Reiner-rivlin nanofluid behavior with gyrotactic microorganisms and slip conditions in a rotating disk configuration

1. Introduction

Fluid mechanics is a scientific study focused on how external forces affect the structure within fluids. It represents a division of classical physics through significant features in zoology, aerospace engineering, chemical engineering, and mechanical engineering. Two sorts of fluids are non-Newtonian and Newtonian fluids. Newtonian fluids adhere to linear relationships between stress and strain, meaning the fluid’s resistance to deformation is constant; meanwhile, non-Newtonian fluids exhibit more complex behaviors, often requiring sophisticated models for accurate description. The Reiner-Rivlin fluid was one of the non-Newtonian fluids, which (Reiner 1945) and (Rivlin 1948) described as unique characteristic of dilatancy. This characteristic refers to the viscoelastic tendency to expand in volume when agitated. (Selvi and Shukla 2022) presented a theoretical investigation to notice an analytical procedure of a micropolar fluid through a sphere by adopting the Mehta-Morse boundary condition. Their work separated the flow into two parts: the outer region fills with the micropolar fluid, while the Reiner-Rivlin fluid regulates the inner flow. They found that the drag is less resistant to a Reiner-Rivlin liquid sphere than to an impermeable sphere. The study on the rate of energy and concentration flux through a revolving stretch disc using the Cattaneo-Christov model adopting the Stefan blowing was published by (Moatimid et al., 2022). The model equations were solved through Homotopy perturbation (HPM). They observed that the rate of heat and mass transport is enhanced through Stefan-blowing particle diffusion. Some medical therapies were able to benefit from this finding. (Puspa Nathan et al., 2024) indicated the impact of Reiner-Rivlin due to the shrinking/stretching rotating disk. They noticed that a dual solution occurs in some range of parameters when the effects are added to the flow. A few series of literature based on the Reiner-Rivlin fluid with several effects over different surfaces can be seen in (Arain et al., 2021; Hiremath et al., 2023; Hayat T et al., 2023; Alarabi et al., 2023; Yasin et al., 2025).

The concept of rotating flow was pioneered by Kármán et al. (1921) by postulating an infinitely large disk spinning at a constant angular velocity within a quiescent fluid medium. Centrifugal compressors and fans are based on this design, which uses a rotating disk to push fluid. However, the centrifugal forces are sufficiently weak to cause the fluid to depart radially due to the disk’s surface. Consequently, the displaced fluid is replenished by the fluid layer located overhead the disk, facilitated by a disk pumping result characterized by descending spiralling motion. The replication of the flow of fluid due to a revolving disk was first described by Cochran (1934). Since then, the work on the flow over rotating disks has sparked curiosity among researchers, especially in the application of power turbines, circulating lubrication, the geothermal industry, and so on. Followed by Smith (1947), who was the first to investigate the boundary layer transition using hot-wire techniques. In his study, he observed that the sinusoidal disturbances appear in the disk boundary layer at sufficiently large Reynolds numbers. Meanwhile, (Imayama et al., 2016) constructed an experimental study with surface roughness. (Thomas et al., 2023) studied the role of wall slip across the linear stability of the spinning disc. The heat transport of inclined flow of the rotating disc under magnetohydrodynamics (MHD) was inspected by Mahmud et al. (2023) using the shooting method. They observed that the strengthening magnetic field and the increase in the rotating disc enhance the heat transfer of the flow. (Faizan et al., 2024) considered the stagnant flow of the Maxwell nanomaterial across a stretching/shrinking spinning disk and solved it using bvp4c in MATLAB.

