Buoyancy Ratio Research Articles

Influence of location of twin-adiabatic blocks on magnetohydrodynamics-based double-diffusive convection and entropy generation for a liquid metal

This research deals with a rectangular cavity encompassing two adiabatic rectangular and impermeable obstacles at various positions. This study serves as a platform to explore the interplay between diverse flow-governing parameters, such as the buoyancy ratio (N = −1, 0, and +1), Hartmann number (Ha = 0, 50, and 100), Lewis number (Le = 1, 5, and 10), Rayleigh number (Ra = 103 and 104), and geometric arrangements of twin-blocks (C1, C2, and C3) to help in developing insights into such complex transport phenomenon driven under the influence of buoyancy and concentration. The arrangements are chosen such that C1 and C3 represent the off-center position of the first obstacle, while C2 represents the in-line position with the second obstacle. The influence of liquid sodium–potassium alloy (Pr = 0.054) on fluid flow, heat, and mass transfer, and entropy generation characteristics due to double-diffusive natural convection in the twin obstacle-filled rectangular enclosure are observed using the lattice Boltzmann method. The results reveal that the maximum amount of heat and mass transfer occurs at the C2 position, making it the most efficient for heat and mass transfer among all. In contrast, the C2 configuration is a thermodynamically inefficient arrangement as entropy generation is maximum, while the C3 configuration is obtained to be more efficient thermodynamically. Furthermore, the results reveal that the average total entropy generation is directly related to the Lewis number, while it has an inverse relation with the Hartmann number.

Couple-stress nanofluid flow comprising gyrotactic microbes subject to convective boundary conditions: Numerical solution

Couple-stress nanofluids have multiple potential applications in numerous industrial and engineering sectors, such as energy production, medical diagnostics, thermal control systems, and the aerospace industry. Couple-stress nanofluids have the ability to improve the heat exchange properties and elevate the performance of nuclear power plants, solar panels, and other renewable energy sources. Therefore, in the current analysis, a non-homogeneous nanofluid model is considered to examine the non-Newtonian Casson nanofluid flow across a prolonging sheet. The flow has been studied under the significance of generalized Fourier’s and Fick’s laws, convective boundary conditions, and the heat source/sink. The modeled equations are simplified into a dimensionless lowest-order system of ordinary differential equations by using similarity transformation. The numerical outcomes are achieved by using the “ND-Solve” approach. It has been noticed that the energy field decreases because of the Prandtl number’s impacts, whereas it increases with the increase in the heat radiation parameter. The couple-stress nanoliquid’s velocity decreases vs increasing values of the magnetic field and mixed convection parameter. The influence of thermal relaxation and couple-stress parameters falls off the energy field. Furthermore, the intensifying effect of Rayleigh number and buoyancy ratio increases the fluid temperature.

MHD double diffusive mixed convection and heat generation / absorption in a lid-driven inclined wavy enclosure filled with a ferrofluid

The magneto-hydrodynamics (MHD) double-diffusive mixed convection and heat generation/absorption in a lid-driven inclined wavy enclosure filled with (Fe3O4/ water) ferrofluid is quantitatively investigated in this paper. The present study focuses on improving the efficiency of mass and thermal performance of a system. Both the left and right sidewalls of the cavity are allowed to move with a constant velocity in the upward and downward directions, respectively. The finite difference approach was applied to discretize the subsequent governing equations followed by the Bi-Conjugate Gradient Stabilized (Bi-CGStab) method to solve them. The numerical simulation was performed for a variety of parameters, including the Hartmann number varied as 0 ≤ Ha ≤ 45, the inclination angle of the enclosure varied as 0° ≤ δ ≤ 180°, the buoyancy ratio varied as -2 ≤ N ≤ 2, heat generation or absorption parameter varied as −10 ≤ Qo ≤ 10, Richardson number varied as 0.01 ≤ Ri ≤ 10, and solid volume fraction varied as 0 ≤ ϕ ≤ 0.06. The numerical simulation results were presented in terms of streamlines, isotherms, isoconcentrations, average Nusselt number, and average Sherwood number. The heat and mass transfer rates were found to decrease with the increase in Ha but increase with N, Ri, and Φ. Also, both of them reach their peak values at Ri = 10. In addition, the heat generation parameter enhances both thermal and mass performance as they reach their maximum values at Qo = 10. Increasing the heat generation factor from Qo = 5 to Qo = 10 increases the Nusselt number by 3.5 times. The outcomes of the study have significant importance for modern industrial applications specifically in the discipline of electronic device cooling.

