Heat and Mass Transfer in Visco-Elastic Fluid through Rotating Porous Channel with Hall Effect

This paper examined the hydromagnetic boundary layer flow of viscoelastic fluid with heat and mass transfer in a vertical channel with rotation and Hall current. A constant suction and injection is applied to the plates. A strong magnetic field is applied in the direction normal to the plates. The entire system rotates with uniform angular velocity (Ω), about the axis perpendicular to the plates. The governing equations are solved by perturbation technique to obtain an analytical result for velocity, temperature, concentration distributions and results are presented graphically for various values of viscoelastic parameter (K2), Prandtl number (Pr), Schmidt number (Sc), radiation parameter (R), heat generation parameter (Qh) and Hall parameter (m).

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Cite this paper

Gaur, P. and Jha, A. (2016) Heat and Mass Transfer in Visco-Elastic Fluid through Rotating Porous Channel with Hall Effect. Open Journal of Fluid Dynamics, 6, 11-29. doi: 10.4236/ojfd.2016.61002.

Received 28 March 2015; accepted 8 March 2016; published 11 March 2016

1. Introduction

2. Mathematical Formulation

The constitutive equations for the rheological equation of state for an elastico-viscous fluid (Walter’s liquid B') are

. (1)

. (2)

in which

. (3)

N(t) is the distribution function of relaxation times. In the above equations pik is the stress tensor, p an arbitrary isotropic pressure, gik is the metric tensor of a fixed co-ordinate system xi and is the rate of strain tensor. It was shown by Walter’s [33] that Equation (2) can be put in the following generalized form which is valid for all types of motion and stress

(4)

where is the position at time t' of the element that is instantaneously at the print xi at time “t”. The fluid with equation of state (1) and (4) has been designated as liquid B'. In the case of liquids with short memories, i.e. short relaxation times, the above equation of state can be written in the following simplified form

. (5)

In which is the limiting viscosity at small rates of shear, and de-

notes the convected time derivative. We consider Oscillatory free convective flow of a viscous incompressible and electrically conducting fluid between two insulating infinite vertical permeable plates. A strong transverse magnetic field of uniform strength is applied along the axis of rotation by neglecting induced electric and magnetic fields. The fluid is assumed to be a gray, emitting, and absorbing, but non scattering medium. The radiative heat flux term can be simplified by using the Rosseland approximation. It is also assumed that the chemically reactive species undergo first-order irreversible chemical reaction.

The equations governing the flow of fluid together with Maxwell’s electromagnetic equations are as follows.

Equation of Continuity

. (6)

Momentum Equation

. (7)

Energy Equation

. (8)

Concentration Equation

. (9)

The generalized Ohm’s law, in the absence of the electric field [34] , is of the form.

(10)

where and pe are velocity, the electrical conductivity, the magnetic permeability, the cyclotron frequency, the electron collision time, the electric charge, the number density of the electron and the electron pressure, respectively. Under the usual assumption, the electron pressure (for a weakly ionized gas), the thermoelectric pressure, and ion slip are negligible, so we have from the Ohm’s law.

. (11)

From which we obtain that

. (12)

The solenoidal relation for the magnetic field where gives (con-

stant) everywhere in the flow, which gives. If are the component of electric current density, then the equation of conservation of electric charge gives.

Since the plates are infinite in extent, all the physical quantities except the pressure depend only on and. A cartesian coordinate system is assumed and -axis is taken normal to the plates, while -axis and -axis are in the upward and perpendicular directions on the plate (origin), respectively .The velocity components, , are in the -, -, -directions respectively. The governing equations in the rotating system in presence of Hall current, thermal radiation and chemical reaction are given by the following equations.

(13)

(14)

(15)

(16)

(17)

where is the Hall parameter, and are coefficients of thermal and solutal expansion, cp is the specific heat at constant pressure, is the density of the fluid, v is the kinematics viscosity k is the fluid thermal conductivity, is the acceleration of gravity, is the additional heat source, is the radiative heat flux, Dm is the molecular diffusivity, K1 is the chemical reaction rate constant .The radiative heat flux is

given by, in which and are the Stefan-Boltzmann constant and the mean absorp-

tion coefficient, respectively.

The initial and boundary conditions as suggested by the physics of the problem are

(18)

where e is a small constant.

We now introduce the dimensionless variables and parameter as follows:

. (19)

After combining (14) and (15) and taking, then (14)-(17) reduce to

(20)

where is the modified Grashof number, is the Prandtl number,

is the modified solutal Grashof number, is the Hartmann

number, is the radiation parameter, is the Schmidt number, and is the reaction parameter.

