# Couple Stress Hybrid Nanofluid Flow through a Converging-Diverging Channel

Authors: Malik Zaka Ullah; Dina Abuzaid; M. Asma; Abdul Bariq
Journal: Journal of Nanomaterials
 (2021)

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## Abstract

This research work is aimed at scrutinizing the mathematical model for the hybrid nanofluid flow in a converging and diverging channel. Titanium dioxide and silver are considered solid nanoparticles while blood is considered as a base solvent. The couple stress fluid model is essentially used to describe the blood flow. The radiation terminology is also included in the energy equation for the sustainability of drug delivery. The aim is to link the recent study with the applications of drug delivery. It is well-known from the available literature that the combination ofTiO2 with any other metal can vanish more cancer cells than TiO2 separately. Governing equations are altered into the system of nonlinear coupled equations the similarity variables. The Homotopy Analysis Method (HAM) analytical approach is applied to obtain the preferred solution. The influence of the modeled parameters has been calculated and displayed. The confrontation to wall shear stress and hybrid nanofluid flow growth as the couple stress parameter rises which improves the stability of the base fluid (blood). The percentage (%) increase in the heat transfer rate with the variation of nanoparticle volume fraction is also calculated numerically and discussed.


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## Body

## 1. Introduction

The flow of fluids in converging/diverging channels has particularly significant applications in science and technology, such as flows in cavities and channels. The converging/divergent channels also relate to the blood flow in the arteries and capillaries. The stretching converging and diverging channels are also very significant to the blood flow due to the occurrence of stress effects. The researcher has worked in the same model for other industrial applications. Sheikholeslami et al. [1] demonstrated the effect of nanoparticles considering Jeffery fluid. Turkyilmazoglu [2], Dogonchi and Ganji [3], Xia et al. [4], and Mishra et al. [5] have considered the same model for the fluid flow using the concept of shrinking/stretching in converging/diverging channels.Nanotechnology has refined and expanded the horizons of today’s scientific world owing to its unpredicted results occurring in the field of energy, biotechnology, drugs, and therapeutics. It has also been demonstrated that stenosis is a damaging and potentially fatal disease, so researchers attempted to eliminate the problem using nanotechnology. Researchers believe that nanotechnology can deliver innovation in treating these kinds of problems since nanoparticles can pass through tissues and cells. Following that, there is a noticeable increase in research related to the advanced progress of nanoparticles in drugs [6–9].Shahzadi and Bilal [10] pioneered nanoparticles by revealing their dynamic and abnormal properties. Nadeem and Ijaz [11] described the use of nanoparticles to transport blood through a stenosis artery with a permeable wall. Ellahi et al. [12] reported blood flow to arteries consisting of the composite when nanoparticles were used. Nadeem and Ijaz [13] studied the effect of nanoparticles on stenotic artery hemodynamics and found them to be very helpful in reducing wall pressure with a shear rate.There is dispersion of more nanoparticles with different thermophysical properties from hybrid nanofluids that have attracted researchers because they are widely used in the fields of energy and medicine [14]. The case of bionanotechnology, which is a renovation and an open and innovative horizon in medicine, is one of the most auspicious applications of hybrid fluids. Numerous studies have demonstrated the effectiveness of nanoparticles in tumor targeting, therapy, and diagnosis process. Many studies have shown how effective nanoparticles are in tumor targeting, diagnosis, and treatment. It should be noted that nanoparticles have eliminated some of the shortcomings of traditional chemotherapy [15]. Liu et al. [16] investigated the use of Pt/TiO2 and Au/TiO2 nanocomposites, which are useful for cancer cell treatment. It was observed that the combination of TiO2 with any other metal can vanish more cancer cells than TiO2 separately. Silver has a wide range of biomedical uses due to its exclusive properties. The product containing silver is usually used for antimicrobial activity versus a broad spectrum of microorganisms. Moreover, experimental data suggest that Ag nanoparticles are a more ecological and biocompatible substitute to standard anticancer medicines [17].Blood, the most important biological fluid, is a liquid composed of various cell types suspended in a matrix of aqueous fluid (the plasma). It should be noted that red blood cells in plasma contribute to rotary motion in the occurrence of a velocity gradient. Body tissues have an angular gyration moment as well as an angular orbital moment. As a result, blood may be assumed a non-Newtonian fluid with a constant density. Stokes’ theory is one of several polar fluid theories that take into consideration [18].Couple stress fluid applications in biological problems are gaining popularity, and they are critical from both a theoretical and practical standpoint. Blood flow can be controlled with adequate couple stress. The theory of couple stress is first time introduced by Stokes [19] in the blood flow and claimed that blood is very reasonably flowing in the vessels due to the occurrence of the couple stresses. Devakar and Iyengar [20] suggested using a couple-stress term to regulate blood flow through the human system. Similarly, the idea was further extended by Devakar and Iyengar using the isothermal conditions and have found the exact solution. Recently, Saeed et al. [21], Ahmad et al. [22], and Gul et al. [23, 24] have used the couple stress fluid terminology in the hybrid nanofluids for drug transport and medication. They have also studied the heat transfer enhancement effect on the blood flow in various geometries.In the light of the above discussion, the novelty of this study is highlighted as follows:(i)
According to the best of the author’s knowledge, no one has tried to investigate the flow through a converging/diverging stretchable/shrinkable channel with blood as the base fluid andTiO2‐Ag as nanoparticles(ii)
This article examines a suitable background of couple stress hybrid nanofluid flow through converging/diverging stretchable/shrinkable channels(iii)
Heat absorption/omission and thermal radiation terminologies also strengthen the novelty of the work(iv)
The system of equations is then analytically solved by HAM(v)
The statistical analysis is also performed and presented through bar charts


