Meccanica 39: 271–283, 2004. © 2004 Kluwer Academic Publishers. Printed in the Netherlands.

Unsteady MHD Flow on a Rotating Cone in a Rotating Fluid S. ROY, H.S. TAKHAR1,∗ and G. NATH2

Department of Mathematics, Indian Institute of Technology Madras, Chennai 600 - 036, India; 1 Department of Engineering, Manchester Metropolitan University, Manchester M1 5GD, U.K.; 2 Department of Mathematics, Indian Institute of Science, Bangalore 560 - 012, India (Received: 15 July 2002; accepted in revised form: 19 June 2003) Abstract. Unsteady flow over an infinite permeable rotating cone in a rotating fluid in the presence of an applied magnetic field has been investigated. The unsteadiness is induced by the time-dependent angular velocity of the body, as well as that of the fluid. The partial differential equations governing the flow have been solved numerically by using an implicit finite-difference scheme in combination with the quasi-linearization technique. For large values of the magnetic parameter, analytical solutions have also been obtained for the steady-state case. It is observed that the magnetic field, surface velocity, and suction and injection strongly affect the local skin friction coefficients in the tangential and azimuthal directions. The local skin friction coefficients increase when the angular velocity of the fluid or body increases with time, but these decrease with decreasing angular velocity. The skin friction coefficients in the tangential and azimuthal directions vanish when the angular velocities of fluid and the body are equal but this does not imply separation. When the angular velocity of the fluid is greater than that of the body, the velocity profiles reach their asymptotic values at the edge of the boundary layer in an oscillatory manner, but the magnetic field or suction reduces or suppresses these oscillations. Key words: Unsteady flows, Magneto-hydrodynamics, Rotating flows

Nomenclature A Cfx Cfy F, G, H Ha L M p ReL t, t ∗ u, v, w x, y, z

a mass transfer parameter. skin friction coefficient in the x-direction. skin friction coefficient in the y-direction. dimensionless similarity variables. Hartman number. characteristic length (m). magnetic parameter. pressure (Pa). Reynolds number based on length, L. dimensional and dimensionless times, respectively. axial velocity components (m s−1 ). axial coordinates.

Greek Letters α η, ξ λ µ ν ρ

semi-vertical angle of the cone. similarity variables. constant = 1 /(1 + 2 ). dynamic viscosity (kg m−1 s−1 ). kinematic viscosity (m2 s−1 ). density (kg m−3 ).

∗ Author for correspondence: e-mail: [email protected]

272 S. Roy et al. φ(t∗ ) ψ 1 , 2

a continuous function of time t ∗ . stream function. angular velocities.

Subscripts i, w 0 ∞

initial conditions and conditions at the wall, respectively. ambient conditions. conditions at infinity.

1. Introduction The rotating flows over a stationary or a rotating body find applications in meteorology, in geophysical and cosmical fluid dynamics, in gaseous and nuclear reactors etc. The magnetohydrodynamics of rotating electrically conducting fluids in the presence of a magnetic field is encountered in many important problems in geophysics and astrophysics. It can provide explanations for the observed maintenance and secular variations of the geomagnetic field [1]. It is also relevant in the solar physics involved in the sunspot development, the solar cycle and the structure of rotating magnetic stars [2]. When a rotating flow interacts with a surface, a complicated three-dimensional flow results which is found frequently in external as well as in internal flows. The rotating flow of a viscous incompressible fluid on a stationary infinite disk was first studied by Bodewadt [3]. The flow over an infinite rotating disk in an otherwise ambient fluid was first considered by Von Karman [4]. The effect of the magnetic field on the rotating flow over a stationary infinite disk was investigated by King and Lewellen [5], King and King [6], Stewartson and Troesch [7], and Nath and Venkatachala [8]. The effect of the magnetic field on the rotating infinite disk in an ambient fluid was examined by Sparrow and Cess [9] and Tarek et al. [10]. The interesting aspect of the Bodewadt’s solution was that it showed oscillations in the velocity profiles before reaching their asymptotic values at the edge of the boundary layer, which led to the controversy as to whether a physically realisable flow could contain velocity oscillation without separation or transition. In contrast to the Bodewadt’s solution, Von-Karman’s solution does not show any oscillation in the velocity profiles. The studies of Rogers and Lance [11], Bein and Penner [12], Kuo [13] and Millsaps and Nydahl [14] indicate the validity of Bodewadt’s solution. King and Lewellen [5], King and King [6] and Stewartson and Troesh [7] found that the magnetic field tends to damp or eliminate the oscillations in the velocity profiles. Also Nath and Venkatachala [8] have shown that like the magnetic field, suction also reduces the magnitude and amplitude of these oscillations. A theoretical analysis of forced flow and heat transfer on a rotating cone was carried out by Tien and Tsuji [15], and Koh and Price [16]. Wang [17] has considered the flow and heat transfer on rotating cones, disks and axi-symmetric bodies. Vira and Fan [18] have studied the flow and heat transfer on a rotating cone in a rotating fluid. All the above studies deal with pertain to steady flows. However, in many practical situations, the flow may be unsteady either due to the time-dependent angular velocity of the fluid or the body, or due to the impulsive change in the angular velocity of the fluid. The unsteady boundary layer flow of an impulsively started translating and spinning rotational symmetric body has been investigated by Ece [19] who obtained the solution for small values of time. The above problem including heat transfer rate has been studied by Ozturk and Ece [20]. In

