# Effect of Nanoparticles on Wire Surface Coating Using Viscoelastic Third-Grade Fluid as a Coating Polymer inside Permeable Covering Die with Variable Viscosity and Magnetic Field

Authors: Zeeshan Khan; Ilyas Khan; N. Ameer Ahammad; D. Baba Basha; Mulugeta Andualem
Journal: Journal of Nanomaterials
 (2022)

---

## Abstract

Goal. The parameters of coated wire products are determined by momentum and heat transmission inside dies. As a consequence, it is essential to understand the polymerization movement, heat mass transmission, and wall stress concentration. The wire covering technique necessitates a boost in thermal efficiency. As a result, the goal of this study is to see how nanomaterials affect the heat and mass transfer mechanisms of third-grade liquid in wire coating analysis. The Buongiorno model is adopted for nanofluids. Methodology/approach. Continuity, momentum, energy, and nanoparticle volume fraction concentration is used to establish the governing equations. For highly nonlinear, the numerical methodology bvph2 technique is applied to yield numerical solutions. The impacts of the input parameters on motion, temperature, and volume fraction are examined using pictorial representations. Moreover, using the ND-solve, the numerical results are validated analytically. Findings. In Reynolds Modeling, the stress on the entire wire surface integrated shear forces at the surface dominate Vogel’s model, according to the analytical conclusions of this inquiry. It is observed that the nanomaterials appear to have a favorable impact on wire force throughout the entire surface and shear forces at the surface. The polymer velocity can be increased using a non-Newtonian parameter. The temperature profile is increased in the first half of the segment with larger values of random motion and nonlinear thermal while decreases in the later part. In addition, the Brownian motion component raises the concentration profile, but the thermophoresis factor decreases it. Practical implications. This research could aid in the advancement of wire coating technologies. Originality/value. For the first time, Brownian motion with generation/absorption slippage processes is used to investigate the importance of nanoparticles in wire coating assessment. Two different models are utilized for time-dependent viscosity: Reynolds and Vogel models.


