Numerical Simulation of Bioconvection Maxwell Nanofluid Flow due to Stretching/Shrinking Cylinder with Gyrotactic Motile Microorganisms: A Biofuel Applications

Abstract

The bioconvection effects with nanofluid are major application in biofuels. This analysis aimed to observe the bioconvection effect in unsteady two-dimensional Maxwell nanofluid flow containing gyrotactic motile microorganisms across a stretching/shrinking cylinder evaluating the consequences of thermal radiation and activation energy. The Cattaneo-Christov double diffusion theory is also observed. Nanofluids are quickly perceptive into many solicitations in the latest technology. The current research has noteworthy implementations in the modern nanotechnology, microelectronics, nano-biopolymer field, biomedicine, biotechnology, treatment of cancer therapy, cooling of atomic reactors, fuel cells, and power generation. By using the proper similarity transformation, the partial differential equations that serve as the basis for the current study are gradually reduced to a set of highly nonlinear forms of ordinary differential equations, which are then numerically, approached using a well-known shooting scheme and the bvp4c tool of the MATLAB software. Investigated is the profile behavior of the flow regulating parameters for the velocity field, thermal field, and volumetric concentration of nanoparticles and microorganisms. From the results, it is concluded that velocity is reduced with a larger bioconvection Rayleigh number. The thermal field is increased with a larger amount of thermal Biot number and thermal radiation. The concentration of nanoparticles increases with an increment in the thermophoresis parameter. Furthermore, the microorganism’s field is decreased with a larger Lewis number. The findings demonstrate that by optimizing the concentration of nanoparticles and microorganisms, the thermal efficiency of biofuels can be significantly improved. This leads to more sustainable and efficient energy production. By optimizing the concentration of nanoparticles and microorganisms in biofuels, the thermal properties can be significantly improved, leading to more efficient combustion processes. This can reduce the overall cost and increase the yield of biofuels. Improved cooling systems for medical imaging devices such as MRI machines can be developed using nanofluids, ensuring better performance and patient safety.

1 Introduction

Maxwell fluid is a non-Newtonian fluid that exhibits both viscous and elastic properties. Unlike Newtonian fluids (like water or air), which have a constant viscosity, the viscosity of Maxwell fluids can change under different flow conditions. They are often used to model polymeric solutions and biological fluids. Maxwell nanofluid describes the flow behavior of a Maxwell fluid that has been enhanced with suspended nanoparticles, leading to improved thermal conductivity and other modified properties. Nonmaterials that are dissolved in a base liquid and employed to speed up the combined heat and mass transformation process are referred to as “nanofluids” in this context. It is of significant significance in the fields of biology and medication. Nanofluid is created by dispersing nano-sized solid nonmaterial in normal base fluids. Maxwell [1] advocated distributing these nano-powders in host fluids to enhance the heat transfer rate, but there are several issues to consider, such as channel erosion and sedimentation, among others. Ganvir et al. [2] highlight the numerous features and viscoelastic aspects of particles and nanoparticles, which are required for nanofluid to recover heat transfer rate. Furthermore, Ma and Banerjee [3] expressed a numerical solution to examine the behavior of thermal transport inside nanofluid employing several phases of filtration sculpted by nonmaterial. Recently, Khan et al. [4] addressed some nanofluid applications as well as the use of gold nanoparticles in blood. The thermophoresis and Brownian movement factors in nanofluid are successfully reported in Buongiorno’s research [5,6,7,8,9,10,11,12]. Saeed et al. [13] scrutinized the hybrid nanoparticle Darcy-Forchheimer flow passing through a stretchable permeable cylinder. Khan et al. [14] used the Prandtl technique to investigate enhanced characteristics of Oldroyd-B nanofluid in the occurrence of radiation. Abdelsalam and Sohail [15] investigated the impact of viscous dissipation on the nanoliquid transport of microorganisms. Nadeem et al. [16] investigate the unstable viscous liquid’s three-dimensional forced bioconvection movement. Raizah et al. [17] created a study of unsteady mixed bioconvection flow arrangement in a horizontal channel with an upper plate moving upward. Plesset and Winet [18] conducted research on bioconvection and its applications. Kuznetsov [19] examined the existence of nanofluid bioconvection a combination of microbes and nanotechnology in suspensions. A comparison was made between the continuous incompressible magnetic-hydrodynamic mass transition bio-convective fluid flow and the deferment point of tiny particles over delayed sheets by Mamatha et al. [20]. Khan et al. [21,22,23,24,25,26] used gyrotactic microorganisms to examine the activation energy properties of a micro-polar nanofluid. Zhang et al. [27] heat transmission was examined in a bioconvection flow of nonmaterial and gyrotactic bacteria passing through an elastic surface.