Unsteady MHD Flow on a Rotating Cone 273 all the above studies, the flow was assumed to be axi-symmetric. The axi-symmetric flow in rotating fluids has been studied only by a few investigators [21–24]. The aim of this study is to consider the unsteady laminar boundary layer flow of a viscous incompressible electrically conducting fluid over a rotating infinite electrically non-conducting cone in a rotating fluid. The unsteadiness in the flow field is due to the time-dependent angular velocity of the cone as well as of the fluid. The fluid and the cone rotate either in the same or in the opposite direction. The problem is formulated in such a way that the magnetic counterpart of both the Von-Karman and Bodewadt flows become a particular case of the present problem. The coupled non-linear parabolic partial differential equations governing the unsteady boundary layer flow have been solved numerically by using an implicit finitedifference scheme in combination with the quasi-linearization technique [25, 26]. For large magnetic fields analytical solutions have been obtained for the steady-state case. The results have been compared with those of King and Lewellen [5], Stewartson and Troesch [7], Nath and Venkatachala [8], and Sparrow and Cess [9]. 2. Formulation We consider the unsteady laminar viscous incompressible electrically conducting fluid flowing over an infinite porous rotating cone in a rotating fluid. Both the cone and the fluid are rotating about the axis with time-dependent angular velocities either in the same direction or in the opposite direction. This introduces unsteadiness in the flow field. Figure 1 shows the coordinate system and the physical model. We have taken the rectangular co-ordinate system (x, y, z), where x is measured along a meridional section, the y-axis along a circular section and the z-axis normal to the cone surface. Let u, v and w be the velocity components along x (tangential), y (circumferential or azimuthal) and z (normal) directions, respectively. The cone surface is assumed as electrically insulated and the flow is taken to be axi-symmetric. The magnetic field B is applied in the normal direction that is, in the z-direction. The magnetic Reynolds number Rem = µ0 σ VL  1, where µ0 and σ are the magnetic permeability and electrical conductivity, respectively, and V and L are the characteristic velocity and length, respectively. Under this condition, it is possible to neglect the induced magnetic field in comparison to the applied magnetic field. Since there is no applied or polarisation voltage imposed on the flow field, the electrical field E = 0. Hence only the applied magnetic field contributes towards the

Figure 1. Co-ordinate system and the physical model.

274 S. Roy et al. Lorentz force. Under the above assumptions the boundary layer equations governing the flow on a rotating cone in a rotating fluid can be expressed as [5–7, 27] Continuity:

ux + x −1 u + wz = 0.

(1)

ut + uux + wuz − x −1 v 2 = νuzz − ρ −1 σ B 2 u,

(2)

vt + uvx + wvz + x −1 uv = νvzz − ρ −1 σ B 2 (v − Ve ) + (Ve )t .

(3)

Momentum:

The initial conditions are given by u(x, z, 0) = ui (x, z), v(x, z, 0) = vi (x, z), w(x, z, 0) = wi (x, z).

