Thermophoresis and Brownian Model of Pseudo-Plastic Nanofluid Flow over a Vertical Slender Cylinder

'is study focuses on the industrial and engineering interest quantities, such as drag force and rate of transmission of heat, for pseudo-plastic nanofluid flow. 'e attributes of natural convection of the pseudo-plastic nanofluid flow model over a vertical slender cylinder are explored.'e pseudo-plastic flow is studied under the influence of concentration of nanoparticles, rate of heat transmission, and drag force. For the first time, the pseudo-plastic nanofluid flow model has been implemented over a vertical slender cylinder which is not yet investigated.'e acquiredmodel is based on thermophoresis and Brownianmotionmechanisms. 'e governing equations of pseudo-plastic nanofluid in cylindrical coordinates are modelled. 'e developed system of nonlinear equations is tackled by boundary layer assumptions and similarity transformations. Moreover, the solution of the acquired system exhibited by using a new powerful numerical technique. A comprehensive debate on drag force and transmission of heat under the influence of various emerging parameters is illustrated in the table. Furthermore, the effects of dimensionless parameters over the velocity profile, temperature profile, and concentration of nanoparticle profile have been exhibited graphically.


Introduction
Investigation of the non-Newtonian fluids gains prodigious attention of researchers over a half-century because naturally, most of the fluids used in the industrial applications are non-Newtonian fluids. is is the main cause of increased applications of non-Newtonian fluids in the industrial field and engineering such as petroleum production, molten plastics, food engineering, automobiles, polymer solution industry, chemical engineering, and power engineering. A solitary established equations cannot pronounce the attributes of such non-Newtonian fluids because these kinds of fluids have a nonlinear relationship among the rate of stress and strain. erefore, several scientists and engineers have pronounced models for non-Newtonian fluids [1][2][3][4][5][6][7][8][9][10]. Ellahi et al. [11] explored thermally charged MHD biphase flow coating with non-Newtonian nanofluid over the slippery walls. e sustainable features of MHD Jaffrey fluid for biobi-phase flow are carried out by Zeeshan et al. [12]. Rehman et al. [13] examined MHD flow of the nanoparticle influenced near a stagnation point over an exponential stretched surface. Parand et al. [14] described the boundary layer flow of Powell-Eyring fluid for a stretching sheet. Generation/ absorption of heat through the flow of axisymmetric Casson fluid over a stretched cylinder is addressed by Javed et al. [15]. Additional appropriate studies in this way are in [16,17].
Over the last few years, we have adopted a pioneering procedure for refining the transmission of heat by utilizing ultrafine solid particles in the fluids, and these particles have been used widely called as nanofluid. e label "nanofluid" was used for the first time in 1995 by Choi and Eastman [18]. He cited that it is conceivable to boost convection of transmission of heat and thermal conductivity efficiency by using nanofluids. To handle nanofluids, Buongiorno [19] introduced a mathematical model and explored numerous transport mechanisms and applications about the nanofluids. Miscellaneous purposes of nanofluids have been uncovered by Das et al. [20]. Zhang et al. [21] scrutinized the nanofluid for MHD radiative flow over a variable heat flux and chemical reaction surface. Transport of heat for the ferromagnetic fluid with thermal stratification over a stretching sheet was examined by Muhammad et al. [22]. Saini and Sharma [23] extended the application of nanofluids through the investigation of double-diffusive bioconvection. Further extensive research and applications of nanofluids across numerous fields are heat transmission, energy, microequipment, and boiling applications [24][25][26][27][28].
Williamson's fluid model has non-Newtonian behaviour in nature like pseudo-plastic fluid which defines the flow of shear thinning. e industrial, engineering, and biological fluids which observe as Williamson's fluid are blood, glue, paint, ketchup, polymer solutions, and nail polish. Williamson [29] who discovered the model to communicate pseudo-plastic attributes along with features of extreme points of viscosity. Due to this invention, innovated researchers are motivated to discover more upfront classifications of non-Newtonian fluid. Ramzan et al. [30] analyzed the numerical solution of MHD flow over a stretched surface with convective boundary conditions using the shooting method. Two-dimensional flow of Williamson's fluid film with heat diffusion under the influence of thermal radiation was inspected via an optimal approach by Shah et al. [31].
Rate of transmission of heat, the transmission of mass rate, and skin friction coefficient play a dynamic character during coating of wires or polymer fibre coating. As wires have a thin cylindrical shape, miscellaneous researchers have launched several mathematical models. In axial incompressible flow, Seban and Bond [32] premeditated the attributes of drag force and rate of transmission of heat for a cylinder in 1951. Under the uniform surface heat flux, Mucoglu and Chen [33], for the first time, scrutinized the slender cylinder with the help of mixed convection regime. Nadeem et al. [34] analyzed the boundary layer flow and transmission of heat of a nanofluid in a vertical slender cylinder. Patil et al. [35] examined the mixed convection nanofluid boundary layer flow under the effect of surface roughness with a moving slender cylinder. Reddy et al. [36] explored the natural convection for the Casson fluid flow past over a hollow slender cylinder. With the help of Bejan's heat function concept, Reddy et al. [37] investigated the unsteady MHD micropolar fluid flow in a homogeneously thermal radiative hollow slender cylinder with the radiative transmission of heat effect. Further latest stimulating work in this area can be found in [38][39][40][41][42].
e literature review replicates that, generally, the researchers engaged themselves to study non-Newtonian fluids' behaviour by assuming different effects. e attributes of natural convection of Williamson's nanofluid model over a vertical slender cylinder are not explored until now.
erefore, the developed model is simplified by the boundary layer approximation and similarity transformations. e governing coupled nonlinear system of equations is then solved by a new powerful numerical technique. Furthermore, physical behaviour for the industrial interests of the fluid will be examined through the table.

