. Oyelakin et al. [16] discussed the numerical solution of the 3-D magnetohydrodynamic nanofluid flow of a tangent hyperbolic by using the Buongiorno model. They also considered the velocity slip effect. One of the key observations was the nanofluid velocity is reduced with the increment of the Weissenberg number. Gharami et al. [17] used bvp4c to examine the numerical solution of the magnetohydrodynamic nanofluid flow of tangent hyperbola over a cylinder with chemical reaction effects. They noted that the velocity of the nanofluid is an increasing function of the thermophoresis, the Grashof number, the Brownian parameter, and the heat source while an opposite behaviour was noticed for the Wissenberg number, heat source, and magnetic parameters. Shafiq et al. [18] presented numerical results of bioconvection tangent hyperbolic nanofluid flow toward the exponentially stretching surface. They also examined the impacts of various physical parameters such as bioconvection Rayleigh number, bioconvection Pecelt parameter, buoyancy force parameter, and bioconvection Lewis parameter. Their concluding remarks show that for increasing the values of bioconvection Lewis and bioconvection Pecelt numbers, the stretched surface has unique results. Ramzan et al. [19] reported 3-D bioconvection tangent hyperbolic nanofluid with the impacts of iron slip and Hall. They performed the simulations by using non-Fourier heat flux model. Their reported results show that fluid velocity is increased with the Hall and iron slip effect. Ali Abbas et al. [20] presented the 3-D peristaltic flow of a tangent hyperbolic nanofluid in a channel. They showed that the velocity of the nanofluid is minimum near the wall, while an opposite trend is noted in the center of the channel. Atif et al. [21–23] studied the radiative tangent hyperbolic nanofluid flow past a stretching sheet, a wedge, and a paraboloid surface of revolution. Khan et al. [24] ascertained the double stratification effect on tangent hyperbolic nanofluid in a porous medium. Their study reveals that the undesirable skin friction is declined as the buoyancy ratio parameter is hiked. By using the quasilinearization method, Srinivas Reddy et al. [25] presented the entropy generation analysis of tangent hyperbolic fluid in quadratic Boussinesq approximation. This study shows that the entropy generation is increased as the nonlinear convection and pressure gradient are increased.

Nanofluids are important because of their significant implementation in heat transfer such as microelectronics, pharmaceutical processes, fuel cell, and hybrid-powered engines. By considering the convective surface, Makinde et al. [26] reported the stagnation point flow of chemically reacting nanofluid. One of the key observations was that the surface temperature is declined as the Richardson number is enhanced whereas it increases as the heat generation parameter is hiked. Thangavelu et al. [27] reported the magnetohydrodynamic (MHD) convection of water-Ag nanofluid flow in an inclined enclosure. They also considered the Rayleigh number, internal heat generation, and inclination angle effects. Their final remarks show that the rate of heat transfer has a direct relation to Hartmann number and inclination angle. Bhatti and Abdelsalam [28] reported the bio-inspired peristaltic propulsion of hybrid nanofluid flow with tantalum (Ta) and gold (Au) nanoparticles under magnetic effects. Aldabesh et al. [29] studied the unsteady transient slip flow of Williamson nanofluid containing gyrotactic microorganism and activation energy. Gul et al. [30] discussed analytical and numerical solutions of the MHD hybrid nanofluid within the conical gap. They noted that the boundary layer is enhanced as both cone and disk are spinning in the same direction, whereas an opposite trend is noted when they are in opposite directions. Furthermore, research related to nanofluids can also be examined [31–36].

After studying the literature, it is noted that no efforts were made for studying 3-D tangent hyperbolic nanofluid flow past a stretching surface. In this numerical study, three aspects have been focused on, first, to address the effect of Maxwell velocity and Smoluchowski temperature jump slip conditions on the tangent hyperbolic fluid, second, to examine the heat transfer in a tangent hyperbolic nanofluid, and third, to examine the impact of nonlinear thermal radiation and magnetic effect. The solution to the arising system of ODEs is achieved with the help of MATLAB built-in function bvp4c. The reliability of the code is done by reproducing the already published results in the literature. The influence of important factors on velocities, temperature, and concentration is visually depicted and described.