Asymptotic analysis of electrohydrodynamic flow through a swarm of porous cylindrical particles

The present article reveals the study of an electrohydrodynamic flow through a membrane composed of a swarm of porous layered cylindrical particles adopting a heat transfer approach. The configuration of the proposed theoretical model is segregated into two regions in which the region proximate to the solid core of the cylindrical particle is a porous region. However, a region surrounded by a porous region is a non-porous (clear fluid) region. The thermal equations are employed under steady-state conditions to establish the temperature distribution when heat conduction prevails over heat convection. The Brinkman and Stokes equations regulate fluid flow through a swarm of porous layered cylindrical particles in porous and non-porous regions, respectively. With the purpose of addressing an electric field in the fluid flow process through a swarm of porous layered cylindrical particles to understand the role of a Hartmann electric number, the momentum equation and the charge density are coupled and nonlinear. The nonlinear second-order differential equation governs the momentum equation and regulates fluid flow through a swarm of porous cylindrical particles. The solutions of the energy equations for both regions are analytically obtained. The asymptotic expansions of velocities for porous and non-porous regions have been derived using the perturbation technique for the small and large values of the nonlinearity parameter α. The effects of various parameters like Hartmann electric number, Grashof number, radiation parameter, viscosity ratio parameter, and porosity of the porous material on the hydrodynamical permeability, Kozeny constant of the membrane, and temperature are analyzed graphically. A noteworthy observation is that a rising Hartmann electric number, the ratio of electric force to the viscous force, enhances the velocity, which is relatively more significant for higher permeability and hence enhances the membrane permeability; however, decay in Kozeny constant is reported with a rising Hartmann electric number. Significant velocity and membrane permeability growth are described with a rising Grashof number, a ratio of thermal buoyancy and viscous forces. The observations from the present study hold promise for advancing our understanding of critical physical and biological applications, including wastewater treatment filtration processes, petroleum reservoir rocks, and blood flow through smooth muscle cells.

Numerical comparison of nonlinear thermal radiation and chemically reactive bio-convection flow of Casson-Carreau nano-liquid with gyro-tactic microorganisms: Lie group theoretic approach

The current research presents a mathematical model to study the flow of a non-Newtonian magnetohydrodynamics (MHD) Casson-Carreau nanofluid (CCNF) over a stretching porous surface, considering mass and heat transport rates with Stefan blowing, non-linear thermal radiation, heat source-sink, chemical reaction, thermophoretic and Brownian motions, convective heating, Joule heating, motile microorganisms, and bio-convection. The presence of microorganisms is utilized to control the suspension of nanomaterials within the nanofluid. The current flow model has been rendered by the boundary layer approximation and we get the highly nonlinear partial differential equations (PDEs). These nonlinear PDEs are simplified by the novel Lie group theoretic method. The one-parameter Lie scaling method simplified the PDEs and convert it into the ordinary differential equations (ODEs). Numerical solutions for these ODEs are obtained using the bvp4c scheme built-in function in MATLAB, ensuring reliable outcomes for temperature, velocity, concentration, and motile microorganism density profiles. The numerical results are presented through graphs and compared with available data, showing good agreement. These numerical outcomes reveal several important flow characteristics. Rates of change for Nr are 0.0007 and 0.0005, and for Ω, they are −0.0754 and −0.0536, respectively. Similarly, the rate of change for Rb in both models is −0.002 and −0.0002. Analysis shows a positive impact of the bioconvection Rayleigh number in both models, notably higher for the Casson fluid compared to the Carreau fluid model. Buoyancy ratio parameter exhibits consistent rates of change, while the reduction in impact is more pronounced for the Casson fluid model in the case of the mixed bioconvection parameter. The mixed bio-convection parameter reduces momentum velocity for both Casson and Carreau fluids, whereas the Darcy parameter boosts fluid velocity. As the Newtonian heating parameter increases, the temperature velocity distribution of both fluids also increases. The concentration profile of both Casson-Carreau fluid phases declines as the heat source-sink parameter and Schmidt number increase. Microbial velocity shows a decrease with increasing R values, whereas the opposite trend is observed for the Peclet number Pe.

