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Open Physics
, Volume 21 (1): 1 – Jan 1, 2023

/lp/de-gruyter/parametric-simulation-of-stagnation-point-flow-of-motile-microorganism-T84fiZL0fA

- Publisher
- de Gruyter
- Copyright
- © 2023 the author(s), published by De Gruyter
- ISSN
- 2391-5471
- eISSN
- 2391-5471
- DOI
- 10.1515/phys-2022-0205
- Publisher site
- See Article on Publisher Site

Nomenclaturecnoddle pointu, v, wvelocity componentkhnf{k}_{hnf}thermal conductivityDhnf{D}_{hnf}mass diffusivity(ρCp)hnf{(\rho {C}_{p})}_{hnf}volumetric heat capacityDnmicroorganism diffusionNNmotile microbes’ densityRebR{e}_{b}Reynolds numberPePeclet numberΘ(η)\Theta (\eta )dimensionless energy profileCw{C}_{w}concentration over the surfaceC∞{C}_{\infty }ambient concentrationCp{C}_{p}heat capacityΦ(η)\Phi (\eta )dimensionless mass transitionScSchmidth numberxe⁎{x}_{e}^{\ast }streamlinesμhnf{\mu }_{hnf}dynamic viscosityρhnf{\rho }_{hnf}densityTTtemperaturePrPrandtl numberKKchemical reactionLeLewis numberh(η)h(\eta )dimensionless motile microorganismf′(η)f^{\prime} (\eta )dimensionless velocity profileϕ1,ϕ2{\phi }_{1},{\phi }_{2}volume frictionT∞{T}_{\infty }ambient temperatureNu\text{Nu}Nusselt numberν\nu kinematic viscosity of fluidη\eta similarity variableCf{C}_{f}skin frictiona⁎{a}^{\ast }, b⁎{b}^{\ast }stream coefficientsβ\beta constant1IntroductionMany engineering operations include flow through a circular cylinder, but far less research has been done on flow over a cylinder in a restricted domain. Several phenomena depend on wall upshots, such as blood flow through medical equipment in veins, flow over cylindrical objects near walls, and so on [1,2,3]. Moreover, the analysis of fluid flow across or over an irregular surface received enough attention. When regular sinusoidal ridges are present, a three-dimensional (3D) printer, owing to desired vibrations throughout the printing procedure, prompts research of the resultant flow consequences [4]. Salahuddin et al. [5] documented the consequence of variously configured nanomaterials on the thermal characteristics and flow efficiency of ferrofluid across rigid and sinusoidal surfaces. Wu et al. [6] utilized several active mechanism wind tunnel to evaluate the aerodynamic workloads of a sinusoidal cylinder. Changing the amplitude and frequency produces a succession that is entirely coherent in the streamwise direction. Bilal et al. [7] described an extending cylinder-induced incompressible Maxwell nanofluid flow accompanied by an unfluctuating suction/injection effect. It was determined that the impacts of the thermophoresis considerably increased the velocity of mass transference, while the consequences of the viscosity factor’s expansion substantially decreased the velocity curve. Seo et al. [8] numerically estimated the free convection in a broad, diagonal domain with a sinusoidal cylinder. In order to outperform a circular cylinder in terms of overall heat exchange, the sinusoidal cylinders were investigated. Bilal et al. [9] addressed the hybrid nanoliquid’s Darcy–Forchhemier mixed convective flow across an angled, expanding cylinder. Alharbi et al. [10] documented the magnetic characteristics of nanoliquid flow with energy flux in a boundary layer incorporating nanocrystals.Special prominence has been paid to the