Magnetic and buoyancy effects on melting from a vertical plate embedded in saturated porous media

The magnetic and buoyancy effects on melting processes about a vertical wall embedded in a saturated porous medium are investigated. The Forchheimer extension is considered in the flow equations, and the magnetic work is included in the energy equation. A similarity solution for the transformed governing equations is obtained, and the combined effect of magnetic field on heat transfer rate is discussed. Numerical results for the velocity and temperature profiles as well as Nusselt number have been presented. The effect of inertial forces on flow and heat transfer in porous media is analyzed. The Nusselt number was found to decrease at the solid–liquid interface as the melting parameter increases.     

MHD dissipative fluid past a stretching sheet in the presence of Soret effect with Newtonian convective heat and mass conditions

This article deliberates a theoretical study on a two‐dimensional magnetohydrodynamic free convection flow of an electrically conducting, heat generation/absorption fluid flowing past a linearly stretching sheet, placed vertical in a non‐Darcian porous medium with Soret effect. As the magnetic Reynolds number of the flow field considered very small (due to noncomparability of the induced and applied magnetic fields), the influence of the induced magnetic field is thus neglected. Again due to weak applied voltage differences at the lateral ends, the influence of the electric current is also ignored. A homotopy analysis method is developed to solve the similarity transformed equations subject to a set of convective heat and mass boundary conditions. Numerical data simulations are made on various fluid variables by using some practical/selected values of the governed parameters and illustrated through graphs and tables. It is found that the Newtonian heating parameter enhanced the velocity, temperature, and concentrations, while the solutal Newtonian heating parameter accelerates the rate of flow of heat and masses but minimizes the temperature gradient. The local Forchheimer and dissipation parameters are found to raise the temperature and concentrations, while the flow rate accelerates due to dissipation parameter but decelerates in presence of Forchheimer parameter.     

Computational modeling of heat transfer in magneto-non-Newtonian material in a circular tube with viscous and Joule heating

Numerous industrial and engineering systems, like, heat exchangers, chemical action reactors, geothermic systems, geological setups, and many others, involve convective heat transfer through a porous medium. The diffusion rate, drag force, and mechanical phenomenon are dealt with in the Darcy–Forchheimer model, and hence this model is vital to study the fluid flow and heat transport analysis. Therefore, numerical simulation of the Darcy–Forchheimer dynamics of a Casson material in a circular tube subjected to the energy losses due to the viscous heating and Joule dissipation mechanisms is performed. The novelty of the present investigation is to scrutinize the convective heat transport characteristics in a circular tube saturated with Darcy–Forchheimer porous matrix by utilizing the non-Newtonian Casson fluid. The flow occurs due to the elongation of the surface of a tube with a uniform heat-based source/sink. The similarity solution of the nonlinear problem was obtained using dimensionless similarity variables. The effects of operating parameters related to the flow phenomena are analyzed. Further, the friction factor and Nusselt number are also analyzed in detail. The present flow model ensures no flow reversal and acts as a coolant of the heated cylindrical surface; the existence of the magnetic field, as well as an inertial coefficient, acts as the momentum-breaking forces, whereas Casson fluidity builds it. The Joule heating phenomenon enhances the magnitude of temperature. The thermal field of the Casson fluid is higher at the surface of the circular pipe due to convective thermal conditions.     

Numerical investigation of natural convection in an inclined enclosure filled with porous medium under magnetic field

In the present study, natural convection of fluid in an inclined enclosure filled with porous medium is numerically investigated in a strong magnetic field. The physical model is heated from left-hand side vertical wall and cooled from opposing wall. Above this enclosure an electric coil is set to generate a magnetic field. The Brinkman–Forchheimer extended Darcy model is used to solve the momentum equations, and the energy equations for fluid and solid are solved with the local thermal non-equilibrium (LTNE) models. Computations are performed for a range of the Darcy number from 10⁻⁵ to 10⁻¹, the inclination angle from 0 to π/2, and magnetic force parameter γ from 0 to 100. The results show that both the magnetic force and the inclination angle have significant effect on the flow field and heat transfer in porous medium.     