---

## Body

## 1. Introduction

Studying and analysing non-Newtonian fluids is of significant curiosity together with theoretical and applied viewpoint [1, 2]. Fluid dynamics and material science awareness related to non-Newtonian fluid motion may have major ramifications in a variety of fields, including polymer preparation, protecting and lacquer, ink-jet printing, aerodynamics, homodynamics, turbulent shear stream, slurry and ingredient suspensions, and blood serum. As a result, there has been a lot of focus on these movements, and the bibliography has a lot of work on statistical, theoretical, and algebraic solutions on the subject [3, 4]. Furthermore, the mobility of such fluids poses significant problems to professionals from a variety of study domains, including computational models, economics, mathematics, and physics. In fact, when contrasted to Newtonian fluids, the equations postulated and produced for non-Newtonian concepts are far more difficult. The modelled equation of non-Newtonian fluids is highly nonlinear, making exact solutions extremely difficult to acquire [5–8]. It is also difficult to obtain extremely precise estimates for viscous liquids regarding the nonlinear and utterly irrelevant nature of the aggregation concept [5–8]. Many scholars have devised analytical and numerical techniques to solve these nonlinear problems for this goal.Wire coating (an extrusion method) is usually applied in the polymerization sector designed for insulating material and protection against mechanical injury. An uncovered warmed wire is immersed and pulled into the melting resin in this technique. Extruding the heated polymer across a rolling wire is also used for this operation. A conventional wire coating machine has five separate components: a pay-off device, a wire preheating instrument, an extrusion, a chilling system, and a pull device. Tubing-type die and pressurized die are the most popular dies used for wire treatments. The latter, which resembles an annulus, is typically utilized wire coating.As a result, flows through such dies resemble flows across an annular region established by a pair of concentric cylinders. The inner cylinder travels axially, while the exterior cylinder remains stationary. Many researchers [9–14] employed power-law and Newtonian frameworks to describe the rheological behavior of the emulsion polymerization flow in early studies. The wire sealant examination employing a pressure form die is presented in [15, 16]. Following then, [17–20] provided more scientific work on the subject. Mitsoulis [21] also provides a comprehensive overview of heat transmission and melted movement in wire veneer. Akter and Hashmi [22] investigated wire coating using a pressurized die. Later, using a cylindrical unit, Akter and Hashmi investigated melting flow throughout the wire coating processes [23, 24].Wire shell is an advanced production method for insulating and protecting wires from the environment. The immersion process, coaxial method, and electromagnetic deposit process are the three categories of wire covering. The dipping progression creates a significantly stronger relationship between the continuums, but it is somewhat sluggish in equated to the other two methods. Han and Rao [25] evaluated the issues associated with coating extrusion using a pressurized type die. The extrusion procedure is made up of three parts: the feeder entity, the chamber, then the cranium through a die. Kozan and Cirak [26] reported on the extensive deliberations of these three unique parts. In particular, Sajid et al. [27] used the HAM method to explain and handle the wire coating operations of Oldroyd 8-constant fluid. Likewise, Shah et al. [28] used the perturbation methodology to examine the wiring layer of viscoelastic third-grade fluid and presented their findings in [28].The Phan-Thein-Tanner (PTT) model, a third-grade non-Newtonian molten, is currently the utmost widely adopted wire coating prototype. Binding et al. [29] investigated the rising wire-coating procedure for viscoelastic fluids in a viscoelastic constitutive model. It also studied the flaws of the actual modeling technique. The wire coating study based on the pipe die was supplied by Multu et al. [30]. Kasajima and Ito [31] investigated the wire coating technique of polymer produced until then. They also focused on how heat transfer affects cooling coatings. Baag and Mishra [32] explored wire coating using the temperature linearly varying at the boundary. Similarly, two-layer coating was investigated by many researchers [33–37].For recent times, the issue of heat transfer improvement has gotten a lot of attention. Thermoelectric scientists have proposed that nanotube metallic or nonmetallic materials be added to conventional fluids to increase thermal properties since nanoparticles have better thermal conductivity than the base liquid. Nanofluid is the resultant mixture that has improved physicochemical properties. Aziz [38] developed first time the terminology of nanofluid in 1995. Furthermore, a special type of nanofluid known as hybrid nanofluid is studied to boost thermal efficiency. Some other related studies can be found in [39–53].This investigation was aimed at studying the effects of nanomaterials proceeding the MHD third-grade fluid in a pressurized sort die during the wire surface layer using Brownian motion in addition to heat conduction.Numerical solutions have been achieved via Runge-Kutta 4th order scheme [48–52]. Reynolds, as well as Vogel’s models, compensates for variable viscosity as well. Such an endeavor has still not been constructed to the aim to contribute. Before being attempted analytically, the relevant resulting equations are made dimensionless by suitable transformation factors. The effect of various parameters accessing the problem is investigated in two situations: (1) the Reynolds model and (2) Vogel’s model.