The fundamental idea behind MHD is that the magnetic field creates forces that might cause magnetic induction owing to fluid flow. Many outcomes for the MHD flow under various situations have been studied by researchers, including heat transmission. The magnetic properties of electric conducting liquids such as seawater, fluid metals, plasmas, and electrolytes are depicted in magneto-hydrodynamics. It is used in different applications such as electrostatic generators, electrical pumps, turbines, and heat producers. Lorentz force is induced by a magnetic field in the spinning motion of a fluid. It lowers fluid flow while raising the concentration and temperature of nanoparticles. Hence, the postponed dispersion of the boundary layer largely depends on the magnetic field. Furthermore, the MHD is useful in medical, manufacturing, physics, chemistry, and metallurgy fields, such as cancer, asthma, and hypertension therapy. MHD is a scientific and mathematical framework that is connected to evidence on magnetic fields as well as the electrically conductive fluids’ characteristics. Many writers, notably Balankin et al. [28], studied comparable difficulties under other situations. There was a presentation of MHD flow through a stretched cylinder with Newtonian heating and homogeneous-heterogeneous reactions by Hayat et al. [29]. Aly et al. [30] investigated the flow of a hybrid nanofluid across a porous surface. Gul et al. [31] studied the combined consequence of a magnetic field and Marangoni convection on carbon nanotubes made from thin layers of diesel fuel oil. Bhuvaneswari et al. [32] using a porous material, Soret and Dufour’s effects on MHD over a vertical plate. Takhar et al. [33] examined impulsive stretching surface motion-induced unsteady three-dimensional MHD boundary layer flow. Sheikholeslami et al. [34] explored MHD convective energy transmission in a wavy chamber strongly affected by a heavily biased magnetic field. Chamkha et al. [35] Soret and Dufour’s consequences on temperature and concentration distributions were evaluated. Radiation has important effects on physics and engineering. In space technology and high-temperature operations, radiation from heat transfer impacts on various flows is especially important. Thermal radiation effects may be important in managing heat transfer in the polymer processing industry since heat-controlling factors influence the finished product’s quality to some extent. Prasher et al. [36] explored the influence of Brownian motion of nanoparticles as the main component for enhancing thermal conductivity. Rahimi-Gorji et al. [37] examined the stability of continuously occurring heat transfer in the presence of hybrid nanofluid-based thermal radiation.

Gyrotactic motile microorganisms are microorganisms that move and orient themselves in a specific pattern influenced by gyrotaxis. Their movement can induce bioconvection, which enhances mixing and transport properties in the fluid. Bioconvection refers to the spontaneous formation of patterns and convection currents within a fluid, driven by the collective movement of motile microorganisms. As these microorganisms swim upwards due to gyrotaxis, they create density variations in the fluid, leading to convection currents that enhance mixing and transport. Xu and Pop [38] considered the flow of a mixed bioconvection fluid including gyrotactic bacteria and nanoparticles between two horizontal parallel plates. Sudhagar et al. [39] explored nanofluid flow in energy and mass transfer using mixed convection.

The Cattaneo-Christov double diffusion theory is an advanced model used to describe heat and mass transfer processes. It addresses some limitations of the classical Fourier’s law for heat conduction and Fick’s law for mass diffusion by incorporating finite propagation speeds for thermal and concentration waves, thus providing a more realistic depiction of the physical phenomena. In the context of this study, the Cattaneo-Christov double diffusion theory is used to model the heat and mass transfer in the Maxwell nanofluid containing gyrotactic motile microorganisms. Recently, some researchers have been working on Cattaneo-Christov heat and mass flux and nanofluids [40,41,42,43,44,45,46,47,48]. Murtaza and Ahmad [49] introduced a novel analysis on clay-based nanofluid under slippage constraints with Caputo derivative. This theory is particularly relevant due to the following reasons:

• The use of this theory allows for a detailed analysis of the bioconvection effects, which is essential for optimizing the performance of nanofluids in biofuel applications. Understanding how these effects influence the flow and distribution of nanoparticles and microorganisms can lead to better control and enhancement of thermal properties.

• The hyperbolic nature of the Cattaneo-Christov equations leads to more realistic simulations, which are important for developing practical applications in modern nanotechnology, microelectronics, biomedicine, and energy systems.

1.1 Objectives

The main application area of this study is in the field of biofuels, where understanding and leveraging bioconvection effects with nanofluids can significantly enhance biofuel production and efficiency. The main objective of the current investigation is to inspect the unsteady radiative flow of Maxwell nanofluid with bioconvection due to microbes over stretching/shrinking cylinder due to its uses in various engineering and biofuels. The Cattaneo-Christov heat and mass fluxes with non-uniform heat source and activation energy are included in the energy and concentration framework. The incorporation of the Cattaneo-Christov double diffusion theory provides a sophisticated approach to modeling these complex phenomena. Key findings from this research have substantial implications for the following areas:

• Biofuels: Enhanced biofuel production through improved thermal properties of nanofluids.

• Nanotechnology: Advanced applications in modern nanotechnology due to improved fluid dynamics and heat transfer properties.

• Microelectronics and Nano-Biopolymers: Better cooling solutions and thermal management.

• Biomedicine and Biotechnology: Applications in cancer therapy and biological processes involving motile microorganisms.

• Energy Systems: Improved cooling techniques for atomic reactors and enhanced performance in fuel cells and power generation systems.

1.2 Overview of Computational Method

The study employed computational methods to observe the bioconvection effect and evaluate the consequences of thermal radiation and activation energy in the context of unsteady two-dimensional Maxwell nanofluid flow containing gyrotactic motile microorganisms. Here is a brief overview of the specific methods used:

• Mathematical modeling:

  • Governing equations: The study began with the formulation of partial differential equations (PDEs) that describe the flow, heat, and mass transfer phenomena. These equations account for the velocity field, thermal field, and concentration profiles of nanoparticles and microorganisms.

  • Incorporation of effects: Thermal radiation and activation energy effects were incorporated into the equations to accurately model the physical scenario. The Cattaneo-Christov double diffusion theory was used to account for the thermal and concentration diffusion effects more realistically than classical Fourier’s and Fick’s laws.

• Similarity transformation:

  • To simplify the complex PDEs, similarity transformations were applied. This technique transforms the PDEs into a set of ordinary differential equations (ODEs) by introducing similarity variables. These variables reduce the number of independent variables, thereby simplifying the problem.

• Numerical methods:

  • Shooting method: This iterative technique was used to handle the boundary value problems (BVPs) resulting from the transformed ODEs. The shooting method guesses the initial conditions to convert the BVPs into initial value problems (IVPs).