(4)

The boundary conditions on the surface are the no-slip conditions and far away from the surface, the conditions are given by the potential flow solution, as follows: u(x, 0, t) = 0, u(x, ∞, t) = 0, u(0, z, t) = 0,

w(x, 0, t) = w0 , v(x, 0, t) = 1 x sin αφ(t ∗ ), v(x, ∞, t) = Ve = 2 x sin αφ(t ∗ ), v(0, z, t) = Ve , z > 0.

(5)

Here t and t ∗ (t ∗ =  (sin α)t) are the dimensional and dimensionless times, respectively; 1 and 2 are the angular velocities of the cone and the fluid far away from the surface, respectively; (= 1 + 2 ) is the composite angular velocity; φ(t ∗ ) is a continuous function of time t ∗ having continuous first-order derivative; the subscripts t, x and z denote partial derivative with respect to t, x and z, respectively. It is possible to reduce the number of independent variables in equations (1)–(3) from three (x, z, t) to two (η, t ∗ ) by applying the following transformations: 
α 1/2 z, t ∗ = ( sin α)t, η =  sin ν u(x, z, t) = −2−1 (x sin α)φ(t ∗ )H  (η, t ∗ ), w(x, z, t) = (ν sin α)1/2φ(t ∗ )H (η, t ∗ ), v(x, z, t) = (x sin α)φ(t ∗ )G(η, t ∗ ), Ha 2 1 L2 sin L σ B 2 L2 , M= , λ= ,  = 1 + 2 , , Ha 2 = ReL = ν ReL µ  w0 . (6) A= (ν sin α)1/2 Using (6) in equations (1)–(3), we find that eqution (1) is identically satisfied and equations (2) and (3) reduce to H  − φHH  + 2−1 φH  − 2φ[G2 − (1 − λ)2 ] − MH  −

  ∂H  dφ  − H = 0, −φ −1 dt ∗ ∂t ∗ 

 dφ ∂G (G − 1 + λ) − = 0. G − φ(HG − H G) − M + φ −1 dt ∗ ∂t ∗ 2

(7) (8)

Unsteady MHD Flow on a Rotating Cone 275 The boundary conditions (5) can be re-written as A , φ(t ∗ ) H  (∞, t ∗ ) = 0, H (0, t ∗ ) =

H  (0, t ∗ ) = 0,

G(0, t ∗ ) = λ,

G(∞, t ∗ ) = 1 − λ.

(9)

Here η and t ∗ are the transformed coordinates; H  , G and H are the dimensionless velocity components along the tangential, azimuthal and normal directions, respectively; λ is the ratio of the angular velocity of the cone to the composite angular velocity at t ∗ = 0; prime denotes derivative with respect to η. If the normal velocity at the wall w0 is taken as a constant, then A is a constant. Also, A < 0 represents suction and A > 0 for injection. Further, λ = 0 implies that the body is stationary and the fluid is rotating, λ = 1 represents the case where the body is rotating in an ambient fluid, and for λ = 0.5, the body and the fluid are rotating with equal angular velocity in the same direction. Thus, for the case λ < 0.5, 1 < 2 and for λ > 0.5, 1 > 2 . The steady-state equations are obtained from equations (7) and (8) by putting t ∗ = φ(t ∗ ) − 1 = dφ/dt ∗ = ∂G/∂t ∗ = ∂H  /∂t ∗ = 0 in them and the resulting equations are given by H  − HH  + 2−1 H  − 2[G2 − (1 − λ)2 ] − MH  = 0,

(10)

G − (HG − H G) − M(G − 1 + λ) = 0

(11)

2

with the boundary conditions H(0) = A,

H (0) = 0,

G(0) = λ,

H (∞) = 0,

G(∞) = 1 − λ. (12)

It may be noted that equations (10)–(12) for λ = 1, A = 0 are identical to those of Sparrow and Cess [9] who studied the flow over a rotating infinite disk in an ambient fluid. Also, equations (10)–(12) for λ = 0 are the same as those of Nath and Venkatachala [8] who investigated the rotating flow over a stationary infinite disk. Further, for λ = 0 equations (10)–(12) reduce to those of King and Lewellen [5] and Stewartson and Troesch [7] if we replace H by −2 F and put n = 1 in the governing equations of King and Lewellen [5] and n = −1 in those of Stewartson and Troesch [7]. The quantities of physical interest are the local skin friction coefficients in the tangential and azimuthal directions and these are given by µ(∂u/∂z)z=0 = Rex −1/2 φ(t ∗ )H  (0, t ∗ ), ρ(x sin α)2 µ(∂v/∂z)z=0 = − = −Rex −1/2 φ(t ∗ )G (0, t ∗ ). ρ(x sin α)2

