Hall Effects on MHD Flow in a Rotating Channel in the Presence of an Inclined Magnetic Field

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1 Journal of Applied Science and Engineering, Vol. 17, No. 3, pp (2014) DOI: /jase Hall Effects on MHD Flow in a Rotating Channel in the Presence of an Inclined Magnetic Field Bhaskar Chandra Sarkar 1, Sanatan Das 2 * and Rabindra Nath Jana 1 1 Department of Applied Mathematics, Vidyasagar University, Midnapore , India 2 Department of Mathematics, University of Gour Banga, Malda , India Abstract MHD flow of a viscous incompressible electrically conducting fluid between two parallel plates in a rotating system in the presence of an inclined magnetic field has been studied on taking Hall currents into account. An exact solution of the governing equations has been obtained in closed form. Numerical results of the fluid velocity components and the shear stresses at the plates are being discussed graphically. It is observed that both Hall currents as well as the angle of inclination of the applied magnetic field have a retarding influence on the primary fluid velocity whereas they accelerate the secondary fluid velocity. The electric field components are being calculated and presented in tabular form. Asymptotic behavior of the solution has been analyzed for small as well as large values of magnetic parameter and rotation parameter. It is interesting to note that either for strong magnetic field or for large rotation there exists a single-deck boundary layer near the upper plate. Key Words: MHD Flow, Hall Current, Magnetic Parameter, Rotation Parameter, Angle of Inclination 1. Introduction MHD flow in a rotating system has received wide attention to many researchers due to its varied and wide applications in many areas of science and technology. The rotating flow of an electrically conducting fluid in the presence of a magnetic field is encountered in geophysical fluid dynamics. It is also important in the solar physics dealing with the sunspot development, the solar cycle and the structure of rotating magnetic stars. It is well known that a number of astronomical bodies possess fluid interiors and magnetic fields. Changes that take place in the rate of rotation, suggest the possible importance of hydromagnetic spin-up. When the magnetic field is strong or the density of the gas is low in the ordinary magnetogasdynamic flows, the Hall effects due to the gyration of the electrons becomes important. The *Corresponding author. tutusanasd@yahoo.co.in current induced in a fluid is usually carried predominantly by electrons, which are considerably more mobile than ions. The electron drift velocity in most cases leads to a second component of the flow velocity, which in turn leads to a secondary force and causes anisotropic electrical conductivity in the flow. The current component created by this anisotropic conductivity is known as the Hall current. In the presence of a uniform magnetic field, this effect modifies the Ohm s law, increasing the order of the differential equations describing the motion and yields a dispersive character to the flow. As a results, there appears various characteristic flow fields in association with the conductivity and compressibility of the flow. Hall currents can have a significant effect on MHD energy systems and plasma flow in accelerators. The presence of longitudinal Hall currents in a flow creates a transverse body force which can lead to transverse pressure gradients, velocity gradients etc. The hydromagnetic flow in a rotating channel with

2 244 Bhaskar Chandra Sarkar et al. Hall effects has been discussed by Seth and Ghosh [1]. Ghosh [2,3] has studied the Hall effects on MHD flow in a rotating channel permeated by an inclined magnetic field in the presence of an oscillator. Hall effects on MHD plasma flow in a rotating environment in the presence of an inclined magnetic field have been analysed by Ghosh and Pop [4,5]. Mondal et al. [6] have studied the Hall effects on MHD plasma Couette flow in a rotating system. The Hall effects in the viscous flow of an ionized gas between parallel plates under transverse magnetic field have been examined by Sato [7]. An analysis has been made on an oscillatory Couette flow in the presence of an inclined magnetic field by Guria et al.[8]. Seth et al. [9] have studied the MHD Couette flow in a rotating system in the presence of inclined magnetic field. Seth and Ghosh [10,11] have initiated the study of the hydromagnetic flow in a rotating channel in the presence of an inclined magnetic field neglecting induced magnetic field. The steady Hartmann flow of a viscous incompressible electrically conducting fluid in a rotating channel in the presence of an inclined magnetic field taking induced magnetic field into account considering different aspects of the problem has been investigated by Ghosh and Bhattacharjee [12]. Seth et al. [13] have studied the Hydromagnetic flow in a rotating channel in the presence of an inclined magnetic field. Hall effects on oscillatory hydromagnetic Couette flow in a rotating system have been observed by Seth et al. [14]. Hall effects on MHD Couette flow between two infinite horizontal parallel porous plates in a rotating system under the boundary layer approximations have been studied by Das et al. [15]. Bég et