  • Bvp4c tool in MATLAB: The well-known bvp4c function in MATLAB was employed to solve the system of nonlinear ODEs. This tool is particularly suited for solving boundary value problems with high accuracy.

• Parameter analysis:

  • The study performed a detailed parametric analysis to investigate how different parameters affect the bioconvection phenomenon. Parameters such as thermal radiation, activation energy, and the characteristics of the gyrotactic microorganisms were varied, and their impacts on velocity, thermal, and concentration profiles were studied.

By combining these methods, the study provided a comprehensive numerical analysis of the bioconvection effects in Maxwell nanofluids. The results offer insights into optimizing the thermal properties and enhancing the efficiency of systems utilizing nanofluids, particularly in biofuel applications.

2 Mathematical Modeling

In this terminology, consider unsteady, radiative flow over a stretching cylinder of bio-convective Maxwell nanofluid of radius R and including motile swimming gyrotactic microorganisms. The current approach, activation energy, and non-uniform heat sink/source characteristics are also taken into account. The Brownian motion and the nanofluid are investigated using the thermophoresis diffusion theory. The cylindrical polar coordinates are taken in such a direction that the $z$-axis along the surface of the cylinder and along the $r$-axis is restrained along the radial way of the cylinder.

It is implicit that the temperature on the surface of the cylinder is expressed by $T_{w}$, and the surface nanoparticles concentration $C_{w}$, surface microorganisms denoted by $N_{w}$, while the ambient temperature, microorganisms, and nanoparticles concentration are $T_{\infty }$, $N_{\infty }$, and $C_{\infty }$. Figure 1 illustrates the geometry of a stretched cylinder under certain conditions.

Fig. 1

Schematic of the problem and graphical abstract

2.1 Governing Equations

$$\frac{{\partial \left( {ru} \right)}}{\partial z} + \frac{{\partial \left( {rw} \right)}}{\partial r} = 0,$$

(1)

$$\begin{gathered} \frac{\partial u}{{\partial t}} + u\frac{\partial u}{{\partial z}} + w\frac{\partial u}{{\partial r}} = \nu \left( {\frac{{\partial^{2} u}}{{\partial r^{2} }} + \frac{1}{r}\frac{\partial u}{{\partial r}}} \right)\, - \lambda_{1} \left( \begin{gathered} \frac{{\partial^{2} u}}{{\partial t^{2} }} + 2u\frac{{\partial^{2} u}}{\partial t\partial z} + 2w\frac{{\partial^{2} u}}{\partial r\partial t} \hfill \\ + 2uw\frac{{\partial^{2} u}}{\partial r\partial z} + w^{2} \frac{{\partial^{2} u}}{{\partial r^{2} }} + u^{2} \frac{{\partial^{2} u}}{{\partial z^{2} }} \hfill \\ \end{gathered} \right) \hfill \\ - \frac{{\sigma B_{0}^{2} }}{{\rho_{f} }}\left( {u + \lambda_{1} w\frac{\partial u}{{\partial r}}} \right) + \left[ {\frac{1}{{\rho_{f} }}\left( \begin{gathered} - g^{*} \gamma \left( {\rho_{m} - \rho_{f} } \right)\left( {N - N_{\infty } } \right) + \hfill \\ - \left( {\rho_{p} - \rho_{f} } \right)g^{*} \left( {C - C_{\infty } } \right) + \left( {1 - C_{f} } \right)\left( {T - T_{\infty } } \right)\rho_{f} \beta^{**} g* \hfill \\ \end{gathered} \right)} \right], \hfill \\ \end{gathered}$$

(2)

$$\begin{gathered} \frac{\partial T}{{\partial t}} + u\frac{\partial T}{{\partial z}} + w\frac{\partial T}{{\partial r}} + \lambda_{T} \left( \begin{gathered} \frac{\partial u}{{\partial t}}\frac{\partial T}{{\partial z}} + \frac{{\partial^{2} T}}{{\partial t^{2} }} + 2u\frac{{\partial^{2} T}}{\partial t\partial z} + \frac{\partial w}{{\partial t}}\frac{\partial T}{{\partial r}} + 2w\frac{{\partial^{2} T}}{\partial t\partial r} + 2uw\frac{{\partial^{2} T}}{\partial r\partial z} + w^{2} \frac{{\partial^{2} T}}{{\partial r^{2} }} \hfill \\ + w\frac{\partial u}{{\partial r}}\frac{\partial T}{{\partial z}} + u^{2} \frac{{\partial^{2} T}}{{\partial z^{2} }} + u\frac{\partial u}{{\partial z}}\frac{\partial T}{{\partial z}}w\frac{\partial w}{{\partial r}}\frac{\partial T}{{\partial r}} + u\frac{\partial w}{{\partial z}}\frac{\partial T}{{\partial r}} \hfill \\ \end{gathered} \right) \hfill \\ = \frac{\partial }{\partial r}\left( {K\left( T \right)r\frac{\partial T}{{\partial r}}} \right)\frac{1}{r}\frac{1}{{\left( {\rho c_{p} } \right)}} + \tau \left[ {D_{B} \frac{\partial T}{{\partial r}}\frac{\partial C}{{\partial r}} + \frac{{D_{T} }}{{T_{\infty } }}\left( {\frac{\partial T}{{\partial r}}} \right)^{2} } \right] - \frac{1}{{\left( {\rho c} \right)_{f} }}\frac{{\partial \left( {rq_{r} } \right)}}{\partial r} + \frac{1}{{\left( {\rho c} \right)_{f} }}Q^{\prime\prime\prime}, \hfill \\ \end{gathered}$$

