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Hall Effects on Unsteady Free Convection in a Heated Vertical Channel in Presence of Heat Generation

Sanatan Das*, Rabindra Nath Jana

Department of Mathematics, University of Gour Banga, Malda, India

Abstract: Hall effects on an unsteady free convective flow of a viscous incompressible electrically conducting fluid between two heated vertical plates in the presence of a transverse applied magnetic field and heat generation have been studied. The governing equations are solved analytically using the Laplace transform technique. In order to get the physical insight into the problem, the effects of important parameters on temperature and velocity profiles are shown graphically and discussed in detail.

Keywords: Hall currents, free convection, heat generation, Prandtl number, Grashof number, frequency parameter

Received September 11, 2013; Revised September 24, 2013; Accepted September 26, 2013

Citation: Sanatan Das, Rabindra Nath Jana, Hall Effects on Unsteady Free Convection in a Heated Vertical Channel in Presence of Heat Generation. Applied Mathematics and Physics, 1 (3). 45-59. January 2013. doi:10.12691/amp-1-3-2

1. Introduction

Heat transfer has emerged as a central discipline in contemporary engineering science. There has been growing interest in buoyancy-induced flows and the associated heat over the past three decades, because of the importance of these flows in many different areas, such as cooling of electronic equipment, pollution, materials processing, energy systems and safety in thermal processes. Several books, for instance, Turner [1], Jaluria [2], Kakac et al. [3], Gebhart et al. [4] and Bejan and Kraus [5] may be consulted for detailed discussions on this subject. The study of heat generation or absorption in moving fluids is important in problems dealing with chemical reactions and those concerned with dissociating fluids. Heat generation effects may alter the temperature distribution and this in turn can affect the particle deposition rate in nuclear reactors, electronic chips and semi conductor wafers. Although exact modeling of internal heat generation or absorption is quite difficult, some simple mathematical models can be used to express its general behaviour for most physical situations. Heat generation or absorption can be assumed to be constant, space-dependent or temperature-dependent. The use of magnetic field that influences heat generation/absorption process in electrically conducting fluid flows has many engineering applications. For example, many metallurgical processes which involve cooling of continuous strips or filaments, which are drawn through a quiescent fluid. The properties of the final product depend to a great extent on the rate of cooling. The rate of cooling and therefore, the desired properties of the end product can be controlled by the use of electrically conducting fluids and the applications of the magnetic fields (Vajravelu and Hadjinicalaou [6]). Some processes in which heat is generated or absorbed may proceed in equipment and in the surrounding medium, for example, heat generation in the fuel elements of nuclear reactors due to the deceleration of nuclear fission fragments and slowing down of neutron fluxes, during a number of chemical reactions, etc. In the ionized gases, the current is not proportional to the applied potential except when the field is very weak in an ionized gas where the density is low and the magnetic field is very strong, the conductivity normal to the magnetic field is reduced due to the free spiraling of electrons and ions about the magnetic lines of force before suffering collisions and a current is induced in a direction normal to both electric and magnetic fields. This phenomenon, well known in the literature, is called the Hall effect. The study of hydromagnetic flows with Hall currents has important engineering applications in problems of magnetohydrodynamic generators and of Hall accelerators as well as in flight magnetohydrodynamics. It is well known that a number of astronomical bodies posses fluid interiors and magnetic fields. It is also important in the solar physics involved in the sunspot development, the solar cycle and the structure of magnetic stars. The unsteady hydromagnetic flow of a viscous incompressible electrically conducting fluid through a vertical channel is of considerable interest in the technical field due to its frequent occurrence in industrial and technological applications. The Hall effects on the flow of ionized gas between parallel plates under transverse magnetic field have been studied by Sato [7]. Miyatake and Fujii [8] have discussed the free convection flow between vertical plates - one plate isothermally heated and other thermally insulated. Natural convection flow between vertical parallel plates- one plate with a uniform heat flux and the other thermally insulated has been investigated by Tanaka et al. [9]. Hall effects on the hydromagnetic convective flow through a channel with conducting walls have been investigated by Dutta and Jana [10]. Joshi [11] has studied the transient effects in natural convection cooling of vertical parallel plates. Singh [12] have described the natural convection in unsteady Couette motion. Singh et al. [13] have studied the unsteady free convective flow between two vertical parallel plates. The natural convection in unsteady MHD Couette flow with heat and mass transfers has been analyzed by Jha [14]. Narahari et al. [15] have studied the unsteady free convective flow between long vertical parallel plates with constant heat flux at one boundary. The unsteady free convective flow in a vertical channel due to symmetric heating have been described by Jha et al. [16]. Singh and Paul [17] have studied the unsteady natural convective between two vertical walls heated/cooled asymmetrically. Guria and Jana [18] have discussed the Hall effects on the hydromagnetic convective flow through a rotating channel under general wall conditions. Jha and Ajibade [19] have studied the unsteady free convective Couette flow of heat generating/absorbing fluid. Rajput and Sahu [20] have studied the unsteady free convection MHD flow between two long vertical parallel plates with constant temperature and variable mass diffusion. Das et al. [21] have studied the radiation effects on free convection MHD Couette flow started exponentially with variable wall temperature in presence of heat generation. The effect of radiation on transient natural convection flow between two vertical walls have been described by Mandal et al. [22]. Das et al. [23] have studied the radiation effects on unsteady MHD free convective Couette flow of heat generation/absorbing fluid. Sarkar et al. [24] have studied an oscillatory MHD free convective flow between two vertical walls in a rotating system. Kalita [25] has obtained an exact solution of an unsteady free convection MHD flow and heat transfer between two heated vertical plates with heat source.

