Darcy Brinkman Forchheimer Model Research Articles

Novel Numerical Investigations of Some Problems Based on the Darcy–Forchheimer Model and Heat Transfer

In this study, we introduced a new type of basis function and subsequently a Chebyshev delta shaped collocation method (CDSC). We then use this method to numerically investigate both the natural convective flow and heat transfer of nanofluids in a vertical rectangular duct on the basis of a Darcy–Brinkman–Forchheimer model and the electroosmosis-modulated Darcy–Forchheimer flow of Casson nanofluid over stretching sheets with Newtonian heating problems. The approximate solution is represented in terms of Chebyshev delta shaped basis functions. Novel error estimates for interpolating polynomials are derived. Computational experiments were carried out to corroborate the theoretical results and to compare the present method with the existing Chebyshev pseudospectral method. To demonstrate our proposed approach, we also compared the numerical solutions with analytic solutions of the Poisson equation. Computer simulations show that the proposed method is computationally cheap, fast, and spectrally accurate and more importantly the obtained approximate solution can easily be used by researchers in this field.

Approximation of One-Dimensional Darcy–Brinkman–Forchheimer Model by Physics Informed Deep Learning Feedforward Artificial Neural Network and Finite Element Methods: A Comparative Study

Approximation of One-Dimensional Darcy–Brinkman–Forchheimer Model by Physics Informed Deep Learning Feedforward Artificial Neural Network and Finite Element Methods: A Comparative Study

Hydrodynamic analysis of nanofluids flow over 45° inclined porous square cylinder using Darcy–Brinkman–Forchheimer model

AbstractVarious engineering applications commonly involve the flow of nanofluids over a porous 45° inclined square cylinder. Therefore, the current study is to assess the impact of the Darcy parameter (Da), nanoparticle volume fraction (ϕ), and Reynolds number (Re) on the momentum transport characteristics over the 45° inclined porous square cylinder. The governing equations were solved numerically using the Darcy–Brinkman–Forchheimer model for different values of Da ( ), nanoparticle volume fraction ϕ ( ), and Reynolds number Re ( ). The results for each parameter were visualized using streamline plots, velocity profiles within the porous cylinder, and vorticity contours. Complex flow behaviors were observed between Da = 10−3–10−2, flow separation detached and disappeared at the cylinder's downstream side at critical Darcy number. Drag and pressure coefficents are used to represent the global and local parameter of the flow fields. The pressure coefficient on the surface of the cylinder showed an inverse relationship with the nanoparticles volume fraction and Darcy number. The strength of the drag coefficient decreased with the addition of nanoparticles to the base fluid, with a decreasing trend observed for Da = 10−4–10−2 and no change observed for Da = 10−6–10−4. At last, the comparative analysis has been conducted between a porous square cylinder at 0° inclined and another 45° inclined.

A computational investigation of diffusivities and heat transfer in the flow of viscoelastic fluid through Darcy–Brinkman–Forchheimer medium

The main objective of this work is to examine the effects of non-linear thermal radiation and induced magnetic fields on the behavior of viscoelastic fluid as it flows through a porous material that does not follow Darcy’s law. Hence, the Darcy–Brinkman–Forchheimer model has been employed when formulating the momentum equation. The research postulates the presence of a chemical process involving two chemical species, A and B. The research utilizes the spectral relaxation method (SRM) for the numerical solution of the altered governing equations. The present algorithm is formulated and executed on computational software MATLAB. The findings demonstrate a high level of concurrence with previously known numerical outcomes in the context of a specific scenario. The findings of this research demonstrate that chemical species B functions as a catalyst in a horizontally subsurface environment. Consequently, the relationship between the homogeneous-heterogeneous scheme may be seen in the context of isothermal cubic autocatalytic events and first-order processes, respectively. The use of the heterogeneous reaction parameter offers significant benefits in reducing the concentration of the bulk fluid while increasing the concentration of the catalyst located on the subsurface. Moreover, the results indicate that the impacts of induced magnetic and viscoelastic factors on velocity and induced magnetic fields are similar in their outcomes but differ in their fundamental nature.