Cfx = 2 Cfy

(13)

where Cfx and Cfy are the local skin friction coefficients in the tangential and azimuthal directions, respectively; and Rex (Rex = x 2 sin α/ν) is the local Reynolds number. 3. Numerical Analysis Equations (7) and (8) under the boundary and initial conditions (9)–(12) have been solved numerically by using an implicit finite-difference method in combination with the

276 S. Roy et al. quasi-linearization technique. Since the method is described in complete detail in [25], its description is not presented here. However, for the sake of completeness its outline is given here. The non-linear coupled partial differential equations (7) and (8) were replaced by an iterative sequence of linear equations using quasi-linearization technique [25]. The resulting sequence of linear partial differential equations were approximated through finite differences by using a central-difference scheme in η – direction and the backward difference scheme in the t ∗ – direction. In each iteration step, these equations were then reduced to a system of linear algebraic equations with a block tri-diagonal structure, which were solved by using the Varga’s algorithm [26]. The step sizes in η- and t ∗ -directions have been chosen as &η = 0.05 and &t ∗ = 0.025, respectively. It has been found that a further reduction in &η or &t ∗ or in both does not change the results up to the 4th decimal place and these choices of step sizes are independent of the values of parameters used here. A convergence criterion based on the relative difference between the current and the previous iterations has been used. When this difference becomes 10−5 , the solution is assumed to have converged and the iterative process is terminated. 4. Asymptotic Solution for Large M (2
M
5) In this section, we have obtained approximate closed form solutions to the steady-state equations (10) and (11) under the boundary conditions (12) for large values of the magnetic parameter M (M  2) without mass transfer (A = 0). Our numerical results show that the normal and tangential velocities (H (η), H  (η)) become small for large M. Hence equation (11) reduces to G − M(G − 1 + λ) = 0.

(14)

The solution of equation (14) under the boundary conditions (12) can be expressed as G = 1 − λ + (2λ − 1) exp(−M 1/2 η).

(15)

The surface shear stress in the azimuthal direction is given by G (0) = −(2λ − 1)M 1/2 .

(16)

Using equation (15) and the conditions that H and H  are small, equation (10) reduces to H  − MH  = 2(2λ − 1)[2(1 − λ) exp(−M 1/2 η) + (2λ − 1) exp(−2M 1/2 η)].

(17)

The solution of equation (17) under the boundary conditions (12) for without mass transfer (A = 0) is given by H = 2(2λ − 1)3−1 M −3/2 [{(2 − λ) + 3(1 − λ)M 1/2 η} exp(−M 1/2 η) − 2−1 (2λ − 1) exp(−2M 1/2 η) − 2−1 (5 − 4λ)].

(18)

The surface shear stress in the tangential direction is given by 2(2λ − 1)(λ − 2) . 3M 1/2 From equation (18) the normal velocity far away from the surface is given by H  (0) =

(19)

(2λ − 1)(5 − 4λ) . (20) 3M 3/2 For λ = 1, these analytical results are identical to those obtained by Sparrow and Cess [9]. For M  2 and A = 0, these analytical results are found to be in good agreement with those of the H (∞) = −

Unsteady MHD Flow on a Rotating Cone 277 numerical results. The comparison between the analytical and numerical results is presented in Figure 8. 5. Results and Discussion Equations (7)–(8) under boundary and initial conditions (9)–(12) have been solved by using an implicit finite-difference scheme in combination with the quasi-linearization technique as described earlier. In order to assess the accuracy of our method, we have compared the surface shear stresses in the tangential and azimuthal directions (i.e., in x- and y-directions), −H  (0) and −G (0) for the steady-state case (t ∗ = 0) without mass transfer (A = 0) with those of Sparrow and Cess [9] for λ = 1, with those of King and Lewellen [5], and Stewartson and Troesch [7] for λ = 0, and with those of Nath and Venkatachala [8] for λ = 0, A = 0, −1, −2. Table 1. Comparison of surface shear stresses in the tangential and azimuthal directions (−H  (0), −G (0)) for the steady-state case (t ∗ = 0) when λ = 1 and A = 0 M