Forced convection heat transfer of Williamson fluid flow in porous media over horizontal plate with constant heat flux

Forced convection of Williamson fluid flow in porous media under constant surface heat flux conditions is investigated numerically. A model of Darcy–Forchheimer–Brinkman is used and the corresponding governing equations are expressed in dimensionless forms and solved numerically using bvp4c with MATLAB package. Boundary layer velocity, shear stress, and temperature profiles, in addition to the local Nusselt number parameter over a horizontal plate, are found. The effects of the Forchheimer parameter, Nusselt number, Darcy parameter, porous inertia, and Williamson parameter on the velocity profiles, temperature profiles, coefficient of friction, and coefficient of heat transfer are investigated. The results showed that as the Darcy parameter increases, boundary layer velocity and shear stress increase, while the temperature and Nusselt number decrease. In addition, as Williamson's parameter increases, velocity within the boundary layer, shear stress, and Nusselt number decrease while the temperature profile increases. Also, with larger values of the Forchheimer parameter, the velocity of the boundary layer, shear stress, temperature, and Nusselt number increase. Furthermore, the Nusselt number and the coefficient of friction are obtained on the surface of the horizontal plate.     

Pulsating hydromagnetic flow of Casson fluid in a vertical channel filled with non-Darcian porous medium

This paper analyzes the Joule heating, Dufour number, and Soret number effects on hydromagnetic pulsatile flow of a Casson fluid in a vertical channel filled with a non-Darcian porous medium. The governing partial differential equations (PDEs) of the Casson fluid flow are transformed to ordinary differential equations (ODEs) using perturbation technique and solved by employing shooting method with Runge–Kutta (R–K) fourth-order technique using MATHEMATICA function NDSolve. The influence of Forchheimer number, Casson fluid parameter, Dufour number, radiation parameter, and Soret number on flow variables has been studied and the numerical results obtained are presented. The results reveal that the velocity rises with the rise of Darcy number, whereas it decreases for a given rise in the Forchheimer number. Furthermore, the temperature distribution enhances by increasing the Dufour number.     

Viscoelastic boundary layer analysis of constant surface temperature plate embedded in saturated porous media

Mathematical models and numerical solutions of Williamson fluid flow under influences of various boundary conditions provide important support to experimental studies in the solar energy field. Therefore, the present study is concerned with the effects of forced convection of the viscoelastic boundary layer on a horizontal plate embedded in saturated porous media subjected to constant surface temperature. The study explores the profiles of shear stress, velocity, temperature, and heat transfer coefficient. The governing equations in nondimensional forms are obtained by using a model of Darcy–Forchheimer–Brinkman and finally are solved numerically by using bvp4c with MATLAB package. The results of the numerical solution show an insignificant rise in the distribution of the velocity boundary layer and shear stress profile as the Darcy parameter is increased, while a decrease in the temperature and Nusselt numbers are found. On the other hand, as the viscoelastic parameter is increased, the Darcy parameter shows a reverse response. Finally, insignificant increases in profiles of boundary layer velocity, temperature, shear stress, and Nusselt number are observed at high values of the Forchheimer number.     

Hybrid nanofluid flow over a stretched cylinder with the impact of homogeneous–heterogeneous reactions and Cattaneo–Christov heat flux: Series solution and numerical simulation

The Catteno–Christov heat flux plays a dynamic role in flow of heat enhancement in various manufacturing, industrial, and engineering applications. This present work focuses on the influence of Catteno–Christov heat flux model on Darcy–Forchheimer flow of a hybrid nanofluid placed in a porous medium. The formulation of the mathematical model is done by considering a fluid with two different nanoparticles Al₂O₃ and Cu dispersed in the water as the base fluid. The set of partial differntial equations is reduced by using similarity variables and boundary conditions to obtain ordinary differntial equations. The coupled nonlinear governing differential equations are solved using Runge–Kutta fourth–fifth order (RKF-45). The impact of numerous dimensionless parameters on the velocity, thermal, and concentration profiles are plotted and studied. Furthermore, the coefficient of skin friction for the relevant parameters are analysed through graphs. Result reveals that, increase in the porosity parameter declines the velocity gradient and shoots up the thermal and concentration gradients. Inclination in magnetic parameter declines velocity and concentration profiles due to the Lorentz force. Enhancement in the thermal relaxation parameter declines the thermal profile. Inclination in homogeneous-heterogeneous reaction parameters declines the mass transfer rate. Also, the well-known differential transform method is used for the validity of RKF-45 method and an impressive agreement is noticed between the results of RKF-45 and DTM.     