al. [16] have presented the magnetohydrodynamic viscous plasma flow in rotating porous media with Hall currents and inclined magnetic field influence. Effects of Hall currents on oscillatory hydromagnetic Couette flow of class-ii in a rotating system in the presence of an inclined magnetic field have been investigated by Seth et al. [17]. Recently, Seth et al. [18] have studied the unsteady hydromagnetic Couette flow of a viscous incompressible electrically conducting fluid in a rotating system in the presence of an inclined magnetic field taking Hall currents into account. Ghosh and Pop [19] have studied the MHD plasma behavior of a rotating environment between two infinite parallel plates channel in the presence of inclined magnetic field. In the formulation of the problem they have assumed that for continuous medium j 0, that is, the conservation of the electric current does not hold. This assumption is seemed to be incorrect. The conservation of electric current must hold good for any conducting fluid. Further, the equation (2.4) of Ghosh and Pop [19] shows that the electric field is non-zero as E ( Ex, Ey, Ez). But equations (2.5)(2.7) of Ghosh and Pop [19] show that electric field as well as induced magnetic field are zero. Hence the Eq. (2.4) of Ghosh and Pop [19] are not compatible with the fundamental equations of MHD as they have stated. In the present study we have reconsidered the problem studied by Ghosh and Pop [19]. We have obtained the solution of the problem where electric field need not be equal to zero ( E 0) and the conservation of electric current ( j 0 ) holds. The effects of the Hall currents, the rotation and the angle of inclination of the magnetic field on both the velocity distribution have been studied. It is seen that the Hall currents and the angle of inclination retard the primary velocity while they have an accelerating influence on the secondary velocity. The rotation has retarding influence on both the primary velocity and the secondary velocity. Asymptotic behavior of the solution is analyzed for small as well as large values of magnetic parameter M 2 and rotation parameter K 2 to gain some physical insight into the flow pattern. It is found that a thin boundary layer is formed near the upper plate, for large values of rotation parameter K 2. The thickness of this boundary layer decreases with increase in magnetic parameter M Mathematical Formulation and Its Solution The basic equations of magnetohydrodynamics for steady flow are (1) (2) (3) (4) (5)

3 Hall Effects on MHD Flow in a Rotating Channel in the Presence of an Inclined Magnetic Field 245 (6) together with generalised Ohm s law on taking Hall currents into account is (see Cowling [20]) (7) where q, B, E, j and D are respectively the velocity vector, the magnetic field vector, the electric field vector, the current density vector and the displacement vector. Also,,, e,, e, p, B 0, e and e are respectively the electric conductivity, kinematic viscosity, magnetic permeability, fluid density, charge density, modified fluid pressure including centrifugal force, applied magnetic field, cyclotron frequency and electron collision time. Consider the steady hydromagnetic flow of a viscous incompressible electrically conducting fluid between two horizontal parallel plates at z = L, rotating with a uniform angular velocity about an axis perpendicular to the plates. The plates and the fluid rotate in unison with uniform angular velocity. A uniform magnetic field B 0 is applied at an angle with the vertical as shown in Figure 1. Since the plates are infinitely long along x and y-directions, all physical quantities, except pressure, will be function of z only. It is assumed that the induced magnetic field may be neglected in comparison to the applied magnetic field since the magnetic Reynolds number is very small for liquid metals for partially ionized gases. The solenoid relation B 0 gives B z = B 0 cos everywhere in the fluid where B (B 0 sin, 0,B 0 cos ). The conservation of electric current j 0 yields ' jz = constant where ' j ( jx, jy, jz). This constant is zero since j z = 0 at the plates which are electrically non-conducting. ' Hence, j z = 0 everywhere in the flow. Since the motion is steady, we have from Maxwell equation E 0 which in turn yields E x = constant and E y = constant everywhere in the flow. In view of the above assumptions, Eq. (7) yields values of m, B 0 is upwards and the electrons of the conducting fluid gyrate in the same sense as the rotating system. For negative values of m, B 0 is downwards and the electrons gyrate in an opposite sense to the rotating system. In general, for an electrically conducting fluid, Hall currents affect the flow in the presence of a strong magnetic field. The effect of Hall currents gives rise to a force in the y-direction, which induces a cross flow in that direction. To simplify the problem, we assume that there is no variation of flow quantities in y-direction. This assumption is considered to be valid if the surface be of infinite extent in the y-direction. ' ' Solving for j x and j y from (8) and (9), we have (11) (12) On the use of (11) and (12), equations of motion in a rotating frame of reference are (13) (14) (8) (9) (10) where m = e e is the Hall parameter. For positive Figure 1. Geometry of the problem.

