# Magneto-Burgers Nanofluid Stratified Flow with Swimming Motile Microorganisms and Dual Variables Conductivity Configured by a Stretching Cylinder/Plate

Authors: Hassan Waqas; Umair Manzoor; Zahir Shah; Muhammad Arif; Meshal Shutaywi
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

Background. The study of nanofluid gains interest of researchers because of its uses in treatment of cancer, wound treatment, fuel reserves, and elevating the particles in the bloodstream to a tumour. This artefact investigates the magnetohydrodynamic flow of Burgers nanofluid with the interaction of nonlinear thermal radiation, activation energy, and motile microorganisms across a stretching cylinder. Method. The developed partial differential equations (PDEs) are transformed into a structure of ODEs with the help of similarity transformation. The extracted problem is rectified numerically by using the bvp4c program in computational software MATLAB. The novelty of analysis lies in the fact that the impacts of bioconvection with magnetic effects on Burgers nanofluid are taken into account. Moreover, the behaviours of thermal conductivity and diffusivity are discussed in detail. The impacts of activation energy and motile microorganism are also explored. No work has been published yet in the literature survey according to the authors’ knowledge. The current observation is the extension of Khan et al.’s work [51]. Results. The consequences of the relevant parameters, namely, thermophoresis parameter, Brownian motion parameter, the reaction parameter, temperature difference parameter, activation energy, bioconvection Lewis number and Peclet number against the velocity of Burgers nanofluid, temperature profile for nanoliquid, the concentration of nanoparticles, and microorganisms field, have been explored in depth. The reports had major impacts in the development of medications for the treatment of arterial diseases including atherosclerosis without any need for surgery, which may reduce spending on cardiovascular and postsurgical problems in patients. Conclusions. The current investigation depicts that fluid velocity increases for uplifting values of mixed convection parameter. Furthermore, it is analyzed that flow of fluid is risen by varying the amount of Burgers fluid parameter. The temperature distribution is escalated by escalating the values of temperature ratio parameter and thermal conductivity parameter. The concentration field turns down for elevated values of Lewis number and Brownian motion parameter, while conflicting circumstances are observed for the thermophoresis parameter and solutal Biot number. Larger values of Peclet number reduce the microorganism’s field. Physically the current model is more significant in the field of applied mathematics. Furthermore, the current model is more helpful to improve the thermal conductivity of base fluids and heat transfer rate.


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## Body

## 1. Introduction

Numerous scholars are fascinated by nanofluids due to their larger thermophysical properties and uses in heavy industrial and engineering technologies. Nanofluids contain fictional features that make them extremely useful and have gained substantial interest owing to their huge variety of applications such as cooling factors in electrical devices, automobiles, industrial-grade engines, and factories to improve efficiency, save energy, and minimize emission levels. Nanofluids are also used as medicines and antibiotics in the biological sciences. Choi rendered the first use of nanofluid in 1995 [1]. Buongiorno [2] suggested two key features, thermophoresis and Brownian motion, to advance the convection of nanofluids. The hybrid nanofluids in the various kinds of heat structure for specific boundary constraints and physical scrutinized by Humic [3]. The two-dimensional Darcy-Forchheimer nanofluid flow over the curved stretching layer was analyzed by Hayat et al. [4]. Ellahi et al. [5] explored the role of the slip in the two-phase nanofluid flow. The analysis of mixed convection flow over a vertical sheet having hybrid nanoparticles with porous medium was observed by Waini et al. [6]. Ahmed et al. [7] addressed the entropy generation in magnetohydrodynamics Eyring–Powell nanoliquids flow with the consequences of dissipation, nonlinear mixed convection, and Joule heating. Lahmar et al. [8] reported the flow and heat transfer of a squeezing unsteady nanofluid between two parallel plates. First and second law study of aqueous (NF) including suspended Ag nanoadditives in two new microchannel heat sinks is analyzed by Yang et al. [9]. The steady nanofluid flow over a permeable stretch/shrink cylinder was evaluated by Roşca et al. [10]. Khan et al. [11] explored the behaviour of magnetic dipole on non-Newtonian fluid including nanoparticles. Abbas et al. [12] analyzed the characteristics of thermal conductivity on magnetized Carreau nanofluid. Abbas et al. [13] analyzed Wu’s slip impacts on the magnetohydrodynamic flow of nanofluid with