The term and phenomenon of bioconvection were used by Loefer et al. (1952) and Platt (1961). According to Pedley et al. (1988), bioconvection is the phenomenon where the fluids exhibit macroscopic motion driven by density gradients generated by the coordinated movement of motile microorganisms. Additionally, research on microorganisms has garnered significant attention from researchers due to their widespread applications in commercial, industrial, and microbiological systems. These applications span a variety of fields, including biofertilizers, bioinformatics, alcohol production, biotechnology, biofuel generation, and biosensors. For instance, microalgae represent a rapidly growing biomass that can be utilized in the production of bio-based materials, alcohol, and biofuels. (Kuznetsov 2010) deliberates the problem of convection with nanofluid considering gyrotactic microorganisms. The main aim of microorganisms is to enhance or induce nanofluid convection. Later, Kuznetsov (2021) continued the study of the nanofluid alongside motile (oxytactic) microorganisms to understand the behavior while inspecting its stability when it occupies a shallow horizontal layer. (Othman et al., 2023) solved the problem comprising the gyrotactic microbe through a stretchable inclined cylinder with heat and mass transfer characteristics. In their study, the thermal layer was enhanced with larger values of heat source sink while reduced with larger magnitudes of Prandtl number. Nabwey et al. (2022) investigated the problem of bioconvection involving gyrotactic microorganisms in a porous surface. In their problem, the problem is numerically solved using the Runge-Kutta procedure and shooting technique. Li et al. (2023) scrutinized the environmental transport of gyrotactic microorganisms in an open-channel flow. Flow shear’s impact remains negligible on the overall shape of the asymptotic concentration distribution. Nevertheless, it has a pronounced effect in significantly augmenting both the transient drift velocity and dispersity. Few studies based on bioconvective gyrotactic microorganisms can be seen from these aforementioned works (Dawar et al., 2021; Algehyne et al., 2023; Fatima et al., 2023; Sarma et al., 2024; Abbas et al., 2024).

The influence of boundary slip in fluids holds significant implications, particularly in features that involve refining artificial cardiac chambers and inside cavities. Speed and energy play crucial roles in governing the fluid flow at the micro-scale. Wall slip behavior plays a great role due to its examination for drag reduction in aerodynamics and hydrodynamics. Navier made his debut with the Navier-Stokes algorithm, which is a popular tool for enhancing fluid flows. (Ramya et al., 2018) discovered the MHD past a non-linear stretchable surface through momentum and thermal slip. They found that the rate of heat and mass transport decreases with the growing magnitude of thermal slip. Yang et al. (2020) explored the effect of viscoelastic second-order slip for the flow of Maxwell fractional fluid. The finite difference method was employed to solve the governing fluid model. The findings indicate that the fractional Maxwell fluid demonstrates enhanced viscosity or elasticity across various fractional parameters. The impact of Navier’s slip and MHD over a porous surface for non-Newtonian fluid is investigated by Maranna et al. (2023). In their work, they chose Ag-Cu, which is presented in the blood as a viscous fluid. It is found that the rising concentration of Ag-Cu intensification the thermal layer. Sharma et al. (2023) studied MHD micropolar fluid through a stretching surface with melting and slip effects. The impact on buoyancy and heat sources over a separation of the unsteady stagnation point was explored by Mahmood et al. (2024). The unsteadiness parameter for opposing flow improved skin friction.

By addressing the limitation, this paper aims to explore von Kármán flow and heat transfer phenomena within a MHD flow of Reiner-Rivlin non-Newtonian fluid over a rotating surface, considering thermal radiation effects and multiple slip conditions with heated and concentration boundary conditions. The previous study was explained by Khan et al. (2024). We have extended their work, added multiple slip effects, and thermal/solutal convective conditions past a rotating disk. Incorporating multiple slip effects offers advantages, particularly in engineering applications, enabling more accurate predictions of flow behavior as well as in the optimization process. Additionally, the inclusion of thermal radiation in the analysis of Reiner-Rivlin fluids will provide a comprehensive understanding of heat transfer performance and fluid behavior. These findings have the potential to enhance simulation accuracy and facilitate the optimization of industrial processes.

2. Physical Problem and Model Formulation

The steady, 3D flow of Reiner-Rivlin Nano liquid containing gyrotactic bacteria that move on a disk that is turning. Nanofluid rotates around the $z-axis$, which is parallel to the surface, at an angle $\Omega$, with $u, v,$ and $w$ being the speeds along $r,$ $\varphi$, and $z$ has been deliberated in Fig. 1(a). It is considered that $T_f,C_f$ and ${\chi }_w$, and $T_{\infty }$, $C_{\infty }$, ${\chi }_{\infty }$ are the fluid temperature, concentration, and the number of moving microbes near and far from the surface. The flow problem’s highly nonlinear solutions are changed into normal ones that are similar enough. This is what we did with the bvp4c plan. We can find the tensor of the Reiner-Rivlin fluid by Reiner (1945) and Rivlin (1948).

