Machine learning analysis of thermo-bioconvection in a micropolar hybrid nanofluid-filled square cavity with oxytactic microorganisms

Nomenclature

( ρ C p ) hnf

heat capacity of hybrid nanofluids

C p

specific isobaric heat per unit mass ( J kg − 1 K − 1 )

D c

diffusivity of oxygen

D m

diffusivity of microorganisms

k hnf

thermal conductivity of hybrid nanofluids

ρ hnf

density of hybrid nanofluids

Δ C

difference between oxygen concentrations ( C 0 − C min )

Δ ρ

difference between the density of cells and fluid ( ρ cell − ρ hnf )

B ₀

magnetic field strength

C

dimensional oxygen concentration

C ₀

concentration of oxygen at walls of enclosure

C _min

minimum concentration of oxygen essential for microorganisms to be active

g

gravitational acceleration ( m s − 2 )

Ha

Hartmann number

J

micro-inertial density ( m − 2 )

k

thermal conductivity ( Wm − 1 K − 1 )

K ₀

vortex viscosity number

n

density of motile microorganisms

N

density of motile microorganisms in dimensionless form

p, P

dimensional and dimensionless pressure

Pe

Peclet number

Pr

Prandtl number

Ra_b

bioconvection Rayleigh number

Re

Reynolds number

Ri

Richardson number

Sc

Schmidt number

T

temperature of the fluid

u , v

dimensional velocity components in x-y directions

U,V

dimensionless velocity components in x-y directions

κ

vortex viscosity parameter

1 Introduction

The growing global demand for clean energy requires focus on reducing energy wastage and optimizing thermal exchange efficiency. Therefore, the mounting need for thermally effective devices has inspired researchers seeking to enhance the thermal conductivity (TC) of base liquids exhibiting poor heat transfer characteristics, employing different techniques. Choi [1] initiated a pioneering experimental study intending to enhance the TC of base liquids through the incorporation of nanoparticles (NPs) of metal/metal oxides resulting in nanofluids (NFs). The introduction of NFs offers a promising avenue for developing advanced thermal devices. Therefore, substantial work has already been attempted, both experimentally – e.g., studies by Xu et al. [2], Saghir et al. [3], Mansour et al. [4] – and computationally – e.g., studies by Khanafer et al. [5], Uddin et al. [6], and Saleem et al. [7] – highlighting NP’s size, shape, and composition affects NF behavior. Mezaache et al. [8] examined the thermal performance of a wavy channel partially embedded with granular/foam porous material structure and filled with NFs found that granular porous structures enhance thermal performance more than foam. Suresh et al. [9] initiated an experiment incorporating a minor proportion, i.e., 10% of Cu NPs into the Al₂O₃-H₂O NF and found that this composite formulation markedly improves TC relative to the single NP NF, i.e., Al₂O₃-H₂O. This novel approach highlights the capabilities of hybrid nanofluids (HNFs), which integrate various NPs into base liquids to enhance heat transfer efficiency. HNFs are an innovative category of NFs that offer a vast spectrum of possible applications in nearly every sector such as microelectronics, microfluidics, manufacturing, transportation, healthcare, defense, acoustics, marine systems, and propulsion. Yasmin et al. [10] explored how the stability of HNFs relies on factors such as stirring duration and speed, volume, and density of the base fluid, NP characteristics (size, type, concentration), surfactant type and amount. Recently, Madheswaran et al. [11], Olabi et al. [12], Vijayalakshmi and Sivaraj [13], Kim et al. [14], and Alktranee et al. [15] carried out numerous numerical studies on the behavior of Newtonian/non-Newtonian fluids within various cavities infused with different NPs owing to their significance and various technical applications such as melting and recrystallization procedures, fluid crystal growth, solar collectors, heat exchangers, nuclear reactors, polymer industries, and fuel cells. Mixed convection (MC) within lid-driven cavity flow presents a fascinating challenge across diverse thermal engineering applications from innovative lubrication solutions and advanced cooling techniques for optimizing electronic device performance to solar power collectors, food processing methods, glass production techniques, food processing methods, and advanced multi-layered shielding systems for nuclear reactors, etc., as illustrated by Ouertatani et al. [16]. Recently, remarkable studies on lid-driven flow within enclosures of various shapes filled with Newtonian and non-Newtonian fluids, incorporating the effects of obstacles of diverse configurations such as triangular, circular, and square on heat convection were conducted by Alruwaele and Gajjar [17], Hasan et al. [18], Ahlawat et al. [19], Akter et al. [20], and Bakar et al. [21].