Analysis of Flow and Mathematical Formulation
Suppose an incompressible Williamson's nanofluid flow along with a permeable vertical slender cylinder with radius a having uniform ambient temperature T ∞ . e coordinates (x, r) are used, whereas x is acting along the surface of the cylinder and r along the radial direction. e velocity profile, temperature profile, and concentration profile are as follows: where (w(x, r), u(x, r)) are velocity components along the surface and radial direction. e boundary layer equations of motion, energy, and the nanoparticle concentration are e accompanied boundary conditions are Here, U(x) � x/l ( )U ∞ is the mainstream velocity, υ is called kinematic viscosity and is defined as υ � μ/ρ , Γ > 0 articulated the material constant for Williamson's fluid, and ρ denotes the density of the fluid. Now, the nondimensional variables and similarity transformations are defined as follows: in which T w − T ∞ � ΔT x/l ( ) and ϕ w − ϕ ∞ � Δϕ x/l ( ) operated for the characteristic temperature ΔT and nanoparticle concentration Δϕ. By using the above transformations, equation (2) is identically satisfied, and equations (3)-(5) will be articulated as Dimensionless attached boundary conditions are where ) means the buoyancy ratio, Pr � υ/α is identified as the Prandtl number, corresponds to the thermophoresis parameter, and Le � α/D B is known as the Lewis number.

Physical Quantities of the Industrial Interest
For the industrial interest, the physical quantities, i.e., drag force and transmission of heat, are stated as where s w stands for the shear stress tensor over the surface of the slender cylinder, while q w is called wall heat flux. ese physical quantities can be articulated as According to similarity transformation, equation (15) is transformed as

Numerical Solution
e solution of the nonlinear system of ODEs (10)-(12) over the accompanied condition equations (13) and (14) is tackled through the numerical algorithm of MATLAB inherent scheme bvp4c. In order to apply this scheme, first of all, higher-order differential equations are transformed into first-order ODEs. e procedure is as follows.

