**1. Introduction**

The thermal energy transportation and the fluid boundary layer motion over stretching (shrinking) sheets are the areas of immense importance due to its broad range industrial and technological applications. Some of the applications consist of: growing crystals structures,

**Citation:** Khan, S.; Selim, M.M; Khan, A.; Ullah, A.; Abdeljawad. T.; Ikramullah; Ayaz. M.; Mashwani, W.K. On the Analysis of the Non-Newtonian Fluid Flow Past a Stretching/Shrinking Permeable Surface with Heat and Mass Transfer. *Coatings* **2021**, *11*, 566. https://doi.org/10.3390/ coatings11050566

Academic Editor: Eduardo Guzmán

Received: 4 April 2021 Accepted: 29 April 2021 Published: 12 May 2021

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plastic sheets preparation, manufacturing of electronic chips and materials, paper industry, cooling process, and so on [1,2]. The basic work in this regard was started by Crane [3]. Andersson et al. [4], and Vajravelu [5] discussed the different aspects of fluids flowing over stretching surfaces. It is important to mention here that the gradients' existence are essential for the growth of various fluxes and flows. In fluids, there are two important effects named as Soret and Dufor effects. In Soret effect, the existence of temperature gradient results in thermal diffusion which governs the thermal energy flow. The mass transfer is mainly governed by Dufor effect, which gives rise to the diffusion-thermo effect. These effects have an influential role in governing the natural convective flow, which is one of the modes by which thermal energy can transfer due to the aggregate motion of the heated fluid. The term cross diffusion refers to the process in which the existence of concentration gradient of one specie develops the flux of the other. This means that the cross diffusion is associated with both thermal and mass diffusion. The heat energy exchangers, steel processing, cooling of nuclear power plant, etc. are the well-known technological sectors in which the convection thermal energy transportation plays an important role. The different aspects during the heat energy flow over a 3D exponentially stretching surface are investigated by Liu et al. [6]. Hayat et al. [7] examined the boundary layer Carreau fluid motion and obtained that the presence of suction depreciates (enhances) the Carreau fluid speed (boundary layer thickness). Further detail analysis about the boundary layer flow can be accessed in refs [8–10].

The Magneto-Hydro-dynamics (MHD) studies the evolution of the macroscopic behaviors of fluids in the ambient magnetic field presence. The MHD flow finds its applications in Astrophysics and Astronomy, nuclear reactors cooling, engineering and technology, Plasma Physics, etc. Nazar et al. [11] investigated the thermal energy transformation during the magnetized flow over a vertical and stretchable surface. During their investigation, they found that the enhancing B-field magnitude reduces the coefficient of skin friction and the thermal energy loss. The analytic investigation of heat energy transfer during the 3D MHD migration over a stretchable plate is carried out by Xu et al. [12] using the series solution approach. The MHD stagnation flow toward an extendable surface is examined by Ishak et al. [13]. A more recent study on stagnation point flow can be found in references [14,15]. The heat energy transfer through convection during the magnetized 3D motion on an extendable surface is worked out by Vajravelu et al. [16]. Pop and Na [17] examined the impacts of B-field on the fluid flowing through a porous and stretchable surface. The recent developments on the magnetized boundary-layer motion can be found in references [18–21].

