Porous Regime Research Articles

MATHEMATICAL MODELING FOR POROUS MEDIA TRANSPORT IN NEWTONIAN RADIATING/CHEMICALLY REACTING FLUID OVER AN IMPULSIVELY-STARTED VERTICAL PLATE: A FINITE DIFFERENCE APPROACH

Theoretical investigation is presented here for the model of unsteady MHD thermal convection flow of a viscous incompressible absorbingemitting optically thin gray gas along an impulsivelystarted semi-infinite vertical plate adjacent to the Darcian porous regime in the presence of a first order chemical reaction and significant thermal radiation effects. The conservation equations are nondimensionalized and are solved by an accurate and unconditionally stable implicit finite difference scheme of Crank-Nicholson type. The flow is found to be accelerated with increasing porosity parameter (K), whereas the temperature and concentration distributions are reduced in the Darcian regime. Flow velocity and Temperature are found to be depressed with progression of thermal radiation (Ra) contribution, but enhanced the concentration distribution. Applications of the model arise in solar energy collector analysis, magneto-fluid dynamics and industrial materials processing.

Finite element and network electrical simulation of rotating magnetofluid flow in nonlinear porous media with inclined magnetic field and hall currents

A mathematical model is presented for viscous hydromagnetic flow through a hybrid non-Darcy porous media rotating generator. The system is simulated as steady, incompressible flow through a nonlinear porous regime intercalated between parallel plates of the generator in a rotating frame of reference in the presence of a strong, inclined magnetic field A pressure gradient term is included which is a function of the longitudinal coordinate. The general equations for rotating viscous magnetohydrodynamic flow are presented and neglecting convective acceleration effects, the two-dimensional viscous flow equations are derived incorporating current density components, porous media drag effects, Lorentz drag force components and Hall current effects. Using an appropriate group of dimensionless variables, the momentum equations for primary and secondary flow are rendered nondimensional and shown to be controlled by six physical parameters-Hartmann number (Ha), Hall current parameter (Nh), Darcy number (Da), Forchheimer number (Fs), Ekman number (Ek) and dimensionless pressure gradient parameter (Np), in addition to one geometric parameter-the orientation of the applied magnetic field (? ). Several special cases are extracted from the general model, including the non-porous case studied earlier by Ghosh and Pop (2006). A numerical solution is presented to the nonlinear coupled ordinary differential equations using both the Network Simulation Method and Finite Element Method, achieving excellent agreement. Additionally very good agreement is also obtained with the earlier analytical solutions of Ghosh and Pop (2006). for selected Ha, Ek and Nh values. We examine in detail the effects of magnetic field, rotation, Hall current, bulk porous matrix drag, second order porous impedance, pressure gradient and magnetic field inclination on primary and secondary velocity distributions and also frictional shear stresses at the plates. Primary velocity is seen to decrease with an increase in Hall current parameter (Nh) with the converse observed for the secondary velocity.

Numerical study of free convective MHD flow past a vertical cone in non-Darcian porous media

A numerical study of buoyancy-driven unsteady natural convection boundary layer flow past a vertical cone embedded in a non-Darcian isotropic porous regime with transverse magnetic field applied normal to the surface is considered. The heat and mass flux at the surface of the cone is modeled as a power-law according to qw(x) = xm and q*w (x) = xn respectively, where x denotes the coordinate along the slant face of the cone. Both Darcian drag and Forchheimer quadratic porous impedance are incorporated into the two-dimensional viscous flow model. The transient boundary layer equations are then non-dimensionalized and solved by the Crank-Nicolson implicit difference method. The velocity, temperature and concentration fields have been studied for the effect of Grashof number, Darcy number, Forchheimer number, Prandtl number, surface heat flux power-law exponent (m), surface mass flux power-law exponent (n), Schmidt number, buoyancy ratio parameter and semi-vertical angle of the cone. Present results for selected variables for the purely fluid regime are compared with the non-porous study by Hossain and Paul [9] and are found to be in excellent agreement. The local skin friction, Nusselt number and Sherwood number are also analyzed graphically. The study finds important applications in geophysical heat transfer, industrial manufacturing processes and hybrid solar energy systems.

