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.     

Couple stress effects on the MHD oscillatory flow in a fluid-saturated porous layer

In this paper, the oscillatory flow of hydromagnetic couple stress fluid-saturated porous layer with inhomogeneous wall temperatures is studied. The flow is modeled using the modified Darcy equation. The fluid is subjected to a transverse magnetic field and the velocity slip at the lower plate is taken into deliberation. The governing coupled partial differential equations of the flow are transformed to coupled ordinary differential equations and are solved analytically. The impact of the physical parameters such as the Grashof number, Prandtl number, Darcy number, Hartmann number, and couple stress parameters on velocity profiles, temperature, rate of heat transfer, and skin friction are emphasized. The velocity field increased as either the Grashof number, the Darcy number, the suction/injection parameter, and Prandtl number increased nevertheless reverse growth can be seen by increasing the Hartmann number and the couple stress parameter. The temperature field in the channel increases with increasing the suction/injection parameter and Prandtl number but a conflicting development can be seen with increasing the oscillation amplitude. It is interesting to note that skin friction increases on both channel plates as injection increases on the heated plate.     

Effects of non-Darcian on forced convection heat transfer over a flat plate in a porous medium-with temperature dependent viscosity

The non-Darcian effect on forced convection heat transfer over a flat plate in a porous medium is examined. The fluid viscosity is assumed to vary as an inverse linear function of temperature. The effects of inertia forces and the distance from the leading edge of the plate on the velocity and temperature fields as well as on the skin friction and heat transfer coefficients in the boundary layer over a semi-infinite plate are studied. The nonlinear boundary layer equations, governing the problem under consideration, are solved numerically by applying an efficient numerical technique based on the Keller box method. The velocity profiles, temperature profiles and the skin friction components on the plate are computed and discussed in detail numerically for various values of the variable viscosity parameter, the modified Reynolds number, the stream wise coordinate and the Prandtl number.     

Numerical solution for natural convection flow near a vertical porous plate having convective boundary condition with nonlinear thermal radiation

This article presents the theoretical study of the effects of suction/injection and nonlinear thermal radiation on boundary layer flow near a vertical porous plate. The importance of the convective boundary condition as regards the heat transfer rate is taken into account. The coupled nonlinear boundary layer equations are translated into a set of ordinary differential equations via a similarity transformation. The consequences of the active parameters like the suction parameter, injection parameter, convective heat transfer parameter, nonlinear thermal radiation parameters, and Grashof number dictating the flow transport are examined. The numerical result obtained shows that with suction/injection, the heat transfer rate could be increased with nonlinear thermal radiation parameter augment whereas decays with the convective heat transfer parameter and Grashof number. In the presence of suction/injection, the wall shear stress generally increases with nonlinear thermal radiation parameter, convective heat transfer parameter, and Grashof number. The suction has an increasing effect on Nusselt number and shear stress whereas a decreasing effect on Nusselt number and skin friction is seen with injection augment. The nonlinear thermal radiation is an increasing function of the temperature gradient far away from the plate whereas a decreasing function near the porous plate.     

Hydromagnetic flow and heat transfer past a continuously moving porous boundary

The effect of magnetic field on the flow and heat transfer past a continuously moving porous plate in a stationary fluid has been analysed. The governing boundary layer equations have been reduced to a set of nonlinear ordinary differential equations using similarity transformations. The resulting boundary value problem has been solved numerically. The effects of magnetic and suction (or injection) parameters on the velocity and temperature profiles as well as on the skin friction and heat transfer coefficients have been studied. It has been observed that the effect of magnetic field is to increase the wall skin friction while the reverse occurs in the case of Nusselt number.     

Mixed convection plume about a line heat source in micropolar fluid

This paper analyses the flow and heat transfer characteristics of the mixed convection in the boundary layer flow of micropolar fluids about a line heat source embedded on the edge of a plate. The dimensionless forms of boundary layer equations and their associated boundary conditions have been derived and investigated numerically in order to characterize the behaviors of the mixed convection wall plume. The numerical results have been obtained using the method of cubic spline collocation and the finite difference scheme. The micropolar parameter reduces the velocity but increases the temperature in the boundary layer, whereas the effects of buoyancy parameter trend conversely. Furthermore, the micropolar parameter decreases the skin friction parameter and the wall couple stress but increases the wall temperature, whereas the effects of buoyancy parameter trend conversely. Finally, the higher the value of Prandtl number, the greater the skin friction parameter, the wall couple stress and the wall temperature.     