The thermal energy radiations and its analysis are extremely important in the solar energy, fission reactors, engines, propulsion equipment for speedy aircrafts, and various chemical phenomena operating at extreme temperatures. Gnaneswara Reddy [22] studied the magnetized nanofluid motion by incorporating the impact of thermal energy radiation. The mixed convection MHD fluid flow through a perforated enclosure is examined by Gnaneswara Reddy [23]. He investigated the effects produced due to chemical reaction, Ohmic dissipation, and heat energy source. Emad [24] investigated the different impacts that arose due to the inclusion of thermal radiations in a conducting fluid flow. The influence of thermal radiations on the thermal energy transfer through convection in an electrically conducting fluid of varying viscosity moving over an extending surface is worked out by Abo-Eldahab and Elgendy [25]. Gnaneswara Reddy [26] investigated the various impacts arose due to Joule-heating, thermo-phoresis and viscous nature of a magnetized fluid flowing over an isothermal, perforated and inclined surface. A more recent and detailed investigation of the MHD flow can be found in references [27–30]. Yulin et al. [31] numerically studied the natural convection flow of nanofluid over inclined enclosure. They investigated the different impacts due to constant heat energy source and temperature. Zhe et al. [32] performed an experimental analysis of water, ethylene glycol, and copper oxides mixture by employing statistical techniques for the multi-walled-carbonnanotubes (MWCNTs). Shah et al. [33–35] analytically scrutinized the micropolar fluid

in different frames. Hayat et al. [36] analyzed the Cu-water MHD nanofluids flow in the rotating disks. Dat et al. [37] have recently studied numerically the *γ*−AlOOH nanoliquid by using different shaped nanoparticles within a wavy container. The recent studies about the nanaofluids along with different advantages can be studied in refs. [38–46].

Fluids are categorized broadly as non-Newtonian and Newtonian. The Newton viscosity relation which shows that the shear stress and strain are directly related, is applicable in the Newtonian fluid. The non-Newtonian fluid can not be described by this simple direct relation between stress and strain. The non-Newtonian fluids, for example manufactured and genetic liquid organisms, blood, polymers, liquids, etc., have central importance in this advance technological world. The non-Newtonian fluids are very hard to be analytically and numerically treated, as compared to the Newtonian fluids, due to its nonlinear behavior. The credit goes to Carreau [47], who developed a relation that describes both, the viscoelastic and nonlinear properties of of such type of complex fluids. Ali and Hayat [48] worked out the Carreau fluid peristaltic motion through an asymmetrical enclosure. Goodarzi et al. [49] analyzed the simultaneous impact of slip and temperature jump over the Non-Newtoinian nanofluid (alumina + carboxy-methyl cellulose) motion through microtube, and investigated the impacts of pertinent parameters over the nanofluid state variables. Maleki et al. [50] analyzed the impacts of heat generation (absorption), suction (injection), nanoparticles type (volume fraction), thermal and velocity slip parameter, and radiation on the temperature and velocity fields of four different types of nanofluids moving over a perforated flat surface. Hayat et al. [51] studied the impacts of induced magnetic field on the flow of Carreau fluid. Tshehla [52] examined the Carreau fluid migration past an inclined surface. Elahi et al. [53] analyzed the Carreau fluid 3D migration from a duct. Gnaneswara Reddy et al. [54] studied the effects due to Ohmic heating during the MHD viscous nanofluid motion through a nonlinear, permeable, and extending surface. Jiaqiang et al. [55] employed the wetting models in order to explain the working procedures of different surfaces found in nature. Khan et al. [56,57] employed the fractional model to Casson and Brinkman types fluids. The impacts produced due to the incorporation of thermal radiations in the presence of suction (injection) on the MHD flow of fluids are investigated by researchers [58–60]. Maleki et al. [61] analyzed the impact of heat generation (absorption) and viscous dissipation on the heat transfer during the non-Newtonian pseudoplastic nanomaterial motion over a perforated flate. Gheynani et al. [62] examined the turbulent motion of a non-Newtonian Carboxymethyl cellulose copper oxide nanofluid in a 3D microtube by investigating the impacts of nanoparticle concentration and diameter over the temperature and velocity fields. Maleki et al. [63] studied the heat transfer characteristics of pseudo-plastic non-Newtonian nanofluid motion over a permeable surface in the presence of suction and injection. The system of governing PDEs is converted to ODEs by using similarity solution technique, and then solved numerically by employing Runge–Kutta–Fehlberg fourth–fifth order (RKF45) method. The numerical investigation of (water + alumina) nanofluid mixed flow through a 2D square cavity having porous medium is carried out by Nazari et al. [64] employing a Fortran Code.

