Forchheimer Number Research Articles

Heat and mass flux dynamics of tangent hyperbolic nanofluid flow with unsteady rotatory stretching disk over Darcy-Forchheimer porous medium

Abstract The purpose of the research is to examine a tangent hyperbolic nanofluid flowing in three dimensions (3D) axisymmetrically on an unsteady rotatory stretching disk over a Darcy-Forchheimer porous medium. First order initial value problems (IVPs) are generated from the governing partial differential equations (PDEs) through the use of similarity transformation and linearization. The Runge-Kutta sixth order (RK6) is utilized to solve the IVP system using the shooting technique and the built-in Python program ‘fsolve model10’. Articles that have already been published are used to validate the implemented approach. Graphs are used to examine how various parameters affect velocity, temperature, and concentration. Additionally, the behavior of heat, mass flux, and skin friction in response to different parameters is investigated. The study’s findings showed that as the Forchheimer number and velocity slip parameter increased, the nanofluid’s radial and tangential velocities decreased as well. As temperature and concentration slip parameters increase, correspondingly, thicker and thinner boundary layer structures are seen. The drag force in the tangential and radial direction behaves in the same manner. Both the rates of heat and mass transfers are initiated for an increase Eckert and Prandtl numbers and demotivated for power-law index number. The dissipation effect with radiation and chemical reaction plays a major role in heat and mass fluxes, respectively. The study can be used in various computer storage, coatings, lubricants, and coolants.

A Hybrid Approach of Buongiorno's Law and Darcy–Forchheimer Theory Using Artificial Neural Networks: Modeling Convective Transport in Al2O3‐EO Mono‐Nanofluid Around a Riga Wedge in Porous Medium

ABSTRACTThe inspiration for this study originates from a recognized research gap within the broader collection of studies on nanofluids, with a specific focus on their interactions with different surfaces and boundary conditions (BCs). The primary purpose of this research is to use an artificial neural network to examine the combination of Alumina‐Engine oil‐based nanofluid flow subject to electro‐magnetohydrodynamic effects, within a porous medium, and over a stretching surface with an impermeable structure under convective BCs. The flow model incorporates Thermophoresis and Brownian motion directly from Buongiorno's model. Accounting for the porous medium's effect, the model integrates the Forchheimer number (depicting local inertia) and the porosity factor developed in response to the presence of the porous medium. The conversion of governing equations into non‐linear ordinary differential systems is achieved by implementing transformations. A highly non‐linear ordinary differential system's final system is solved using a numerical scheme (Runge–Kutta fourth‐order). Findings indicate that the porosity factor positively impacts the skin friction and the momentum boundary layer. The influence suggests an increment in the frictional force and a decline in the velocity profile. The volume fraction, Prandtl number, and magnetic number significantly impact the flow profiles. The skin friction data is tabulated with some physical justifications.

Thermal transport analysis for ternary hybrid nanomaterial flow subject to radiation and convective condition

Ternary nanomaterials are now recognized as useful not only for thermal efficiency importance but also to enhance physical characteristics of liquids. Insertion of three nanoparticles in base liquid is characterized as ternary hybrid nanomaterial. Such materials have better magnet properties, electrical conducting and mechanical resistance. Here we discuss three-dimensional magnetized stretched flow of ternary nanofluid through porous space is addressed. Porous space is scrutinized by Darcy-Forchheimer relation. Convective condition is imposed. Nanoparticles here include copper, polyphenol coated and aluminum oxide and water as conventional liquid. Energy equation consists of radiation, magnetohydrodynamics, dissipation and heat generation. Resulting dimensionless system is computed by Newton built in-shooting technique. Outcomes of velocity, temperature, skin friction and Nusselt number for emerging parameters of interest are organized. Comparison of results for ternary nanofluid (CoFe2O3+Cu+Al2O3/H2O), hybrid (CoFe2O3+Cu/H2O) and nanofluid (CoFe2O3/H2O) is examined. It is found that temperature and velocity against Hartman number have reverse trends. Temperature and Nusselt number for radiation have increasing features. Similar characteristics for temperature through heat generation and Eckert number is noticed. Larger thermal Biot number corresponds to rise the Nusselt number and temperature. An increase in temperature through Forchheimer number is witnessed while reverse trend holds for velocity.