Numerical computation of nonlinear oscillatory two‐immiscible magnetohydrodynamic flow in dual porous media system: FTCS and FEM study

The transient Hartmann magnetohydrodynamic flow of two immiscible fluids flowing through a horizontal channel containing two porous media with oscillating lateral wall mass flux is studied. A two‐dimensional spatial model is developed for two fluids, one of which is electrically conducting and the other is electrically insulating. Both the fluid regimes are driven by a common pressure gradient. A Darcy‐Forchheimer drag force model is used to simulate the porous media effects on the flow in both the fluid regimes. Special boundary conditions are imposed at the interface. The governing second‐order nonlinear partial differential dimensionless equations are obtained for each region using a set of transformations. The resulting transport equations are controlled by the Hartmann hydromagnetic parameter (Ha), viscosity ratio parameter (α), two Darcy numbers (Da ₁ and Da ₂), two Forchheimer numbers (Fs ₁ and Fs ₂), two Reynolds numbers (Re ₁ and Re ₂), frequency parameter ( εA) associated with the transpiration (lateral wall flux) velocity and a periodic frequency parameter ( ω*t*). Numerical forward time/central space finite‐difference solutions are obtained for a wide range of the governing parameters. Bench marking is performed with a Galerkin finite‐element method (MAGNETO‐FEM), and the results are found to be in excellent agreement. Applications of the model include magnetic cleanup operations in coastal/ocean seabed oil spills and electromagnetic purification of petroleum reservoir fluids.     

Darcy–Benard–Marangoni convection in porous media

The onset of coupled Darcy–Benard–Marangoni convection in a liquid saturated porous layer of high permeability of practical importance is investigated by employing the Brinkman–Forchheimer– Lapwood-extended Darcy flow model with fluid viscosity different from effective viscosity. The lower boundary is taken to be rigid and insulating to temperature perturbations, while the upper surface is open to atmosphere and subject to a general thermal condition. The critical eigenvalues are obtained numerically, in general, using Galerkin method. However, closed form solution is also obtained using regular perturbation technique for insulated boundaries. Besides, the eigenvalue problem is solved exactly for pure Darcy–Marangoni convection. The numerical and analytical results are found to be in excellent agreement with each other. It is observed that the effect of buoyancy is destabilizing, while an increase in the permeability parameter is to delay the onset of convection. The Biot number and the ratio of effective viscosity to fluid viscosity are found to increase the critical conditions. Some known results are recovered as special cases.     

幂律流体饱和多孔介质通道中的粘性耗散效应

针对Darcy-Brinkman-Forchheimer流动模型,分析了幂律型非牛顿流体在填充多孔介质平板通道中强迫对流传热过程充分发展的黏性耗散效应,并比较了三个不同的黏性耗散项Darcy项、Al-Hadhrami项和Forchheimer项对流动传热率的影响。推导出了无量纲轴向流速分布和无量纲温度分布的计算表达式,并在恒热流边界条件下,利用经典Runge-Kutta法进行数值求解。模拟结果表明,布林克曼数Br、达西数Da、综合惯性参数F和幂律指数n等重要参数对无量纲温度分布有着较大的影响,同时发现不同的黏性耗散效应对流动传热特性也有着重要的影响。     

Dissipative heat for the Casson fluid flow past an expanding cylindrical surface

A non-Newtonian model is developed by considering the flow of non-Newtonian Casson fluid past an expanding cylinder embedded in a porous medium. The novelty arises because of the conjunction of dissipative heat, and the additional heat source that enriches the heat transport phenomenon significantly. The application of the study is vital due to the flow of blood through the artery, a physiological study. Therefore, the study of Casson fluid plays an important role. The nonlinear partial differential equations that appeared in the formulation are now renovated to the coupled nonlinear ordinary differential equations. However, a numerical technique associated with shooting-based followed by Runge–Kutta fourth-order is employed for the solution of these transformed equations. The uniqueness of diverse pertinent parameters on the flow phenomena is scrutinized through graphs and numerically simulated results presented in tables. The important observations are as follows; the magnetic parameter and permeability augment the shear rate coefficients, whereas the Casson parameter rendered the opposite impact. Furthermore, the non-Newtonian Casson parameter retards the fluid temperature, and the curvature parameter significantly enhances it.     

