# Zero and Nonzero Mass Flux Effects of Bioconvective Viscoelastic Nanofluid over a 3D Riga Surface with the Swimming of Gyrotactic Microorganisms

Authors: T. S. Karthik; K. Loganathan; A. N. Shankar; M. Jemimah Carmichael; Anand Mohan; Mohammed K. A. Kaabar; Safak Kayikci
Journal: Advances in Mathematical Physics
 (2021)

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## Abstract

This work addresses 3D bioconvective viscoelastic nanofluid flow across a heated Riga surface with nonlinear radiation, swimming microorganisms, and nanoparticles. The nanoparticles are tested with zero (passive) and nonzero (active) mass flux states along with the effect of thermophoresis and Brownian motion. The physical system is visualized via high linearity PDE systems and nondimensionalized to high linearity ordinary differential systems. The converted ordinary differential systems are solved with the aid of the homotopy analytic method (HAM). Several valuable and appropriate characteristics of related profiles are presented graphically and discussed in detail. Results of interest such as the modified Hartmann number, mixed convection parameter, bioconvection Rayleigh number, and Brownian motion parameter are discussed in terms of various profiles. The numerical coding is validated with earlier reports, and excellent agreement is observed. The microorganisms are utilized to improve the thermal conductivity of nanofluid, and this mechanism has more utilization in the oil refinery process.


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## Body

## 1. Introduction

“Bioconvection” is known to be the convective movement within sight of swimming microorganisms. In this convective mode, the cells with bottom-heavy cells tend to swim at an angle to vertical, and this process is known as gyrotactic [1]. Therefore, the gyrotactic microorganisms are stable in the upper layer of the fluid, and consequently, stratification of the top-heavy fluid layer will become unbalanced. Thus, the system, which consists of a gyrotactic microorganism, induces one of the exciting characters in heat transfer that is “stability.” The reason is that nanofluids that have higher stability tend to improve the thermal efficiency of the heat exchanger (any energy systems). Hosseinzadeh et al. [2] examined the gyrotactic microorganism influence over a cylindrical surface with cross fluid flow. Mogharrebi et al. [3] present the MHD nanofluid flow towards a rotating cone with motile oxytactic microorganisms. Nowadays, the research on nanofluid through a Riga plate becomes an exciting area of research. For instance, mixed convective nanofluid flowed a Riga plate is studied numerically and analytically in [4]. It is shown that the desired size of the nanoparticle influences the skin friction coefficient. Ahmad et al. [5] studied the role of nanofluid past a heated vertical Riga plate numerically. Influence on viscous dissipation and thermal radiation of nanofluid flow between the Riga plate is explored numerically [6]. They have used carbon nanotubes as the nanoparticle, and it is shown that by varying the radiation parameter, the local heat transfer rate elevates. The application of the Cattaneo-Christov approach heat generation and absorption for the second-grade fluid that passed through the Riga plate is presented numerically [7]. In line with the application of the Riga plate, the stagnation flow on the vacillating Riga plate