#### Abstract

The 3D Prandtl fluid flow through a bidirectional extending surface is analytically investigated. Cattaneo–Christov fluid model is employed to govern the heat and mass flux during fluid motion. The Prandtl fluid motion is mathematically modeled using the law of conservations of mass, momentum, and energy. The set of coupled nonlinear PDEs is converted to ODEs by employing appropriate similarity relations. The system of coupled ODEs is analytically solved using the well-established mathematical technique of HAM. The impacts of various physical parameters over the fluid state variables are investigated by displaying their corresponding plots. The augmenting Prandtl parameter enhances the fluid velocity and reduces the temperature and concentration of the fluid. The momentum boundary layer boosts while the thermal boundary layer mitigates with the rising elastic parameter () strength. Furthermore, the enhancing thermal relaxation parameter ()) reduces the temperature distribution, whereas the augmenting concentration parameter () drops the strength of the concentration profile. The increasing Prandtl parameter declines the fluid temperature while the augmenting Schmidt number drops the fluid concentration. The comparison of the HAM technique with the numerical solution shows an excellent agreement and hence ascertains the accuracy of the applied analytical technique. This work finds applications in numerous fields involving the flow of non-Newtonian fluids.

#### 1. Introduction

Non-Newtonian fluids are those fluids that do not obey Newton’s law of fluid motion. The Newtonian fluids have constant viscosity as evidenced by the direct relationship between the shear stress and the resulting strain. The majority of applications of non-Newtonian fluids in different fields such as petroleum production, bio-chemicals preparation, pharmaceutical industry, food, and power engineering, have been thoroughly investigated by various researchers and investigators. The non-Newtonian fluids, for example, genetic and manufactured liquid organisms, polymers, emulsions, paints, blood, oil, toothpaste, and ketchup, have a crucial and important role in this advanced scientific, industrial, and technological arena. The non-Newtonian fluids due to their complex and nonlinear nature are very difficult to be handled both numerically as well as analytically as compared to Newtonian fluids. Carreau [1, 2] for the first time introduced an expression for the stress tensor that models the nonlinear and viscoelastic features of non-Newtonian fluids. The Carreau relation for the stress tensor has opened the way to investigate the non-Newtonian fluids both analytically and numerically.

Non-Newtonian fluids may be further divided into many types based on their physical characteristics. There exists an important type called shear thinning or pseudoplastic fluids. The viscosity of such fluids displays a decreasing tendency to the applied shear stress. The most important and common example of such kind of fluid is the human blood. Researchers have developed different models to describe the behavior and properties of shear-thinning fluids. The importance of these models is the Prandtl fluid model, which is capable to describe the shear-thinning nature of such kinds of fluids. Akbar et al. [3] inspected the outcomes of hydromagnetic stagnation-point flow of Prandtl fluid through stretching flat surface embedded in a porous medium. Bilal et al. [4] analyzed the consequences of double diffusion on the MHD Prandtl nanoliquid moving through an extended sheet taking into account the effects of external magnetic influence. Nadeem et al. [5] performed the analytical treatment of a Prandtl fluid model with heat transfer characteristics. Khan et al. [6] briefly investigated the impacts of homogenous and heterogeneous reactions on the 2D MHD flow of Prandtl fluid through a nonlinear heated surface. Amanullah et al. [7] investigated the hydromagnetic flow of Prandtl-Eyring fluid through an isothermal permeable spherical surface, considering the magnetic and slip effects. The effect of slip-on electrically conductive hydromagnetic boundary layer flow through an exponentially extended sheet with viscous dissipation and thermal radiation has been investigated by Mukhopadhyay [8]. Ali et al. [9] described the unique features of unsteady Eyring-Powell nanofluid flow near the stagnation point through a convectively heated stretching surface. Further relevant studies about viscoelastic Prandtl fluid can be studied in references [10–19]. Abbas et al. [20] addressed the entropy optimized Darcy–Forchheimer nanofluid flow through an elongated curved surface with MHD and temperature-dependent viscosity effects. Sreedevi and Reddy [21] examined the impacts of thermophoresis and Brownian motion on the hydromagnetic 3D Maxwellian nanoliquid migration through a stretching sheet in the presence of thermal radiation and chemical reaction effects. Recently, enormous research on the dynamic features of nanofluid motion has been undertaken in references [22–27].