activation energy. Abbas et al. [14] discussed the thermal dependent viscosity on nanofluid with entropy generation. Abbas et al. [15] scrutinized the micropolar nanofluid with magnetic field under three-dimensional flows. There are many investigators who investigated the nanofluid behaviours [16–20].The theory of heat and mass transfer in magnetohydrodynamic flow has many applications in a wide range of industrial applications, including geophysics, magnetic material processing, crude oil purification, and cooling rate control. The behaviour of energy equations under the Joule heating effect was discussed by Khan et al. [21]. The unstable flow of viscous fluid via magnetohydrodynamics was scrutinized by Ghalib et al. [22]. Khan et al. [23] premeditated 2-dimensional flow over a stretched surface of a non-Newtonian liquid with entropy optimization. Mohamad et al. [24] studied mixed convection of unstable noncoaxial viscous fluid rotational flow past an accelerated vertical disk. The effect of binary chemical reactions and activation energy on 3rd-grade hydromagnetic nanofluid streams combined with convective boundary layers was evaluated by Hayat et al. [25]. Iskender et al. [26] analyzed the steady flow of nanomaterials with melting heat phenomenon. Khedr et al. [27] explored the micropolar fluid with the impact of the magnetic field and heat source sink. Zaraki et al. [28] discussed the analysis of heat and mass transfer on nanomaterials liquid. Chamkha [29] analyzed the MHD flow under a porous medium over a vertical plate. Chamkha and Khaled [30] explored the coupled thermal and mass transfer over a permeable surface. Reddy et al. [31] scrutinized the heat and solutal transfer in nanofluid past a disk embedded in a porous medium. Many researchers [32–35] discussed the nanofluid properties over a different surface.Bioconvection induced by variations in density of motile microorganisms is efficiently coordinated in the fields of ecological classifications, biogas, and production. The existence of such microorganisms improves the main density of the liquid and produces a gradient of density by floating that contributes to bioconvection. This fascinating discovery ultimately leads to an unstable, low-density layer. For example, microorganisms are spontaneously driven and move in the liquids to the atmosphere, while nanoparticles are directed by thermophoresis and Brownian motion in the surface liquid. Adding motile microorganisms to dilute nanomaterials suspensions is tremendously helpful for improving mass transport. Microorganisms were usually categorized as gyrotactic and gravitational microorganisms based on the pulsating force of different forms. There are still some different and related properties of nanostructures and motile microorganisms. The bioconvective stagnation point movement of nanoliquid, like swimming microorganisms, through a nonlinear stretching surface, is viewed by Mondal and Pal [36]. The Carreau-Yasuda nanofluid bioconvection flow in the appearance of microorganisms has been reported by Waqas et al. [37]. Two-dimensional generalized second-grade nanoliquid flow pasta Riga plate is investigated by Waqas et al. [38]. Khan et al. [39] analyzed the rheology of nanofluid stress couples using activation energy, thermal radiation, porous material, and convective boundary conditions of the field. The magnetized bioconvection movement of a nanofluid with microorganisms across a nonlinear inclined stretch sheet is intentional by Beg et al. [40]. Bhatti and Michaelides [41] inspect the activation energy of Arrhenius via the Riga plate for nanofluid thermobioconvection. Zadeh et al. [42] digitally observe the movement, heat, and mass transfer of nanofluids through a vertical stretch layer under the effect of motile microorganisms. Many researchers scrutinize the bioconvection aspects with nanofluids [43–48].The main aspiration of the current inquiry is to evaluate the MHD flow of Burgers nanofluid configured by a stretching cylinder/plate. The consequences of thermal radiation, motile microorganisms, and chemical reactions are also taken into account. The governing equations of the flow problem are reduced by eminent shooting technique and cracked numerically with the bvp4c method via MATLAB commercial software. The effects of prominent parameters of the flow equation against velocity concentration and temperature profile are extracted numerically and graphically through graphs and tables. The considered problem may be helpful in industrial sectors. The current model is more useful in the field of technology to improve the heat transfer rate of heat storage devices. The proposed model is useful in automobiles, industrial-grid engines, cancer treatment, medicine, biosciences, biotechnology, pharmaceutical science, mechanical engineering, nuclear reactor, cooling of devices, and electrical engineering, as well as in many more fields. Bioconvectional model is more faithful in the biosensors, oil refineries, drugs delivery, medicines, chemotherapy, and military sectors.