investigation of the hybrid nanoliquid with energy and mass transmission. Because of its critical significance in engineering and innovation, it has attracted the attention of numerous scientists and experts [11,12,13]. Propagation of the hybrid nanoliquid flow, as well as energy transference, plays crucial roles in biotechnology, nuclear sectors, paper manufacturing, geophysics, chemical plants, and unusual lubricants are only some of the uses in industry [13–15]. Commonly used fluids cannot fulfill the global demands, in the era of scientific and technological advancement. However, comparable base liquids with the deposition of small particles showed a significant enhancement in thermal characteristics [16]. Currently, we have employed the Ag and MgO nanoparticles (NPs). The compound MgO is composed of the ions Mg2+ and O2, which are joined by a special interaction. It is more useful for metalworking and electrical procedures. Similar to this, the antiseptic capabilities of Ag NPs could be used to change bioactivity in a diverse range of settings, including dental procedures, surgery, wound care, and pharmaceutical equipment [17]. A 3D numerical estimation of Ag–MgO-based nanofluid flow across a curved whirling disc, is investigated by Ahmadian et al. [18]. The hybrid composite was made by the addition of Ag–MgO NPs. By employing the numerical procedure parametric continuation method (PCM), the solution was found. Ag–MgO was considered to be more useful in addressing insufficient energy transport. Broad-spectrum antibacterial activities in metal and metal oxide nanomaterials have been extensively demonstrated for silver and MgO [19]. Anuar et al. [20] added MgO and Ag nanoparticulates, to produce a hybrid nanoliquid to evaluate boundary layer flow and temperature distribution. The consequences exposed that growing the weight fraction of Ag nanomaterials in a base fluid reduces the Nusselt number. Gangadhar et al. [21] have conducted a quantitative analysis of the properties of a nanofluid combining MgO and Au nanoparticles for convective heating. The influence of increased slip condition massively enhances the energy conduction ratio in the saddle and nodal point regions, according to the conclusions. Hiba et al. [22] inspected the thermal performance and fluidity of a water-based hybrid nanofluid made of MgO and Ag over a cylindrical, highly permeable region. Rasool et al. [15] examined numerical study of electro-magnetohydrodynamic nanoliquid flows through a Riga pattern inserted diagonally in a Darcy–Forchheimer permeable media. Shah et al. [23] quantitatively investigated the convective fluxes of a remarkable non-homogeneous nanofluids mixture across an impenetrable longitudinal electrostatic substrate. Ashraf et al. [24] documented the nanoliquid flow using the generalized numerical approach.The analysis of gyrotactic microbes in free surface flows has recently gained the attention of the scientific community. A microorganism is a living organism that can reproduce, evolve, react to environmental stimuli, and maintain a structured order. It can be utilized to improve oil recovery, which involves adding micronutrients and microorganisms to fuel layers to balance out permeability discrepancies [25–27]. The benefits of motile microbe interruption include nanofluid