MHD and electrically conducting flows of non-Newtonian fluids through stretching surfaces have enormous applications in the different fields of engineering and technology. Some of them are hot rolling, glass fiber, nuclear reactor cooling, plastic sheet extrusion, metal casting, glass blowing, and metallurgical casting. Abel et al. [28] analyzed the effects of heat transport and MHD motion over 2D UCM fluid flow through a stretching surface in the presence of induced magnetic parameter. Vishnu et al. [29] performed the Lie symmetry analysis of the magnetic field effects on the free convection nanofluid flow over a semi-infinite stretchable surface. Das et al. [30] demonstrated the hydromagnetic nanofluid flow through a stretching surface embedded in permeable source through a rotating frame. Ibrahim et al. [31] numerically analyzed the impacts of heat source and applied magnetic field on the mixed convective MHD dissipative Casson nanoliquid stagnation-point flow through a stretching surface by taking into account the convective boundary condition and velocity slip impacts. Kumar et al. [32] thoroughly examined the hydromagnetic flow of chemically reactive non-Newtonian viscoelastic Williamson liquid through a curved/flat sheet with generation of variable heat and radiation effects. Das and Zheng [33] studied the impacts of melting and external magnetic force on the stagnation-point flow of conducting viscoelastic Jeffrey fluid through a curved surface with Newtonian heating. Aziz and Afify [34] evaluated numerically the MHD and boundary layer flow through a curved surface in the presence of magnetic field and viscous dissipation. Recently, Abbas et al. [35] studied 3D MHD flow of micropolar nanofluid through a rotating permeable exponentially elongated surface using convective boundary conditions. In [36–38], homotopy analysis method (HAM) has applied for solving various types of nonlinear differential equations. In [39], the influence of thermal radiation, inclined magnetic field, and stratification on the third-grade nanofluid with swimming gyrotactic microorganisms have been mainly analysed by using the optimal homotopy analysis method (OHAM). The fluid is flowing past a horizontal cylinder exposed to an inclined magnetic field. The boundary layer system of equations is reconstructed into ordinary differential equations using similarity transformations. The system of equations is resolved using the optimal homotopy analysis method (OHAM) and the results are visualized numerically and graphically. The Galerkin finite element scheme has employed for developing new system of equations to investigate the generation of entropy in viscoelastic Sisko nanofluid over a cylinder. The analysis is performed in the existence of nanofluid and motile microorganisms. The magneto-hydrodynamic flow including the inclined magnetic field is considered. Fluxes for heat, mass, the concentration of nano-sized particles, and the concentration of swimming microorganisms are computed and scrutinized [40]. Recently in [41], a simulation model has been employed in order to observe knowledge flow within the regional innovation system under relationships of varying strength. The results show that when the relationship between the subjects of the regional innovation system reaches certain strength, the system will exhibit high module independence and high network integrity, forming a loosely coupled system. The knowledge flow in the system exhibits the emergence of a fast flow rate, a high mean value, and little variance. When relationship strength is at other levels, the emergence of knowledge cannot be identified.

Motivated through the above literature, it is planned to investigate the various thermal features during the three-dimensional Darcy–Forchheimer flow of Prandtl fluid through a stretching and permeable surface. The thermal analysis of the flow is performed using the Cattaneo–Christov model. The main aim of the investigation is to explore the thermal energy transport features of the non-Newtonian Prandtl fluid migrating through a porous and stretching surface. The structure of the article is organized in the following manner.