Non-Newtonian liquids are often utilized in different engineering processes, including polymers and paints. The micropolar fluid (MF) theory, introduced by Eringen [22,23], extends the classical Navier–Stokes theory by incorporating the microrotation of particles and coupled stresses, offering a more comprehensive model for fluid behavior at the microscale. It is widely used in liquid crystals, polymer suspensions, animal blood, colloidal suspensions, bubbly liquids, and biofluids. Additionally, it is used in clay coating, mud, vaccines, shampoo, body lotions, syrups, and emulsions. Convective heat transfer within cavities filled by micropolar NFs is used in diverse technological implementations, i.e., heat transfer control and chemical fields. Recently, considerable investigations have been conducted on the behaviors of MF enriched with NPs within the cavities. Yan et al. [24] delved into the effects of inclined Lorentz forces and thermal radiation on micropolar NF flow within a porous enclosure. Batool et al. [25] examined the flow and heat transfer of lid-driven micropolar NFs in the presence of a magnetic field. Ali et al. [26] examined the significance of mixed convective flow of micropolar NF/HNF subject to heat source and Lorentz force. They noticed that temperature and velocity profiles achieved higher values by considering an HNF. Manaa et al. [27] computationally investigated micropolar HNF flow and thermal features within a cubic cavity. They noted that mass and heat convection rates are reduced when considering the micropolar NF rather than the NF model. Ahlawat and Sharma [28] carried out numerical analysis to investigate the influence of a porous layer and heated block on heat convection and EG within an enclosure saturated with micropolar HNF. Their findings reveal that as the vortex viscosity parameter enhances, heat transfer irreversibilities become more dominant than frictional irreversibilities. Ahlawat et al. [29] conducted numerical simulations to optimize thermal convection and reduce entropy generation in steady magnetohydrodynamic flow of a micropolar HNF (Ag–Al₂O₃/H₂O) within an annular domain featuring partially insulated and heated walls, revealing that entropy generation decreases as the vortex viscosity parameter increases. Ahlawat et al. [30] performed an investigation within a porous hexagonal enclosure containing a heated obstacle, revealing a 4.135% reduction in the heat transfer rate as the vortex viscosity parameter increased from 2 to 7.5. Riaz et al. [31] studied triple diffusion in micropolar NFs with radiation and viscous dissipation effects.

Bioconvection is a captivating phenomenon in various practical scenarios, occurring when numerous swimming motile microorganisms collectively move in a fluid domain with a slightly higher density than water, all swimming in a specific direction. Bioconvection can improve mass transport and induce mixing, especially in micro-volumes, impacting the stability of NFs crucial in various microsystems. In recent times, transport phenomena related to thermo-bioconvection have shown potential applications in microfluidic devices, medical fields, biological waste treatment, catalytic converters, thermal energy storage, and oil reservoirs. Sheremet et al. [32] computationally studied the bioconvective thermal and mass transfer analysis of NF flow containing gyrotactic microorganisms within a square cavity considering the impact of an inclined magnetic field. Mandal et al. [33] explored MHD mixed bioconvection in a Cu–water NF with oxytactic microorganisms, showing that an undulating curved surface optimizes heat transfer at specific undulation magnitudes and operating conditions. Biswas et al. [34] studied mixed thermo-bioconvection in a Cu-NP magnetized fluid with oxytactic bacteria inside a W-shaped porous cavity, revealing superior mass and heat transfer rates compared to trapezoidal and square enclosures. Biswas et al. [35] investigated bioconvective mass and heat transfer in a Cu-H₂O NF with oxytactic microorganisms inside a magnetized complex wavy-porous cavity. Recently, Rahman et al. [36,37] analyzed entropy generation and heat convection in bioconvective NF flow over a rotating porous disk and stretchable cylinder. The practical interest in the thermo-bioconvective flow and thermal transfer features within the cavity by utilizing micropolar-NF with oxytactic microorganisms has significantly increased in the past few years. It has been discovered in many investigations over the deformable surface by combining micropolar-HNF with oxytactic microorganisms due to the broad range of applications. Hussain et al. [38] investigated the heat transfer features in a three-dimensional porous frame containing a micropolar-based NF with eukaryotic microbes. Yasmin et al. [39] recently explored the flow dynamics of a water-based HNF infused with Cu and Al₂O₃ NPs and gyrotactic microorganisms over a stretching sheet and found that enhanced magnetic effects, thermophoresis, thermal radiation, heat source, and Brownian motion boost the heat transfer rate.

