Hartmann Flow in a Rotating System in the Presence of Inclined Magnetic Field with Hall Effects

Transcription

1 Tamkang Journal of Science and Engineering, Vol. 13, No. 3, pp (2010) 243 Hartmann Flow in a Rotating System in the Presence of Inclined Magnetic Field with Hall Effects G. S. Seth, Raj Nandkeolyar* and Md. S. Ansari Department of Applied Mathematics, Indian School of Mines University, Dhanbad , India Abstract Hartmann flow of a viscous incompressible electrically conducting fluid in a rotating system in the presence of an inclined magnetic field is studied. Solution for the velocity and induced magnetic field, in dimensionless form, contains four pertinent parameters viz. M 2 (square of Hartmann number), K 2 (rotation parameter), m (Hall current parameter) and (angle of inclination of magnetic field). Asymptotic behavior of the solution is analyzed for small and large values of K 2 and M 2 to gain some physical insight into the flow-pattern. For large values of K 2 and M 2, the flow field is divided into two regions, namely, (1) boundary layer region and (2) central core region. The expressions for the shear stress at the plates of channel due to the primary and secondary flows and mass flow rates in the primary and secondary flow directions are derived. It is found that the maxima of velocity profiles occur near the walls of channel which indicate the formation of boundary layers near the walls. It is noticed that the magnetic field decreases fluid temperature whereas rotation, Hall current, angle of inclination and Prandtl number increase it. Key Words: Coriolis Force, Hall Current, Modified Hartmann Boundary Layer, Modified Hydromagnetic Ekman Boundary Layer, Viscous and Joule Dissipations 1. Introduction Magnetohydrodynamics of rotating fluids is highly important due to its varied and wide applications in the areas of Geophysics, Astrophysics and Fluid Engineering. An order of magnitude analysis shows that in the basic field equations, the effects of Coriolis force are more significant as compared to that of inertial forces. Furthermore, it may be noted that the Coriolis and Magnetohydrodynamic forces are comparable in magnitude and Coriolis force induces secondary flow in the fluid. It is well known that, in an ionized fluid where the density is low and/or the magnetic field is strong, the effects of Hall current become significant as stated by Cowling [1]. Keeping in view this fact, Datta and Jana [2], Seth and Ghosh [3], Nagy and Demendy [4] and Hayat et al. [5] considered the effects of Hall current on hydromagnetic *Corresponding author. rajnandkeolyar@gmail.com flow in a rotating channel under different conditions. In all these investigations, magnetic field is applied parallel to the axis of rotation. However, in the problems of interest, it may not be possible to have applied magnetic field always parallel to the axis of rotation. Seth and Ghosh [6] studied steady hydromagnetic flow of a viscous incompressible electrically conducting fluid in a rotating channel with perfectly conducting walls in the presence of an inclined magnetic. Subsequently, Ghosh and Bhattacharjee [7] extended this problem to take into account the effects of Hall current. In their investigation they have not studied the effects of Hall current, magnetic field, rotation and angle of inclination of magnetic field on the shear stress at the walls of channel due to the primary and secondary flows, mass flow rates in the primary and secondary flow directions and fluid temperature and the effects of magnetic field and rotation on the velocity and induced magnetic field for large values of magnetic parameter M 2 and rotation parameter K 2. Also