is studied numerically [8]. The variable thicked Riga plate for melting heat transfer application is explored numerically, and it is reported that for higher values of modified Hartmann number, the velocity profile distribution increases [9]. Recently, several research papers have been devoted to the study of nanofluid and their applications in practical situations [10–13].Further, the study on heat transfer characteristics of nanofluid is widely studied. For instance, viscoelastic fluid with Newtonian heating is numerically studied [14–17]. It is shown that Al2O3/water nanofluid exhibits higher performance evaluation criteria. Besides, the study of viscoelastic nanofluid through the Riga plate is extensively studied. A nonuniform heat flux unsteady viscoelastic fluid that is unsteady is numerically investigated [18]. The viscoelastic nanofluid over an unsteady surface that is stretchable is evaluated numerically [19]. Also, the magnetic field effect on the Maxwell viscoelastic nanofluid over a plate that is moving at a uniform velocity is numerically assessed [20]. The temperature and velocity relaxation time influencing the heat transfer rate of the nanofluid are concluded by them. Besides that, the nonlinear effects on the MHD stagnation flow of viscoelastic nanofluid are explored numerically [21]. The Homotopy Analysis Method (HAM) is employed by Hayat et al. [22] to solve the 3D flow of a viscoelastic nanofluid over a stretching surface. The same research group extended their work towards the viscoelastic model for various applications [23–25]. A slew of researchers studies the gyrotactic microorganism impact. For example, mixed convection of nanofluid containing third-grade nanomaterial containing gyrotactic microorganisms is figured out numerically [26]. Walters B nanofluid with the incorporation of gyrotactic microorganisms is evaluated numerically [27]. Acharya et al. [28] reported the effects of solar radiation bioconvection nanofluid with gyrotactic microorganisms. The suspension of microorganisms induced with the effect of the magnetic field is reported [29, 30]. Also, numerous researches on gyrotactic microorganisms are studied and explored [30–35]. Analysis of active and passive controls with the chemical reaction of nanofluid is analyzed [36]. Using the homotopy perturbation method, unsteady nanofluid with active and passive controls is developed by Acharya et al. [37]. Over a bent surface, the active-passive control of dihydrogen monoxide nanofluid is explored numerically [38]. These controls are studied in various nanofluids with diverse applications [39–44].By considering the earlier reports, it is concluded that no studies were found to analyze the bioconvection of viscoelastic nanofluid on a 3D Riga surface with a comparison of active and passive control. The present study explores the 3D viscoelastic nanofluid flow across the heated Riga surface with nonlinear radiation and heat generation/absorption effects. The effects of several parameters are compared with the active and passive control model of nanoparticles. The homotopy analytic method [45–51] is employed to study the present nonlinear ODE systems. The results are discussed in terms of various profiles. The numerical coding of the present study is validated with earlier reports. The relevant research is applied to multiple engineering streams like bioengineering, chemical, nuclear, thermal, and mechanical.