#### 2. Mathematical Description and Flow Field Analysis

##### 2.1. Physical Modeling

For this analysis, three-dimensional shear-thinning fluid in the presence of nanoparticles with Maxwell velocity slip and Smoluchowski thermal conditions at the boundary conditions has been considered. It is assumed that the temperature differences are large; therefore, nonlinear thermal radiation is implemented. The fluid flow is assumed in the region , and the surface coincides with the plane . The stretching velocities are assumed as and along and axes. Maxwell velocity slip [37] is acting along and axes on the stretching surface and is given by and . Smoluchowski temperature slip length [38] has also been considered.

##### 2.2. Mathematical Modeling

Under the boundary layer approximation, the governing equations [16–24] are as follows:

##### 2.3. Boundary Conditions

The relevant boundary conditions [37–39] arewhere is the specific heat, represents the electrical conductivity, is the density of nanofluid the thermal conductivity of the tangent hyperbolic nanofluid, is the power-law index, and represent thermophoresis diffusion and the Brownian coefficients, respectively. represents a molecular mean free path, , are momentum and temperature accommodation coefficient. The Rosseland approximation [8, 9] is defined as

In the above modelled problem, we have considered the nonlinear thermal radiation; therefore, can be represented in the form . Therefore, takes the form

##### 2.4. Similarity Transformations

For the conversion of the dimensional equations, the following set of similarity variables [40] is introduced.

##### 2.5. Nondimensional Equations

After the application of similarity transformation, the equations (1)–(5) are transformed into the following form:

##### 2.6. Nondimensional BCs

The transformed boundary conditions are

##### 2.7. Nondimensional Parameters

In the above equations, dimensionless parameter are the Weissenberg number in direction, the Eckert number in direction, the Schmidt number, the Weissenberg number in direction, the magnetic number, the Eckert number in direction, the thermophoresis parameter, the thermal radiation parameter, the temperature ratio parameter, the Eckert number in direction, the stretching rate parameter, denotes the Prandtl number, the velocity slip parameter, the temperature jump, and is the Brownian motion parameter.

#### 3. Quantities of Interest

In this numerical study, we are interested in the skin friction coefficients and Nusselt number . The skin friction coefficients for the above modelled problem arewhere and are given by

In nondimensional form,

The heat transfer rate coefficients are given by

In nondimensional form,

#### 4. Numerical Procedure

For the solution of the modelled problem represented in equations (10)–(13) along with BCs (12), an effective numerical approach bvp4c has been utilized. A finite domain has been considered for the solutions of these equations. The results are calculated for this finite domain and for such that the computational findings are unchanged.

Introducing the new variables, , , , , , , , , and . Equations (10)–(13) have been converted into the following system of ten first-order ODEs:with boundary conditions

##### 4.1. Code Validity

For the authenticity of the MATLAB code, the results of the skin friction coefficients as published by Wang [41] and Kumar et al. [42] have been reproduced. The results obtained by our simulations have excellent agreement with them as shown in Table 1.

#### 5. Results and Discussion

To analyze the modelled problem, the influence of the important parameters on surface drag coefficients , , and Nusselt number has been computed and discussed in detail. The results of Table 2 show the behaviour of surface drag coefficients along horizontal and vertical directions due to the influence of the important parameters. Both skin frictions are enhanced as the stretching ratio parameter , power-law index , and magnetic parameter are upsurged. However, both are decreased as Weissenberg numbers , and velocity slip parameter are hiked. From Table 3, it is noticeable that the Nusselt number is boosted as each of the power law index , thermal radiation parameter , temperature ratio parameter is gradually boosted, whereas it is depressed as temperature slip parameter is upsurged.

##### 5.1. Discussion on Graphical Results

Graphical results of velocity, temperature, and concentration for the variation of the governing parameters are presented in this section. All the parameters are assigned fixed values, and the fluctuation parameter is represented in the respective figure. This fixed value is as