## 2. Formulation

Assume the steady, laminar, incompressible, and couple stressTiO2‐Ag hybrid nanofluid, while the fluid motion is caused by the thermal radiation and a source or sink among the binary contracting/expanding channel, such that 2α is the angle between them. The walls of the channel are also assumed to be stretchable along the radial direction.
(1)u=sr=uw.Here,u=ur,θ,s stands for the velocity of the hybrid nanofluids and extending/contracting phenomena, respectively. The conditions α>0,α<0 are used to show that the channels are divergent and convergent correspondingly. The velocity for the fluid motion is the function of both r,θ. The couple stress terminology is imposed to the flow field whereas the other assumptions of [3–5] are used; the basic constituent dimensional equations of the hybrid nanofluid are taken into account.
(2)ρhnfur+∂u∂r=0,(3)ρhnfu∂u∂r+∂P∂r=μhnf∂∂rur+∂u∂r+1r2∂2u∂θ2−η0∂4u∂r4,(4)∂P∂θ−2μhnf1r2∂u∂θ=0,(5)ρCphnfu∂T∂r=khnf1r∂∂rr∂T∂r+1r2∂2T∂θ2−1r1r∂∂θqθ,rad+∂∂rrqr,rad.The pressure of fluid, electromagnetic field, and radiative heat flux are presented byP,B0,qr,rad,qθ,rad.The radiation terms are further written as(6)qθ,rad=−16σ∗T033kf∗∂T∂θ,(7)qr,rad=−16σ∗T033kf∗∂T∂r.Here,knf∗andσ∗are the absorption term and Stefan-Boltzmann constants.Putting the values of equation (7) into equation (5), we have
(8)ρCphnfu∂T∂r=khnf+16σ∗T∞33k∗f1r∂∂rr∂T∂r+1r2∂2T∂θ2.In the above equations,η0 is the couple stress term; also, ρhnf,μhnf,ρCphnf,andkhnf represent density, viscosity, density, and specific heat; the thermal conductivity of the hybrid nanofluids such that hnf stands for hybrid nanofluid.