Comparative analysis of non‐newtonian fluids over an exponentially stretching sheet with convective boundaries

AbstractThe primary objective of this research is to explore the behavior of several non‐Newtonian fluids, including Casson, Williamson, and Maxwell fluids, over an exponentially stretching sheet considering an inclined magnetic field into account. Non‐Newtonian fluids can be encountered in many industrial and biological applications. They differentiate themselves by their sophisticated viscosity and flow properties. For the purpose of optimizing applications like coating technologies, bioengineering, and polymer processing, it is essential to comprehend their flow dynamics. This work illuminates these fluids' behaviors under convective boundary conditions, which is important for numerous industrial processes. Due to the convective boundary conditions, heat transfer at the sheet is affected by the surrounding fluid. The dominating flow field's nonlinear partial differential equations (PDEs) are synthesized into a system of coupled nonlinear ordinary differential equations (ODEs) using appropriate similarity transformations, and the resulting equations are then numerically solved using the BVP4C solver in MATLAB software. To demonstrate the behaviors of velocity, temperature, and concentration, a parametric research has been attempted. Significant parameter's effect on velocity, temperature, concentration, skin friction coefficient, Nusselt number, and Sherwood number have been investigated, and numerical findings are shown graphically. The numerical data is validated to previously published results on a variety of particular instances and found to be in great agreement. For varying values of the porosity parameter, Darcy‐Forchheimer parameter, and magnetic field, the fluid velocity falls. It is claimed that as the Forchheimer number and the porosity parameter expands, the momentum boundary layer thickness and the velocity profile diminishes. Furthermore, at the maximum levels of Schmidt number, the temperature profile rises synchronously, although the concentration profile reflects the flip pattern.

Comparative numerical study between MHD Forchheimer nano and hybrid nanofluid flows over stretching sheet under aligned magnetic field in the presence of radiation absorption

ABSTRACT This study investigates the mass and heat transfer properties of Ag-H2O nanofluid MHD Forchheimer flows and Ag-MoS2/H2O hybrid nanofluid over two-dimensional linear stretching surfaces in aligned magnetic fields with absorption of radiation. The fluid experiences rotational motion around the vertical axis at a consistent angular speed. The system of nonlinear ODEs is derived from the set of nonlinear PDEs by use of similarity transformations. The BVP-5C shooting approach in MATLAB is then employed to solve the associated system of ODEs. As the Forchheimer number, porosity parameter, rotation parameter and aligned magnetic field exhibit an ascent, the velocity profile along the x-axis and y-axis experiences a decrement. Simultaneously, an augmentation in the magnetic parameter, thermal radiation and rotation parameter induces a heightened temperature profile. Additionally, greater values of the Forchheimer number, rotation parameter and magnetic parameters correspond to an increase in concentration profile. Furthermore, the results indicated an increased temperature profile in the Ag-MoS2/H2O hybrid nanofluid compared to that in the Ag-H2O nanofluid for magnetic, rotation and radiation parameters. Also, the concentration profile of magnetic and rotation parameters along with the Forchheimer number were noted to be higher in the Ag-MoS2/H2O hybrid nanofluid compared to that in the Ag-H2O nanofluid.

Passive control of bio-convective flow on Eyring–Powell nanofluid over a slippery surface with activation energy and magnetic impact

The current communication deliberates the consequences of the Darcy–Forchheimer flow of Eyring–Powell nanofluid past a slippery surface containing activation energy and motile microorganisms. The flow is influenced by the consequences of Brownian motion, thermal radiation, the Cattaneo–Christov heat-mass flux theory, and thermophoresis. The framed flow models are transformed into ordinary derivative equations by adopting appropriate conversion variables. The transformed equations are numerically tackled by using the bvp4c scheme in MATLAB. The study is remarkable for its comprehensive analysis of the interplay of several flow factors, such as the Forchheimer number, Richardson number, bioconvection Rayleigh number, radiation, thermophoresis, Brownian motion, thermal and mass relaxation time parameters. The outcomes are visualized through tables and diagrams, which provide significant insights into the intricate physical mechanisms involved in this multifaceted subject. Evidently, the velocity profile declines when there is a rise in the buoyancy ratio parameter and the opposite trend is obtained for the Richardson number. The temperature grows when there is a larger magnitude of the thermophoresis parameter and it reduces for greater values of the time relaxation parameter. The activation energy and mass relaxation parameters enhance the concentration profile. The microbe density increases when enhancing the quantity of Peclet number and it declines for bioconvection Lewis number.