Free convection of composite porous medium in a vertical channel

A fully developed free convection flow of immiscible fluids in a vertical channel filled with a porous medium is analyzed in the presence of source/sink. The flow is modeled using the Darcy–Brinkman–Forchheimer equation model. The viscous and Darcy dissipation terms are included in the energy equation. The channel walls are maintained at two different constant temperatures. The transport properties of both fluids are assumed to be constant. Continuous conditions for velocity, temperature, shear stress, and heat flux of both fluids at the interface are employed. The resulting coupled nonlinear equations are solved analytically using regular perturbation method and numerically using finite difference method. The velocity and temperature profiles are obtained in terms of porous parameter, Grashof number, viscosity ratio, width ratio, conductivity ratio, and heat generation or heat absorption coefficient. It is found that the presence of porous matrix and heat absorption reduces the flow field. © 2011 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.20340     

Numerical study of Williamson fluid flow and heat transfer over a permeable stretching cylinder with the effects of Joule heating and heat generation/absorption

The present study aims to discuss the Williamson fluid flow and heat transfer across a permeable stretching cylinder with heat generation/absorption effects. The effects of viscous dissipation, Joule heating, and magnetic field are also taken into account. The BVP-4C numerical solver in MATLAB is adopted for all the numerical simulations in the present study. For this, the modeled partial differential equations are translated into dimensionless ordinary differential equations using some well-developed similarity transformations. A good agreement between the numerical results of the present study and existing literature is exhibited. The dimensionless physical parameters being investigated are Reynolds number, magnetic field parameter, suction parameter, heat source/sink parameter, Williamson fluid parameter, and mixed convection parameter. The numerical calculations are also performed for the skin friction coefficient and local Nusselt number to get an understanding of the shear stress rate and heat transfer rate, respectively. Furthermore, the impact of all these physical parameters on the velocity and temperature profiles is investigated and represented throughout the literature.     

An exact analysis of radiative heat transfer and unsteady MHD convective flow of a second‐grade fluid with ramped wall motion and temperature

The influence of simultaneously applied ramped boundary conditions on unsteady magnetohydrodynamic natural convective motion of a second‐grade fluid is investigated and analyzed in this study. The motion of the fluid is considered near an infinite upright plate that is nested in a porous medium subject to nonlinear thermal radiation effects. The Laplace transformation technique is utilized to acquire the exact solutions of momentum and energy equations. To effectively examine the rate of heat transfer and shear stress, the Nusselt number and skin friction coefficient are also established. The outcomes of mathematical computations are elucidated through tables and figures to highlight some physical aspects of the problem. Some limiting models of the present problem are also deduced and presented. On comparison, it is observed that the fluid exhibits lower temperature and velocity profiles under ramped boundary conditions. It is also found that wall shear stress can be controlled by choosing large values of the magnetic parameter (M) and Prandtl number (Pr). In addition, the heat transfer rate specifies inverse trends for growing values of radiation parameter (Nr) and Prandtl number (Pr), while it increases rapidly under a ramped surface condition and decreases slowly under a constant surface condition.     

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.     

Transient MHD convective flow of a micropolar fluid past a moving vertical plate in the presence of thermal diffusion

In this article, we investigate a transient magnetohydrodynamic convective micropolar fluid flow over a semi-infinite vertical plate embedded in a porous medium in the presence of chemical reaction and thermal diffusion. The dimensionless governing equations are solved by adopting the regular perturbation technique. The impact of various parameters on the velocity, microrotation, temperature, concentration profiles, skin friction, Sherwood number, and Nusselt number over the boundary layer is analyzed using graphs. The fluid velocity and microrotation reduce under the effect of thermal diffusion and chemical reaction. Furthermore, concentration rises due to thermal diffusion (Soret) effect, but concentration falls under the effect of chemical reaction. It is found that the velocity and skin friction fall with enhancing value of magnetic parameter. But Sherwood number increases as the magnetic parameter increase.     

