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

Authors: Faizan Hussain; Azad Hussain; Sohail Nadeem
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

This study focuses on the industrial and engineering interest quantities, such as drag force and rate of transmission of heat, for pseudo-plastic nanofluid flow. The attributes of natural convection of the pseudo-plastic nanofluid flow model over a vertical slender cylinder are explored. The 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. The acquired model is based on thermophoresis and Brownian motion mechanisms. The governing equations of pseudo-plastic nanofluid in cylindrical coordinates are modelled. The 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.


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## Body

## 1. 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. This 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. Therefore, several scientists and engineers have pronounced models for non-Newtonian fluids [1–10]. Ellahi et al. [11] explored thermally charged MHD biphase flow coating with non-Newtonian nanofluid over the slippery walls. The sustainable features of MHD Jaffrey fluid for bio-bi-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. The 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 [24–28].Williamson’s fluid model has non-Newtonian behaviour in nature like pseudo-plastic fluid which defines the flow of shear thinning. The 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 [38–42].The literature review replicates that, generally, the researchers engaged themselves to study non-Newtonian fluids’ behaviour by assuming different effects. The attributes of natural convection of Williamson’s nanofluid model over a vertical slender cylinder are not explored until now. Therefore, the developed model is simplified by the boundary layer approximation and similarity transformations. The 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.


## 2. Analysis of Flow and Mathematical Formulation

Suppose an incompressible Williamson’s nanofluid flow along with a permeable vertical slender cylinder with radiusa having uniform ambient temperature T∞. The coordinates x,r are used, whereas x is acting along the surface of the cylinder and r along the radial direction. The velocity profile, temperature profile, and concentration profile are as follows:(1)Vx,r=w,0,u,T=Tx,r,ϕ=ϕx,r,where wx,r,ux,r are velocity components along the surface and radial direction. The boundary layer equations of motion, energy, and the nanoparticle concentration are(2)∂∂rrw+r∂u∂x=0,(3)u∂u∂x+w∂u∂r=−1ρ∂p∂x+υ∂2u∂r2+1r∂u∂r+Γ1r2∂u∂r2+2Γ∂2u∂r2∂u∂r+ρ∗−ρϕ−ϕ∞ρ+1−ϕ∞T−T∞βthggr,(4)w∂T∂r+u∂T∂x=α∂2T∂r2+1r∂T∂r+ρ∗cp∗ρcpDB∂ϕ∂r∂T∂r+DTT∞∂T∂r2,(5)w∂ϕ∂r+u∂ϕ∂x=DB∂2ϕ∂r2+1r∂ϕ∂r+DTT∞∂2T∂r2+1r∂T∂r.The accompanied boundary conditions are(6)ux,a=0,Tx,a=Twx,(7)ux,r⟶Ux,asr⟶∞,Tx,r⟶T∞,asr⟶∞,(8)ϕx,a=ϕwx,ϕx,r⟶ϕ∞,asr⟶∞.Here,Ux=x/lU∞ 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:(9)u=U∞xlf′η,w=−arυU∞l1/2fη,θ=T−T∞Tw−T∞,ψ=ϕ−ϕ∞ϕw−ϕ∞,η=r2−a22aU∞υl1/2,in which Tw−T∞=ΔTx/l and ϕw−ϕ∞=Δϕx/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(10)2ηγc+1f‴+2γcf″+3λγc2ηγc+11/2f″2+2λ2ηγc+13/2f″f‴+ff″−f′2+λN1−ϕ∞θ+Nrψ+1=0,(11)2ηγc+1θ″+2γcθ′+Prfθ′+f′θ+2ηγc+1Tpθ′+2Bpθ′ψ′=0,(12)2ηγc+1ψ″+2γcψ′+PrLefψ′−f′ψ+TpBp2ηγc+1θ″+2γcθa′=0.Dimensionless attached boundary conditions are(13)f0=c0,f′0=0,f′⟶1,asη⟶∞,(14)θ0=1,ψ0=1,θ⟶0,ψ⟶0,asη⟶∞,where γc=υl/U∞a2 illustrates the curvature parameter, λ=ΓxU3/2/2υl3/2 symbolizes Williamson’s dimensionless parameter, λN=lΔTggrβth/U∞2 pronounces as the parameter of natural convection, Nr=ϕw−ϕ∞ρ∗−ρ/ρβth1−ϕ∞Tw−T∞ means the buoyancy ratio, Pr=υ/α is identified as the Prandtl number, Bp=ρ∗cp∗DBϕw−ϕ∞/ρcpα stands for the Brownian motion parameter, Tp=ρ∗cp∗DTTw−T∞/αT∞ρcp corresponds to the thermophoresis parameter, and Le=α/DB is known as the Lewis number.