The object of the present paper is to study the Hall effects on the unsteady MHD free convective flow of a viscous incompressible electrically conducting fluid in a vertical channel. The problem is solved analytically for the velocity, temperature, rate of heat transfer and shear stress. The effects of various emerging parameters on the velocity and temperature distributions are shown and discussed with the help of graphs.

2. Formulation of the Problem and Its Solution

Consider the unsteady magnetohydrodynamic flow of a viscous incompressible electrically conducting fluid between two infinitely long vertical parallel plates with variable temperature separated by a distance . The flow is set up by the buoyancy force arising from the temperature gradient. Choose a Cartesian co-ordinates system with the - axis along the vertical plates at , the - axis perpendicular to the plates, -axis is normal to the -plane and the origin of the axes at the middle point between the plates [See Figure 1]. Initially, at time , the fluid is rest and the temperature of the fluid and the plates are same as . At time , the temperature of the plates at is raised to , where is the frequency of the temperature oscillations and being the reference temperature. A uniform transverse magnetic field is applied perpendicular to the plates. We assume that the flow is laminar and the pressure gradient term in the momentum equation can be neglected. It is assumed that the effect of viscous and Joule dissipations are negligible. As the plates are infinitely long, the velocity field and temperature distribution are functions of and only.

Figure 1. Geometry of the problem

Under the usual Boussinesq approximation, the flow is governed by the Navier-Stokes equations

[picture] | (1)

[picture] | (2)

[picture] | (3)

where is the fluid density, the fluid temperature, is the plate temperature, the kinematic viscosity, and are fluid velocity components, the coefficient of thermal expansion, the acceleration due to gravity, the thermal conductivity, the specific heat at constant pressure and a constant.

The initial and boundary conditions for the velocity and temperature distributions are

[picture] | (4)

The generalized Ohm's law, on taking Hall currents into account and neglecting ion-slip and thermo-electric effect, is (see Cowling [26])

[picture] | (5)

where is the current density vector, the velocity vector, the magnetic field vector, the electric field vector, the cyclotron frequency, the electrical conductivity of the fluid and the collision time of electron.

We assume that the magnetic Reynolds number for the flow is small so that the induced magnetic field can be neglected. This assumption is justified since the magnetic Reynolds number is generally very small for partially ionized gases (Shereliff [27]). The solenoidal relation for the magnetic field gives constant everywhere in the fluid where . Further, if be the components of the current density , then the equation of the conservation of the current density gives . This constant is zero since at the wall which is electrically non-conducting. Thus everywhere in the flow. Since the induced magnetic field is neglected, the Maxwell's equation becomes which gives and . This implies that and constant everywhere in the flow. We choose this constants equal to zero, i.e. .

In view of the above assumption, the equation (5) gives

[picture] | (6)

[picture] | (7)

where is the Hall parameter.

Solving (6) and (7) for and , we have

[picture] | (8)

[picture] | (9)

On the use of (8) and (9), the momentum equations (1) and (2) along - and -directions become

[picture] | (10)

[picture] | (11)

Introducing non-dimensional variables

[picture] | (12)

equations (10), (11) and (3) become

[picture] | (13)

[picture] | (14)

[picture] | (15)

where is the magnetic parameter, the Grashof number, the Prandtl number and the heat generation parameter.

The initial and boundary conditions (4) become

[picture] | (16)

where is the frequency parameter.

Combining equations (13) and (14), we get

[picture] | (17)

where

[picture] | (18)

Taking the Laplace transform of equations (17) and (15), we have

[picture] | (19)

[picture] | (20)

where

[picture] | (21)

The initial and boundary conditions for and are

[picture] | (22)

Solutions of equations (19) and (20) subject to the boundary conditions (22) are given by

[picture] | (23)

[picture] | (24)

The inverse Laplace transforms of equations (23) and (24) give the solution for the temperature distribution and the velocity field as

[picture] | (25)

[picture] | (26)

where

[picture] | (27)

In the absence of Hall currents (), the equation (26) reduces to

[picture] | (28)

[picture] | (29)

Equation (28) is not identical with the equation (15) of Kalita [25] due to the mathematical error done by Kalita. It is also seen from figures of Kalita [25] that the temperature and velocity boundary conditions do not satisfy.

