Generalized slip impact of Casson nanofluid through cylinder implanted in swimming gyrotactic microorganisms

Nomenclature

1. Introduction

Nobody can decline the important utilization of non-Newtonian fluids by different fields. That is because of the singular physical characteristics of non-Newtonian fluids that differentiate them from others. The non-Newtonian fluid which performs like elastic solids is named as “Casson fluid”. It is the shear-thinning fluid pretended by the infinite viscosity at zero rates on shear (i.e., shear stress was never used), provided stress under that did not flow, and the zero viscosity at the infinite rate of shear. Few applied physical models on Casson fluid are constituted sauces/ketchup, soup, honey, jellies and human blood. Nazeer et al.¹ displayed the important chemical utilization of Casson fluid. We implicitly find that elastic interruption can be excellently utilised for manufacturing and coating processes. Sanjalee et al.² extended the work and discussed the common and chaotic Rayleigh–Benard convection in the absence of the non-uniform within heat source in free–free and rigid–rigid boundary conditions. Upreti et al.³ discussed the mass transfer and nature of heat in three-dimensional (3D) flow by the Casson nanofluid to consist on gyrotactic microorganisms through the Riga plate into changed magnetic field. Lanjwani et al.⁴ examined the magnetic boundary-layer Fe-Casson nanofluid flow persuaded into the radiated flat plate. Akhtar et al.⁵ scrutinized the peristaltic flow on elliptic duct by mass and heat transfer effect. Mahesh et al.⁶ explored the impact ofsuction and injection through the Marangoni boundary layer flow on thermal radiation and magnetohydrodynamic (MHD) Casson fluid by the permeable surface. Hussain et al.⁷ studied the EMHD stagnation point flow of Casson nanofluids into thermal radiations and heat source or sinks. Waqas et al.⁸ examined the Casson nanofluid by the porous solar collector flow of the infinite surface.

Bioconvection was induced by self-propelled motion of microorganisms, which compressed the fluid. Although the microorganisms swim upward, this great layer on microorganisms above the surface of the liquid acquired by microorganisms. Microorganisms, like algae, were assumed to accrue on the layer above the fluid, generating the irregular most superior into more-density stratifications. This Brownian motion and thermophoresis effect build into a nanostructured motion. This nanomaterial’s motion does not affect the movement of the motile microorganism. Accordingly, the combined intersections among nanofluids and bioconvection direction are significantly into microfluidic device. Being the interruption on swimming microorganism, Kuznetsov and Avramenko⁹ analyzed bioconvections. Bees et al.¹⁰ introduced the word “bioconvection”, which represents the dynamic strength of blood plasma and fluids in life patterns. Khan et al.¹¹ marked that convective boundary layer flows on Oldroyd-B nanofluid constructed over the stretching cylinder. Ali et al.¹² extended the work and examined the enhanced thermal transportation of bioconvection and mixed convection on tangent hyperbolic nanofluid flow into stagnation regions by rotating sphere. This effect of Stefan blowing on bioconvection by microorganism over popular edge ablation or accretion is analyzed by Gangadhar et al.¹³ Asjad et al.¹⁴ inspected the bioconvection flow on the MHD viscous fluid over a vertical surface.