This work is motivated by the mounting need to improve heat exchange as well as microbial activity in enclosed fluid structures that are significant for industrial and medical fields. Therefore, this research combines classical hydrodynamics, bio-convection mechanisms, and computational developments to provide results for designing effective and ecologically sound heat control processes in commercial sectors such as energy, medicine, and ecology. Moreover, machine learning enhances real-time analysis of complex thermophysical behaviors, optimizing fluid flow, heat transfer, and microorganism dynamics efficiently. To the best knowledge of the authors, the bioconvective micropolar HNF lid-driven flow within a square cavity under the influence of a magnetic field employing machine learning has not been previously reported. From a practical standpoint, designing this type of model presents a significant challenge. The governing equations are solved using the finite difference technique. The numerical results obtained are validated to ensure the accuracy of the present study against existing literature. This investigation is conducted for various ranges of relevant parameters, which are discussed graphically. The current study can aid in understanding the design of numerous engineering and industrial systems, such as nano-bioconvective systems, bio-microsystems, and microbial fuel cells.

2 Mathematical formulation

The present study examines the thermo-bioconvection behavior of lid-driven flow inside a square enclosure infused with MHD MF, integrating NPs (titanium dioxide and graphene oxide [GO]) and oxytactic microorganisms, as depicted in Figure 1. It is considered that the enclosure’s walls have equal length and width, denoted as “H” with no-slip boundary conditions (BCs) applied along the walls of enclosure. The upper and lower walls of enclosure are insulated while upper wall moves in the direction of positive x-axis with a uniform velocity ( u 0 ) . The enclosure’s left and right walls are supposed to be thermally heated (T _h) and cooled (T _c), respectively. A steady magnetic field is imposed along the direction of flow. Thermophysical characteristics of TiO 2 , GO, and water are detailed in Table 1.

Figure 1

(a) Physical model of the considered problem and (b) sample of grid generation.

Key assumptions framing the problem are as follows:

1. Flow is considered laminar, 2D, steady, and follows the micropolar continuum theory, accounting for microrotational effects, which indicates that each particle has rotational and translational motion.

2. Bioconvective flow and oxytactic microorganism transport are governed by the continuum models of Hillesdon and Pedley [42] and Pedley and Kessler [43].

3. The HNF is treated as an incompressible, electrically conducting, and homogeneous viscous fluid.

4. Thermophysical properties are considered independent of temperature, aside from density in the body force term, calculated via the Boussinesq approximation [44].

5. Joule heating and viscous dissipation effects are deemed negligible.

6. A state of local thermal equilibrium is assumed between the water, NPs, and oxytactic microorganisms.

3 Numerical solution and code validation

The detailed mechanism of heat transfer inside an enclosure can be explained utilizing a number of advanced numerical methods. This study employs the finite difference discretization technique to discretize the governing Eqs. (14)–(19). The resultant discretized equations are as follows:

Here, h is the grid space in x and y directions and indices (i, j) denote a grid point in the domain. The diffusion terms are determined employing second-order central differencing, whereas the upwind scheme is employed to manage the convective terms in the governing equations. The resulting equations are tackled employing a combination of the SOR, SUR, and Gauss–Seidel iteration methods. Iterations continued until the condition ∑ i , j ∣ φ i , j k + 1 − φ i , j k ∣ ∑ i , j ∣ φ i , j k + 1 ∣ ≤ 10 − 7 was satisfied, where φ stands for any measured value from ψ , ω , N , M , Φ ,  and  θ . The desired results are obtained using self-developed MATLAB codes. The computation was performed over a maximum of 10⁴ iterations, necessitating approximately 5 min of the CPU time, with an error threshold of 10⁻⁷ using MATLAB software. Figure 2 illustrates the steps of the computational program, providing a concise overview of the problem-solving methodology. Furthermore, the numerical technique utilized in this work was validated both qualitatively and quantitatively by comparing it with the specific results of Sheremet and Pop [52], as depicted in Figure 3 and Table 2. The strong correlation between the results further validates the accuracy of our simulation. The grid independency of the MATLAB codes was tested by calculating Nu avg , Sh avg , and Nn avg at the hot wall, as outlined in Table 3. Therefore, to achieve the desired outcomes, a grid of 181 × 181 was employed.

Figure 2

Stepwise representation of the computational program.

Figure 3

Validation of self-developed MATLAB codes with the findings of Sheremet and Pop [52] by comparing contours of oxygen concentrations, microorganism concentrations, streamlines, and isotherms at Ra = 100, Rb = 10, Le = 1, and Pe = 0.1.

4 Artificial neural network (ANN)

A model for processing data that draws inspiration from biological nerve systems, such as the brain, is known as an ANN. The core components of this system for solving specific problems are a large number of neuronal connections that work in tandem with one another. Neuronal networks are mathematical structures designed to perform various tasks. Neural networks are capable of data mining, classification, pattern recognition, prediction, and process modeling, among other things, depending on the configuration in which they are trained. ANNs learn, similar to humans, by seeing and imitating examples. A neural network (ANN) can be set up to do a certain job, like making predictions or sorting data, by learning how to do it. Because of their key role in the learning process, biological systems have to be fine-tuned to synaptic connections. Similar steps are taken by ANNs as well. There are two main categories of ANNs based on their topology: feed-forward networks and feedback networks. One well-known type of feed-forward network is the multilayer perceptron (MLP). Although all of the topologies have their uses, MLP is best suited for engineering applications. When coupled with the backpropagation learning technique, MLP networks can adapt their networks to fit the model.

