Effect of gravity modulation on weakly nonlinear bio-thermal convection in a porous medium layer

$a$

Total wave number

$D a$

Darcy number

$D m$

Diffusivity of micro-organisms

$e$

Vertically upward unit vector

$G$

Dimensionless orientation parameter

$g$

Gravity vector

$h$

Dimensional layer depth

$i$

Unit vector in the $x$ direction

$j$

Unit vector in the $y$ direction

$K$

Permeability of the porous medium

$k m$

Effective thermal conductivity

$L b$

Bioconvection Lewis number

$l$

Unit vector of movement of micro-organisms

$m ^ ′$

Perturbation unit vector of swimming micro-organisms

$Nu$

Nusselt number

$n$

Micro-organism concentration

$P$

Fluid pressure

$Pe$

Bioconvection Peclet number

$Pr$

Prandtl number

$Ra$

Rayleigh–Darcy number

$Ra B$

Modified bioconvection Rayleigh–Darcy number

$R b$

Bioconvection Rayleigh–Darcy number

$V a$

Modified Vadasz number

$V D$

Darcy velocity

$W c$

Constant micro-organisms velocity

Greek

$α 0$

Measure of the cell eccentricity

$α m$

Coefficient of thermal diffusivity

$β$

Thermal expansion coefficient

$δρ$

Difference between the cell density and the fluid density

$ε p$

Porosity of the porous medium

$μ$

Viscosity of fluid

$μ ~$

Brinkman effective viscosity

$ρ 0$

Density of the nanofluid at the reference temperature $T u$

$( ρ c ) f$

Heat capacity of fluid

$( ρ c ) m$

Effective heat capacity

$σ ~$

Thermal capacity ratio

The investigation of fluid flow through a porous medium is of significant practical significance in fields, such as soil mechanics, groundwater hydrology, oil production, and industrial filtration. In recent times, there has been a growing interest in a new area of research known as bioconvection in porous media. The study of bacterial movement and biofilm growth holds significant relevance in the field of microbiological oil production technologies. Thus, there is a need for theoretical investigations to explore the interactions between bioconvection and natural convection. Understanding the dynamics of this interaction is significant for further advancements in the field.

Prominent monographs authored by Ingham and Pop¹ as well as Nield and Bejan² extensively cover the subject of thermal instability concerning a fluid layer within a porous medium. Additionally, Vadasz³ provides a comprehensive review focusing on the detailed study of fluid flow and heat transfer in rotating porous media. These monographs provide in-depth analyses and discussions of the various aspects and challenges associated with thermal instability in such systems. The study of natural convection in fluid-saturated porous media has been expanded through a series of investigations that take into account additional factors, including rotation,³ magnetic fields,^4,5 anisotropy,⁶ heterogeneity,⁷ gravity modulation,⁸ and other related effects. These studies aim to explore the influence and implications of these factors on the convective flow dynamics within porous media. The classical Darcy’s law was modified by Brinkman through the inclusion of a Laplacian term in the Stokes’ equation, which accounts for the Darcy resistance within porous media. The Darcy–Brinkman equation is commonly used to investigate flow in porous media with high porosity.⁹ In this regard, a number of problems arise related to the study of bioconvection in media with high porosity based on the Darcy–Brinkman model.

The term “bioconvection” refers to the phenomenon where convective patterns are formed as a result of the presence of self-propelled micro-organism species that are denser than the surrounding fluid medium.^10–12 The responsive movement of organisms to various stimuli, such as gravity, light, chemicals, or the presence of food, is referred to as taxis. Taxis can be classified based on the type of stimulus and whether the organism moves toward or away from it. Positive taxis, also known as attraction, occurs when an organism or cell moves in the direction of the stimulus source. On the other hand, negative taxis, or repulsion, describes the movement of an organism or cell away from the stimulus source. The term gravitaxis is used to describe an organism’s directional movement in response to gravitational forces. Magnetotaxis refers to an organism’s ability to detect and respond to a magnetic field by moving. Chemotaxis is the organism’s response to a gradient in chemical concentration, while phototaxis is the movement of an organism in response to light. In this paper, our focus will be on gravitactic micro-organisms. Childress et al.¹³ were the pioneers in developing a comprehensive theory and mathematical model for the bioconvection of gravitactic micro-organisms. Hill et al.¹¹ introduced a theoretical bioconvective model specifically for gravitactic micro-organisms. Pedley and Kessler¹² developed linear stability theory for analyzing the stability of bioconvection involving gyrotactic micro-organisms within a shallow layer of a regular fluid. These studies identified the conditions necessary for the initiation of bioconvective flow.