4 246 Bhaskar Chandra Sarkar et al. (15) The boundary conditions are (23) Introducing the non-dimensional variables (16) where E 0 = e y ie x, (17) (24) Eqs. (13)(15) become (18) (19) and E 0 is the complex conjugate of E 0. On the use of (17) and combining Eqs. (11) and (12), we have where J = j y ij x, F = u + iv. 3. Results and Discussion (25) where M B L (20) is the magnetic parameter, K 2 = L 2 * the rotation parameter, R p x the pressure 2 Lp gardient and p * 2. The boundary conditions (16) become (21) Eqs. (18) and (19) can be solved subject to the boundary conditions (21) and the solution can be written as (22) To study the effects of Hall currents, rotation and the angle of inclination of the magnetic field on the velocity field, the numerical values of the fluid velocity components are shown graphically against for several values of Hall parameter m, rotation parameter K 2 and angle of inclination of the magnetic field in Figures 24 when M 2 = 10 and R = 1. For M 2 = 0, the hydromagnetic drag force vanishes and we have purely hydrodynamic channel flow in a rotating system. It is seen from Figure 2 that the primary velocity u decreases with an increase in rotation parameter K 2. The primary velocity u is more near the plates than that at the middle of the channel. Further, the hump near the channel plates exhibits the formation of the boundary layer in the vicinity of the plates. It is also seen that the magnitude of the secondary velocity v decreases for increasing values of K 2. This means that the rotation has retarding influence on both the primary and secondary velocities. Ekman number expresses the relative significance of viscous hydrodynamic and rotational (Coriolis) forces. The rotation parameter K 2 is inversely proportional to Ekman number. For K 2 = 1, the viscous and rotational forces are of the same order of

5 Hall Effects on MHD Flow in a Rotating Channel in the Presence of an Inclined Magnetic Field 247 Figure 2. Primary and secondary velocities for different K 2 when =45 and m = 0.5. magnitude. For K 2 > 1, the rotational effects clearly dominate viscous effects. For K 2 < 1, the rotational effects are dominated by viscous effects. Coriolis body forces arise in both the primary (main flow) and secondary (cross flow) momentum Eqs. (18) and (19) via -2K 2 v and 2K 2 u, respectively. Large values of Ekman number imply lesser rotational effects. This rotational drag force is a positive body force which therefore accelerates the flow i.e. boosts momentum. However as K 2 increases the magnitude of this force is reduced which causes a reduction in the fluid velocities as shown in Figure 2. A key objective of the present study has been to widen the sensitivity of the analysis for small values of Ekman number, to ascertain exactly how plasma rotating channel flow responds to strong rotational effects. Our attention in the present study has ignored convective inertial effects and dwelled on viscous effects, since we are concerned with low speed transport, as opposed to very high speed flow. It is observed from Figure 3 that the primary velocity u decreases while the magnitude of secondary velocity v increases with an increase in Hall parameter m. It means that Hall current has a retarding influence on the primary velocity while it accelerates the secondary velocity. In the momentum Eq. (19), the term, 2 M cos ( v mucos ), contributes effectively to m cos the primary velocity u, according to the expression 2 2 mm cos uindicating that a rise in m causes a direct m cos rise in the hydromagnetic drag force term affecting the primary velocity u. This secondary effect is the major factor contributing to a decrease in primary flow velocity with an increase in Hall parameter m. On the other hand, the Hall parameter m has a marked effect on the secondary velocity profiles. This is because the effective conductivity /(1 + m 2 ) decreases as m increases. Thus, an increasing in Hall parameter reduecs the resistive Lorentz force and the motion of the fluid particles is reinforced and hence the secondary velocity component is enhanced. This is a new phenomenon, which appears as Figure 3. Primary and secondary velocities for different m when =45 and K 2 =4. Figure 4. Primary and secondary velocities for different when m = 0.5 and K 2 =4.