The geometry and mathematical model of the problem under investigation are presented in Section 2. The model equations are transformed to simple form by using similarity relations. The solution methodology of HAM applied to solve the reduced model is explained in Section 3. The convergence analysis of the applied procedure is discussed in Section 4 by displaying graphs and tables of the state variables. The results are displayed through different graphs describing the impacts of physical parameters over the state functions in Section 5. The work is concluded finally in the last section.

#### 2. Problem Framework

We consider the 3D Darcy–Forchheimer flow of Prandtl fluid through a permeable stretching surface as displayed in Figure 1. The surface temperature and concentration are expected to remain constant. The Cattaneo–Christov model of heat and mass flux is employed to analyze the heat and mass flow rates. The stretching velocities along x- and *y*-axes are, respectively, and . The *z*-axis is taken normal to the fluid flow.

The governing model of the flow consists of the following equations:

The flow satisfies the following conditions:

Here, , and are the velocity Cartesian components, is the kinematic viscosity, is the fluid dynamic viscosity, is the fluid density, and are the material constants for the Prandtl liquid, is the relaxation time of heat flux, is the inertia coefficient, is the relaxation of mass flux, is the thermal diffusivity, is the thermal conductivity, is the heat capacity of working liquid, is the effective heat capacity of nanoparticles, is the temperature, is the concentration, is the surface temperature, is the surface concentration, is the ambient temperature, is the ambient concentration, and is the mass diffusion coefficient.

The similarity transformations are defined as

Using these transformations, the continuity equation is satisfied identically, whereas equations (2)–(6) are transformed aswhere , , and are respectively the ratio parameter, Prandtl fluid parameter, elastic parameter, concentration parameter, thermal relaxation parameter, Prandtl number, Schmidt number, inertia parameter, and porosity parameter. These are given by

The skin frication coefficients, heat transfer rate, and mass flow rate are defined aswhere and indicate the local Reynolds numbers. The present fluid reduces to the Newtonian fluid when and .

#### 3. Solution Methodology

In order to solve the nonlinear system of equations (8)–(11) subjected to the conditions in equation (12), we employ the homotopic scheme. The following steps are needed to execute the HAM procedure. The solution having the auxiliary constraint adjusts and regulates the convergence of the analytical series solution.

The initial guesses are

The linear operators are represented bywith the following properties:where all are numbers in the general solution.

The subsequent nonlinear operators are given as

The basics of HAM have been explained in references [36]. The 0^{th}-order problem from the system of equations (8)–(11) is

The extreme values arewhere is the embedding parameter, and are used to control the convergence of the obtained solution.

When we have

The Taylor’s series expansions of about are given bywhereas

The secondary constraints are chosen in such a way that series (27) converges at Substituting in equation (27), we get

The problem satisfies the following relations:

The corresponding boundary conditions are

The different symbols in equation (30) represent the following expressions:with

#### 4. Convergence Analysis

The right selection of auxiliary constraints , and well affects the convergence of a series solution of a nonlinear governing problem by homotopic technique. The precise choice of these parameters plays a significant role in regulating the convergence zone. In this regard, we computed and sketched the velocity, temperature, and concentration plots in Figure 2. The interval of convergence is determined by the flat part of these plots. Tables 1 and 2 show the ranges of the permissible auxiliary parameter values and fruitful convergence. It is found that as the order of approximation upsurges, the average squared residual error diminishes, as shown in Figure 3. The comparison of the HAM solution with the shooting technique (RK-4) has been made to confirm the effectiveness of the implemented method. Results are shown graphically in Figures 3–6 and tabulated in Tables 3–6.