Free Convective MHD Flow Past a Vertical Cone with Variable Heat and Mass Flux

A numerical study of buoyancy-driven unsteady natural convection boundary layer flow past a vertical cone embedded in a non-Darcian isotropic porous regime with transverse magnetic field applied normal to the surface is considered. The heat and mass flux at the surface of the cone is modeled as a power law according to qwx=xm and qw*(x)=xm, respectively, where x denotes the coordinate along the slant face of the cone. Both Darcian drag and Forchheimer quadratic porous impedance are incorporated into the two-dimensional viscous flow model. The transient boundary layer equations are then nondimensionalized and solved by the Crank-Nicolson implicit difference method. The velocity, temperature, and concentration fields have been studied for the effect of Grashof number, Darcy number, Forchheimer number, Prandtl number, surface heat flux power-law exponent (m), surface mass flux power-law exponent (n), Schmidt number, buoyancy ratio parameter, and semivertical angle of the cone. Present results for selected variables for the purely fluid regime are compared with the published results and are found to be in excellent agreement. The local skin friction, Nusselt number, and Sherwood number are also analyzed graphically. The study finds important applications in geophysical heat transfer, industrial manufacturing processes, and hybrid solar energy systems.

Unsteady MHD double-diffusive convection boundary-layer flow past a radiate hot vertical surface in porous media in the presence of chemical reaction and heat sink

This paper presents an analytical study of the unsteady MHD free convective heat and mass transfer flow of a viscous, incompressible, gray, absorbing-emitting but non-scattering, optically-thick and electrically conducting fluid occupying a semi-infinite porous regime adjacent to an infinite moving hot vertical plate with constant velocity. We employ a Darcian viscous flow model for the porous medium the Rosseland diffusion approximation is used to describe the radiative heat flux in the energy equation. The homogeneous chemical reaction of first order is accounted in mass diffusion equation. The governing equations are solved in closed form by Laplace-transform technique. A parametric study of all involved parameters is conducted and representative set of numerical results for the velocity, temperature, concentration, shear stress function \(\frac{\partial u}{\partial y} \vert_{y=0}\), temperature gradient \(\frac{\partial \theta }{ \partial y}\vert_{y=0}\), and concentration gradient \(\frac{ \partial \phi }{\partial y}\vert_{y=0}\) is illustrated graphically and physical aspects of the problem are discussed.

Numerical Study of Free Convection Flow past a Vertical Cone with Variable Heat and Mass Flux

ABSTRACT A numerical study of buoyancy-driven unsteady natural convection boundary layer flow past a vertical cone embedded in a non-Darcian isotropic porous regime with transverse magnetic field applied normal to the surface is considered. The heat and mass flux at the surface of the cone is modeled as a power-law according to () m q x x w  and * () n q x x w  respectively, where x denotes the coordinate along the slant face of the cone. Both Darcian drag and Forchheimer quadratic porous impedance are incorporated into the two-dimensional viscous flow model. The transient boundary layer equations are then non-dimensionalized and solved by the Crank-Nicolson implicit difference method. The velocity, temperature and concentration fields have been studied for the effect of Grashof number, Darcy number, Forchheimer number, Prandtl number, surface heat flux power-law exponent (m), surface mass flux power-law exponent (n), Schmidt number, buoyancy ratio parameter and semi-vertical angle of the cone. Present results for selected variables for the purely fluid regime are compared with the published work nonand are found to be in excellent agreement. The local skin friction, Nusselt number and Sherwood number are also analyzed graphically. The study finds important applications in geophysical heat transfer, industrial manufacturing processes and hybrid solar energy systems.

