Investigation of suspended nanoliquid flow of Eyring–Powell fluid with gyrotactic microorganisms and density number

The aim of the current investigation is to discuss the behavior of mixed convection magnetohydrodynamic flow of Eyring–Powell nanoliquid subjected to gyrotactic microorganisms over a stretchable cylinder. Energy communication is developed through the first law of thermodynamics and deliberated in the manifestation of viscous dissipation. Furthermore, Brownian motion and thermophoresis effects are also considered. Nonlinear system of partial differential equations is altered into ordinary one due to employing transformations. The given systems are then solved through ND-solve technique. Impact of influential variables on velocity, motile microorganism’s temperature, and concentration is deliberated graphically. Skin friction coefficient, mass transfer rate, density number, and Nusselt number are numerically computed versus different influential variables. Velocity and temperature have opposite impact for curvature parameter. For higher estimation of fluid parameter, temperature and velocity fields boost up.


Introduction
Study of rheological characteristics of non-Newtonian liquids is very monotonous than compared to viscous liquids. A single constitutive equation is not appropriate to scrutinize the non-Newtonian liquids because of their complex and diverse behaviors. Recently, numerous investigators and scientists have focused their consideration on non-Newtonian liquids. It is because of their vast applications in engineering, biology, and industries like fiber sheets, wire drawing, glass formation, paper production, and crystal growth. Common examples of non-Newtonian liquids are shampoo, ketchup, yogurt, mud, greases, pasta, certain oils, paints, and so on. Initially, in 1944, Eyring and Powell proposed a Eyring-Powell model which is based on kinetic theory of fluids. Influence of radiation on magnetohydrodynamic (MHD) flow of Eyring-Powell liquid flow due to a stretchable surface is highlighted by Hayat et al. 1 Hosseinzadeh et al. 2 worked on hybrid nanoparticles in a hexagonal triplex latent heat with fine effects. Impact of heat flux on an Eyring-Powell liquid flow due to shrinking surface is examined by Ara et al. 3 Entropy optimization in reactive flow of Eyring-Powell liquid with variable thermal conductivity by a stretchable surface is discussed by Salawu et al. 4 Some fruitful researches about nanomaterials in flow of non-Newtonian and Newtonian fluids are highlighted in some of the studies. [5][6][7][8][9][10][11][12][13][14][15][16] Colloidal suspension of nano-sized particles (oxides, metals, carbides, or carbon nanotubes) and conventionally working materials (water, oil, and ethylene glycol) are known as nanomaterial. Nanomaterials have innovative behaviors that make them more significant in various applications in thermal transmission like medicinal procedures, domestic refrigerators, hybrid-powered engines, fuel cells, heat exchangers, and automobile thermal management. Choi and Eastman 17 are the first who theoretically proved that heat conduction phenomenon of conventionally working materials can be increased by inserting nano-sized particles. Khan et al. 18 discussed the behavior of thermophoresis and Brownian diffusion in Prandtl-Eyring nanoliquids with entropy optimization and cubic autocatalysis chemical reaction. Some investigations made by numerous researchers are presented in some of the studies. 2,[19][20][21][22][23][24][25][26] Bioconvection is a phenomenon in which microorganisms are inserted in nanoliquids for nanoparticles movement. It is presumed that nanoparticles have no impact on the spinning direction and motion of microorganisms. Initially, Kuznetsov 27 studied fluid layers of finite depth with suspended gyrotactic microorganisms. Impact of magnetic field, stratification phenomenon, and gyrotactic microorganisms on Maxwell nanoliquid flow is highlighted by Khan et al. 28 Khan et al. 29 explored the behavior of Darcy-Forchheimer mixed convective over a curved sheet with activation energy and entropy generation. Heat and mass transport over a convective stretched sheet with gyrotactic microorganisms and stratification phenomena is examined by Alsaedi et al. 30 Khan et al. 31 investigated the effect of gyrotactic microorganisms and activation energy on natural bioconvective flow of Sisko nanofluids.
In this article, we investigated the behavior of mixed convective MHD flow of Eyring-Powell nanoliquid subjected to gyrotactic microorganisms over a stretchable cylinder. Energy attribution is developed through the first law of thermodynamics. Brownian diffusion and thermophoretic effects are also accounted. The gyrotactic microorganisms concept is used to control the random motion of fluid nanoparticles. Heat, motile microorganisms, and mass transfer rates are examined subjected to stratification effects. Partial differential system is altered to ordinary system by suitable transformations and then tackle through numerical built in ND-solve method. [32][33][34][35][36][37][38][39] Features of influential variables on velocity, motile microorganisms, temperature, and concentration are examined through graphs. Surface drag force, gradient of temperature, Sherwood, and density numbers are numerically computed and discussed.