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Fig. 1.

(a) Physical geometry of the rotating flow, (b) Numerical method Bvp4c flow chart.

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The stress tensor of the Reiner-Rivlin model is described as:

Here, $e_{ij}=\left(\frac{\partial u_i}{\partial x_j}\right)+\left(\frac{\partial u_j}{\partial x_i}\right)$, $p,\mu ,{\delta }_{ij},e_{jj},{\mu }_c$ is pressure, dynamic viscosity of coefficient, Kronecker delta, deformation rate tensor, and cross-viscosity coefficient.

The constitutive equations are explained as

The stress tensor for the deformation is (Rivlin 1948)

Stress tensor components are considered as (Selvi and Shukla, 2022, and Sabu et al., 2016):

Using Eqns. (3) – (5)

Boundary conditions for the above problem are (Imamaya et al., 2016)

Here, $T$, $C$, $\chi$ represent the temperature, concentration, and motile density of the fluid, respectively. Then, ${\left(\rho c\right)}_p$ is nanoparticle tendency, ${\alpha }_m$ is thermal diffusivity, ${\left(\rho c\right)}_f$ is base fluid tendency,$C_{\infty }\text{is}\text{ambient}\text{concentration},$ $k$ denotes thermal conductivity, ${\chi }_w$ is motile microorganism, $T_f$ is temperature of the fluid, $C_f$ is concentration of the surface. $T_{\infty }\text{is}\text{ambient}\text{temperature},{\chi }_{\infty }$ is motile microorganism, $D_B$ is Brownian diffusivity, and $D_T$ is thermal diffusivity,

The suitable transformation is given below (Arain et al., 2021)

Utilizing Eq. (20), Eqns. (14) - (18) reduces to

The converted boundary conditions are

Where, ${\beta }_1$ shows radial slip,${\beta }_2$ azimuthal slip, $M$ is a magnetic number, $\lambda$ is Reiner-Rivlin fluid parameter, $Nr$ is the buoyancy ratio variable, $Nc$ is the bio covection Rayleigh number, $Nb$ is Brownian motion $Rd$ represents thermal radiation, $Bit$ exhibits the thermal Biot number,$Bic$ exhibits the solutal Biot number, ${{\rm A}}_e$ is the activation energy, $Cr$ is a chemical reaction, $Pr$ is Prandtl number, $Sc$ exhibits Schmidt number,$Nt$ is a thermophoretic parameter $\omega$ microorganism difference parameter, $Lb$ is bio connection Lewis number, $Pe$ is the Peclet number.

$\begin{array}{c}\lambda =\left(\frac{{\mu }_c\Omega }{\mu }\right),{\beta }_1=\left(\rho \sqrt{\Omega v}{\delta }_1\right),{\beta }_2=\left(\rho \sqrt{\Omega v}{\delta }_2\right),M=\left(\frac{\sigma B_0^2}{\rho }\right),\\ Nr=\frac{\left({\rho }_p-{\rho }_f\right)\left(C_f-C_{\infty }\right)}{\beta \left(1-C_f\right)\left(T_f-T_{\infty }\right){\rho }_f}\\ Nc=\frac{\left({\rho }_m-{\rho }_f\right)\left({\chi }_f-{\chi }_{\infty }\right)}{\beta \left(1-C_f\right)\left(T_f-T_{\infty }\right){\rho }_f},Rd=\left(\frac{16\sigma \text{*}{T}_{\infty }^3}{3k\text{*}{k}_0}\right),Bit=\frac{h_f}{k1}\left(\sqrt{\frac{v_f}{\Omega }}\right),\\ Bic=\frac{h_f}{k2}\left(\sqrt{\frac{v_f}{\Omega }}\right),\\ A_e=\left(\frac{E_0}{k{T}_{\infty }}\right),Nb=\left(\frac{\tau D_B\left(C_f-C_{\infty }\right)}{v}\right),Nt=\left(\frac{\tau D_T\left(T_f-T_{\infty }\right)}{{\tilde{T}}_{\infty }v}\right),\\ \mathrm{Pr}=\left(\frac{\mu c_p}{k}\right),Sc=\left(\frac{v}{D_B}\right),\omega =\left(\frac{{\chi }_{\infty }}{{\chi }_w-{\chi }_{\infty }}\right),Lb=\left(\frac{v}{D_N}\right),Pe=\left(\frac{bWc}{v}\right)\end{array}$