## 2. Problem Development

We have considered bioconvective viscoelastic nanofluid flow withuw=ax in the x direction vw=by in the ydirection over a 3D Riga surface with gyrotactic microorganisms swimming. The surface is expanding in all three directions, namely, x,y, and z. It is assumed that nanoparticles do not influence the swimming microorganism; also, the nanoparticles are assumed to be stable in the fluid layer. Also, the nanoparticles have no effect on the velocity and temperature of the swimming microorganisms. The geometrical configuration is shown in Figure 1. By considering the above assumptions, the governing equations are described below [42]:
(1)∂u∂x+∂v∂y+∂w∂z=0,u∂u∂x+v∂u∂y+w∂u∂z=υ∂2u∂z2−αu∂3u∂x∂z2+w∂3u∂z3−∂u∂x∂2u∂z2−∂u∂z∂2w∂z2−2∂u∂z∂2u∂x∂z−2∂w∂z∂2u∂z2+1ρf1−C∞ρfβgT−T∞−ρp−ρfgC−C∞−n−n∞gωρm−ρf+πj0M0exp−π/ez8ρ,u∂v∂x+v∂v∂y+w∂v∂z=υ∂2v∂z2−αv∂3v∂y∂z2+w∂3v∂z3−∂v∂y∂2v∂z2−∂v∂z∂2w∂z2−2∂v∂z∂2v∂y∂z−2∂w∂z∂2v∂z2,ρcpu∂T∂x+v∂T∂y+w∂T∂z=k∂2T∂z2+τDB∂C∂z∂T∂z+DTT∞∂T∂z2−∂qr∂z+Q0ρcpT−T∞,u∂C∂x+v∂C∂y+w∂C∂z=DB∂2C∂z2+DTT^∞∂2T∂z2−KmC−C∞,u∂N∂x+v∂N∂y+w∂N∂z−DN∂2N∂z2=−dWcCw−C∞∂∂zN∂C∂z.Figure 1
Geometry of flow problem.With boundary conditions(2)ux,y,0=ax,vx,y,0=by,wx,y,0=0,−k∂T∂zx,y,0=hfTfx,y,0−Tx,y,0,Cx,y,0=Cwx,AC,DB∂C∂zx,y,0+DTT∞∂T∂zx,y,0=0,PCNx,y,0=Nwx,(3)ux,y,∞=0,vx,y,∞=0,∂u∂zx,y,∞=0,∂v∂zx,y,∞=0,Tx,y,∞=T∞,Cx,y,∞=C∞,Nx,y,∞=N∞.Now, we state the nondimensional similarity variables below:(4)f1′η=uax,η=aν0.5z,g1′η=vay,f1η+g1η=−waν0.5,θη=T−T∞Tf−T∞,ϕη=C−C∞Cw−C∞AC,ϕη=C−C∞C∞PC,ξη=N−N∞Nw−N∞.Using the transformations in Equation (1), we get
(5)f1‴−f1′2+f1+g1f1″+γf1+g1f1iv+f1″−g1″f1″−2f1′+g1′f1‴+λθ1−Nrϕ1−Rbξ1+Qe−d1η=0,g1‴−g1′2+f1+g1g1″+γf1+g1g1iv+f1″−g1″g1″−2f1′+g1′g1‴=0,1Prθ1″+RdPrθ1θw−1+123θ1′2θw−1+θ1θw−1+1θ1″+f1+g1θ1′+Nbϕ1′θ1′+Ntθ1′2+Hgθ,ϕ1″+Lef1+g1ϕ1′+NtNbθ1″−LeCrϕ1=0,ξ1″+Lbf1+g1ξ1′−Peϕ1″ξ1+Ω+ϕ1′ξ1′=0.Boundary condition (2) and (3) in expressions of f1,g1,θ1,ϕ1,andξ1 is developed:
(6)f10=0,g10=0,f1′0=1,g1′0=ϵ,f1′∞=0,g1′∞=0,f1″∞=0,g1″∞=0,θ1′0=−Bi1−θ10,θ1∞=0,Nbϕ1′+Ntθ1′=0,ϕ1∞=0PC,ϕ10=1,ϕ1∞=0AC,ξ10=1,ξ1∞=0.The dimensionless variables are(7)γ=αaν,ϵ=ba,Pr=ρcpk,Bi=hfkνa,Nb=τDBνC∞,Nt=τDTνTf−T∞,Cr=Kma,Hg=Q0ρcpa,Q=πj0M08a3xρ,θw=TwT∞,Lb=νDm,Pe=dWcDm,λ=βω1−C∞Tw−T∞auw,Nr=ρp−ρfβρfCw−C∞Tf−T∞,Rb=ωNw−N∞ρm−ρfβρf1−C∞Tf−T∞.The nondimensional structure of surface drag forceCfx&Cfy and heat transfer rate Nu, mass transfer rate Sh, and microorganism Nn density number is stated as
(8)CfxRe0.5=f1″+γ2f1′f1″−f1‴f1+g1+f1′+g1′f1″η=0,CfyRe0.5=g1″+γ2g1′g1″−g1‴f1+g1+f1′+g1′g1″η=0,NuRe−0.5=−1+43Rdθw3η=0,ShRe−0.5=−ϕ1′η=0AC,ShRe−0.5=NtNbθ1′η=0PC,NnRe−0.5=−ξ1′η=0.