Typically, an MLP neural network has three layers: input, hidden, and output. Neural connections in the input layer gather data, which is then processed and sent to the hidden layer, which does additional processing before delivering it to the output layer. Each layer is fully linked to every other layer, and weights are assigned to each connection. In order to activate the neurons, activation functions are used in both the hidden and output layers. To handle more complicated jobs, more hidden layers might be added as needed. The jth hidden layer neuron’s input signal X _i is added together after multiplying it by the connection weight. Thereafter, an activation function is used to excite the neurons using the acquired sum. It follows the same pattern from the hidden layer to the output layer when processing data. The expression for the jth neuron’s output is as follows: y i j = A j ∑ k = 1 N j − 1 w k i j y j − 1 + b i j .

There is an activation function f and a weight W _ji that represents the link from the ith neuron in the previous layer in the given equation. Reducing errors between the generated and required output is the main emphasis of the ANN training process. The variable f can be used to represent various types of activation functions, such as exponential, hyperbolic, or logistic functions. Neural network training frequently makes use of the error propagation algorithm. The weights between the layers of the network are adjusted during this supervised learning process. By contrasting the actual results produced by the neural network with the predicted results from the training dataset, one can ascertain the learning error rate. The configuration of the MLP network has a noticeable effect on its performance. In order to find the best network design for minimizing the error rate, iterative testing is usually employed. The procedure comprises running tests with varying values for the parameters, such as the hidden layer’s neuron count, the activation functions used in the output and hidden layers, and so on. The model’s error metric is computed for various parameter values throughout these experiments. One output parameter (Nu_avg/Sh_avg/Nn_avg) and eight input parameters make up the ANN prediction model presented in this work. To obtain he most out of an ANN model, the number of neurons is crucial. The optimal number of neurons to be included in the ANNs is not something that everyone agrees upon. Due of this, various models with variable neuron counts were tested and assessed for predictive performance throughout the design phase of both MLP networks. The study data have discovered the ideal number of neurons that provide the accurate prediction act. Neural networks are composed of basic building blocks known as neurons. An activation function is present in neurons. A vector is used to weight the input signal, and a bias is added on top of that. The activation function receives this signal and takes it to the neuron’s output. Levenberg–Marquardt, often known as damped least-squares, is the learning method used in this study.The hidden layer uses a sigmoid activation function as follows:

5 Results and discussion

The computations were performed for the lid-driven flow of water-based micropolar HNFs containing gyrotactic microorganisms within a porous cavity. The computational results are illustrated through contours of streamlines, isotherms, and iso-concentration contours for microrotations, oxygen, and microorganisms along with local and average values of Sherwood and Nusselt numbers, as well as the density of motile microorganisms, are provided under various relevant parameters. The calculations were executed for relevant parameters, encompassing a wide spectrum of values, as follows: 10 ≤ Re ≤ 50, 0 ≤ Ha ≤ 20, 0 ≤ K ₀ ≤ 7.5, 0.1 ≤ Ri ≤ 10, 0.1 ≤ Ra_b ≤ 10, 0.1 ≤ Pe ≤ 1, 0% ≤   ϕ hnf   ≤ 4%, and 0.3 ≤ Sc ≤ 1. The default values are K ₀ = 2.5, Pr = 6.26, Ha = 10, Ri = 1, Ra_b = 0.1, χ 1 = 0.1; Sc = 1, σ 1   = 1, χ   = 1, Pe = 0.1, ϕ hnf = 4 % , and Re = 25.