## 2. Mathematical Formulation

In this article, we analyzed the consequences of Burgers nanofluid with the impacts of thermal radiation and motile microorganisms past a stretching cylinder/plate. Burgers nanofluid flow model over an expanding cylinder/plate is constructed with the strength of uniform magnetic fieldB=B0,0,0 which is vertical to the flow path. Also, the temperature and concentration at the surface of the cylinder are supposed to be Tw,Cw each (see Figure 1).(1)∂u∂r+ur+∂w∂z=0,(2)u∂w∂u+w∂w∂z+β1u2∂2w∂r2+w2∂2w∂z2+2uw∂2w∂z∂r+β2u3∂3w∂r3+w3∂3w∂z3+2u2∂u∂r∂2w∂r2+∂w∂r∂2w∂r∂z−u2∂w∂r∂2u∂r2+∂w∂z∂2w∂r2+2w2∂u∂z∂2w∂r∂z+w2∂w∂z∂2w∂z2−∂w∂r∂2u∂z2+3uwu∂3w∂r2∂z+w∂3w∂z2∂r+2uw∂u∂r∂2w∂z∂r+∂u∂z∂2w∂r2+∂w∂r∂2w∂z2−∂w∂r∂2u∂r∂z=we∂we∂r+vβ3u3∂3w∂r3+w∂3w∂r2∂z+ur∂2w∂r2−∂w∂r∂2u∂r2+wr∂2w∂r∂z−1r∂u∂r∂w∂z−1r∂w∂r∂w∂z−∂w∂r∂u2∂z+v∂2w∂r2+1r∂w∂r−σB02ρw−weβ1u∂w∂r+β2w∂u∂z∂2w∂r2−u∂w∂z∂2w∂r2+uw∂2w∂r∂z+u2∂2w∂r2+1ρf1−Cfρfβ∗∗g∗T−T∞−ρp−ρfg∗C−C∞−N−N∞g∗γρm−ρfsinδ2,(3)u∂T∂r+w∂T∂z=1ρcf∂∂rKT∂T∂r+α11r∂∂rr∂T∂r−1ρcf∂qr∂r+τDb∂D∂r∂T∂r+DTT∞∂T∂r2,(4)u∂C∂r+w∂C∂z=∂∂rDC∂C∂r+DBr∂∂rr∂C∂r+DTT∞1r∂∂rr∂T∂r−Kr2C−C∞TT∞nexp−EakT,(5)u∂N∂r+w∂N∂z+∂∂rN∂C∂rbWcCf−C∞=Dm∂∂r∂N∂r.Figure 1
Physical view of flow.In expressions (3) and (5), the thermal diffusivity and solutal diffusivity are read as kT=k∞1+∈1T−T∞/Tf−T∞ and DC=D∞1+∈2C−C∞/Cf−C∞.With boundary conditions(6)w=wS=U0zl,u=0,−k∂T∂r=hfTf−T,−DB∂C∂r=hgCf−C,−Dm∂N∂r=hnNf−N,atr=R,w⟶we=U∞zl,∂w∂r⟶0,T⟶Tf,C⟶Cf,N⟶Nf,asr⟶∞.The following similarities are introduced for obtaining dimension system of the current