## 3. Solution Approach: Homotopy Analysis Method (HAM)

The primary assumptions of the homotopy analytic method are stated as follows:(9)f10=1−exp−η,g10=ϵ∗1−exp−η,θ10=Bi∗exp−η1+Bi,ϕ10=exp−ηAC,ϕ10=−NtNb∗Bi∗exp−η1+BiPC,ξ10=exp−η.The auxiliary linear operatorsLf1, Lg1, Lθ1, Lϕ1, and Lξ1 are derived as
(10)Lf1=f1‴η−f1′η,Lg1=g1‴η−g1′η,Lθ1=θ1″η−θ1η,Lϕ1=ϕ1″η−ϕ1η,Lξ1=ξ1″η−ξ1η.The above linear operators satisfying(11)Lf1S1+S2eɳ+S3e−ɳ=0,Lg1S4+S5eɳ+S6e−ɳ=0,Lθ1S7eɳ+S8e−ɳ=0,Lϕ1S9eɳ+S10e−ɳ=0,Lξ1S11eɳ+S12e−ɳ=0.The appropriate solutionsf1m∗,g1m∗,θ1m∗,ϕ1m∗,ξ1m∗ are
(12)f1mη=f1m∗η+S1+S2eɳ+S3e−ɳ,g1mη=g1m∗η+S4+S5eɳ+S6e−ɳ,θ1mη=θ1m∗η+S7eɳ+S8e−ɳ,ϕ1mη=ϕ1m∗η+S9eɳ+S10e−ɳ,ξ1mη=ξ1m∗η+S11eɳ+S12e−ɳ,where Sjj=1−12 denote the arbitrary conditions.


## 4. Convergence Analysis

To analyze the convergence of the numerical code, the convergence analysis is carried out for the value ofh. The variable h has control over the convergence and divergence of the numerical code, and it is shown in Figure 2. In the case of active and passive control, the values of hf1,hg1,hφ1,hθ1,hξ1 in the range of −2.0≤hf1,hg1,hφ1,hθ1,hξ1≤0.5. For the convergence study in the direction of x and y axes, the value of hf1,hg1 is in the range of -2.0 to 0.5, respectively. The order of approximation is shown in Tables 1 and 2. Table 3 reports that the value of f1′′0andg1′′0 is compared with earlier results with Qayyum et al. [14].Figure 2
Convergence plots of the HAM solution.
(a)(b)(c)Table 1
Order of approximation for−f1″0 and −g1″0.
Order of approximation−f1″0−g1″0ActivePassiveActivePassive11.10621.06421.00621.043751.14101.09970.99001.0467101.14091.10190.98921.0456151.14081.10230.98921.0455201.14081.10250.98921.0455251.14081.10250.98921.0454301.14081.10250.98921.0454351.14081.10250.98921.0454Table 2
Order of approximation for−θ1′0, −ϕ1′0, and −ξ1′0.
Order of approximation−θ1′0−ϕ1′0−ξ1′0ActivePassiveActivePassiveActivePassive10.31410.32410.975−0.21610.9170.636350.30100.31960.9274−0.21310.85850.4737100.30030.31930.9334−0.21290.86530.4657150.30020.31930.9336−0.21290.86570.4645200.30020.31930.9336−0.21290.86570.4641250.30020.31930.9336−0.21290.86570.4641300.30020.31930.9336−0.21290.86570.4641350.30020.31930.9336−0.21290.86570.4641Table 3
Comparison of−f1″0 and −g1″ (0) for various values of ϵ with limiting conditionsQ=d1=λ=Nr=Rb=0.
ϵ−f1″0−g1″(0)Qayyum et al. [14]PresentQayyum et al. [14]Present0.01.0000001.000000.0000000.000000.11.0202591.020260.0669470.066850.21.0394951.039500.1487360.148740.31.0579541.057950.2433590.243360.41.0757881.075790.3492080.349210.51.0930951.093090.4652040.465210.61.1099461.109950.5905280.590530.71.1263971.126400.7245310.724530.81.1424881.142490.8666820.866680.91.1582531.158251.0165381.016541.01.1737201.173711.1737201.17371