## 2. Modeling of the Problem

Figure1 depicts the flow problem’s geometry, wherein the wire is pulled within a cylinder fed with third-grade nanofluid as a coating substance. The electrical field is presented towards the fluid in a normal direction. Because of the low Reynolds number, the reto magnetic field is expected to be insignificant, which really is adequate in the experiment. The location of the continuum is believed to be concentrically situated. (Rw,θw,ϕw) and (Rd,θd,ϕd) are the radius, temperature, and volume fraction of the wire as well as die, compatibly.Figure 1
Two-phase coating geometry.Uw is also the speed of the said wire as it is inserted along the central path of the die. The emulsion polymerization flow should be axisymmetric, continuous, and homogeneous. Han and Rao [12] evaluate the velocity, additional stress tensor, heating rate, and volume fraction of nanomaterials:
(1)w⟶=0,0,wr,S=Sr,T=Τr.Substance to constraints(2)w=Uw,θ=θw,ϕ=ϕwatr=Rww=0,θ=θd,ϕ=ϕdatr=Rd.Regarding third-grade liquid, the stress tensorS is described as
(3)S=ηA1+α1A2+α2A1+τ1A2+τ2A1A2+A2A1+τ3trA2A1.The governing parameters that apply are as follows [5–12]:
(4)∇.w=0,(5)ρfDqDt=−∇p+F+J×B,(6)ρcpnfDθDt=k∇2θ+ϕ+ρcpnfDB∇θ.∇ϕ+DTθd∇θ.∇θ,(7)DϕDt=Db∇2ϕ+DTθd∇2θ.The parameters involve in the exceeding balances are well-defined in the nomenclature given at the end of the article.The electrical field is presented in a positive radially normal direction towards the wire, and the resultant magnetic force is believed to be insignificant. As a result, effective body force is determined by(8)J×B=0,0,−σB02w.The dissipation factor with tensor components are as regards:(9)Srz=μdwdr+2β2+β3dwdr3,(10)Srr=α2+2α1dwdr2,(11)Szz=α2dwdr2,(12)ϕ=μdwdr2+2β2+β3dwdr4.In light of the foregoing relationships, the equation of motion (5) yields(13)2β2+β3ddrrdwdr3+ηrddrrdwdr−σB02u=dpdz,(14)−2α2+α3ηrddrrdwdr=dpdr.The flow is caused by the pressure difference, as shown by expression (13). Because there is just pull of a wire after it leaves the die, the pressure difference in the axially is insignificant. As a result, the expression (13) can be reduced to
(15)2β2+β3ddrrdwdr3+ηrddrrdwdr−σB02u=0.In view of equation (10), the energy equation (7) becomes
(16)kd2θdr2+1rdθdr+μdwdr2+2β2+β3dwdr4+ρcpfDBdθdrdϕdr+DTθddθdr2=0,(17)DBd2dr2+1rddrϕ+DTθdd2θdr2+1rdθdr=0.The shear stress on the wire surface is calculated as follows:(18)Srzr=Rw=μdwdr+2β2+β3dwdr3r=Rw.The force acting on the die total wire exterior is as described in the following:(19)Fw=2πRwLSrzr=Rw.Furthermore, the Nusselt numberNw has the following definition:
(20)Nwr=rqwKθd−θw,where qd=−kdθ/drr=Rw is the heat flow at the wire’s surface. We propose to explore temperature dependent viscosity in this work, as previously stated. As a result, the two additional cases are investigated.


### 2.1. Case 1: Reynolds Model

Nondimensional viscosity is incorporated in the study as follows [12]:
(21)η=exp−βΩθ≈1−βΩθ,whereΩ is the Reynolds model parameter.In light of (21, 22), the expressions (15)–(20) should be read as follows (without the asterisks):
(22)r∗=rRw,w∗=wUw,β0=β2+β3,RdRw=δ>1,β∗=β0∗ηRw2μ0/Uw2,M=σB02Rw2μ0,Κ=Rw2VwΚ∗,θ∗=θ−θwθd−θw,Br=μ0Uw2kθd−θw,μ∗=μμ0,ϕ∗=ϕ−ϕwϕd−ϕw,Nb=DBρcpϕd−ϕwk,Nt=DTρcpfθd−θwθdk.Dimensionless velocity and heat equations with BCs ignoring asterisks are
(23)1−βΩθrd2dr2+ddrw+2β3rd2wdr2dwdr2+dwdr3−βΩrdθdrdwdr−Mwr=0,(24)d2dr2+1rddrθ+1−βΩθBrdwdr2+2Brβdwdr4+Nbdθdrdϕdr+Ntdθdr2=0,d2ϕdr2+1rdϕdr+NtNbd2dr2+1rddrθ=0′,w1=1,θ1=0,ϕ1=0,wδ=0,θδ=1,ϕδ=1.(25)Srzr=Rw=μSrzUwμ0Rwr=1=1−βΩθdwdr+2βdwdr3r=1,(26)Fw=Fw2πRwLr=1=1−βΩθdwdr+2βdwdr3r=1,(27)Nwr=−dθdrr=1,where M,Br,β,Nb, and Nt are the magnetic factor, Brinkmannnumber, non-Newtonianffactor, Brownianmmotion factor, and thermophoresis factor, respectively. The definition of each variable or parameter is given in Table 1.Table 1
Nomenclature.
A1, A2, A3: Kinematic tensorsB0: Magnetic field strengthDT: Thermophoretic diffusion coefficientL1: Velocity vectorJ: Current densityB: Magnetic fieldcp: Specific heatF: ForceL: Lengthk: Thermal conductivityBr: Brinkman numberDB: Brownian diffusion coefficientD, B, B∗: Vogel’s model viscosity parametersr: Radial directionw: Velocity componentM: Magnetic parameterNt: Thermophoresis parameterp: PressureS: Stress tensorUw: Velocity of the wireNb: Brownian motion parameterm: Reynolds model viscosity parameterq: Velocity field;T: Transpose of the matrixα1,α2, β1,β2,β3: Material constantsμ: Viscosityμ0: Reference viscosityθ; Temperature fieldϕ: Nanoparticle volume fraction fieldΦ: Dissipation functionσ: Electrical conductivityβ: non-Newtonian parameterΩ: Vogel’s based viscosity parameterρ: Densityz: Axial direction