Unsteady Magnetohydrodynamic Heat Transfer in a Semi-Infinite Porous Medium with Thermal Radiation Flux: Analytical and Numerical Study

The unsteady, buoyancy-induced, hydromagnetic, thermal convection flow in a semi-infinite porous regime adjacent to an infinite hot vertical plate moving with constant velocity, is studied in the presence of significant thermal radiation. The momentum and energy conservation equations are normalized and then solved using both the Laplace transform technique and Network Numerical Simulation. Excellent agreement is obtained between both analytical and numerical methods. An increase in Hartmann number (M2) strongly decelerates the flow and for very high strength magnetic fields (M2=20), the flow is reversed after a short time interval. The classical velocity overshoot is also detected close to the plate surface for low to intermediate values of M2 at both small and large times; however this overshoot vanishes for larger strengths of the transverse magnetic field (M2=10). An increase in radiation-conduction parameter (Kr) significantly increases temperature throughout the porous regime at both small and larger times, adjacent to the plate, but decreases the shear stress magnitudes at the plate. Temperature gradient is reduced at the plate surface for all times, with a rise in radiation-conduction parameter (Kr). Shear stress is reduced considerably with an increase in Darcian drag parameter (Kp).

Natural convection heat and mass transfer in a micropolar fluidsaturated non-Darcy porous regime with radiation and thermophoresis effects

An analysis is presented for the steady thermal convection heat and mass transfer in a micropolar-fluid-saturated non-Darcian porous medium in the presence of radiation and thermophoresis effects. The governing boundary layer equations for momentum, energy, species transfer and angular momentum (micro-rotation) are transformed from an (x,y), coordinate system into (?), coordinate system. The influence of Darcy number (Da), Forchheimmer number (Fs), local Grashof number (Gr), Prandtl number (Pr), Schmidt number (Sc), radiation (R) and thermophoresis (k), surface parameter (s), on the velocity, temperature, concentration profiles and angular velocity (micro-rotation) are studied graphically. Applications for the problem arise in chemical engineering systems and geothermal energy systems.

UNSTEADY HYDROMAGNETIC NATURAL CONVECTION OF A SHORT-MEMORY VISCOELASTIC FLUID IN A NON-DARCIAN REGIME: NETWORK SIMULATION

A mathematical model for the unsteady magnetohydrodynamic (MHD) laminar natural convection flow of a viscoelastic fluid from an infinite vertical porous plate to an isotropic, homogeneous, non-Darcian porous regime, with time-dependent suction, in the presence of a uniform transverse magnetic field, is studied. The generalized Beard-Walters rheological model is employed, which introduces a mixed third-order derivative into the momentum conservation equation. The transformed conservation equations are solved using the robust, well-tested computational procedure known as network simulation method (NSM). The NSM computations have shown that with an increase in viscoelasticity parameter (S) the flow accelerates considerably with time. Increasing magnetic field (M), however, retards the flow strongly with time. An increase in the Darcy number (Da) serves to augment the velocity (w) profiles, i.e., accelerate the flow in both the conducting (M ≠ 0) and nonconducting (M = 0) cases. Velocities also increase in value over time (τ). A velocity overshoot is identified close to the plate. A rise in the Forchheimer number (Fs), corresponding to an accentuation in the quadratic porous drag effect, induces a strong deceleration in the flow, in particular near the plate surface, for both conducting and nonconducting cases. Increasing buoyancy effects, as simulated via a rise in the thermal Grashof number (Gr), leads to a substantial retardation in the flow; this effect is enhanced with Lorentzian magnetic drag force. An increase in the suction parameter (A) causes a stronger adherence of the hydrodynamic boundary layer to the plate and leads to a reduction in velocities along the entire plate regime. A similar decrease in temperature (θ) is caused with increasing suction parameter (A). The results are of relevance in, for example, magneto-rheological materials processing operations and advanced hybrid magnetohydrodynamic energy systems exploiting non-Newtonian fluids.

A numerical solution of unsteady MHD convection heat and mass transfer past a semi-infinite vertical porous moving plate using element free Galerkin method

Abstract In this paper, the unsteady MHD free convection heat and mass transfer of viscous fluid flowing through a Darcian porous regime adjacent to a moving vertical semi-infinite plate under Soret and Dufour effect have been examined. Viscous dissipation effects are included in the energy equation. A uniform magnetic field is applied transversely to the direction of the flow. The differential equations governing the problem have been transformed by a similarity transformation into a system of non-dimensional differential equations which are solved numerically by element free Galerkin method. The influence of Grashof number (Gr), magnetic parameter (M), heat absorption parameter (Q), permeability parameter (K), Schmidt number (Sc), Soret number (Sr), and Dufour number (Du) on the velocity, temperature and concentration profiles are shown graphically. Some of the result has been compared with finite element method. Finally, the numerical values of local skin friction, local rate of heat transfer parameter and local mass transfer parameter are also presented in tabular form. The present problem finds significant applications in chemical engineering, materials processing, solar porous wafer absorber systems and metallurgy.