problem:(7)u=−RrU0vlfζ,w=U0zlf′ζ,θζ=T−T∞Tf−T∞,ϕζ=C−C∞Cf−C∞,ϕζ=N−N∞Nf−N∞,ζ=U0vlr2−R22R,(8)1+2λζ2γ12ff′f″−f2f‴−1+2λζλγ1f2f″−1+2λζ2γ23f2f″2+2ff′2f″−f3fiv−4λ2γ2f‴f″+1+2λζγ23f2f′f″+f3f‴+1+2λζ3γ3f″2−ffiv−4λγ31+2λζ2ff‴+1+2λζ3f‴+1+2λζ22f″+ff″−f′2+1+2λζ2A2−1+2λζ2M2−A+γ2ff‴−γ1ff″+f′+Sθ−Nrϕ−Ncχsinδ2=0,(9)Rd1+θθw−13θ″1+2λζ′+∈1θθ″+∈1θ′2+Prθ′λf+PrNbϕ′θ′1+2λζ+PrNtθ′21+2λζ=0,(10)1+2λζ+∈2ϕϕ″+∈2ϕ′2+2αϕ′+LePrfϕ′+1+2λζNtNbθ″+2λNtNbθ′−PrLeσ∗1+δθnexp−E1+δθϕ=0,(11)1+2λζχ″+2λχ′+Lbfχ′−Peϕ″χ+δ1+χ′ϕ′=0,(12)f=0,f′=1,θ′=−α11−θζ,ϕ′=−α21−ϕζ,χ′=−α31−χζ,atζ=0,f′⟶A,f″⟶0,θ⟶0,ϕ⟶0,χ⟶0,asζ⟶∞.We have that velocity ratio parameter isA=U∞/U0, U∞ denotes the velocity of free stream, U0 stands for stretching velocity, S denotes mixed convection parameter, Nr stands for buoyancy ratio parameter, Nc stands for bioconvection Rayleigh number, α is the curvature parameter, γ1 and γ3 are Deborah numbers, Burgers fluid parameter is represented by γ2, M is the magnetic parameter, Pr denotes Prandtl number, Lewis number is denoted by Le, Rd is radiation parameter, θw denotes temperature ratio parameter, thermophoresis parameter is Nt, Brownian motion parameter is expressed as Nb, σ∗ is the reaction parameter, δ is temperature difference parameter, E represents activation energy, bioconvection Lewis number is denoted by Lb, Pe is Peclet number, microorganisms difference parameter is δ, α1 is the thermal Biot number, α2 is the concentration Biot number, and α3 denotes the microorganisms Biot number, which are given as follows:(13)S=l2β∗∗g∗1−C∞Tf−T0zU02ρf,Nr=ρp−ρfCf−Co1−C∞Tf−T0,Nc=γ∗ρm−ρfNf−N01−C∞Tf−T0β∗∗,λ=1RνlU0,γ1=β1U0l,γ3=β3U0l,γ2=β2U0l2M=σlB02ρfU01/2,Le=α1DB,Pr=να1,θw=TfT∞,Rd=16σT∞33kk∗,Nt=τDTTf−T∞νT∞,Nb=τDBCf−C∞ν,σ∗=lKr2U0,δ=Tf−T0T∞,E=EakT,Lb=νDm,Pe=bWcDm,δ1=N∞Nf−N0,α1=hfkνlU0,α2=hgDBνlU0,α3=hnDmνlU0.The physical quantities of interest are defined as follows:(14)Nuz=zqmkTw−T∞,Shz=zjmDBCw−C∞,Snz=zqmDmNw−N∞,qm=−k∂T∂rr=R−16σT33k∗∂T∂rr=R,jm=−DB∂C∂rr=R,qm=−Dm∂N∂rr=R,NuzRe−1/2=−1+43Rd1+θw−1θ03θ′0,ShzRe−1/2=−ϕ′0,SnzRe−1/2=−χ′0.


