Comparison of the current and recent numerical values of the radial and tangential skin-friction coefficients and the rate of heat transfer coefficient for various of with .
In general, the thermal radiation and thermal diffusion effects over an electrically conducting, Newtonian fluid in a steady laminar magnetohydrodynamic convective flow over a porous rotating infinite disk with the consideration of heat and mass transfer in the presence of Soret and Dufour’s diffusion effects have been obtained and studied numerically. The governing continuity, momentum, energy and concentration equations are converted into a system of non-linear ordinary differential equations by means of similarity transformation. The resulting system of coupled non-linear ordinary differential equations is solved numerically. In this chapter, numerical results were presented for velocity (radial, axial and tangential), temperature, concentration and pressure profiles for different parameters of the problem Also, the effects of the pertinent parameters on the radial and tangential skin friction, the rate of heat and mass transfer are obtained and discussed numerically and illustrated graphically.
- heat and mass transfer
- thermal radiation
- slip flow
- porous medium
- rotating disk
The flow due to rotating disks is of great interest in many practical and engineering aspects. Rotating disk flows of electrically conducting fluids have practical applications in many areas, such as rotating machinery, lubrication, oceanography, computer storage devices, viscometer and crystal growth processes etc. Also, the study is interesting from the mathematical point of view. During the last two decades, research on renewable energy sources, as for example, solar, wind energy or energy from hydro-power and the preparation of oxygenated additives to blend diesel fuel, has been intensified.
Pioneering study of fluid flow due to an infinite rotating disk was carried by authors [1, 2, 3]. Chemical reactions usually accompany a large amount of exothermic and endothermic reactions. These characteristics can be easily seen in a lot of industrial processes, it has been realized that it is not always permissible to neglect the convection effects in porous constructed chemical reactors . The reaction produced in a porous medium was extraordinarily in common, such as the topic of PEM fuel cells modules and the polluted underground water because of discharging the toxic substance, etc.
Fourier’s law, for instance, described the relation between energy flux and temperature gradient. In other aspects, Fick’s law was determined by the correlation of mass flux and concentration gradient. Moreover, it was found that energy flux can also be generated by composition gradients, pressure gradients, or body forces. The energy flux caused by a composition gradient was discovered in 1873 by Dufour and was correspondingly referred to the Dufour effect.
It was also called the diffusion-thermo effect. On the other hand, mass flux can also be created by a temperature gradient, as was established by Soret. This is the thermal-diffusion effect. In general, the thermal-diffusion and the diffusion-thermo effects were of a smaller order of magnitude than the effects described by Fourier’s or Fick’s law and were often neglected in heat and mass transfer processes. There were still some exceptional conditions. The thermal-diffusion effect has been utilized for isotope separation and in mixtures between gases with very light molecular weight and of medium molecular weight , the diffusion-thermo effect was found to be of a magnitude such that it may not be neglected in certain conditions .
The first traceable interest in magnetohydrodynamics (MHD) flow was in 1907, when Northrop built an MHD pump prototype [6, 7]. Since then, analysis of the effects of both rotation and magnetic fields on fluid flows has been an active area of research. While technology expanded in many directions, the subject of MHD has developed in the use of magnetic fields and the range of fluid and thermal processes by [8, 9, 10, 11, 12, 13]. This study considers the effect of slip as a result of rarefied effect, a type of flow commonly encountered in many engineering tasks such as high altitude flight, micro-machines, vacuum technology, aerosol reactors, etc. In this study, the slip and no-slip regimes that lie in the range 0.1 > > 0 are considered. A completely different extension of von Karman’s one-disk problem is the analysis of Sparrow et al. . They considered the flow of a Newtonian fluid due to the rotation of a porous-surfaced disk and for that purpose replaced the conventional no-slip boundary conditions at the disk surface with a set of linear slip flow conditions. A substantial reduction in torque then occurred as a result of surface slip. Recently Frusteri and Osalusi  studied the effects of variable properties on MHD and slip flow over a porous rotating disk.
In all these studies Soret and Dufour effects were assumed to be negligible. Such effects are significant when density differences exist in the flow regime. For example, when species are introduced at a surface in fluid domain, with different (lower) density than the surrounding fluid, both Soret (thermo-diffusion) and Dufour (diffusion-thermal) effects can be influential. An analytical study of convection along a horizontal cylinder for a Helium-air system was reported subsequently by Sparrow et al. . In view of the importance of above mentioned effects, Maleque  studied Soret effect on convective heat and mass transfer past a rotating porous disk and he neglected the Dufour effect. Ahmed  investigated the Dufour and Soret effects on free convective heat and mass transfer over a stretching surface considering suction or injection. Recently, numerical study of free convection magnetohydrodynamic heat and mass transfer due to a stretching surface under saturated porous medium with Soret and Dufour effects was also discussed by Anwar Beg et al. .