### 2.1. Properties of the Materials

Initially, nanoparticles (titanium) are dispersed in the bloodstream (base fluid) to produce one (mono-nano fluid). (Silver) is then distributed as an additional nanoparticle to form the (hybrid nanofluid). On this occasion,TiO2 represents (titanium dioxide nanomaterial) and silver (Ag nanoparticles) and subscript f describes blood (base fluid). In Tables 1 and 2, ϕ1 and ϕ2 state the volume fraction of TiO2 and Ag nanoparticles, where ϕ1=ϕ2=0 refers to the base fluid.Table 1
Properties ofTiO2 and blood nanofluid [18].
Viscosityμnf=μf/1−ϕ12.5Densityρnf=ρf1−ϕ1+ρfϕ1ρTiO2Specific heatρCpnf=ρCpf1−ϕ1+ϕ1ρCpTiO2Thermal conductivityknf=kfkTiO2+2kf+2ϕ1kf−kTiO2−1kTiO2+2kf−2ϕ1kf−kTiO2Table 2
Various thermophysical properties ofTiO2‐Ag are stated as [18].
Viscosityμhnf=μf1−ϕ1−2.51−ϕ2−2.5Densityρhnf=1−ϕ2ϕ1ρTiO2+1−ϕ1ρf+ϕ2ρTiO2Specific heatρCphnf=ρCpf1−ϕ21−ϕ1+ϕ1ρCpTiO2+ϕ2ρCpAgThermal conductivitykhnf=kfkTiO2+2knf+2ϕ2knf−kTiO2−1kTiO2+2knf−2ϕ2knf−kTiO2×kAg+2kf−2ϕ1kf−kAg−1kAg+2kf−2ϕ1kf−kAg


### 2.2. Initial and Boundary Conditions

The auxiliary conditions at boundaries are(9)u=r−1uc,∂T∂θ=u∂T∂r=0,atθ⟶0,r≠0u=r−1s=uw,T=r−2Twasθ⟶±α.


### 2.3. Introduction of Nondimensional Variables

In the case of the radial flow, equation (1) reduced to
(10)Fθ=rur,θ.The nondimensional transformation is defined as(11)Fη=uc−1Fθ,Θη=r2TTw−1,η=θα−1.The use of (10) and (11) and thermophysical properties alter equations (3)–(5) in the simplified form as
(12)F‴+2αReρhnfρfμfμhnfFF′+4α2F′−24α2k∗F′=0,(13)khnfkf+RdΘ″+α2ρCphnfρCpf2PrFΘ+4+4RdΘ=0.The simplified form of the physical conditions are stated as(14)F=1,Θ′=F′=0,atη=0F=λ,Θ=1,atη=±1.Here,λ=s/uc>0 is the stretching parameter, λ=s/uc<0 is a shrinking parameter, Rd=16σ∗T3∞/3kfk∗f is radiation parameter, Re=rαuc/υf is the Reynolds number, Pr=μCpf/kf is the Prandtl number, and k∗=η0/μr2 is the couple stress parameter.


### 2.4. Drag Force and Heat Transfer Rate

The significant parameters of curiosity are defined as(15)uc2ρfCf=μhnf1r∂u∂θθ=±α,(16)TwNu=16σ∗T3∞3kfk∗f+khnfkf∂T∂θθ=±α.The alteration used for equation (16) and the simplified form is attained as
(17)Cf=1Reμhnfμff′±1,Nu=1αRd+khnfkfΘ′±1.


## 2.1. Properties of the Materials

Initially, nanoparticles (titanium) are dispersed in the bloodstream (base fluid) to produce one (mono-nano fluid). (Silver) is then distributed as an additional nanoparticle to form the (hybrid nanofluid). On this occasion,TiO2 represents (titanium dioxide nanomaterial) and silver (Ag nanoparticles) and subscript f describes blood (base fluid). In Tables 1 and 2, ϕ1 and ϕ2 state the volume fraction of TiO2 and Ag nanoparticles, where ϕ1=ϕ2=0 refers to the base fluid.Table 1
Properties ofTiO2 and blood nanofluid [18].
Viscosityμnf=μf/1−ϕ12.5Densityρnf=ρf1−ϕ1+ρfϕ1ρTiO2Specific heatρCpnf=ρCpf1−ϕ1+ϕ1ρCpTiO2Thermal conductivityknf=kfkTiO2+2kf+2ϕ1kf−kTiO2−1kTiO2+2kf−2ϕ1kf−kTiO2Table 2
Various thermophysical properties ofTiO2‐Ag are stated as [18].
Viscosityμhnf=μf1−ϕ1−2.51−ϕ2−2.5Densityρhnf=1−ϕ2ϕ1ρTiO2+1−ϕ1ρf+ϕ2ρTiO2Specific heatρCphnf=ρCpf1−ϕ21−ϕ1+ϕ1ρCpTiO2+ϕ2ρCpAgThermal conductivitykhnf=kfkTiO2+2knf+2ϕ2knf−kTiO2−1kTiO2+2knf−2ϕ2knf−kTiO2×kAg+2kf−2ϕ1kf−kAg−1kAg+2kf−2ϕ1kf−kAg


## 2.2. Initial and Boundary Conditions

The auxiliary conditions at boundaries are(9)u=r−1uc,∂T∂θ=u∂T∂r=0,atθ⟶0,r≠0u=r−1s=uw,T=r−2Twasθ⟶±α.