The boundary conditions (9) can be expressed in complex form as:

. (21)

3. Method of Solution

The set of partial differential Equations (20) cannot be solved in closed form. So it is solved analytically after these equations are reduced to a set of ordinary differential equations in dimensionless form. We assume that

(22)

where R stands for q or or and which is applicable for small perturbation.

Substituting (22) into (20) and comparing the harmonic and non harmonic terms, we obtain the following ordinary differential equations:

(23)

where and dashes denote the derivatives w.r.t.

The Transformed boundary conditions are

. (24)

The solutions of (23) under the boundary conditions are

(25)

(26)

(27)

(28)

(29)

(30)

(31)

(32)

. (33)

Equation (25) corresponds to the steady part, which gives 0 as the primary and V0 as secondary velocity components. The amplitude (resultant velocity) and phase difference due to these primary and secondary velocities for the steady flow are given by

(34)

where

Equation (26) and (27) together give the unsteady part of the flow. Thus unsteady primary and secondary velocity components and, respectively, for the fluctuating flow can be obtained from the following.

. (35)

The amplitude (resultant velocity) and the phase difference of the unsteady flow are given by

(36)

where

The amplitude (resultant velocity) and the phase difference are given by

(37)

where u = Real part of q and v = Imaginary part of q.

3.2. Amplitude and Phase Difference of Shear Stresses Due to Steady and Unsteady Flow at the Plate

The amplitude and phase difference of shear stresses at the stationary plate (η = 0), the steady flow can be obtained as

. (38)

For the unsteady part of flow, the amplitude and phase difference of shear stresses at the stationary plate () can be obtained as.

where

(39)

(40)

where

. (41)

The amplitude and phase difference of shear stresses at the stationary plate (η = 0) can be as

(42)

where Real part of and Where Imaginary part of.

The Nusselt number

. (43)

The rate of heat transfer (i.e. heat flux) at the plate in terms of amplitude and phase difference is given by

. (44)

The Sherwood number

. (45)

The rate of mass transfer (i.e. mass flux) at the plate in terms of amplitude and phase difference is given by.

. (46)

4. Results and Discussion

The system of ordinary differential Equation (23) with boundary condition (24) is solved analytically using perturbation technique. The solutions are obtained for the steady and unsteady velocity fields from (25)-(27), temperature fields from (28)-(30) and concentration fields are given by (31)-(33). The influences of each of the parameters on the thermal mass and hydrodynamic behaviors of buoyancy-induced flow in a rotating vertical channel are studied. The results are presented graphically. Temperature of the heated wall (left wall) is a function of time as given in the boundary conditions and the cooled wall at is maintained at a constant temperature. Further it is assumed that the temperature difference is small enough so that the density changes of the fluid in the system will be small. When the injection/suction parameter l is positive, fluid is injected through the hot wall into the channel and sucked out through the cold wall. The effect of various physical parameters on flow, heat, concentration fields, skin-friction Nusselt number, and Sherwood number are presented graphically in Figures 1-14. The profiles for resultant velocity Rn for the flow are in Figures 1-4 for suction/injection parameter l, rotation parameter W, viscoelastic parameter K2, and e respectively. Figure 1 shows that increase in the suction parameter l leads to an increase of Rn within the stationary plates. Similarly the

Figure 1. Resultant velocity Rn due to u and v versus l for different values of h at t = π/4.

Figure 2. Resultant velocity Rn due to u and v versus h for different values of W at t = π/4.

Figure 3. Resultant velocity Rn due to u and v versus h for different values of e at t = π/4.

Figure 4. Resultant velocity Rn due to u and v versus h for different values of K2 at t = π/4.

Figure 5. Phase difference a due to u and v versus h for different values of l at t = π/4.

Figure 6. Phase difference a due to u and v versus h for different values of W at t = π/4.

Figure 7. Phase difference a due to u and v versus h for different values of K2 at t = π/4.

Figure 8. Phase difference a due to u and v versus h for different values of Gr, Gm, M and m at π/4.

Figure 9. Concentration profiles against f for different values of x at t = π/4.

Figure 10. Concentration profiles against f for different values of Sc at t = π/4.

Figure 11. Temperature profiles q against h for different values of QH at t = π/4.

Figure 12. Temperature profiles q against h for different values of R at t = π/4

Figure 13. Nusselt number against for different values of QH with l = 0.5, Pr = 0.71, R = 2, e = 0.01 at t = π/4.

Figure 14. Sherwood number against w for different values of x with l = 0.5, e = 0.01, Sc = 0.3 at t = π/4.

resultant velocity increases with increasing values of rotation parameter W. This is due to the fact that the rotation effects being more dominant near the walls, so when W reaches high values secondary velocity component v decreases with increases in W as shown in Figure 2. From Figure 3, it is observed that the increase in the e leads to an increase of Rn within the stationary plates. From Figure 4, it inferred that resultant velocity Rn goes on increasing with increasing value of viscoelastic parameter K2.