Significance of Darcy–forchheimer Casson fluid flow past a non-permeable curved stretching sheet with the impacts of heat and mass transfer

Casson fluid flow based on a curved surface has been mathematically modeled using Brownian motion and the Darcy-Forchheimer equation of porosity. Similarity variables are applied to convert flow-anchored partial differential equations into basic ordinary differential equations. The MATLAB solver “bvp4c” is utilized to derive numerical responses for the problem under consideration. During the numerical approach, all parameters have been set to their values based on reviewed literatures and numerical solutions for all flow fields have been obtained against the concerned parameters. Additionally, a variety of graphs are created utilizing numerical extractions to discuss the results. The curvature parameter (K=1.5,2.5,3.5,4.5) results in an improvement in velocity distribution (f'η=1=0.32564,0.360142,0.378247,0.385401). The reason is that for higher values of the curvature parameter, the radius of a curved sheet decreases. Hence, a smaller region of sheet will be in contact, which produces a small amount of resistance towards fluid particles, so that the velocity profiles show an enhancement. Increasing inputs of Forchheimer number imply stronger inertial effects and it introduce an additional resistance to flow that's why fluid velocity decreased. Additionally, for a curved stretched sheet, a higher drag force is required compared to a flat plate because curved sheet has a larger surface area while flat plate has a minimal surface area in contact with the fluid.

Predicting Water Inflow in Tunnel Construction: A Fracture Network Model with Non-Darcy Flow Considerations

Fractures are widely distributed in karst areas, and when flow rates are high, they exhibit complex nonlinear behavior that cannot be accurately described by Darcy’s law. In this work, a hydro-mechanical coupling model based on a discrete fracture network is proposed to predict tunnel water inflow, accounting for the impact of non-Darcy flow. The model’s feasibility has been validated by comparing it with experimental results and the field measurements of flow rates at the Bodaoling Tunnel in Guizhou, China. The results show that Darcy flow tends to overestimate water inflow by approximately 25% compared to non-Darcy flow. The non-Darcy effect grows with the increase in initial fracture width and empirical constant q. When q exceeds 8.77 × 10−6, the growth rate of the Forchheimer number along the fracture width slowed down, and the inhibitory effect of non-Darcy flow on flow became gentle. Additionally, in a complex fracture network, the inflow rate limited by non-Darcy flow at one point drives the water flow through a connect fracture to another point, which increases the difficulty in water inflow prediction. This work highlights the importance of non-Darcy flow and fracture networks when accurately predicting water inflow in tunnels.

Numerical tackling for MHD Darcy-Forchheimer flow of water-based CNTs in a rotating frame with homogeneous-heterogeneous reactions: An artificial neural network approach

Numerous scientific fields rely heavily on rotating flows, such as jet engine design, geophysical flows, turbine engineering, vacuum cleaner design, and many others. Prior research on rotational flow has largely disregarded the impact of viscous dissipation and homogeneous-heterogeneous responses. The present study investigates the Darcy-Forchheimer flow of carbon nanotubes (CNTs) in a rotating frame caused by a biaxial stretched plate with heat consumption and radiation. Furthermore, the mass transport mechanism is treated in terms of homogeneous-heterogeneous chemical reactions. The equations governing the flow are reduced into ordinary differential equations by using appropriate transforms. An advanced computational technique, bvp4c, has been used to provide precise numerical solutions.The computation of the resulting systems is also performed using an artificial neural network (ANN) and also the machine learning study shows a high correlation between the numerical and ANN findings, with a mean squared error (MSE) of 0.000053 for SWCNT and 0.000268 for MWCNT. A comprehensive investigation of several parameters yields both graphical and numerical outcomes. It has been discovered that larger values of the Forchheimer number and porosity parameter result in the decay of fluid velocities. The x − direction skin friction coefficient reduced 50% and the y − direction skin friction coefficient improved 46% when the magnetic field enhanced 200%. The 200% porosity parameter leads to reducing the local Nusselt number by 22% for both CNTs.