stability [28]. Hydrodynamic convection is created by oxytactic microbes, which forms a flow system that moves cells and oxygenation from the highest to the lowest fluid regions. The nanostructure mobility is regulated by molecular diffusion. The motile bacteria’s motility seemed to be independent of nanomaterial gestures [29,30]. The working mechanism of gyrotactic microorganisms in nanofluid was first discussed in refs [29,31]. Kuznetsov [32] extended the idea of suspensions by including Buongiorno’s conceptualization of bio-convection in nanoliquids. Xu et al. [33] analyzed a ferrofluid flow, consisting of motile microbes that flowed across parallel surfaces and transmitted energy. The velocity curve enhances with the mounting upshot of bioconvection [34]. Sohail et al. [35] examined the varied thermophysical characteristics of the 3D flow of a liquid with mass and energy conveyance in the presence of peristaltic transport of motile microbes over a curved elongated substrate. Despite the fact that the earlier analyses have indeed focused on understanding nanoliquid convection, there has been no effort in the literature to inspect the mass and motile microorganism transition under the influence of chemical reaction through the cylinder with sinusoidal radius (Figure 1).Figure 1Physical configuration of the fluid flow.The objective of this analysis is to scrutinize the features of the 3D stagnation point flow of Ag–MgO based hybrid nanoliquids traveling through a spherical cylinder with a harmonic radius. The electrostatic source consequences are evaluated in the stream flow. Our second priority is to elaborate also the uses and applications of silver and magnesium particles for medical and industrial purposes. After depersonalization, the computational technique PCM is used to calculate the powerful nonlinear systems. Furthermore, a pictorial assessment of the outstanding characteristics is performed on the velocity, mass, temperature, and motile microbes’ profiles.2Mathematical formulationWe have presumed the 3D stream of Ag–MgO water-based hybrid nanofluids through a circular cylinder of the periodic surface. It is important to note that there exists a motionlessness point at each point M, N, and O. The u, v, and w are the velocities terms in the path of x, y, and z, respectively. Whereue⁎=xa⁎,ve⁎=yb⁎.{u}_{e}^{\ast }=x{a}^{\ast },\hspace{.25em}{v}_{e}^{\ast }=y{b}^{\ast }.Here, a⁎{a}^{\ast }, b⁎{b}^{\ast }are free stream coefficients. The streamlines are defined as xe⁎=β1/c,{x}_{e}^{\ast }={\beta }^{1/c},where β\beta is constant and c=b⁎/a⁎c={b}^{\ast }/{a}^{\ast }. The nodal stagnation point appendix-lines span is 0<c<1.0\lt c\lt 1.The flow pattern is shown in Figure 1(a). In Figure 1, point N is a saddle point, where points M and O are identified as nodal points. The streamlines in this respect are specified in Figure 1(b). Based on the above presumption, the basic equations can be described as [36,37]:(1)∂u∂r+ur+∂w∂z=0,\frac{\partial u}{\partial r}+\frac{u}{r}+\frac{\partial w}{\partial z}=0,(2)u∂u∂x+v∂u∂y+w∂u∂z=xa⁎2+νhnf∂2u∂z2,u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+w\frac{\partial u}{\partial z}=x{a}^{\ast 2}+{\nu }_{hnf}\frac{{\partial }^{2}u}{\partial {z}^{2}},(3)u∂v∂x+v∂v∂y+w∂v∂z=yb⁎2+νhnf∂2u∂z2v,u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}+w\frac{\partial