Non-Darcy natural convection of a non-Newtonian fluid in a porous cavity

A numerical study of non-Darcy natural convection in a porous enclosure saturated with a power-law fluid is presented. Hydrodynamic and heat transfer results are reported for the configuration in which the enclosure is heated from a side-wall while the horizontal walls are insulated. The flow in the porous medium is modeled using the modified Brinkman–Forchheimer-extended Darcy model for power-law fluids, which accounts for both inertia and boundary effects. The results indicate that when the power law index is decreased, the circulation within the enclosure increases leading to a higher Nusselt number and these effects are enhanced as the Darcy number is increased. Consequently as the power law index decreases, the onset of the transitions from Darcy regime to Darcy–Forchheimer–Brinkman regime to asymptotic convection (boundary layer) regime shift to higher corresponding values of the Darcy number. An increase in Rayleigh number produces similar effects as a decrease in power law index.     

Unsteady boundary-layer flow and heat transfer of a nanofluid over a permeable stretching/shrinking sheet

The unsteady boundary layer flow of a nanofluid over a permeable stretching/shrinking sheet is theoretically studied. The governing partial differential equations are transformed into ordinary ones using a similarity transformation, before being solved numerically. The results are obtained for the skin friction coefficient, the local Nusselt number and the local Sherwood number as well as the velocity, temperature and the nanoparticle fraction profiles for some values of the governing parameters, namely, the unsteadiness parameter, the mass suction parameter, the Brownian motion parameter, the thermophoresis parameter, Prandtl number, Lewis number and the stretching/shrinking parameter. It is found that dual solutions exist for both stretching and shrinking cases. The results also indicate that both unsteadiness and mass suction widen the range of the stretching/shrinking parameter for which the solution exists.     

Effect of uniform suction on laminar filmwise condensation on a finite-size horizontal flat surface in a porous medium

The theoretical analysis of filmwise condensation outside a finite-size horizontal flat surface embedded in a porous medium filled with a dry saturated vapor has been solved by a boundary layer treatment. The Newton-Raphson scheme was employed to solve the finite-size horizontal flat plate in porous medium. Results turns out that the average Nusselt number for condensation heat transfer is expressed in terms of Darcy number, Jakob number, film liquid Prandtl number, Darcy-modified Rayleigh number and the parameter of suction, as well as are given for the condensate layer thickness profiles.     

Effect of radiation on natural convection flow on a sphere in presence of heat generation

Effect of radiation on natural convection flow around a sphere in presence of heat generation has been investigated. The governing equations are transformed into dimensionless non-similar equations by using a set of suitable transformations and solved numerically by the finite difference method along with Newton's linearization approximation. We have focused our attention on the evaluation of velocity profiles, temperature profiles, shear stress in terms of local skin friction and the rate of heat transfer in terms of local Nusselt number for different values of heat generation parameter, radiation parameter and the Prandlt number and the numerical results have been shown graphically.     

A study on entropy generation and heat transfer in a magnetohydrodynamic flow of a couple-stress fluid through a thermal nonequilibrium vertical porous channel

The effect of local thermal nonequilibrium (LTNE) on the entropy generation and heat transfer characteristics in the magnetohydrodynamic flow of a couple-stress fluid through a high-porosity vertical channel is studied numerically using the higher-order Galerkin technique. The Boussinesq approximation is assumed to be valid and the porous medium is considered to be isotropic and homogeneous. Two energy equations are considered one each for solid and fluid phases. The term involving the heat transfer coefficient in both equations renders them mutually coupled. Thermal radiation and an internal heat source are considered only in the fluid phase. The influence of inverse Darcy number, Hartmann number, couple-stress fluid parameter, Grashof number, thermal radiation parameter, and interphase heat transfer coefficient on velocity and temperature profiles is depicted graphically and discussed. The entropy generation, friction factor, and Nusselt number are determined, and outcomes are presented via plots. The effect of LTNE on the temperature profile is found to cease when the value of the interphase heat transfer coefficient is high, and in this case, we get the temperature profiles of fluid and solid phases are uniform. The physical significance of LTNE is discussed in detail for different parameters' values. It is found that heat transport and friction drag are maximum in the case of LTNE and minimum in the case of local thermal equilibrium. We observe that LTNE opposes the irreversibility of the system. The corresponding results of a fluid-saturated densely packed porous medium can be obtained as a limiting case of the current study.     