Porous silicon/metal nanocomposite with tailored magnetic properties

AbstractPorous silicon (PS) templates in the meso/macro porous regime with oriented pores clearly separated from each other and filled in a galvanic deposition process with various metals, especially ferromagnetic ones are magnetically investigated. The employment of different metals (e.g. Ni, Co, NiCo) together with the variation of the electrochemical deposition parameters modifies the structural characteristics of the PS/metal nanocomposite and thus leads to distinct magnetic properties of the hybrid system. Furthermore the use of different PS‐templates which means a change of the pore‐diameter and interpore spacing results also in various magnetic characteristics, especially influences on the magnetic interactions among the deposited metal nanostructures. Therefore the specimens show tailored magnetic properties like coercivity, squareness and magnetic anisotropy. The achieved nanocomposite merges electronic properties of a semiconductor with nanomagnetism and therefore opens the possibility of integrated spin‐based electronic devices. (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Network simulation solutions for laminar radiating dissipative magneto-gas dynamic heat transfer over a wedge in non-Darcian porous regime

We study the steady-state, magnetohydrodynamic, optically thick, dissipative gas boundary layer flow and heat transfer past a non-isothermal porous wedge embedded in a scattering, homogenous, isotropic Darcy–Forchheimer porous medium, with significant thermal radiation effects in the presence of heat sink/sources and surface transpiration, in an ( x , y ) coordinate system. The Rosseland diffusion approximation is employed for which radiative flux can propagate only small distances prior to scattering or absorption. Joule electric dissipation, viscous heating and also stress work are incorporated in the boundary layer equations and a temperature-dependent heat source/sink term utilized. Following a transformation to a ( ξ , η ) coordinate system, the transformed coupled, nonlinear pseudosimilar equations for momentum and energy are solved using a powerful computational method based on thermoelectric analogy, viz the Network Simulation Method. The effects of magnetism (Hartmann number), Darcy number, Forchheimer number, Rosseland radiation–conduction parameter, pressure gradient parameter, Eckert number, transpiration (wall mass transfer) parameter, and heat source/sink parameter on velocity and temperature profiles are depicted graphically. Numerical solutions are compared where possible with earlier non-porous and non-dissipative studies and found to be in excellent agreement. The current study has potential applications in simulating laminar radiative-magnetohydrodynamic heat transfer over astronautical bodies in debris-laden (porous) regimes and also in geothermal physics and magnetic materials processing.

Numerical Study of Chemically Reactive Buoyancy-Driven Heat and Mass Transfer across a Horizontal Cylinder in a High-Porosity Non-Darcian Regime

We investigate the free convection boundary layer flow and heat and mass transfer across an isothermal cylinder embedded in an isotropic, homogenous, saturated porous regime with a first-order chemical reaction in the diffusing species. A Darcy-Forchheimer drag force model is implemented to simulate porous impedance effects in high-porosity media, which are encountered in various industrial and geophysical applications. The partial differential conservation equations are nondimensionalized and solved using a network simulation methodology. The effects of Darcy number, Forchheimer number, Schmidt number, and reaction parameter on dimensionless velocity, temperature, and species concentration distributions are studied in detail for the case of water of relevance to geohydraulic flows. Computations are also provided for the variation of local Nusselt number and local Sherwood number with various thermophysical parameters. Concentration is found to decrease continuously with distance into the boundary layer (y-coordinate) with an increase in chemical reaction parameter; values are markedly higher for the non-Darcian case than for the Darcian case. Temperatures are however increased by an increase in reaction parameter. Applications of the study include electrolysis processes, chemical filtration treatment systems, natural convection from buried waste canisters in geomaterials, geothermal systems, etc.