(3)

$$\begin{gathered} w\frac{\partial C}{{\partial r}} + u\frac{\partial C}{{\partial z}} + \frac{\partial C}{{\partial t}} + \lambda_{C} \left( \begin{gathered} \frac{{\partial^{2} C}}{{\partial t^{2} }} + \frac{\partial w}{{\partial t}}\frac{\partial C}{{\partial r}} + \frac{\partial C}{{\partial z}}\frac{\partial u}{{\partial t}} + 2u\frac{{\partial^{2} C}}{\partial t\partial z} + 2w\frac{{\partial^{2} c}}{\partial t\partial r} + 2uw\frac{{\partial^{2} C}}{\partial r\partial z} + w^{2} \frac{{\partial^{2} C}}{{\partial r^{2} }} \hfill \\ + u\frac{\partial w}{{\partial z}}\frac{\partial C}{{\partial r}} + w\frac{\partial w}{{\partial r}}\frac{\partial C}{{\partial r}}u^{2} \frac{{\partial^{2} C}}{{\partial z^{2} }} + u\frac{\partial u}{{\partial z}}\frac{\partial C}{{\partial z}} + w\frac{\partial u}{{\partial r}}\frac{\partial C}{{\partial z}} \hfill \\ \end{gathered} \right) \hfill \\ = D_{B} \left( {\frac{{\partial^{2} C}}{{\partial r^{2} }} + \frac{\partial C}{{\partial r}}\frac{1}{r}} \right) + \left( {\frac{{\partial^{2} T}}{{\partial r^{2} }} + \frac{\partial T}{{\partial r}}\frac{1}{r}} \right)\frac{{D_{T} }}{{T_{\infty } }} - \left( {\frac{T}{{T_{\infty } }}} \right)^{n} k_{r}^{2} \left( {C - C_{\infty } } \right)\exp \left( { - \frac{{E_{a} }}{kT}} \right), \hfill \\ \end{gathered}$$

(4)

$$\frac{\partial N}{{\partial t}} + w\frac{\partial N}{{\partial r}} + u\frac{\partial N}{{\partial z}} + \frac{{bW_{c} }}{{\left( {C_{w} - C_{\infty } } \right)}}\left( {\frac{\partial }{\partial r}\left( {N\frac{\partial C}{{\partial r}}} \right)} \right) = D_{m} \left( {\frac{{\partial^{2} N}}{{\partial r^{2} }}} \right),$$

(5)

$$\left. \begin{gathered} \lambda u_{w} \left( {t,z} \right) = \lambda \frac{az}{{\left( {1 - \gamma t} \right)}} = u\left( {t,z,r} \right),\,\,\,\,\,0 = w\left( {t,z,r} \right) \hfill \\ - k\frac{\partial T}{{\partial r}} = h_{f} \left( {T_{w} - T} \right),\frac{{D_{T} }}{{T_{\infty } }}\frac{\partial T}{{\partial r}} + D_{B} \frac{\partial C}{{\partial r}} = 0,N = N_{w} \,at\,\,R_{1} = r \hfill \\ u \to u_{w} \left( {t,z} \right),\,\,N \to N_{\infty } ,T \to T_{\infty } ,\,\,C \to C_{\infty } ,\,\,as\,\,r \to \infty \hfill \\ \end{gathered} \right\}$$

(6)

Here, velocities of Maxwell fluid are u and w in $r$ and $z$-direction, respectively, thermal diffusivity of nanoliquid represents with $\alpha_1$, $\lambda_{1}$ is the coefficients of thermal relaxation to time, $\nu$ simplify the kinematic viscosity, $\rho_f$ denotes for density of nanofluid, $\rho_p$ express the nanoparticles density, $\beta^\ast$ denotes coefficients of volume suspension, $\rho_m$ is the gyrotactic motile microorganism density, σ^* be the Stefan-Boltzmann constant, k^* denotes coefficient of Rosseland mean spectral, τ defines heat capacity ratio to the fluid of the host heat capacity, $\left(\frac T{T_\infty}\right)^n\exp\;\left(\frac{-E_a}{kT}\right)$ is modified Arrhenius function, activation energy coefficient is represented by $E_a$ and for fitted rate constant n, Kr² be the chemical reaction constant, T and C denoted the temperature of fluid and concentration of nanoparticles etc., N denotes microorganisms, D_B and D_T express the Brownian diffusion coefficients and thermophoresis diffusion coefficients, microorganism coefficient is D_m, and b is chemotaxis and the maximum cell swimming speed is symbolized by W_c. The non-uniform heat source sink Q‴ is addressed as

$$\left[h\left(T_w-T_\infty\right)f'+h\ast\left(T-T_\infty\right)\right]\frac{kW\left(z\right)}{zv}=Q'''$$

2.1.1 Similarity Transformation

The similarity transformations mentioned below are reduced to the PDE equation to the system of ordinary differential equations.

$$\begin{array}{ll}\varsigma=\sqrt{\frac a{\upsilon\left(1-\gamma t\right)}}\left(\frac{r^2-R_1^2}{2R_1}\right),u=\frac{az}{1-\gamma t}f'\left(\zeta\right),w=-\frac{R_1}r\sqrt{\frac{a\upsilon}{\left(1-\gamma t\right)}}f\left(\zeta\right)\\ \chi\left(\varsigma\right)=\frac{N-N_\infty}{N_w-N_\infty},\theta\left(\varsigma\right)=\frac{T-T_\infty}{T_w-T_\infty},\phi\left(\zeta\right),w=\frac{C-C_\infty}{C_w-C_\infty}, \end{array}$$

(7)

