Activation energy and Coriolis force impact on three-dimensional dusty nanofluid flow containing gyrotactic microorganisms: Machine learning and numerical approach

Abbreviations

C & C p

concentration of the fluid and dust particles

C A x Re x 1 / 2 / 2 & C A y Re x 1 / 2 / 2

skin friction coefficients

c m

specific heat dust particles (J/kg K)

c pf

specific heat of fluid particles (J/kg K)

C ∞

ambient nanoparticles concentration

D B

Brownian diffusion coefficient,

D T

thermophoretic diffusion coefficient

E

non-dimensional activation energy

I

mass concentration of dusty particles

j

micro-inertia

K = 6 π μ f a

Stokes drag constant

κ

vortex viscosity

k

thermal conductivity (W m⁻¹ k⁻¹)

k ⁎

mean absorption coefficient

Lb

bio convective Schmidt number

m

mass concentration of dust particles

N

dust particle’s number density

N B

Brownian motion parameter

N T

thermophoresis parameter

Nu Re x − 1 / 2

Nusselt number

Pe

Peclet number

Pr

Prandtl number

Re x

local Reynolds number

Rd

thermal radiation

Sc

Schmidt number

T & T p

temperature of the fluid and dust particles

T w

wall temperature (K)

T ∞

temperature of the ambient fluid

v ₁ , v ₂ & v ₃

velocity component along x, y, & z-direction

x, y & z

Cartesian coordinates (m s⁻¹)

τ T

thermal equilibrium time

Greek symbols

α

rotational parameter

β v

fluid–particle interaction parameter

τ 2

temperature jump parameter

ρ

density (kg m⁻³)

γ

ratio of specific heat

η

similarity variable

μ

dynamic viscosity (kg m⁻¹ s⁻¹)

υ

kinematic viscosity (m² s⁻¹)

θ

dimensionless temperature

β t

fluid–particle interaction parameter for temperature

β_c

fluid–particle interaction parameter for concentration

Subscripts

f

fluid

1 Introduction

Bioconvection is a naturally occurring phenomena that mostly includes the dispersion of self-propelled microorganisms in the field of fluid dynamics. Bioconvection is distinct from conventional multi-phase flows because it is characterized by molecules that are not self-propelled, but rather propelled by the fluid flow, in contrast to typical multi-phase flows. These microorganisms have a tendency to move upwards in the top section of the fluid, where there is a concentration that leads to instability because of the large difference in density. These self-propelled motile organisms have a tendency to convene at the apex of the liquid stratum, causing the top layer to become more denser than the bottom region. This imbalance in density ultimately results in system instability [1,2,3,4]. The nanoparticles in the nanofluid are moved by the Brownian motion and thermophoresis, unlike motile microorganisms, which are capable of self-propulsion. Consequently, the movement of the nanoparticles is not connected to the motion of the bacteria. The motile microorganisms can be classified into various types based on their impellent, such as oxytocic, gyrotactic, and negative gravities microorganisms. Microorganisms are of great importance in the fields of engineering, biology, medicine, and the environment, both in theoretical and applied study. Examples of applications in environmental and biological sciences include the ability to absorb CO₂ more efficiently than agricultural crops, waste treatment plants, heterogeneous catalysis, the production of biodiesel, ethanol, and fertilizers, bacterially optimized petroleum and natural gas retrieval systems, production of food, chemotherapy for cancer, and the utilization of enzyme biomaterials in bio-microsystems [5,6,7,8,9,10,11]. Ahmad et al. [12] examined the unique characteristics of hybrid nanoparticles, such as manganese zinc ferrite and nickel zinc ferrite, in the movement of motile gyrotactic bacteria in a bio-convective flow. This flow occurs in a media known as Darcy Forchheimer. Dey and Chutia [13] examined the flow of a nanofluid including dust particles across a vertically stretched flat surface, considering the presence of a two-phase bio-convective system. Agarwal et al. [4] investigated mass and heat transfer in nanofluid flow inside the Powell-Eyring model across an exponentially stretched sheet containing motile microorganisms. Rajeswari and De [14] investigated the point at which tetra-hybrid nanofluid blood flow reaches a state of standstill on a vertical stretched cylinder using bioconvection. Loganathan et al. [15] investigated the bioconvective gyrotactic microorganisms in third-grade nanofluid flow over a Riga surface, focusing on stratification and entropy formation. Loganathan et al. [16] investigated the dynamics of Ree-Eyring nanofluid under the influences of free convection, bioconvection, heat source, and thermal radiation on a convection-heated Riga plate. Rashed et al. [17] used thermophoresis and Brownian motion to investigate the bioconvective flow around a tiny surgical needle in blood containing ternary hybrid nanoparticles.