3. Quantity of Engineering Importance

The least moment of torque is represented by $C_{m,r}$ (Selvi and Shukla 2022):

$T_r$ signifies with uniform disk rotation

The physical quantities of $C_f,Nu,Sh$ and Nh is given as:

Using Eq. (19), Eq. (28) becomes

4. Numerical Solution Methodology

The Bvp4c has been used for solving the configuration of nonlinear equations (21) - (25) with particular boundary condition (26) by MATLAB software. The nonlinear equations are converted into first order equations by assigning the new variables.

Subsequently, boundary conditions are

It is important to carefully choose the starting assumptions in these numerical simulations to make sure that the solution will meet the initial conditions. Many times, of the iterations have been done to get to a convergent result. Compared to other ways of doing analysis, these answers are very good. In most cases, the above-mentioned convergence criteria are used as a help to decide what values should be assigned to the given parameters. Each iteration of the process is run with a new parameter number as its starting point.

the use of a finite value for ${\xi }_{max}$ using the condition in Eqn. (36) is calculated by

To get a numerical solution, convergent criteria are preferred, and the step size is set at $\Delta \xi = 0.0001$. The numerical flow chart has been described in Fig. 1(b).

5. Numerical validation of the BVP4c code

A comparison of the skin friction coefficient $-F\text{'}\text{'}\left(0\right)$ and $-G^{\prime }\left(0\right)$ are compared with wall slip parameters ${\beta }_1$ and ${\beta }_2$ values are compared to those published in previous studies to assess the validity of the present numerical code. Tables 1 and 2 exhibit this by comparing the validity of the current research to previous investigations by (Abdal et al., 2021). The data confirms that the present results validate and reinforce the findings of previous studies, indicating strong agreement and reliability in the observed trends.

6. Results and Discussion

An incompressible mixed convection flow of Reiner-Rivlin nanoparticles with multiple effects across a rotating extended warmed disk is investigated in the presence of microorganisms. Nondimensional ordinary differential equations (ODEs) (20) – (24) with boundary conditions equation (25) are analyzed using the Bvp4c technique. The results are presented to show how various limits impact the problem, which helps to physically explain it. Among these factors are the following: the volume fraction slip parameter ${\beta }_1,{\beta }_2$, magnetic field M, Reiner-Rivlin fluid factor $ϵ$, mixed convective parameter $\gamma$, buoyancy ratio parameter $Nr$, bioconvection Rayleigh number $Nc$, biot number $Bit$, the Brownian movement limit $Nb$, solutal biot number $Bi$, the thermophoresis limitation $Nt$, the, the Schmidt numeral $Sc$, activation energy $Ae$, Peclet number $Pe$. The effects of these restrictions on the velocities, temperatures, nanoparticle profiles, skin friction $Cf$, Nusselt Nu, and Sherwood Sh variables are the primary focus of the current investigation. The data described are used to plot these distributions are seen in Figs. 2-9. For the entire study, we kept the variables $\begin{array}{c}{\beta }_1=0.3,{\beta }_2=0.3,M=0.4,\epsilon =0.1,\gamma =0.2,Nr=0.2,Nc=0.2,Nb=0.2, \\ Nt=0.1,Bit=0.2,Bic=0.2,Sc=1.0,Pe=1.0,Lb=1.0,A_e=\mathrm{1.0.}\end{array}$

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Fig. 2.

