Vertical Permeable Plate Research Articles

NUMERICAL AND ANALYTICAL SOLUTIONS FOR NATURAL CONVECTION FLOW WITH THERMAL RADIATION AND MASS TRANSFER PAST A MOVING VERTICAL POROUS PLATE BY DQM AND HAM

In this paper, the boundary-layer natural convection flow on a permeable vertical plate with thermal radiation and mass transfer is studied when the plate moves in its own plane. A uniform temperature with uniform species concentration at the plate is affected and the fluid is considered to be a gray, absorbing–emitting. A viscous flow model is presented using boundary-layer theory comprising the momentum, energy, and concentration equations, which is solved analytically by means of an excellent method called homotopy analysis method (HAM). First, a comparison between HAM results and those obtained by means of a higher-order numerical method, namely differential quadrature method (DQM), is done. Close agreement of two sets of results indicates the accuracy of the HAM. The velocity, temperature, and concentration distributions are displayed graphically, and a parametric study is performed in which the effect of various parameters on the skin friction, the local Nusselt number (Nn), and the local Sherwood number (Mu) are investigated.

Effect of Hall current on MHD natural convection flow from vertical permeable flat plate with uniform surface heat flux

The effect of the Hall current on the magnetohydrodynamic (MHD) natural convection flow from a vertical permeable flat plate with a uniform heat flux is analyzed in the presence of a transverse magnetic field. It is assumed that the induced magnetic field is negligible compared with the imposed magnetic field. The boundary layer equations are reduced to a suitable form by employing the free variable formulation (FVF) and the stream function formulation (SFF). The parabolic equations obtained from FVF are numerically integrated with the help of a straightforward finite difference method. Moreover, the nonsimilar system of equations obtained from SFF is solved by using a local nonsimilarity method, for the whole range of the local transpiration parameter ζ. Consideration is also given to the regions where the local transpiration parameter ζ is small or large enough. However, in these particular regions, solutions are acquired with the aid of a regular perturbation method. The effects of the magnetic field M and the Hall parameter m on the local skin friction coefficient and the local Nusselt number coefficient are graphically shown for smaller values of the Prandtl number Pr (= 0.005, 0.01, 0.05). Furthermore, the velocity and temperature profiles are also drawn from various values of the local transpiration parameter ζ.

Ingham problem for free convection near a continuously moving vertical permeable plate

Ingham problem for free convection near a continuously moving vertical permeable plate

Unsteady MHD free convection flow past a vertical permeable flat plate in a rotating frame of reference with constant heat source in a nanofluid

The unsteady magnetohydrodynamic flow of a nanofluid past an oscillatory moving vertical permeable semi-infinite flat plate with constant heat source in a rotating frame of reference is theoretically investigated. The velocity along the plate (slip velocity) is assumed to oscillate on time with a constant frequency. The analytical solutions of the boundary layer equations are assumed of oscillatory type and they are obtained by using the small perturbation approximations. The influence of various relevant physical characteristics are presented and discussed.

MHD flow of a micropolar fluid towards a vertical permeable plate with prescribed surface heat flux

The problem of a steady mixed convection stagnation point flow towards a permeable vertical plate with prescribed surface heat flux immersed in an incompressible micropolar fluid is studied numerically. The governing partial differential equations are first transformed into a system of ordinary differential equations using a similarity transformation, before being solved numerically by a finite-difference scheme known as the Keller-box method and the Runge–Kutta–Fehlberg method with shooting technique. The effects of the material parameter, buoyancy parameter, suction/injection parameter and the Prandtl number on the fluid flow and heat transfer characteristics are discussed. It is found that dual solutions exist for both assisting and opposing flows. The skin friction coefficient and the local Nusselt number increase in the presence of suction and magnetic field. Moreover, suction as well as fluids with larger Prandtl number widens the range of the buoyancy parameter for which the solution exists.