Convective heat transfer over a heated square porous cylinder in a channel

A numerical study is made of the unsteady flow and convection heat transfer for a heated square porous cylinder in a channel. The general Darcy–Brinkman–Forchheimer model is adopted for the porous region. The parameters studies including porosity, Darcy number, and Reynolds number on heat transfer performance have been explored in detail. The results indicate that the average local Nusselt number is augmented as the Darcy number increases. The average local Nusselt number increases as Reynolds number increases; in particular, the increase is more obvious at a higher Darcy number. In contrast, the porosity has slight influence on heat transfer.     

Influence of Viscous Dissipation on non‐Darcy Mixed Convection Flow of Nanofluid

In this paper, the steady fully developed non‐Darcy mixed convection flow of a nanofluid in a vertical channel filled with a porous medium with different viscous dissipation models is analyzed. The Brinkman‐Forchheimer extended Darcy model is used to describe the fluid flow pattern in the channel. The transport equations for a nanofluid are solved analytically using the seminumerical‐analytical method known as differential transformation method, and numerically with the Runge‐Kutta shooting method. Finally, the influence of pertinent parameters, such as solid volume fraction, different nanoparticles, mixed convection parameter, Brinkman number, Darcy number, and inertial parameter on the velocity and temperature fields are shown graphically. The results show that velocity and temperature are enhanced when the mixed convection parameter, Brinkman number, and Darcy number increases whereas solid volume fraction and inertial parameter decreases the velocity and temperature fields. The obtained results show that the nanofluid enhances the heat transfer process significantly.     

Laminar film condensation on an elliptical tube embedded in porous media

Laminar film condensation of saturated vapor flowing over an isothermal elliptical tube embedded in a porous medium is analyzed for conditions of free and forced convection. The flow field in the porous medium is described by the Darcy–Brinkman–Forchheimer model. The effect of vapor shear on condensation is determined by simultaneous solution of the two phase vapor boundary layer and condensate film momentum equations. The numerical results, which are presented in the form of local film thickness and local Nusselt number, show a dependence of these physical parameters on practical dimensionless parameters such as Reynolds number, Darcy number, Bond number and eccentricity.     

Heat and mass transfer analysis for boundary layer stagnation-point flow towards a heated porous stretching sheet with heat absorption/generation and suction/blowing

A similarity analysis is performed to investigate the structure of the boundary layer stagnation-point flow and heat transfer over a stretching sheet in a porous medium subject to suction/blowing and in the presence of internal heat generation/absorption. A scaling group of transformations is applied to get the invariants. Using the invariants, a third and a second order ordinary differential equations corresponding to the momentum and energy equations are obtained respectively. Boundary layer velocity and temperature profiles are determined numerically for various values of the ratio of free stream velocity and stretching velocity, the permeability parameter, suction/blowing parameter, heat source/sink parameter, Prandtl number. It is found that the horizontal velocity increases with the increasing value of the ratio of the free stream velocity (ax) and the stretching velocity (cx). The temperature decreases in this case. At a particular point of the porous stretching sheet, the non-dimensional fluid velocity decreases with the increase of the permeability of the porous medium and also with the increasing suction parameter when the free stream velocity is less than stretching velocity whereas fluid velocity increases with the increasing injection parameter. But when the free stream velocity is greater than the stretching velocity the opposite behaviour of horizontal velocity is noticed. The dimensionless temperature at a point of the sheet decreases due to suction but increases due to injection. The temperature at a point is found to decrease with the increasing Prandtl number.