Analysis of a reactive porous separation effects on depollution and indoor air quality: Application of LBM-MRT to heat and mass transfers

The reduction of energy demand and the indoor air quality associated with energy demand are the main goals of thermal buildings. This work is devoted to the study of the effect of a reactive porous separation on the ventilation (cooling) and depollution capacity in a rectangular room ventilated by air displacement. The model is considered as a cavity heated on its right verticle wall and thermally isolated by the other three walls. A porous separation divides the room into two compartments. The system of equations was solved using the Lattice Boltzmann method with multiple relaxation times. The extended Darcy Brinkman-Forchheimer model was used to simulate the porous material. An additional linear term is added to the standard transport equations (material diffusion) to account for reaction effects, this term was derived from Arrhenius' law. Over a wide range of Richardson and Darcy numbers, the results of the computations show the influence of these parameters on the flow structure, making it possible to categorize the different convection phenomena (natural, forced and mixed). The most important point to note is that the addition of reaction (fixing reaction) improves indoor air quality and can achieve a 55 % reduction in air renewal time, thus saving on energy costs. However, this reactif effect has no influence on the thermal efficiency of the proposed model.

The influence of magnetic field on entropy generation in a wavy cavity equipped with internal heated plate using Darcy–Brinkman–Forchheimer model

In this study, entropy accumulation during thermal convection of a hybrid nanosuspension in an undulating cavity having an interior heated plate is investigated in relation to magnetic fields. The Darcy-Brinkman-Forchheimer approach is employed to simulate the liquid circulation and energy transport parameters, while the impact of the Lorentz force on the system's thermodynamic properties is analyzed. To better understand the characteristics of this complex phenomenon, best compromise combination of different interacted parameters is needed to be set on. The geometrical parameters are number of cavity waves (N), length (s) and inclination angle of heated plate (γ) and the inclination angle of Lorentz force (δ). These parameters are studied with different volume fractions of compound nanofluid (φ = 0 − 0.02), different Rayleigh numbers (103 − 106), Darcy number changes between (Da = 10−5 − 10−1) and various Hartmann numbers (Ha = 0 − 60). The mean Nusselt number rises with a growth of the Rayleigh number by about three times (≈300%), while the average skin friction parameter reduces with a growth of the number of wall waves approximately to the half (≈50%). Additionally, the average Bejan number reaches its maximum value when the parameter of number of waves equals to (N = 5). Taking into consideration an objective function of maximizing average Nu and minimizing skin frication coefficient (Cf), the best scenario is determined at (N = 3, s = 0.4, γ = π/3 and δ = π/2).

A study of Cu/Ag nanoparticle shape-augmented heat transfer in Darcy–Brinkman–Forchheimer flow in a square cavity

ABSTRACT The present work is an analysis of heat transfer flow phenomena due to the natural convection process in a square enclosure filled with a porous medium containing Cu/Ag water-based nanofluid. Multiphysics faceted Darcy–Brinkman–Forchheimer model is used to govern the flow in a porous medium. Different shapes (sphere, blades, bricks, cylinders, and platelets) of the nanoparticles are considered to examine their influence on heat transfer. The model has been solved by the finite element method. The numerical results have been validated against results from the literature. Computed results have been presented graphically by streamlines, isotherms, local and cumulative Nusselt number plots for various physical parameters, Rayleigh number (Ra), inverse Darcy number (), Brinkman effect (Λ), the porosity of the porous medium and solid volume fraction of the nanoparticles. The study pioneers the investigation on the significance of the combined influence of effective viscosity and nanofluid viscosity through the Brinkman effect and reports its importance on nanoparticle shape-based heat transfer augmentation. Cost-effective Cu nanoparticles are shown to augment the heat transfer as achieved by the conventionally preferred Ag nanoparticles. While spherical nanoparticles outperform other shapes in heat transfer augmentation for Brinkman effect , the blade-shaped nanoparticles dominate when Brinkman effect . Higher heat transfer rates are noticed for lesser values of the Brinkman effect, high porosity, the maximum volume fraction of the nanoparticles, high Rayleigh number, and low permeability with spherical-shaped nanoparticles than blade-shaped nanoparticles.