This implies that

$$\begin{array}{ll}\left(1+2\zeta\lambda\right)f''' -\zeta\frac S2f''+2.\lambda f''+ff''-sf'-f'^2-\frac74\beta_1S^2\eta f''\frac{\beta_1}4\zeta^2S^2\beta_1f'''\\ \quad \quad \quad \quad \quad \ -2\beta_1Sf'^2-\beta_1S\zeta f'f''+S\beta_1S\zeta ff'''+2\beta_1ff'f''+3\beta_1ff''-\frac{\lambda\beta_1}{\left(1+2\lambda\zeta\right)}f''f'^2\\ \quad \quad \quad \quad \quad \ -\beta_1f^2f'''-\left(f'-\beta ff''\right)M^2+\alpha\left(\theta-Nr\varphi-Nc\chi\right)=0 \end{array}$$

(8)

From the above equations, the unsteadiness parameter is $S = \left( {\frac{\gamma }{a}} \right)$, the curvature parameter is $\lambda \left( { = \frac{1}{R}\sqrt {\frac{{v\left( {1 - \gamma t} \right)}}{a}} } \right)$, the Maxwell parameter is $\beta_{1} = \frac{{\lambda_{1} a}}{{\left( {1 - \gamma t} \right)}}$, the magnetic parameter $Nc = \frac{{\left( {\rho_{m} - \rho_{f} } \right)\left( {N_{w} - N_{\infty } } \right)\gamma **}}{{\beta^{*} \left( {1 - C_{f} } \right)\left( {T_{w} - T_{\infty } } \right)\rho_{f} }}$ signifies the mixed convection parameter $\lambda = \frac{Gr}{{\left( {\text{Re}} \right)^{2} }}$, the buoyancy parameter is denoted by $\sigma * = \frac{{kr^{2} \left( {1 - \gamma t} \right)}}{a}$, and $Nc = \frac{{\gamma **\left( {N_{w} - N_{\infty } } \right)\left( {\rho_{m} - \rho_{f} } \right)}}{{\rho_{f} \beta^{*} \left( {T_{w} - T_{\infty } } \right)\left( {1 - C_{f} } \right)}}$ is the bioconvection Rayleigh number.

$$\begin{array}{ll}\left(1+2\lambda\zeta\right)\theta''+pr\left(f\theta'-\frac S2\zeta\theta'\right)+\left(1+2\lambda\zeta\right)\left(\theta\theta''-\theta'^2\right)\varepsilon+2\lambda\theta'+2\lambda\varepsilon\theta\theta''\\-pr\beta_t\left[\frac34s^2\zeta\theta'-\frac32sff'-\frac12s\zeta f'\theta'+\frac14s^2\zeta^2\theta''-s\zeta f\theta''+\theta''f^2+\theta''ff'\right]\\ \left(1+2\lambda\zeta\right)prNb\theta'\phi'+\left(1+2\lambda\zeta\right)prNt\theta'^2+\left(1+\frac43Rd\right)\left[\left(1+2\lambda\zeta\right)\theta''+2\lambda\theta'\right]\\+pr\left(Q^\ast f'+Q\theta\right)=0, \end{array}$$

(9)

where the Prandtl number is $pr\left( { = \frac{\nu }{{\alpha_{1} }}} \right)$ and $\beta_{t} = \frac{{\lambda_{t} a}}{{\left( {1 - \gamma t} \right)}}$ is the thermal relaxation time parameter. $Q^{*} = \frac{{A^{*} \alpha }}{{v\left( {\rho c} \right)_{f} }}$ denotes space-dependent heat source parameter, and $Q = \left( {\frac{{\alpha B^{*} }}{{v\left( {\rho c} \right)_{f} }}} \right)$ is time-dependent heat source constant. Thermophoresis parameter is represented with $Nt = \frac{{\tau D_{T} \left( {T_{w} - T_{\infty } } \right)}}{{vT_{\infty } }}$, Brownian motion parameter diffusion is denoted with $Nb = \frac{{\tau D_{T} \left( {C_{w} - C_{\infty } } \right)}}{{vT_{\infty } }}$, and radiation parameter is represented by $Rd=\frac{4{\sigma }^{*}{T}_{\infty }^{3}}{kk*}$.

$$\begin{array}{ll}\left(1+2\lambda\zeta\right)\phi''+pr\left(f\theta'-\frac S2\zeta\theta'\right)+\left(1+2\lambda\zeta\right)\left(\theta'\theta''-\theta'^2\right)\varepsilon+2\lambda\phi'+2\lambda\varepsilon\theta'\theta''\\-pr\beta_c\left[\frac34s^2\zeta\theta'-\frac32sff'-\frac12s\zeta f'\theta'+\frac14s^2\zeta^2\theta''-s\zeta f\theta''+\theta''f^2+\theta''ff'\right]\\ \left(1+2\lambda\zeta\right)\frac{Nt}{Nb}\theta''-Lepr\sigma\left[1+\delta_0\theta\right]^n\exp\left(-\frac E{1+\delta_0\theta}Rd\right)\phi+2\mathrm\lambda\frac{Nt}{Nb}\theta''=0 \end{array}$$

(10)

From the above equations, $S = \left( {\frac{\gamma }{a}} \right)$ is the unsteadiness parameter, $\lambda = \left[ {\left( \frac{1}{R} \right)\sqrt {\frac{{\left( {1 - \gamma t} \right)v}}{a}} } \right]$ the curvature parameter, $\beta_{c} = \left( {\frac{{a\lambda_{c} }}{{\left( {1 - \gamma t} \right)}}} \right)$ the mass relaxation time parameter, and $Le\left( { = \frac{{\alpha_{1} }}{{D_{B} }}} \right)$ the Lewis number. $E = \frac{{E_{a} }}{kT}$ is the activation energy parameter, $\frac{{T_{w} - T_{\infty } }}{{T_{\infty } }}$ is the temperature difference parameter, and $\sigma * = \frac{{kr^{2} \left( {1 - \gamma t} \right)}}{a}$ represents the chemical reaction parameter.