## 3. Numerical Scheme

The numerical limitations of the dimensionless flow system (8)–(11), along with boundary restriction (12), are tackled numerically. As established equations are highly nonlinear, it is difficult to get an accurate solution. Therefore, we use a famous numerical scheme through bvp4c via MATLAB computational software. So, we have to renovate the higher-order BVP to 1st-order IVP.Let(15)f=h1,f′=h2,f″=h3,f‴=h4,fiv=h4′,θ=h5,θ′=h6,θ″=h6′,ϕ=h7,ϕ′=h8,ϕ″=h8′,χ=h9,χ′=h10,χ″=h10′,h4′=1+2λζ2γ12h1h2h3−h12h4−1+2λζλγ1h12h3−1+2λζ2γ23h12h32+2h1h22h3−4λ2γ2h4h3+1+2λζλγ23h12h2h3+h13h4+1+2λζ3γ3h32−4λγ31+2λζ2h1h4+1+2λζ3h4+1+2λζ22λh3+h1h3−h22+1+2λζ2A2−1+2λζ2M2−A+γ2h1h4−γ1h1h3+h2−Sh5−Nrh7−Nch9sinδ/21+2λζ3γ3h1−1+2λζ2γ2h13h6′=−Prh6h1−∈1h62−PrNbh8h61+2λζ−PrNth621+2λζRd1+h5θw−131+2λζ+∈1h5′,h8′=−2λh8−LePrh1h8−1+2λζNt/Nbh6′−2λNt/Nbh6+PrLeσ∗1+δh5nexp−E/1+δh5h7−∈2h821+2λζ+∈2h7,h10′=−2λh10−Lbh1h10+Peh8′h9+δ1+h10h81+2λζh1=0,h2=1,h6=−α11−h5ζ,h8=−α21−h7ζ,h10=−α31−h9ζ,atζ=0,h2⟶A,h3⟶0,h5⟶0,h7⟶0,h9⟶0,asζ⟶∞.


## 4. Results and Discussion

Salient features of mixed convection parameterS versus the velocity of fluid f′ are illuminated in Figure 2. It is viewed that the velocity field f′ is enlarged for rising values of mixed convection parameter for both values λ=0&0.3. Figure 3 reflects the consequence of the buoyancy ratio parameter Nr against the velocity profile f′. The buoyancy ratio parameter reduced the velocity of Burgers nanofluid f′ for both values λ=0&0.3. Figure 4 displays the valuation in the velocity profile f′ for bioconvection Rayleigh number Nc. It is scrutinized through the figure that the mounting magnitude of bioconvection Rayleigh number decays the velocity distribution for both cases λ=0&0.3. The behaviour of Burgers fluid parameter γ2 against the velocity field f′ is clarified in Figure 5. Here velocity field f′ reduced as the escalating value of the Burgers fluid parameter for both values λ=0&0.3. Figure 6 elucidates the effect of Deborah numbers γ3 versus the velocity field f′. It is noticed that the velocity of fluid f′ is increased by increasing values of Deborah numbers for both cases λ=0&0.3. Physically Deborah numbers γ3 depend upon retardation to time. As a result improvement in retardation times the Deborah number increase. The acceleration is induced in the flow of fluid and velocity field is boosted up. The influence of the magnetic parameter M versus the velocity field f′ is explicated in Figure 7. It is to be observed that velocity of fluid f′ decays by enhancing the variation in magnetic parameter for both cases λ=0&0.3. Consequently, the Lorentz forces are introduced via a larger magnetic parameter, so the flow of fluid reduces.Figure 2
Disparity ofS via f′.Figure 3
Disparity ofNr via f′.Figure 4
Disparity ofNc via f′.Figure 5
Disparity ofγ2 via f′.Figure 6
Disparity ofγ3 via f′.Figure 7
Disparity ofM via f′.Figure8 expounded the inspiration of Deborah numbers γ1 on the velocity of the fluid f′. It is perceived that the velocity of fluid f′ diminishes by enhancement in the magnitude of Deborah numbers for both cases of plate and cylinder λ=0&0.3. Deborah number is the ratio of relaxation to observation times; thus, relaxation time rises with increment in Deborah number, and as a result confrontation in liquid motion swells which leads to reducing the flow of fluid. Prominent attribution of temperature ratio parameter θw versus temperature distribution θ is shown in Figure 9. The temperature field θ is enlarged for a larger