Free Convective Flow

Singh and Singh(2010) presented the transient MHD free convective flow near a semi infinite vertical wall having ramped temperature. The effects of heat transfer and viscous dissipation on MHD free convection flow past an exponentially accelerated vertical plate with variable temperature was investigated by Kishore et al.(2010). The motivation of our present investigation is to study the unsteady free convective flow of a viscous incompressible fluid past an exponentially accelerated vertical plate with ramped wall heat flux. Initially, at time t  0 , the plate and the fluid are at the same constant temperature T  in a stationary condition. At time t > 0 , the

A theoretical analysis is performed to study the effects of Thermal Diffusion and Permeability of the medium on three dimensional MHD free convective flow through a porous plate. A Perturbation technique is used to solve the equations governing the flow. The solutions for velocity, temperature and concentration are obtained and also illustrated the results of flow characteristics for the fluid fields. The analysis is made and the results are summarized as follows:

Consider a steady laminar, two-dimensional free convective flow of chemically re- acting and viscous incompressible and electrically conducting along an inclined non- conducting plate kept at uniform temperature T w 0 . It is assumed that the x 0 -axis is along the plate, the y 0 -axis is normal to it, and the flux is uniformly in the y 0 direction. Initially, the fluid and the plate are assumed to have the same temperature, and for t 0 > 0, the plate temperature is raised to T w 0 , and the concentration level close to the plate is also raised to C w 0 . The physical model and the coordinate system are illustrated in Figure 1.

The unsteady free convective flow past a vertical porous plate with Newtonian heating has been studied. The governing equations have been solved numerically by Crank-Nicolson implicit finite-difference scheme. The variations of velocity and fluid temperature are presented graphically. It is found that the fluid velocity decreases with an increase in Prandtl number. Both the fluid velocity and the fluid temperature increase with an increase in suction parameter. An increase in Grashof number leads to rise in the fluid velocity. Further, it is observed that the shear stress and the rate of heat transfer at the plate increase with an increase in either Prandtl number or suction parameter or time.

Sastry and Rao [3] have focused the numerical solution of a micropolar fluid flow in a channel with porous walls. Further, Bhargava and Rani [4] have found the numerical solution on the heat transfer phenomenon of the microp- olar fluid flowing in a channel having porous walls. Agarwal and Dhanapal [5] have analyzed numerically the free convective flow of the micropolar fluid between two parallel porous vertical plates while Gorla et al. [6] have studied on mixed convective flow of a micropolar fluid. Kim [7] considered unsteady convection flow of micropolar fluids past a vertical porous medium. Srinivasacharya et al. [8] have investigated the unsteady Stokes flow of a micropolar fluid between two parallel porous plates. Kim [9] has studied the unsteady MHD convective flow of the polar fluids past a vertical moving porous plate in a porous medium. Further, Kim and Fedorov [10] have discussed on the transient mixed radiative convection of the micropolar fluid past a moving semi-infinite vertical porous plate while Bhargava et al. [11] have studied the numerical solution of MHD free convective flow of the micropolar fluid flow between two parallel porous vertical plates. Chamkha et al. [12] reported the solution of fully developed free con- vection of a micropolar fluid in a vertical channel.

Recently the study of free convective mass transfer effects are dominant features in many engineering applications such as rocket nozzles, cooling of nuclear reactors, high sinks in turbine blades high speed aircrafts and their atmospheric re-entry, chemical devices and process equipments. The unsteady free convective flow with radiation effect was investigated by Cogley [3]. Unsteady effect on MHD free convective and mass transfer flow through porous medium with constant suction and constant heat flux in rotating system studied by Sharma [4].But in all these papers thermal diffusion effects have been neglected, where as in a convective fluid when the flow of mass is caused by a temperature difference, thermal diffusion effects cannot be neglected. Gebhart and Mollendorf [5] have shown that when the temperature difference is small or in high prandtl number fluids or when the gravitational field is of high intensity viscous dissipative heat should be taken into account in free convection flow past a semi infinite vertical plate. In recent years progress has been considerably made in the study of heat and mass transfer in magneto hydrodynamic flows due to its application in many devices, like the MHD power generator and Hall accelerator. The influence of magnetic field on the flow of an electrically conducting viscous fluid with mass transfer and radiation absorption is also useful in planetary atmosphere research. Yih [6] numerically analyzed the effect of transportation velocity on the heat and mass transfer characteristics of mixed convection about a permeable vertical plate embedded in a