2 244 G. S. Seth et al. they have not discussed the asymptotic behavior of the solution for the velocity and induced magnetic field for small as well as large values of K 2 and M 2. The aim of the present paper is to reconsider the problem studied by Ghosh and Bhattarcharjee [7]. The asymptotic behavior of the analytical solution is analyzed for small as well as large values of K 2 and M 2 to gain some physical insight into the flow pattern. The expressions for the shear stress at the plates of channel due to the primary and secondary flows and mass flow rates in the primary and secondary flow directions are derived. Heat transfer characteristics of the fluid flow is considered taking viscous and Joule dissipations into account. 2. Formulation of the Problem and its Solution Consider steady fully developed flow of a viscous incompressible electrically conducting fluid within a parallel plate channel z = L with perfectly conducting walls under the influence of a constant pressure gradient, which is applied in x-direction. The fluid is permeated by a uniform magnetic flux density B 0 applied in a direction, which is inclined at an angle with the positive direction of z-axis in xz-plane. Both the fluid and channel are in a state of rigid body rotation with uniform angular velocity about z-axis. The Physical model of the problem is presented in Figure 1. The hydromagnetic equations governing the fluid flow in a rotating frame of reference are (6) where q, B, J, E, t, k,,, e,, e, e and p are, respectively, fluid velocity, magnetic induction vector, current density, electric field, time, unit vector along z-axis, fluid density, kinematic coefficient of viscosity, magnetic permeability, electrical conductivity of the fluid, cyclotron frequency, electron collision time and fluid pressure including centrifugal force. On eliminating E and J from equations (3), (4) and (6), we obtain induction equation for magnetic field taking Hall current into account as (7) Since plates of the channel are very large in x and y directions in comparison to the width of the channel in z-direction and the flow is steady so all physical quantities, except pressure, depend on z only. Therefore equation of continuity (2) reduces to w 0 which, because z w = 0 at the plates, implies that w = 0 everywhere in the (1) (2) (3) (4) (5) and Ohm s law for a moving conductor taking Hall current into account is Figure 1. Physical model of the problem.

3 Hartmann Flow in a Rotating System in the Presence of Inclined Magnetic Field with Hall Effects 245 fluid. The solenoidal relation for magnetic field. B 0 leads to B z = constant. Since a uniform magnetic field B 0 cos acts along z-direction (see Figure 1), therefore, we assume Bz B0 cos. Thus the velocity and induced magnetic field are given by (8) [7] with slight change of notations (15) (16) Under the above assumptions equation of momentum (1) and induction equation for magnetic field (7), in dimensionless form, reduce to (17) (9) where and are real and imaginary parts of respectively and are given by (10) (11) (18) (12) (13) where = z/l, = x/l, u = u L/, v = v L/, B x = B x / e B 0, B y = B y / e B 0, p = L 2 p / 2, R = p/ is non-dimensional pressure gradient which is a constant, M 2 = B 2 0 L 2 / is magnetic parameter which is square of Hartmann number, m = e e is Hall current parameter, P m = e is magnetic Prandtl number and K 2 = L 2 / is rotation parameter which is reciprocal of Ekman number. The boundary conditions for the problem, in dimensionless form, are given by 2.1 Shear Stress at the Plates The non-dimensional shear stress components x and y at the upper and lower plates due to the primary and secondary flows, respectively, are given by (19) 2.2 Mass Flow Rates The non-dimensional mass flow rates Q x / and Q y / in the primary and secondary flow directions, respectively, are given by (20) (14) The solution of equations (9), (10), (12) and (13) subject to the boundary conditions (14) is obtained and expressedinthefollowingformwhichisinagreement with the solution obtained by Ghosh and Bhattacharjee 3. Asymptotic Solutions We shall now discuss the asymptotic behavior of the solution given by (15) to (18), for small and large values of K 2 and M 2, to gain some physical insight into the flow pattern.

4 246 G. S. Seth et al. Case I: M 2 << 1 and K 2 << 1 Since K 2 and M 2 are very small, neglecting squares and higher powers of K 2 and M 2 in (15) to (18), we obtain velocity and induced magnetic field distributions as (21) (22) (23) (28) The equations (25) to (28) reveal that, in the absence of Hall current, primary velocity u and primary induced magnetic field Bx are independent of rotation whereas secondary velocity v and secondary induced magnetic field By are unaffected by the applied magnetic field which is in agreement with the result obtained by Seth and Ghosh [6]. Case II: K 2 >> 1 and M 2~O(1) When K 2 is large and M 2 is of small order of magnitude, the fluid flow becomes boundary layer type. For the boundary layer flow near the upper plate h = 1, introducing boundary layer coordinate x = 1 - h, we obtain velocity and induced magnetic field from (15) to (18) as (29) (24) It is evident from the expressions (21) to (24) that in a slowly rotating system when the conductivity of the fluid is low, primary velocity u is independent of rotation whereas secondary velocity v and primary and secondary induced magnetic fields Bx and By respectively have considerable effects of rotation, Hall current, magnetic field and angle of inclination. This is due to the fact that Hall current induces secondary flow in the flow-field similar to that of Coriolis force. It may be noted that when the applied magnetic field is weak the effects of Hall current become insignificant and it can be neglected. Therefore, in the absence of Hall current i.e. when m = 0, the equations (21) to (24) reduce to (25) (26) (27) (30) (31) (32) where (33) The expressions (29) to (32) demonstrate the existence of a thin boundary layer of thickness O(1/a1) near upper plate of the channel. This boundary layer may be recognized as modified hydromagnetic Ekman boundary layer