Now, the analysis on nanofluids’ important features is the major topic among the study fraternities. Nanofluids are used in all facets of life. These fluids are generally used in lubricants to improve their capabilities and efficiency. That was experimentally noted in high fluids and it was utilized by the heat-producing or lubricant agents (like machine lubricants and oil) which had less thermal conductivities. Hence, investigators had applied more cases to improve thermal conductivities on the fluids. We modified the shape on geometry, boundary conditions, and convinced the various measures of metallic particles because we do not carry out correct outcomes. Choi and Eastman¹⁵ primly added nanosized particle in the fluid and observed that the heat capacity on the mixture was greater, although it is related into comparable base fluids. The data concrete the reason of the modern field on study. Lately, investigators are frequently employing to illustrate the important features of nanofluids and other information along with theoretical methods. These investigators analyzed those fluids and marked into those mixtures and found they were much efficient by coolant agents to enhance the efficiency of several pieces of equipment, such as the electronic cooling system, heat exchanger and radiators. In addition, the nanoliquid-based ultrasensitive optical sensor was used in environmental and industrial models to trace toxic captions. Therefore, this fluid was much helpful for both industrialists and scientists. Buongiorno¹⁶ carefully inspected the complete effort on nanofluid flow. Again, he considered the influence of various thermophysical phenomena such as Magnus effect, inertia, thermophoresis, diffusiophoresis, fluid drainage, gravity of nanofluid flows and Brownian diffusion. This Brownian motion and thermophoresis had important effect on nanofluids mechanisms. Hence, it is suggested that continuums form about nanofluid flow into alluring thermophoresis and Brownian motion impact is taken into account. Hence, newly these fluids were analyzed by several investigators.^{17,18,19,20,21,22}

This MHD flow, identical to the stretched surface, has critical applications in agricultural techniques, geophysical problem, petroleum engineering, casting, metal working, and cooling on the surfaces inside the confinement containers in an atomic reactor. This concept of magnetic field was completely appreciable when investigating the relationship between the magnetic fluid particles within blood flows. Salahuddin et al.²³ extended the work and discussed the flow of cross-fluid form into mass and heat transmission through the linearly stretched sheet. Butt et al.²⁴ observed the MHD impact of the two-dimensional nanofluid boundary layer flows and the effect on radiations. Sivasankaran et al.²⁵ reported the heat and mass transfer characteristics of MHD copper–water nanofluid flows cramped inside both parallel plates. Waini et al.²⁶ utilized the steady magneto-nanofluid flows along radiative heat transfer induced over the shrinking cylinder. Eegunjabi and Makinde²⁷ inspected the irreversibility of hydromagnetic mixed convection by a radiating nanofluid among both concentric apt cylindrical pipes. Sharma et al.²⁸ studied the nanoparticles through the rotating disk effecting downward into viscous dissipation. Abd-Alla et al.²⁹ discussed the peristaltic transport on the MHD fractional fluid through the porous medium on the nonuniform channel. Patil and Shankar³⁰ investigated the connected convection flow over the yawed cylinder. The important studies are listed in Refs. 31–63.

We carefully inspected the recorded research that was established to generalise slip flows by this gyrotactic swimming microorganism on the Casson nanoliquid through the stretching cylinder into a strong magnetic field and modern mass flux condition was not considered here. Hence, the present flow composition of this radiative nanofluid flows by stretching cylinder on the absence of gyrotactic swimming microorganism is analyzed. Nield conditions were studied in generalized slip conditions. This non-Newtonian particulate interruption was considered for applying Casson fluids. Therefore, non-Newtonian suspensions into nanoparticles is the innovative idea, against the perspective of coating processes.

2. Mathematical Formulation

Consider the Casson nanoliquid flow, two-dimensional, axisymmetric, frequently stretching cylinder along velocity $U_0(x_0)=\frac{U_1{x}_0}{l}l$ by axial direction. The magnetic field (strength $B_0$) was required in the normal direction. Generalized slip conditions were studied including the modern mass flux condition and convective heat transfer by that wall on surface, i.e., $r_0=R_0$. Additionally, these gravity-driven microorganisms were introduced on the bioconvection and nanoliquids which isinduced wbythe self-propelled movement oofthis microorganism. These radiations may only move to a short distance inside optically thick nanofluids; moreover, Rosseland approximation was given to amount toradiative heat transfer.

This rheological equation of the isotropic and incompressible flow by the Casson fluid is as follows³¹ :

where $e_{ij}$ performs on $(i,j)$th components on the deformation rate, ${\tau }_{ij}$ signifies the stress tensor, ${\pi }_0=e_{ij}{e}_{ij}$, i.e., the product of this component on clear-cutting into itself, ${\pi }_k$ represents the crucial value of this product subject to the non-Newtonian form, ${\mu }_{0B}$ signifies the plastic dynamic viscosity on this Casson fluid, and $p_y$ represents the stress provided by the fluid.

