Magnetized and non-magnetized Casson fluid flow with gyrotactic microorganisms over a stratified stretching cylinder

This study presents the magnetized and non-magnetized Casson fluid flow with gyrotactic microorganisms over a stratified stretching cylinder. The mathematical modeling is presented in the form of partial differential equations and then transformed into ordinary differential equations (ODEs) utilizing suitable similarity transformations. The analytical solution of the transformed ODEs is presented with the help of homotopy analysis method (HAM). The convergence analysis of HAM is also presented by mean of figure. The present analysis consists of five phases. In the first four phases, we have compared our work with previously published investigations while phase five is consists of our new results. The influences of dimensionless factors like a magnetic parameter, thermal radiation, curvature parameter, Prandtl number, Brownian motion parameter, Schmidt number, heat generation, chemical reaction parameter, thermophoresis parameter, Eckert number, and concentration difference parameter on physical quantities of interests and flow profiles are shown through tables and figures. It has been established that with the increasing Casson parameter (i.e. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta \to \infty$$\end{document}β→∞), the streamlines become denser which results the increasing behavior in the fluid velocity while on the other hand, the fluid velocity reduces for the existence of Casson parameter (i.e. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta = 1.0$$\end{document}β=1.0). Also, the streamlines of stagnation point Casson fluid flow are highly wider for the case of magnetized fluid as equated to non-magnetized fluid. The higher values of bioconvection Lewis number, Peclet number, and microorganisms’ concentration difference parameter reduces the motile density function of microorganisms while an opposite behavior is depicted against density number.

www.nature.com/scientificreports/ a three-dimensional frame was addressed by Gupta and Bhattacharyya 6 . The micropolar fluid flow with Joule heating and magnetic impacts is represented by Hakiem et al. 7 . The mass and heat transfer characteristics of MHD micropolar fluid flow through a circular cylinder was analyzed by Mansour et al. 8 . The thermal transmission characteristics of MHD fluid flow in the existence of thermal radiation were proposed by Sadeek 9 . The flow of micropolar fluid with MHD and constant suction impacts was presented by Amin 10 . The MHD flow of Oldroyd-B fluid was addressed by Hayat et al. 11 . The MHD flow of Casson fluid through a shrinking surface was discussed by Nadeem et al. 12 . Thammanna et al. 13 addressed the chemically reactive MHD flow of Casson fluid through an unsteady stretching sheet with convective boundary conditions. Ramesh et al. 14 presented the MHD Casson fluid flow with Cattaneo-Christov heat theory heat absorption/generation. The MHD flow of nanofluid past an unsteady contracting cylinder with convective condition and heat generation/absorption was investigated by Ramesh et al. 15 . Bilal et al. 16 presented the numerical analysis of MHD viscoelastic fluid past an exponentially extending sheet. The related studies towards this development can be seen in Refs.  .
Because of its useful applications in engineering and industry, a study of stratification phenomena in non-Newtonian fluids has got a lot of interest. Temperature fluctuations, composition variations, or a combination of different liquids of varying thicknesses causes stratification of the medium. For instance, thermal rejection into the atmosphere, storage systems of heat energy, geophysical flows, etc. In short, stratification takes place in both natural and industrial phenomena. Using thermal and solutal stratifications many attempts have been performed like Yang et al. 47 analyzed the convective fluid flow over a thermally stratified medium. The buoyance flow over a stratified medium was discussed by Jaluria and Gebhart 48 . The mixed convection fluid flow over a stratified medium was introduced by Ishak et al. 49 . The thermally stratified flow of micropolar fluid over a vertical plate constant and uniform heat flux was examined by Chang and Lee 50 . The incompressible and electrically conducting viscous fluid flow through an inclined plate was presented by Singh and Makinde 51 . The MHD flow of dissipative fluid with heat generation and second-order chemical reaction was introduced by Malik and Rehman 52 . The incompressible and mixed convective flow of viscous fluid through a thermally stratified stretching cylinder was introduced by Mukhopadhyay and Ishak 53 . Cheng et al. 54 investigated the mass and heat transmission in a power-law fluid through a stratified medium. The heat transmission in a boundary layer flow through a stratified vertical plate was presented by Ibrahim and Makinde 55 . Furthermore, related analyses are mentioned in Refs. [56][57][58][59][60] .
Our contribution to the field of non-Newtonian fluids consists of Casson fluid flow containing gyrotactic microorganisms through a stratified stretching cylinder. Furthermore, stagnation point, Joule heating, heat absorption/generation, thermal stratification, mass stratification, motile stratification, nonlinear thermal radiation, magnetic field, and chemical reaction are taken into account. Also, the fluid flow is treated for magnetized and non-magnetized conditions. The present analysis consists of five phases. In the first four phases, we have compared our work with previously published investigations while phase five is consists of our new results.
At the end of this analysis, we will be able to answer that: • How the streamlines behave for non-Newtonian (Casson) and Newtonian fluid?