The phenomenon in which the application of an external magnetic-field to a conducting fluid produces potential difference, is termed the Hall effect. The impacts due to the inclusion of Hall effect are examined by various researchers due to its relevance with a variety of technological and industrial applications. Biswal and Sahoo [65] investigated the impacts of Hall current on the magnetized fluid motion over a vertical, permeable and oscillating surface. Raju et al. [66] worked out the Hall current impacts on the MHD flow over an oscillatory surface having porous upper wall. Datta and Janna [67] analyzed the magnetized and oscillatory fluid motion on a flat surface in the presence of Hall current. Aboeldahab and Elbarbary [68] analyzed the impacts due to Hall current during the MHD fluid dynamics through a semi-infinite and perpendicular plate. Khan et al. [69] used the finite element method for the Newtonian fluid past a semi-circular cylinder. The variation in temperature and mass diffusion in the MHD fluid flow considering the inclusion of Hall

effect is examined by Rajput and Kanaujia [70]. Further studies on similar footings are performed by Shah et al. [71–73] employing semi-analytical calculations. The magnetized and peristaltic fluid dynamics of Carreau–Yasuda fluid through a channel is numerically investigated by Abbasi et al. [74] taking into account the Hall effect impact. Abdeljawad et al. [74] investigated the 3D magnetite Carreau fluid migration through a surface of parboloid of revolution by incorporating mass transfer and thermal radiations. The impacts of Hall current and cross diffusion on the two dimensional (2D) MHD Carreau fluid flow through a perforated and stretchable (shrinkable) surface is recently investigated in [75].

Here, we extend the previous work [75] to 3-dimensional space in order to analyze what actually happens in the most general situation. The novelty of the current investigation is to examine analytically the thermal energy and mass transfer properties of the MHD Carreau fluid 3D motion through a perforated stretching sheet by considering the effects of Hall current and cross diffusion. This research work has potential applications in problems involving motion of the non-Newtonian fluid over perforated stretching (shrinking) surfaces. The research work carried out is organized in the following manner:

The geometrical description and model equations of the current investigation are presented in Section 2. The obtained results are discussed and explained by plotting various graphs in Section 3. The comparison and the computation of engineering-based related quantities are discussed through different tables in Section 4. The work is finally concluded in Section 5.

## **2. Mathematical Modeling**

The 3D magnetized Carreau fluid is considered along a linear stretching and contracting permeable sheet by incorporating the impacts of thermal radiations and Hall current. The flow is assumed to be incompressible, laminar, steady, and electrically conducting. The external magnetic field *B*<sup>0</sup> is applied in the *y*-direction. The thermal energy and mass diffusion impacts due to the existence of temperature gradient and concentration gradient are considered as well. The geometry is chosen in such a way that the sheet velocities along *x*-and *y*−axis are respectively *uw* and *vw*, whereas the flow is restricted to the positive *z*−axis, as can be seen in Figure 1. Furthermore, convective heat energy flow and mass transfer are considered on the sheet, such that the assumed liquid below the sheet has temperature *Tf* and concentration *Cf* in order to make them consistent with the heat and mass conversion coefficients *h*<sup>1</sup> and *h*2.

**Figure 1.** Geometrical description of the study.

The Carreau fluid flow is governed by the relation [47,76]:

$$\eta = \left[ \eta\_{\infty} + (\eta\_0 - \eta\_{\infty}) \left( 1 + (\lambda \dot{\gamma})^2 \right)^{\frac{n-1}{2}} \right],\tag{1}$$

where *η*<sup>0</sup> (*η*∞) denotes the zero (infinite) shear-rate viscosity, *n* is the index of power law, *λ* denotes the time constant of the material. The symbol *γ*˙ is given by [76]:

$$
\dot{\gamma} = \sqrt{\frac{1}{2} \sum\_{i} \sum\_{j} \dot{\gamma}\_{ij} \dot{\gamma}\_{ji}} = \sqrt{\frac{1}{2} \prod\_{\prime}} \tag{2}
$$