MHD FREE CONVECTION FLOW OF A NANOFLUID PAST A VERTICAL PLATE IN THE PRESENCE OF HEAT GENERATION OR ABSORPTION EFFECTS

This work is focused on the numerical solution of steady natural convection boundary-layer flow of a nanofluid consisting of a pure fluid with nanoparticles along a permeable vertical plate in the presence of magnetic field, heat generation or absorption, and suction or injection effects. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. The governing boundary-layer equations of the problem are formulated and transformed into a non-similar form. The obtained equations are then solved numerically by an efficient, iterative, tri-diagonal, implicit finite-difference method. Comparisons with previously published work are performed and are found to be in excellent agreement. Representative results for the longitudinal velocity, temperature, and nanoparticle volume fraction profiles as well as the local heat transfer rates for various values of the physical parameters are displayed in both graphical and tabular forms.

Effects of thermophoresis on hydromagnetic mixed convection and mass transfer flow past a vertical permeable plate with variable suction and thermal radiation

The present paper deals with the effect of surface mass transfer on MHD mixed convection flow past a heated vertical flat permeable surface in the presence of thermophoresis, radiative heat flux and heat source/sink. Usual similarity transformations are introduced to obtain similarity solution, using regular perturbation technique. To observe physical insight and interesting aspects of the problem in the presence of thermophoresis, the non-dimensional velocity, temperature and concentration field are numerically studied and displayed graphically for pertinent parameters. It is observed that the thermophoresis has dominant effect on mass transfer mechanism in particle deposition process. The model finds applications in studying particulate deposition on turbine blades, removing small particles from gas streams and determining chemical vapor deposition rate in chemical industries.

Mixed Convection Boundary Layer Flow over a Permeable Vertical Flat Plate Embedded in an Anisotropic Porous Medium

An analysis is performed to study the heat transfer characteristics of steady mixed convection flow over a permeable vertical flat plate embedded in an anisotropic fluid-saturated porous medium. The effects of uniform suction and injection on the flow field and heat transfer characteristics are numerically studied by employing an implicit finite difference Keller-box method. It is found that dual solutions exist for both assisting and opposing flows. The results indicate that suction delays the boundary layer separation, while injection accelerates it.

Perturbation analysis of unsteady magnetohydrodynamic convective heat and mass transfer in a boundary layer slip flow past a vertical permeable plate with thermal radiation and chemical reaction

An analytical study for the problem of unsteady mixed convection with thermal radiation and first-order chemical reaction on magnetohydrodynamics boundary layer flow of viscous, electrically conducting fluid past a vertical permeable plate has been presented. Slip boundary condition is applied at the porous interface. The classical model is used for studying the effect of radiation for optically thin media. The non-linear coupled partial differential equations are solved by perturbation technique. The results obtained show that the velocity, temperature and concentration fields are appreciably influenced by the presence of chemical reaction, thermal stratification and magnetic field. It is observed that the effect of thermal radiation and magnetic field decreases the velocity, temperature and concentration profiles in the boundary layer. Also, the effects of the various parameters on the skin-friction coefficient and the rate of heat transfer at the surface are discussed.

MHD mixed convection of a viscous dissipating fluid about a permeable vertical flat plate

The problem of steady laminar magnetohydrodynamic (MHD) mixed convection heat transfer about a vertical plate is studied numerically, taking into account the effects of Ohmic heating and viscous dissipation. A uniform magnetic field is applied perpendicular to the plate. The resulting governing equations are transformed into the non-similar boundary layer equations and solved using the Keller box method. Both the aiding-buoyancy mode and the opposing-buoyancy mode of the mixed convection are examined. The velocity and temperature profiles as well as the local skin friction and local heat transfer parameters are determined for different values of the governing parameters, mainly the magnetic parameter, the Richardson number, the Eckert number and the suction/injection parameter, fw. For some specific values of the governing parameters, the results agree very well with those available in the literature. Generally, it is determined that the local skin friction coefficient and the local heat transfer coefficient increase owing to suction of fluid, increasing the Richardson number, Ri (i.e. the mixed convection parameter) or decreasing the Eckert number. This trend reverses for blowing of fluid and decreasing the Richardson number or decreasing the Eckert number. It is disclosed that the value of Ri determines the effect of the magnetic parameter on the momentum and heat transfer.