Theoretical Analysis of Radiative Effects on Transient Free Convection Heat Transfer past a Hot Vertical Surface in Porous Media

The purpose of the present investigation deals with the unsteady free convective flow of a viscous incompressible gray, absorbing-emitting but non-scattering, optically-thick fluid occupying a semi-infinite porous regime adjacent to an infinite moving hot vertical plate with constant velocity. We employ a Darcian viscous flow model for the porous medium. The momentum and thermal boundary layer equations are non-dimensionalized using appropriate transformations and then solved subject to physically realistic boundary conditions using the Laplace transform technique. Thermal radiation effects are simulated via a radiation-conduction parameter, Kr, based on the Rosseland diffusion approximation. The influence of Grashof (free convection) number, radiation-conduction parameter (Kr), inverse permeability parameter (Kp) and dimensionless time (t) are studied graphically. We observe that increasing thermal radiation parameter causes a noticeable increase in the flow velocity, u. Temperature, θ, is significantly increased within the boundary layer with a rise in Kr since the latter represents the relative contribution of thermal radiation heat transfer to thermal conduction heat transfer. Increased radiation therefore augments heat transfer, heats the fluid and increases the thickness of the momentum and thermal boundary layers. Velocity is found to decrease with an increase in Kp (inverse permeability parameter) as are shear stress function ( ∂u/∂y | y=0) magnitudes owing to greater resistance of the porous medium for lower permeabilities, which decelerate the flow. An increase in Kr however boosts the shear stress function magnitudes i.e. serves to accelerate the flow. Temperature gradient, ∂θ/∂y | y=0 is also positively affected by an increase in thermal radiation (Kr) and with time. The present study has applications in geological convection, forest fire propagation, glass heat treatment processes at high temperature, metallurgical processing etc.

Network Numerical Simulation of Impulsively-Started Transient Radiation-Convection Heat and Mass Transfer in a Saturated Darcy-Forchheimer Porous Medium

We study the effects of thermal radiation and porous drag forces on the natural convection heat and mass transfer of a viscous, incompressible, gray, absorbingemmitting fluid flowing past an impulsively started moving vertical plate adjacent to a non-Darcian porous regime. The governing boundary-layer equations are formulated in an (X∗, Y∗, t∗) coordinate system with appropriate boundary conditions. The Rosseland diffusion approximation is employed to analyze the radiative heat flux and is appropriate for non-scattering media. The model is non-dimensionalized and solved with the network simulation model. We study the influence of Prandtl number, radiation-conduction parameter, thermal Grashof number, species Grashof number, Schmidt number, Darcy number and Forchheimer number on the dimensionless velocity, temperature and species function distributions. Additionally we compute the variation of the local skin friction, Nusselt number and Sherwood number for selected thermophysical parameters. Increasing Darcy number is seen to accelerate the flow; the converse is apparent for an increase in Forchheimer number. Thermal radiation is seen to reduce both velocity and temperature in the boundary layer. The interactive effects of second order porous drag and thermal radiation are also considered. The model finds applications in solar energy collection systems, porous combustors, transport in fires in porous media (forest fires) and also the design of high temperature chemical process systems.

NSM analysis of time-dependent nonlinear buoyancy-driven double-diffusive radiative convection flow in non-Darcy geological porous media

A network numerical simulator is developed and described to simulate the transient, nonlinear buoyancy-driven double diffusive heat and mass transfer of a viscous, incompressible, gray, absorbing–emitting fluid flowing past an impulsively started moving vertical plate adjacent to a non-Darcian geological porous regime. The governing boundary-layer equations are formulated in an (X *, Y *, t *) coordinate system with appropriate boundary conditions. An algebraic diffusion approximation is used to simplify the radiation heat transfer contribution. The non-dimensionalized transport equations are solved in an (X, Y, t) coordinate system using the network simulation model (NSM) and the computer code, Pspice. A detailed discussion of the network design is provided. The effects of Prandtl number, radiation–conduction parameter (Stark number), thermal Grashof number, species Grashof number, Schmidt number, Darcy number and Forchheimer number on the transient dimensionless velocities (U, V), non-dimensional temperature (T) and dimensionless concentration function (C) are illustrated graphically. Additionally, we have computed plots of U, V, T, C versus time and average Nusselt number and Sherwood number versus X, Y coordinate, for various thermophysical parameters. The model finds applications in geological contamination, geothermal energy systems and radioactive waste-repository near-field thermo-geofluid mechanics.