6 248 Bhaskar Chandra Sarkar et al. a result of including the Hall term. The case m = 0 corresponds to the neglect of the Hall effects. The mechanism by which Hall currents influence hydromagnetic channel flow (whether translational or rotational) is therefore via secondary effects and coupling in the momentum equations. This has important applications in practical MHD energy systems where better performance or control can be achieved of flows using Hall currents. Figure 4 reveals that the primary velocity u decreases while the magnitude of the secondary velocity v increases with an increase in the angle of inclination of the magnetic field. This implies that the angle of inclination has a retarding influence on the primary velocity while it accelerates the secondary velocity. The influence of is experienced via the term 2 M cos ( ucos ). Maximum drag force is correspond to the maximum value of cos. Hence the mini m cos mum hydromagnetic drag force corresponds to =90 for which there is be no hydromagnetic drag, explaining the maximum magnitude of the secondary velocity for this case. Maximum primary velocity in the channel is shown when =0 (magnetic field along the positive z-axis); however the maximum secondary velocity corresponds to =90 (magnetic field directed along the negative x-axis). It is inferred that a variable inclination of the applied magnetic field is the best control mechanism for the flow velocity. In consistency with classical MHD channel flow, all profiles are symmetrical about the centre line of the channel i.e. about = 0. In all cases the values of the secondary velocity v are negative indicating the presence of flow reversal (backflow) in the secondary field. Inclination however is important in elucidating the efficiency of MHD plasma devices, accelerators, energy systems and also in studying more realistic geophysical flows. To determine the electric field E 0, we impose the condition (26), integrating and rearranging, we get (27) (28) It is observed from Tables 1 and 2 that the electric field component e x first decreases, reach a minimum and then increases while the electric field component e y decreases with an increase in Hall parameter m. Itisalso seen that both e x and e y decrease with an increase in either rotation parameter K 2 or the angle of inclination of the magnetic field. Case I: When M 2 << 1 and K 2 << 1, then Eqs. (22) and (23) become (29) (30) It is interesting to note that for small values of M 2 and K 2, the primary velocity u() is unaffected by rotation whereas the secondary velocity v() is affected by both the magnetic field and rotation. If e x = e y = 0, Eqs. (29) and (30) become (31) (26) which is equivalent to the assumption that the total current flowing between the plates is zero. On the use of (25), together with (22) and (23) in (32) Further, if M 2 =0andK 2 = 0, then Eqs. (31) and (32)

7 Hall Effects on MHD Flow in a Rotating Channel in the Presence of an Inclined Magnetic Field 249 Table 1. Electric field components e x and e y when M 2 =10and =45 e x m\k e y Table 2. Electric field components e x and e y when M 2 =10andK 2 =4 e x m\ e y reduce to (33) Eq. (33) gives the hydrodynamical flow between infinite parallel plates channel in the presence of an applied pressure gradient. Case II: When M 2 << 1 and K 2 >> 1. In this case Eqs. (22) and (23) yield (34) boundary layer near the plate = 1. The thickness of this boundary layer is of the order ofo( 1 1 ) where 1 is given by (36). The thickness of this boundary layer decreases with an increase in the magnetic parameter while it increases with an increase in rotation parameter, since 1 increases with an increase in the magnetic parameter but decreases with an increases in rotation parameter. If e x = e y = 0 then the Eqs. (34) and (35) become (38) (39) where (35) It is seen that the exponential terms in (34) and (35) damp out quickly as increases. When 1/ 1,i.e. outside the boundary layer region, the primary and secondary velocities become (36) (40) (37) (41) It is seen from Eqs. (34) and (35) that there exists a thin Further, if e x = e y = 0 then from Eqs. (40) and (41), we have