MHD convective flow of Ag–TiO[formula omitted] hybrid nanofluid in an inclined porous annulus with internal heat generation

The current article deals with the computational study of buoyant convection and heat dissipation processes of hybrid nanoliquid saturated in an inclined porous annulus. The fluid flow movement in the porous annular region is modeled using Darcy–Brinkman–Forchheimer model. The vertical boundaries of the cylinder are subjected to uniform but different heating profiles and horizontal surfaces are maintained adiabatic. In the current investigation, for the conservation laws which govern the considered physical process, numerical simulations have been performed using the time-splitting ADI (Alternating Direction Implicit) and line over-relaxation methods. Computations have been performed for broad range of physical and geometric parameters, such as Hartmann number (0≤Ha≤50), geometric inclination angle (−60∘≤γ≤60∘), Darcy number (10−5≤Da≤10−1), aspect ratio (0.5≤Ar≤2) and internal heat generation (0≤Q≤20) to address their impacts on hybrid nanofluid movement and associated heat dissipation rate in the annulus. In addition, heat transfer rate has also been estimated by considering the impact of concentration of each nanoparticle present in the hybrid nanofluid pair. The outcome of numerical computations reveal that an increment in Darcy number enhances the average Nusselt number. Additionally, it has been noticed that the geometric tilt angle of 30° results in dissipating maximum amount of thermal energy in the system. Through this investigation, it is also noticed that shallow annular enclosure exhibits greater amount of heat transport compared to other aspect ratios. Also, significant impact of magnetic field on fluid flow and thermal transport rate has been noticed from the detailed numerical simulations. Further, an enhancement in internal heat generation deteriorates the heat transfer rate and this reduction becomes more steep as the internal heat generation increases.

Experimental and Numerical Analysis of Forced Convection in a Horizontal Tube Partially Filled with a Porous Medium under Local Thermal Equilibrium Conditions

The objective of the present work is to analyze experimentally and numerically the laminar forced convection flow in a horizontal pipe partially filled with a porous medium under constant heat flux and to study the influence of the eccentricity of the porous medium on the results. In a numerical analysis, the governing equations are solved in three dimensions. To simplify the grid generation and the satisfaction of the boundary conditions, conformal mapping is applied to convert the cross-section of the tube in the fluid domain (space between two eccentric circles) into a rectangle, and the equations are solved in a computational domain in this domain. The Darcy–Brinkman–Forchheimer model is applied to simulate the hydrodynamic behavior of the flow in the porous region. Thermal equilibrium between solid and fluid is assumed for the energy equation. A FORTRAN program was developed to solve the equations using the finite volume method and the SIMPLE algorithm. Velocity profile, pressure drop and average Nusselt number are studied in a wide range of Darcy numbers, thickness of porous mediums and eccentricities. The results show that the eccentricity of the porous material reduces the heat transfer coefficient and the pressure drop simultaneously; of course, the reduction in the heat transfer coefficient is less noticeable when the thickness of the porous medium is smaller. For example, at RP = 0.5, when the eccentricity of the porous medium increases up to E = 0.4, the average Nusselt number decreases by 66%, and this reduction for a smaller porous thickness decreases to 11%. The maximum pressure drop reduction for Da = 10−5 and E = 0.4 is 25%.

Convective-radiative magnetized dissipative nanofluid (CNTs-water) transport in porous media, using Darcy–Brinkman–Forchheimer model

The main objective of this investigation is to deliberate the novel analysis of buoyancy-driven nanofluid flow across a vertical stretching surface embedded in a porous medium with the consideration of an inclined magnetic field and heating effects caused by viscosity, thermal radiations, and heat source factor. A material made of glass ball is applied as the porous medium. Water is regarded as a base fluid, while carbon nanotubes are termed as the nanoparticles. The governing equations are formulated by employing fundamental laws. With the application of appropriate non-similar transformations, the emerging flow system is translated into dimensionless differential form. The obtained coupled, non-similar system of nonlinear partial differential equations (PDEs) is tackled by employing local non-similarity technique up to second level of iterations in conjunction with the Lobatto III technique in MATLAB. According to the findings, increasing the Hartmann number diminishes fluid velocity while augmentation in radiation parameter and nanoparticle volume fraction raises the temperature profile. Moreover, nanofluids contain MWCNTs as such nanoparticles exhibit larger estimations of Nusselt number than SWCNTs-water nanofluid. Authors introduced appropriate transformations for considered problem and argued the local non-similarity approach for simulating the dimensionless structure. To the best of authors' observations, no such study is yet published in literature.