Entropy optimized radiative flow conveying hybrid nanomaterials [formula omitted] with melting heat characteristics and Cattaneo-Christov theory: OHAM analysis

Here flow for radiative hybrid-nanomaterial (MgO+MoS2) satisfying Darcy-Forchheimer relation is addressed. Ethylene glycol C2H6O2 is utilized as a base fluid. Magnesium oxide (MgO) and Molybdenum disulfide (MoS2) are considered as distinct nanoparticles. Thermal expression is examined through heat generation and viscous dissipation. Analysis for Cattaneo-Christov theory is carried out. Condition for melting heat is deliberated. Entropy generation is examined. Variable thermal conductivity of hybrid nanomaterial is taken. Suitable transformations are implemented to obtain nonlinear dimensionless systems. Optimal homotopy analysis method (OHAM) is implemented for computations. Temperature, velocity and entropy generation are scrutinized physically. Nusselt number and skin friction are discussed. Conclusion synthesis salient points. Present work is relevant in metallurgical engineering and polymer processing. The findings conclude that Forchheimer number correspond to decline in velocity. Entropy rate and thermal field have similar trend for heat generation variable. A decrease in velocity against melting variable is noted. Temperature increases for variable thermal conductivity and thermal relaxation parameters. Entropy optimization improved for Brinkman number. Skin friction decays for porosity parameter and Forchheimer number. Skin friction improves for melting variable and Forchheimer number. Entropy augments against higher thermal relaxation time. Higher melting variable lead to an enhancement for Nusselt number. There is an improvement for radiation through Nusselt number and temperature.

Non-similar analysis of suction/injection and Cattaneo-Christov model in 3D viscoelastic non-Newtonian fluids flow due to Riga plate: A biological applications

Bioconvective non-Newtonian fluids with motile microorganisms have diverse applications in biology and medicine, offering numerous research opportunities. This article presents a comprehensive investigation into the non-similar analysis of bio-convective micropolar flow induced by a Darcy-Forchheimer exponentially stretchable sheet, considering the effects of suction/injection and thermal radiation in a three-dimensional framework. The fluid under scrutiny is a viscoelastic tangent hyperbolic nanofluid, the impact of Cattaneo-Christov heat subjected to Riga plate is considered for novelty. The role of activation energy and heat source are also computed. The governing equations of viscoelastic tangent hyperbolic nanofluids are transformed into couple of ordinary differential equations via similarity approximations. The resulting ODEs are tackled numerically via bvp4c tool of MATLAB. The impression of appropriate parameters like magnetic parameter, bio-convection parameter, viscoelasticity parameter suction/injection parameter etc. on involved profiles are computed via graphical and tabulated trends. It is observed that velocity layer f′ is dwindled for growing value of Forchheimer number Fr, while reverse relation in velocity curve is noted for developing value of mixed convection parameter δ. Micropolar fluids find biomedical engineering applications in modeling blood flow in microvessels, understanding cerebrospinal fluid dynamics, and optimization of biomedical devices and therapies and simulating drug delivery in tissues.

Analytical and numerical study of water-based silver nanofluid (Ag) across a Riga plate with nonlinear radiation and viscous dissipation: A three-dimensional study

The primary endeavor of this investigation is to determine the Darcy–Forchheimer flow of water-based silver nanofluid (Ag) across a bidirectional Riga plate with viscous dissipation and nonlinear radiation. By manipulating the pertinent variables, the governing equations undergo a transformation into nonlinear ordinary differential equations. The remodeled flow problems are investigated numerically and analytically using the MATLAB (bvp4c built-in function) algorithm and the homotopy analysis method scheme. The repercussions of the appropriate factors on the x− and y− direction velocity profiles, temperature profile, skin friction coefficient of the x− and y− directions, and local Nusselt number are examined via tables and graphs. It is found that both directional fluid velocities decline in the presence of the suction/injection parameter. Increasing the radiation parameter and the Eckert number exalts the fluid temperature. The surface shear stress decays as the suction/injection parameter and the Forchheimer number grow larger. In addition, a greater heat transfer rate appears on the Riga plate compared to the stationary plate.

