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*versión impresa* ISSN 0327-0793

### Lat. Am. appl. res. v.38 n.4 Bahía Blanca oct. 2008

**Baro-diffusion and thermal-diffusion in a binary fluid mixture confined between two parallel discs in presence of a small axial magnetic field**

**B. R. Sharma ^{1} and R. N. Singh^{2}**

^{1} *Department of Mathematics, Dibrugarh University, Dibrugarh-786004, India bishwaramsharma@yahoo.com*

^{2}

*Marwari Hindi High School, Dibrugarh-786001, India*

nirojsingh@yahoo.co.in

nirojsingh@yahoo.co.in

*Abstract* — The effect of a small uniform axial magnetic field on separation of a binary mixture of incompressible viscous thermally and electrically conducting fluids sheared between two parallel discs of which one is stationary and the other is rotating with a constant angular velocity, is examined. It is assumed that one of the components, which is rarer and lighter, is present in the mixture in a very small quantity. Neglecting the induced electric field the equations governing the motion, temperature distribution and concentration distribution are solved in cylindrical polar coordinate by expanding the flow parameters as well as the temperature and the concentration in powers of Reynolds number. The solution obtained for concentration distribution is plotted against the width of the channel for various values of non-dimensional parameters. It is found that the pressure gradient and the temperature gradient favour the separation of the components of binary fluid mixture but the axial magnetic field retards the same. Near the rotating disc the rarer and lighter component of the binary mixture increases with the increase in the value of the Reynolds number but reverse effect is found in the central part as well as near the stationary discs.

*Keywords* — Binary Mixture. Incompressible Fluid. Thermal Diffusion. Baro Diffusion. Magnetic Field.

**I. INTRODUCTION**

Separation processes of components of a fluid mixture wherein one of the components is present in extremely small proportion are of much interest due to their applications in science and technology. Separation of isotopes from their naturally occurring mixture is one of such examples. It is well known that only one part of heavy water which is an isotope of water is found in 25000 parts of water in normal occurrence (Arnikar, 1963; Rastogy *et al.*,1992) but it is required for use as a (i) moderator in nuclear reactions for slowing down the neutrons, (ii) tracer compound for studying the mechanism of many chemical reactions and (iii) heat transport medium *i.e.*, a coolant in atomic power plant. Because of their small relative mass difference isotopes of heavier molecules offer the greatest practical challenge in attempts to isolate the rarer component. Electromagnetic method of separation (Srivastava, 1999a) works only at relatively higher values of concentrations.

In a binary fluid mixture the diffusion of individual species takes place by three-mechanisms namely ordinary diffusion, pressure diffusion (or baro-diffusion) and thermal diffusion. The diffusion flux **i** of lighter and rarer component is given by Landau and Lifshitz (1960) as:

(1) |

where ρ is the density of the binary fluid mixture, D is the diffusion coefficient, c_{1} is the ratio of mass of the lighter component to the total mass of the fluid, k_{p}D is the pressure diffusion coefficient, p is pressure, k_{T}D is the thermal diffusion coefficient and T is temperature. The ordinary diffusion contribution to the mass flux is seen to depend in a complicated way on the concentration gradients of the components present in the mixture. The baro-diffusion indicates that there may be a net movement of the components in a mixture if there is a pressure gradient imposed on the system. An example of baro –diffusion is the process of diffusion in the binary mixture of different kinds of gases present in the atmosphere. By reasons of variation of forces of gravity with height thereby causing a density gradient, different constituents in the atmosphere tend to separate out. The pressure gradient created by the gravity as well as the rotation of the earth separates various components of air. The tendency for a mixture to separate under a pressure gradient is very small but use is made of this effect in centrifuge separations in which tremendous pressure gradient is established. Thermal diffusion describes the tendency for species to diffuse under the influence of a temperature gradient. In many practical problems dealing with flows in porous media one encounters with a multiple component electrically conducting fluids e.g. molten fluid in the earth's crust, crude oil in the petroleum. It is customary to consider one of the components as solvent and the other component as solute. It is shown in reference Groot and Mazur (1962) that if separation due to thermal diffusion occurs then it may even render an unstable system to stable one. This effect is also quite small, but devices can be arranged to produce very steep temperature gradients so that separations of mixtures are effected.

Sarma (1973) perhaps was the first to study the problem of baro-diffusion in a binary mixture of incompressible viscous fluids set in motion due to an infinite disk rotation. He obtained results on separation action in this configuration for small baro –diffusion number taking the Schmidt number to be of the order of unity and including the effect of separation at the disk. Hurle and Jakeman (1971) have discussed the effect of a temperature gradient on diffusion of a binary mixture.