Numerous publications have investigated the impact of gyrotactic micro-organisms on fluid flows in bounded porous media. Nield, Kuznetsov, and Avramenko^14–18 have made significant contributions to understanding the dynamics of biological processes in porous media. In their work,¹⁴ it was established that if the permeability is below a critical value, the system remains stable and bioconvection does not occur. Conversely, when the permeability exceeds the critical value, bioconvection can develop. They further studied the occurrence of bioconvection in a horizontal layer filled with a saturated porous medium.¹⁵ Critical Rayleigh numbers were determined for different values of the Peclet number, the gyrotaxis number, and cell eccentricity. The influence of vertical flow on the onset of bioconvection in a suspension of gyrotactic micro-organisms in a porous medium was investigated in.¹⁶ A linear analysis was employed to obtain an equation for the critical Rayleigh number, and it was demonstrated that vertical throughflow stabilizes the system. A continuum model of thermobioconvection, focusing on oxytate bacteria in a porous medium, was presented in Ref. 17. This study examined the effect of heating micro-organisms from below on the stability of a horizontally layered fluid saturated with a porous medium. By utilizing the Galerkin method to solve the linear stability problem, a relationship between the critical value of the Rayleigh number and the thermal Rayleigh number was obtained. Avramenko¹⁸ developed a nonlinear theory of bioconvection for gyrotactic micro-organisms in a layer of ordinary liquid based on the Lorenz approach.¹⁹ This work¹⁸ delineated the boundaries of various hydrodynamic regimes observed in two-dimensional bioconvection.

Hwang and Pedley²⁰ conducted a study on the role of uniform shear in the bioconvective instability of a shallow suspension containing swimming gyrotactic cells. They introduced shear by implementing a flat Couette flow, which significantly counteracted the influence of gravity on the cells. The research revealed that bioconvective instability in a dilute suspension arises from three distinct physical processes: gravitational overturning, cell gyrotaxis, and negative cross-diffusion flow. Shear at sufficiently high velocities acts as a stabilizing factor, analogous to Rayleigh–Bénard convection. However, at low shear rates, it destabilizes these perturbations through the overstability mechanism discussed by Hill, Pedley, and Kessler.¹¹ Dmitrenko²¹ provides a comprehensive review of the main aspects of bioconvection in nanofluids and porous media, presenting a mathematical model based on Darcy’s law for porous media. Sharma and Kumar²² investigated the influence of high-frequency vertical vibration on the onset of bioconvection in a dilute solution of gyrotactic micro-organisms through analytical and numerical methods. Their findings demonstrated that high-frequency, low-amplitude vertical vibration and the bioconvection Peclet number have a stabilizing effect on the system. A more detailed analysis of the stability of vibrational systems consisting of shallow layers filled with randomly swimming gyrotactic micro-organisms was conducted by Kushwaha et al.²³ Recently, Garg et al.²⁴ considered the stability of the thermo-bioconvection flow of a Jeffery fluid containing gravitactic micro-organisms in an anisotropic porous medium.