## 2.3. Introduction of Nondimensional Variables

In the case of the radial flow, equation (1) reduced to
(10)Fθ=rur,θ.The nondimensional transformation is defined as(11)Fη=uc−1Fθ,Θη=r2TTw−1,η=θα−1.The use of (10) and (11) and thermophysical properties alter equations (3)–(5) in the simplified form as
(12)F‴+2αReρhnfρfμfμhnfFF′+4α2F′−24α2k∗F′=0,(13)khnfkf+RdΘ″+α2ρCphnfρCpf2PrFΘ+4+4RdΘ=0.The simplified form of the physical conditions are stated as(14)F=1,Θ′=F′=0,atη=0F=λ,Θ=1,atη=±1.Here,λ=s/uc>0 is the stretching parameter, λ=s/uc<0 is a shrinking parameter, Rd=16σ∗T3∞/3kfk∗f is radiation parameter, Re=rαuc/υf is the Reynolds number, Pr=μCpf/kf is the Prandtl number, and k∗=η0/μr2 is the couple stress parameter.


## 2.4. Drag Force and Heat Transfer Rate

The significant parameters of curiosity are defined as(15)uc2ρfCf=μhnf1r∂u∂θθ=±α,(16)TwNu=16σ∗T3∞3kfk∗f+khnfkf∂T∂θθ=±α.The alteration used for equation (16) and the simplified form is attained as
(17)Cf=1Reμhnfμff′±1,Nu=1αRd+khnfkfΘ′±1.


## 3. Solution Methodology

The series solution is one of the valued methods to handle nonlinear problems. Nonlinear problems usually arise in the field of science and engineering. HAM is one of the latest and fast convergence techniques and is frequently used in the solution of nonlinear and coupled equations. The BVPh 1.0 and BVPh 2.0 are the latest packages of HAM that enhance the convergence of the proposed problems. These packages are very helpful in the rapid convergence, and one can use the BVPh 2.0 package up to the 100th iterations easily. The idea of HAM was first introduced by Liao [25]. The idea is further improved by the same author by introducing the new packages [26]. These packages are frequently used like [27–32].The feedback problem (12)–(18) was resolved by the HAM-BVPh 2.0 technique. The estimate of the iterations is utilized up to the 30th order. The trial solution or initial solution is required for the HAM solution. The zeroth-order solution is obtained as
(18)F0η=1−η21−λ&Θ0η=1.Equations (12)–(14) are set under the planned packaging and presented as
(19)ƛpF=1p+1∑k=1pΦF∑j=1lFηη=kjp2,ƛpΘ=1p+1∑k=1p∑j=1lFηη=kjpΦΘ∑j=1lΘηη=kjp2.The sum of the two components in the form of square residual errors is displayed as(20)ƛpTotal=ƛpF+ƛlpΘ.The numerical results of the converging parameter are obtained as(21)0.130021≤hf≤−1.203417,0.120432≤hθ≤−0.8992310.The range of convergence control parameters is used to find out the physical and numeric results.