Figure 4 displays the contours of streamlines (ψ), isotherms (θ), iso-concentrations of oxygen ( Φ ), and microorganisms (N), as well as the isolines of microrotation (M), highlighting the effects of varying Reynolds number. Physically, at a low Re value, viscous forces dominate, resulting in smooth streamlines, uniform isotherms, and homogeneous distributions of microorganisms. While an increase in Re weakens viscous forces, resulting in vortices, disrupted gradients (θ, Φ), and intensified microrotation (M) owing to shear forces. The heated left wall warms the fluid in contact with it, causing it to rise due to density differences. As this warmer fluid ascends, it is sheared by the moving top wall toward the colder right wall, cooling it, and causing it to descend. This cyclical process generates a continuous clockwise circulation within the cavity, effectively filling the entire space. At low Reynolds numbers, viscous forces exert a dominant influence over buoyancy forces, resulting in streamlines that are relatively sparse and less densely packed. The streamline contours become more pronounced with an increase in the Reynolds number due to enhanced buoyancy forces. As the Reynolds number increases from 10 to 25, the magnitudes of the circulation cells also increase, and circulation cells are pushed toward the bottom of the cavity. An additional increase in the Reynolds number from 25 to 50 leads to notable changes in the development of circulation cells, causing the inner circulation cell to diminish significantly. In analyzing the isothermal contours at Re = 10, it is evident that the isotherm lines are more widely spaced, with the highest temperature concentration found near the heated wall and the lowest temperature located in the central region of the cavity. As the Reynolds number increases from 10 to 25, the temperature contours shift toward the right wall and become more densely packed near to the heated wall. This shift suggests that the increasing wall velocity is causing the thermal boundary layer to thin, resulting in enhanced heat transfer efficiency. However, as the Reynolds number increases from 25 to 50, the thermal distribution in the upper section of the cavity shows marked improvement, resulting in elevated temperature levels in that region. Consequently, as the Re enhances from 10 to 50, the heated wall of the cavity exhibits pronounced temperature gradients, which contribute to a 67.64% increase in the Nu_avg, as shown in Table 4. The distributions of isoconcentrations of oxygen and microorganisms are significantly influenced by the increasing Reynolds number. In both scenarios, the outer circulation cells remain unaffected. Notably, the influence of the inner circulation cells is particularly pronounced in the central region of the cavity and low concentration zone shifts to upper left corner of the cavity. Meanwhile, the data presented in Table 4 reveal a significant increase of 3.34% in Sh_avg and 8.42% in Nn_avg as the Reynolds number increases from 10 to 50. Furthermore, Figure 4 illustrates the influence of Reynolds number on the isoconcentrations of microrotations. The effects observed closely resemble those of the streamlines, although with differing magnitudes. This suggests that as the Reynolds number increases, the angular momentum intensifies due to the enhanced microrotational effects.

Figure 4

Effect of Re on ψ , θ , Φ , N ,  and  M when K ₀ = 2.5, Pr = 6.26, Ha = 10, Ri = 1, Ra_b = 0.1, χ 1 = 0.1, Sc = 1, σ 1 = 1, χ = 1, Pe = 0.1, and ϕ hnf = 4 % .

Figure 5 depicts the influence of Richardson number (Ri) on the contours of ψ, θ, Φ, and N, as well as on isolines of microrotation (M). The Richardson number quantitatively describes the relative importance of buoyancy forces compared to inertial forces in determining the convective heat transfer regime. Physically, at low Ri values, characterized by shear-dominated flows, streamlines are densely grouped, indicating significant velocity gradients, while isotherms and oxygen iso-concentrations are more widely distributed due to enhanced turbulent mixing. However, at higher Ri values, specifically in buoyancy-assisted flow, stratified layers emerge, exhibiting smoother streamlines, sharply delineated isotherms and oxygen contours, along with vertically segregated distributions of microorganisms. This study analyzes the aforementioned contour profiles across forced, mixed, and natural convective regimes. At low Richardson number (Ri = 0.1), characteristic of forced convection, the flow is predominantly driven by the motion of the top wall, causing streamlines to cluster tightly near the top wall. However, as the Richardson number increases to Ri = 10, indicative of natural convection dominance, the streamlines spread more evenly throughout the enclosure, signifying the increasing influence of natural convection across the domain, overpowering the shear forces generated by the top wall’s movement. Therefore, it is evident that the Richardson number enhances the flow velocity within the cavity. The Richardson number exerts a substantial effect on isotherm contours near the heated wall as it regulates the balance between forced and natural convection within the flow domain. However, its influence on the isotherms in the rest of the domain remains minimal. In contrast to forced and mixed convective regimes, the natural convective regime leads to a more compact isotherm lines near the heated wall. This is attributed to a reduction in the thickness of the thermal boundary layer, which in turn enhances the overall thermal distribution. Thus, as Ri boosts from 0.1 to 10, there is a 14.03% bump in Nu_avg. Moreover, while the distribution of oxygen isoconcentrations is minimally affected by Ri, the contours of microorganism isoconcentrations are significantly affected as Ri enhances from 0.1 to 10. The inner circulation cells shrink when the Richardson number increases from Ri = 0.1 to Ri = 1. At higher Richardson numbers, specifically from Ri = 1 to Ri = 10, two inner eddies form within the cavity, indicating low concentration circulation zone affected with increase in Ri values. Consequently, with an increase in Ri from 0.1 to 10, both Sh_avg and Nn_avg decrease by 2.97 and 3.50%, respectively. Additionally, the isolines of microrotation display behavior akin to that of streamlines w.r.t. Ri, despite with differing magnitudes. As Ri enhances from 0.1 to 10, the angular momentum strengthens due to augmented microrotational effects.