This Prandtl theory was used by the governing modeled systems, later using Prandtl theory, the governing equations on nanoliquid concentration, mass, energy, momentum and microorganism density are taken by this consequent model as follows^7,31 :

along with the boundary conditions In this upper system, the velocity components are denoted by $u_0$ and $v_0$ in $x_0$- and $r_0$-directions correspondingly.

For an optically thick nanofluid, the linear thermal radiation $q_r$ may be reported as

This following stream utilization assures the governing flow equations : The governing partial differential structure was modified by nondimensional ordinary differential problem with the help of accurate scalar transfer as follows : We present $\xi$ as the nondimensional self-reliant variable. $\theta ,\mathrm{\Gamma },\omega$ and $\chi$ are dimensionless temperature, velocity, motile microorganism density and nanoparticle concentration correspondingly. Applying Eq. (11) into Eqs. (1)–(8), this suggested stream functions converted the continuity equation equally for the temperature, velocity, nanoparticle concentration, and motile microorganism density equations transformed by this model The two-point conditions become In Eqs. (12)–(17), slip parameters were represented by $\alpha$ and ${\beta }_1$, M shows the Hartmann number, ${\mathrm{\text{Re}}}_x$ denotes the Reynolds number, $\mathrm{\text{Rd}}$ performs the radiation parameter, $\mathrm{\text{Rb}}$ signifies the bio-convection Rayleigh number, $\mathrm{\text{Nr}}$ is the buoyancy ratio parameter, Gr is the Grashof number, $\mathrm{\text{Ri}}=\mathrm{\text{Gr}}/{\mathrm{\text{Re}}}_x^2$ is the Richardson number, $\gamma$ is the curvature parameter, $\mathrm{\text{Nt}}$ is the thermophoresis parameter, $\mathrm{\text{Nb}}$ denotes the Brownian motion parameter, Pr is the Prandtl number, $\beta$ signifies the Casson parameter, Lb and Le are the bio-convection and traditional Lewis numbers, respectively, $\sigma$ is the density parameter and $\mathrm{\text{Pe}}$ is the Peclet number. These dimensionless parameters are accurate as follows This system on practical concern, especially, coefficient of wall frictions ${Cf}_x$, density of motile microorganism by wall ${Nn}_x$ and wall heat transfer ${Nu}_x$, are definite as follows : where ${\tau }_w$, $q_n$, and $q_w$, represent the wall heat transfer, wall microorganisms density and wall shear stress, respectively. Those quantities are accurate by Utilizing the scaling transformations by Eqs. (18) and (19), these quantities are suitable for the following model :

3. Numerical Solution Procedure

New programming languages create balance among behavior, certain capability and safety, and display the valuable potential about high-behavior calculating utilizations. Greater-level applications on these computational sources were meeting the question about high-performance and large data analytic scientific making. In addition to imitation facility, the resolver’s info may offer advice in representing the results on the actual mode situation. Commonly, the inexpensive tool was subject to mathematical forms or algebraic relation on less complication. By the effort, they employed bvp4c Lobatto-III formula. This combination problem considered is well-coded into MATLAB script to give cost-effective and reliable mathematical outcomes by the more-performed calculating environments.

The attractive differential explanation (12)–(15) by the boundary conditions (16) and (17) was given by greatly nonlinear models. These greatly nonlinear equations were employed by the MATLAB computational software. Since opening the method, greater ODEs were transmuted by first-order equations using variables, like

4. Validation of the Numerical Code

To validate this analysis, the allegations are proven from the past work of Hussain and Malik³¹; the comparisons are recorded in Table 1. The agreement with past efforts gives assurance to this analysis.