Effect of a Uniform Horizontal Throughflow on the Darcy Free Convection over a Permeable Vertical Plate with Volumetric Heat Generation

This article deals with the steady Darcy free convection adjacent to a heated or cooled permeable vertical flat plate of constant temperature, which is embedded in a fluid-saturated porous medium of uniform ambient temperature T∞. There is a uniform horizontal throughflow of the fluid and a volumetric heat generation q′′′ takes place, which is considered to be a power-law function of the local temperature difference T − T∞, i.e., q′′′ ~ (T − T∞)n. To be specific, two cases of this type of volumetric heat generation are considered in the analysis in some detail, namely, the linear and the quadratic cases, n = 1 and n = 2, respectively.

Thermo-micropolar fluid flow along a vertical permeable plate with uniform surface heat flux in the presence of heat generation

A two-dimensional steady convective flow of thermo-micropolar fluid past a vertical permeable flat plate in the presence of heat generation with uniform surface heat flux has been analyzed numerically. The local similarity solutions for the flow, microrotation (angular velocity) and heat transfer characteristics are illustrated graphically for various material parameters entering into the problem. The effects of the pertinent parameters on the local skin friction coefficient, plate couple stress, and the rate of heat transfer are also calculated and displayed graphically. The results show that skin friction coefficient (viscous drag) and the rate of heat transfer (Nusselt number) in micropolar fluid are less compared to that of the Newtonian fluid.

Transient MHD Double‐Diffusive Natural Convection over a Vertical Surface Embedded in a Non‐Darcy Porous Medium

The problem of transient, laminar, MHD double‐diffusive free convection over a permeable vertical plate embedded in Darcy and non‐Darcy porous medium is numerically investigated. Nonsimilarity solutions are obtained for constant wall temperature and concentration with a specified power law of mass flux parameter. The effects of the magnetic parameter, the inertial coefficient, Lewis number, the buoyancy ratio, and the lateral mass flux on heat and mass transfer coefficients are presented and discussed.

Effects of free convection on the oscillatory flow of a polar fluid through a porous medium in the presence of variable wall heat flux

The purpose of this work is to study a laminar two-dimensional free convective oscillatory flow of an incompressible polar fluid through a saturated porous medium occupying a semi-infinite region of the space which is bounded by an infinite vertical permeable plate in the presence of oscillating suction and variable wall heat flux. The governing equations are based on the local volume averaging technique. The generalized Darcy equation accounting for polar effects is employed. Analytical expressions for the linear momentum, angular momentum, and energy fields are obtained by using the two-term perturbation technique. The numerically computed results are shown graphically and compared with the corresponding ones for a viscous (Newtonian) fluid. Distribution of the mean velocity of a polar fluid is found to increase as compared to the Newtonian fluid. The analysis reveals multiple boundary-layer structure for velocity.

Radiation effect on MHD mixed convection flow about a permeable vertical plate

This study investigates mixed convection heat transfer about a permeable vertical plate in the presence of magneto and thermal radiation effects. The effects of the mixed convection parameter, the radiation–conduction parameter, the surface temperature parameter, the magnetic parameter and the suction/injection parameter on the local skin friction and local heat transfer parameters are presented and analyzed.

The Entrainment Theorem for the Darcy free convection over a permeable vertical plate

The Cheng–Minkowycz model of the Darcy free convection boundary layer flow over a permeable vertical plate with prescribed power-law temperature distribution T w ( x ) = T ∞ + A ⋅ x λ and an applied lateral mass flux is revisited in this paper. The relationship between the wall heat flux and the entrainment velocity (the similar transversal velocity at the outer edge of the boundary layer) as function of the mass transfer parameter f w is examined analytically by using the Merkin transformation method. It is shown that at the value of f w where the Nusselt number becomes zero and changes sign, the entrainment velocity passes through its minimum value ( Entrainment Theorem). The converse statement is also true, and holds for all the surface temperature distributions with power-law exponent in the range − 1 < λ < 0 . It also applies to the Darcy free convection over a permeable vertical plate with exponential temperature distribution when the effect of viscous dissipation is significant.