Buoyant heat transfer of nanofluids in a vertical porous annulus: a comparative study of different models

PurposeThis study aims to numerically study the buoyant convective flow of two different nanofluids in a porous annular domain. A uniformly heated inner cylinder, cooled outer cylindrical boundary and adiabatic horizontal surfaces are considered because of many industrial applications of this geometry. The analysis also addresses the comparative study of different porous media models governing fluid flow and heat transport.Design/methodology/approachThe finite difference method has been used in the current simulation work to obtain the numerical solution of coupled partial differential equations. In particular, the alternating direction implicit method is used for solving transient equations, and the successive line over relaxation iterative method is used to solve time-independent equation by choosing an optimum value for relaxation parameter. Simpson’s rule is adopted to estimate average Nusselt number involving numerical integration. Various grid sensitivity checks have been performed to assess the sufficiency of grid size to obtain accurate results. In this analysis, a general porous media model has been considered, and a comparative study between three different models has been investigated.FindingsNumerical simulations are performed for different combinations of the control parameters and interesting results are obtained. It has been found that the an increase in Darcy and Rayleigh numbers enhances the thermal transport rate and strengthens the nanofluid movement in porous annulus. Also, higher flow circulation rate and thermal transport has been detected for Darcy model as compared to non-Darcy models. Thermal mixing could be enhanced by considering a non-Darcy model.Research limitations/implicationsThe present results could be effectively used in many practical applications under the limiting conditions of two-dimensionality and axi-symmetry conditions. The only drawback of the current study is it does not include the three-dimensional effects.Practical implicationsThe results could be used as a first-hand information for the design of any thermal systems. This will help the design engineer to have fewer trial-and-run cases for the new design.Originality/valueA pioneering numerical investigation on the buoyant convective flow of two different nanofluids in an annular porous domain has been carried out by using a general Darcy–Brinkman–Forchheimer model to govern fluid flow in porous matrix. The results obtained from current investigation are novel and original, with numerous practical applications of nanofluid saturated porous annular enclosure in the modern industry.

Periodic Flow Through Regenerator Fillers Made from Packed Spherical Microparticles

Periodic flow of gases at cryogenic temperatures through microporous structures occur in the regenerators of Stirling-type cryocoolers. For computational fluid dynamics simulation of such flows using the Darcy–Brinkman–Forchheimer model, permeability and inertial coefficients are needed. Experimental pore-level characterization of such regenerators is difficult due to their microporous structure, and the occurrence of near-zero instantaneous flow rates and extraction of these parameters from experimental data can involve significant uncertainty. In this work, by three-dimensional direct simulation, the hydrodynamic parameters are found for flow through a metallic fill powder common to some recent cryocooler regenerators that operate at very low temperatures. The intrinsic permeability and the Forchheimer coefficients are found for parameter ranges of interest to pulse-tube cryocoolers and are compared with reported extracted experimental values. The permeability and Forchheimer coefficients are correlated to be suitable for application in computational fluid dynamics codes.

Simulating Nanofluid Forced Convection Flow by Thermal Lattice Boltzmann Approach

Improving heat transfer using nanofluids has proven to be a promising option with many practical applications. However, the behavior of particles conveying energy for thermal transport depends closely on the dimensions of systems and channels where the flow evolves. Thereby, any fine thermal analysis should lean on a mesoscale approach applied at a microscale level. To this end, the multi-distribution functions–thermal lattice Boltzmann method has been taken to deal with convective heat flow and entropy generation in a channel with isothermal top–bottom walls and filled with a nanofluid (Cu/water). It was extended to simulate the flow governed by the Brinkman–Forchheimer Darcy model using the local thermal equilibrium assumption. The effects of nanoparticles’ volume fraction, Darcy number, porosity, heat capacity ratio and thermal conductivity ratio on heat transfer, entropy generation, average Nusselt number, and Bejan number are investigated. Among the salient results, it can be stated that the nanoparticles’ volume fraction increases heat transfer and entropy generation, but such a propensity can be affected by the porous medium permeability used. To sum up, the findings confirm the potential of the multi-distribution functions–lattice Boltzmann formalism to tackle forced nanofluid flows with heat transfer in porous media.

Numerical study of natural convection in an annulus between two concentric cylinders provided with metal foam fins

Natural convection in an annular space provided with metal foam fins attached to the inner cylinder is studied numerically. The metal foam fins made of copper were inserted in different axial sections with three fins in each section. The temperature of the inner cylinder is kept constant while the annular outer surface is adiabatic. The thickness effect of the inner pipe wall was considered. Naiver Stokes equation with Boussinesq approximation is used for the fluid regime while Brinkman-Forchheimer Darcy model is used for metal foam. In addition, the local thermal non-equilibrium condition in the energy equation of the porous media is presumed. The effect of Rayleigh number and number of foam fins in the axial direction, on fluid flow and heat transfer characteristics, were examined. The current model was valid with the available published results and good agreement is noticed. Results showed that as the Rayleigh number increases the dominated of convection mode increases and average Nusselt increases. It was found that at Rayleigh of 106 Nusselt reached its higher value which is 4.6 for the case of adding seven axial metal foams. A comparison between adding foam fins and copper fins was established for a range of Rayleigh numbers between 104 and 106. It showed a good enhancement in Nusselt number and the greatest enhancement percentage was 45.9% at Rayleigh equal 106 for the case of using seven sections of foam fins.