5. Results and Discussion

In this section, calculations were performed to analyze the effects of Casson parameter $\beta$, slip parameters $\alpha ,{\beta }_1$, radiation parameter $\mathrm{\text{Rd}}$, Brownian motion impact $\mathrm{\text{Nb}}$, thermophoresis effect $\mathrm{\text{Nt}}$, Lewis number Le, bioconvection Rayleigh number $\mathrm{\text{Rb}}$, Hartmann number M, curvature parameter $\gamma$, bioconvection Lewis number Lb, Richardson number Ri on fluid motion, temperature variation, concentration variation and motile microorganism density. The outcomes are illustrated in Figs. 2–9. Furthermore, Tables 2 and 3 document the impact of temperature gradient and governing parameters on skin friction by the boundary. This default parameter values are given as follows: $\beta =0.4,$ $\gamma =0.1$, $M=0.2$, $\mathrm{\text{Ri}}=2.0$, $\mathrm{\text{Nr}}=0.4$, $\mathrm{\text{Rb}}=2.0$, $\mathrm{\text{Rd}}=0.2$, $\mathrm{\text{Pr}}=7.1$, $\mathrm{\text{Nt}}=0.1$, $\mathrm{\text{Le}}=1.0$, $\mathrm{\text{Nb}}=0.1$, $\mathrm{\text{Lb}}=5.0$, $\mathrm{\text{Pe}}=0.1$, $\sigma =0.1$, $\alpha =0.2$, ${\beta }_1=0.2$.

The impact of Casson fluid parameter $\beta$ and Hartmann number M on the flow profile ${\omega }^{\prime }(\xi )$ is presented in Fig. 2. It is noted that developing the intensity of the magnetic fields completely change the performance of the flow in the Casson fluid. For this essential flow on electromagnetism, the magnetic field and the interaction of electricity with the fluid are compared with the dimensionless ratio (Hartmann number). One may observe from the graph that the strong magnetic effects restrict the fluid flow interruption by nanoparticles. The phenomenon is named as Lorentz force, by magnetic fields performing on this transverse direction of this flow. Moreover, this utilization on magnetic fields reacts equally into the resistive force, never aiding this nanofluid flow. Furthermore, this momentum of the fluid reduces evenly. Similarly, Fig. 2 shows that the velocity of the nanofluid reduces with the increase in the Casson fluid parametric quantity $\beta$ and viscosity of the fluid augment, and that is due to the fact that the increase in viscosity was authorized into acquiesce stress, that opposes the fluid motion confined into the fluid stratum abutting to boundary.

Figure 3 illustrates the effect of curvature parameter ($\gamma$) and slip parameter ($\alpha$) against flow distribution ${\omega }^{\prime }(\xi$). It was delayed using this figure until the flow curves on the fluid increased for curvature parameter. The radius of the cylinder shrinks with the increasing curvature and the interaction region on fluid along the geometry. Thus, the diminution was produced on the resistance outside and finally the fluid velocity and the composed thickness on boundary layer developed. Influence of the velocity slip parameter ($\alpha$) by the fluid velocity is again illustrated in Fig. 3. The velocity slip was assumed on this wall velocity boundary conditions provided in Eq. (16). This velocity slip parameter changes from 0.6 to 1.8 in this analysis. The fluid velocity decreases with the increasing values of the velocity slip parameter. Especially, low value of flow was drawn, as the slip gets stronger. Increasing velocity slip, the reduction in this infiltration by the immovable surface over a boundarye in the momentum boundary layer thickness on this flow was slowdown into developing slip in case to that skin friction was lesser additional by this wall.

The thermal profile $\theta (\xi )$ is displayed in Fig. 4 which shows the impact of the Casson parameter $(\beta )$ and Hartmann number $(M)$. The thermal distribution profile increased with the increase of $\beta$ during the uplift on plastic dynamic viscosities, which imposed friction by this fluid motion, consequently increasing the temperature. Further, the thermal profile increases with the increase in M. It was the Lorentz force power, reverse to that flow and gives high opposition into transit phenomenon. This interaction by liquid motion and then acting on the used magnetic fields results in the refusal by velocity gradient and the raise on frictional heating, that cases thermal gradient inside the fluid layers to improve. Similar consequences are observed in Ref. 32.