The Darcy–Brinkman model has been extensively utilized in porous media research over the past few decades. Zhao et al.²⁵ extended the application of this model by studying biothermal convection²⁶ within a highly porous medium, considering a suspension of gyrotactic micro-organisms. They conducted a stability analysis to examine the behavior of biothermal convection under the influence of heating from below. Using the Darcy–Brinkman model, Kopp et al.²⁷ studied biothermal instability in a porous medium saturated by a water-based nanofluid containing gyrotactic micro-organisms in the presence of a vertical magnetic field. In Ref. 27, it was established that an increase in the concentration of gyrotactic micro-organisms enhances the onset of magnetic convection. In addition, as shown in Ref. 27, spherical gyrotactic micro-organisms more effectively contribute to the development of biothermal instability. Furthermore, Kopp and Yanovsky²⁸ investigated the impact of the rotation effect, specifically the Coriolis force, on bio-thermal convection in a layer of porous medium saturated with suspension containing gyrotactic micro-organisms.

Of great interest is also a relatively new direction: bioconvection in nanofluids. In several recent studies,^29–33 two-dimensional flows of a nanofluid containing nanoparticles and gyrotactic micro-organisms were considered within the framework of the boundary flow problem. The examination of these flows has been undertaken through the employment of pertinent similarity transformations for essential variables, such as velocity, temperature, nanoparticle volume fraction, and the density of mobile micro-organisms. Through these transformations, the original equations, given the prescribed boundary conditions, undergo a conversion into ordinary differential equations. The resolution of these equations has been accomplished analytically via the utilization of the homotopy analysis method. This approach stands as a crucial tool in addressing the intricacies of the bioconvection phenomena in nanofluids, enabling a more comprehensive understanding of these complex interactions.

In engineering and technical applications, the control of heat and mass transfer is a crucial concern. Various methods exist to manipulate convective processes through external parametric or modulation effects on the system. Investigating the impact of modulation on convection is essential for understanding how external disturbances or parameter variations can influence the flow and transport phenomena within the system. Modulation techniques commonly employed include temperature modulation, gravity modulation, rotation modulation, and magnetic field modulation. In this study, we will employ a convection control method based on the modulation of the gravity field. Prior to presenting our detailed rationale for selecting gravitational modulation, we will provide a concise overview of relevant literature that explores the utilization of gravity modulation in diverse convective systems. The use of gravity modulation to enhance the stability of a heated fluid layer heated from below was initially introduced by Gresho and Sani.³⁴ Subsequently, numerous researchers delved into examining the impact of gravity modulation on the initiation of convection. Malashetty and Begum³⁵ extended these investigations by considering additional physical conditions and non-Newtonian fluids, and they studied the effect of small amplitude gravity modulation on the onset of convection in both fluid layers and fluid-saturated porous layers. Govender³⁶ conducted an analysis of natural convection in porous layers subjected to gravity modulation. The study included a linear stability analysis and a weak non-linear analysis, considering both synchronous and subharmonic solutions. The analysis determined the exact transition point from synchronous to subharmonic solutions. It was found that increasing the excitation frequency leads to rapid stabilization of convection until the transition point is reached. Beyond the transition point, increasing the frequency slowly destabilizes the convection. The weak non-linear analysis demonstrated that increasing the excitation frequency results in a decay of the convection amplitude. An analogy was drawn between an inverted pendulum with an oscillating pivot point and the gravity-modulated porous layer, showing that the wavelength of the convection cell is related to the length of the pendulum. Kiran³⁷ studied the nonlinear thermal instability in a viscoelastic, nanofluid-saturated porous medium under gravitational modulation. In recent years, Kiran conducted a series of studies^38–40 on the impact of g-jitter (gravity modulation) on Rayleigh–Bénard convection (RBC) and Darcy convection. In particular, Kiran et al.⁴¹ focused on the effect of g-jitter on RBC in nanofluids using the Ginzburg–Landau (GL) model. Nonlinear analysis was employed to calculate the thermal and concentration Nusselt numbers, taking into account various physical parameters. Furthermore, Manjula et al.⁴² investigated the combined effects of gravity modulation and rotation on thermal instability in a horizontal layer of a nanofluid.