Discussion
Natural convection of Williamson's nanofluid over a slender cylinder has been analyzed. Numerical solution is obtained by using MATLAB scheme bvp4c. Physical properties of the nanofluid are portrayed by using the Buongiorno model. Likewise, transmission of heat and mass is differentiated by means of assorted parameters. (1) and (2) show the attributes of curvature parameter c c and Williamson's dimensionless parameter λ for f ′ (η). It is well defined from Figures (1) and (2) that intensifying the values of c c tends to devalue f ′ (η); however, equivalent behaviour is anticipated for Williamson's parameter λ. is is because after intensifying the curvature parameter c c , the radius of the cylinder with the fluid declines. Similarly, after intensifying Williamson's parameter λ which will cause a decline in velocity because fluid opposes more resistance.

Attributes of λ N and
N r for f ′ (η). e attribution of λ N and N r is shown in Figures (3) and (4). From Figures (3) and (4), it defines that, by intensifying natural convection λ N and buoyancy ratio N r , the buoyancy force will cause the higher velocity attained by the fluid.

Attributes of c c and Pr for θ(η).
e behaviour of temperature profile θ(η) for distinct values of curvature parameter c c and Prandtl number is reported in Figures (5) and (6). Evidently, from Figure 5, it can be described that enhancement in the curvature parameter yields intensifying in θ(η). Moreover, in Figure 6, the temperature profile and the boundary layer thickness decrease due to enhancement in Prandtl number (Pr). Which uncovers the truth that, enlarging in Pr cause the reduction in the thermal diffusivity of the fluids accordingly.

Attributes of T p and B p for θ(η).
e contribution of thermophoresis parameter T p and Brownian motion B p is described in Figures (7) and (8). Evidently, larger T p and B p produce higher θ(η). Practically, movement of a small number of particles from higher temperature to the lower one is defined as the thermophoresis phenomenon. Hence, a greater number of nanoparticles are shifted from the hot region which raises the fluid temperature. However, due to the increase in Brownian motion parameter B p in the result, random motion of the nanoparticles will be raised which causes intensifying in the temperature of the fluid. Figure 9 exhibits the consequences of curvature parameter c c on nanoparticle concentration ψ(η). It is clear from the figure that intensifying curvature parameter c c diminishes the nanoparticle concentration of the fluid. According to the industrial view, after increase in the curvature parameter, it caused the reduction in the radius of the slender cylinder; hence, concentration of nanoparticles also reduced.

Attributes of Lewis Number Le and Prandtl Number on ψ(η).
e behaviour of Lewis number Le is depicted for ψ(η) through Figure 10. It is scrutinized that ψ(η) is diminished by increasing Lewis number Le. Physically, due to the levitation of diffusivity of heat and mass, nanoparticle concentration ψ(η) diminished. Furthermore, scrutinization of the attribution of Prandtl number Pr for ψ(η) is portrayed through Figure 11. Physically, intensifying in Prandtl number will cause the more heat convection occur because thickness of the thermal boundary layer is greater than the velocity boundary layer. Hence, nanoparticle concentration diminished.

Attributes of T p and B p on ψ(η).
e impact of thermophoresis parameter T p and Brownian motion parameter B p on ψ(η) is depicted in Figures (12) and (13). Due to the increase of T p , boundary layer thickness of nanoparticle concentration intensified. As a matter of fact, thermophoresis force increases for higher estimation of T p , which yields nanoparticle relocations from higher to lower temperature; hence, ψ(η) intensifies (see Figure 12). However, ψ(η) and its concentration boundary layer diminish when B p is amplified (see Figure 13).   Table 1. It is evidently clear from the table that drag force is intensified by enhancing c c , λ, λ N , and N r . Table 2 expresses the behaviour of physical parameters on the rate of

Final Remarks
is exploration scrutinized the natural convection of Williamson's nanofluid for boundary-layer stagnation point flow over a vertical slender cylinder. Hence, thermophoresis, Brownian motion, natural convection, Williamson's parameter, and curvature parameter are utilized for modelling and analysis. e governing coupled nonlinear ODEs are then solved numerically by using MATLAB technique bvp4c.
is research reveals the following outcomes:

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.  Mathematical Problems in Engineering 9