Impact of Shape Factor on a Couple Stress Flow of a Ternary Hybrid Nanofluid over an Inclined Flat Plate when Non-Uniform Heat Source and Thermal Radiation are Significant: A Darcy-Forchheimer Model

Abstract The study of fluid flow over an inclined flat plate finds applications in a diverse range of engineering fields including aerodynamics, energy production and automotive design. This study theoretically investigates the steady and radiative flow of a ternary hybrid nanofluid (Water + TiO2 + CoFe2O4 + MgO) with couple stress, using the Darcy-Forchheimer model. The flow occurs through a tilted flat plate and is subjected to irregular heat source parameter and entropy generation. The problem’s equations have been transformed into a collection of ordinary differential equations (ODEs), which has been skillfully resolved using the bvp4c solver. Graphs are utilized to elucidate outcomes for two instances of shape components, namely platelet and spherical. An escalation in the couple stress parameter (S) is demonstrated to be inversely related to the fluid velocity, resulting in a drop. Specifically, when 0.5 ≤ S ≤ 3, the friction factor exhibits a decline, with rates of 0.306201851 (for Platelet shape) and 0.304466755 (for Spherical shape). An intriguing observation reveals an augmentation in the generation of entropy as the volumetric fraction of TiO 2 rises. Upon investigation, it has been determined that when the Eckert number (Ecn) increases within the range of 0 ≤ Ecn ≤ 0.3, there is a significant reduction in the Nusselt number. Specifically, the decline is measured to be 0.328685192 for the platelet shape and 0.308939422 for the spherical shape. The utility of the Forchheimer number in regulating the fluid’s motion has been unveiled.

Exploration of Hall current effect and multiple convections in a Darcy Forchheimer flow of hybrid nanofluid over rotating permeable disk: a Stefan blowing approach

This article explores the slip boundary layer electrically conducting hybrid-nanofluid ( GO - Ti O 2 / H 2 O ) flow over a rotating disk. Flow is considered to be steady and incompressible type. Hall current effect and thermal radiation are considered in flow. We carried out the analysis of heat and mass transmissions in the flow through multiple convections. The surfaces of disk subjected to the influence of Stefan blowing. By completing similarity assessment, we renovate the primary PDEs of presumed flow model into ODEs. The resulting equations are solved numerically by recalling Runge–Kutta–Fehlberg fourth–fifth order scheme through shooting system. Discussions on significance of flow features on flow specific are structured precisely via graphs and charts. The acknowledged numerical concerns are novel and distinguishing. For Forchheimer number heat transmission declines in suction and in injection both cases and rate of heat transmission is higher in injection than suction which supposes in cooling exercise, flow through rotating disk suction stands more appropriate. Process of fluid flow over rotating disk and transportation via porous forms has engineering and scientific applications like gas turbines, oil condensers, nano-bio-materials processing.

Mathematical modeling of entropy generation in MHD mixed convective Cu–Ag-Al2O3/H2O tri-hybrid nanofluid over an exponential permeable shrinking surface with radiation and slip impacts: Multiple solutions with stability analysis

The present investigation aims to determine the existence of a dual solution in magnetohydrodynamic (MHD) mixed convective radiative flow of Cu-Ag- A l 2 O 3 / H 2 O tri-hybrid nanofluid in a permeable Darcy–Forchheimer porous medium over an exponentially shrinking surface by considering velocity, thermal slips, and suction effects at the surface. This study focuses on assessing entropy production in novel coolant applications. To streamline the analysis, the complex nonlinear partial differential equations (PDEs) are simplified by converting them into a set of ordinary differential equations (ODEs) through the utilization of the similarity transformation technique. These ODEs are then solved utilizing the bvp4c numerical MATLAB function. Graphical and tabular analyses are conducted to investigate the impact of emerging variables on entropy generation, as well as velocity, temperature, skin friction coefficient, and Nusselt number. Due to the contraction of the surface, dual solutions are found, but dual solutions cannot be found beyond the critical values. The critical values of S C are 1.58928, 1.48700, and 1.66058, but ξ C are − 1.239499 , − 1.416999 , and − 1.130030 for Cu / H 2 O nanofluid, Cu − Ag / H 2 O hybrid nanofluid, and Cu − Ag − A l 2 O 3 / H 2 O tri-hybrid nanofluid, respectively. A positive minimum eigenvalue γ 1 signifies the existence of the upper stable solution branch, while a negative minimal eigenvalue indicates the presence of the lower unstable solution branch. The tri-hybrid nanofluid exhibits superior thermal properties compared to nanofluid and hybrid fluids. Adding nanoparticles to traditional fluids is perceived as improving their ability to transmit heat. The analysis has demonstrated that the thermal radiation and temperature slip parameters significantly influence the heat transfer rate. Thermal radiation is found to speed up the creation of entropy. Additionally, in the case of a stable solution, an increase in the Forchheimer number results in a deceleration of the liquid flow, an effect which is further amplified by the presence of the magnetic field and velocity slip parameter.