The formation of the dusty fluid (DF) occurs as a result of the interaction between the base liquid and the minute particles, which are commonly measured in microns or millimeters. Airborne dust consists of little desiccated particles. When a fluid with lower viscosity has solid particles dispersed in it, the movement of this fluid through the system will lead to the creation of a DF. A prominent illustration of the issue of dusty air is the phenomenon of fluidization, which transpires when minuscule dust particles aggregate, resulting in the formation of precipitation. When referring to the pace at which dust velocity shifts, the word “relaxation time” is the term that is employed. To determine the relaxation time, it is essential to first get the particle scale. Furthermore, Fortov et al. [18] produced evidence that demonstrated that the forces exerted by dust on gas particles and on other dust particles are comparable. To stress the fact that dusty plasma is a sort of ionized gas that contains tiny charged condensed particles, it is essential to accentuate this property. Because of the significance of dust particles that are carried in the air, a significant number of experts have made contributions to the database of information that is now available. In in the context of the present situation and the increasing concerns about needs in the mechanical, industrial, technical, and medical domains, experts in these fields have called for the use of DF. This fluid has many real-world uses, such as in cancer detection and treatment, in assessing its interactions with living things, in studying lunar thermal plant ash, in analyzing metal objects coated with polymers, and in helping to dry solids, especially for the cultivation of beneficial microbes. To mention a few of its various applications, it is used in fuel transportation, air conditioner cooling, crude oil purification, and for the manufacturing of granulated polyethylene power station pipes [19,20,21,22,23,24,25,26,27]. Imitaz et al. [28] evaluated the moving mechanics of blood flow in a cylinder with magnetic dust particles. The effect of a fluctuating pressure gradient allowed this to be achieved. Nawaz and colleagues [29] investigated the transmission of thermal energy that results from the dispersion of both single and blended nanoparticles in a hyperbolic fluid with a tangent that contains dust particles. When it comes to dust particles, the hybrid nanofluid is cooler than that nanofluid. All of this was based on their findings.

The fundamental criterion for mass transmission is characteristically not satisfied in chemical reactions that occur at the species level and have constrained Arrhenius activation energies (AAE). Within the context of this discussion, the term “Activation Energy” (AAE) refers to the minimal quantity that must be present in order to initiate a chemical reaction between atoms or molecules that are present in a chemical system. A Swedish scientist named Svante Arrhenius is credited with coining the term “activation energy” in the year 1889. When the fundamental equation for mass conservation is solved, it is possible to easily determine the impact that mass transfer has on the reaction. The use of chemical processes, including AAE, is commonplace in a variety of domains, including oil reservoir engineering, food production, and geothermal theory [30,31,32]. Ansys CFX software is a commercially available computational fluid dynamics (CFD) tool that is highly successful in tackling difficult chemical kinetics problems. The development of this system was specifically intended to address complex computational issues related to chemical processes [33]. Khan et al. [34] employed electromagnetic hydrodynamics to study the effect of an AAE chemical reaction on the Casson fluid flow at a point of stagnation on an extended surface. Khan et al. [35] did a study to examine how in the case of the Casson nanofluid, the nonlinear radiative mixed convective flow over a stretched surface was influenced by both the chemical process and the AAE. Shafique et al. [36] conducted a research examining transport of Maxwell nanofluid in three dimensions throughout a stretched medium. The researchers considered the presence of Arrhenius activation energy and chemical reaction. The observed activation energy (AAE) rose, resulting in a decline in solute concentration and therefore reducing the mass flow across the wall. The magnitude of these drops becomes more severe as the distance between the wall and the temperature of the surrounding environment decreases. The investigation conducted by Reddy et al. [37] showed that the activation energy has an impact on exothermic chemical processes involving the 3D-MHD slip flow of a Powell-Eyring fluid across a thin surface. Gomathi and Poulomi [38] investigated the dual Casson Williamson nanofluid including radiation and chemical processes on a stretching/shrinking sheet. Mabrouk et al. [39] investigated how motile microorganisms affect the bioconvection of unstable nonhomogeneous hybrid nanofluids.