The motivation for this study is based on the fact that gravitaxis and gyrotaxis play crucial roles in the phenomenon of bioconvection. Earlier, these cellular responses to gravity forces were observed to have a significant influence on the behavior of micro-organisms in fluid-filled porous media.¹⁵ Therefore, we were driven by the desire to gain a deeper understanding of the interplay between gravitaxis, gyrotaxis, and thermal convection under gravity modulation in order to understand the dynamics of the system. The primary objective of this article is to investigate the behavior of weakly nonlinear biothermal convection in a porous medium that is filled with a Newtonian fluid containing gyrotactic organisms. The study focuses on the effects of modulation in the gravitational field, and it is carried out using the Ginzburg–Landau (GL) model.

This study on weakly nonlinear bio-thermal convection in a porous medium with gyrotactic organisms under the influence of gravitational field modulation can be useful in environmental, biotechnological, medical, geoscientific, and materials-related domains.

Obviously, the linear theory provides information about the beginning of convection but does not provide information about the final amplitude of convection. This amplitude occurs when there is an interaction between several perturbation modes. Therefore, it is important to understand the physical mechanism of nonlinear effects and quantify heat and mass transfer in terms of finite amplitudes. At present, the perturbation method developed by Malkus and Veronis⁴³ is widely used to construct a nonlinear theory of convection. They demonstrated that the initial heat transfer through convection has a direct linear relationship with the Rayleigh number. However, as convective instability progresses, the heat transfer deviates slightly from the linear scenario—a phenomenon termed as weakly non-linear behavior. This concept of weakly nonlinear convective instability was further developed by Kiran,^37–41 Manjula et al.,⁴² and Bhadauria et al.⁴⁴ They provided a more detailed description of the weakly nonlinear convection phase using the autonomous Ginzburg–Landau equation, considering finite amplitude conditions.

In the weakly nonlinear theory of convective instability, the interaction between small amplitude convective cells can be described as follows. Assuming the amplitude of these convective cells aligns with the scale of $O( ε 1)$⁠, their mutual interaction yields effects of the second harmonic and nonlinear nature at the magnitude of $O( ε 2)$⁠. Subsequently, this progression leads to nonlinear effects of $O( ε 3)$ and beyond. In this context, the nonlinear elements within Eq. (34) are treated as perturbations when compared to the linear convection problem. The Rayleigh parameter $Ra$ governing the convection is presumed to be in proximity to the critical value $Ra c$⁠. Additionally, it is postulated that the amplitude of the oscillating gravitational field, denoted as $ε 2δ g 0$⁠, falls within the realm of secondary smallness, i.e., $O( ε 2)$⁠. Consequently, its impact on the nonlinear interaction among convective cells is anticipated to emerge at the third order, specifically $O( ε 3)$⁠. Given the minimal impact exerted by unstable modes, our goal is to deduce equations that depict the interplay among these modes. The overarching procedure for formulating a theory of weak nonlinearity is outlined as follows.

The Ginzburg–Landau equation, obtained from perturbation analysis, gives an idea of the behavior of the system at finite amplitudes. By setting the initial amplitude $A(0)=0.3$ and the force of gravity modulation $(δ,Ω)$ in the Mathematica software environment, we can simulate the evolution of the system over time. The graphical representation of the results of our numerical calculations is illustrated in Figs. 3–5. We analyzed the dependencies of heat transfer $Nu$ on the dimensionless time parameter $τ$⁠. By varying the mixed fluid parameters, such as $V a, R b, PeG 0$⁠, and $α 0$⁠, as well as the modulation parameters $(δ,Ω)$⁠, we could investigate their impact on the heat transfer characteristics. We make the following assumptions in this study regarding the Vadasz number $V a$⁠, amplitude modulation $δ$⁠, and the frequency of gravity modulation $Ω$⁠:

1. Assuming that the fluid layer’s viscosity is not high, which allows for the exploration of convection and heat transfer phenomena with reasonable fluid mobility and responsiveness to external influences.