0 0.5 1 2 3 4

Present results

Sparrow and Cess [9]

−H  (0)

−G (0)

−H  (0)

−G (0)

1.0207 0.7703 0.6194 0.4613 0.3813 0.3308

0.6159 0.8488 1.0692 1.4418 1.7477 2.0097

1.0210 0.7700 0.6190 0.4610 0.3810 0.3310

0.6160 0.8490 1.0690 1.4420 1.7480 2.0100

Table 2. Comparison of surface shear stresses in the tangential and azimuthal directions (−H  (0), −G (0)) for the steady case (t ∗ = 0) when λ = 0 M

0 0 0 1 1 1 2 2 2 3 3 3

−A

0 1 2 0 1 2 0 1 2 0 1 2

Present results

Nath and Venkatachala [8]

−H  (0)

−G (0)

−H  (0)

−G (0)

1.8838 1.6703 1.2427 1.2208 1.1638 0.9873 0.9184 0.9062 0.8153 0.7606 0.7603 0.7066

0.7727 1.3866 2.1554 1.1119 1.7241 2.4798 1.4606 2.0484 2.7685 1.7582 2.3308 3.0233

1.8839 (1.8840)a,b 1.6700 1.2424 1.2198 (1.2200)a,b 1.1634 0.9870 0.9183 (0.9180)a 0.9062 0.8152 0.7604 0.7601 0.7062

0.7729 (0.7730)a,b 1.3863 2.1550 1.1119 (1.1120)a,b 1.7237 2.4794 1.4602 (1.4600)a 2.0481 2.7682 1.7580 2.3306 3.0230

a King and Lewellen [5]. b Stewartson and Troesch [7].

278 S. Roy et al. The results are found to be in very good agreement. The comparisons are given in Tables 1 and 2. Figures 2–3 show the effect of λ, which is the ratio of the angular velocity of the cone to the composite angular velocity, on the velocity profiles in the tangential and azimuthal directions (H  (η, t ∗ ), G(η, t ∗ )) for φ(t ∗ ) = 1 + εt ∗2 , ε = 0.2, t ∗ = 1, M = 1, A = 0. The ratio of the angular velocities λ strongly affects the velocity profiles. The effect of λ on H  (η, t ∗ ) is more pronounced for λ < 0.5 (1 < 2 ) than for λ > 0.5 (1 > 2 ). For λ = 0.5, the velocity gradient vanishes and it does not imply flow separation because we are dealing with a rotating flow over a rotating body. In Figure 3, when λ > 0.5 the fluid is being dragged by

Figure 2. Effect of λ on the velocity profiles in the tangential direction H  (η, t ∗ ) for φ(t ∗ ) = 1 + εt ∗2 , ε = 0.2, t ∗ = 1, M = 1, A = 0.

Figure 3. Effect of λ on the velocity profiles in the azimuthal direction G(η, t ∗ ) for φ(t ∗ ) = 1 + εt ∗2 , ε = 0.2, t ∗ = 1, M = 1, A = 0.

