American Journal of Applied Mathematics
Volume 4, Issue 3, June 2016, Pages: 114-123

Effects of Mass Transfer on Unsteady Hydromagnetic Convective Flow Past an Infinite Vertical Rotating Porous Plate with Heat Source

Thomas Mwathi Ngugi*, Mathew Ngugi Kinyanjui, David Theuri

Department Pure and Applied Mathematics, Jomo Kenyatta University of Agriculture and Technology, Nairobi, Kenya

(T. M. Ngugi)

*Corresponding author

Thomas Mwathi Ngugi, Mathew Ngugi Kinyanjui, David Theuri. Effects of Mass Transfer on Unsteady Hydromagnetic Convective Flow Past an Infinite Vertical Rotating Porous Plate with Heat Source. American Journal of Applied Mathematics. Vol. 4, No. 3, 2016, pp. 114-123. doi: 10.11648/j.ajam.20160403.11

Received: February 11, 2016; Accepted: February 23, 2016; Published: April 25, 2016

Abstract: The effect of mass transfer on unsteady Hydromagnetic convective flow, of an incompressible electrically conducting fluid, past an infinite vertical rotating porous plate in presence of constant injection and heat source has been investigated. The non-linear partial differential equations governing the flow are solved numerically using the finite differences method. The effect of Hartmann's number, Grashof number for heat transfer, Grashof number for mass transfer, permeability parameter, Schmidt number, Heat source parameter, Prandtl number, Eckert number and rotational parameter on the flow field are presented graphically. A change on the parameters is observed to either increase, decrease or to have no effect on the profiles. The study has some useful information to engineers in the field of oil exploration, geothermal reservoirs, in petroleum and mineral industries, MHD generators, among many other areas.

Keywords: Magnetohydrodynamics (MHD), Porous Medium, Mass Transfer, Heat Source, Injection

Contents

1. Introduction

2. Formulation of the Problem

Consider the unsteady free convective flow of a viscous incompressible electrically conducting fluid past an infinite vertical rotating porous plate in presence of constant injection and heat source and transverse magnetic field. Let the x-axis be taken in vertically upward direction along the plate and y-axis normal to it. The plate is infinite in X-direction and is non-conducting. The fluid and the plate are in a state of rotation about y-axis with uniform angular velocity .The plate is maintained at a uniform temperature. The free stream temperature and concentration are andrespectively. A magnetic field is applied perpendicular to the plate. Neglecting the induced magnetic field and the Joulean heat dissipation and applying Boussinesq’s approximation the governing equations of the flow are given by:

Fig. 1. Configuration of the problem.

Continuity equation:

(1)

Momentum along x-axis:

(2)

Momentum along z-axis:

(3)

Energy equation:

(4)

Concentration equation:

(5)

With initial and boundary conditions:

,  at

(6)

The following non dimensional quantities are introduced,

(7)

Where,  are acceleration due to gravity, density, electrical conductivity, coefficient of kinematic viscosity, volumetric coefficient of expansion for heat transfer, volumetric coefficient of expansion for mass transfer, angular frequency, coefficient of viscosity, thermal diffusivity at constant pressure, temperature, temperature at the plate, temperature at infinity, concentration, concentration at the plate, concentration at infinity, specific heat at constant pressure, molecular mass diffusivity respectively.

The governing equations in non-dimensional form are:

(8)

(9)

(10)

(11)

With initial and boundary conditions as,

(12)

3. Method of Solution

The set of differential equations (8) – (11) subject to the boundary conditions (12), are highly nonlinear, coupled and therefore they cannot be solved analytically. Hence, the Crank-Nicolson method is used to obtain an accurate and efficient solution to the boundary value problem under consideration. Setting the finite difference averages for velocity, temperature and concentration as:

(13)

(14)

(15)

We Substitute (13), (14) and (15) in equations (8) to (11). Equation (8) becomes,

(16)

Multiplying all through by  simplifying, and letting the coefficients of interior nodes to be:

(17)

We have,

(18)

Equation (18) can be represented in a tridiagonal matrix form as follows. For i=2, 3, 4….. (N-1)

(19)

Equation (9) becomes,

(20)

Multiplying all through by  simplifying, and letting the coefficients of interior nodes to be:

(21)

We have,

(22)

Equation (22) can be represented in a tridiagonal matrix form as follows. For i=2, 3, 4….. (N-1)

(23)

Equation (10) becomes:

(24)

Multiplying all through by  simplifying, and letting the coefficients of interior nodes to be:

(25)

We have,

(26)

Equation (26) can be represented in a tridiagonal matrix form as follows. For i=2, 3, 4….. (N-1)