2. The selection of moderate Vadasz number $V a$ values indicates that the fluid has a balanced ratio between its thermal diffusivity and kinematic viscosity in a high-porosity medium.

3. We investigate the system’s behavior under small perturbations, where the influence of gravity field modulation is not overly dominant.

4. The assumption of a low frequency of gravity modulation suggests that lower frequencies maximize the impact of gravity modulation on the system’s behavior.

In Fig. 3(b), the stationary Rayleigh–Darcy number is plotted against the dimensionless wavenumber for different values of the modified bioconvection Rayleigh–Darcy number: $Ra B=(0,15,25)$⁠. The fixed parameters in this plot are $D a=0.5$ and $α 0=0.4$⁠. The red curve represents the dependency of the stationary Rayleigh–Darcy number on the wavenumber in the absence of micro-organism bioconvection (⁠ $Ra B=0$⁠). A similar dependence is observed when there is no gyrotaxis (⁠ $G=0$⁠), represented by $Ra B=0$⁠. From Fig. 3(b), it can be observed that as the $Ra B$ parameter increases, the threshold for the occurrence of bio-thermal convection decreases. This means that the movement of micro-organisms leads to a redistribution of the fluid’s density, which in turn reduces the efficiency of heat transfer in the fluid. As a result, the swimming motion of the micro-organisms can destabilize the stationary bio-thermal convection process.

In Fig. 3(c), the stationary Rayleigh–Darcy number is plotted against the dimensionless wavenumber for different values of cell eccentricity: $α 0=(0,0.4,0.8)$⁠. The fixed parameters in this plot are $D a=0.8$ and $Ra B=15$⁠. From the graph, it can be observed that as the cell eccentricity increases, the threshold for the onset of bio-thermal convection decreases. This means that the spherical shape of micro-organisms has a stabilizing effect on the onset of biothermal convection. In other words, micro-organisms with a more elongated or non-spherical shape tend to enhance the development of bio-thermal convection compared to micro-organisms with a spherical shape. A similar conclusion was made in Refs. 15 and 18. This finding suggests that the morphology or shape of micro-organisms can play a significant role in the behavior and stability of the bio-thermal convection process in the studied system.

Using the expression for the critical Rayleigh number (41), one can numerically determine the minimum values of the wave number $k c$⁠. This value impacts the size of the convective cells, which is influenced by the eccentricity of micro-organism cells $α 0$⁠. According to Fig. 3(d), as parameter $α 0$ increases, i.e., the shape of micro-organisms becomes less spherical, the size of the convective cell $l c∼ ( k c min ) − 1$ also increases.

As can be seen from Fig. 4(a), an increase in the Vadasz number leads to an increase in the heat transfer over a short time interval. Since the Vadasz number is proportional to the Prandtl number, a similar effect was observed in studies by Kiran et al.^40–42 and Bhadauria and Agarwal.⁴⁶ 

The effect of the modified bioconvection Rayleigh–Darcy number on the thermal Nusselt number is shown in Fig. 4(b). The red curve in Fig. 4(b) represents the case of the absence of bioconvection of micro-organisms (⁠ $Ra B=0$⁠); i.e., heat transfer is carried out due to the temperature gradient. As the modified bioconvection Rayleigh–Darcy number increases due to the growth of micro-organism concentrations, the intensity of the convective flow induced by the micro-organisms also intensifies. This stronger flow leads to more efficient mixing and dispersion of heat within the fluid, reducing the temperature gradients and, consequently, decreasing the Nusselt number, which represents the heat transfer rate. Therefore, the increased activity of gyrotactic micro-organisms in the system leads to a decrease in the Nusselt number as more heat is dispersed throughout the fluid.