## 4. Results and Discussion

The flow of the blood-based hybrid nanofluid consisting ofTiO2andAg has been considered in the converging and diverging channel. The heat transfer mechanism and medication are the main purposes of the proposed model. The main finding of the obtained results is shown physically and numerically. The geometry of the problem and convergence controlling sketches are demonstrated in Figures 1(a) and 1(b). The thermophysical properties of the materials are presented in Tables 1–3.Figure 1
(a) The geometry of the problem and (b) the HAM method.
(a)(b)Table 3
Numerous thermophysical properties are defined as [18].
Solid material and base fluidcpJ/kgKkW/mKρKg/m3TiO2 (titanium dioxide)686.28.9544250Silver:Ag23542910500Blood35940.4921063Table4 shows the assessment of the current work with the available literature and the closed agreement to authenticate the validation of the problem. The drag force on the upper and lower walls is calculated for the embedding parameters and demonstrated in Table 5. The accumulative growth in the values of the constraints is used to keep the convergent range of the proposed problem. The drag force rises with the increment in these parameters (ϕ1,ϕ2,Re,andk∗) for both nanofluids and hybrid nanofluids. The calculated increase shows that the resistive force is more effective by using the hybrid nanofluid TiO2+Ag at both the lower and upper walls of the channels. Furthermore, the friction force is efficiently working in the converging channel as compared to the other one.Table 4
Comparison between the present work with previous work considering common parameters only.
ReF″±1,α=503F″±1α=504F″±1α=505F″±1α=50PresentF″±1,α=−503F″±1α=−504F″±1α=−505F″±1α=−50Present11.864201.864311.864121.867010.774200.774320.774110.7770321.886441.886521.886311.889120.795310.795420.795200.7982131.904221.904341.904121.9070.802140.802230.802050.80501Table 5
Influence of parameter versus−ReCf.
ϕ1,ϕ2Rek∗−ReCfα>0TiO2+AgFη−ReCfα<0TiO2+Ag−ReCfα<0TiO20.000.10.10.395860.373691.294951.275310.010.418710.406511.271221.245310.010.4349320.4137311.287121.26610.20.496380.474161.385421.363210.40.5974930.5752941.496151.424020.20.467350.445131.356211.130320.40.5154010.5040211.404871.38612The heat transfer rate is calculated numerically using the embedded parameters, and the results are exhibited in Table6. The augmentation in the values of the parameters Rd,ϕ1,andϕ2 progresses the heat transfer rate ultimately. The attained results show that the heat transfer rate is more immediate by using the TiO2+Ag hybrid nanofluids. The heat transfer rate stimulates fluid motion by controlling the viscous effect. The TiO2 material works as the treatment material in cancer therapy while the stability in the blood is controlled through silver. The (%) wise increase in the heat transfer rate versus the nanoparticle volume fraction has been calculated and displayed in Table 7. The hybrid nanofluid improves the heat transfer analysis as compared to the other traditional fluids.Table 6
Nusselt numberNux versus physical parameters.
Rdϕ1,ϕ2−NuTiO2&Agα>0−NuTiO2α<0−NuTiO2&Agα>0−NuTiO2α<00.20.019.415719.3310211.3925311.291230.49.473219.3732111.5032011.321340.69.536319.4124111.6310311.531610.029.446459.2460911.4570811.381630.039.654359.5847911.7879011.67849Table 7
% analysis versusNux.
ϕ1,ϕ2−Nuα>0TiO2&Ag%α>0−Nuα>0TiO2%α>0−Nuα<0TiO2&Ag%α<0−Nuα<0TiO2%α<00.09.23445……9.23445…….11.14515……11.14515…….0.019.418711.9979.334020.90911.395532.24911.304231.4290.029.524033.1389.42612.07811.545843.59811.443632.680.039.634424.3349.505112.93311.687214.86611.576233.870.049.748325.5689.616124.13611.806285.93511.681984.82Figures1–4 describe the influence of the physical parameters ϕ1,ϕ2,Re,k∗, on the velocity Fη considering both converging and divergent channels. The parts (a, b) and (c, d) of each figure show the same effect in 2D and 3D expressions. The parameters ϕ1,ϕ2 decline the fluid motion Fη for its higher values using the extending/convergent and contracting/divergent channels as revealed in Figures 2(a)–2(d). The nanoparticle dispersion in the base fluid enhances the viscous effect of the base solvent and improves the cohesive forces among the fluid molecules to resist the fluid motion.Figure 2
Fη versus ϕ1,ϕ2 in stretching/convergent circumstance with α=−5∘,5∘.