#### 5. Results Analysis

This part is devoted to understand the physical consequences through the graphical display of the state variables of the investigated fluid flow. The system of nonlinear equations (8)–(11) with the extreme conditions in equation (12) are tackled via the famous analytical method of HAM. The impact on velocity, energy, and concentration profiles is explored graphically with the variation of various dimensionless parameters in Figures 2–10. The computational upshots for the drag force, heat, and mass flow rates are tabularized respectively in Tables 7–9. Table 7 represents the variation in the drag force coefficient with varying values of , , and . It has been perceived that the drag force diminishes with the increasing values, while enhances with the augmenting values. Table 8 displays the heat transfer features through variation of Nusselt number with varying and . It is noticed that Joule heating increases with higher Prandtl number values and drops with the increasing thermal relaxation parameter. Table 9 displays the impact of varying strength of and on the mass flow rate. The tale shows that the mass flow rate enhances with the increasing Schmidt number values and drops with the augmenting inertia parameter strength.

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Figures 8(a)–8(c) demonstrate the impact of Prandtl fluid parameter variation over fluid velocity, thermal field, and concentration field, respectively. One can perceive that is the growing function of . Thus, the fluid velocity and associated boundary layer thickness enhance with the rise in Prandtl fluid parameter. Here, the thermal field and concentration profile drop with the increasing , and therefore, the boundary layer thickness diminishes in both cases. Figures 9(a)–9(c) depict the effect of elastic parameter variation on the flow field, temperature distribution, and concentration, respectively. Figure 9(a) shows that the thickness of the momentum boundary layer boosts with the larger elastic parameter strength. The boosting is more prominent in the intermediate region. The increasing elastic parameter diminishes the heat conduction and escalates the heat convection. In consequence, the thermal field curves and boundary layer thickness dwindle with the enhancing as displayed by Figure 9(b). The concentration field diminishes with the rising as displayed by Figure 9(c). Figure 10(a) illustrates the variation in the thermal field () with the different values of thermal relaxation parameter . This graph reveals that the temperature field decreases with the higher relaxation parameter strength. In fact, the working fluid dissipates extra heat with the augmenting thermal relaxation parameter. In consequence, the fluid temperature diminishes. Figure 10(b) delineates the impact of on the concentration profile . We see that higher estimation of retards the concentration and the related boundary layer thickness mitigates. In consequence, reduces. Figures 11(a) and 11(b) present the outcome of porosity parameter and inertia parameter on the stream flow profiles. As anticipated, the velocity of material particles diminishes with the larger porosity and inertia parameters. Figure 12(a) shows the impact of variation of Prandtl number on the thermal field distribution. Clearly, the thermal field decreases with the higher strength of . An upsurge in the Prandtl number declines the thermal diffusion in the fluid layers. In consequence, a thinner thermal boundary layer forms and the temperature field diminishes. Figure 12(b) portrays the Schmidt number effect on the concentration profile . As we know that the Schmidt number and mass diffusion are related inversely with each other, hence an upsurge in results in a decay in the concentration field as displayed in the figure.

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#### 6. Conclusions

This section concludes the findings of the undertaken research. In the current mathematical framework, the 3D Darcy–Forchheimer flow of Prandtl fluid motion over a nonlinear permeable surface with heat conduction and mass diffusion is considered. The Prandtl motion is modeled through coupled nonlinear PDEs. The system of PDEs is converted to a simplified set of coupled nonlinear ODEs using suitable similarity relations. HAM is applied to solve the reduced system of equations. The convergent analysis of the applied procedure shows that the solution is convergent over a wide range. The comparison of HAM with the shooting technique through graphs and tables displays tremendous agreement. This research work has a wide range of applications in industrial and technological fields. The key points of the current work are outlined as follows: With the increase in Prandtl fluid parameter, the momentum boundary layer enhances, while the thermal and concentration boundary layers diminish The momentum boundary layer upsurges for larger elastic parameters, whereas both temperature and concentration fields dwindle The rising thermal relaxation and concentration parameters depict similar reducing behavior for the thermal and concentration fields The augmenting Schmidt number causes drop in the fluid concentration The drag force coefficient decreases with the enhancing Prandtl fluid *t* and elastic parameters

#### Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

#### Conflicts of Interest

The authors declare no conflicts of interest regarding the study in this paper.

#### Authors’ Contributions

All authors have equally contributed in this paper.