v}{\partial z}=y{b}^{\ast 2}+{\nu }_{hnf}\frac{{\partial }^{2}u}{\partial {z}^{2}}v,(4)(ρCp)hnfu∂T∂x+v∂T∂y=khnf∂2T∂z2,{(\rho {C}_{p})}_{hnf}\left(u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}\right)={k}_{hnf}\frac{{\partial }^{2}T}{\partial {z}^{2}},(5)u∂C∂x+v∂C∂y=Dhnf∂2C∂z2−k(C−C0),u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial y}={D}_{hnf}\frac{{\partial }^{2}C}{\partial {z}^{2}}-k(C-{C}_{0}),(6)u∂N∂x+v∂N∂y=Dn∂2N∂z2.u\frac{\partial N}{\partial x}+v\frac{\partial N}{\partial y}={D}_{n}\frac{{\partial }^{2}N}{\partial {z}^{2}}.Here, νhnf{\nu }_{hnf}, Dhnf{D}_{hnf}, and (ρCp)hnf{(\rho {C}_{p})}_{hnf}are the kinematic viscosity, mass diffusivity, and volumetric heat capacity, where Dnis the microorganism diffusion and NNis the motile microbe density.The boundary conditions are:(7)u=0,v=0,w=0,N=Nw,C=Cw,T=Twatz=0,u→ue⁎,v→ve⁎,T→T∞,N→N∞,C→C∞whenz→∞.\left.\begin{array}{c}u=0,\hspace{.25em}v=0,\hspace{.25em}w=0,\hspace{.25em}N={N}_{w},\hspace{.25em}C={C}_{w},\hspace{.25em}T={T}_{w}\hspace{.25em}\text{at}\hspace{.25em}z\text{=0,}\\ u\to {u}_{e}^{\ast },\hspace{.25em}v\to {v}_{e}^{\ast },\hspace{.25em}T\to {T}_{\infty },\hspace{.25em}N\to {N}_{\infty },\hspace{.25em}C\to {C}_{\infty }\hspace{.25em}\text{when}\hspace{.25em}z\to \infty .\end{array}\right\}The thermophysical properties of hybrid nanoliquid and model are [9]:υhnf=μhnfρhnf,{\upsilon }_{hnf}=\frac{{\mu }_{hnf}}{{\rho }_{hnf}},μhnf=μf(1−ϕAg)5/2(1−ϕMgO)5/2,{\mu }_{hnf}=\frac{{\mu }_{f}}{{(1-{\phi }_{\text{Ag}})}^{5/2}{(1-{\phi }_{\text{MgO}})}^{5/2}},(ρ)hnf(ρ)f=(1−ϕMgO)1−1−ρAgρfϕAg+ϕMgOρMgOρf,\frac{{(\rho )}_{hnf}}{{(\rho )}_{f}}=(1-{\phi }_{\text{MgO}})\left(1-\left(1-\frac{{\rho }_{\text{Ag}}}{{\rho }_{f}}\right){\phi }_{\text{Ag}}\right)+{\phi }_{\text{MgO}}\left(\frac{{\rho }_{\text{MgO}}}{{\rho }_{f}}\right),(ρCp)hnf(ρCp)f=(1−ϕMgO)1−1−(ρCp)Ag(ρCp)fϕA+(ρCp)MgO(ρCp)fϕMgO,\frac{{(\rho {C}_{p})}_{hnf}}{{(\rho {C}_{p})}_{f}}=(1-{\phi }_{\text{MgO}})\left\{1-\left(1-\frac{{(\rho {C}_{p})}_{\text{Ag}}}{{(\rho {C}_{p})}_{f}}\right){\phi }_{A}\right\}+\frac{{(\rho {C}_{p})}_{\text{MgO}}}{{(\rho {C}_{p})}_{f}}{\phi }_{\text{MgO}},khnfknf=kMgO+2knf−2ϕMgO(knf−kMgO)kMgO+2knf+ϕCu(knf−kMgO),\frac{{k}_{hnf}}{{k}_{nf}}=\frac{{k}_{\text{MgO}}+2{k}_{nf}-2{\phi }_{\text{MgO}}({k}_{nf}-{k}_{\text{MgO}})}{{k}_{\text{MgO}}+2{k}_{nf}+{\phi }_{\text{Cu}}({k}_{nf}-{k}_{\text{MgO}})},knfkf=kAg+2kf−2ϕAg(kf−kAg)kAg+2kf+ϕAg(kf−kAg).\hspace{.25em}\frac{{k}_{nf}}{{k}_{f}}=\frac{{k}_{\text{Ag}}+2{k}_{f}-2{\phi }_{\text{Ag}}({k}_{f}-{k}_{\text{Ag}})}{{k}_{\text{Ag}}+2{k}_{f}+{\phi }_{\text{Ag}}({k}_{f}-{k}_{\text{Ag}})}.In order to diminish the system of partial differential equations (PDEs) to the system of nonlinear ordinal differential equations (ODEs), we defined the following variables:(8)u=a⁎xf′(η),w=−a⁎νf(cg(η)+f(η)),v=b⁎yg′(η),Θ(η)=T−T∞Tw−T∞,Φ=C−C∞Cw−C∞,h=n−n∞nw−n∞,η=zνfa⁎.\begin{array}{c}u={a}^{\ast }x\hspace{.25em}f^{\prime} (\eta ),\hspace{.25em}w=-\sqrt{{a}^{\ast }{\nu }_{f}}(cg(\eta )+f(\eta )),\hspace{.25em}v={b}^{\ast }y\hspace{.25em}g^{\prime} (\eta ),\hspace{.25em}\Theta (\eta )=\frac{T-{T}_{\infty }}{{T}_{w}-{T}_{\infty }},\\ \Phi =\frac{C-{C}_{\infty }}{{C}_{w}-{C}_{\infty }},\hspace{.25em}h=\frac{n-{n}_{\infty }}{{n}_{w}-{n}_{\infty }},\hspace{.25em}\eta =z\sqrt{\frac{{\nu }_{f}}{{a}^{\ast }}}.