The primary purpose of this model is to explore the heat transmission properties of a nanofluid that is classified as dusty and contains gyrotactic microorganisms on a surface that is three-dimensional. This inquiry will occur in the context of linear thermal radiation and non-uniform heat sources and sinks. No previous studies have investigated the impact of Coriolis force and activation energy on a three-dimensional porous surface when exposed to a dusty nanofluid (DNF) containing gyrotactic microorganisms. This has been substantiated by previous study. In order to solve this model, the equations governing the transfer of flow temperature and concentration are transformed into an ordinary differential system by implementing the necessary self-similarity transformations. This method goes beyond the conventional numerical and empirical methodologies by using a multi-layer artificial neural network (ANN) with MLP feed-forward backpropagation and the Levenberg-Marquardt algorithm. Using the state-of-the-art bvp4c solution in MATLAB, the complicated system of nonlinear equations controlling skin friction, heat transfer rate, velocity, and temperature is solved.

• What effects do the Coriolis force and activation energy have on a DF temperature, concentration, and velocity profiles?

• How do the characteristics of the fluid-particle interaction affect the rates of mass and heat transfer?

• What is the impact of thermophoresis, Brownian motion, and permeability parameters on the thermal and microbiological properties of nanofluids?

• Can ANNs enhance the efficiency and precision of solving nonlinear differential equations in fluid dynamics relative to traditional numerical methods?

• What is the impact of gyrotactic microorganisms on the stability and heat transfer properties of the DNF system?

2 Formulation of the problem

The study focuses on a stable, three-dimensional, incompressible nanofluid containing dust particles. The nanofluid is subjected to a non-uniform heat source/sink and thermal radiation while being stretched on a surface. A constant velocity speed, denoted by the letter Ω , is maintained during the rotational motion of the fluid all over z -axis (Figure 1). The various characteristics of the Coriolis force are represented by the first and second momentum equations, which are denoted by the letters − 2 Ω v 2 and 2 Ω v 1 , respectively. The Buongiorno nanofluid model incorporates the influences of thermophoresis and Brownian motion. The considerations of activation energy, heat radiation, and chemical reactions are also included. It is believed that the stream of dust consists of spheres that are of the same size and density. T ∞ for the fluid around the extended sheet and T w for the sheet itself are the temperatures, respectively, of this system. In this case, the nonlinear differential equations may be derived by taking into account the problem constraints and the accurate factors for the surface flow [40].

3 Methods of solution

The MATLAB bvp4c solver, along with the shooting technique, is employed to obtain numerical solutions to the coupled ordinary differential equations (ODEs) (16)–(24) subject to boundary conditions (26). The initial step in transforming the set of coupled nonlinear ODEs into a system of first-order differential equations is as follows:

3.1 Dusty phase

(35) f 13 ' = ( ( f 13 ) 2 − β ν ( f 2 − f 13 ) ) / f 12 ,

(36) f 14 ' = ( f 13 f 14 − β ν ( f 4 − f 14 ) ) / f 12 ,

(37) f 15 ' = ( − γ β t ( f 6 − f 15 ) ) / f 12 ,

(38) f 16 ' = ( − γ β c ( f 8 − f 16 ) ) / f 12 .