In these two papers [15, 20] they have studied the effect of the magnetic field on the equations of motion that I found them used in the cylindrical coordinate, although the magnetic field parameters are in the Cartesian coordinate, which is wrong.
2. Mathematical formulation
Consider the steady, axially symmetric, incompressible flow of an electrically conducting fluid with heat and mass transfer flow due to a rotating porous disk in the presence of radiation has been considered. Assume that the fluid is infinite in extent in the positive z-direction. Let be the set of cylindrical polar coordinates and let the disk rotate with constant angular velocity and is placed at . The components of the flow velocity are in the directions of increasing , respectively. , are the pressure, the temperature and the concentration distribution, respectively. The surface of the rotating disk is maintained at a uniform temperature and uniform concentration . Far away from the surface, the free stream is kept at a constant temperature , at a constant concentration and at a constant pressure . The fluid is assumed to be gray, emitting and absorbing heat, but not scattering medium and is assumed to be Newtonian. The physical model and geometrical coordinates are shown in Figure 1.
The MHD body forces the Maxwell’s equations:
where is the electric current density, is the total magnetic field, is the magnetic permeability and is the induced magnetic field. The external uniform magnetic field is imposed in the direction normal to the surface of the disk which is assumed unchanged by taking small magnetic Reynolds number, so that the flow induction distortion of the applied magnetic field can be neglected as in the case with most of conducting fluids. In addition, a uniform suction is applied at the surface of the disk for the entire range.
The magnetic body force takes the form , therefore, , where is the electrical conductivity of fluid and is velocity vector and . The Lorentz force (MHD body force) has two components:
Under these assumptions, the governing equations for the continuity, momentum, energy and concentration in laminar incompressible flow can be written as follows:
When the mean free path of the fluid particle is comparable to the characteristic dimensions of the flow field domain, Naiver-Stokes equations break down since the assumption of continuum media fails. In the range of Knudsen Number, the higher order continuum equation, for example, Burnett equation should be used. For the range of , no slip boundary conditions cannot be used and should be replaced with the following expression (Gad-el-Hak ):
Where is the tangential velocity, is the normal direction to the wall, is the tangent momentum accommodation coefficient and is the mean free path. For , the no-slip boundary condition is valid, therefore, the velocity at the surface is equal to zero. In this study the slip and the no-slip regimes of the Knudsen number that lies in the range is considered. By using Eq. (7), the appropriate boundary conditions for the flow induced by an infinite disk which rotates with constant angular velocity subjected to uniform suction through the disk are given by
where is the thermal conductivity, is the kinematic viscosity of the ambient fluid, is the electrical conductivity, is the permeability of the porous medium, is the gravitational acceleration, are the expansion coefficients of temperature and concentration, are the density, dynamic viscosity and the specific heat at constant pressure, respectively, is the volumetric heat generation/absorption rate, is the molecular diffusion coefficient, are the thermal-diffusion rate, concentration susceptibility, the mean fluid temperature and the radiative heat flux. Using the Rosseland approximation (Rashed ), the radiative heat flux could be expressed by
Where the represents the Stefan-Boltzman constant and is the Rosseland mean absorption coefficient.
Assuming that the temperature difference within the flow is sufficiently small such that could be approached as the linear function of temperature
For the flow under study, it is relevant to assume that the applied magnetic field has the form cobble  , where is constant magnetic flux density.
To obtain the non-dimensional form of the above equations, the following dimensionless variables are introduced.
The boundary conditions (8) are reduced to
The governing equations are obtained by introducing a dimensionless normal distance from the disk, along with the von-Karman transformations,
With the appropriate boundary conditions:
Where the slip is factor and represents a uniform suction at the disk surface. The boundary conditions given by Eq. (26) imply that the radial , the tangential components of velocity, temperature and concentration vanish sufficiently far away from the disk, whereas the axial velocity component is anticipated to approach a yet unknown asymptotic limit for sufficiently large
3. Skin-friction coefficient, Nusselt number and Sherwood number
The parameters of engineering interest for the present problem are the local skin-friction coefficients and the local rates of heat and mass transfer to the surface are calculated. The radial shear stress and tangential shear stress are given by:
The tangential and radial skin-friction coefficient are, respectively, given by, Eq. (27):
Now the heat flux and the mass flux at the surface are given by.
Where is the rotational Reynolds number.