MHD Flow of a Hybrid Nano-Fluid in a Triangular Enclosure with Zigzags and an Elliptic Obstacle.

The current study uses the multi-physics COMSOL software and the Darcy–Brinkman–Forchheimer model with a porosity of ε = 0.4 to conduct a numerical study on heat transfer by Cu-TiO2/EG hybrid nano-fluid inside a porous annulus between a zigzagged triangle and different cylinders and under the influence of an inclined magnetic field. The effect of numerous factors is detailed, including Rayleigh number (103 ≤ Ra ≤ 106), Hartmann number (0 ≤ Ha ≤ 100), volume percent of the nano-fluid (0.02 ≤ ϕ ≤ 0.08), and the rotating speed of the cylinder (−4000 ≤ w ≤ 4000). Except for the Hartmann number, which decelerates the flow rate, each of these parameters has a positive impact on the thermal transmission rate.

Analysis of Thermal Performances in a Ventilated Room Using LBM-MRT: Effect of a Porous Separation

This article demonstrates the feasibility of porous separation on the performance of displacement ventilation in a rectangular enclosure. A jet of fresh air enters the cavity through an opening at the bottom of the left wall and exits through an opening at the top of the right wall. The porous separation is placed in the center of the cavity and its height varies between 0.2 and 0.8 with three values of thickness, 0.1, 0.2, and 0.3. The heat transfer rate was calculated for different intervals of Darcy (10−6 ≤ Da ≤ 10), Rayleigh (10 ≤ Ra ≤ 106), and Reynolds (50 ≤ Re ≤ 500) numbers. The momentum and the energy equations were solved by the lattice Boltzmann method with multiple relaxation times (LB-MRT). Schemes D2Q9 and D2Q5 were chosen for the velocity and temperature fields, respectively. For porous separation, the generalized Darcy–Brinkman–Forchheimer model was adopted. It is represented by a term added in the standard LB equations. For the dynamic domain, numerical simulations revealed complex flow structures depending on all control parameters. The results showed that the thermal field, mainly in the second compartment, is very dependent on the size and permeability of the porous separation. However, they have no influence on the transfer rate.

Natural Convection in Annulus Between Two Concentric Cylinders Partially Filled with Metal Foam Distributed with New Suggested Design

The investigation of natural convection in an annular space between two concentric cylinders partially filled with metal foam is introduced numerically. The metal foam is inserted with a new suggested design that includes the distribution of metal foam in the annular space, not only in the redial direction, but also with the angular direction. Temperatures of inner and outer cylinders are maintained at constant value in which inner cylinder temperature is higher than the outer one. Naiver Stokes equation with Boussinesq approximation is used for fluid regime while Brinkman-Forchheimer Darcy model used for metal foam. In addition, the local thermal equilibrium condition in the energy equation of the porous media is presumed to be applicable for the present investigation. CFD ANSYS FLUENT software package (version 18.2) is used as a solver to this problem. Various parameters are examined; Rayleigh number, Darcy number, and thermal conductivity ratio to study the effect of them on fluid flow and heat transfer inside the annuli space in the suggested design of metal foam layer. current model is compared with the available published results and good agreement is noticed. Results showed that as Rayleigh number increases the dominated of convection mode increases and Nusselt increases. Also, Nusselt is larger at the higher Darcy and thermal conductivity ratio. It was found that at Rayleigh of 106 and thermal conductivity ratio of 104 Nusselt reach its higher value which is 6.69 for Darcy of 0.1 and 6.77 for Darcy of 0.001. A comparison between this design and the traditional design was established for Darcy 0.001 and thermal conductivity ratio 102, and its showed a good enhancement in Nusselt number and the greatest enhancement percentage was 44% at Rayleigh equal 5*104 while the lowest percentage is 6% for Rayleigh equal106.