The effect on thermal radiation factor $(\mathrm{\text{Rd}})$ and thermophoresis factor $(\mathrm{\text{Nt}})$ on thermal profile is displayed in Fig. 5. Increase in thermal radiation means high heat is approached by the system, and the temperature on nanofluid increases during this. Figure 5 certifies this fact and the increase in the radiation parameter $\mathrm{\text{Rd}}$ improves the nanofluid temperature inside the boundary layer. Actually, the strengthening by this parameter $\mathrm{\text{Rd}}$ transfers high heat into the fluid and advances the thermal boundary layer thickness. Further, it was noted that the thermal profile increases due to the higher estimation of $\mathrm{\text{Nt}}$. The greater values of thermophoretic parameter signify higher thermophoretic force; it shows interruption in mixtures of fluids and particles like nanofluids. The phenomenon is detailed into temperature gradient, which increases the momentum of the fluid particles in the hottest area against the cold regions. Thus, the heat transfer possibilities increase.

The variation of nanoparticle concentration profile against the variation in thermophoresis force $\mathrm{\text{Nt}}$ and Brownian movement $\mathrm{\text{Nb}}$ is shown in Fig. 6. The concentration distribution increases for larger Brownian diffusion. Actually, this in-predictive motion on the nanoparticles increase differences in outcomes of high dilution into the nanoparticles near the surface compared to the outside of the surface. Also, Fig. 6 provides the effect of the expanded values of the thermophoretic forces by this concentration of nanoparticles. This greatest thermophoretic force improves the in-predictive movement on this fluid particle outcome at high distance about the nanoparticles.

Figure 7 illustrates the difference observed in the concentration distribution by the fluid for clarification of the increased values of Lewis number Le and Richardson number Ri. Rechardson number Ri details the reduction on this concentration by nanoparticle on the boundary layers. Also, this graph validates that the Lewis number Le decreases mass diffusivity and thus this nanoparticle concentration essentially on magnitude reason.

The variation in microorganism mass density $\chi (\xi )$ is depicted in Figs. 8 and 9 using the physical parameters $(\mathrm{\text{Pe}},\mathrm{\text{Lb}},\sigma ,\mathrm{\text{Nt}})$. Figure 8 displays the fluctuations on microorganisms density profile changing the Peclet number $\mathrm{\text{Pe}}$ and bioconvection Lewis number $\mathrm{\text{Lb}}$. That structure conformed to the case of bioconvection Lewis number Lb reduces into density on motile microorganisms being bioconvection. Lewis numbers had opposite relations into difusivities by microoganism’s density. This figure also shows that the Peclet number $\mathrm{\text{Pe}}$ reduces into microorganisms’ density profiles. That ratio on maximal cell swimming speed into this diffusivity on microorganisms is termed as the Peclet number. The increase in Peclet number is followed by the rise in cell swimming speed. By the reason, this rise in the bioconvection Peclet number reduces density of motile microorganism’s profile.

This effect of density parameter $\sigma$ and thermophoresis parameter $\mathrm{\text{Nt}}$ is examined through Fig. 9. The density parameter $\sigma$ decreases into the motile microorganism densities on boundary layer regions. The lower density parameter is correlated with the lower median microorganism densities and the overall densities. This graph shows that thermophoresis parameter $\mathrm{\text{Nt}}$ raised the density of the motile microorganisms.