\end{array}Now, by using Eq. (8) in Eqs. (1)–(6) and (7), we get:(9)A1f‴+cgf″−f′2+ff″+1=0,{A}_{1}f\prime\prime\prime +cgf^{\prime\prime} -{f^{\prime} }^{2}+ff^{\prime\prime} +1=0,(10)A1g‴+fg″+cgg″−cg′2+c=0,{A}_{1}g\prime\prime\prime +fg^{\prime\prime} +cgg^{\prime\prime} -c{g^{\prime} }^{2}+c=0,(11)khnfkf1B1PrΘ″+fΘ′+cgΘ′=0,\frac{{k}_{hnf}}{{k}_{f}}\frac{1}{{B}_{1}\hspace{.25em}\text{Pr}}\Theta ^{\prime\prime} +f\Theta ^{\prime} +cg\Theta ^{\prime} =0,(12)Φ″C1−Sc(f′Φ′)−K=0,\Phi ^{\prime\prime} {C}_{1}-\text{Sc}\hspace{.25em}(f^{\prime} \Phi ^{\prime} )-K=0,(13)h″+(2Scfh′+(h′Φ′−hΦ″)Pe)Re=0.h^{\prime\prime} +(2\text{Sc}fh^{\prime} +(h^{\prime} \Phi ^{\prime} -h\Phi ^{\prime\prime} )\text{Pe})\text{Re}=0.Where,(14)A1=(1−ϕAg)2.5(1−ϕMgO)2.51−ϕMgO1−ϕAg+ϕAg(ρCp)Ag(ρCp)f+ϕMgO(ρCp)MgO(ρCp)f,B1=1−ϕMgO1−ϕAg+ϕAg(ρCp)Ag(ρCp)f+ϕMgO(ρCp)MgO(ρCp)f,C1=(1−ϕAg)2.5(1−ϕMgO)2.5.\begin{array}{c}{A}_{1}=\left[{(1-{\phi }_{\text{Ag}})}^{2.5}{(1-{\phi }_{\text{MgO}})}^{2.5}\left(1-{\phi }_{\text{MgO}}\left(1-{\phi }_{\text{Ag}}+{\phi }_{\text{Ag}}\frac{{(\rho {C}_{p})}_{\text{Ag}}}{{(\rho {C}_{p})}_{f}}\right)\right)+{\phi }_{\text{MgO}}\frac{{(\rho {C}_{p})}_{\text{MgO}}}{{(\rho {C}_{p})}_{f}}\right],\hspace{.25em}\\ {B}_{1}=\left[\left(1-{\phi }_{\text{MgO}}\left(1-{\phi }_{\text{Ag}}+{\phi }_{\text{Ag}}\frac{{(\rho {C}_{p})}_{\text{Ag}}}{{(\rho {C}_{p})}_{f}}\right)\right)+{\phi }_{\text{MgO}}\frac{{(\rho {C}_{p})}_{\text{MgO}}}{{(\rho {C}_{p})}_{f}}\right],\\ {C}_{1}={(1-{\phi }_{\text{Ag}})}^{2.5}{(1-{\phi }_{\text{MgO}})}^{2.5}.\end{array}The renovated conditions are:(15)f′(0)=0,f(0)=0,g′(0)=0,g(0)=0,Θ(0)=1,Φ(0)=1,h(0)=1atη=0,f′(∞)=1,Θ(∞)=0,g′(∞)=1,Φ(∞)=0,h(∞)=0asη→∞.\left.\begin{array}{c}f^{\prime} (0)=0,\hspace{.03em}f(0)=0,\hspace{.25em}g^{\prime} (0)=0,\hspace{.25em}g(0)=0,\hspace{.25em}\Theta (0)=1,\hspace{.25em}\Phi (0)=1,\hspace{.25em}h(0)=1\hspace{.25em}\text{at}\hspace{.25em}\eta =0,\\ f^{\prime} (\infty )=1,\hspace{.03em}\Theta (\infty )=0,\hspace{.25em}g^{\prime} (\infty )=1,\hspace{.25em}\Phi (\infty )=0,\hspace{.25em}h(\infty )=0\hspace{.25em}\text{as}\hspace{.25em}\eta \to \infty .\end{array}\right\}The physical interest quantities are:(16)Cfx=τwxρfUw2,Cfy=τwyρfUw2,Nux=xqxkf(Tw−T∞).{C}_{fx}=\frac{{\tau }_{wx}}{{\rho }_{f}{U}_{w}^{2}},\hspace{.25em}{C}_{fy}=\frac{{\tau }_{wy}}{{\rho }_{f}{U}_{w}^{2}},\hspace{.25em}N{u}_{x}=\frac{x{q}_{x}}{{k}_{f}({T}_{w}-{T}_{\infty })}.Where, τwx{\tau }_{wx}, τwy{\tau }_{wy}, and qw{q}_{w}are defined as:(17)τwx=(μhnf)∂u∂zz=0,qw=−(khnf)∂T∂zz=0,τwy=(μhnf)∂v∂zz=0.{\tau }_{wx}={\left(({\mu }_{hnf})\frac{\partial u}{\partial z}\right)}_{z=0},\hspace{.25em}{q}_{w}={\left(-({k}_{hnf})\frac{\partial T}{\partial z}\right)}_{z=0},\hspace{.25em}{\tau }_{wy}={\left(({\mu }_{hnf})\frac{\partial v}{\partial z}\right)}_{z=0}.The dimensionless form of Eq. (16) is:(18)RexCfx=1(1−ϕ1)2.5(1−ϕ2)2.5f″(0),(x/y)RexCfy=c(1−ϕ1)2.5(1−ϕ2)2.5g″(0),NuxRex=−khnfkfθ′(0).\begin{array}{c}\sqrt{{\text{Re}}_{x}{C}_{fx}}=\left(\frac{1}{{(1-{\phi }_{1})}^{2.5}{(1-{\phi }_{2})}^{2.5}}\right)f^{\prime\prime} (0),\hspace{.25em}(x/y)\sqrt{{\text{Re}}_{x}{C}_{fy}}=\left(\frac{c}{{(1-{\phi }_{1})}^{2.5}{(1-{\phi }_{2})}^{2.5}}\right)g^{\prime\prime} (0),\\ \frac{N{u}_{x}}{\sqrt{{\text{Re}}_{x}}}=-\left(\frac{{k}_{hnf}}{{k}_{f}}\right)\theta ^{\prime} (0).