Mathematical modeling
Consider incompressible, two-dimensional, and steady MHD mixed convective flow of Eyring-Powell nanomaterials by a stretchable surface of cylinder. Microorganisms are exploited to control the motion of fluid nanoparticles. Furthermore, dissipation is taken into consideration in modeling of heat equation. Heat, mass, and motile microorganisms transfer rates are discussed in the presence of stratification effects. Let u w ð¼ w 0 z=lÞ be the stretching velocity along z-direction. A constant magnetic field of strength ðB 0 Þ is exerted at an inclination of a to the cylinder. The flow diagram is highlighted in Figure 1.
The governing layer expressions in view of aforementioned assumptions are (2) where u; w show the velocity components in r-and z-direction, respectively; l is the dynamic viscosity; q f is the density of nanoparticles; q f 1 is the ambient density of nanofluid; q m is the density of microorganisms; b and c are the Eyring-Powell fluid parameters; g is the gravitational acceleration; b c is the concentration expansion coefficient; b T is the thermal expansion coefficient; q p is the density of nanoparticles; T is the temperature; T 1 is the ambient temperature; T 0 is the reference temperature; T w is the surface temperature; C is the concentration; C 1 is the ambient concentration; C 0 is the reference concentration; C w is the surface concentration; n is the concentration of microorganisms; n 1 is the ambient concentration of microorganisms; n 0 is the reference concentration of microorganisms; n w is the surface concentration of microorganisms; b is the chemotaxis constant; c is the average volume of microorganisms; D B is the coefficient of Brownian diffusion; D T is the thermophoresis; W c is the maximum speed of microorganisms cells; r is the electrical conductivity; R is the radius of the cylinder; a 1 , b 1 , c 1 , a 2 , b 2 , and c 2 are the dimensionless constants; and P is the pressure vector. Considering One can get with where c Ã ð¼ 1=Rð ffiffiffiffiffiffiffiffiffiffiffiffi ffi t f l=w 0 p ÞÞ is the curvature parameter; k ð¼ w 3 0 z 2 =2c 2 t f l 3 Þ and b Ã ð¼ 1=lbcÞ are the Eyring-Powell fluid material parameters; Ha ð¼ rB 2 0 l=w 0 q f Þ is the magnetic parameter; Pr ð¼ t f ðqc p Þ f =k f Þ is the Prandtl number; G t ð¼ gb T ðl 2 q f 1 =q f w 2 0 zÞðT w À T 0 ÞÞ is the thermal Grashof number; G c ð¼ gb c ðl 2 =q f w 2 0 zÞðq p À q 1 ÞðC w À C 0 ÞÞ is the concentration Grashof number; Rb ð¼ b c ðl 2 gc=q f w 2 0 zÞðn w À n 0 Þðq m À q f 1 ÞÞ is the bioconvection Rayleigh number; Nt ð¼ sD T ðT w À T 0 Þ= t f T 1 Þ is the thermophoresis parameter; Nb ð¼ sD B ðC w À C 0 Þ=t f Þ is the Brownian motion parameter; Ec ð¼ w 2 0 =c p ðT w À T 0 ÞÞ is the Eckert number; Pe ð¼ bW c =D n Þ is the bioconvection Peclet number; S 1 ð¼ a 2 =a 1 Þ is the thermal stratification parameter; S 2 ð¼ b 2 =b 1 Þ is the mass stratification parameter; S 3 ð¼ c 2 =c 1 Þ is the motile density stratification parameter; Lb ð¼ t f =D n Þ is the bioconvection Lewis number; Sc ð¼ t f =D B Þ is the Schmidt number; and X ð¼ n 1 =n w À n 0 Þ is the concentration difference of microorganisms.

Physical quantities
Coefficient of skin friction ðCf z Þ, Nusselt number ðNu z Þ, Sherwood number ðSh z Þ, and density number ðNn z Þ are expressed as where s r z is the shear stress, q w is the heat flux, q m is the mass flux, and q n is the density flux expressed as s rz ¼ l @w @r þ 1 bc @w @r À 1 6bc 3 @w @r 3 ; q w ¼ À k @T @r r¼R ; q m ¼ À D B @C @r r¼R ; q n ¼ À D n @n @r Finally, we can write Nu z Re À0:5 where Re z ð¼ z 2 w 0 =t f lÞ shows the local Reynolds number.

Results and discussions
In this article, we have employed Newton built-in shooting method to progress numerical results for the obtained nonlinear differential system. Furthermore, the salient effect of pertinent parameters on velocity, temperature, and concentration of microorganisms is examined through graphs. In this section, skin friction coefficient ðCf z Þ, gradient of temperature ðNu z Þ, Sherwood number ðSh z Þ, and density number ðNn z Þ are numerically computed through various interesting parameters.