Problem formulation
The mathematical model for Casson fluid containing gyrotactic microorganisms through a stretching cylinder is modeled under the effects of various parameters like stagnation point, Joule heating, heat absorption/generation, thermal stratification, mass stratification, motile stratification, thermal radiation, magnetic field, and chemical reaction.
For an isotropic and incompressible flow of Casson fluid, the rheological equation is stated as 61 : where τ ij is the i, j th component of stress tensor, π = e ij e ij and e ij are the i, j th component of the deformation rate, p y is the fluid yield stress, and µ B is the plastic dynamic viscosity of the non-Newtonian fluid, π is the product of component of deformation rate with itself and π c is the critical value of this product. According to these assumptions the leading equations are 38,57,60 : The conversion parameters are defined as: Using (7), (2)(3)(4)(5)(6) are reduced as: with transformed boundary conditions: aρ is the magnetic parameter,A = a * a is the ratio of veloci- νT ∞ is the thermophoresis parameter,δ 1 = c b is the temperature stratification parameter,δ 2 = e d is the concentration stratification parameter, δ 3 = h g is the motile density stratification parameter,Q = Q 0 L ρc p U 0 is the heat generation parameter,Sc = ν D B is the Schmidt number,R c = LR 0 U 0 is the chemical reaction parameter, Le = α D n is bioconvection Lewis number, and Ec = xU 0 is the bioconvection Peclet number, = N ∞ N ∞ −N 0 is the concentration difference parameter.

Engineering quantities of interests
Skin friction (SF). At the cylindrical surface, the SF is specified by:  (7), the dimensionless form of SF is written as: where Re x = U 0 x 2 νL is the local Reynolds number.

Nusselt number (NN).
At the surface of cylinder, the NN is given by: In view of Eq. (7), the dimensionless form of NN is written as:

Sherwood number (SN).
At the surface of cylinder, the SN is given by: In view of Eq. (7), the dimensionless form of SN is written as: Density number (DN). At the surface of cylinder, the DN is given by: In view of Eq. (7), the dimensionless form of SN is written as:

HAM solution and convergence
An analytical of the modeled problem is solved with the help of HAM. The initial guesses and linear operators are taken as: with where are the arbitrary constants. The convergence analysis of HAM is presented in Fig. 1a Fig. 1a. Similarly, the convergence areas for θ ′ (0) , φ ′ (0) and χ ′ (0) are −1.75 ≤ θ ≤ 1.50 , −0.75 ≤ φ ≤ 1.60 and −1.75 ≤ χ ≤ 1.20 respectively, as shown in Fig. 1b.

Phases of study
This paper contains the generalized mathematical formulation for both magnetized and non-magnetized Casson fluid flow with gyrotactic microorganisms over a stratified cylinder. So the present analysis divides the proposed study into sub-phases in order to analyze the Casson fluid flow under different situations. It should be noted that in the current investigation, we have ignored the mixed convection phenomena. The present analysis consists of Rd θ ′ (0). x Dn x = −χ ′ (0).

Phase 1.
Ignoring the suspended nanoparticles assumptions, stagnation point, Joule heating, thermal radiation, concentration distribution, and motile distribution the comprehensive formulation compact to the problem described in Ref. 38 . In this study, the comparative analysis of MHD mixed convective flow of Casson fluid through the stratified flat and cylindrical stretching surfaces are analyzed. The obtained variations were found to be surprisingly large for cylindrical geometry as particularly in comparison to plane surface geometry. The SF is found the heightening function of κ,Pr,β and δ 1 whereas a reducing function of Q . The NN is found the heightening function of κ and Pr whereas a diminishing function of Q,δ 1 and β . By applying the assumptions reported in Ref. 38 , an analytical solution is proposed in order to verify the current model. An outstanding comparison is found with Ref. 38 as shown in Table 1. The mathematical formulation designed in Ref. 38 is given as: with transformed boundary conditions:

Phase 2.
In the absence of suspended nanoparticles assumptions, Joule heating, thermal radiation, stagnation point, and motile distribution the comprehensive formulation compact to the problem described in Ref. 56 .
In this analysis, a stratified flow of Casson fluid through an inclined cylinder was presented. It was concluded that the SF is reduced with K,β,Pr and δ 1 whereas the opposite trend in SF was found due to δ 2 and Q * . Furthermore, the NN has reduced with β,δ 1 ,Q * however the NN has increased with K and Pr . Additionally, it was also concluded that the higher K,δ 2 ,R c and Sc have increasing impacts on SN. By applying the assumptions reported www.nature.com/scientificreports/ in Ref. 56 , an analytical solution is proposed in order to verify the current model. An outstanding comparison is found with Ref. 56 as shown in Tables 2 and 3. The mathematical formulation designed in Ref. 56 is given as: Phase 3. In the absence of suspended nanoparticles assumptions, Joule heating, thermal radiation, stagnation point, magnetic field, concentration distribution, motile distribution, and with the condition when β → ∞ , the comprehensive formulation condensed to the problem reported in Ref. 53 . It was found that the buoyancy force escalates the SF and the thermal stratification parameter increases the NN. In addition, it was claimed that the SF and NN are highly impressed for a cylinder in contrast with plate. The mathematical formulation designed in Ref. 53 is given as: with transformed boundary conditions: Phase 4. By ignoring the Joule heating and motile distribution, the comprehensive formulation condensed to the problem reported in Ref. 57 . In this examination, the authors found that the SF reduces with K and β . Furthermore, the NN increases with K and Pr whereas as reduces with δ 1 . Also, the SN reduces with K,δ 2 and increases  Refs. 38,56,57 with gyrotactic microorganisms through a stratified stretching cylinder is presented in this phase. The generalized mathematical modeling is given in (8)(9)(10)(11) with boundary conditions (12). Here, we are interested to study the behavior of swimming microorganisms in a Casson fluid flow and to study the variation in Casson fluid flow through a stratified medium due to bioconvection Peclet number, bioconvection Lewis number, and concentration difference parameter. Furthermore, to analyze the flow of both magnetized and non-magnetized  Table 7. Here we have found that the DN increases with K , Pe, Le and while the opposite trend in found via δ 3 .
Graphical results. Figure 2a shows the variations in the streamlines of Newtonian fluid velocity for magnetized stratified medium under the stagnation point. Here, with the increasing Casson parameter (i.e.β → ∞ ), we have observed that the streamlines become denser which results the increasing behavior in the fluid velocity, while on the other hand the fluid velocity reduces for the existence of Casson parameter (i.e.β = 1.0 ) as shown in Fig. 2b. Figure 3a-f show the variations in the streamlines of Casson fluid velocity for both magnetized and non-magnetized stratified medium under the region of stagnation point. Figure 3a,b are plotted for both nonmagnetized and magnetized stratified medium when A = 0.5 respectively. We can see that the streamlines are wider for the case of magnetized Casson fluid as equated to non-magnetized fluid. Figure 3c,d are plotted for both non-magnetized and magnetized stratified medium when A = 1.0 respectively. Here, we observed that the streamlines are stifled for the case for the case of magnetized Casson fluid as equated to non-magnetized case. This conduct is due to the Lorentz force which always provides the contrasting force to the fluid flow. Figure 3e,f are plotted for both non-magnetized and magnetized stratified medium when A = 1.5 respectively. A similar impact of magnetic field is depicted as observed in Fig. 3a-d. Thus, we have concluded that the magnetic field   Figure 4 shows the influence of bioconvection Lewis number Le on motile density function. Here, we concluded that the higher values of Le reduce the motile density function. Figure 5 shows the influence of bioconvection Peclet Pe number on motile density function. The greater Peclet number reduces the diffusivity of microorganisms which conclude a reducing influence in motile density function. Thus, the greater Pe reduces the motile function of the microorganisms. Figure 6 shows the effect of microorganisms' concentration difference parameter on motile function. The increase in microor- www.nature.com/scientificreports/ ganisms' concentration difference parameter heightens the microorganisms' in the ambient liquid which consequently reduces the motile density function.

Final comments
The mathematical model for both magnetized and non-magnetized Casson fluid containing gyrotactic microorganisms through a stretching cylinder is modeled under the effects of various parameter like stagnation point, Joule heating, heat absorption/generation, thermal stratification, mass stratification, motile stratification,   www.nature.com/scientificreports/ nonlinear thermal radiation, magnetic field, and chemical reaction. We have studied the Casson fluid under different situations in order to analyze the flow behavior. The final comments are listed as: • The increasing Casson parameter (i.e.β → ∞ ) the streamlines become denser which results the increasing behavior in the fluid velocity, while on the other hand the fluid velocity reduces for the existence of Casson parameter (i.e.β = 1.0). • The streamlines of stagnation point Casson fluid flow are highly wider for the case of magnetized fluid as equated to non-magnetized fluid. • The motile function of microorganisms is reducing with higher values of bioconvection Lewis number, Peclet number, and microorganisms' concentration difference parameter. • The DN increases with higher values of curvature parameter, bioconvection Lewis number, Peclet number, and microorganisms' concentration difference parameter while reduces with motile density stratification parameter.