An analysis of the effect of viscous dissipation on a mag- netic hydrodynamic free convective flow past an infinite vertical porous plate has been carried out. In all the cases considered, the velocity was resolved in two components and the work was restricted to the laminar boundary layer. In this case the Grash of number, Gr > 0, implying that the temperature of the plate was greater than that of the fluid in the free stream region hence heat was transferred from the plate to the fluid which led to the convective cooling of the plate by free convection currents.

5. M.Umamaheswar, S.V.K.Varma and M.C.Raju, Unsteady MHD free convective visco-elastic fluid flow bounded by an infinite inclined porous plate in the presence of heat source, viscous dissipation and ohmic heating, International Journal of Advanced Science and Technology, 61 (2013) 39-52.

Here we investigate the two-dimensional free convection flow of a thermally stratified viscous fluid through a porous medium bounded by a heated vertical plate taking into account both t[r]

4.1.4. Effect of Soret number : The thermophoresis factor has been calculated from molecular interaction potentials derived from known molecular models. The ratio of diffusion coefficient is small compared to thermo diffusion coefficient, and hence the values of in our discussion is greater than one. Thus increase in thermo diffusion brings about increase in concentration. As in the case for thermal distribution, increase in Soret number decreases temperature constitution of the flow. It could be confirm from figure 9 that effect of Soret number on temperature is significant when and slightly significant when outside which the temperature becomes insignificant. Soret number, increases the velocity with maximum velocity in the body of the fluid close to the surface as shown in figure 15. The maximum for each of Soret number tilt to the right of as Soret number increases.

Hall effect on free convective flow in a rotating fluid. It was observed by Agarwal et al. [7] that the primary and secondary shear-stresses increases and decreases, respectively, with the increase in magnetic field and Hall parameters. Mazumdar et al. [8] worked on flow with heat transfer in the hydrodynamic Ekman layer on a porous plate with Hall effects. Further, Jaimala et al. [9] analysed the effect of magnetic field and Hall current on an electrically conducting couple-stress fluid layer heated from below; and they concluded that in the presence or absence of Hall current, magnetic field has a stabilizing effect on the thermal convection. In 2014, Reddy [10] worked on heat and mass transfer effects on unsteady MHD radiative flow of a chemically reacting fluid past an impulsively started vertical plate. He [10] observed that the presence of chemical reaction retards the fluid velocity, decreases the concentration and increases the thermal boundary layer thickness. Recently Mahmoudpour Molaei et al. [11] analysed the MHD free convection flow of a non-newtonian power-law fluid over a vertical plate with suction effects.

T hermal radiation effects on unsteady free convective flow of a viscous incompressible flow past an exponentially accelerated infinite vertical plate with variable temperature and mass diffusion, in the presence of magnetics field has been considered. The fluid considered here is a gray, absorbing-emitting radiation but a non-scattering medium. The plate temperature is raised linearly with time and the concentration level near the plate is also raised linearly with time . An exact solution to the dimensionless governing equations has been obtained by the Laplace transform method, when the plate is exponentially accelerated with a velocity u = u 0 exp ( a ′ t ′ ) in its own plane against gravitational field.

However when the strength of magnetic field is very strong, One can not neglect the effect of Hall current. Also the rotating flows of viscous, incompressible and electrically conducting fluid have attracted attention of investigators due to their abundant geophysical and astrophysical applications. It is also important in the solar physics dealing with the sunspot development, the solar cycle and the structure of rotating magnetic stars. It is well known that a number of astronomical bodies possess fluid interiors and magnetic fields. Many scholars have studied such model for instance Mazumdar et al. [20] studied the hydrodynamic study flow with the effect of hall current. Agarwal et al. [21] analysed the combined influence of dissipation and Hall Effect on free convective flow in a rotating fluid and they analysed that the primary shear-stress increases and secondary shear-stress decreases with increase in magnetic and Hall parameters.