Figure 5

Effect of Ri on ψ , θ , Φ , N , and M when K ₀ = 2.5, Pr = 6.26, Ha = 10, Re = 25, Ra_b = 0.1, χ 1 = 0.1, Sc = 1, σ 1   = 1, χ   = 1, Pe = 0.1, and ϕ hnf = 4 % .

Figure 6 illustrates the impact of Ha on ψ, θ, Φ, N, and M when K ₀ = 2.5, Pr = 6.26, Ri = 1, Re = 25, Ra_b = 0.1, χ 1 = 0.1, Sc = 1, σ 1 = 1, χ = 1; Pe = 0.1, and ϕ hnf = 4 % . It is well known that augmenting the magnetic field generates the Lorentz force, which leads to the suppression of flow velocity. The motion of the top wall develops a single circulating cell within the cavity. Moreover, Figure 6 demonstrates that as Ha increases from 0 to 20, the flow intensity diminishes, reflecting a significant reduction in fluid velocity and indicating a notable deceleration in flow movement. A potential explanation for this finding is that the existence of a magnetic field yields Lorentz force, consequently impeding the fluid’s velocity. An increase in Ha from 0 to 10, there is a slight reduction in the intensity of the outer circulating cell. However, at higher Hartmann number, i.e., Ha = 20, the flow intensity experiences a considerable decrease, leading to a noticeable contraction of the outer circulating cell. This indicates that increasing the uniform magnetic field strength decreases fluid flow within the cavity. Although the isotherm lines exhibit minimal variation as Ha increases from 0 to 10, a further increment from 10 to 20 induces a significant change in the isotherm lines, with reduced curvature and diminished density of the isotherms near to the heated wall. This change signifies a substantial decrease in heat convection, with conduction emerging as the leading form of heat transfer. This trend is further supported by the data presented in Table 4, which show that as the Ha enhances from 0 to 20, Nu_avg experiences a decline of 3.25%. Moreover, the isoconcentration profiles of oxygen show no significant changes when applying magnetic strengths from 0 to 10. However, when the magnetic strength is further increased from 10 to 20, the intensity of the primary circulation cells is slightly enhanced, with a small primary vortex forming at the center of the cavity. This suggests that the impact of the magnetic field on mass transport is relatively weak. In the isoconcentration of microorganism profiles, without the influence of the magnetic field, two inner, weaker circulation cells are formed. Increasing the magnetic field causes these weaker circulating cells to merge, indicating an enhancement of the microorganisms. With the application of high magnetic strength, the magnitudes of the circulating cells improve throughout the cavity. Based on these observations, the impact of the magnetic field affects the weaker circulation cells in the isoconcentration of both oxygen and microorganism contours. Furthermore, Figure 6 displays a drop in the magnitude of microrotation isolines as Ha increases. The increasing Ha limits circulation by amplifying the Lorentzian force, which substantially influences flow dynamics and, in turn, affects angular momentum. Consequently, augmenting Ha intensifies the Lorentz force, thereby diminishing the angular momentum distribution within the cavity.

Figure 6

Effect of Ha on ψ , θ , Φ , N , and M when K ₀ = 2.5, Pr = 6.26, Ri = 1, Re = 25, Ra_b = 0.1, χ 1 = 0.1, Sc = 1, σ 1 = 1, χ = 1, Pe = 0.1, and ϕ hnf = 4 % .

Figure 7 illustrates the impact of the bioconvection Rayleigh number (Ra_b) on ψ, θ, Φ , N, and M. Physically, the Rayleigh bioconvection number quantifies the balance between buoyancy-driven flows induced by microbial up-swimming and thermal gradients. Therefore, the enhancement of the bioconvective Rayleigh number implies the boost of the bioconvection, which occurs due to the accumulation of microorganisms on the surface, leading to the production of top-heavy density stratification. Higher Rayleigh number values promote enhanced flow within the cavity. Additionally, the deformation of isotherms remains consistent with the increase in the bioconvection Rayleigh number, with a slight intensification observed near the heated wall due to the power of bioconvection circular flow becomes adequately strong to compensate for the detrimental impact of the contraction that takes place between bioconvection and MC flows. The circulation cells show minimal changes in the isoconcentration profiles of oxygen and microorganisms. The reduced circulation cells suggest that microorganisms can swim freely within the enclosure, particularly at lower values. Moreover, increasing the Rayleigh number from 1 to 10 does not significantly influence this behavior.