(a) Variation of $M$ against Axial velocity $F\left(\xi \right)$, (b) Variation of $M$ against Radial velocity $F\text{'}\left(\xi \right)$, (c) Variation of $M$ against tangential velocity $G\left(\xi \right)$.

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Fig. 3.

(a) Variation of $ϵ$ against Axial velocity $F\left(\xi \right)$, (b). Variation of $ϵ$ against Radial velocity $F\text{'}\left(\xi \right)$, (c) Variation of $ϵ$ against tangential velocity $G\left(\xi \right)$.

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Fig. 4.

(a) Variation of $\gamma$ against Radial velocity $F\text{'}\left(\xi \right)$, (b). Variation of ${\beta }_1,{\beta }_2$against Radial velocity $F\text{'}\left(\xi \right)$.

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Fig. 5.

(a). Variation of $ϵ$ against temperature profile $\vartheta \left(\xi \right)$, (b). Variation of $M$ against temperature profile $\vartheta \left(\xi \right)$, (c). Variation of $Nb$ against temperature profile $\vartheta \left(\xi \right)$, (d). Variation of $Nt$ against temperature profile $\vartheta \left(\xi \right)$, (e). Variation of Pr against temperature profile $\vartheta \left(\xi \right)$, (f). Variation of Rd against temperature profile ϑ(ξ)

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Fig. 6.

(a) variation of $Nb$ against concentration profile $\Phi \left(\xi \right)$, (b) variation of $Nt$ against concentration profile $\Phi \left(\xi \right)$, (c) Variation of $Sc$ against concentration profile $\Phi \left(\xi \right)$, (d) variation of $Cr$ against concentration profile $\Phi \left(\xi \right)$

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Fig. 7.

(a). Variation of $Pe$ against microorganism density profile ${\rm Y}\left(\xi \right)$, (b). variation of $Lb$ against microorganism density profile ${\rm Y}\left(\xi \right)$.

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Fig. 8.

(a). Variation of $M,\lambda$ against $Cf$, (b) variation of Rd, Pr against Nu, (c) variation of $A_e,Sc$ against $Sh$, (d) variation of $Pe,Lb$ against $Nh$

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Fig. 9.

(a). Plot for grid analysis for $F\left(\xi \right)$, (b). Plot for grid analysis for $F\text{'}\left(\xi \right)$, (c). Plot for grid analysis for $G\left(\xi \right)$, (d). Plot for grid analysis for $\vartheta \left(\xi \right)$, (e). Plot for grid analysis for $\Phi \left(\xi \right)$, (f). Plot for grid analysis for Υ(ξ).

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7. Future Scope

This study presents a detailed analysis of Reiner-Rivlin nanofluid flow with gyrotactic microorganisms over a rotating disk under multiple slip conditions. Future research could explore unsteady and 3D flow to capture transient and realistic dynamics. Incorporating external electric, magnetic, or thermal effects would enhance the model’s relevance to advanced fluid applications. Investigating hybrid nanofluids and varying nanoparticle shapes may improve heat and mass transfer predictions. Bioconvective stability analysis can deepen understanding of microorganism-induced patterns. Finally, experimental and CFD-based validation would strengthen the model’s practical applicability in biomedical and industrial systems.

8. Conclusions

The explanation connects with the gyrotactic microorganism on Reiner-Rivlin nano liquid and activation energy aspects past a disk in rotatory flow through slip effect and thermal solutal convective condition. The converted ODEs are numerically solved by Bvp4c. The effects of various variables have been deliberated in graphical form. The major point of this analysis is highlighted below.

• When the Reiner Rivlin variables change, the radial flow outside the disk slows down. This reduced flow is then improved by centrifugal force.

• Wall slip parameter rises result in a reduction in radial velocity.

• The radial velocity profile goes up as the convective variable goes up.

• As the Brownian motion parameter and radiation parameter enhance, so does the rate of heat movement accelerate.

• When $Sc$ and $Cr$ values upturn, concentration goes down.

• With an increase in $Pe\text{and}Lb$, the motile density profile exhibits a declining trend.