5 Hartmann Flow in a Rotating System in the Presence of Inclined Magnetic Field with Hall Effects 247 and can be viewed as classical Ekman boundary layer modified by Hall current, magnetic field and angle of inclination. Similar type of boundary layer is also formed near lower plate of the channel. The thickness of this boundary layer increases with increase in and decreases with increase in either M 2 or K 2. The exponential terms in the expressions (29) to (32) damp out quickly as increases and when 1/ 1 i.e. outside the boundary layer region, we obtain where (40) (41) (34) (35) (42) The expressions (34) and (35) reveal that, in the central core region i.e. outside the boundary layer region, fluid flows in secondary flow direction only and is unaffected by magnetic field, Hall current and angle of inclination. Also induced magnetic field components B x and B y vary linearly with and have considerable effects of Hall current, rotation and angle of inclination. In the absence of Hall current, secondary induced magnetic field B y persists whereas primary induced magnetic field B x vanishes away which is in agreement with the result obtained by Seth and Ghosh [6]. Case III: M 2 >> 1 and K 2 ~O(1) In this case also boundary layer type flow is expected. For the boundary layer flow near the upper plate = 1, we obtain velocity and induced magnetic field from (15) to (18) as (36) (37) (38) (39) The expressions (36) to (39) show that there arises a thin boundary layer of thickness O(1/ 2 ) adjacent to upper plate of the channel. This boundary layer may be identified as modified Hartmann boundary layer and can be viewed as classical Hartmann boundary layer modified by Hall current, rotation and angle of inclination. A similar type of boundary layer appears near lower plate of the channel. The thickness of this boundary layer decreases with increase in either M 2 or K 2.In the absence of Hall current and angle of inclination there appears classical Hartmann boundary layer of thickness O(1/M) near the plates of the channel. In the central core region i.e. outside the boundary layer region, we obtain from (36) to (39) as (43) (44) It is evident from expressions (43) and (44) that in the central core region fluid flows in both the directions and are affected by magnetic field, Hall current and angle of inclination. In the absence of Hall current, fluid flows in primary flow direction only which is in agreement with the result obtained by Seth and Ghosh [6]. In the central core region, primary induced magnetic field B x persists and varies linearly with whereas secondary induced magnetic field B y vanishes away. Also pri-

6 248 G. S. Seth et al. mary induced magnetic field B x in the central core region is unaffected by Hall current and rotation. 4. Heat Transfer Characteristics We shall now discuss heat transfer characteristics of the flow field when the upper and lower plates are maintained at uniform temperatures T 1 and T 0 respectively. The energy equation for a viscous incompressible electrically conducting fluid taking viscous and Joule dissipations into account is given by (45) where T,, k, andc p are, respectively, fluid temperature, coefficient of viscosity, thermal conductivity of the fluid and specific heat at constant pressure. and J 2 / are viscous and Joule dissipation terms respectively. Under the assumptions made in section 2, the energy equation (45) reduces to (46) where * = k/ c p is thermal diffusivity of the fluid. The plates of the channel are maintained at constant temperatures, therefore, boundary conditions for the temperature field are (47) Using the non-dimensional variables defined in section 2 in equation (46), we obtain energy equation, in nondimensional form, as (48) where T( ) =(T T 0 )/(T 1 T 0 ), P r = / * is Prandtl number and E r = 2 /L 2 c p (T 1 T 0 ) is Eckert number. The equation (48) may be expressed as (49) where F ( ) and B ( ) are complex conjugate of F( ) and B( ) respectively. The boundary conditions (47), in non-dimensional form, become (50) Making use of the solution for velocity and induced magnetic field from equations (15) to (18) in equation (49), the resulting differential equation subject to the boundary conditions (50) is solved numerically with the help of MATLAB software. 5. Results and Discussion To study the effects of magnetic field and rotation on the velocity and induced magnetic field, the numerical values of the velocity components u and v and induced magnetic field components B x and B y, computed from the analytical solution reported in section 2 with the help of MATLAB software, are depicted graphically taking R = 1 in Figures 2 to 5 for various values of magnetic parameter M 2 and rotation parameter K 2 when M 2 and K 2 are large. Figures 2 and 3 show that profiles of the primary velocity u and secondary velocity v are flattened in the central core region and the maxima of profiles exist near the walls of channel which indicate the formation of boundary layers adjacent to the walls of channel for large values of M 2 and K 2. The primary velocity u decreases with the increase in M 2 near the walls of channel whereas it increases with increase in M 2 in the central core region. However, the secondary velocity v decreases with the increase in M 2 throughout the region. The primary as well as secondary velocity decreases with the increase in K 2. Figures 4 and 5 reveal that the primary induced magnetic field B x and secondary induced magnetic field B y decrease with the increase in either M 2 or K 2. The numerical values of the shear stress components