3. Results and Discussion

We have presented the non-dimensional velocity components and temperature distribution for several values of magnetic parameter , Hall parameter , heat generation parameter , Prandtl number , frequency parameter , Grashof number and time against in Figures 2-17. It is seen from Figure 2 that the primary velocity decreases while the magnitude of the secondary velocity increases with an increase in magnetic parameter . This agrees with the expectations, since the magnetic field exerts a retarding force on the free convective flow. Figure 3 reveals that the primary velocity decreases whereas the magnitude of the secondary velocity increases with an increase in Hall parameter . Figure 4 and Figure 5 show that the primary velocity and the magnitude of the secondary velocity increase with an increase in either heat generation parameter or Grashof number . Physically, the heat generation (thermal source) has the tendency to make more heating the fluid. This behaviour is seen from Figure 4 in which the velocity components increase as increases. An increase in Grashof number leads to an increase in velocity, this is because, increase in Grashof number means more heating and less density. It is illustrated from Figure 6 that the primary velocity and the magnitude of the secondary velocity decrease with an increase in Prandtl number . Figure 7 and Figure 8 show that both the primary velocity and the magnitude of the secondary velocity increase with an increase in either frequency parameter or time . It is reveal form Figure 9 that the steady state reaches when time progresses. It is seen from Figure 10 that the fluid temperature increases with an increase in heat generation parameter . This result agrees with expectations, as increases, the fluid is more heated and hence the fluid temperature increases. It is observed from Figure 11 that the fluid temperature decreases with an increase in Prandtl number . Smaller fluids possess higher thermal conductivities so that heat can diffuse away from the plate surface faster than for higher fluids (thicker boundary layers). Our computations show that a rise in depresses the temperature function, a result consistent with numerous other studies. Figure 12 and Figure 13 show that the fluid temperature increases with an increase in either frequency parameter or time .

Figure 2. Velocities and for different when , , , , and

Figure 3. Velocities and for different when , , , , and

Figure 4. Velocities and for different when , , , , and

Figure 5. Velocities and for different when , , , , and

Figure 6. Velocities and for different when , , , , and

Figure 7. Velocities and for different when , , , , and

Figure 8. Velocities and for different when , , , , and

Figure 9. Velocities and for different when , , , , and

Figure 10. Temperature for different when , and

Figure 11. Temperature for different when , and

Figure 12. Temperature for different for, and

Figure 13. Temperature for different for , and

The rate of heat transfer at the plates is and is given by

[picture] | (30)

Table 1. Rate of heat transfer θ’ (1,τ) when N=2 and τ=0.5

Numerical results of the rate of heat transfer at the plate is which is presented in Tables 1-2 for several values of heat generation parameter, Prandtl number , frequency parameter and time . Table 1 shows that the rate of heat transfer increases with an increase in Prandtl number . This may be explained by the fact that frictional forces become dominant with increasing values of and hence yield greater heat transfer rates. On the other hand, the rate of heat transfer decreases with an increase in heat generation parameter . Table 2 shows that the rate of heat transfer increases when time progresses. The rate of heat transfer increases with an increase in frequency parameter . The negative value of physically explains that there is heat flow from the plate to the fluid.

Table 2. Rate of heat transfer θ’ (1,τ) when Q=0.5 and Pr=0.025

The non-dimensional shear stresses at the plate is given by

[picture] | (31)

where , , , , and are given by (31).

Figure 14. Shear stresses and for different when , , , and

Numerical results of the non-dimensional shear stresses and at the plate due to the primary and the secondary flows are plotted in Figures 14-19 against Hall parameter for several values of magnetic parameter , heat generation parameter , Prandtl number , Grashof number , frequency parameter and time . Figure 14 shows that the magnitude of the shear stress due to the primary flow decreases while the shear stress at the plate due to the secondary flow increases with an increase in magnetic parameter . It is seen from Figure 15 that the magnitude of the shear stress and the shear stress increases with an increase in heat generation parameter . Figure 16 that the magnitude of the shear stress and the shear stress decreases with an increase in Prandtl number . It is seen from Figures 17-19 that the magnitude of the shear stress and the shear stress increases with an increase in either Grashof number or frequency parameter or time . These results are in agrement with the fact that the velocity increases with an increase in or or .

Figure 15. Shear stresses and for different when , , , and

Figure 16. Shear stresses and for different when , , ,

Figure 17. Shear stresses and for different when , , , and

Figure 18. Shear stresses and for different when , , , and

Figure 19. Shear stresses and for different time when , , , and

4. Conclusion

Hall effects on an unsteady free convective flow of a viscous incompressible electrically conducting fluid between two vertical heated plates in the presence of transverse applied magnetic field and heat generation have been studied. It is found that magnetic field retards the fluid motion due to the opposing Lorentz force generated by the magnetic field. Hall parameter increases the velocity components. An increase in either heat generation parameter or Grashof number or frequency parameter leads to increase the primary velocity and the magnitude of the secondary velocity. The fluid temperature increases with an increase in either heat generation parameter or frequency parameter. Further, the Hall parameter or heat generation parameter or Grashof number or frequency parameter increases the shear stresses at the right plate. Prandtl number or frequency parameter tends to enhance the heat transfer efficiency. This model is useful to gain a deeper knowledge of the various industrial processes.

Acknowledgements

The authors would like to express thanks to the anonymous referees for their valuable suggestions.

References

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