Figure 7

Effect of Ra_b on ψ , θ , Φ , N , and M when Ha = 10, Pr = 6.26, Ri = 1, Re = 25, K ₀ = 2.5, χ 1 = 0.1, Sc = 1, σ 1 = 1, χ = 1, Pe = 0.1, and ϕ hnf = 4 % .

Figure 8 depicts the influence of ϕ hnf on ψ, θ, Φ, N, and M. The translation of the top wall along the x-axis induces a clockwise vortex that propagates throughout the cavity, as depicted by the streamline contours, effectively occupying the entire domain. The introduction of NPs can potentially hinder the fluid’s flow by creating resistance. Therefore, streamline contours indicate a decrease in the strength of fluid flow within the cavity as the NP volume fraction increases. In contrast, the isothermal contours remain unchanged with variations in NP concentration and consistently displaying uniform isothermal lines adjacent to the heated wall. Furthermore, raising the percentage of ϕ hnf into the conventional fluid raises the k hnf of the resulting solution, leading to enhanced heat convection. As shown in Table 5, an increase in ϕ hnf results in a significant increase of 6.26% in Nu_avg. Additionally, Table 5 demonstrates that hybrid NPs outperform single NPs in terms of heat convection performance. When comparing the concentration of 4% hybrid NPs to single NPs, it is observed that the use of 4% TiO₂ NPs results in an approximate 1% reduction in heat convection. In contrast, employing 4% GO NPs leads to a decrease in heat convection of about 0.16%. Additionally, isoconcentration profiles for oxygen and microorganisms display consistent contours, showing no changes even as the concentration of NPs in the base liquid increases from 0 to 4%. Consequently, the Sh_avg and Nn_avg show negligible variation in response to changes in nanoparticle concentration as demonstrated in Table 5. The isolines of microrotation form a single circulation vortex akin to streamlines, though with differing strengths and magnitudes. Notably, as the concentration of NPs increases, the inner circulation cell diminishes, whereas the outer circulation cell remains predominantly unchanged. Consequently, ϕ hnf exerts a considerable influence on the isolines of microrotation, resulting in a notable reduction in the intensity of microrotation lines.

Figure 8

Effect of ϕ hnf on ψ , θ , Φ , N , and M when Ha = 10, Pr = 6.26, Ri = 1, Re = 25, K ₀ = 2.5, χ 1 = 0.1, Sc = 1, σ 1 = 1, χ = 1, Pe = 0.1, and Ra_b = 0.1.

Figure 9 shows the consequence of the Peclet number (Pe) on ψ, θ, Φ, N, and M. Physically, the Peclet number is a representation of the ratio between the convective and diffusive transport rates. A significant Peclet number suggests that advection predominates, signifying increased fluid flow and potentially leading to pronounced concentration gradients. Improving the Peclet number causes significant changes in the streamline profiles. As the Peclet number increases from 0.1 to 1, new contour lines emerge, shifting downwards, while the density of the streamlines also intensifies. This trend suggests that elevating the Peclet number markedly improves fluid flow. Further, insignificant changes are obtained in the isotherms and isoconcentration of the oxygen profiles. As the Peclet number (Pe) increases from 0.1 to 1, the isotherm contours exhibit a slight curvature toward the cooled wall, leading to a decrease in density of the lines adjacent to the heated wall. This phenomenon signifies a reduction in heat convection, supported by the data presented in Table 4. Consequently, as the Pe enhances from 0.1 to 1, Nu_avg declines by 7.54%. In the isoconcentration profile of oxygen, the inner circulating cell exhibits a slight reduction in size, while the outer circulating cell remains unchanged across the entire domain. Consequently, a notable reduction of 2.18% in Sh_avg is noted as Pe increases from 0.1 to 1. In contrast, the isoconcentration lines of microorganisms are significantly influenced by the Peclet number. Raising the Pe values from 0.1 to 0.5 leads to the division of the inner weaker cell into two parts. This change results in the formation of densely packed primary and secondary cells within the cavity. As the Peclet number is further elevated from 0.5 to 1, additional inner vortices emerge, migrating toward the upper wall of the enclosure. As a result, an increase of 389.27% in Nn_avg is observed when Pe decreases from 0.1 to 1. Moreover, Figure 9 illustrates that the parameter Pe does not significantly influence the isolines of microrotation.

Figure 9

Effect of Pe on ψ , θ , Φ , N , and M when Ha = 10, Pr = 6.26, Ri = 1, Re = 25, K ₀ = 2.5, χ 1 = 0.1, Sc = 1, σ 1 = 1, χ = 1, Ra_b = 0.1, and ϕ hnf = 4 % .