Mixed convection heat and mass transfer of non‐Newtonian fluids from a permeable surface embedded in a porous medium

PurposeTo consider simultaneous heat and mass transfer by mixed convection for a non‐Newtonian power‐law fluid from a permeable vertical plate embedded in a fluid‐saturated porous medium in the presence of suction or injection and heat generation or absorption effects.Design/methodology/approachThe problem is formulated in terms of non‐similar equations. These equations are solved numerically by an efficient implicit, iterative, finite‐difference method.FindingsIt was found that as the buoyancy ratio was increased, both the local Nusselt and Sherwood numbers increased in the whole range of free and mixed convection regime while they remained constant for the forced‐convection regime. However, they decreased and then increased forming dips as the mixed‐convection parameter was increased from the free‐convection limit to the forced‐convection limit for both Newtonian and dilatant fluid situations.Research limitations/implicationsThe problem is limited to slow flow of non‐Newtonian power‐law fluids in porous media. Future research may consider inertia effects of porous media for relatively higher velocity flows.Practical implicationsA very useful source of information for researchers on the subject of non‐Newtonian fluids in porous media.Originality/valueThis paper illustrates simultaneous heat and mass transfer in porous media for power‐law fluids with heat generation or absorption effects.

Effect of Hall current on the MHD laminar natural convection flow from a vertical permeable flat plate with uniform surface temperature

The effect of Hall current on the steady, laminar, natural convection boundary layer flow of MHD viscous and incompressible fluid from a semi-infinite heated permeable vertical flat plate with an applied magnetic field transverse to it has been investigated, assuming that the induced magnetic field is negligible compared to the imposed magnetic field. With appropriate transformations the boundary layer equations are reduced to local nonsimilarity equations and the solutions are obtained employing four distinct methods, namely, (i) regular perturbation method for small transpiration parameter; (ii) asymptotic solutions for large transpiration rate; (iii) implicit finite difference method together with Keller-box scheme and the (iv) local nonsimilarity method for any transpiration rate. Effects of the magnetic field, M, and the Hall parameter, m, on the local skin friction and local rate of heat transfer groups are shown graphically for smaller values of the Prandtl number, Pr ( = 0.1 , 0.01 ) , that represent liquid metals.

Natural convection of air and water with variable thermophysical properties about a vertical isothermal flat plate embedded in a Darcy porous medium

We study the steady laminar flow along an isothermal vertical impermeable and permeable plate embedded in a saturated porous medium for which the Darcy law is valid. The investigation concerns air and water taking into account the variation of their physical properties with temperature in both upward and downward flow. The results are obtained with the direct numerical solution of the governing equations and cover relatively large temperature differences. The Prandtl number varied between 0.54 and 0.86 for air and between 3.83 and 13.18 for water. It is found that the variation of physical properties with temperature play an important role on wall heat transfer and wall shear stress and large departures from the constant properties values are observed. The largest differences of wall heat transfer and wall shear stress from the corresponding constant properties values reach 75% and 357% in air and 42% and 388% in water.

Free convection flow with thermal radiation and mass transfer past a moving vertical porous plate

The combined free convection boundary layer flow with thermal radiation and mass transfer past a permeable vertical plate is studied when the plate moves in its own plane. The plate is maintained at a uniform temperature with uniform species concentration and the fluid is considered to be gray, absorbing–emitting. The coupled unsteady non-linear momentum, energy and concentration equations governing the problem is obtained and made similar by introducing a time-dependent length scale. The similarity equations are solved numerically using superposition method. The resulting velocity, temperature and concentration distributions are shown graphically for different values of parameters entering into the problem. The numerical values of the local wall shear stress, local surface heat and mass flux are shown in tabular form.