4. Numerical results and discussion
The system of non-linear ordinary differential Eqs. (20)–(25) together with the boundary conditions (26) are locally similar and solved numerically by using the Keller-box method. In order to gain physical insight, the velocity (radial, axial and tangential), temperature, concentration and pressure profiles have been discussed by assigning numerical values to the parameter, encountered in the problem which the numerical results are tabulated and displayed with the graphical illustrations.
In order to verify the accuracy of our present method, we have compared our results with those of Frusteri and Osalusi , Osalusi and Sibanda  and Maleque and Sattar . Table 1 shows the values of for various values of . The comparisons in all above cases are found to be excellent and agreed, so it is good.
|Frusteri et al. ||Osalusi et al. ||Maleque et al. ||Present study||Frusteri et al. ||Osalusi et al. ||Maleque et al. ||Present study||Frusteri et al. ||Osalusi et al. ||Maleque et al. ||Present study|
Figures 2, 3 and 11a–d display the velocity (radial, axial and tangential), temperature, concentration and pressure profiles under the magnetic field parameter, porosity parameter and Schmidt number. The (radial, axial and tangential) components of the velocity and pressure profile decrease with increase of magnetic field due to the inhibiting influence of the Lorentz force and increasing of all porosity parameter and Schmidt number, while the temperature and the concentration profiles increase with increasing of all magnetic field parameter, porosity parameter and Schmidt number. In Figures 4 and 5a–d, it is clear that the (radial and axial) components of the velocity and pressure profiles increase with increase of temperature buoyancy parameter and increasing concentration buoyancy parameter, while the tangential component of the velocity, the temperature and the concentration profiles decrease with increasing temperature buoyancy parameter and concentration buoyancy parameter.
The effects of on the (radial and axial) components of the velocity, the temperature, concentration and pressure profiles are shown in Figure 6a–d, respectively. It is observed that both the (radial and axial) components of the velocity, the temperature and pressure profiles decrease with the increase of Prandtl number. While, the concentration profile increase. Physically, it means that thermal boundary layer thickness gets decreased. In fact, it is well known that the thermal boundary layer thickness is inversely proportional to the square root of Prandtl number. Hence, the decrease of temperature profile with increasing is straightforward.
Figures 7 and 10a–d show the effects of radiation parameter and heat source parameter on the velocity (radial and axial), temperature, concentration and pressure profiles, also, we found that the (radial and axial) components of the velocity, temperature and pressure profiles increase with the increase of radiation parameter and increasing heat source parameter. While the concentration profile decrease. The effects of on the velocity (radial and axial), temperature, concentration and pressure profiles, are shown in Figure 8a–d, respectively. It is observed that the (radial and axial) components of the velocity, temperature, concentration and pressure profiles increase with the decreasing the Dufour’s number and increasing Soret number.
The effects of on the velocity (radial and axial) and pressure profiles, are shown in Figure 9a and b, respectively. We found that the (radial and axial) components of the velocity and pressure profiles decrease with the increase Joule heating parameter. In Figure 12a–c, it is clear that the (radial and axial) components of the velocity, concentration and pressure profiles decrease with increase of chemical reaction parameter. Figure 13a–d displays the velocity (radial, axial and tangential), temperature, concentration and pressure profiles under the suction parameter. The (radial, axial and tangential) components of the velocity, temperature, concentration and pressure profiles decrease with increase of suction parameter. The effects of on the velocity (radial, axial and tangential) and pressure profiles are shown in Figure 14a and b, respectively. It is observed that the (radial and axial) components of the velocity, and pressure profiles increase with the increasing slip parameter. While the tangential component of the velocity profile decrease.
The radial and tangential skin frictions and the heat and mass transfer coefficients are tabulated in Table 2 for various values of , , , , , , , , , , , and . We observed that increase for all magnetic field parameter and porosity parameter leads to an decrease in the all tangential skin friction , heat transfer rate and mass transfer rate , while an increase in the radial skin friction , the increase for all radial skin friction, tangential skin friction, heat transfer rate and mass transfer rate, with increasing of the temperature buoyancy parameter and the concentration buoyancy parameter . We found that the radial skin friction, tangential skin friction, mass transfer rate decreases while heat transfer rate increase with increasing of Prandtl number, Dufour number decreases and Soret number increases. It can that be seen that the radial skin friction, tangential skin friction and mass transfer rate increase while heat transfer rate decrease with increasing values of and . It is observed that an increase in the Joule heating parameter, results in a decrease in the tangential Skin-friction coefficient, Nusselt number and Sherwood number. The tangential skin friction and heat transfer rate decrease but the radial skin friction and mass transfer rate increase with increasing the Schmidt number. It also can be seen from this table that increasing the chemical reaction parameter to decrease in the radial skin friction, tangential skin friction and heat transfer rate while increase the mass transfer rate. We found also the tangential skin friction increase but the radial skin friction, Nusselt number and Sherwood number decrease with increasing the suction parameter. Finally, the radial skin friction and the tangential skin friction decrease but Nusselt number and Sherwood number increase with increasing slip parameter.