The phase difference a for the flow is shown graphically in Figures 5-8. Figure 5 shows phase angle for various positive values of suction/injection parameter l. The figure shows that the phase angle a decreases with the increases of suction parameter. Figure 6 is the phase angle for various values of rotation parameter W. From this figure, it is observed that the phase angle a decreases with an increase in rotation parameter. From Figure 7, it is observed that the phase angle a increases with an increase in visco elastic parameter. Figure 8 shows the variation of a against h for different values of thermal Grashof number Gr, Solutal Grashof number Gm, and Hartmaan number. From this figure, it is found that the values of phase a decreases with increasing values of Gr, Gm and M while reversed effect is observed for the Hall parameter m.

The concentration profile f for the flow is shown graphically in Figure 9, Figure 10. From Figure 9, it is observed that with the increasing the value of the chemical reaction parameter decreases the concentration of species in the boundary layer; this due to the fact that destructive chemical reduces the solutal boundary thickness and increases the mass transfer. Opposite trend is seen in the case when Schmidt number is increased as noted in Figure 10. It may also be observed from this figure that the effect of Schmit number Sc is to be increases the concentration distribution in the solutal boundary layer.

The temperature profiles q are shown graphically in Figure 11 and Figure 12. Figure 11 has been plotted to depict the variation of temperature profiles against h for different values of heat absorption parameter QH by fixing other physical parameters. From this figure, we observe that temperature q decreases with increase in the heat absorption parameter QH because when heat is absorbed, the buoyancy force decreases the temperature profile. Figure 12 represents graph of temperature distribution with h for different values of radiation parameter. From this figure, we note that initial temperature q = 1 decreases zero satisfying boundary condition at h = 1.0 Further, it is observed from this figure that increase in the radiation parameter decreases the temperature distribution in the thermal boundary layer due to decreases in the thickness of the thermal boundary layer with thermal radiation parameter R. This is because large values of radiation parameter correspond to an increase in dominance of conduction over radiation, thereby decreasing the buoyancy force and temperature in the thermal boundary layer.

Figure 13 and Figure 14 show the amplitude of skin-friction, Nusselt number, and Sherwood number against frequency parameter w for different values of QH, and, respectively. The amplitude of Nusselt number decreases with increasing the value of heat source parameter QH which is shown in Figure 13. Figure 14 shows the variation of Sherwood number with and w. From this figure, it is observed that the Sherwood number decreases with increasing the value of chemical reaction parameter, and opposite trend is seen with increasing the values of w.

Appendix

, ,

,

,

Nomenclature

Dimensional concentration

x-component of current density

Concentration at the left plate

Mean absorption coefficient

Specific heat at constant pressure

Chemical reaction rate constant

Specific heat at constant pressure

Hall parameter

Distance of the plate

Hartmann number

Chemical molecular diffusivity

Nusselt number

Electric charge

Number density of the electron

Acceleration due to gravity

Dimensional pressure

Modified Grashof number for mass transfer

Electron pressure

Modified Grashof number for heat transfer

Prandtl number

Magnetic field

Magnetic field of uniform strength

Dimensional heat source

x-Component of magnetic field

Heat source parameter

Current density

Temperature at the right wall

Resultant velocity

Dimensional time

Schmidt number

Non zero constant mean velocity

Sherwood number

Dimensional temperature

Electron velocity

Temperature at the left wall

Visco-elastic parameter

Dimensional injection /suction velocity

Primary velocity component for unsteady flow

Primary velocity component for steady flow

Secondary velocity component for steady flow

Secondary velocity component for unsteady flow

Velocity components are in the directions respectively.

Greek Symbols

Injection/suction parameter

Dynamic viscosity

Phase difference for the flow

Magnetic permeability

Phase difference of mass flux

Kinematic viscosity

Small positive constant

Oscillation parameter

Dimensionless distance

Dimensional parameter

Phase difference of heat flux

Angular velocity

Fluid thermal conductivity

Cyclotron frequency

Electric conductivity

Amplitude of mass flux

Stefan-Boltzmann constant

Amplitude of heat flux

Non dimensional concentration

Density

Non-dimensional temperature

Coefficient of thermal expansion

Coefficient of thermal expansion

Coefficient of solutal expansion

Phase difference of shear stresses for the steady flow

Phase difference of shear stresses for the unsteady flow

Phase difference of shear stresses for the flow

Amplitude of shear stresses for the flow

Amplitude of shear stresses for the steady flow

Amplitude of shear stresses for the unsteady flow

Conflicts of Interest

The authors declare no conflicts of interest.