Soundalgekar and Gupta [8] have studied free convection effects on the flow past an accelerated vertical plate. Raptis and Singh [9] have analyzed MHD free convection flow past an accelerated vertical plate. Muthucumaraswamy and Visalakshi [10] have considered thermal radiation effects on the unsteady free convective flow of a viscous incompressible flow past an exponentially accelerated infinite vertical plate with variable temperature and uniform mass diffusion. A radiation effect on an exponentially accelerated vertical plate with a uniform mass diffusion was studied by Sathappan and Muthucumaraswamy [11]. Singh and Kumar [12] have investigated analytically the free- convection flow of an incompressible and viscous fluid past an exponentially accelerated infinite vertical plate. Rajput and Kumar [13] analyzed with the Laplace- transform technique the rotation and radiation effects on a MHD free convection flow past an impulsively started vertical plate with variable temperature. Hussain and Takhar [14] have considered radiation effects on mixed convection along a vertical plate with uniform surface temperature. Effects of radiation in and optically thin gray gas flowing past a vertical infinite plate in the presence of magnetic field was studied by Rapits et al [15]. Raju et al. [16], studied radiation and mass transfer effects on a free convection flow through a porous medium bounded by a vertical surface.

We consider an unsteady hydromagnetic, chemically reacting free convective flow of an incompressible and electrically conducting fluid past an infinite vertical porous plate in the presence of constant suction and heat absorbing sink. Let x′ - axis be taken in the vertically upward direction along the infinite vertical plate and y′ - axis normal to it. The magnetic field of uniform strength is applied and induced magnetic field is neglected. Boussineq’s approximation within the boundary layer, the governing equations of continuity, momentum, energy and diffusion are as follows: Continuity equation

investigated the effects of simultaneous heat and mass transfer, thermal radiation and Hall currents on the unsteady free convective flow over a vertical porous plate; [11] studied the effects of chemical reaction, convection, thermal radiation, cross-diffusion and magnetic field on the transient flow over a porous vertical plate. With the effects of magnetic field in view, for the moderate or less interactive flow, [12-22] studied the free convective flow over porous plates in the presence of suction, thermal radiation, chemical reaction rate and viscous dissipation. Importantly, for a partial in-depth review of literature, [23] studied the unsteady natural convective flow over a porous vertical plate using numerical approach, and noticed that the velocity is increased by the convective currents and Darcy number but is decreased by the magnetic field parameter, Prandtl and Schmidt numbers. [24] investigated the effect of chemical reaction rate in a transient free convective flow over a vertical plate in the presence of oscillating temperature and variable suction, and found that the magnetic field parameter increases the skin friction; the PrandtL number decreases the velocity, temperature, skin friction and Nusett number; the Schmidt number decreases the velocity, concentration, skin friction and Sherwood number; the convective currents increase the skin friction; chemical reaction rate decreases the velocity. [25] examined the unsteady mixed convective flow over a porous plate using similarity transformation and a numerical approach, and saw that the velocity is increased by the convective currents but it is decreased by the suction and magnetic field parameters; the heat source source/sink parameter increases the temperature and skin friction.

In view of several industrial and technological importances, Pattabhi Ramacharyulu [6] studied the problem of the exact solutions of two dimensional flows of a second order incompressible fluid by considering the rigid boundaries. Later, Lekoudis et.al [7] presented a linear analysis of the compressible boundary layer flow over a wall. Subsequently, Shankar and Sinha [8] studied the problem of Rayleigh for wavy wall. The effect of small amplitude wall waviness upon the stability of the laminar boundary layer had been studied by Lessen and Gangwani [9]. Ramana Murthy et.al [10] discussed the flow of an elastico viscous fluid past an infinite plate with variable suction wherein the effects of various flow entities have been discussed. Further, the problem of free convective heat transfer in a viscous incompressible fluid confined between vertical wavy wall and a vertical flat wall was examined by Vajravelu and Shastri [11] and thereafter by Das and Ahmed [12]. The free convective flow of a viscous incompressible fluid in porous medium between two long vertical wavy walls was investigated by Patidar and Purohit [13]. Rajeev Taneja and Jain [14] had examined the problem of MHD flow with slip effects and temperature dependent heat source in a viscous incompressible fluid confined between a long vertical wall and a parallel flat plate. Recently, Ramana Murthy et.al [15] studied on the class of exact solutions of an incompressible second order fluid flow by creating sinusoidal disturbances, where different situations and effects have been examined.