Many investigators (Sarma, 1973, 1975; Sharma and Hazarika, 2002; Sharma, 2003, 2004; Sharma and Shah, 1996, 2002; Sharma and Gogoi, 2000; Sharma and Singh, 2003, 2004, 2007; Srivastava, 1979, 1985,1991, 1999a, 1999b, 1999c; Zimmermann *et al.*, 1992; Zimmermann and Muller, 1992) analyzed the effect of baro-diffusion and thermal diffusion on separation of a binary mixture of electrically non-conducting fluids under various geometry. In all these investigations it has been found that an increase in the pressure gradient and/or the temperature gradient can enhance the separation. In many cases the fluid mixture is found to be electrically conducting and so to study the effect of magnetic field on separation we have considered in this paper a binary mixture of incompressible viscous thermally and electrically conducting fluids sheared between two parallel discs one of which is rotating in its own plane in presence of a small axial magnetic field. The heat transfer for the same geometry has been addressed by Srivastava and Sharma (1964).

**A. Governing equations and boundary conditions.**

We consider here the case when one of the components of the binary mixture of incompressible thermally and electrically conducting viscous fluids is present in small quantity, hence the density and viscosity of the mixture is independent of the distribution of the components. The concentration c_{2} of heavier and more abundant component is given by c_{2} =1- c_{1}. The flow problem of the binary mixture is identical to that of a single fluid but the velocity is to be understood as the mass average velocity **V** = (ρ_{1}**V**_{1} + ρ_{2}**V**_{2})/ρ and the density ρ =ρ_{1} + ρ_{2}, where the subscripts 1 and 2 denote the rarer and the more abundant components respectively. The equation of continuity and the equation of motion of an incompressible fluid in steady case are, respectively,

(2) |

and

(3) |

where µ is the coefficient of viscosity of the binary fluid mixture and **B** is the magnetic flux density vector. In steady motion the Maxwell equations are given by

curl H = 4πJ, | (4) |

curl E = 0, | (5) |

div H =0, | (6) |

and Ohm's law, on neglecting Hall current, is given by

(7) |

where **J** is the current density vector and

(8) |

**H** is the magnetic field vector, **E** is the electric field vector, σ is the electric conductivity and µ_{e} is the magnetic permeability.

The energy equation in steady case is given by

(9) |

where c_{p} is the specific heat at constant pressure, k is the thermal conductivity of the fluid mixture, Φ represents the heat due to viscous dissipation and the last term **J**^{2}/σ represents heat due to Joulean dissipation.

The equation for species conservation of the first component is given by (see Landau and Lifshitz, 1959)

(10) |

where **i** is given by (1). The coefficients k_{p} and k_{T} may be determined from the thermodynamic properties alone. Landau and Lifshitz (1959) have given the explicit expression for the baro-diffusion ratio k_{p} as

(11) |

where p_{∞} denotes the working pressure of the medium and m_{1}, m_{2} are masses of two kinds of particles. Neglecting c_{1}^{2} (since concentration of rarer and lighter component c_{1} is very small) (2.10) becomes

(12) |

where

(13) |

The expression for k_{T} has been suggested by Hurle and Jakeman (1971) as

(14) |

where s_{T} is the Soret coefficient. For small values of c_{1} (14) becomes

(15) |

Substituting the expression for **i** from (1), k_{p} from (12) and k_{T} from (15) in (10) we get the Eq. for c_{1} as

(16) |

Boundary conditions for the flow field, temperature field and electromagnetic field are the same as in the usual magneto-hydrodynamic problems. The boundary conditions for the concentration c_{1} are different in different cases. At the surface of a body insoluble in the fluid mixture the total mass flux as well as the individual species flux normal to the surface should vanish (see Srivastava, 1979) *i.e.*,

(17) |

where **n** is the unit normal drawn at the solid surface directed outwards. Substituting the expression for **i** from (1) into (17), we get,

(18) |

If, however, there is diffusion from a body that dissolves in the fluid, equilibrium is rapidly established near its surface, and the concentration in the fluid adjoining the body in this case is the saturation concentration c_{0} (say); the diffusion out of this layer takes place more slowly than the process of solution. The boundary condition at such surface is, therefore,

c = c_{0}. | (19) |

**II. METHODS**

**A. Formulation of the problem.**

We consider here the steady flow of a binary mixture of thermally and electrically conducting viscous incompressible fluids by using cylindrical polar coordinate system (r, θ, z). The binary fluid mixture is sheared between two infinite circular discs at z = 0 and z = d. The discs are maintained at different constant temperatures T_{0} and T_{1} (T_{0}>T_{1}) at z = 0 and z = d respectively. An axial magnetic field of uniform strength B_{0} is applied. It is assume that the disc at z = 0 rotates with a constant angular velocity Ω in its own plane and the other disc at z = d is at rest. Let u, v, w be the velocity components in the directions of r, θ, z respectively.