With boundary conditions

(39) f 2 ( 0 ) = 1 , f 4 ( 0 ) = 0 , f 1 ( 0 ) = 0 , f 6 ( 0 ) = 1 , f 8 ( 0 ) = 1 , f 10 ( 0 ) = 1 , f 2 ( ∞ ) → 0 , f 4 ( ∞ ) → 0 , f 6 ( ∞ ) → 0 , f 8 ( ∞ ) → 0 , f 10 ( ∞ ) → 0 , f 12 ( ∞ ) → 0 , f 14 ( ∞ ) → 0 , f 12 ( ∞ ) → f 1 ( ∞ ) , f 15 ( ∞ ) → 0 , f 16 ( ∞ ) → 0 .

The variable step size η ∞ = 0.001 was used until the results asymptotically reached the tolerance limit of 10 − 8 .

Figure 1

Schematic diagram and coordinate system.

4 Construction of ANN

ANN is a kind of architecture that interprets data in parallel by using numerous tiers of basic supplies that are intimately connected to one another and are referred to as cells. An ANN is a mathematical structure and a technologically sophisticated computer system that functions via the use of neuro-computing in order to efficiently manage data. Not only do ANNs have the ability to create networks of neurons, but they can also duplicate patterns. This model, which derives its cues from the human nervous system, has functional qualities that are indistinguishable from the characteristics of the genuine neuron. It comes from the human nervous system. Each and every one of the billions of sensory cells that are found in our skulls is used by the human brain, which is capable of doing enormous computations of any kind. For the purpose of remembering information, ANNs make use of weighted links, in contrast to genuine neurons, which keep their information in preservation. The calculation load that is connected with the generation of structural characteristics may be sped up and simplified with the help of this technique, which provides for the possibility of acceleration. The evident input and output levels are not the only levels that are present; there is also a hidden layer that contains components that are responsible for the information transformation. With this, the layer that is accountable for producing leads is able to use the output for decision-making, which is one of the mechanisms that constitutes ANNs. Artificial neurons have the astonishing capacity to easily recognize patterns and generate unique qualities that are difficult for both computers and humans to grasp. They are also able to extract important insights from incorrect data, which is a remarkable talent. In a broad sense, this may help to minimize the number of errors that take place [41].

Figure 2 shows the output of the jth hidden neuron.

where W1 _ji indicates the coupling weight between the two layers, x _i is the ith node’s input layer, and a _j is the jth node’s hidden layer.

Figure 2

The back propagation of a neural network is shown schematically.

The result showing the jth hidden node is shown below:

This section considers the layer’s kth output node

The weight W2 _kj of a connection between two nodes in the hidden layer and a node in the output layer indicates the strength of that connection. And you could see the word “bias” (or “b _k ” for short) in the kth output layer node.

Multiple datasets were used to train, test, and verify the ANN algorithm. The model was trained using about 70% of the data, with the remaining 30% split evenly between validation and testing. Figure 3 shows the ANNs’ layered architecture. The scenario HNF model’s ANN training technique ran a regression analysis, and the results are shown in Figure 4.

Figure 3

An ANN model with several layers is shown schematically.

Figure 4

Statistical analysis with data from ( 1 / 2 C f Re x 1 / 2 X ) , ( Sh Re x − 1 / 2 ) , and ( N x Re x − 1 / 2 X − 1 ) .

According to Tables 1–4, the ANN algorithm outperformed the computational approaches in producing good outcomes. Previous research has shown that ANNs can reliably predict the rates of skin friction and heat transfer.