Effect of inclined Lorentzian force on radiated nanoflow Williamson model under asymmetric energy source/sink: Keller box method

An asymmetric energy source/sink can be designed to efficiently convert ambient energy into usable forms; this could have applications in micro-/nanoscale power generation, i.e., energy harvesting. The asymmetric energy source/sink and inclined Lorentzian force could be used to control the flow of fluids within these devices. This study numerically investigates the model of a Williamson nanofluid influenced by an angled magnetic force and an asymmetric energy input/output on a stretching surface with a convective wall boundary condition. The partial differential equations connected to the momentum, energy, and concentration equations are transformed into nonlinear ordinary differential equations (ODEs) by applying relevant similar variables. The obtained ODEs are handled by the Thomas algorithm and a finite difference in the Keller box method. A thorough examination of a change in velocity, temperature, and concentration is done for all the relevant parameters. A higher buoyancy ratio parameter lowers the streamline density. As far as the numerical method is concerned, the Keller box method gives the highest convergence value when compared to other methods, so we use this method to investigate the sleeping behavior of the Williamson nanofluid. The energy source decreases the non-Newtonian passing surface friction. The concentration gradient increases for an increasing value of the chemical reaction parameter. A decreased diffusion rate is seen for increasing Brownian number, while the opposite behavior is noticed for the thermophoretic parameter. The wall friction coefficient increases for augmenting We but decreases for the angled Lorentzian force. Except for radiation, energy transfer is high in all other flows, affecting parameters such as A, B, Nb, Nt, and Pr. By controlling the magnetic field, MHD heat exchangers can manipulate heat transfer rates for various industrial applications. In fusion reactors, strong magnetic fields confine hot plasma, and understanding the interaction between the field and heat sources is crucial for efficient energy generation.

A semi-analytical strategy for mixed convection non-Newtonian nanofluid flow on a stretching surface using Cattaneo-Christov model

This study investigates flow of non-Newtonian fluid containing nano particles and gyrotactic micro-organisms on stretching surface considering magnetic factor and thermal radiations. Cattaneo-Christov model is employed to analyze flow characteristics. The governing Partial Differential Equations (PDEs) along with associated boundary conditions describing the model are converted into Ordinary Differential Equations (ODEs) by suitable transformations. Homotopy Analysis Method (HAM), a semi-analytic solution technique is employed to obtain the solutions. The inspiration of important embedding variables on velocity, temperature, and concentration profiles are presented in tabular and graphical form to elaborate flow properties. It is deduced that the convective parameter and Weissenberg number, both have positive effect on dimensionless velocity whereas buoyancy ratio factor, bio-convective Rayleigh number, and magnetic force have inverse relationship with velocity profile. The presence of radiations and Brownian motion parameter boost energy transfer while it diminishes for higher Prandtl number values and mixed convection factor. Concentration intensifies for larger Prandtl number, activation energy, and thermophoresis parameters whereas it decreases for increasing temperature difference, Brownian motion, Schmidt number, and mixed convection factor. Bio-convective Peclet number, Lewis number, and microorganism concentration gradient factor, all depreciate microorganism concentration panel.

Double-diffusive convection of non-newtonian power-law fluids in an inclined porous layer

This paper presents a numerical study of Double-Diffusive convection within an inclined porous medium saturated by a non-Newtonian fluid. The power-law model is utilized for modelling the behavior of the flow in the porous layer. The given statement implies that the long side of the cavity experience thermal and solutal flux rates, whereas the other walls are impermeable and thermally isolated. The issue is characterized by a set of tightly linked non-linear differential equations, termed governing equations, encompassing the mass conservation equation (known as the continuity equation), the momentum equation, the energy equation, and the species equation. The relevant factors that govern the problem being investigated are the Rayleigh number, R_T, the power-law index, n, the angle of inclination, Φ, the cavity aspect ratio, A, the Lewis number, Le, the normalized porosity, ξ, and the buoyancy ratio, N, two types of cavity configuration have been studied: inclined cavity (i.e. Φ≠0°), then we have studied the case of a vertical cavity (i.e. Φ=90°) where the buoyancy forces induced by the thermal and solutal effects are opposing each other and of equal intensity (N=-1). A semi-analytical solution, valid for an infinite layer (A>>1), is derived on the basis of the parallel flow approximation, A numerical approach utilizing the finite differences method was utilized to resolve the governing equations within the porous medium. It is demonstrated that both the inclination of the layer, Φ, and the power-law index, n, have a strong influence on the strength of the intensity of flow, Ψ_0, the heat transfer rate, Nu, and the mass transfer rate, Sh, within the enclosure. A good agreement is found between the predictions of the parallel flow approximation and the numerical results obtained by solving the full governing equations.