In axisymmetric case, the governing Eqs. (2), (3), (9) and (16) for the steady flow of a binary mixture of incompressible thermally and electrically conducting viscous fluids sheared between two electrically non-conducting parallel discs in presence of a uniform axial magnetic field becomes

(20) | |

(21) | |

(22) | |

(23) | |

(24) | |

| (25) |

where is the kinematic coefficient of viscosity,

(26) |

and

(27) |

The boundary conditions of the problem are given by

(28) |

and

(29) |

To reduce the partial differential Eqs. (20)-(25) to ordinary differential equations we take the forms of the velocity components, pressure, temperature and concentration:

(30) |

where η=z/d and c_{0}<<1 is the representative value of the concentration of the rarer and lighter species, say the value in the undisturbed mixture.

For the above form of the temperature distribution and the concentration distribution our solution is valid and meaningful in axial zone.

Using (30) in (20)-(25) and equating the coefficients of like powers of r from both sides, we get

(31) | |

(32) | |

(33) | |

(34) | |

(35) | |

(36) | |

(37) | |

(38) |

and

(39) |

where R=ρΩd^{2}/µ is the Reynolds number, M = µ_{e}H_{0}d(σ/µ)^{1/2} is the Hartmann number, H_{0} is the constant non-dimensional axial magnetic field, P_{r} = µc_{p}/k is the Prandtl number, s_{m =} µ/(ρD) is the Schmidt number, β= AµΩ is the baro-diffusion coefficient, t_{d} = s_{T}(T_{0}-T_{1}) is the thermal diffusion number and E_{c} = νΩ/{c_{p}(T_{0}-T_{1})} is the Eckert number.

The boundary conditions (28)- (29) under the forms in (30) become,

(40) |

and

(41) |

For small values of R, a regular perturbation scheme can be developed by expanding F, G, H, p_{1}, p_{2}, f_{1}, f_{2}, g_{1} and g_{2} in powers of R. We assume,

(42) |

Substituting the expansions (42) in the equations (31)-(39) and equating the coefficients of R^{0} and R^{1} on both sides, we get

(43) | |

(44) | |

(45) | |

(46) | |

(47) | |

(48) | |

(49) | |

(50) | |

(51) | |

(52) | |

(53) | |

(54) | |

(55) | |

(56) | |

(57) |

and

(58) |

Also, the boundary conditions (40)-(41) under the assumptions (42) becomes

(59) |

and

(60) |

where ∀n ∈ W, W represents the set of whole numbers.

**B. Solutions of the problem.**

The solutions of Eqs. (43) to (58) under boundary conditions (59)-(60) are obtained as

(61) | |

(62) | |

(63) | |

(64) | |

(65) | |

(66) | |

(67) | |

(68) | |

(69) | |

(70) | |

(71) |

where

(72) | |

(73) | |

(74) | |

(75) |

To get an estimate of mass concentration of the lighter and rater component of the mixture the average value of concentration is calculated from

(76) |

where 'a' is the radius of the finite discs.

Substituting *c*_{1}(η) from (30) in (76) by making the use of (61)-(71), we get,

(77) |

**III RESULTS**

In absence of magnetic field the Eq. (77) produces a singular solution. So, putting B_{0} = 0 in equations (43)-(58) and solving under the boundary conditions (59)-(60), we get

(78) |

If we put t_{d} = 0 and β = 0 in expressions (77) and (78) for average concentration of the first component of the binary fluid mixture we get = *c*_{0} for all values of η. From this we can conclude that the separation of species ceases to take place if we neglect the combine effect of the thermal diffusion number and baro-diffusion coefficient. Since the Schmidt number s_{m} occurs in terms of order of R^{2} therefore its effect on separation of species of the binary mixture is very small.