channel with permeable boundaries. It is disclose that the fluid velocity is reduced by magnetic field and wall slip. Kumar et al. [34] studied a total dispersion tensor in two dimensional packed beds consisting of randomly placed parallel cylinders for porosities between 38% to 90% Pecklet numbers up to 100 and Reynolds numbers up to 20 based on the cylindrical diameter. Mahdy [42] investigated the magneto-hydrodynamic (MHD) free convection flow of a non-Newtonian power- law fluid over a vertical wavy surface with a uniform free-stream of constant velocity and temperature. Asadi et al. [37] developed rising of a single bubble in a quiescent liquid which under microgravity condition was simulated. They related to unsteady incompressible full Navier- Stokes equations which were solved using a conventional finite difference method with a structured staggered grid. Kumar et al. [38] studied the distribution of transverse velocity which was not symmetrical and for non- Newtonian fluid, large recirculation occured at upper disc in comparison with the recirculation at lower disc on increasing value of Reynolds number. Makinde et al. [39] developed a steady, axi-symmetric, magneto hydrodynamic (MHD) flow of a viscous, Newtonian, incompressible, electrically- conducting fluid through an isotropic, homogenous porous medium located in the annular zone between two concentric rotating cylinders in the presence of a radial magnetic field. Khedr et al.[40] considered a steady, laminar, MHD flow of a micro polar fluid past a stretched semi-infinite, vertical and permeable surface in the presence of temperature dependent heat generation or absorption, magnetic field and thermal radiation effects. Kumar et al. [41] studied a MHD three dimensional free convective flow of viscous incompressible fluid through a porous medium. They are solutions of governing equations obtained by Finite difference technique.

fluid between parallel vertical plates. An exact solution is obtained for the fluid velocity and temperature distribution. Rushikumar et.al (2012) analyzed the influence of heat and mass transfer characteristics of two dimensional steady laminar free convective flow of a viscous incompressible fluid between two parallel porous walls. Lingaraju et.al (2013) discussed the unsteady two layered fluid flow and heat transfer of conducting fluids in a channel between parallel porous plates. The unsteady two dimensional laminar flow of viscous fluid between two parallel porous plates was analyzed by Ganesh et.al (2014). The problem was reduced to a third order nonlinear differential equation depending on a suction Reynolds number and Hartmann number.

Abstract: Investigation was carried out on steady two-dimensional free convective flow and heat transfer of a boundary layer hydromagnetic flow over a stretching porous vertical sheet embedded in an expanse of an incompressible, and electrically conducting fluid awash of uniform transverse magnetic field in the presence of combined influences of stress work, exponentially decaying heat generation, Soret and Dufour alongside the suction/ injection. It was assumed that the thermal conductivity of the fluid was a linear function of temperature. Using appropriate similarity variables, the governing non-linear PDEs are transformed into a set of ODEs together with the considered BCs, which were solved numerically by shooting iteration method coupled with the fourth order classical Runge-Kutta integration scheme through a reliable computation software package. A comparison with previously published results on the similar special cases was accessed, and the results were found to be in perfect agreement. The effects of Prandtl number, Eckert number, buoyancy force, Lorentz force, the thermal heat generation, thermal-diffusion, diffusion-thermo, velocity ratio and mass blowing/fluid suction on the fluid behavior were delineated in physical terms. Finally, numerical values of pertinent physical quantities, such as the skin-friction coefficient, the local Nusselt and Sherwood numbers were sorted out and presented in tabular form while those of the dimensionless fluid velocity, temperature and concentration were addressed and discussed by graphs. Our findings reveal that all the basic emerging flow parameters significantly influence the heat and mass characteristics of the flow.