\end{array}3Problem solutionDifferent steps, which are used during applying PCM to system of Eqs. (9)–(13) and (15), are [38–43]:Step 1: Reduced BVPs to the first order ODEs(19)f=ζ1,f″=ζ3,g′=ζ5,Θ=ζ7,Φ=ζ9,h=ζ11,f′=ζ2,g=ζ4,g′′=ζ6,Θ′=ζ8,Φ′=ζ10,h=ζ12,\left.\begin{array}{c}\hspace{.25em}f={\zeta }_{1},\hspace{.25em}f^{\prime\prime} ={\zeta }_{3},\hspace{.25em}g^{\prime} ={\zeta }_{5},\hspace{.25em}\Theta ={\zeta }_{7},\hspace{.25em}\Phi ={\zeta }_{9},\hspace{.25em}h={\zeta }_{11},\hspace{.25em}\\ f^{\prime} ={\zeta }_{2},\hspace{.25em}g={\zeta }_{4},\hspace{.25em}g^{\prime} ^{\prime} ={\zeta }_{6},\hspace{.25em}\Theta ^{\prime} ={\zeta }_{8},\hspace{.25em}\Phi ^{\prime} ={\zeta }_{10},\hspace{.25em}h={\zeta }_{12},\end{array}\right\}By employing Eq. (19) to Eqs. (9)–(13) and (15), we get:(20)A1ζ′3+(cg+ζ1)ζ3−ζ22+1=0,{A}_{1}{\zeta ^{\prime} }_{3}+(cg+{\zeta }_{1}){\zeta }_{3}-{{\zeta }_{2}}^{2}+1=0,(21)A1ζ′6+(f+cζ4)ζ6−cζ52+c=0,{A}_{1}{\zeta ^{\prime} }_{6}+(f+c{\zeta }_{4}){\zeta }_{6}-c{{\zeta }_{5}}^{2}+c=0,(22)khnfkf1B1Prζ′8+(f+cg)ζ8=0,\frac{{k}_{hnf}}{{k}_{f}}\frac{1}{{B}_{1}\hspace{.25em}\text{Pr}}{\zeta ^{\prime} }_{8}+(f+cg){\zeta }_{8}=0,(23)ζ′10C1−Sc(f′ζ10)−K=0,{\zeta ^{\prime} }_{10}{C}_{1}-\text{Sc}\hspace{.25em}(f^{\prime} {\zeta }_{10})-K=0,(24)ζ′12+Re((2Scζ1+Peζ10)ζ12−Peζ11ζ′10)=0.{\zeta ^{\prime} }_{12}+\text{Re}((2Sc{\zeta }_{1}+\text{Pe}{\zeta }_{10}){\zeta }_{12}-\text{Pe}{\zeta }_{11}{\zeta ^{\prime} }_{10})=0.boundary conditions are:(25)ζ1(0)=0,ζ2(0)=0,ζ4(0)=0,ζ5(0)=0,ζ7(0)=1,ζ9(0)=1,ζ11(0)=1,ζ2(∞)=1,ζ5(∞)=1,ζ7(∞)=0,ζ9(∞)=0,ζ11(∞)=0.\begin{array}{c}{\zeta }_{1}(0)=0,\hspace{.25em}{\zeta }_{2}(0)=0,\hspace{.03em}{\zeta }_{4}(0)=0,\hspace{.25em}{\zeta }_{5}(0)=0,\hspace{.25em}{\zeta }_{7}(0)=1,\hspace{.25em}{\zeta }_{9}(0)=1,\hspace{.25em}{\zeta }_{11}(0)=1,\\ {\zeta }_{2}(\infty )=1,\hspace{.03em}{\zeta }_{5}(\infty )=1,\hspace{.25em}{\zeta }_{7}(\infty )=0,\hspace{.25em}{\zeta }_{9}(\infty )=0,\hspace{.25em}{\zeta }_{11}(\infty )=0.\end{array}Step 2: Familiarizing the embedding constraint p to Eqs. (20)–(24):(26)A1ζ′3+(cg+ζ1)(ζ3−1)p−ζ22+1=0,{A}_{1}{\zeta ^{\prime} }_{3}+(cg+{\zeta }_{1})({\zeta }_{3}-1)p-{{\zeta }_{2}}^{2}+1=0,(27)A1ζ′6+(f+cζ4)(ζ6−1)p−cζ52+c=0,{A}_{1}{\zeta ^{\prime} }_{6}+(f+c{\zeta }_{4})({\zeta }_{6}-1)p-c{{\zeta }_{5}}^{2}+c=0,(28)khnfkf1B1Prζ′8+(f+cg)(ζ8−1)p=0,\frac{{k}_{hnf}}{{k}_{f}}\frac{1}{{B}_{1}\hspace{.25em}\text{Pr}}{\zeta ^{\prime} }_{8}+(f+cg)({\zeta }_{8}-1)p=0,(29)ζ′10C1−Scf′(ζ10−1)p−K=0,{\zeta ^{\prime} }_{10}{C}_{1}-\text{Sc}f^{\prime} ({\zeta }_{10}-1)p-K=0,(30)ζ′12+Re((2Scζ1+Peζ10)(ζ12−1)p−Peζ11ζ′10)=0.{\zeta ^{\prime} }_{12}+\mathrm{Re}((2\text{Sc}{\zeta }_{1}+\text{Pe}{\zeta }_{10})({\zeta }_{12}-1)p-\text{Pe}{\zeta }_{11}{\zeta ^{\prime} }_{10})=0.\hspace{1em}4Results and discussionThe results are exposed through Figures (2–4), and Tables and discussion on the obtained results are categorized as:Figure 2Exhibition of velocity (f′(η),g′(η))(f^{\prime} (\eta ),\hspace{.25em}g^{\prime} (\eta ))curve against volume fraction of silver ϕ1{\phi }_{1}and magnesium oxide ϕ2{\phi }_{2}, while keeping c = −0.5 (dishes) and c = 0.5 (solid line).Figure 3Exhibition of energy Θ(η)\Theta (\eta )curve against (a) volume fraction of silver ϕ1{\phi }_{1}(b) magnesium oxide ϕ2{\phi }_{2}(c) Prandtl number (d) noddle point, while keeping c = −0.5 (dishes) and c = 0.5 (solid line).Figure 4Behavior of mass transition Φ(η)\Phi (\eta )and motile microbes h(η)h(\eta )profile against (a) chemical reaction K, (b) Schmidth number Sc, (c) Lewis number, (d) Peclet number, while keeping c = −0.5 (dishes) and c = 0.5 (solid line).4.1Velocity profileFigure 2(a)–(d) highlights the presentation of axial f′(η)f^{\prime} (\eta )and radial velocity g′(η)g^{\prime} (\eta )profile against volume fraction of silver ϕ1{\phi }_{1}and magnesium oxide ϕ2{\phi }_{2}, while keeping c = −0.5 (dishes) and c = 0.5 (solid line), respectively. Both axial and radial velocities significantly enhance with the effect of hybrid nanoparticles (Ag–MgO). The heat absorbing capacity of water as compared to MgO and Ag nanomaterial is higher, so, adding more nanoparticles (ϕ1=ϕ2=0.01−0.04)({\phi }_{1}={\phi }_{2}=0.01-0.04)to water reduces its average heat capacity. That is why, such trend has been noticed.4.2Energy profileFigure 3(a)–(d) uncovers the behavior of energy Θ(η)\Theta (\eta )profile against the volume fraction of silver ϕ1{\phi }_{1}, magnesium oxide ϕ2{\phi }_{2}, Prandtl number, and noddle point, while keeping c = −0.5 (dishes) and c = 0.5 (solid line). Figure 3(a) and (b) manifest that the energy outline boosts with the increment of nanoparticulate in the base liquid. As we have discoursed before that the heat absorbing capacity of water as compared to MgO and Ag nanomaterial is higher, so adding more nanoparticles to water reduces its specific heat capacity. Figure 3(a) demonstrates that the energy distribution diminishes with the effect of Prandtl number. High effect Prandtl fluid has always low thermal diffusivity, so as a result of the current analysis, the Prandtl effect reduces the energy propagation rate. The energy transfer rate also improves at the noddle point as shown in Figure 3(d).4.3Mass and motile microorganism profileFigure 4(a)–(d) emphasizes the performance of mass transition Φ(η)\Phi (\eta )and motile microbes h(η)h(\eta )profile against chemical reaction K, Schmidth number Sc, Lewis number, and Peclet number, while keeping c = −0.5 (dishes) and c = 0.5 (solid line). Figure 4(a) and (b) show that the upshot of chemical reaction enhances the mass transition because the chemical reaction parameter influence exercises the molecules inside the fluid, which encourages the mass transfer. On the other hand, the Schmidth number drops the mass transfer, because the kinetic viscosity of fluid augments with the action of the Schmidth number as reported in Figure 4(b). Figure 4(c) and (d) describes that the motile microorganism outline elevated with the increment of Lewis and Peclet number.Table 1 presents the experimental values of Ag, MgO, and water. Table 2 discovers the comparative valuation of bvp4c package and PCM with the arithmetical outcomes of skin friction and the local Nusselt number. The variation of both Ag and MgO nanoparticles boosts the drag force and energy transference rate. From Table 2, it can be noticed that the PCM is fast approaching technique than bvp4c. Tables 3 and 4 report the comparison of simple and hybrid nanofluid behavior for skin fraction and energy transition. As compared to nanofluid, hybrid fluid has tremendous tendency for energy propagation.Table 1Experimental values of Ag, MgO, and