Hall effects on the hydromagnetic free convective flow and mass transfer through a porous medium bounded by an infinite vertical porous plate with constant heat flux

Effects of Hall current on free convection and mass transfer flow through a porous medium bounded by a vertical surface when a uniform magnetic field acts in a plane which makes an angle x with the plane transverse to the plate have been analysed. An analytic solution of the problem is obtained and the effects of the Hall parameter and the permeability parameter, as well as the other parameters entering into the problem, are discussed and shown graphically.     

Non‐Newtonian squashed flow simulation across Darcy‐Forchheimer sensor

The current mathematical formulation is dedicated to investigate the Darcy‐Forchheimer boundary layer–squeezed hydromagnetic flow of a Casson fluid passing through a sensor surface. The flow phenomenon is occurring in a locally free stream under the combined sway of heat generation and thermic radiation. The energy equation is deliberated with the assistance of Cattaneo‐Christov theory rather than using Fourier's law for conduction of heat. Here, the thermic conductivity is being presumed as a function of temperature. The governing mathematical structure consists of highly nonlinear terms, so a set of regulatory parameters is being accomplished to attain the unpretentious dimensionless equations. This nondimensional structure is then treated numerically to attain the nearly converging results. The significance of substantial parameters such as magnetic factor, radiation parameter, Casson fluid parameter, heat origination, and thermal relaxation time on the flow phenomenon is estimated and presented graphically. Besides this, the factors of engineering interest like the Prandtl number and squeezed flow index with vacillating thermic conductivity have strong effects on the flow behavior of the fluid. It is observed that the magnetic effect causes an expansion in the velocity curve while a reduction is found for squeezed flow index parameter.     

Numerical solution for Sisko nanofluid flow through stretching surface in a Darcy–Forchheimer porous medium with thermal radiation

The heat transfer assessments in a Sisko nanofluid flow over a stretching surface in a Darcy–Forchheimer porous medium with heat generation and thermal radiation are studied. The numerical analysis technique is used to assess the governing nonlinear equations of the model. The influence of Forchheimer number, porosity, heat generation, radiation, and material parameters is examined. The outlines of Nusselt number and skin friction coefficient corresponding to pertinent parameters are revealed. The comparison of Nusselt number outlines of working fluid and Newtonian fluid is depicted. From the analysis, it has been examined that with the increase in Forchheimer number and material parameter values, heat transfer function decreases, whereas heat transfer characteristics of Sisko nanofluid increase with heat generation and material parameters. Moreover, working fluid velocity outlines depreciate when there is an increase in porosity parameter for both shear-thinning and shear-thickening. The comparison of this study with previous research has been conducted.     

Darcy–Forchheimer flow of Ree–Eyring fluid over an inclined plate with chemical reaction: A statistical approach

In spite of various reports on non-Newtonian fluids, little is known on the impact of chemical reaction on the Darcy–Forchheimer flow of Ree–Eyring fluid when Cattaneo–Christov (C-C) heat flux (HF) is significant. The inclusion of porous medium occurs in various procedures which include heat transfer, geophysics design, and so forth. It also influences oil production recovery, energy storage units, solar receivers, and many others. The Darcy–Forchheimer flow model is important in the fields where a high flow rate effect is a common phenomenon, for instance, in petroleum engineering. In this study, we aim to analyze the dissipative Darcy–Forchheimer flow of Ree–Eyring fluid by an inclined (stretching) plate with chemical reaction. We have included the C-C HF model to investigate the heat transfer characteristics of the fluid. Equations in the mathematical model are metamorphosed as ordinary differential equations and then unriddled with the aid of shooting strategy. The main advantage of the shooting method is that it is easy to apply. The shooting method requires good initial guesses for the first derivative and can be applied to both linear and nonlinear problems. Results are explicated through graphs. We took the help of a statistical tool, that is, correlation coefficient to analyze the impression of crucial parameters on surface friction drag (skin friction coefficient), heat and mass transfer rates. The main inferences of this study are porosity parameter and Forchheimer numbers deprecate the fluid velocity, Eckert number ameliorates fluid temperature and concentration minifies with larger chemical reaction parameter. It is discovered that the Forchheimer and Weissenberg numbers deprecate the surface friction drag. Mass transfer rate has a substantial positive relationship with Schmidt number and chemical reaction. Furthermore, the heat transfer rate has a substantial positive correlation with the thermal relaxation parameter and a substantial negative correlation with the Eckert number.     