7 Hartmann Flow in a Rotating System in the Presence of Inclined Magnetic Field with Hall Effects Figure 2. Velocity profiles for large M 2 when K 2 = 25, m = 1 and q = Figure 3. Velocity profiles for large K 2 when M 2 = 81, m = 1 and q = 45. Figure 4. Induced magnetic field profiles for large M 2 when K 2 = 25, m = 1 and q = 45. Figure 5. Induced magnetic field profiles for large K 2 when M 2 = 81, m = 1 and q = 45. at the upper plate due to the primary and secondary flows and mass flow rates in the primary and secondary flow directions, computed with the help of MATLAB software, are presented in tabular form in Tables 1 to 4 for various values of M 2, K 2, m and q taking R = 1. It is evident from Table 1 that the shear stress t x h =1 due to primary flow and shear stress t y h =1 due to secondary flow decrease with the increase in either M 2 or K 2. It is found from Table 2 that t x h =1 decreases while t y h =1 increases with the increase in m. t x h =1 and t y h =1 increase with the increase in q. It is observed from Table 3 that, mass flow rate in primary flow direction Qx /ru increases and mass flow rate in secondary flow direction Qy /ru decreases with the increase in M 2 whereas Qx /ru and Qy / ru decrease with the increase in K 2. It is evident from Table 4 that Qx /ru decreases with the increase in m whereas Qy /ru increases with the increase in m except when q = 60. Qx /ru increases, attains a maximum and then decreases with the increase in q except when m = 1.5. For m = 1.5, it increases with the increase in q. Qy / ru increases with the increase in q. To study the effects of magnetic field, rotation, angle of inclination and Hall current on temperature field the numerical solution for the fluid temperature T is presented graphically in Figures 6 to 9 for various values of M 2, K 2, q and m taking Pr = 0.71 (ionized air), Er = 2 and R = 1. It is found from Figures 6 to 9 that fluid temperature T decreases with the increase in M 2 whereas it in-

8 250 G. S. Seth et al. Table 1. Shear stress at the upper plate when m =1and =45 M 2 K 2 x 1 y Table 2. Shear stress at the upper plate when M 2 =81and K 2 =25 m x 1 y Table 3. Mass flow rates when m =1and =45 M 2 Q x / Q y / K Table 4. Mass flow rates when M 2 =81andK 2 =25 m Q x / Q y / Figure 6. Temperature profiles when K 2 =9,m =1and = 45. Figure 8. Temperature profiles when M 2 =9,K 2 =9andm =1. Figure 7. Temperature profiles when M 2 =9,m = 1 and = 45. Figure 9. Temperature profiles when M 2 =9,K 2 = 9 and = 45.