magnitude of fluid temperature ratio parameter θw for both plate and cylinder λ=0&0.3. The temperature ratio parameter improves the thermal state of liquid; therefore, the temperature field is improved. Figure 10 is deliberating the outcome of thermophoresis parameter Nt for the temperature concentration profile θ. It is pragmatic that augmentation in Nt boosted the temperature profile θ for both values λ=0&0.3. Figure 11 shows the variations in temperature profile θ for swelling values of the Prandtl number Pr. It is regarded through drafts that the growing variations of Prandtl number Pr fall off the temperature field θ for both cases λ=0&0.3. Physically the escalating value of Prandtl number causes a reduction in thermal diffusivity. Hence, the temperature field falls. Figure 12 is apprehended to examine the behaviour of thermal conductivity ε1 against temperature distribution θ. It is detected that the swelling variation of the thermal conductivity causes an upsurge in the temperature field θ. The impact of thermal stratification Biot number α1 against temperature distribution θ is accomplished in Figure 13. From the figure, it is initiated that the enhancing values of thermal stratification Biot number improved the temperature distribution for both plate and cylinder λ=0&0.3.Figure 8
Disparity ofγ1 via f′.Figure 9
Disparity ofθw via θ.Figure 10
Disparity ofNt via θ.Figure 11
Disparity ofPr via θ.Figure 12
Disparity of∈1 via θ.Figure 13
Disparity ofα1 via θ.Figure14 illuminates the significance of the Prandtl number Pr for the concentration field ϕ. The increasing valuation of the Prandtl number reduces the concentration field ϕ. Figure 15 is captured to perceive the nature of thermophoresis parameter Nt against the volumetric concentration field ϕ. The approximation in the thermophoresis parameter results in a boost in the volumetric concentration of nanoparticles ϕ. Physically the solid particles transfer from hot section to cold region due to developed thermophoresis valuation. Outstanding features of the Brownian motion parameter Nb for the concentration field ϕ are sketched in Figure 16. Concentration field ϕ dwindles by raise in the magnitude of the Brownian motion parameter for both values λ=0&0.3. Physically Brownian motion controls the diffusion of the solid particles in the system away from the boundary. Hence, improvement of Brownian motion parameter results in a decline of concentration field. Figure 17 explicates the consequences of Lewis number Le for the concentration field of nanoparticles ϕ. By the escalation of Lewis number Le, the concentration field ϕ diminishes. Figure 18 illuminates the significance of solutal conductivity ε2 against the concentration field ϕ of nanoparticles. It proposed that the concentration field of nanoparticles increased by the positive valuation of solutal conductivity for both values λ=0&0.3. Figure 19 describes the consequence of activation energy E versus the concentration profile of nanoparticles ϕ. It is anticipated that the concentration field of nanoparticles is amplified by the growing values of activation energy for both cases λ=0&0.3. The upshot of solutal Biot number α2 on the concentration field ϕ of nanomaterials is portrayed in Figure 20. It is estimated that the enhancing values of activation energy augmented the concentration field of nanoparticles for both cases λ=0&0.3. Figure 21 finds the outcome of Peclet number Pe against microorganism’s concentration field χ. It can be scrutinized