### 2.2. Case 2: Vogel’s Model

In this case, the temperature-dependent viscosity is taken as(28)μ=μ0expHF+θ−θw.After using the expansion we have(29)μ=m1−HF2θ,where m=μ0expH/F−θw, and H and F are the viscosity parameters associated with Vogel’s model.So the nondimensional momentum and energy equations with boundary conditions omitting asteriks are(30)m1−HF2θrd2wdr2+dwdr+2β3rd2wdr2dwdr2+dwdr3−μΗF2rdθdrdwdr−Mwr=0,(31)d2θdr2+1rdθdr+m1−HF2θBrdwdr2+2Brβdwdr4+Nbdθdrdϕdr+Ntdθdr2=0,(32)d2ϕdr2+1rdϕdr+NtNbd2dr2+1rddrθ=0,(33)w1=1,θ1=0,ϕ1=0,wδ=0,θδ=1,ϕδ=1.Also equations (26) and (27) become
(34)Srzr=Rw=SrzUwμ0Rwr=1=1−βmθDF2θdwdr+2βdwdr3r=1,(35)Fw=Fw2πRwLr=1=1−βmθDF2θdwdr+2βdwdr3r=1.


## 2.1. Case 1: Reynolds Model

Nondimensional viscosity is incorporated in the study as follows [12]:
(21)η=exp−βΩθ≈1−βΩθ,whereΩ is the Reynolds model parameter.In light of (21, 22), the expressions (15)–(20) should be read as follows (without the asterisks):
(22)r∗=rRw,w∗=wUw,β0=β2+β3,RdRw=δ>1,β∗=β0∗ηRw2μ0/Uw2,M=σB02Rw2μ0,Κ=Rw2VwΚ∗,θ∗=θ−θwθd−θw,Br=μ0Uw2kθd−θw,μ∗=μμ0,ϕ∗=ϕ−ϕwϕd−ϕw,Nb=DBρcpϕd−ϕwk,Nt=DTρcpfθd−θwθdk.Dimensionless velocity and heat equations with BCs ignoring asterisks are
(23)1−βΩθrd2dr2+ddrw+2β3rd2wdr2dwdr2+dwdr3−βΩrdθdrdwdr−Mwr=0,(24)d2dr2+1rddrθ+1−βΩθBrdwdr2+2Brβdwdr4+Nbdθdrdϕdr+Ntdθdr2=0,d2ϕdr2+1rdϕdr+NtNbd2dr2+1rddrθ=0′,w1=1,θ1=0,ϕ1=0,wδ=0,θδ=1,ϕδ=1.(25)Srzr=Rw=μSrzUwμ0Rwr=1=1−βΩθdwdr+2βdwdr3r=1,(26)Fw=Fw2πRwLr=1=1−βΩθdwdr+2βdwdr3r=1,(27)Nwr=−dθdrr=1,where M,Br,β,Nb, and Nt are the magnetic factor, Brinkmannnumber, non-Newtonianffactor, Brownianmmotion factor, and thermophoresis factor, respectively. The definition of each variable or parameter is given in Table 1.Table 1
Nomenclature.
A1, A2, A3: Kinematic tensorsB0: Magnetic field strengthDT: Thermophoretic diffusion coefficientL1: Velocity vectorJ: Current densityB: Magnetic fieldcp: Specific heatF: ForceL: Lengthk: Thermal conductivityBr: Brinkman numberDB: Brownian diffusion coefficientD, B, B∗: Vogel’s model viscosity parametersr: Radial directionw: Velocity componentM: Magnetic parameterNt: Thermophoresis parameterp: PressureS: Stress tensorUw: Velocity of the wireNb: Brownian motion parameterm: Reynolds model viscosity parameterq: Velocity field;T: Transpose of the matrixα1,α2, β1,β2,β3: Material constantsμ: Viscosityμ0: Reference viscosityθ; Temperature fieldϕ: Nanoparticle volume fraction fieldΦ: Dissipation functionσ: Electrical conductivityβ: non-Newtonian parameterΩ: Vogel’s based viscosity parameterρ: Densityz: Axial direction