Velocity
Characteristics of pertinent parameters like curvature parameter ðc Ã Þ, Eyring-Powell fluid material parameters ðb Ã Þ, magnetic or Hartmann number ðHaÞ, thermal Grashof number G t ð Þ, solutal Grashof number ðG c Þ, and bioconvection Rayleigh number ðRbÞ on velocity ðf 0 ðgÞÞ are delineated in Figures 2-7, respectively. Characteristic of c Ã on f 0 ðgÞ is portrayed in Figure 2. For larger c Ã , the radius of the cylinder decreases and consequently fluid contact area with cylinder decreases and as a result velocity decreases. Figure 3 Figure 5. Clearly, f 0 ðgÞ is a decreasing function of G t . Figure 6 discusses the effect of G c on velocity ðf 0 ðgÞÞ. As expected, velocity increases when an enhancement occurs in the concentration of Grashof number. Impact of bioconvection Rayleigh number ðRbÞ on f 0 ðgÞ is portrayed in Figure 7. One can find that f 0 ðgÞ decreased with rising values of bioconvection Rayleigh number ðRbÞ.  Figures 8-14, respectively, show the behaviors of Prandtl number ðPrÞ, thermal stratification parameter ðS 1 Þ, Brownian diffusion parameter ðNbÞ, thermophoresis parameter ðNtÞ, curvature parameter ðc Ã Þ, fluid material parameter ðb Ã Þ, and Eckert number ðEcÞ on temperature ðhðgÞÞ. Impact of Pr on hðgÞ is displayed in Figure 8. As expected, hðgÞ increases when an increment occurs in the Prandtl number. Figure 9 shows the impact of S 1 on hðgÞ. For higher values of S 1 , the hðgÞ decreases. Figures 10 and 11, respectively, examine the effect of Nb and Nt on temperature ðhðgÞÞ. Here, we observed temperature enhancing through Brownian diffusion variable ðNbÞ and thermophoresis parameter ðNtÞ.       Figure 12 shows the behavior of curvature parameter on hðgÞ. Clearly note that hðgÞ decreases with increasing c Ã . Behavior of b Ã on hðgÞ is portrayed in Figure 13. Here, it is noted that hðgÞ increases when an increment occurs in b Ã . Characteristic of Ec on hðgÞ is highlighted in Figure 14. Here, increasing values of Ec leads to an increment in hðgÞ. Figure 15 is plotted to study the behaviors of Sc on concentration ðuðgÞÞ. For higher Sc, mass diffusivity decreases and thus uðgÞ is diminished. Figure 16 is sketched to examine the impact of S 2 on concentration ðuðgÞÞ. One can observe that concentration declines with S 2 . Figure 17 depicts the effect of Brownian      movement variable ðNbÞ on concentration. This figure manifests that with the increasing value of Nb, the concentration decreases. Figure 18 shows the effect of Nt on concentration ðuðgÞÞ. Clearly, uðgÞ increases with higher estimation of Nt.

Motile density
Salient behaviors of Pe, Lb, S 3 , and X on vðgÞ are discussed in Figures 19-22, respectively. Impact of Pe on vðgÞ is depicted in Figure 19. As expected, motile density decreased when an increment occurs in the bioconvection Peclet number. Behavior of Lb on vðgÞ is displayed in Figure 20. It is observed that motile density decreases against bioconvection Lewis number. Figure 21 shows the behavior of S 3 on vðgÞ. Here, one can find that motile density decreases for larger S 3 . Characteristic of X on vðgÞ is sketched in Figure  22. It is scrutinized that an increase in X increases the

Engineering quantities
In this article, the influence of influential parameters on Cf, Sh z , Nu z , and Nn z is examined. The behavior of Cf z versus involved parameters is displayed in Table 1.
Here, skin friction coefficient increases with curvature variable ðc Ã Þ, fluid parameter ðb Ã Þ, Hartmann number ðHaÞ, thermal Grashof number ðG t Þ, and bioconvection Rayleigh number ðRbÞ, while decreases for higher concentration of solutal Grashof number ðG c Þ. From Table 2, one can find that Nu z diminishes with

Conclusion
From this study, the following conclusions can be drawn: • The temperature field, mass concentration, and motile density decrease with increasing curvature variable, while reverse effect is observed in the case of velocity; • Velocity has opposite behavior for c Ã and Ha; • For larger b Ã , velocity ðf 0 ðgÞÞ decreases; • f 0 ðgÞ decreases with increasing values of G t , G c , and Rb; • Nt and Nb have similar effects on temperature; • The temperature field has opposite effect for curvature parameter and fluid material parameter; • Concentration decreases through Sc and S 2 ; • For larger Nt, the uðgÞ increases, whereas opposite effect is observed for Nb; • Motile density decreases through Lb, Pe, and X; • Cf z increases with higher estimation of c Ã and Ha; • Nu z boosts up with c Ã , Nt, and Nb; • Sh z has opposite effect for Nt and Nb; • For higher estimation of Pe and Lb, the Nn z increases.

Declaration of conflicting interests
The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding
The author(s) received no financial support for the research, authorship, and/or publication of this article.