Magnetohydrodynamics thermosolutal convection of hybrid MWCNT-Fe[formula omitted]O[formula omitted]-water nanofluid along a vertical wavy surface with wall heat and mass flux

This numerical analysis investigates the behavior of magnetohydrodynamics (MHD) thermosolutal convection in a hybrid nanofluid made up of multi-wall carbon nanotube (MWCNT)-Fe3O4 and water along a vertical wavy surface with wall heat and mass flux conditions. The mathematical model is developed using the conservation equations of mass, momentum, energy, and energy. The governing equations are converted into a non-dimensional form using the appropriate transformation, and the implicit finite difference technique is used to solve them. The effects of various parameters, such as the Schmidt number (Sc), nanoparticles volume fraction (ϕ), wavy amplitude (a), magnetic parameter (M), buoyancy ratio (Br), on the velocity, temperature, concentration distribution (Φ), streamlines, isotherms, local skin friction coefficient (Cf), local Nusselt (Nu) number, local Sherwood number (Sh) are investigated. Results show that higher amplitude and the addition of hybrid nanofluids volume fraction slow down the heat and mass transport rate, as indicated by declining local Nusselt and Sherwood numbers. However, enhancing the buoyancy ratio increases the fluid flow’s skin friction, heat, and mass transport. The velocity, temperature, concentration distribution, skin friction, and convective heat transport within the wavy surface is weakened by the addition of the magnetic parameter M.

A numerical investigation of the thermal characteristics of supercritical carbon dioxide (sCO2) flowing through mini serpentine channels

In this study, we numerically investigate the heat transfer characteristics of CO2 during phase transitions in its supercritical state in serpentine channels. Four different inclination angles are applied to three different radii of curvature flow channels with the same diameter of 2 mm. The supercritical state is maintained by assuming that the mass flux is approximately 191 kg/m2s, the inlet temperature is 298.15 K, and the operating pressure is 7.65 MPa. To ensure a consistent total heat input, different heat flux values are applied to the three different cases, eliminating the potential impact of the increased surface area on the heat transfer performance and allowing a clearer investigation of the geometric effects within the same temperature range. In a channel with a constant radius of curvature, the centrifugal force has a dominant influence on the heat flow compared to gravitational buoyancy, showing a flow phenomenon independent of the angle. In contrast, in a flow channel with a varying radius of curvature, the gravitational buoyancy fluctuates depending on the local area. As a result, the ratio of the centrifugal force to the gravitational buoyancy decreases, causing more variations with changes in the inclination angle. As the centrifugal force contributes significantly to turbulent mixing, the average heat transfer coefficient is highest in the flow channel with the shortest wavelength. As the wavelength increases, the ratio of centrifugal force to buoyancy begins to affect density stratification, especially in gas-like regions, leading to buoyancy. The local heat transfer coefficient at points around the circumference of the cross-section affected by buoyancy and in the film is analyzed, which is closely related to the centrifugal buoyancy and gravitational buoyancy, and their ratio. To express this influence in a correlation equation, a new Nu correlation with an error of less than 25 % is proposed using De, which combines the centripetal force and Reynolds number, ratio of centrifugal buoyancy and gravitational buoyancy, Ф, and Pr as.Nu=0.849De0.541∅-0.154Pr-0.160

Thermal analysis of 3D viscoelastic micropolar nanofluid with cattaneo-christov heat via exponentially stretchable sheet: Darcy-forchheimer flow exploration

This article presents a comprehensive study on the thermal behavior of a three dimensional viscoelastic micropolar nanofluid under the inspiration of Cattaneo-Christov heat conduction model. The nanofluid is subject to an exponentially stretchable sheet and the flow is characterized by Darcy-Forchheimer formulation. Similarity variables are used to convert viscoelastic nanofluids PDEs into ODEs. MATLAB platform with bvp4c command is used to solve resulting ODE. The impact of involving parameters such that velocity ratio, Schmidt number, Prandtl number, Peclet number, Forchheimer number, buoyancy ratio parameter, Brownian motion, Heat source, Magnetic parameter on velocity profiles, rotation, temperature concentration and density profiles via graphs and tables. Temperature curve is improved up for magnetic parameters, temperature exponent parameter and thermopherosis parameter while significance loss is noted for increasing value of Prandtl number Pr and relaxation time λT. This research bears significance in applications related to advanced heat exchangers, microfluidic systems, and other emerging technologies where the interplay of nanofluids, viscoelasticity, and non-local heat conduction is of paramount importance.