water [9]ρ(kg/m3)\rho \hspace{.5em}(kg/{m}^{3})Cp(J/kgK){C}_{p}\hspace{.5em}(\text{J}/kg\hspace{.5em}K)k(W/mK)k\hspace{.5em}(W/m\hspace{.1em}K)β×105(K−1)\beta \times {10}^{5}\hspace{.5em}({\text{K}}^{-1})Pure water997.14,1790.61321Magnesium oxide3,560955451.80Silver10,5002354291.89Table 2Comparative assessment of PCM and bvp4c package for skin friction and the Nusselt number(ϕ1,ϕ2)({\phi }_{1},\hspace{.25em}{\phi }_{2})RexCfx\sqrt{{\text{Re}}_{x}}{C}_{fx}x/yRexCfyx/y\sqrt{{\text{Re}}_{x}}{C}_{fy}NuxRex{\text{Nu}}_{x}\sqrt{{\text{Re}}_{x}}PCMbvp4cPCMbvp4cPCMbvp4cAg0.001.26831.26820.49940.49921.33041.33000.011.94881.93850.76350.76291.61871.61830.022.69742.69701.06211.06151.92961.9290MgO0.001.26861.26820.49990.49911.33211.32990.011.66671.66590.65650.65561.49771.49540.022.15412.15350.84820.84741.73961.7383Table 3Numerical outcomes of skin friction for the nano and hybrid nanoliquidϕAg{\phi }_{\text{Ag}}ϕMgO{\phi }_{\text{MgO}}Ag–MgO/waterAg/waterMgO/waterRexCfx\sqrt{{\text{Re}}_{x}}{C}_{fx}x/yRexCfyx/y\sqrt{{\text{Re}}_{x}}{C}_{fy}RexCfx\sqrt{{\text{Re}}_{x}}{C}_{fx}x/yRexCfyx/y\sqrt{{\text{Re}}_{x}}{C}_{fy}RexCfx\sqrt{{\text{Re}}_{x}}{C}_{fx}x/yRexCfyx/y\sqrt{{\text{Re}}_{x}}{C}_{fy}0.010.011.3000.4501.2310.4231.2310.4240.020.021.4510.5131.3000.4501.3030.4530.030.031.6470.5911.3800.4821.3880.4850.040.041.8820.6871.4710.5181.4820.523Table 4Numerical outcomes of Nusselt number for the nanoliquid and hybrid nanoliquidϕAg{\phi }_{Ag}ϕMgO{\phi }_{MgO}Ag–MgO/waterAg/waterMgO/waterNuxRexCfxN{u}_{x}\sqrt{R{e}_{x}}{C}_{fx}NuxRexCfxN{u}_{x}\sqrt{R{e}_{x}}{C}_{fx}NuxRexCfxN{u}_{x}\sqrt{R{e}_{x}}{C}_{fx}0.010.010.5370.5180.5210.020.020.5730.5370.5410.030.030.6080.5560.5610.040.040.6430.5760.5815ConclusionWe have observed the features of 3D stagnation point flow of Ag–MgO-based hybrid nanoliquids traveling through a spherical cylinder of sinusoidal radius. The facts have been formulated in the form of system of PDEs. After transformation, the computational technique PCM is used to estimate the nonlinear systems of differential equations. Furthermore, a graphical assessment of the physical characteristics is accomplished on the velocity, mass, temperature, and motile microbes’ profiles. The key conclusions are as follows:Both axial and radial velocities significantly augment with the intensifying effect of hybrid nanoparticulates (Ag–MgO).The energy transmission profile also boosts with the increment of nanoparticulate in the base liquid.The energy transfer rate also improves at the noddle point.The upshot of chemical reaction enhances the mass transition because the chemical reaction parameter influence exercises the molecules inside the fluid, whose encourages the mass transfer.The motile microorganism outlines elevated with the increment of Lewis and Peclet number.As compared to nanoliquid, hybrid nanoliquid is more convenient for heat and mass transmission.The skin friction coefficient augments with the rising numbers of nanoparticulates in the base fluid.The addition of MgO and Ag nanomaterials to the base fluid also enhances the Nusselt number.

Open Physics – de Gruyter

**Published: ** Jan 1, 2023

**Keywords: **hybrid nanofluid; motile microorganism; parametric continuation method; circular cylinder; sinusoidal radius

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