(27)

Equation (11) becomes:

(28)

Multiplying all through by  simplifying, and letting the coefficients of interior nodes to be:

(29)

We have,

(30)

Equation (30) can be represented in a tridiagonal matrix form as follows. For i=2, 3, 4….. (N-1)

(31)

Matrices (19), (23), (27) and (31) are simulated using MATLAB algorithm subject to initial and boundary conditions (12)

4. Results and Discussions

The effect of Mass transfer on unsteady free convective flow of a viscous incompressible electrically conducting fluid past an infinite vertical rotating porous plate with constant injection and heat source in presence of a transverse magnetic field has been studied. The governing equations of the flow problem have been solved using Finite Difference Method. The effects of the flow parameters are analyzed and discussed with the help of velocity profiles, temperature profiles and concentration distribution. During numerical calculations we have chosen the values of Pr =0.71 representing air at 25°C, Sc =0.60 representing H2O vapor, Grc>0 corresponding to cooling of the plate and S>0 representing heat source. The figures below represent effects of variation of various parameters on velocity, temperature and concentration profiles. The block lines represent primary velocity profiles while the broken lines represent secondary velocity profiles.

Velocity Profiles

Fig. 2. Velocity profiles for different values of Hartmann number (M).

Fig. 3. Velocity profiles for different values of Grc.

Fig. 4. Velocity profiles for different values of Grt.

Fig. 5. Velocity profiles for different values of Pr.

Fig. 6. Velocity profiles for different values of X.

Fig. 7. Velocity profiles for different values of Ec.

Fig. 8. Velocity profiles for different values of S.

Fig. 9. Velocity profiles for different values of Sc.

Fig. 10. Velocity profiles for different values of R0.

Temperature Profiles

Fig. 11. Temperature profiles for different values of Pr.

Fig. 12. Temperature profiles for different values of Ec.

Fig. 13. Temperature profiles for different values of S.

Concentration Profiles

Fig. 14. Concentration profiles for different values of Sc.

From Fig. 2, it is observed that, an increase in Hartmann number M, leads to a decrease in both the primary and secondary velocity. This is because when M increases it means that, electromagnetic force increases and when transverse magnetic field is applied to an electrically conducting fluid, it gives rise to a force called the Lorentz force which acts against the flow if applied in the normal direction as in the present study. This resistive force has a tendency to slow down the motion of the fluid in the boundary layer. A weak magnetic field does not have much effect on velocity. When M = 0 means that magnetic force is so small compared to viscous force.

From Fig. 3 increase in the Grashof number for heat transfer , causes an increase in the primary velocity profiles and an increase in the magnitude of the secondary velocity profiles respectively. The Grashof number for heat transfer  represents the effects of free convection currents and physically  corresponds to heating of the fluid (or cooling of the surface). Velocity of the fluid increases because the fluid flow is assisted by the free convection currents. As expected, increase in the velocity profiles is partly due to the enhancement of thermal buoyancy force.

Fig. 4 shows that increase in Grashof number for mass transfer  causes an increase in the primary velocity profiles. It also causes increase in the magnitude of the secondary velocity profiles. The velocity distribution attains a distinctive maximum value near the porous plate and then decays smoothly to approach a free stream value. The Grashof number for mass transfer  defines the ratio of the species buoyancy force to the viscous hydrodynamic force hence, as expected, the fluid velocity increases due to increase in the species buoyancy force. Increase in species buoyancy force results into a higher species transportation rate away from the rotating plate, resulting into lower concentration.

Fig. 5 shows the effect of the Prandtl number on both primary and secondary velocity profiles. The values of the Prandtl number are chosen Air at  and one atmospheric pressure (= 0.71), and water at  (= 7.00). It is observed that increasing the values of the Prandtl number, results in a decrease in the fluid velocity. This is because, an increase in Prandtl number means that the viscous forces are increasing as the thermal forces increase hence decreasing the velocity of the fluid particles.

Fig. 6 shows that increase in permeability parameter  causes an increase in both the primary and secondary velocity profiles. It is observed that the fluid velocity increases and a peak value is attained near the plate then decays continuously to approach the free stream. Increasing  decreases the resistance of the porous medium since permeability physically becomes more with an increase in. This increases the magnitude of the flow velocity.

Fig. 7 shows that increase in Eckert number , causes an increase in both the primary and secondary velocity profiles. It is observed that the fluid velocity increases sharply and obtains a distinctive maximum value near to the wall of the porous plate and then decays continuously with increasing y distance. This is because when  is large, it implies that the kinetic energy dominates the boundary layer enthalpy which means that the particles or molecules of the fluid have high velocities. When the Ec number is small, it implies that the kinetic energy is small and hence the particles have low velocities, hence when  is increased, the velocity also increases.