## 2.2. Case 2: Vogel’s Model

In this case, the temperature-dependent viscosity is taken as(28)μ=μ0expHF+θ−θw.After using the expansion we have(29)μ=m1−HF2θ,where m=μ0expH/F−θw, and H and F are the viscosity parameters associated with Vogel’s model.So the nondimensional momentum and energy equations with boundary conditions omitting asteriks are(30)m1−HF2θrd2wdr2+dwdr+2β3rd2wdr2dwdr2+dwdr3−μΗF2rdθdrdwdr−Mwr=0,(31)d2θdr2+1rdθdr+m1−HF2θBrdwdr2+2Brβdwdr4+Nbdθdrdϕdr+Ntdθdr2=0,(32)d2ϕdr2+1rdϕdr+NtNbd2dr2+1rddrθ=0,(33)w1=1,θ1=0,ϕ1=0,wδ=0,θδ=1,ϕδ=1.Also equations (26) and (27) become
(34)Srzr=Rw=SrzUwμ0Rwr=1=1−βmθDF2θdwdr+2βdwdr3r=1,(35)Fw=Fw2πRwLr=1=1−βmθDF2θdwdr+2βdwdr3r=1.


## 3. Numerical Procedure and Validations of the Method

In several physical problems, the consequent differential equations are significantly nonlinear. It is challenging for investigators and scientists in computing analytical or numerical approaches to such situations. For estimating the numerical solution of nonlinear partial and ordinary differential equations, the RK4 method is one of the most successful computational methods. For the numerical analysis, Runge-Kuta fourth order method is used built in package in MATHLAB SOFTWARE by taking step size∆η=0.01. The existing work’s calculation is determined by the comparison between existing and published work [28] as given in Table 2. The iteration procedure was stopped until all of the nodes in the η-direction met the convergence condition 10-5. Additionally, the HAM method is also applied for confirmation of the method as given in Figures 2(a) and 2(b).Table 2
Numerical comparison of HAM, RK4 Methods, and published work [28].
rBvph2ND-solvePublished work1.01111.20.573523650.573523550.573523551.40.403254910.403254800.403254911.60.321093230.321093220.321093211.80.210362710.210366010.210362702.000.0131∗10-210.0020∗-25Figure 2
(a) Comparison of RK4 and HAM methods. (b) Comparison of bvph2 and ND-solve methods.
(a)(b)The Runge-Kutta-Fehlberg strategy is used to solve the multidegree differential equation system specified in equations (23 and 24) and (31–33), for this purpose, following transformations are applied:(36)ς1=w,ς2=w′,ς3=θ,ς4=θ′,ς5=ϕandς6=ϕ′.As a result, we get the following.(37)ζ1′=ς2,ς2′=Mς1r+βΩrς2ς4+βΩς3−1ς2−2βς22r1+M1+6βς23−βΩς3,ς3′=ς4,ς4′=−1rς4+Br1−βΩς3ς22+2βΩBrς24+Nbς4ς6+Ntς42,ς5′=ς6,ς6′=−1rς6+NtNbς4′+1rς4.Transferred boundary conditions are(38)ς11=1,ς21=α1,ς31=0,ς41=α2,ς51=0,ς61=α3.The best guess estimates for the uncertaintiesα1,α2, and α3 are determined, and afterwards, the shooting mechanism is used to determine them.