Double-diffusive mixed convection in an inclined square cavity filled with nanofluid: A numerical study with external magnetic field and heated square blockage effects

This study explores the influence of Fe3O4 – water nanofluid, magnetic forces, and double-diffusive mixed convection characteristics on a steady-state, two-dimensional, laminar flow within an inclined square cavity containing four heated square blockages. The governing equations are solved using the Finite Volume Method (FVM). In this configuration, the temperature and concentration inside the square blockages are higher than those on the left and right walls, which are moving in opposite directions to each other. The remaining two are adiabatic and impermeable. A wide range of parameters is investigated, including cavity nanoparticle volume fractions (φ = 0.0, 0.02, 0.04, 0.06), inclination angle (γ = 0°, 30°, 60°), Richardson number (Ri = 0.1, 1.0, 10), Hartmann number (Ha = 0, 50, 100), heated block ratio (B = 1/8), Lewis number (Le = 5.0), Prandtl number (Pr = 0.71), and Buoyancy ratio (N = 2.0). The study extensively analyzes temperature, concentration, and streamline contours based on these parameters. Furthermore, the research examines the heat and mass transfer rates on the heated block surfaces by analyzing local and average Nusselt (Nuavg) and Sherwood numbers (Shavg). The results suggest that introducing nanofluid has a more pronounced influence on the flow fields than the temperature and concentration patterns. Both the inclination angle and the Hartmann number have a significant impact on both the flow and temperature patterns. At higher Ri, an increase in φ reduces the Nuavg and Shavg. However, the total Nuavg and Shavg decrease at any Ri and γ in a magnetic field.

Impact of Moving Walls and Entropy Generation on Doubly Diffusive Mixed Convection of Casson Fluid in Two-Sided Driven Enclosure.

In this study, numerical simulations are conducted with the goal of exploring the impact of the direction of the moving wall, solute and thermal transport, and entropy production on doubly diffusive convection in a chamber occupied by a Casson liquid. Wall movement has a significant impact on convective flow, which, in turn, affects the rate of mass and heat transfer; this sparked our interest in conducting further analysis. The left and right (upright) walls are preserved with constant (but different) thermal and solutal distributions, while the horizontal boundaries are impermeable to mass transfer and insulated from heat transfer. Numerical solutions are acquired using the control volume technique. Outcomes under a variety of Casson fluid parameters, including Ri, Gr, buoyancy ratio, and direction of the moving wall(s), are explored, and the influences of entropy generation are comprehensively investigated. While the flow field consists of a single cell in case I, it is dual-cellular in case III for all values of the considered parameters. Comparing the three cases, the average heat and mass transport presented lower values in case III due to the movement of an isothermal (left) wall against the buoyant force, while these values are enhanced in case I. The obtained results are expected to be useful in thermal engineering, material, food, and chemical processing applications.

Thermochemical convection in a slowly rotating spherical shell: a transition between prograde and retrograde flows

We numerically investigate thermochemically-driven convection in a spherical shell at a high Ekman number for a range of Rayleigh numbers, Prandtl numbers and buoyancy ratios. Rotating convection displays a nearly axisymmetric large-scale structure and appears mostly unaffected by the nature (thermal or chemical) of the driving buoyancy source. We observe both precession directions, prograde and retrograde, of convection structures and identify a snap-through transition between the prograde and retrograde flows in the parameter space spanned by the Rayleigh and Prandtl numbers. Equatorially wall-attached and spiralling columnar convection structures, which typically emerge in rapidly rotating shells, are not observed.