For the study of the influence of emerging parameters on dimensionless velocities, concentration, and temperature fields, Figures 1–6 have been drawn. Figures 1(a)–1(d) are exhibited to analyze the effect of on dimensionless velocity profiles, dimensionless temperature distribution, and concentration distribution. In the presence of the electric field, the magnetic effect produces a Lorentz force which is a resistive force due to which the velocities are depressed as presented in Figures 1(a) and 1(b). Due to the slowness of the fluid, the dimensionless concentration and energy distributions are increased as presented in Figures 1(c) and 1(d). An increase in the value of causes an enhancement in the viscosity of the fluid due to which and are the diminished as reported in Figures 2(a), 2(b). An enhancement in the value of *n* means an increment in the viscosity of the fluid. Due to this reason, both velocities of the fluid are declined. Figure 3(a) demonstrates the dynamics of velocities against the velocity slip parameter . Both velocity profiles are declined because the fractional resistance between the fluid particles and the surface is increased due to which the motion of the fluid is slowed down. Figure 3(b) is portrayed to analyze the impact of on velocity and . An enlargement in the stretching ratio parameter causes a decline in momentum boundary layer thickness. It is due to the fact that the -component of the velocity is reduced as the ratio parameter is upsurged. On the other hand, is hiked with an increment in the increase in the ratio parameter. It is because when acquires values greater than zero, the lateral surface starts to move in the vertical direction that results an increase in and decrease in as shown in Figure 3(b). The temperature of the object depends upon the speed of the molecules. The increment in the speed of the nanoparticles results an enhancement of the random motion of the nanoparticles. To support this phenomenon, Figure 4(a) is demonstrated. An increment in the Brownian motion results an increment in the motion of the small particles inside the flow region. This enhanced chaotic movement intensifies the velocity of the particles, which causes the increment in the kinetic energy, due to which the temperature is increased as displayed in Figure 4(b). Figures 5(a) and 5(b) are displayed to study of on temperature and concentration. Increment in results a decrement in the temperature and increment in . Physically, in thermophoresis, the particles apply force on the other particles due to which particles from the hotter region move towards the colder region. Larger values of means more application of the force on the other particles, and as a result, more fluid moves from the higher temperature region to the less hotter region. Due to this fact, a decrement in the thermal and an increment in the nanoparticles concentrations are noticed. Figure 6(a) is demonstrated to analyze the effect of the temperature ratio parameter on energy profile. The graphs of this figure show that the temperature is increased as is upsurged. Figure 7 is chalked out to visualize the effect of on . An increment in the Schmidt number increases the viscous diffusion due to which the concentration profile is upsurged.

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#### 6. Concluding Remarks

In this numerical investigation, shear-thinning tangent hyperbolic nanofluid has been investigated numerically. The energy transport phenomenon is featured with Joule heating and nonlinear thermal radiation. Some of the key features are as follows:(i)The fluid motion along the horizontal direction is decreased whereas it is increased along the vertical direction as the ratio parameter is boosted(ii)Both velocities are declined as each of the magnetic parameter , and the velocity slip parameter and power-law index are hiked(iii)The temperature is increased for larger values of the Brownian motion parameter and the temperature ratio parameter (iv)The concentration profile is hiked as the thermophoresis parameter gets bigger values(v)Both skin friction coefficients are enhanced as the stretching ratio parameter and magnetic number are hiked(vi)The Nusselt number is boosted as the temperature ratio parameter is increased; however, it is declined for the temperature slip parameter

#### 7. Future Work

The study is extensible in the following direction for future consideration:(i)Other non-Newtonian fluids can be considered in similar flow situations(ii)Microstructures inclusion in the base fluid(iii)Effects such as induced magnetic field, viscous dissipation, chemical reactions, and bioconvection can be incorporated into the mathematical model to explore their effect

#### Nomenclature

: | Applied magnetic field |

: | Fluid concentration inside the boundary layer |

: | Specific heat |

: | Fluid concentration outside the boundary layer |

: | Skin friction coefficient |

: | Concentration at wall surface |

: | Brownian diffusion parameter |

: | Coefficient of mass diffusion |

: | Thermophoresis diffusion parameter |

: | Eckert number in direction |

: | Eckert number in direction |

: | Dimensionless velocities |

: | Local mass flux |

: | Constants |

: | Magnetic number |

: | Power law index |

: | Brownian motion parameter |

: | Thermophoresis parameter |

: | Nusselt number |

: | Prandtl number |

: | Heat transfer rate |

: | Radiative heat flux |

: | Thermal radiation parameter |

: | Local Reynolds number |

: | Schmidt number |

: | Sherwood number |

: | Boundary layer temperature |

: | Surface temperature |

: | Ambient temperature |

: | Time |

: | Velocity components |

: | Characteristics velocity |

: | Stretching rate |

: | Weissenberg number in direction |

: | Weissenberg number in direction |

: | Kinematic viscosity |

: | Temperature ratio parameter |

: | Dynamic viscosity |

: | Velocity slip parameter |

: | Dimensionless concentration |

: | Dimensionless temperature |

: | Fluid density |

: | Similarity variable |

: | Heat capacity of the fluid |

: | Heat capacity of the nanoparticles |

: | Time constant |

: | Thermal conductivity |

: | Temperature jump parameter. |

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding this publication.