Sensitivity analysis and response surface methodology for entropy optimization in the exponentially stretching stratified curved sheet for Casson–Williamson nanofluid flow

The current study explains the parametric entropy optimization by using response surface methodology and sensitivity analysis. It also reveals the effects of thermal radiation and exponential heating on Casson–Williamson nanofluid flow over an exponentially stretching curved sheet. The Darcy–Forchheimer model is used and stratification, Navier slip, and suction-injection are used as the boundary constraints. The 4–5th ordered Runge–Kutta Fehlberg approach is enacted to exhibit the modeled flow. Response surface methodology is used to build the statistical design for the Weissenberg number, magnetic parameter, and Brinkmann number. The thermal stratification parameter, according to the findings, lowers temperature. Entropy grows as Brinkmann number and magnetic parameter increase. When the thermal buoyancy factor assumes the minimum value, little shear stress is seen for high Forchheimer number. The squared co-efficient is confirmed to be 99.99 %. The Pareto chart confirms that point 2.2 is the important point. Entropy surface diagrams in three dimensions demonstrate that with high magnetic parameter, high Brinkman number levels and low Weissenberg number levels, high entropy is obtained. For low and medium levels of magnetic parameter, Weissenberg number exhibits positive sensitivity, whereas for high and medium levels of magnetic parameter, it exhibits negative sensitivity.

Viscous dissipation and Joule heating effects of Carreau nanofluid axisymmetric flow past unsteady radially stretching porous disk

In this piece of communication, a theoretical investigation of two dimensional boundary layer flow of Magnetohydrodynamics (MHD) Carreau nanofluid axisymmetric flow past unsteady radially stretching porous disk is made. The investigation includes the effects of viscous dissipation, Joule heating, thermal radiation, suction/injection, Darcy–Forchheimer porosity model and binary chemical reaction with activation energy, etc. To examine the impacts of the aforementioned effects, the conservation laws are formulated with strongly nonlinear partial differential equations (PDEs), which are then transformed into a system of initial value problems (IVPs) using appropriate similarity transformations and techniques. The 5th order Runge–Kutta with the shooting technique is used to find numerical solutions aided by Python programming with built in program “fsolve”. The method is validated with other published articles under common assumptions and it agrees nicely. The impacts of the pertinent parameters on velocity, temperature, concentration, skin friction coefficient and the rates of mass and heat transfers with in the flow regime are displayed using graphs and tables. The main outcomes include the motivation of local Nusselt and Sherwood numbers for shear thickening (n=1.5) than shear thinning (n=0.5) cases of the Carreau nanofluid. The momentum boundary layer of the Carreau nanofluid gets thinner with an increase in Forchheimer number, unsteady and suction parameters. A dual nature is observed for concentration of the Carreau nanofluid with a rise in Eckert number, i.e increasing in concentration around the wall and decreasing far away from the wall. The rise in destructive and constructive chemical reaction parameters disperse and accumulate the concentration of the system, respectively.

Computation of hydromagnetic tangent hyperbolic non-Newtonian flow from a rotating non-isothermal cone to a non-Darcy porous medium with thermal radiative flux