Triple-diffusive free convection enhancement at the stagnation point on moving sheet under the influence of hall effect and mass flux

This study seeks to examine the impact of convective heat transfer, buoyancy ratios, hall current effect, nonlinear thermal radiation, Schmidt number, Prandtl number and mass flux condition on the temperature profiles, velocity profiles and concentration profiles. The research explores into mass and heat transfer characteristics of a stagnation point flow of a free convective triple diffusion with considerations for convective boundary constraints and nonlinear thermal radiation over a mobile vertical plate. To elucidate the aims and methodology, the study utilizes similarity transformation to convert the governing partial differential equations into a set of nonlinear ordinary differential equations. Numerical solutions are obtained employing a fourth-order Runge-Kutta shooting strategy. The findings, showcased through graphical representations, unravel the intricate interplay of parameters, shedding light on flow distribution, temperature, and velocity profiles. These quantitative results not only enhance our scientific understanding of fluid dynamics but also hold practical implications across diverse sectors. Notably, the acquired insights are poised to benefit fields such as environmental science and engineering, where optimizing heat and mass transfer processes is paramount. This research thus contributes valuable perspectives to both the theoretical framework of fluid dynamics and its real-world applications.

Inhomogeneous flow model of natural convection, entropy generation, and heatlines visualization in wavy I‐shaped enclosure with rectangular heater

AbstractThe present investigation is on examination of the natural convection and entropy generation considering the heatlines visualization of nanofluid I‐shaped enclosure with two corrugated walls considering inner rectangular heater of three different heights. The influence of Brownian motion along with thermophoresis had been implemented using Inhomogeneous two‐phase model of nanofluid. The governing equations were solved numerically using COMSOL software. Influence of Rayleigh number , Buoyancy ratio number , Lewis number , heater length . The results indicate that the influence of Lewis number on heat transfer bettering is stronger at high Rayleigh number while its impact is negligible at a lower value of Rayleigh number (conduction mode). In addition, the total entropy generation gets its highest value at Lewis number . Bejan number, fluid flow strength and heat rate increase as the rectangular heater height increases. Also, higher heat transfer augmentation is taken when the heater height is while increasing the heater height to leads to more total entropy generation. The impact of heater height on total entropy generation is highly affected by Rayleigh number as increasing the heater height from into , total entropy generation increases by at while it increases by at .

Numerical computations for Darcy–Forchheimer-based dual convection reactive flow considering Casson nanomaterial by convected permeable surface

The use of nanoparticles in heat transmission is an interesting research subject and many scholars have shown an interest in this topic. In accordance with the motivating relevance of nanomaterials, the current research advocates for an evaluation of Casson nanofluid using porous medium thermal analysis in conjunction with vibrant implementations of mixed convection through a chemical reaction. Additionally, a nonlinear suction/injection phenomenon is taken into consideration. A consistent flow pattern is created by the stretched porous structure. The extended Casson nanofluid model is used to identify the heat production of non-Newtonian fluids. The essential incentives for choosing the Casson nanofluid model are validated as it simultaneously achieves Casson fluid, nanofluid and porous medium outcomes. The convective transport of nanofluid has been addressed using convective temperature boundary conditions and convective flow. A dimensionless form with similarity variables is used to simplify the fundamental equations for the provided flow model. The fourth-order Runge–Kutta (RK) strategy is deployed to arrive at a numerical solution for simulated flow. The physical conveyance of flow variables is accessed visually. It is calculated from engineering quantities (i.e., skin friction, Nusselt number and Sherwood number) to generate the numerical values. The velocity declines with increasing porosity and material factors while opposite outcomes are witnessed for buoyancy factors (mixed convection and buoyancy ratio) and injection factor.

Optimizing thermosolutal and hydrothermal performance of radiative hybrid ferrofluid and entropy generation in a wavy porous enclosure

This research conducts a comprehensive investigation to enhance the thermosolutal and hydrothermal efficiency of a radiative Fe3O4–Cu–H2O hybrid ferrofluid within a wavy porous enclosure with the enlightenment for heat transfer and material science. The primary focus is on optimizing the thermosolutal performance in this irregular enclosure in which the lower boundary undergoes non-uniform heating and concentration, and the curved lateral boundaries keep cooling, lower concentration. Simultaneously, the upper flat boundary remains adiabatic. Using an efficient Higher Order Compact (HOC) scheme, we solve the governing streamfunction (ψ)-vorticity (ζ) form with the incorporation of energy and species transport equations of the coupled Navier–Stokes equations. Our results are robust and validated against established experimental and numerical data sources. The investigation delves into the impacts of well-defined parameters such as Rayleigh number (Ra), Hartmann number (Ha), Darcy number (Da), Buoyancy ratio (N), Lewis number (Le), Radiation parameter (Rd), shape factor of nanoparticles (m) and solid volume fraction (ϕhnp) of the hybrid ferrofluid. Key findings underscore the consistent enhancement of both average Nusselt number (Nu¯) and average Sherwood number (Sh¯) with higher Buoyancy Ratio Numbers (N). Blades-shaped nanoparticles (m=8.6) demonstrate superior performance in both unitary and hybrid ferrofluid systems. Optimal conditions, identified at ϕhnp=0.04 and Ra=104 by the Bejan number (Be), underscore the positive impact of higher Buoyancy Ratio Numbers (N), resulting in a 25.41% increase in Nu¯ and an impressive 43.81% boost in Sh¯ observed from the shifting of N=0 to N=10.