The graphs in Fig. 4(c) show that the shape of micro-organisms can have a significant effect on the heat transfer in the system, which is reflected in the dependence of the Nusselt number $Nu(τ)$ on time $τ$⁠. Physically, it can be explained by the fact that the shape of micro-organisms affects their movement and interaction with the fluid, thereby affecting the process of convective heat transfer. In the case of spherical micro-organisms [see the red curve on Fig. 4(c)], their symmetric shape allows for a relatively unhindered movement through the fluid. This promotes efficient mixing and heat transfer, resulting in higher convective heat transfer rates and, consequently, higher Nusselt numbers. On the other hand, if the micro-organisms have non-spherical or irregular shapes [see black and blue curves on Fig. 4(c)], their motion through the fluid becomes more complex. The presence of asymmetries and irregularities in their shape can lead to altered flow patterns and hinder fluid mixing. As a result, the convective heat transfer process may be impeded, leading to lower heat transfer rates and lower Nusselt numbers compared to the case of spherical-shaped micro-organisms.

The impact of the modulation frequency $Ω$ is illustrated in Fig. 5(a). Specifically, at lower modulation frequencies, corresponding to the low-frequency case $(Ω=2)$⁠, higher heat transfer is achieved compared to higher vibrational rates $(Ω=5)$ and $(Ω=25)$⁠. Figure 5(b) illustrates the impact of the modulation amplitude $δ$ on heat transfer in the system. The study considers a range of $δ$ values ranging from 0.1 to 0.3, specifically chosen to enhance heat transfer. It is important to note that the frequency of modulation $Ω$ has a diminishing effect on both heat and mass transfer. This observation aligns with the findings reported by Gresho and Sani³⁴ and Kopp et al.⁴⁷ in the context of ordinary (or pure) fluids. These results emphasize the significance of utilizing low-frequency $g$-jitter to optimize the transport process and enhance heat transfer in the system.

Equation (47) presents an analytical expression for the amplitude of convection in the unmodulated case. Figure 5(c) provides a comparison between the modulated and unmodulated systems. The graph clearly shows that in the unmodulated system, there is a sudden increase in the Nusselt number $Nu(τ)$ for low values of the time parameter $τ$⁠, followed by stabilization for higher values of $τ$⁠. In contrast, the modulated system exhibits oscillatory behavior in $Nu(τ)$⁠. This means that the heat transfer undergoes periodic fluctuations as a result of the gravity field modulation.

Through our study, we first developed a weakly nonlinear theory to investigate the effects of gravity modulation on stationary bio-thermal convection in a porous medium layer saturated with the Newtonian fluid containing gyrotactic micro-organisms. Our analysis is based on perturbation theory, specifically focusing on the small supercriticality parameter $ε$⁠, which represents the deviation from the critical Rayleigh number. In our analysis, we consider the small amplitude of the modulated gravity field up to the second order in $ε$⁠. We find that in the first order of $ε$⁠, the parametric modulation does not significantly influence the development of convection, yielding results consistent with linear theory. However, as we move to the third order of $ε$⁠, we obtain a nonlinear Ginzburg–Landau equation with a time-periodic coefficient, indicating the presence of nonlinearity in the system. By performing numerical analysis, we have drawn several conclusions based on the obtained results. These conclusions shed light on the impact of gravity modulation on bio-thermal convection in porous media. We draw the following conclusions based on the obtained results:

1. When the values of the parameters $V a$ are increased, a short-term growth in heat transfer is observed.

2. An increase in the $Ra B$ number leads to a decrease in heat transfer.

3. The spherical shape of the micro-organisms contributes to a more efficient heat transfer process.

4. Increasing the modulation frequency $Ω$ leads to a decrease in the variations of the Nusselt numbers $Nu(τ)$ resulting in suppressed heat transfer.

5. Increasing the modulation amplitude $δ$ enhances heat and mass transfer.

We thank three anonymous reviewers for their valuable suggestions and comments.

The authors have no conflicts to disclose.

M. I. Kopp: Conceptualization (equal); Formal analysis (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). V. V. Yanovsky: Conceptualization (equal); Methodology (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – review & editing (equal).

The data that support the findings of this study are available within the article.