where ∏ is the strain-rate tensor second invariant. Hall effect arises when magnetic field is applied externally to the conducting fluid which can modify the flow pattern. This phenomenon can be studied with the help of Ohm's law [75,77] given as:

$$
\vec{j} + \frac{\omega\_{\text{ef}} t\_{\text{ef}}}{B\_0} \times (\vec{j} \times \vec{B}) + \frac{\sigma P\_{\text{f}}}{e n\_{\text{ef}}} = \sigma (\vec{\mathcal{V}} \times \vec{B} + \vec{E}), \tag{3}
$$

where *j* is the current density, *ω<sup>e</sup>* (*te*) is the angular frequency (collision time interval) of electrons, *σ* denotes the conductivity, - *E* (- *B*) is the electric field (magnetic field), *ne*( *e*) is the number density (charge) of electrons, and *Pe* is the pressure of electrons. The *y*− component of*j* is zero due to the application of external magnetic field in this direction. The *<sup>x</sup>* and *<sup>z</sup>*−components of*j* are expressed in the chosen geometry as:

$$j\_x = \frac{\sigma B\_0^2 (mu - w)}{1 + m^2},\tag{4}$$

$$j\_z = \frac{\sigma B\_0^2 (mw + u)}{1 + m^2},\tag{5}$$

where *m* = *ωete* is the Hall parameter. Using Equations (1)–(5) at *η*<sup>∞</sup> = 0, the Carreau fluid equations are written respectively as [75]:

$$
\frac{\partial \underline{u}}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0,\tag{6}
$$

$$\begin{split} \ln \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} + w \frac{\partial u}{\partial z} - \nu \frac{\partial^2 u}{\partial y^2} \left( 1 + \left( \frac{n-1}{2} \right) \lambda^2 \left( \frac{\partial u}{\partial y} \right)^2 \right) = \nu (n-1) \lambda^2 \frac{\partial^2 u}{\partial y^2} (\frac{\partial u}{\partial y})^2 \times \left( 1 + (\frac{n-3}{2}) \lambda^2 (\frac{\partial u}{\partial y})^2 \right) \\ & - \frac{\sigma B\_0^2 (mw + u)}{\rho (1 + m^2)} - \frac{\nu u}{k}, \end{split} \tag{7}$$

$$\begin{split} u\frac{\partial w}{\partial x} + v\frac{\partial w}{\partial y} + w\frac{\partial w}{\partial z} - \nu \frac{\partial^2 w}{\partial y^2} \left( 1 + \left(\frac{n-1}{2}\right) \lambda^2 \left(\frac{\partial w}{\partial y}\right)^2 \right) = \nu (n-1) \lambda^2 \frac{\partial^2 w}{\partial y^2} (\frac{\partial w}{\partial y})^2 \times \left( 1 + (\frac{n-3}{2}) \lambda^2 (\frac{\partial w}{\partial y})^2 \right) \\ &+ \frac{\nu B\_0^2 (mu-w)}{\rho (1+m^2)} - \frac{\nu w}{k}, \end{split} \tag{8}$$

$$u\frac{\partial T}{\partial x} + v\frac{\partial T}{\partial y} + w\frac{\partial T}{\partial z} = -\frac{1}{\rho c\_p}\frac{\partial q\_r}{\partial y} + u\frac{\partial^2 T}{\partial y^2} + \frac{D\_m K\_T}{c\_s c\_p}\frac{\partial^2 C}{\partial y^2} \tag{9}$$

$$
\mu \frac{\partial \mathcal{C}}{\partial x} + v \frac{\partial \mathcal{C}}{\partial y} + w \frac{\partial \mathcal{C}}{\partial z} = \frac{D\_m K\_T}{T\_m} \frac{\partial^2 T}{\partial y^2} + D\_m \frac{\partial^2 \mathcal{C}}{\partial y^2}. \tag{10}
$$