$$\left(1+2\lambda\zeta\right)\;\chi\left(\zeta\right)-2\lambda\chi'-\frac S2Lb\zeta p_8+Lbf\;\chi'+Pe\left(1+2\lambda\zeta\right)\left[\chi''\left(p_8+\varpi\right)+\phi\chi'\right]=0$$

(11)

$Lb = \frac{\nu }{{D_{m} }}$ is represented by (Lewis number) bioconvection, Peclet number is $pe = \frac{{bW_{c} }}{{D_{m} }}$, and $\varpi = \frac{{N_{\infty } }}{{N_{w} - N_{\infty } }}$ is microorganisms’ difference parameter.

2.1.2 Boundary Conditions

$$\left. \begin{gathered} \lambda^{*} \frac{az}{{\left( {1 - \gamma t} \right)}} = u\left( {t,z,r} \right) = \lambda^{*} u_{w} \left( {t,z} \right),\,\,\,\,\,\,\,\,w\left( {t,z,r} \right) = 0, \hfill \\ - k\frac{\partial T}{{\partial r}} = h_{f} \left( {T_{w} - T} \right),D_{B} \frac{\partial C}{{\partial r}} + \frac{\partial T}{{\partial r}}\frac{{D_{T} }}{{T_{\infty } }} = 0,N_{w} = N\,at\,\,r = R_{1} \hfill \\ u \to u_{w} \left( {t,z} \right),\,\,C \to C_{\infty } ,T \to T_{\infty } ,\,\,N \to N_{\infty } ,\,\,as\,\,r \to \infty \hfill \\ \end{gathered} \right\}$$

(12)

For transmute the boundaries equation into dimensionless ODEs form as:

$$\begin{gathered} \theta^{\prime}\left( 0 \right) = - \frac{{h_{f} }}{k}\sqrt {\frac{{v\left( {1 - \gamma t} \right)}}{a}} \left[ {1 - \theta \left( 0 \right)} \right],\left[ {Nb\phi^{\prime}\left( 0 \right)} \right] + \left[ {Nt\theta^{\prime}\left( 0 \right)} \right] = 0, \hfill \\ \chi \left( 0 \right) = 1,f^{\prime}\left( \infty \right) \to 0,\theta \left( \infty \right) \to 0,\phi \left( \infty \right) \to 0,\chi \left( 0 \right) \to 0 \hfill \\ \end{gathered}$$

(13)

3 Numerical Scheme

The coupled system equations with boundary constraints are solved numerically. The dimensionless ordinary differential equations are solved numerically by applying the shooting scheme via the bvp4c built-in function MATLAB. The error convergence of the current scheme $10^{ - 6}$ is approved. For a complete numerical model, the appropriate step size $\Delta \zeta = 0.01$ is opt with the boundary layer length $\zeta_{\infty } = 5$. For this method, firstly, it is compulsory to convert the higher order of differential equations to first-order differential equations by introducing some new variables as follows:

$$p^{\prime}_{3} = \frac{{\left[ \begin{gathered} - 2\lambda p_{3} + \frac{S}{2}\zeta p_{3} + sp_{2} + p_{2}^{2} - p_{1} p_{3} + \frac{7}{4}\beta_{1} S^{2} \zeta p_{3} \hfill \\ + 2\beta_{1} S^{2} p_{2}^{2} + 2\beta_{1} Sp_{2}^{2} + \beta_{1} S\zeta p_{2} p_{3} - 3\beta_{1} p_{1} p_{3} - 2\beta_{1} p_{1} p_{2} p_{3} \hfill \\ + \frac{{\lambda \beta_{1} }}{{\left( {1 + 2\lambda \zeta } \right)}}p_{1}^{2} p_{3} + M^{2} \left( {p_{2} - \beta p_{2} p_{3} } \right) - \alpha \left( {\theta - Nr\varphi - Nc\chi } \right) \hfill \\ \end{gathered} \right]}}{{\left( {1 + 2\lambda \zeta } \right) - \frac{{\beta_{1} }}{4}\zeta^{2} S^{2} + S\beta_{1} \zeta q_{1} - \beta_{1} q^{2}_{1} }}$$

(14)

$$p^{\prime}_{5} = \frac{{\left[ \begin{gathered} - pr\left( {p_{1} p_{5} - \frac{s}{2}\zeta q_{5} } \right) - \left( {1 + 2\lambda \zeta } \right)\left( { - p_{5}^{2} } \right)\varepsilon - \hfill \\ 2\lambda p_{5} + pr\beta_{t} \left\{ {\frac{3}{4}s^{2} \zeta p_{5} - \frac{3}{2}sp_{1} p_{2} - \frac{1}{2}s\zeta p_{1} p_{5} + p_{5} p_{1} p_{2} } \right\} \hfill \\ - \left( {1 + 2\lambda \zeta } \right)prN_{b} p_{5} \phi^{\prime} + \left( {1 + 2\lambda \zeta } \right)prN_{t} p_{5}^{2} \hfill \\ - pr\left( {Q^{*} p_{2} + Qp_{4} } \right) - 2\left( {1 + \frac{4}{3}Rd} \right)\lambda p_{5} \hfill \\ \end{gathered} \right]}}{{\left( {\left( {1 + 2\lambda \zeta } \right)\left\{ {1 + \left( {1 + \frac{4}{3}Rd} \right) + q_{4} } \right\}} \right) + 2\lambda \varepsilon p_{4} - pr\beta_{t} \left( { + \frac{1}{4}s^{2} \zeta^{2} - s\zeta p_{1} + p_{1}^{2} } \right)}}$$