## 3. Physical Quantities of the Industrial Interest

For the industrial interest, the physical quantities, i.e., drag force and transmission of heat, are stated as(15)Cf=sw1/2ρU2,Nu=xqwα∞Tw−T∞,where sw stands for the shear stress tensor over the surface of the slender cylinder, while qw is called wall heat flux. These physical quantities can be articulated as(16)sw=μ∂u∂r+Γ2∂u∂r2r=a,qw=−α∞∂T∂rr=a.According to similarity transformation, equation (15) is transformed as(17)12CfRe1/2=f″0+λf″02,NuRe−1/2=−θ′0.


## 4. Numerical Solution

The 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. The procedure is as follows.Let us suppose that(18)fη=y1x,f′η=y2x,f″η=y3x,f‴η=y3′x,θη=y4x,θ′η=y5x,θ″η=y5′x,ψη=y6x,ψ′η=y7x,ψ″η=y7′x.Therefore, equations (10)–(14) are transformed as follows:(19)y3′x=−2γcy3−3λγc1+2xγc1/2y32−y1y3+y22−λN1−ϕ∞y4+Nry6−11+2xγc+2λ1+2xγc3/2y3x,y5′x=−2γcy5−Pry1y5+y2y4−1+2xγcBpy5y7+Tpy521+2xγc,y7′x=−2γcy7−LePry1y7−y2y6−Tp/Bc1+2xγcy5′+2γcy51+2xγc,with conditions(20)y10−c0=0,y20=0,y2∞−1=0,y40−1=0,y5∞=0,y60−1=0,y7∞=0.


## 5. 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.


### 5.1. Velocity Profiles

#### 5.1.1. Attributes ofγc and λ for f′η

Figures(1) and (2) show the attributes of curvature parameter γc and Williamson’s dimensionless parameter λ for f′η. It is well defined from Figures (1) and (2) that intensifying the values of γc tends to devalue f′η; however, equivalent behaviour is anticipated for Williamson’s parameter λ. This is because after intensifying the curvature parameter γc, 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.Figure 1
Consequences forγc on f′η.Figure 2
Consequences forλ on f′η.


#### 5.1.2. Attributes ofλN and Nr for f′η

The attribution ofλN and Nr is shown in Figures (3) and (4). From Figures (3) and (4), it defines that, by intensifying natural convection λN and buoyancy ratio Nr, the buoyancy force will cause the higher velocity attained by the fluid.Figure 3
Consequences forλN on f′η.Figure 4
Consequences forNr on f′η.


### 5.2. Temperature Profiles

#### 5.2.1. Attributes ofγc and Pr for θη

The behaviour of temperature profileθη for distinct values of curvature parameter γc 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 θη. 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.Figure 5
Consequences forγc on θη.Figure 6
Consequences for Pr onθη.


#### 5.2.2. Attributes ofTp and Bp for θη

The contribution of thermophoresis parameterTp and Brownian motion Bp is described in Figures (7) and (8). Evidently, larger Tp and Bp produce higher θη. 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 Bp in the result, random motion of the nanoparticles will be raised which causes intensifying in the temperature of the fluid.Figure 7
Consequences forTp on θη.Figure 8
Consequences forBp on θη.


### 5.3. Concentration Profiles

#### 5.3.1. Attributes ofψη on γc

Figure9 exhibits the consequences of curvature parameter γc on nanoparticle concentration ψη. It is clear from the figure that intensifying curvature parameter γc 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.Figure 9
Consequences forγc on ψη.


#### 5.3.2. Attributes of Lewis Number Le and Prandtl Number onψη

The behaviour of Lewis number Le is depicted forψη through Figure 10. It is scrutinized that ψη is diminished by increasing Lewis number Le. Physically, due to the levitation of diffusivity of heat and mass, nanoparticle concentration ψη diminished. Furthermore, scrutinization of the attribution of Prandtl number Pr for ψη 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.Figure 10
Consequences of Lewis number Le.Figure 11
Consequences of Prandtl number Pr.


#### 5.3.3. Attributes ofTp and Bp on ψη

The impact of thermophoresis parameterTp and Brownian motion parameter Bp on ψη is depicted in Figures (12) and (13). Due to the increase of Tp, boundary layer thickness of nanoparticle concentration intensified. As a matter of fact, thermophoresis force increases for higher estimation of Tp, which yields nanoparticle relocations from higher to lower temperature; hence, ψη intensifies (see Figure 12). However, ψη and its concentration boundary layer diminish when Bp is amplified (see Figure 13).Figure 12
Consequences ofTp on ψη.Figure 13
Consequences ofBp on ψη.