9 Hartmann Flow in a Rotating System in the Presence of Inclined Magnetic Field with Hall Effects 251 creases with the increase in either K 2 or or m. We know that Prandtl number P r is a measure of the relative importance of the viscosity and thermal conductivity of the fluid. It is desirable to know the nature of fluid temperature with respect to Prandtl number. The profiles of fluid temperature T are drawn for different values of P r taking M 2 =9,K 2 =9, =45, m =1,E r = 2 and R = 1 in Figure 10. It is evident from Figure 10 that fluid temperature T increases with the increase in P r. Fluid temperature T for liquid metal (P r = 0.01) is smaller than that of ionized air (P r = 0.71) and water (P r = 7). For P r > 1 i.e. when there is lesser contribution of transport energy as compared to that of momentum, the fluid temperature T increases rapidly from the lower plate, attains a maximum value in central core region and then decreases with the increase in channel width to approach its boundary value at the upper plate whereas for P r < 1 i.e. when there is greater contribution of transport energy as compared to that of momentum, the fluid temperature T increases slowly from the lower plate with the increase in to approach its boundary value at the upper plate. 6. Conclusion Figure 10. Temperature profiles when M 2 =9,K 2 =9, =45 and m =1. An analysis of steady Hartmann flow of a viscous incompressible electrically conducting fluid in a rotating system in the presence of an inclined magnetic field is presented. It is found that profiles of primary velocity u and secondary velocity v are flattened in central core region and the maxima of profiles exist near the walls of channel which indicate the formation of boundary layers near the walls of channel. It is also observed that rotation has retarding influence on the primary as well as on the secondary velocity throughout the region whereas magnetic field has retarding influence on the secondary velocity throughout the region and has accelerating influence on the primary velocity in the central core region. Magnetic field and rotation reduce primary and secondary induced magnetic fields B x and B y respectively throughout the region. It is noticed that for large K 2 and M 2, fluid flow is divided into two regions, namely, boundary layer region and central core region. For large K 2, boundary layer region is confined to modified hydromagnetic Ekman boundary layer of thickness O(1/ 1 ) which arises near the walls of channel. For large M 2, boundary layer region is confined to modified Hartmann boundary layer of thickness O(1/ 2 ) which appears adjacent to the walls of channel. It is found that the angle of inclination has tendency to increase both the shear stress components while Hall current reduces shear stress component in primary flow direction and increases it in secondary flow direction. Hall current reduces mass flow rate in primary flow direction and increases it in secondary flow direction except when = 60. Angle of inclination tends to increase mass flow rate in secondary flow direction. It is noticed that the magnetic field reduces fluid temperature whereas rotation, angle of inclination, Hall current and Prandtl number have tendency to increase it. Nomenclature B 0 uniform magnetic flux density B x, B y components of induced magnetic field c p specific heat at constant pressure E r Eckert number k thermal conductivity of the fluid K 2 rotation parameter L half width of the channel m Hall current parameter M 2 magnetic parameter p fluid pressure including centrifugal force P m magnetic Prandtl number P r Prandtl number R non-dimensional pressure gradient T 0, T 1 temperatures of lower and upper plates respectively

10 252 G. S. Seth et al. T fluid temperature u, v components of velocity fluid density electrical conductivity of the fluid uniform angular velocity e cyclotron frequency e electron collision time * thermal diffusivity of the fluid angle of inclination of applied magnetic field viscous dissipation term boundary layer coordinate coefficient of viscosity e magnetic permeability kinematic coefficient of viscosity Superscripts dimensional properties * opposite direction of axis Acknowledgement The authors are grateful to the referees for providing many useful suggestions to improve the paper in its present form. References [1] Cowling, T. G., Magnetohydrodynamics, New York: Interscience Publisher Inc (1957). [2] Datta, N. and Jana, R. N., Hall Effects on Hydromagnetic Flow and Heat Transfer in a Rotating Channel, J Inst Math Applics., Vol. 19, p. 217 (1977). [3] Seth, G. S. and Ghosh, S. K., Effect of Hall Current on Unsteady Hydromagnetic Flow in a Rotating Channel with Oscillating Pressure Gradient, Ind J Pure Appl Maths., Vol. 17, p. 819 (1986). [4] Nagy, T. and Demendy, Z., Effects of Hall Current and Coriolis Force on Hartmann Flow Under General Wall Conditions, Acta Mech., Vol. 113, p. 77 (1995). [5] Hayat, T., Wang, Y. and Hutter, K., Hall Effects on the Unsteady Hydromagnetic Oscillatory Flow of a Second-Grade Fluid, Int J Non-linear Mech., Vol. 39, p (2004). [6] Seth, G. S. and Ghosh, S. K., Hydromagnetic Flow in a Rotating Channel in the Presence of Inclined Magnetic Field, Proc Math Soc BHU, Vol. 11, p. 111 (1995). [7] Ghosh, S. K. and Bhattacharjee, P. K., Hall Effects on Steady Hydromagnetic Flow in a Rotating Channel in the Presence of an Inclined Magnetic Field, Czech J Phys., Vol. 50, p. 759 (2000). Manuscript Received: Feb. 20, 2009 Accepted: Aug. 14, 2009