(15)

$$p^{\prime}_{7} = \frac{{ - 2\lambda p_{7} - pr\left( {p_{1} p_{7} - \frac{s}{2}\zeta p_{7} } \right) + pr\beta_{c} \left[ \begin{gathered} \frac{3}{4}s^{2} \zeta p_{7} - \frac{3}{2}sp_{1} p_{2} - \frac{1}{2}s\zeta p_{2} p_{7} + p_{7} p_{1} p_{2} \hfill \\ + \left( {1 + 2\lambda \zeta } \right)p_{7}^{2} \varepsilon - 2\lambda p_{7} \hfill \\ \end{gathered} \right]}}{{\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {1 + 2\lambda \zeta } \right)\left\{ {1 + 2\varepsilon p_{7} - p_{7}^{2} \varepsilon } \right\} + 2\lambda \varepsilon p_{7} + pr\beta_{c} \left( {\frac{1}{4}s^{2} \zeta^{2} - s\zeta p_{1} + p_{1}^{2} } \right)}}$$

(16)

$$p^{\prime}_{9} = \frac{{2\lambda p_{8} - \frac{S}{2}Lb\zeta p_{8} - + Lbp_{1} p_{8} - Pe\left( {1 + 2\lambda \zeta } \right)\left[ {p^{\prime}_{7} \left( {p_{8} + \varpi } \right) + p_{7} q_{8} } \right]}}{{\left( {1 + 2\lambda \zeta } \right)}}$$

(17)

4 Result Discussion and Representation

This section discusses the sound effects of varying important parameters on the velocity field, species temperature field, nanoparticle concentration profile, and microorganism field.

4.1 Effect of Velocity Field

The importance of several important parameters compared to the performance profile is shown in Figs. 2 and 4. Figure 2a characterizes the estimation of $M$ magnetic parameter via velocity profile $f^{\prime}$ for stretching case. Here, the velocity profile $f^{\prime}$ declined when $M$ the magnetic parameter increases. Physically, when an exterior magnetic field is applied to several electrically conductive fluids, then a repulsive force identified as Lorentz force emerged which apprise to strongly oppose to the flow. Figure 2b shows the behavior of the velocity field for the magnetic parameter in the case of the shrinking cylinder. Here, the velocity is also reduced. Figure 3 captures the consequence of mixed convection on the velocity distribution. It is concluded that the fluid velocity is boosted up via larger magnitudes of the mixed convection. Figure 4a is drawn to observe the performance of the velocity field via larger variations in the buoyancy ratio parameter. It is analyzed that the velocity field decreased by augmentation of the buoyancy ratio parameter. Figure 4b illustrates the curve lines of the velocity field via the buoyancy ratio parameter in the case of the shrinking cylinder. In shrinking case, the flow of fluid is also reduced. Figure 5a, b expresses the velocity field’s bioconvection Rayleigh number including both stretching and contracting conditions. It is noticed that the velocity gradient declines due to an increment in the values of the bioconvection Rayleigh number for both cases. Figure 6 is built to investigate Deborah number’s significance on relaxation number using velocity profile. Here, it is examined that the higher amounts of Deborah number cause a reduction in the velocity field.

Fig. 2

A Sway of $f^{\prime}$ for MNF due to $M$. b Sway of $f^{\prime}$ for shrinking case due to $M$

Fig. 3

Sway of $f^{\prime}$ for MNF due to $\alpha$

Fig. 4

A Sway of $f^{\prime}$ for MNF due to $Nr$. b Sway of $f^{\prime}$ for shrinking case due to $Nr$

Fig. 5

A Sway of $f^{\prime}$ for MNF due to $Nc$. b Sway $f^{\prime}$ for shrinking case due to $Nc$

Fig. 6

Sway of $f^{\prime}$ for MNF due to $\beta_{1}$

4.2 Effect of Temperature Distribution

In this portion, the behavior of different flow and controlling parameters for the thermal field of species are captured through Figs. 7, 8, 9, 10, 11, 12, 13, 14, 15, and 16. The temperature gradient is employed to demonstrate how the Prandtl number affects things in Fig. 7. Here, it is observed that a temperature field declines by increasing the magnitudes of the Prandtl number. Physically, the Prandtl number (Pr) is inversely related to the thermal diffusivity. An increase in the Prandtl number results in weaker thermal diffusion, causing the temperature field to decrease. Figure 8 indicates thermophoresis is a parameter $Nt$ against the thermal field. In Fig. 9, the effects of the magnet parameter temperature field were described. It is obviously evident that an increased magnetic parameter causes an upsurge in the heat field. Figure 10 is charted to estimate the characteristics of the buoyancy ratio parameter against the temperature gradient. It is concluded that the temperature reports improve by mounting the value of the buoyancy ratio parameter. The nature of the thermal radiation parameter versus temperature gradient is shown in Fig. 11. From the curve’s lines of the thermal field, we observed that a larger thermal radiation parameter causes an increment in the estimations of temperature dispersion. Physically internal energy of fluid increased due to thermal radiation as results thermal field is increased. Figure 12 is designed to analyze the effect of heat distribution vs Rayleigh number on bioconvection. It is observed that the energy gradient is boosted up via a larger bioconvection Rayleigh number. The influence of the space-based heat source parameter against thermal distribution is captured in Fig. 13. By changing the stability of the space-based heat source parameter, the temperature distribution is improved. Figure 14 is drawn to investigate how the temperature distribution is affected by a time-dependent heat source constant. Here, it is clear that the gradient temperature is enlarged via greater values of the time-dependent heat source constant. Figure 15 demonstrates how temperature gradients react in relation to the different thermal Biot number values. It can be noticed that the temperature gradient is boosted up via a greater thermal Biot number. Figure 16 explores the quality of the differential convection parameter in the existence of a thermal field. It is established that a higher mixed convection value reduces the thermal field of species.