Fig. 8 shows the primary and secondary velocity profiles for different values of heat source . It is observed that an increase in the heat source parameter , results to an increase in the fluid velocity. The presence of a heat source produces a heating effect that increase velocity of the convection currents that move next to the surface of the rotating plate, leading to higher velocity profiles.

Fig. 9 shows that increase in Schmidt number  causes a decrease in primary profiles and in the magnitude of the secondary velocity profiles respectively. The values of are chosen for the gases so that: Hydrogen ), Helium (), Water vapor (). The Schmidt number  signifies the ratio of the momentum to mass diffusivity. It quantifies the relative effectiveness of momentum and mass transport by diffusion in the hydrodynamic (velocity) and concentration (species) boundary layers. An increase in leads to thinning of the velocity and the concentration boundary layers respectively. A large value of  means a presence of a heavier fluid and this implies a lower velocity of the fluid.

Fig. 10 shows that that as rotation parameter  increases; the primary velocity u decreases whereas secondary velocity w increases. This indicates that rotation retards fluid flow in the primary flow direction, but it accelerates fluid flow in the secondary flow direction. This is due to the fact that the Coriolis force acts as a constraint in the main fluid flow when the plate is suddenly set into motion. It can be said that Coriolis force ends fluid flow in the primary flow direction to induce cross flow and secondary flow in the flow field. Absence of rotation translates to absence of the secondary velocity profiles. This means rotation can be used to control emergence of the secondary velocity profiles in a rotating system.

The temperature of the flow suffers a substantial change with the variation of the flow parameters such as Prandtl number , Eckert number  and Heat source parameter S. The temperature profiles are in good agreement with those of Das et al. (2010). From fig. 11, it is observed that an increase in Prandtl Number leads to a decrease in temperature profiles. This is because the viscous forces dominate over thermal forces as Prandtl Number is raised. An increase in the Prandtl Number results to a decrease of the thermal boundary layer thickness and in general lowers the average temperature in the boundary layer. Smaller values of are equivalent to increase in the thermal conductivity of the fluid and therefore heat is able to diffuse away from the heated surface more rapidly than for higher values of . Thus, the temperature of water at  (= 7.00) falls more rapidly compared to Air at  and one atmospheric pressure (= 0.71). Prandtl number controls the relative thickness of the momentum and thermal boundary layers.

Fig. 12 shows that an increase in Eckert number  leads to an increase in temperature profiles. Hence the rate at which the fluid loses heat decreases as the Eckert Number is increased. This observation can be attributed to the viscous dissipation which increases with kinetic energy of the fluid particles. Increase in means the fluid absorbs more heat energy that is released from the internal viscous forces. This in turn increases the temperature of the convection currents due to increased thermal buoyancy forces.

Fig. 13 shows the variation of different values of the heat source parameter to the temperature. It is observed that as increases, the temperature increase. Increase in heat source produces a heating effect hence increase in the temperature. The thermal boundary layer is weakened when heat source is present hence the increase in temperature.

Fig. 14 shows that an increase in Schmidt number Sc causes decrease in the concentration of the fluid. The values of are chosen for the gases so that: Hydrogen ), Helium () and Water vapor (). The concentration falls gradually and progressively for hydrogen in distinction to other gases. This is because an increase of  mean a decrease of molecular diffusivity, which results in decrease of concentration boundary layer. Hence, the concentration of species is smaller for higher values of .

5. Conclusion

A summary of effects of varying different flow parameters on the velocity, temperature and the concentration distribution of the flow is given below

1. An increase in Hartmann's number, Prandtl number  and Schmidt number  retards both the primary and secondary velocity of the fluid at all points.

2. The effect of increasing Grashof number for heat transfer , Grashof number for mass transfer , permeability parameter , Heat source parameter , and Eckert number  is to accelerate both the primary and secondary velocity profiles at all points.

3. Rotation parameter  retards fluid flow in the primary flow direction, but it accelerates fluid flow in the secondary flow direction. Absence of rotation translates to absence of the secondary velocity profiles. This means rotation can be used to control emergence of the secondary velocity profiles in a rotating system.

4. An increase in Eckert number  and heat source parameter , increases the temperature of the flow field at all points while a growing Prandtl number  retards the temperature of the flow. The temperature of the flow grows rapidly for small values ) and for higher values the effect reverses.

5. The effect of increasing Schmidt numberis to reduce the concentration boundary layer thickness of the flow field at all points. The concentration of species is smaller for higher values of .

References

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