Dynamic interaction of gravity currents in a confined porous layer

We study the dynamic interaction of two gravity currents in a confined porous layer, one heavier and one lighter, partly inspired by the practice of geological $\mathrm {CO}_2$ sequestration in oil fields. Two coupled nonlinear advective-diffusive equations are derived to describe the time evolution of the profile shape of both the upper (lighter) and lower (heavier) currents. At early times, the upper and lower currents remain separated and propagate independently. As time progresses, the currents approach each other and start to interact. We have identified eight different regimes of gravity current interaction at late times, impacted by four dimensionless parameters, representing the flow rate partition, ratio of buoyancy over the injection force, and the viscosity contrasts between the two injecting and displaced fluids. By defining appropriate similarity variables at either the early or late times, the governing partial differential equations (PDEs) reduce to different ordinary differential equations (ODEs), corresponding to the classic nonlinear diffusion solutions at early times and eight different self-similar solutions at late times when the currents attach to each other. It is of interest to note that in four of the eight regimes of late-time interaction (regimes 2, 6–8), self-similar solutions can be constructed by combining appropriately the three basic solutions (i.e. shock, rarefaction and travelling wave solutions) identified during single fluid injection in confined porous layers. In the four other regimes (regimes 1, 3–5), implicit solutions in the form of logarithm or error functions are obtained due to the influence of flow confinement and interaction of gravity currents. Potential implications of the model and solutions are also briefly discussed in the context of ${\rm CO}_2$ -water co-flooding for simultaneous ${\rm CO}_2$ sequestration and oil recovery.

A proceeding to numerical study of mathematical model of bioconvective Maxwell nanofluid flow through a porous stretching surface with nield/convective boundary constraints

Nanofluids become significant in the mass and heat transfer models, especially in engineering problems. Current proceedings focused on the bioconvective Maxwell nanofluid flow passing through the permeable stretchable sheet contingent to nield boundary conditions involving effects of activation energy and thermal radiation. Various physical quantities are involved in this mechanism like magnetic field, thermophoresis, and Brownian motion. The main objective of the study is to report the heat and mass transport in the existence of motile microorganisms. In a mathematical perspective, this structured physical model is going to govern with the help of partial differential equations (PDEs). These governing PDEs are then converted into dimensionless ordinary differential equations form by utilizing appropriate similarity transformations. For numerical results, the shooting technique with ‘bvp4c’ built-in package of MATLAB was implemented. Computed results are then visualized graphically and discussed effects of involving physical variables on the nano-fluid flow profiles are comprehensively. From results, it has been concluded that the fluid flow velocity, temperature, concentration, and microorganism density profiles show escalation on increasing the numeric values of porosity, thermophoresis, buoyancy ratio, bioconvection Rayleigh, Peclet number parameters and decrement reported due to increasing the counts of Prandtl number, magnetic field, radiation, Brownian motion, Lewis number as evident from figures. The numerical outcomes observed by fixing the physical parameters as 0.1<λ<3.0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0.1 < \\lambda < 3.0$$\\end{document}, 0.1<M<1.5\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0.1 < M < 1.5$$\\end{document}, 0.1<Nr<6.0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0.1 < Nr < 6.0$$\\end{document}, 0.1<Rb<1.5\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0.1 < Rb < 1.5$$\\end{document}, 0.1<Nb<6.0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0.1 < Nb < 6.0$$\\end{document}, 0.1<Nt<1.0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0.1 < Nt < 1.0$$\\end{document}, 2.0<Pr<2.9\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2.0 < \\Pr < 2.9$$\\end{document}, 0.1<Rd<0.4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0.1 < Rd < 0.4$$\\end{document}. Magnetic field and Brownian motion create retardation impact due to the liquid momentum. In tables, the numerical values of Skin friction, Nusselt number, Sherwood number, and microorganisms density number are presented and also comparison table of our computed results and already published results is included for the validation.