Fig. 7

Sway of $\theta$ for MNF due to $\Pr$

Fig. 8

Sway of $\theta$ for MNF due to $Nt$

Fig. 9

Sway of $\theta$ for MNF due to $M$

Fig. 10

Sway of $\theta$ for MNF due to $Nr$

Fig. 11

Sway of $\theta$ for MNF due to $Rd$

Fig. 12

Sway of $\theta$ for MNF due to $Nc$

Fig. 13

Sway of $\theta$ for MNF due to $Q*$

Fig. 14

Sway of $\theta$ for MNF due to $Q$

Fig. 15

Sway of $\theta$ for MNF due to $Bi$

Fig. 16

Sway of θ for MNF due to $\alpha$

4.3 Effect of Volumetric Concentration of Nanoparticles

The behavior of various parameters via volumetric concentration is described in this section as nanoparticles for MNF over cylinder/plate and is shown in Figs. 17, 18, 19, 20, and 21. Figure 17 examines the behavior of Prandtl number versus concentration of particles for MNF over stretching cylinder/plate. It is observed from the concentration of nanoparticle curves that the Prandtl number leads to a decrease in the concentration of species. Figure 18 is plotted to illustrate the behavior of the solutal field via the Brownian motion parameter. From the curve lines, it is clear that the concentration field declines via greater Brownian motion parameter. Brownian motion refers to the random movement of particles suspended in a fluid, caused by collisions with the fast-moving molecules of the fluid. This erratic motion results in particles being randomly displaced within one region of the fluid and then moving to another. Each movement leads to further random disturbances in the new area, characterizing a liquid in thermal equilibrium. In a flowing system, as the value of Brownian diffusion increases, the random movement of suspended nanoparticles also increases due to the rise in the system’s internal energy, resulting in an increase in the thermal field while decrement in concentration. The following chart demonstrates the manipulation of the thermophoresis parameter on the concentration field (Fig. 19). The concentration of species is enhanced by improving the values of the thermophoresis parameter. In thermophoresis phenomena, tiny fluid particles are drawn from warmer areas to cooler areas. Consequently, nanoparticles in the fluid move away from the heated surface, leading to an increase in temperature, enhancement of the thermal boundary layer, and improved nanoparticle volume fraction profiles. From Fig. 20, it is indicated that the solutal field is boosted up via a larger buoyancy ratio parameter. This example shows the importance of the Lewis number in connection to the volumetric concentration of nanoparticles (Fig. 21). Here, the solutal field is reduced by increasing the Lewis number.

Fig. 17

Sway of $\phi$ for MNF due to $\Pr$

Fig. 18

Sway of $\phi$ for MNF due to $Nb$

Fig. 19

Sway of $\phi$ for MNF due to $Nt$

Fig. 20

Sway of $\phi$ for MNF due to $Nr$

Fig. 21

Sway of $\phi$ for MNF due to $Le$

4.4 Effect of Microorganism’s Profile

Figure 22 is drawn to examine the microorganism’s profile via greater magnetic parameter. It is concluded that a higher magnetic parameter leads to a growth in the microorganism’s profile. The relationship between the bioconvection Lewis number and the profile of microorganisms is revealed in Fig. 23. Here, it can be noticed that the microorganism profile is declined via greater bioconvection Lewis number. The diffusivity of microorganisms decreases with larger values of Lb, ultimately reducing the motile density of the fluid. This behavior is associated with the weakened diffusivity of microorganisms. Figure 24 is designed to estimate the estimations in the microorganism’s field via a greater mixed convection parameter. It is apparent that the microorganism field decreases as the mixed convection parameter increases. The nature of the Deborah number for the relaxation parameter versus the microorganisms’ profile is mentioned in Fig. 25. From the curve lines, it was concluded that the microorganism’s field is boosted up for a greater Deborah number for the relaxation parameter.

Fig. 22

Sway of $\chi$ for MNF due to $M$

Fig. 23

Sway of $\chi$ for MNF due to $Lb$

Fig. 24

Sway of $\chi$ for MNF due to $\alpha$

Fig. 25

Sway of $\chi$ for MNF due to $\beta_{1}$

5 Conclusion

The summary of this research work illustrates the MHD of bio-convective Maxwell nanofluid through a stretching/shrinking cylinder/plate. The Cattaneo-Christov double diffusion theory is also used. The implications of activation energy and thermal radiation are investigated. The governing flow equations in the mathematical model also include equations for five equations as continuity, velocity, energy, concentration of nanoparticles, and microorganisms. The major outcomes are given below:

• An increment in mixed convection parameters exhibits a growing nature of velocity profile for cylinder and plate.

• Bioconvection Rayleigh number and buoyancy ratio parameter reduce the flow rate.

• Radiation parameter and Cattaneo-Christov heat flux increased the temperature of the fluid.

• Increasing the value of the thermal Biot number causes an increment in the heat transfer rate.

• Reduction of the concentration of nanoparticles increases the value Brownian motion parameter and Lewis number.

• Thermophoresis is used to increase the concentration of nanoparticles.

• The microorganism’s profile reduces with a larger bioconvection Lewis number.

• The larger Deborah number enhanced the microorganism’s field.