### 5.4. Attributes of Drag Force and Rate of Transmission of Heat

The attributes of physical parameters such as curvature parameterγc, Williamson’s parameter λ, natural convection λN, and buoyancy ratio Nr on drag force are exhibited in Table 1. It is evidently clear from the table that drag force is intensified by enhancing γc, λ, λN, and Nr. Table 2 expresses the behaviour of physical parameters on the rate of transmission of heat. It is inspected that the rate of transmission of heat enhances by increasing γc, Tp, and Pr, while it declines for inclination of λ, Bp, and Le.Table 1
Numerical values of drag forces1/2CfRe1/2.
γcλλNNrf″01/2CfRe1/20.10.10.10.11.72902.02790.2———1.76702.07920.3———1.80302.12810.4———1.83902.1772—1——1.13402.4200—2——0.92522.6372—3——0.84883.0102—4——0.78533.2521—10.1—1.22902.7394——0.2—1.24302.7880——0.4—1.28602.9398——0.7—1.35103.17620.530.10.10.90113.3370———0.20.90313.3499———0.40.90673.3730———0.70.90703.3749Table 2
Numerical values of the rate of transmission of heatNu/Re1/2.
γcλTpBpPrLeNu/Re1/20.10.10.10.10.30.40.81180.2—————0.84460.3—————0.8739—0.5————1.1190—0.7————0.9200—0.9————0.8702——0.3———1.0120——0.5———1.1430——0.7———0.7092——0.10.3——0.8221———0.5——0.7641———0.7——0.7092———0.10.5—1.1140———-0.7—1.3200————0.9—1.5120————0.30.50.8764—————0.60.8700—————0.80.8589


## 5.1. Velocity Profiles

### 5.1.1. Attributes ofγc and λ for f′η

Figures(1) and (2) show the attributes of curvature parameter γc and Williamson’s dimensionless parameter λ for f′η. It is well defined from Figures (1) and (2) that intensifying the values of γc tends to devalue f′η; however, equivalent behaviour is anticipated for Williamson’s parameter λ. This is because after intensifying the curvature parameter γc, 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.Figure 1
Consequences forγc on f′η.Figure 2
Consequences forλ on f′η.


### 5.1.2. Attributes ofλN and Nr for f′η

The attribution ofλN and Nr is shown in Figures (3) and (4). From Figures (3) and (4), it defines that, by intensifying natural convection λN and buoyancy ratio Nr, the buoyancy force will cause the higher velocity attained by the fluid.Figure 3
Consequences forλN on f′η.Figure 4
Consequences forNr on f′η.


## 5.1.1. Attributes ofγc and λ for f′η

Figures(1) and (2) show the attributes of curvature parameter γc and Williamson’s dimensionless parameter λ for f′η. It is well defined from Figures (1) and (2) that intensifying the values of γc tends to devalue f′η; however, equivalent behaviour is anticipated for Williamson’s parameter λ. This is because after intensifying the curvature parameter γc, 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.Figure 1
Consequences forγc on f′η.Figure 2
Consequences forλ on f′η.


## 5.1.2. Attributes ofλN and Nr for f′η

The attribution ofλN and Nr is shown in Figures (3) and (4). From Figures (3) and (4), it defines that, by intensifying natural convection λN and buoyancy ratio Nr, the buoyancy force will cause the higher velocity attained by the fluid.Figure 3
Consequences forλN on f′η.Figure 4
Consequences forNr on f′η.


## 5.2. Temperature Profiles

### 5.2.1. Attributes ofγc and Pr for θη

The behaviour of temperature profileθη for distinct values of curvature parameter γc 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 θη. 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.Figure 5
Consequences forγc on θη.Figure 6
Consequences for Pr onθη.


### 5.2.2. Attributes ofTp and Bp for θη

The contribution of thermophoresis parameterTp and Brownian motion Bp is described in Figures (7) and (8). Evidently, larger Tp and Bp produce higher θη. 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 Bp in the result, random motion of the nanoparticles will be raised which causes intensifying in the temperature of the fluid.Figure 7
Consequences forTp on θη.Figure 8
Consequences forBp on θη.


## 5.2.1. Attributes ofγc and Pr for θη

The behaviour of temperature profileθη for distinct values of curvature parameter γc 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 θη. 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.Figure 5
Consequences forγc on θη.Figure 6
Consequences for Pr onθη.


## 5.2.2. Attributes ofTp and Bp for θη

The contribution of thermophoresis parameterTp and Brownian motion Bp is described in Figures (7) and (8). Evidently, larger Tp and Bp produce higher θη. 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 Bp in the result, random motion of the nanoparticles will be raised which causes intensifying in the temperature of the fluid.Figure 7
Consequences forTp on θη.Figure 8
Consequences forBp on θη.


## 5.3. Concentration Profiles

### 5.3.1. Attributes ofψη on γc

Figure9 exhibits the consequences of curvature parameter γc on nanoparticle concentration ψη. It is clear from the figure that intensifying curvature parameter γc 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.Figure 9
Consequences forγc on ψη.