## 5. Results and Discussions

In this work, viscoelastic nanofluid past a stretching 3D Riga surface with gyrotactic microorganisms is explored numerically. The governing parameters such as velocity (f1,g1) parameters, temperature (θ1) parameter, ξ, modified Hartmann number (Q), viscoelastic parameter (γ), mixed convection parameter (λ), buoyancy ratio parameter (Nr), bioconvection Rayleigh number Rb, bioconvection Peclet number (Pe), heat generation/absorption parameter (Hg), thermophoresis parameter (Nt), and microorganism concentration difference parameter (Ω) are studied.Figure3(a) describes the impact of Q (modified Hartmann’s number) on the velocity profile. It is observed that with an increase in Q, velocity profiles decrease in both directions (x,y). This is due to the increase in Q leading to an increase in Lorentz force and consequently velocity profile decreases. In Figure 3(b), it is noticed that as the viscoelastic parameter (γ) increases, the velocity profile decreases. In general, tensile stress is generated by viscoelasticity of the fluid. This stress opposes the fluid motion, and finally, the velocities of x and y directions are decays when the values of γ are improved. The increase in velocity profile is noted in Figure 3(c) with an enhancement in the mixed convection parameter (λ). By definition, λ is the ratio between the buoyancy force and the viscous force. Figure 3(d) reveals that an increase in buoyancy ratio parameter velocity profile decreases. In this work, the thermal and concentration forces are considered, and those forces provide smaller resistance consequently due to this reason that the velocity profile decreases. Also, it is observed in Figure 3(e) that the extending values of bioconvection Rayleigh number have a tendency to diminish the velocity profile. In Figure 3(f), it is noted that an increase in the bioconvection Peclet number produces the diminishing performance in the swimming speed of microorganisms that cause a decreasing trend in the velocity profile.Figure 3
Velocity profile onx and y directions for various values of Q,γ,λ,Nr,Rb, and Pe.
(a)(b)(c)(d)(e)(f)From Figure4(a), it is evident that the increase in temperature profile increases with an increase in bioconvection Rayleigh number (Rb). In fact, for larger values of Rb, boundary layer thickness increases which increases the buoyancy forces; subsequently, temperature profile increases for both the cases of active and passive controls. The impact of temperature profile with regard to the thermophoresis parameter Nt is discussed in Figure 4(b). With an increase in the thermophoresis parameter, Nt temperature profile is increased. With an increase in Nt, more nanoparticles shifted to the colder place from the hotter one, so the temperature profile increases. An increase in temperature ratio parameter θw increases the temperature profile as observed in Figure 4(c). The ratio of temperature at the surface (Tw) to the temperature at free stream (T∞) is mathematically defined as θw=Tw/T∞. For nonlinear radiation, the value of θw must be higher than 1. Moreover, an increasing trend found in the temperature along the surface is noted for larger θw values. Consequently, thermal boundary and associated layer thickness improve. The effect of the heat generation/absorption parameter, Hg on the temperature profile, is proclaimed in Figure 4(d). As we increase Hg, more movement between the nanoparticles is reported; therefore, the temperature profile increases. The slowing moment of the liquid flow due to the modified Hartmann number Q leads to producing a heat energy, and it is highly pronounced in the lesser value of modified Hartmann number Q, as seen in Figure 4(e).Figure 4
Temperature profiles for various values ofRb,Nt,Hg,θw, and Q.
(a)(b)(c)(d)(e)Figure5 illustrates the role of nanoparticle volume fraction on (a) thermophoresis parameter (Nt) and (b) Brownian motion parameter (Nb), for active and passive control cases. It is noted that for both behaviours of active and passive control, the Nb value increases. As we increase the Nb value, more resistive force is applied to the surface of the nanoparticle; thus, to increase the heat transfer, the nanoparticle volume fraction has to be increased. This increasing trend is reported in Figure 5(a) for both the cases of active and passive controls. As overturn to the above discussion, it is noted in Figure 5(b) that the volume fraction of nanoparticles decreases with an increase in the Brownian motion parameter, Nb. The fact is that the larger values of the Brownian motion parameter increase the length of the mean free path of the nanoparticle, which incredibly decreases the volume fraction of the nanoparticles.Figure 5