This performance on wall friction factor versus flow parameters $(\alpha ,\beta ,{\beta }_1,\gamma ,\mathrm{\text{Ri}},\mathrm{\text{Rb}})$ was investigated to be $M=0$ (hydrodynamic flows) and $M=2.0$ (MHD flows). Table 2 demonstrates that the MHD fluid flows had higher wall frictions than hydrodynamic fluid flows. The table shows that the curvature parameter $\gamma$ increases coefficient of wall friction by 21.4349%, bioconvection Rayleigh number $\mathrm{\text{Rb}}$ increases friction coefficient by 2.65864% and Casson parameter increases by 2.83577% while slip parameters $\alpha ,{\beta }_1$ decrease it by 32.389% and 53.8235% in the presence of strong magnetic field.

The variation of local Nusselt number i.e., ${\mathrm{\text{Nu}}}_x$ alters thermophysical parameters $(\beta ,\mathrm{\text{Rd}},M,\mathrm{\text{Nt}},\mathrm{\text{Nb}},\mathrm{\text{Ri}},\mathrm{\text{Rb}},\mathrm{\text{Pr}})$ for $\gamma =0.0$ (stretching sheet fact) and $\gamma =1.0$ (stretching cylinder fact) is considered in Table 3. The table displays this local Nusselt number having greater values on stretching cylinder. In addition, it is found that parameters $(\beta ,M,\mathrm{\text{Nt}},\mathrm{\text{Rb}})$ decrease the local Nusselt number by 4.15787%, 2.59037%, 26.7341%, 0.28141% while rest of controlling parameters $(\mathrm{\text{Rd}},\mathrm{\text{Nb}},\mathrm{\text{Ri}},\mathrm{\text{Pr}})$ enhance it by 41.1893%, 0.0026%, 1.84555%, 83.2765% in the case of stretching cylinder.

6. Conclusion

In this paper, modern mass flux condition and generalized slip flow on the movement of gyrotactic microorganism and thermophysical phenomena on magnetic field over a Casson nanoliquid with linear thermal radiation are investigated. Even though few works on the thermophysical properties by Casson nanoliquid were usable in this paper, the mathematical evaluation by this magnetized flow on the non-Newtonian fluid form into that development on gyrotactic motile microorganisms had never up to now grown. Bvp4c application MATLAB was utilized to obtain the mathematical result by this dimensionless conducting system of equations. The following basic results were recordedas follows:

(1)	Increasing Casson fluid and Hartmann number parameters reduces the velocity profile of the fluid. Moreover, this thermal boundary layer thickness enhances significantly when Hartmann number and Casson parameters increase.
(2)	The nanofluid velocity profile decreases with the increase of slip parameter while it increases for curvature factor.
(3)	Due to the Casson parameter, tThe thermal boundary layer thickness and radiation parameter increase. The nanoparticle concentration profile decreases although the Lewis number was raised.
(4)	Greater thermal boundary layer thickness was composed into larger Brownian motion parameter value, although higher concentration on boundary layer thickness was acquired to lesser Brownian motion parameter values.
(5)	It is obvious that increase in Peclet number and thermophoresis parameter correspond to increase in microorganism density profile. In addition, increasing bioconvection Lewis number values will reduce the density profile of microorganisms.
(6)	Wall skin friction coefficient enhances into developing values of curvature factor and bioconvection Rayleigh number. For example, the curvature effect increases by 21.4349% in the absence on strong magnetic field.
(7)	These increasing values of Casson fluid parameter and slip factor lead to decrease in wall shear stress. For example, the Casson parameter decreases by 53.8235% in the case of strong magnetic field.
(8)	It was noted that the wall temperature gradient decreases with the increase in Casson parameter and Brownian motion or decrease in the radiation parameter and Prandtl number. For example, the radiation effect increases by 35.177% for stretching sheet case as well as 41.1893% for stretching cylinder case.

We will extend this work for Carreau nanofluid and Maxwell fluids in future.

ORCID

Kotha Gangadhar  https://orcid.org/0000-0002-0264-2512

T. Sujana Sree  https://orcid.org/0000-0003-2477-2512

Abderrahim Wakif  https://orcid.org/0000-0003-2477-8442