### 5.3.2. Attributes of Lewis Number Le and Prandtl Number onψη

The behaviour of Lewis number Le is depicted forψη through Figure 10. It is scrutinized that ψη is diminished by increasing Lewis number Le. Physically, due to the levitation of diffusivity of heat and mass, nanoparticle concentration ψη diminished. Furthermore, scrutinization of the attribution of Prandtl number Pr for ψη 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.Figure 10
Consequences of Lewis number Le.Figure 11
Consequences of Prandtl number Pr.


### 5.3.3. Attributes ofTp and Bp on ψη

The impact of thermophoresis parameterTp and Brownian motion parameter Bp on ψη is depicted in Figures (12) and (13). Due to the increase of Tp, boundary layer thickness of nanoparticle concentration intensified. As a matter of fact, thermophoresis force increases for higher estimation of Tp, which yields nanoparticle relocations from higher to lower temperature; hence, ψη intensifies (see Figure 12). However, ψη and its concentration boundary layer diminish when Bp is amplified (see Figure 13).Figure 12
Consequences ofTp on ψη.Figure 13
Consequences ofBp on ψη.


## 5.3.1. Attributes ofψη on γc

Figure9 exhibits the consequences of curvature parameter γc on nanoparticle concentration ψη. It is clear from the figure that intensifying curvature parameter γc 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.Figure 9
Consequences forγc on ψη.


## 5.3.2. Attributes of Lewis Number Le and Prandtl Number onψη

The behaviour of Lewis number Le is depicted forψη through Figure 10. It is scrutinized that ψη is diminished by increasing Lewis number Le. Physically, due to the levitation of diffusivity of heat and mass, nanoparticle concentration ψη diminished. Furthermore, scrutinization of the attribution of Prandtl number Pr for ψη 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.Figure 10
Consequences of Lewis number Le.Figure 11
Consequences of Prandtl number Pr.


## 5.3.3. Attributes ofTp and Bp on ψη

The impact of thermophoresis parameterTp and Brownian motion parameter Bp on ψη is depicted in Figures (12) and (13). Due to the increase of Tp, boundary layer thickness of nanoparticle concentration intensified. As a matter of fact, thermophoresis force increases for higher estimation of Tp, which yields nanoparticle relocations from higher to lower temperature; hence, ψη intensifies (see Figure 12). However, ψη and its concentration boundary layer diminish when Bp is amplified (see Figure 13).Figure 12
Consequences ofTp on ψη.Figure 13
Consequences ofBp on ψη.


## 5.4. Attributes of Drag Force and Rate of Transmission of Heat

The attributes of physical parameters such as curvature parameterγc, Williamson’s parameter λ, natural convection λN, and buoyancy ratio Nr on drag force are exhibited in Table 1. It is evidently clear from the table that drag force is intensified by enhancing γc, λ, λN, and Nr. Table 2 expresses the behaviour of physical parameters on the rate of transmission of heat. It is inspected that the rate of transmission of heat enhances by increasing γc, Tp, and Pr, while it declines for inclination of λ, Bp, and Le.Table 1
Numerical values of drag forces1/2CfRe1/2.
γcλλNNrf″01/2CfRe1/20.10.10.10.11.72902.02790.2———1.76702.07920.3———1.80302.12810.4———1.83902.1772—1——1.13402.4200—2——0.92522.6372—3——0.84883.0102—4——0.78533.2521—10.1—1.22902.7394——0.2—1.24302.7880——0.4—1.28602.9398——0.7—1.35103.17620.530.10.10.90113.3370———0.20.90313.3499———0.40.90673.3730———0.70.90703.3749Table 2
Numerical values of the rate of transmission of heatNu/Re1/2.
γcλTpBpPrLeNu/Re1/20.10.10.10.10.30.40.81180.2—————0.84460.3—————0.8739—0.5————1.1190—0.7————0.9200—0.9————0.8702——0.3———1.0120——0.5———1.1430——0.7———0.7092——0.10.3——0.8221———0.5——0.7641———0.7——0.7092———0.10.5—1.1140———-0.7—1.3200————0.9—1.5120————0.30.50.8764—————0.60.8700—————0.80.8589


## 6. Final Remarks

This 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. The governing coupled nonlinear ODEs are then solved numerically by using MATLAB technique bvp4c. This research reveals the following outcomes:(i)
The velocity profile is declining due to upsurge in the curvature parameter and Williamson’s parameter(ii)
Inclination of the Prandtl number and curvature parameter causes decline in the temperature profile, whereas it is intensified by increments inγc, Bp, and Tp(iii)
The concentration of nanoparticles is dropped via amplifyingLe, Pr, and Bp(iv)
The greater value of Williamson’s parameter and curvature parameter causes the drag force to increase(v)
Rate of transmission of heat is reduced for larger Williamson’s parameter



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