that the microorganism’s field χ declined for higher variations of Peclet number Pe. The result of the microorganism stratification Biot number α3 against microorganism’s concentration field χ is sketched in Figure 22. The microorganism’s field χ is heightened for advanced values of microorganism stratification Biot number for both values λ=0&0.3. Figure 23 designates the conclusion of bioconvection Lewis number Lb against microorganism’s concentration field χ. It is inspected that the microorganism’s field χ is deteriorated for higher values of bioconvection Lewis number for both plate and cylinder λ=0.0&0.3.Figure 14
Disparity ofPr via ϕ.Figure 15
Disparity ofNt via ϕ.Figure 16
Disparity ofNb via ϕ.Figure 17
Disparity ofLe via ϕ.Figure 18
Disparity ofε2 via ϕ.Figure 19
Disparity ofE via ϕ.Figure 20
Disparity ofα2 via ϕ.Figure 21
Disparity ofPe via χ.Figure 22
Disparity ofα3 via χ.Figure 23
Disparity ofLb via χ.The numerical outcomes of the skin friction coefficient, local Nusselt number, local Sherwood number, and local density number of motile microorganisms against developed parameters are captured through Tables1–5. In Table 1, we accomplished the comparison. From Table 2, the local skin friction coefficient succeeds with M while reducing for S. In Table 3, it can be noticed that the local Nusselt number turns down for a larger amount Rd. From Table 4, it is scrutinized that the local Sherwood number boosted up with various values of α2. The rescaled density number of motile microorganisms is diminishing function of γ3, which is shown in Table 5.Table 1
Comparison table for variation of−f″0 for distinct values of M in special case when Nt=Nb=Pe=Lb=λ=0.
MShehzad et al. [49]Hayat et al. [50]Khan et al. [51]Present study0.01.000001.000001.000001.000000.21.019801.019801.0198011.0198070.51.118031.118031.1180291.1180300.81.280631.280631.2806331.2806341.01.414211.414211.4142211.4142221.21.562051.562051.5620481.5620481.51.803031.803031.8030441.803045Table 2
Numerical result of local skin friction−f″0 Versus M, S, Nr, Nc, γ3, γ1, and γ2.
Parameters−f″0MSNrNcγ3γ1γ2λ=0.0λ=0.30.10.20.10.10.30.11.00.82410.98380.60.96431.08331.21.09661.18370.50.10.10.10.30.11.00.84140.85181.00.97721.31682.01.10151.46370.50.20.20.10.30.11.00.84930.85611.01.31271.40612.01.67141.71790.50.20.10.20.30.11.00.83840. 86901.01.01430.98212.01.02441.29690.50.20.10.10.10.11.00.95841.08530.60.91040.98291.20.85480.97390.50.20.10.10.30.11.00.91011.03390.61.00931.12491.21.11511.21500.50.20.10.10.30.10.10.91721.04070.60.97311.21151.21.04521.3665Table 3
Numerical results of−θ′0 corresponding to M, S, Nc, Nr, Nt, NbPr,Rd, λ1, and Le.
MSNcNrNtNbPrRdα1Le−θ′0λ=0.0λ=0.30.10.20.10.10.30.21.20.80.32.00.18270.17720.60.17880.17461.20.17540.17210.50.20.10.10.30.21.20.80.32.00.18400.17581.60.18820.17672.00.18280.17690.50.20.20.10.30.21.20.80.32.00.18480.17591.00.17200.16552.00.16830.15490.50.20.10.20.30.21.20.80.32.00.18480.17591.00.17320.17542.00.16190.17470.50.20.10.10.10.21.20.80.32.00.18140.17670.60.17650.17071.20.17030.16320.50.20.10.10.30.11.20.80.32.00.18020.17600.60.17710.17231.20.17350.16810.50.20.10.10.30.22.00.80.32.00.20050.19633.00.21580.21234.00.22530.22230.50.20.10.10.30.21.20.10.32.00.19960.19630.60.18430.18081.20.17150.17950.50.20.10.10.30.21.20.80.12.00.08210.08130.60.25290.24381.20.31580.30130.50.20.10.10.30.21.20.80.31.00.17960.17511.60.17950.17502.20.17940.1749Table 4
Numerical results of local Sherwood number−ϕ′0 corresponding to M, S, Nc, Nr, Nt, Nb, Pr, γ1, γ2, Le, and α2.