Local thermal non-equilibrium (LTNE) effects on thermal-free convection in a nanofluid-saturated horizontal elliptical non-Darcian porous annulus

In this investigation, a comprehensive and accurate numerical analysis of the local thermal non-equilibrium effects on the natural convection characteristics in a horizontal elliptical porous annulus saturated with nanofluid has been carried out using finite volume technique. The internal surface is isothermally heated while the external one is fixed at uniform lower temperature. The nanofluid phase and the solid phase of the porous structure are out of local thermal equilibrium situation. The heat and the hydrodynamics equations in their dimensionless form under Darcy–Brinkman Forchheimer model were solved computationally by the finite volume technique using the standard SIMPLER algorithm. The results have been examined for various porous medium properties under the LTNE condition. The standard 0.05 significance level was used to identify the local thermal equilibrium within the system. According to the obtained results, the higher the Darcy number and the smaller the non-dimensional heat transfer coefficient for the solid/nanofluid interface, the modified thermal conductivity ratio and the media’s porosity, the higher are the LTNE effects. In addition, the spatial distribution contours of the LTNE sources within the porous annulus were schematized.

Spectral computation of reactive bi-directional hydromagnetic non-Newtonian convection flow from a stretching upper parabolic surface in non-Darcyporous medium

This paper presents a mathematical model for bi-directional convection magnetohydrodynamic (MHD) tangent hyperbolic nanofluid flow from the upper horizontal subsurface of a stretching parabolic surface to a non-Darcian porous medium, as a simulation of nanocoating. Chemical reaction, activation energy and thermo solutal buoyancy effects are included. The Darcy–Brinkman–Forchheimer model is deployed which permits the analysis of inertial (second order) porous drag effects. The Buongiorno nanoscale model is deployed which includes Brownian motion and thermophoresis effects. The dimensionless, transformed, nonlinear, coupled ordinary differential equations are solved by implementing the spectral relaxation method (SRM). Validation with previous studies is included. The numerical influence of key parameters on transport characteristics is evaluated and visualized graphically. Velocity is elevated (and momentum boundary layer thickness is reduced) with increasing wall thickness parameter, permeability parameter, Forchheimer parameter, Weissenberg (rheological) parameter and modified Hartmann (magnetic body force) number. Velocity enhancement is also computed with increment in stretching rate parameter, rheological power-law index, thermal Grashof number, and species (solutal) Grashof number, and momentum boundary layer thickness diminishes. Temperature is suppressed with increasing stretching rate index and Prandtl number whereas it is substantially elevated with increasing Brownian motion and thermophoresis parameters. Velocity and temperature profiles are reduced adjacent to the parabolic surface with larger wall thickness parameter for stretching rate index [Formula: see text]1, whereas the reverse behavior is observed for stretching rate index [Formula: see text]1. Nanoparticle concentration magnitude is depleted with larger numeric of Lewis number and the Brownian motion parameter, whereas it is enhanced with greater values of the stretching index and thermophoresis parameter. The nanoparticle concentration magnitude is reduced with an increase in chemical reaction rate parameter whereas it is boosted with activation energy parameter. Skin friction, Nusselt number and Sherwood number are also computed. The study is relevant to electromagnetic nanomaterials coating processes with complex chemical reactions.

Darcy–Brinkman–Forchheimer Model for Nano-Bioconvection Stratified MHD Flow through an Elastic Surface: A Successive Relaxation Approach

The present study deals with the Darcy–Brinkman–Forchheimer model for bioconvection-stratified nanofluid flow through a porous elastic surface. The mathematical modeling for MHD nanofluid flow with motile gyrotactic microorganisms is formulated under the influence of an inclined magnetic field, Brownian motion, thermophoresis, viscous dissipation, Joule heating, and stratifi-cation. In addition, the momentum equation is formulated using the Darcy–Brinkman–Forchheimer model. Using similarity transforms, governing partial differential equations are reconstructed into ordinary differential equations. The spectral relaxation method (SRM) is used to solve the nonlinear coupled differential equations. The SRM is a straightforward technique to develop, because it is based on decoupling the system of equations and then integrating the coupled system using the Chebyshev pseudo-spectral method to obtain the required results. The numerical interpretation of SRM is admirable because it establishes a system of equations that sequentially solve by providing the results of the first equation into the next equation. The numerical results of temperature, velocity, concentration, and motile microorganism density profiles are presented with graphical curves and tables for all the governing parametric quantities. A numerical comparison of the SRM with the previously investigated work is also shown in tables, which demonstrate excellent agreement.