Unsteady MHD Flow on a Rotating Cone 279 the rotating cone and the velocity gradient is negative. On the other hand, when λ < 0.5 the cone is dragged by the fluid which results in positive velocity gradients. For λ
0, the velocity profiles H  (η, t ∗ ) and G(η, t ∗ ) reach their asymptotic values at the edge of the boundary layer in an oscillatory manner. Physically, these oscillations are caused by surplus convection of angular momentum present in the boundary layer. Similar trend has been noticed by Bodewadt [3], King and Lewellen [5], Stewartson and Troesch [7], and Nath and Venkatachala [8]. For λ > 0.5 (1 > 2 ), these profiles do not show any oscillation as observed by Von Karman [4] and Sparrow and Cess [9] when λ = 1. In Figures 4 and 5, the effect of the magnetic parameter M on the local skin friction coefficients in x- and y-directions (Rex 1/2 Cfx , Rex 1/2 Cfy ) for φ(t ∗ ) = 1 + εt ∗2 , ε = ±0.2, λ = 0.25, A = 0 is shown. The effect of the time variation on the skin friction coefficients is more pronounced for t ∗ > 1 and the results for ε = −0.2 (i.e., when the angular velocities are decreasing with increasing time) are not the mirror reflection of ε = +0.2. The skin friction coefficient in the tangential direction (Rex 1/2 Cfx ) decreases with increasing M, but the skin friction coefficient in the azimuthal direction (Rex 1/2Cfy ) increases. The reason for this behaviour can be explained as follows: The magnetic field induces a magnetic force in the tangential direction which tends to directly oppose the tangential velocity H  (η, t ∗ ). So the tangential velocity is reduced with increasing M and the tangential velocity gradient is also decreased with increasing M. On the other hand, the azimuthal velocity G(η, t ∗ ) is increased with increasing M because the magnetic force induced by the magnetic field supports the motion. This results in an increase in the skin friction coefficient in the azimuthal direction (Rex 1/2 Cfy ). The effect of the mass transfer parameter A on the velocity profiles (H  (η, t ∗ ), G(η, t ∗ )) for φ(t ∗ ) = 1 + εt ∗2 , ε = ±0.2, λ = 0.25, M = 1 is shown in Figures 6 and 7. Since suction (A < 0) reduces the boundary layer thickness, the tangential velocity profile (H  ) is reduced but the velocity profile in the azimuthal direction (G) are increased. Injection (A > 0) produces a

Figure 4. Effect of magnetic parameter M on the local skin friction coefficient in the tangential direction Rex 1/2 Cfx for φ(t ∗ ) = 1 + εt ∗2 , ε = ±0.2, A = 0, λ = 0.25.

280 S. Roy et al.

Figure 5. Effect of magnetic parameter M on the local skin friction coefficient in the azimuthal direction Rex 1/2 Cfy for φ(t ∗ ) = 1 + εt ∗2 , ε = ±0.2, A = 0, λ = 0.25.

Figure 6. Effect of the mass transfer parameter A on the velocity profiles in the tangential direction H  (η, t ∗ ) for φ(t ∗ ) = 1 + εt ∗2 , ε = ±0.2, t ∗ = 1, M = 1, λ = 0.25.

deeper inflow layer and de-stabilises the motion, while suction or the magnetic field suppresses inflow layer and stabilises the motion. The surface shear stresses in the x- and y-directions (H  (0), G (0)) for the steady-state case ∗ (t = 0) without mass transfer (A = 0) obtained by both the analytical and numerical methods are shown in Figure 8. For M  2, the results are found to be in very good agreement.

Unsteady MHD Flow on a Rotating Cone 281

Figure 7. Effect of the mass transfer parameter A on the velocity profiles in the azimuthal direction G(η, t ∗ ) for φ(t ∗ ) = 1 + εt ∗2 , ε = ±0.2, t ∗ = 1, M = 1, λ = 0.25.

Figure 8. Comparison of the results (H  (0), G (0)) obtained by analytical and numerical methods for t ∗ = 0 and A = 0.

6. Conclusion The results reveal that in the absence of the magnetic and surface mass transfer, the velocity profiles tend to their asymptotic values at the edge of the boundary layer in an oscillatory manner and the effect of the magnetic field or suction is to reduce or remove these oscillations. The time variation of the local skin friction coefficients is pronounced for t ∗ > 1. When the

282 S. Roy et al. cone and the fluid rotate with the same angular velocity in the same direction, the surface shear stresses vanishes, but this does not imply flow separation. The magnetic field reduces the skin friction coefficient in the tangential direction but increases the skin friction coefficient in the azimuthal direction. Suction increases the skin friction coefficients but the effect of injection is just the opposite. Injection produces a deeper inflow layer and de-stabilises the motion while the magnetic field or suction reduce the inflow layer and produce stability. For the steady flow, analytical solutions have been obtained for large magnetic parameter (M  2) and the results are found to be in good agreement with those obtained by using the numerical method. References 1. 2. 3. 4. 5. 6.

7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.