8 250 Bhaskar Chandra Sarkar et al. (42) (48) Case III: When M 2 >> 1 and K 2 << 1, then Eqs. (22) and (23) become (43) (44) In this case also the fluid flow behavior is similar to case II. The shear stresses at the channel plates are important physical characteristics in the context of MHD plasma accelerator and energy system configurations. The nondimensional shear stress components x and y at the plates ( = -1) and ( = 1) due to the primary and secondary flows are respectively given by where (45) (49) Like case II, in this case also there exists a thin boundary layer near the plate = 1. The thickness of this boundary layer is of order ofo( 1 2 ) where 2 is given by (45). The thickness of this boundary layer decreases with an increase in magnetic parameter M 2 whereas it increases with an increase in Hall parameter m. It is interesting to note that the boundary layer thickness is unaffected by the rotation when magnetic field is very strong. In the absence of e x and e y (e x = e y = 0), Eqs. (43) and (44) become (46) where, are given by (24) and A, B are given by (37). Numerical results of the shear stress components x and y are being presented in the Figures 56 for several values of m, K 2 and when M 2 = 10 and R = 1. Figures 5 and 6 show that both the magnitude of the shear stresses x and y decrease with an increase in either K 2 or.itis illustrated from Figure 6 that the magnitude of the shear stress x first increases, reaches a maximum and then decreases while the shear stress y decreases with an increase in Hall parameter m. It is shown that primary shear stresses at the channel plates are always negative indicating significant backflow. However flow separation does not occur as shear stresses are never zero. 4. Conclusions (47) Further, the exponential terms in Eqs. (46) and (47) damp out quickly as increases near 1/ 2 where 2 is given by (45). The fluid flow outside the boundary layer is given by Hall effects on MHD flow permeated by an inclined magnetic field in a rotating system have been investigated. An exact solution of the governing equations describing the flow has been obtained in closed form. Hall currents and angle of inclination of the applied magnetic field have a retarding influence on the primary velocity while they have an accelerating influence on the secondary velocity. Asymptotic behavior of the solution has

9 Hall Effects on MHD Flow in a Rotating Channel in the Presence of an Inclined Magnetic Field 251 Authores are thankful to the reviewers for their valuable comments regarding revision of this manuscript. References Figure 5. Shear stresses x and y for different K 2 when = 45. Figure 6. Shear stresses x and y for different when K 2 =4. been analyzed for small as well as large values of magnetic parameter M 2 and rotation parameter K 2 to gain some physical insight into the flow pattern. The thickness of this boundary layer decreases with an increase in magnetic parameter M 2. Further, both the electric field components decrease with an increase in inclination of the applied magnetic field with the axis of rotation. The magnitude of the shear stresses x and y at the plates decrease with an increase in either rotation parameter K 2 or the angle of inclination of the applied magnetic field. Acknowledgment [1] Seth, G. S. and Ghosh, S. K., Hydromagnetic Flow in a Rotating Channel with Hall Effects, Acta Ciencia Indica, Vol. XXV M, No. 4, pp (1999). [2] Ghosh, S. K., Hall Effect on Unsteady Hydromagnetic Flow in a Rotating Channel