**Figure 1** reveals that the species separation increases with the increase of the values of parameters β and Fig. 2 reveals that the separation of species increases with the decrease in the value of the thermal diffusion number t_{d}. Our these conclusions are found to be good agreement with the conclusions derived by the researchers (Sarma, 1973, 1975; Sharma, 2003, 2004; Sharma and Hazarika, 2002; Sharma and Shah, 1996, 2002; Sharma and Gogoi, 2000; Sharma and Singh, 2003, 20004, 2007; Srivastava, 1979, 1991, 1999a, 1999b; Zimmermann et al., 1992; Zimmermann and Mulller, 1992).

Fig. 1. The graph of concentration distribution against the width of the channel η for various values of baro-diffusion number by taking t_{d}=0.001, M=0.2, R=0.1, a/d=250, P_{r}E_{c}=0.001.

Fig. 2. The graph of concentration distribution against the width of the channel η for various values of the thermal diffusion number by taking β=0.0025, M=0.2, R=0.1, a/d=250, P_{r}E_{c}=0.001.

**Figure 3**, Fig. 4 and Fig. 5 reveal that the values of that decreases with increase in the value of M, a / d and P_{r}E_{c} . Hence we can conclude that the effect of decrease in the values of magnetic parameter M, the aspect ratio a/d, the product of Prandtl number P_{r} and the Eckert number E_{c} is to reduce the process of species separation.

Fig. 3. The graph of concentration distribution against the width of the channel η for various values of the Hermann number by taking t_{d}=0.001, β=0.0025, R=0.1, a/d=250, P_{r}E_{c}=0.001.

Fig. 4. The graph of concentration distribution against the width of the channel η for the various values of the aspect ratio a/d by taking t_{d}=0.001, M=0.2, R=0.1, β=0.0025, P_{r}E_{c}=0.001.

Fig. 5. The graph of concentration distribution against the width of the channel η for various values of the product of Prandtl number and Eckert number by taking t_{d}=0.001, M=0.2, R=0.1, a/d=250, β=0.0025.

**Figure 6** reveals that the species separation increases by increasing the values of Reynolds number but the reverse effect is observed in the region 0.06 < η < 1. In all of the figures, we observe that the concentration of lighter component is more at η=1 than its value at η=0.

Fig. 6. The graph of concentration distribution against the width of the channel η for various values of the Reynolds number by taking t_{d}=0.001, M=0.2, β=0.0025, a/d=250, P_{r}E_{c}=0.001.

**IV. CONCLUSIONS**

The problem of mass transfer due to the flow of an electrically and thermally conducting, viscous incompressible binary fluid mixture between two infinite parallel discs, one rotating and the other at rest, under the influence of a magnetic field acting perpendicular to the discs, has been investigated under the assumption that one of the components, which is rarer and lighter, is present in the mixture in a very small quantity. Analytical solutions of the governing equations have been obtained by expanding the flow parameters as well as the temperature and the concentration in powers of Reynolds number. Different analytic expressions are obtained for non-dimensional velocity, temperature and concentration profiles in presence of the magnetic field. An analytic expression is also obtained for the concentration profile in absence of magnetic field. The specific conclusions derived from this study can be listed as follows:

- the combined effect of the thermal diffusion number and the barodiffusion coefficient is to separate the species of the binary fluid mixture i.e. in absence of the barodiffusion coefficient and the thermal diffusion number separation of species ceases to take place;

- the species separation increases with the increase of the barodiffusion coefficient and with the decrease of the thermal diffusion number;

the effect of decrease in the values of the magnetic parameter, the aspect ratio, the Prandtl number and the Eckert number is to reduce the - separation of species of the binary fluid mixture;

- the effect of increase in values of the Reynolds number is to increase the species separation in the region 0<η<0.5 but a reverse effect is observed in the rest of the region;

- it is observed that the concentration of the lighter component of the binary mixture is more at the stationary disc than at the rotating disc.

Thus the effect of the pressure gradient and the temperature gradient is to separate the components of the binary fluid mixture by throwing the lighter component towards the stationary disc and collect the heavier component near the surface of the rotating disc and thus affecting the process of separation. The influence of the axial magnetic field is to retard the process of separation.

**ACKNOWLEDGEMENT**

The Authors would like to record their sincere thanks to, the referees for their helpful suggestions which contributes in improving the work and Prof. Adrian Posteinicu, Head, Dept. of Fluid Mechanics and Thermal Engineering, Transilvania University of Brasov, Bdul Eroilor. No 29, Brasov, 500036, Romaia for providing references which helped us to explain the queries made by the referees.

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**Received: March 11, 2007. Accepted: Jannary 27, 2008. Recommended by Subject Editor: Walter Ambrosini**