Nanoparticle concentration profile for various values ofNtandNb.
(a)(b)The influence of the bioconvection Rayleigh number on the concentration of microorganisms is shown in Figure6(a). With the increase in the Rayleigh number, the movement of microorganisms is increased; consequently, the concentration of microorganisms also increases. The role of the microorganism’s concentration difference parameter Ω in concentration is explored in Figure 6(b). For both the cases of active and passive controls, the trend is reported to be the same. In Figure 6(c), it is seen that the concentration of microorganisms increases as the bioconvection Lewis number Lb decreases. It is due to the variation in the temperature difference between the nanoparticles. It is noted in Figure 6(d) that the trend of active and passive control is in an opposite direction for the variation in the bioconvection Peclet number, Pe. As we increase the value of Pe, the diffusivity of microorganisms is decreased. Therefore, the concentration of microorganisms is decreased for passive control. Whereas the scenario is in active control, the trend is increased. The thermophoresis parameter Nt effect on concentration of microorganisms is plotted in Figure 6(e). With the increase of Nt, the low heated nanoparticle is moved into the high heated surface which is induced by the increase in Nt. Hence, the concentration on microorganism is increased. In final, the variation in mixed convection parameter λ is analyzed in Figure 6(f). The increase in λ and the concentration on microorganisms is decreased. While for larger values of λ, temperature variation between the nanoparticles increases; therefore, this strongly affects the concentration of microorganisms.Figure 6
Motile density profile for various values ofRb,Ω,Lb,Pe,Nt, and λ.
(a)(b)(c)(d)(e)(f)Figures7 and 8 depict the heat transfer rate (Nux) for different combinations of pertinent parameters. From Figure 7, the heat transfer rate enhances for higher values of modified Hartmann number (Q), and it decays for radiation parameter (Rd). Figure 8 elucidates that heat transfer rates reduce for larger values of γ and Hg. Figure 9 explores the combination of Nb and Nt in the mass transfer rate (Shx). It is noted from this figure that the mass transfer rate enhances for the Brownian motion parameter for active control while an opposite behaviour is noticed for the thermophoresis parameter for both cases. Figures 10 and 11 portray the motile microorganism’s density number rate (Nnx) for different related parameters. From these figures, we found that the combined parameters RbandΩ and PeandLb show the inverse effects on the microorganism’s density number rate for active and passive controls.Figure 7
The effect of Nusselt number for combined parametersRdandQ.
(a)(b)Figure 8
The effect of Nusselt number for combined parametersHgandγ.
(a)(b)Figure 9
The effect of Sherwood number for combined parametersNtandNb.
(a)(b)Figure 10
The effect of motile microorganism’s density number for combined parametersRbandΩ.
(a)(b)Figure 11
The effect of motile microorganism’s density number for combined parametersLbandPe.
(a)(b)


## 6. Conclusion

In this work, the active and passive control of viscoelastic nanofluid flow over a 3D Riga plate is investigated analytically by the homotopy technique. Effects are studied with the influence of the gyrotactic microorganism in bioconvective heat transfer. The following outcomes are observed:(i)
An increase in the modified Hartmann number produces the diminishing impact on velocity profile for bothx and y directions(ii)
Fluid velocity decays in bothx and y directions for the augmentation in bioconvection Rayleigh number(iii)
The microorganism profile is an increasing function of bioconvection Peclet number for active control, and it is decreasing for passive control(iv)
Mass transfer rate enhances for Brownian motion parameter while an opposite behaviour is noticed for thermophoresis parameter for both cases(v)
An increase in bioconvection Peclet number produces the inverse phenomena in microorganism profile for active and passive controls



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