## 4. Results and Discussion

For the two scenarios, RM and VM, the inspiration of essential factors on speed, heat, and nanoparticle concentration outlines is explored in the occurrence and nonappearance of attractive field. The shear stress happening in the superficial of the total wire and the size of the Nusselt number on the external are estimated both for Reynolds and Vogel’s model situations. The shear force happening in the surface of the total wire is proportional tow′(r), as shown by equations (26), (27), (34), and (35). As a result, shear tress on a total wire surface has the same characteristic as w′1.Figures3–5 show the effect of m, viscosity factor on the velocity profile (wr), temperature profile (θr), and concentration profile (ϕr) distributions for Reynolds model, respectively. The higher values of m indicate an upsurge in velocity field but decreases in heat and concentration eprofiles. Because increasing the variable viscosity component lowers the bulk viscosity, the velocity of the fluid rises as a consequence, and the temperature and concentration profile decreases. Both the existence and absence of magnetism produce the same descriptive trend as shown in Figures 3 to 5. It is price mentioning that the in fallouts of the current study, the movement and thermal measurements match those of Shah et al. [28] study on the effect of the friction factor.Figure 3
Consequence ofm on velocity for RM.Figure 4
Consequence ofm on temperature in RM.Figure 5
Consequence ofm on concentration in RM.Figures6–8 show a pictorial representation of the variances of Ω scheduled velocity, temperature, and concentration outlines. Figure 6 shows that the fluid velocity grows in the constituency 1≤r<1.5, but this behavior observed retreat in the lasting of the region. It is observed from graphs 7 and 8 that the temperature profile increases while the concentration profiles decrease both in the presence and absence of magnetic field. Furthermore, while comparing the effects of the Reynolds model case and Vogel model case on the velocity field, we revealed that the melting velocity across the die enhances for the Reynolds model but is constrained somewhat for the Vogel model, especially near the die boundary.Figure 6
Consequence ofΩ on velocity in VM.Figure 7
Consequence ofΩ on temperature in VM.Figure 8
Consequence ofΩ on concentration in VM.Figures9 and 10 show the variation of temperature profiles for various values of Nb. It is observed that the heat transfer inside the die increases with the increasing vales of Nb. It is also investigated that the temperature profiles increase in the region 1≤r<1.4 for Reynolds and Vogel situations; however, the behavior in the rest of the state is the total opposite. Furthermore, in the RM situation, the temperature field, θr, overcomes the Vogel case. With increasing Nb, the stochastic collision among nanoparticles and liquid molecules increases, causing a flow to become heated and the nanoparticle’s concentration field to decrease (see Figures 11 and 12). Furthermore, the magnetism has no discernible effect on the ϕr field at any location on the die.Figure 9
Consequence ofNb on temperature in RM.Figure 10
Consequence ofNb on temperature in VM.Figure 11
Consequence ofNb on concentration in RM.Figure 12
Consequence ofNb on concentration in VM.As shown in Figures13 and 14, the significance of Nt on heat transfer rate is similar to that of Nb. The convective heat transfer force is a force that causes nanomaterials to spread into the surrounding fluid as a conclusion of a temperature difference. The enhancement of thermophoretic force causes nanoparticles to transfer deeper into the polymer. As a result, the temperature distribution increases fashionable nearly half of a domain. Figures 15 and 16 show that the concentration profile declines as Nt increases. This is true in both cases, i.e., in the presence and absence of a magnetic field for Reynolds and Vogel models.Figure 13
Consequence ofNt on temperature in RM.Figure 14
Consequence ofNt on temperature in VM.Figure 15
Consequence ofNt on concentration in RM.Figure 16
Consequence ofNt on concentration in VM.Figures17 and 18 show the effect of viscoelastic parameter β on velocity profile (wr) both for Reynold and Vogel models, respectively. In non-Newtonian parameter, denominator contains rheological properties. So, the fluidity of the polymers decreases with increasing β. As a consequence, the melting polymer moment increases as the non-Newtonian parameter increases. However, the effect of β on velocity filed is additionally noticeable in the Reynolds model as compare to Vogel model. The non-Newtonian feature implies that the coating polymer movement can be increased.Figure 17
Consequence ofNt on velocity in RM.Figure 18
Consequence ofβ on velocity in VM.The influence ofBr upon θr viscosity is seen in Figures 19 and 20 for the Reynolds and Vogel’s model models, respectively. A larger amount of Br enhances the θr profile. Brinkman number denotes the relative value of viscous heating by conduction of heat. Furthermore, in Vogel’s case, the thermal profile varies substantially more for Br than in the RM case. This is validated by the results of studies published by Shah et al. [28].Figure 19
Consequence ofBr on temperature in RM.Figure 20
Consequence ofBr on temperature in VM.Figures21–23 show the effect of β, M, Br, Nb, and Nt on θ′1 designed for RM and VM case. It is observed from Figures 21(a) and 21(b) that θ′1 is larger with greater values of β and decreases in increasing M in VM case. Figures 22(a), 22(b) and 23 indicate the effect of Br, Nb, and Nt on θ′1 for VM and RM. It is observed that θ′1 decreases for Br, Nb, and Nt. This is true in both RM and VM scenarios. Additionally, in the situation of RM, force at the surface of the total wire plus shear and force at the surface area are greater than VM.Figure 21
Impact ofβ and M on −θ′1 in VM.
(a)(b)Figure 22
Impact ofBr and Nb on −θ′1 in RM.
(a)(b)Figure 23
Impact ofNt on θ′1 in RM case.