A Study of Steady Buoyancy-Driven Dissipative Micropolar Free Convection Heat and Mass Transfer in a Darcian Porous Regime with Chemical Reaction

In the present paper we examine the steady double-diffusive free convective heat and mass transfer of a chemically-reacting micropolar fluid flowing through a Darcian porous regime adjacent to a vertical stretching plane. Viscous dissipation effects are included in the energy equation. Assuming incompressible, micro-isotropic fluid behaviour the transport equations are formulated in a two-dimensional coordinate system (x, y) using boundary-layer theory. The influence of the bulk porous medium retardation is modeled as a drag force term in the translational momentum equation. Transformations render the conservation equations into dimensionless form in terms of a single independent variable, η, transverse to the stretching surface. A simplified first order homogenous reaction model is also used to simulate chemical reaction in the flow. Using the finite element method solutions are generated for the angular velocity field, translational velocity field, temperature and species transfer fields. The effects of buoyancy, porous drag and chemical reaction rate are studied. Chemical reaction is shown to decelerate the flow and also micro-rotation values, in particular near the wall. Mass transfer is also decreased with increasing chemical reaction rate. Increasing Darcy number is shown to accelerate the flow. Applications of the study include cooling of electronic circuits, packed-bed chemical reactors and also the near field flows in radioactive waste geo-repositories.

Self-Peeling of Porous Nickel Foam from the Electrochemically Etched Porous Silicon

AbstractA low-doped p-type silicon wafer was wet-etched to form macropores in a high-aspect-ratio, straight and parallel manner along the Si (100) direction. It was then plated in aqueous alkaline solution containing Ni2+. Metallic Ni was rapidly deposited in the macropores on the sidewall surface without using a reducing agent or activation treatment at slightly elevated temperature. After being immersed for certain duration, the single crystalline Si of sidewalls was replaced by polycrystalline Ni while the initial porous structure was still maintained. When Ni became dominant in the entire porous regime, the porous film more than 200 μm thick was discovered to be able to peel off very easily from the Si substrate beneath. In this way, a piece of nickel foam with straight pores of very high aspect ratio is self-formed and self-peeled.

Predictive and Diagnostic Simulation of In Situ Electrical Heating in Contaminated, Low-Permeability Soils

By in situ heating, contaminants can be mobilized by vaporization, modified into a harmless chemical state, or entrapped by soil vitrification. However, effective heating may be difficult or impossible by depending solely upon conduction of heat externally applied to a contaminated target layer. Heating a layer internally by passing an electric current through it has been employed in several different in-situ remediation field experiments. The authors present a three-dimensional computer model of groundwater flow and transport in an electrically heated, partially saturated, porous regime. Simulations are used to show that the uniformity of heating can be enhanced by increasing the number of electrical phases employed in supplying current to the electrodes. However, a more serious concern for the field-scale application of this internal heating method is the nonuniform heating of the target layer because of large heating-rate differences between the near-electrode region and the center of the array. Finally, for a dissolved solvent like TCE, the authors show that ohmic heating is exceptionally effective at clearing the layer by vapor-phase partitioning. Once expelled from the low-permeability layer, the solvent that is not already thermally destroyed by a pyrolytic reaction is available for extraction. (A)

Non-Darcy effects on convective boundary layer flow past a semi-infinite vertical plate in saturated porous media

The composite effects of viscosity, porosity, buoyancy parameter, thermal conductivity ratio and non-Darcy effects of Brinkman friction and Forscheimmer quadratic drag on the mixed convection boundary layer flow past a semi-infinite plate in a fully-saturated porous regime are theoretically and numerically investigated using Keller’s implicit finite-difference technique and a double-shooting Runge-Kutta method. The Brinkman Forcheimer-extended Darcy model is implemented in the hydrodynamic boundary layer equation. The effects of the various non-dimensional thermofluid parameters, viz Grashof number, Darcy number, and Forchheimer number, and also porosity, thermal conductivity and viscosity parameters on the velocity and temperature fields are discussed. Computations for both numerical schemes are made where possible and found to be in excellent agreement.