|0.1||2.0||0.5||0.5||1.0||5.0||0.25 & 0.2||0.5||0.0||0.64||0.5||−1.0||0.2||0.3857||1.6257||0.3144||1.0486|
|0.1||2.0||0.5||0.5||1.0||5.0||1.0 & 0.05||0.3910||1.6266||0.2324||1.0591|
|0.25 & 0.2||0.3857||1.6257||0.3144||1.0486|
|0.2 & 0.25||0.3856||1.6256||0.3195||1.0454|
|0.1||2.0||0.5||0.5||1.0||5.0||0.25 & 0.2||0.1||0.3858||0.3157||1.0487|
|0.1||2.0||0.5||0.5||1.0||5.0||0.25 & 0.2||0.5||0.3795||1.6247||0.3938||1.0370|
|0.1||2.0||0.5||0.5||1.0||5.0||0.25 & 0.2||0.5||0.0||0.22||0.6083||1.4121||0.3451||0.5590|
|0.1||2.0||0.5||0.5||1.0||5.0||0.25 & 0.2||0.5||0.0||0.64||0.2||0.3927||1.6268||0.3193||0.9130|
|0.1||2.0||0.5||0.5||1.0||5.0||0.25 & 0.2||0.5||0.0||0.64||0.5||−1.0||0.3857||1.6257||0.3144||1.0486|
|0.1||2.0||0.5||0.5||1.0||5.0||0.25 & 0.2||0.5||0.0||0.64||0.5||−1.0||0.2||0.3857||1.6257||0.3144||1.0486|
In this work, thermal radiation and thermal diffusion effects over an electrically conducting, Newtonian fluid in a steady laminar magnetohydrodynamic convective flow over a porous rotating infinite disk with the consideration of heat and mass transfer in the presence of Soret and Dufour’s diffusion effects have been obtained and studied numerically. Magnetic field parameter, porosity parameter, temperature buoyancy parameter, concentration buoyancy parameter, Prandtl number, radiation parameter, Soret and Dufour’s number, Joule heating parameter, heat source parameter, Schmidt number, chemical reaction parameter, suction parameter, slip parameter effects were considered in the separate cases. The subsequent outcome may be drawn as:
The components (radial, axial and tangential) of the velocity, temperature, concentration and pressure profiles under the porosity parameter. The (radial, axial and tangential) components of the velocity and pressure profile decrease with increasing porosity parameter, while the temperature and the concentration profiles increase with increasing porosity parameter. And also, that increase porosity parameter leads to a decrease in the all tangential skin friction , heat transfer rate and mass transfer rate , while an increase in the radial skin friction .
The (radial and axial) components of the velocity and pressure increase with increasing of temperature buoyancy parameter and concentration buoyancy parameter, while the tangential component of the velocity, the temperature and the concentration profiles decrease with increasing temperature buoyancy parameter and concentration buoyancy parameter. We found that the increase for all radial skin friction, tangential skin friction, heat transfer rate and mass transfer rate, with increasing of the temperature buoyancy parameter and the concentration buoyancy parameter .
The (radial and axial) components of the velocity, temperature and pressure profiles increase with the increase of heat source parameter. While the concentration profile decrease. And also, the radial skin friction, tangential skin friction and mass transfer rate increase while heat transfer rate decrease with increasing value of .
The (radial and axial) components of the velocity, concentration and pressure profiles decrease with increase of chemical reaction parameter. And in Table 2, an increase the chemical reaction parameter, results in an decrease in the (radial and tangential) Skin-friction coefficient and Nusselt number, while Sherwood number increases.
external uniform magnetic field
constant magnetic flux density
induced magnetic field
specific heat at constant pressure
tangential skin-friction coefficient
radial skin-friction coefficient
molecular diffusion coefficient
rate of chemical reaction
permeability of the porous medium
Rosseland mean absorption coefficient
radial , tangential and axial components of dimensionless velocity
magnetic field parameter
concentration buoyancy parameter
normal direction to the wall
electric current density
Joule heating parameter
volumetric heat generation/absorption rate,
radiative heat flux,
mean fluid temperature
components of the flow velocity are in the directions of increasing , respectively,
tangent momentum accommodation coefficient
mean free path
kinematic viscosity of the ambient fluid,
heat source parameter
temperature buoyancy parameter
dimensionless temperature distribution
dimensionless concentration distribution
constant angular velocity
chemical reaction parameter
coefficient of temperature
coefficient of concentration
conditions at the surface and in the free stream