The influence of the Schmidt number (Sc) on ψ , θ , Φ ,  and  N ,  and  M is depicted in Figure 10. The Schmidt number (Sc) is defined as the ratio of the kinematic viscosity (ν) of a fluid to its mass diffusivity (D), i.e., Sc = ν/D. Physically, Sc quantitatively characterizes the dispersion of a particle within a fluid in relation to the fluid’s momentum flow. The results indicate that the outer circulation cells of the streamlines exhibit minimal variation as the Schmidt number increases, while the inner circulation cells undergo significant changes. As the Schmidt number is increased, the intensity of the inner weaker circulation cells at the upper part of the cavity is enhanced, causing enhanced fluid flow. Moreover, substantial modifications are observed in the isotherms and the isoconcentration’s profiles of oxygen. At low Sc values, the isotherm lines are tightly packed near the heated right wall. However, as the Schmidt number increases to 1, these lines become more dispersed and migrate toward the cooled wall, reflecting a decrease in temperature distribution close to the heated wall. Consequently, as the value of Sc enhanced from 0.3 to 1, there was a significant decrease of 13.32% in Nu_avg. In the oxygen isoconcentration profile, the inner, weaker circulating cell expands, signifying a growing low-concentration zone within the cavity. In contrast, the outer circulating cell remains unchanged across the entire domain. As a result, Sh_avg experiences an increase of 0.57% when Sc increases from 0.3 to 1. Furthermore, notable changes are observed in the isoconcentration profiles of microorganisms. A relatively small Schmidt number (Sc < 1) suggests that the diffusion of microorganism concentration takes place significantly than momentum diffusion, indicating rapid dispersion of microorganisms within the fluid. Therefore, as the Peclet number increased from 0.3 to 0.7, the inner weaker cell expands along the middle section of the cavity, although it experiences a slight contraction. Furthermore, at Sc = 1, the size of the secondary inner circulation cell increases, while it simultaneously shrinks within the cavity. As the Sc increases, signifying an increased momentum diffusion in comparison to mass diffusion, the microrotation isolines become thinner, suggesting finer and larger rotational influences.

Figure 10

Effect of Sc on ψ , θ , Φ , N , and M when Ha = 10, Pr = 6.26, Ri = 1, Re = 25, K ₀ = 2.5, χ 1 = 0.1, Pe = 0.1, σ 1   = 1, χ = 1, Ra_b = 0.1, and ϕ hnf = 4 % .

Figure 11 shows the profiles of Nu_local, Sh_local, and Nn_local for varying Reynolds and Richardson numbers, respectively. An increase in Re from 10 to 50 improves the profiles of Nu_local, Sh_local, and Nn_local. As Re increases, the motion of the upper wall boosts, resulting in a greater volume of fluid being sheared toward the cooler wall of the enclosure; consequently, the flow changes to turbulence, allowing improved mixing of the fluid layers. This mixing disturbs the thermal boundary layer, diminishing its thickness and elevating the Nu_local. Consequently, elevated Reynolds numbers typically result in increased local Nusselt numbers, enhancing convective heat transfer efficacy. Furthermore, Figure 11 illustrates that the Richardson number (Ri) substantially affects the profiles of the Nu_local, Sh_local, and Nn_local. At a relatively small Ri = 0.1, buoyancy effects are diminished relative to inertial forces, resulting in improved convective heat and mass transfer. As Ri approaches 1, inertial and buoyancy effects become similar, resulting in a balanced convective flow regime. At a large Ri = 10, buoyancy effects prevail, diminishing convective transport and resulting in decreased Nu_local, Sh_local, and Nn_local. The density of motile microorganisms tends to accumulate in regions where buoyancy effects are stronger, resulting in a less uniform distribution. Moreover, Figure 12 depicts the profiles of Nu_local, Sh_local, and Nn_local for different Peclet and Schmidt numbers. Increasing the Peclet numbers from 0.1 to 1 decreases Nu_local and Sh_local, while Nn_local increases. Increasing the Schmidt numbers from 0.3 to 1 declines the profile of Nu_local; while profiles of Sh_local and Nn_local increase.

Figure 11

Variation in Nu_local, Sh_local, and Nn_local w.r.t. Re and Ri when K ₀ = 2.5, Pr = 6.26, Ha = 10, Ra_b = 0.1, χ 1 = 0.1, Sc = 1, σ 1 = 1, χ = 1, Pe = 0.1, and ϕ hnf = 4 % .

Figure 12

Variation in Nu_local, Sh_local, and Nn_local w.r.t. Pe and Sc when Ha = 10, Pr = 6.26, Ri = 1, Re = 25, K ₀ = 2.5, χ 1 = 0.1, σ 1 = 1, χ = 1, Ra_b = 0.1, and ϕ hnf = 4 % .