MSNcNrNtNbPrγ1γ2Leα2−ϕ′0λ=0.0λ=0.30.10.20.10.10.30.21.20.30.32.00.30.14940.14500.60.14740.14391.20.14300.14280.50.20.10.10.30.21.20.30.32.00.30.14820.14761.60.14600.14582.00.14240.14190.50.20.20.10.30.21.20.30.32.00.30.14860.14681.00.14520.14492.00.14380.14250.50.20.10.20.30.21.20.30.32.00.30.14790.14691.00.14600.14512.00.14510.14460.50.20.10.10.10.21.20.30.32.00.30.16100.16160.60.13010.13841.20.12290.11680.50.20.10.10.30.11.20.30.32.00.30.12810.12090.60.16140.15961.20.16470.16840.50.20.10.10.30.22.00.30.32.00.30.15870.15533.00.16620.16364.00.17080.16870.50.20.10.10.30.21.20.10.32.00.30.14880.14430.80.14760.14341.60.14610.14240.50.20.10.10.30.21.20.30.12.00.30.14830.14440.80.14760.14331.60.14610.14220.50.20.10.10.30.21.20.30.31.00.30.12210.11581.60.14060.13582.20.15100.14730.50.20.10.10.30.21.20.30.32.00.10.08040.07880.60.23680.23091.20.49450.4899Table 5
The numerical results of−χ′0 for M,S, Nr, Nc, γ3, Pe, Lb, and α3.
Parameters−χ′0MSNrNcγ3PeLbα3λ=0.0λ=0.30.10.20.10.10.30.11.00.30.20360.19970.60.20010.19751.20.19770.19530.50.10.10.10.30.11.00.30.20140.19181.00.19270.18482.00.18150.18370.50.20.20.10.30.11.00.30.20390.19301.00.20270.19162.00.20130.19970.50.20.10.20.30.11.00.30.20480.29291.00.20430.18122.00.20340.18890.50.20.10.10.10.11.00.30.20090.19750.60.20150.19521.20.20230.19310.50.20.10.10.30.21.00.30.20270.19941.00.21370.21092.00.22520.22270.50.20.10.10.30.11.20.30.20840.20552.00.22710.22513.00.23950.23820.50.20.10.10.30.11.00.10.08600.08170.60.30320.29931.20.4047As a significance, from these tables, we are assured that the recent results are very accurate.


## 5. Concluding Remarks

The thermal and solutal conductivity phenomena for magneto-Burgers nanofluid with swimming motile microorganisms were established. The eminent shooting method is employed to crack the flow problems of magneto-Burgers nanofluid via bvp4c solver in computational software MATLAB. The main outcomes are highlighted as follows:(i)
It is scrutinized that the velocity of magneto-Burgers nanofluid signifies reducing trend for higher buoyancy ratio parameter, Burgers fluid parameter, and bioconvection Rayleigh number.(ii)
An improvement in the mixed convection parameter and Deborah numbers enhances velocity field.(iii)
Temperature distribution declines with a larger Prandtl number, while an inverse trend is shown for thermal Biot number.(iv)
An increment in the variations of solutal conductivity and activation energy enhances the volumetric concentration of nanoparticles.(v)
The rescaled microorganism’s field exaggerates with microorganisms Biot number while it dwindles for Peclet number and bioconvection Lewis number.



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