## 5. Concluded Remarks

Regarding RM and VM situations, the significance of temperature-dependent-viscosity in hydromagnetic heat/mass molecular diffusion of third-grade fluid through nanoparticle concentration is investigated. Variable viscosity has a significant impact on all fluid flow. Viscosity influences can efficiently control the heat transport of resin in a die. For bigger values of random motion and thermal radiation, the temperature gradient is enhanced in the first quarter of the section, but negative behavior occurs in the second half. Furthermore, the Brownian motion factor increases the concentration profile, but the thermophoresis factor shows a decrease. It is perceived that when the viscosity factor increases, the polymer melt flow increases, but the heat as well as concentration profile decreases. Because of increasing the variable viscosity component lowers the bulk viscosity, as a consequence, the fluid velocity rises whereas the temperature and concentration fields decrease. From this study, it is also analyzed that fluid velocity grows in the constituency1≤r<1.5, but this behavior observed retreat in the lasting. It is also investigated that the temperature profile increases while the concentration profiles decrease both in the occurrence and deficiency of magnetic field. The influence of β on velocity is additional perceptible in the Reynolds model than in the Vogel model. The non-Newtonian feature implies that the coating polymer movement can be increased. A larger amount of Br enhances the θr profile since the Brinkman number denotes the relative value of viscous heating by conduction of heat. In Vogel’s model, the thermoelectric field varies more strongly than in the Reynolds models case. In RM, the force happening on the total surface of wire and shear stress at the surface is greater than the VM. When RM prevails over VM, the influence of nanomaterials is positive for force on the entire wire and shear forces at the surface.



---

*Source: 1111657-2022-05-06.xml*