Permeated by an Inclined Magnetic Field in the Presence of an Oscillator, Czechoslovak J. Physics, Vol. 49, No. 4, pp (1999a). doi: /BF [3] Ghosh, S. K., Hall Effect on Unsteady Hydromagnetic Flow in a Rotating Channel Permeated by an Inclined Magnetic Field in the Presence of an Oscillator (Erratum), Czechoslovak Journal of Physics, Vol. 49, No. 12, p (1999b). doi: /BF [4] Ghosh, S. K. and Pop, I., Hall Effects on MHD Plasma Couette Flow in a Rotating Environment, International Journal of Applied Mechanics and Engineering, Vol. 9, No. 2, pp (2004). [5] Ghosh, S. K. and Pop, I., Comments on MHD Plasma Flow in a Rotating Environment in the Presence of an Inclined Magnetic Field, International Journal of Applied Mechanics and Engineering, Vol. 9, No. 4, pp (2004). [6] Mondal, G., Mandal, K. K. and Choudhury, G., Hall Effects on MHD Plasma Couette Flow in a Rotating System, Journal of Physical Society of Japan, Vol. 51, p (1982). doi: /JPSJ [7] Sato, H., The Hall Effects in the Viscous Flow of Ionized Gas between Parallel Plates under Transverse Magnetic Field, Journal of Physical Society of Japan, Vol. 16, No. 7, pp (1961). doi: / JPSJ [8] Guria, M., Das, S., Jana, R. N. and Ghosh, S. K., Oscillatory Couette Flow in the Presence of an Inclined Magnetic Field, Meccanica, Vol. 44, pp (2009). doi: /s [9] Seth, G. S., Nandkeolyar, R., Mahto, N. and Singh, S. K., MHD Couette Flow in a Rotating System in the Presence of Inclined Magnetic Field, Applled Mathematical Science, Vol. 3, No. 59, pp (2009). [10] Seth, G. S. and Ghosh, S. K., Unsteady Hydromagnetic Flow in a Rotating Channel in the Presence of

10 252 Bhaskar Chandra Sarkar et al. Oblique Magnetic Field, Int. J. Engng. Sci., Vol. 24, p (1986). doi: / (86) [11] Seth, G. S. and Ghosh, S. K., Hydromagnetic Flow in a Rotating Channel in the Presence of Inclined Magnetic Field, Proceeding of the Mathematical Society Banaras Hindu University, Vol. 11, p. 111 (1995). doi: /BF [12] 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, Czechoslovak Journal of Physics, Vol. 50, p. 759 (2000). [13] Seth, G. S., Ansary, M. S., Mahto, N. and Singh, S. K., Hydromagnetic Flow in a Rotating Channel in the Presence of an Inclined Magnetic Field, Acta Ciencia Indica, Vol. XXXIV M, No. 3, pp (2008). [14] Seth, G. S., Nandkeolyar, R. and Ansari, M. S., Hall Effects on Oscillatory Hydromagnetic Couette Flow in a Rotating System, International Journal of Academic Research, Vol. 1, pp. 617 (2009). [15] Das, S., Sarkar, B. C. and Jana, R. N., Hall Effects on MHD Couette Flow in Rotating System, International Journal of Computer Application, Vol. 35, No. 13, pp (2011). [16] Bég, O. A., Sim, L., Zueco, J. and Bhargava, R., Numerical Study of Magnetohydrodynamic Viscous Plasma Flow in Rotating Porous Media with Hall Currents and Inclined Magnetic Field Influence, Commun Nonlinear Sci Numer Simulat, Vol. 15, pp (2010). doi: /j.cnsns [17] Seth, G. S., Mahato, G. K. and Nandkeolyar, R., Effects of Hall Current on Oscillatory Hydromagnetic Couette Flow of Class-II in a Rotating System in the Presence of an Inclined Magnetic Field, International Journal of Advanced Scientific and Technical Research, Vol. 1, No. 1, pp (2011). [18] Seth, G. S., Nandkeolyar, R. and Ansari, M. S., Effects of Hall Current and Rotation on Unsteady MHD Couette Flow in the Presence of an Inclined Magnetic Field, Journal of Appllied Fluid Mechanics, Vol. 5, No. 2, pp (2012). [19] Ghosh, S. K. and Pop, I., An Analytical Approach to MHD Plasma Behavior of a Rotating Environment in the Presence of an Inclined Magnetic Field as Compared to Excitation Frequency, International Journal of Applied Mechanics and Engineering, Vol. 11, No. 4, pp (2006). [20] Cowling, T. G., Magnetohydrodynamics, Intersscience, New York (1957). doi: / Manuscript Received: Jul. 8, 2012 Accepted: Aug. 20, 2014