(a)(b)(c)(d)Figures3(a)–3(d) show the effect of Re on Fη in the case of extending/convergence and narrowing/diverging. It can be witnessed that the increases in Reynolds number cause an increase in extending/convergent case, which can be observed from Figures 3(a) and 3(b) while a reverse result is obtained in the contracting/divergent case because the growth in Reynolds number causes a decline in the fluid motion, in this case, that can be seen in Figures 3(c) and 3(d). Figures 3(a)–3(d) show the effect of Re on Fη in the case of extending/convergence and narrowing/diverging. It can be witnessed that the increases in Reynolds number cause an increase in extending/convergent case, which can be observed from Figures 3(a) and 3(b) while a reverse result is obtained in the contracting/divergent case because the growth in Reynolds number causes a decline in the fluid motion, in this case, that can be seen in Figures 3(c) and 3(d).Figure 3
(a–d)Fη versus Re stretching/convergent circumstance with α=−5∘,5∘.
(a)(b)(c)(d)Figures4(a)–4(d) indicate the influence of k∗ on Fη in extending/convergent and contracting/divergent cases. The fluid motion decays as increasing the value of k∗, that is, fluid motion decline with the improvement in the values of k∗.Figure 4
(a–d)Fη versus k∗ in stretching/convergent circumstance with α=−5∘,5∘.
(a)(b)(c)(d)Figures5(a)–5(d) and 6(a)–6(d) represent the special effects of αandϕ1,ϕ2 temperature distribution Θη for extending/converging and contracting/divergent cases.Figure 5
(a–d)Θη versus α.
(a)(b)(c)(d)Figure 6
(a–d)Θη versus ϕ1,ϕ2.
(a)(b)(c)(d)The larger values ofα augmented the temperature distribution in each case as shown in Figures 5(a)–5(d). The enlarging values of the parameters ϕ1andϕ2 in the specific domain enhance the temperature distribution and are revealed in Figures 6(a)–6(d). In each case, the accumulative growth provides the increasing effect, and this improvement is more effective using the hybrid nanofluids.The comparison of the obtained results is compared with the available literature [3–5] and displayed in Figures 7(a) and 7(b) considering diverging and converging cases of the channel. The closed agreement has been achieved while choosing the common parameter Re. The influence of the nanoparticle volume fraction versus the skin friction has been shown in Figures 7(c) and 7(d) for both cases. The augmentation in the values of ϕ1,ϕ2 improves the resistive force to rise the drag force at the upper and lower walls. The influence is relatively strong using the hybrid nanofluids.Figure 7
(a–d) Matching of the current work with published literature [3–5] and Cf versus ϕ1,ϕ2.
(a)(b)(c)(d)The percentage increase in the heat transfer rate has been revealed in Figures8(a)–8(d). The values of the nanoparticle volume fraction are used up to 3% as ϕ1,ϕ2=0.0,0.01,0.02,0.03. The comparative analysis of the nanofluid and hybrid nanofluid is shown in Figures 8(a) and 8(c) for the diverging and converging cases of the channel, while the % analysis has been performed in Figures 8(b) and 8(d) for the same cases, respectively. The % increase is more appropriate by using the hybrid nanofluids in both α>0 and α<0.Figure 8
(a–d) Matching of the current work with published literature [3–5] and Cf versus ϕ1,ϕ2.
(a)(b)(c)(d)


## 5. Conclusions

The current article explores the blood flow across a converging/diverging channel with stretchable/shrinkable walls with couple stress for the application of drug delivery. The consequences of the converging/diverging parameter, couple stress parameter, and solid nanoparticles are incorporated.To the best of our knowledge and belief, the converging/diverging channel including blood as a base fluid does not exist in the existing literature. Furthermore, the work also extended using theAgandTiO2 hybrid nanofluid. Couple stress terminologies are also used as a novelty in the current problem.The key conclusions of the existing study are as follows:(i)
The rising values of solid nanoparticlesϕ1,ϕ2 enhance the energy transmission rate, and the impact is relatively larger in the case of hybrid nanofluid(ii)
The velocity field declines with the accumulative values of the parametersϕ1,ϕ2,andRe(iii)
The couple stress parameterk∗ has a significant role in blood flow analysis and declines the hybrid nanofluid motion(iv)
TiO2+Ag hybrid nanofluids have an important role in the Escherichia coli culture to evaluate their antibacterial strength(v)
The % analysis shows that hybrid nanofluids are more efficient for heat transfer analysis(vi)
The pH values improve with the increment in heat transfer. That is why the purpose of the recent study is to use theTiO2+Ag hybrid nanofluids for medication



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