A theoretical and numerical study is conducted on nonlinear, steady-state thermal convection boundary layer flow of a magnetized incompressible Tangent Hyperbolic non-Newtonian fluid from a rotating cone to a non-Darcy porous medium. Power-law variation in temperature on the cone surface is considered and thermal radiation heat transfer is also present. The Brinkman-Darcy-Forchheimer model is deployed for the porous medium. The study is motivated by rotational (spin) coating with new emerging magnetic rheological polymers, a process which often utilizes filtration media and high temperatures. The transformed non-dimensional conservation equations are solved numerically subject to physically appropriate boundary conditions using a second-order accurate implicit finite-difference Keller Box technique. The numerical code is validated with previous studies. Extensive visualization of axial, tangential velocity components and temperature distributions with variation in key parameters including Rosseland radiative number, Darcy number, Forchheimer number (non-Darcy inertial parameter), magnetic interaction parameter, tangent-hyperbolic non-Newtonian power-law index and Weissenberg (non-Newtonian) number is included. Additionally, axial and tangential (circumferential) skin friction and Nusselt number values are tabulated for variation in key control parameters. With increasing Weissenberg number, axial velocity is depleted near the cone surface, tangential velocity is suppressed throughout the boundary layer regime and temperature is strongly enhanced. Axial flow is strongly decelerated further from the cone surface with increasing tangent-hyperbolic power-law index and there is also a significant depletion in tangential (swirl) velocity. Temperature is however boosted throughout the boundary layer transverse to the cone surface with a rise in tangent-hyperbolic power-law index. Both tangential and axial velocity are suppressed with increment in magnetic interaction parameter whereas temperature and thermal boundary layer thickness are enhanced. With larger Darcy number (i.e. greater permeability), axial velocity is strongly increased near the cone surface with no tangible modification further from the cone; However tangential velocity is consistently elevated throughout the boundary layer with greater Darcy number whereas temperature is depleted. An increase in Forchheimer number substantially damps both axial and tangential velocity whereas it elevates temperature. Increasing radiative flux strongly energizes the magnetic polymer and elevates temperature but suppresses the axial and tangential velocities. With elevation in non-isothermal wall exponent, axial skin friction is suppressed whereas Nusselt number are elevated at the cone vertex. Further along the cone surface a similar response is observed but there is also a reduction in magnitudes of tangential skin friction.

Darcy Forchhiemer imposed exponential heat source-sink and activation energy with the effects of bioconvection over radially stretching disc

The Darcy–Forchheimer model is a commonly used and accurate method for simulating flow in porous media, proving beneficial for fluid separation, heat exchange, subsurface fluid transfer, filtration, and purification. The current study aims to describe heat and mass transfer in ternary nanofluid flow on a radially stretched sheet with activation energy. The velocity equation includes Darcy–Fochheimer porous media effects. The novelty of this study is enhanced by incorporating gyrotactic microorganisms which are versatile and in nanofluid can greatly improve the thermal conductivity and heat transfer properties of the base fluid, resulting in more efficient heat transfer systems. Furthermore, the governing PDEs are reduced to ODEs via appropriate similarity transformations. The influence of numerous parameters is expanded and physically depicted through the graphical illustration. As the Forchheimer number escalates, so do the medium's porosity and drag coefficient, resulting in more resistive forces and, as a result, lowering fluid velocity. It has been discovered that increasing the exponential heat source/sink causes convective flows that are deficient to transport heat away efficiently, resulting in a slower heat transfer rate. The concentration profile accumulates when the activation energy is large, resulting in a drop in the mass transfer rate. It is observed that the density of motile microorganisms increases with a rise in the Peclet number. Further, the results of the major engineering coefficients Skin-friction, Nusselt number, Sherwood number, and Microorganism density number are numerically examined and tabulated. Also, the numerical outcomes were found to be identical to the previous study.

Numerical illustration of diffusive flow of blood-based tri-hybrid nanofluid generated by a curved stretching sheet using law of porosity

The study of flow of blood through the blood vessels carries nanoparticles which opens new ways in the research. This study explores one such flow of trihybrid nanoparticles suspended in blood over a curved stretching surface which undergoes the Darcy-Forchheimer porosity model. The analysis employs Cattaneo-Christov heat flux model along with chemical reaction, linear heat source allowing the surface to slip with heat and mass convection. The trihybrid nanofluids consist of a mixture of titanium dioxide, iron oxide, and silica nanoparticles mixed with base fluid (blood) to generate TiO 2 - Fe 3 O 4 - SiO 2 /blood hybrid nanofluids. Runge-Kutta-Fehlberg 4th–5th order technique, a robust numerical technique, was employed to obtain mathematical solutions. Throughout the numerical process, all parameters, except those under examination, were kept at their default values. Additionally, various plots were generated based on numerical results to illustrate the obtained data. The findings indicate that an increase in the Forchheimer number and porosity results in a reduction in the velocity profile. Furthermore, an increase in the radiation parameter and thermal Biot number leads to an increase in the temperature profile, while the temperature profile decreases with an increase in the thermal relaxation parameter. Additionally, an increase in the chemical reaction parameter and concentration Biot number results in an increase in the concentration profile. Moreover, higher Brinkman and Prandtl numbers enhance the Nusselt number.