Effects of buoyancy and conjugate heat transfer on non-Darcy mixed convection about a vertical slender hollow cylinder embedded in a porous medium with high porosity

This study investigates mixed convection heat transfer about a vertical slender hollow cylinder in the buoyancy and conjugate heat transfer effects in the porous medium with high porosity. The non-similar solutions using the Keller box method are obtained. The wall conduction parameter p, the porous medium parameter k₁, the Forchheimer parameter F∗ and the Richardson number are the main parameters. For various values of these parameters the local skin friction and local heat transfer parameters are determined. The validity of the methodology is checked by comparing the results with those available in the open literature and a fairly good agreement is observed. Finally, it is determined that the local skin friction and the local heat transfer coefficients increase with an increase buoyancy parameter Ri, porous medium parameter k₁, Forchheimer parameter F∗ and decrease with conjugate heat transfer parameter p.     

Brinkman‐Forchheimer slip flow subject to exponential space and thermal‐dependent heat source in a microchannel utilizing SWCNT and MWCNT nanoliquids

This communication examines the impact of carbon nanotubes (single‐wall carbon nanotubes [SWCNT] and multi‐wall carbon nanotubes [MWCNT]) on magnetohydrodynamic Brinkman and Forchheimer flow in a planar microchannel with multiple slips. Flow through a porous medium is modeled via Brinkman and Forchheimer theory. The impacts of thermal‐dependent heat source (THS) and exponential space‐dependent heat source (ESHS) are deployed. Aspects of Joule and viscous dissipations are also retained. The dimensionless equations are solved using the Runge‐Kutta‐Fehlberg joint with shooting methodology. The significance of various nondimensional parameters on the flow distributions as well as skin‐friction and Nusselt number is illustrated and analyzed. Closed form solution of momentum quantity is developed for a particular case. Obtained numerical results are in perfect agreement with analytical results. Further, the results of SWCNT and MWCNT are compared.     

Natural convection heat transfer from a horizontal cylinder embedded in a porous medium

This paper presents the results of an experimental investigation of heat transfer by natural convection from a horizontal cylinder embedded in porous media consisting of randomly packed glass spheres saturated by either water or silicone oil. It is shown that the overall range of the Rayleigh number, Ra, can be divided into two subregions, called ‘low’ and ‘high’, in each of which the Nusselt number, Nu, behaves differently. It is demonstrated that the low Ra region corresponds to Darey flow and the high to Forchheimer flow. Correlation equations for Nu for the Darcy regime are presented that account for viscous dissipation, and others for the Forchheimer regime that involve the first and second Forchheimer coefficients. The variation of properties with temperature and the wall effect on porosity (and consequently on heat transfer) are considered. The paper includes information concerning the resistance to flow in porous media that was obtained in conjunction with the heat transfer study.     

Nonlinear thermal radiation and chemical reaction effects on Maxwell fluid flow with convectively heated plate in a porous medium

This paper investigated the chemically reactive radiating flow by using a two‐dimensional Darcy‐Forchheimer model with the convectively heated plate. The nonlinear thermal radiation is described by Joule heating and heat generation. Also, Darcy‐Forchheimer equation is related to porous medium flows. For the solution of equations, we used the numerical method. Further, more physical interpretation of the parameters was demonstrated with figures. It is found that an increase in the Prandtl number had a direct effect on the Nusselt number and temperature, whereas the opposite scenario was observed in the Eckert number.