Hide, R., Roberts, P.H., ‘The origin of the main geomagnetic field’, in: Physics and Chemistry of the Earth. Vol. 4, Pergamon Press, New York, 1961, pp. 27–98. Dieke, R.H., ‘Internal rotation of the sun’, in: L. Goldberg (Ed.) Annual Reviews of Astronomy and Astrophysics. Vol. 8, Annual Review Inc., New York, 1970, pp. 297–328. Bodewadt, U.T., ‘Die Drehstromung Uber festern Grand’, ZAMM 20 (1940) 241–253. Von Karman, Th., ‘Uber laminare und turbulente Reibung’, ZAMM 1 (1921) 233–252. King, W.S. and Lewellen, W.S., ‘Boundary layer similarity solutions for rotating flow with or without magnetic interaction’, Phys. Fluids 7 (1964) 1674–1680. King, W.S. and King, W.S., ‘The boundary layer of a conducting vortex flow over a disk with an axial magnetic field’, in: Ostrach, S. and Scanlan, R.H. (eds) Development in Mechanics, Vol. 2, 1965, pp. 107–125. Stewartson, K. and Troesch, B.A., ‘On a pair of equations occurring in swirling viscous flow with an applied magnetic field’, ZAMP 28 (1977) 951–963. Nath, G. and Venkatachala, B.J., ‘The effect of suction on boundary layer for rotating flows with or without magnetic field’, Proc. Ind. Acad. Sci. 85 (1977) 332–337. Sparrow, E.M. and Cess, R.D., ‘Magneto-hydrodynamic flow and heat transfer about a rotating disk’, J. Appl. Mech. 29 (1962) 181–187. Tarek, M.A., Mishkawy, El., Hazem, A.A. and Adel, A.M., ‘Asymptotic solution for the flow due to an infinite rotating disk in the case of small magnetic field’, Mech. Res. Comm. 25 (1998) 271–278. Rogers, M.H. and Lance, G.N., ‘The rotationally symmetric flow of a viscous fluid in the presence of an infinite rotating disk’, J. Fluid Mech. 7 (1960) 617–631. Bien, F. and Penner, S.S.: ‘Velocity profiles in steady and unsteady rotating flow for a finite cylindrical geometry’, Phys. Fluids 13 (1970) 1665–1671. Kuo, H.L., ‘Axi-symmetric flows in the boundary layer of a maintained vortex’, J. Atmos. Sci. 28 (1971) 20–41. Millsaps, K. and Nydahl, J.E., ‘Heat transfer in a laminar cyclone’, ZAMM 53 (1973) 241–246. Tien, C.L. and Tsuji, I.J., ‘A theoretical analysis of laminar forced flow and heat transfer about a rotating cone’, J. Heat Transfer 87 (1965) 184–190. Koh, T.C.Y. and Price, J.F., ‘Non-similar boundary layer heat transfer of a rotating cone in forced flow’, J. Heat Transfer 89 (1967) 139–145. Wang, C.Y., ‘Boundary layers on rotating cones, disks and axi-symmetric surfaces with concentrated heat source’, Acta Mechanica 81 (1990) 245–251. Vira, N.R. and Fan, D.N., ‘Heat transfer from a cone spinning in a co-rotating fluid’, J. Heat Transfer 103 (1981) 815–817. Ece, M.C., ‘An initial boundary layer flow past a translating and spinning rotational symmetric body’, J. Eng. Math. 26 (1992) 415–428. Ozturk, A. and Ece, M.C., ‘Unsteady forced convection heat transfer from a translating and spinning body’, J. Heat Transfer 117 (1995) 318–323. Berker, R., ‘A new solution of the Navier–Stokes equations for the motion of a fluid confined between two parallel plates rotating about the same axis’, Arch. Mech. 31 (1979) 265–280. Porter, S.V. and Rajagopal K.R., ‘Swirling flow between rotating plates’, Arch. Rat. Mech. Anal. 86 (1984) 305–315.

Unsteady MHD Flow on a Rotating Cone 283 23. 24. 25. 26. 27.

Lai, C.Y., Rajagopal, K.R. and Szeri, S.Z., ‘Asymmetric flow between parallel rotating disks’, J. Fluid Mech. 146 (1984) 203–225. Lai, C.Y., Rajagopal, K.R. and Szeri, S.Z., ‘Asymmetric flow above a rotating disk’, J. Fluid Mech. 157 (1985) 471–492. Inouye, K. and Tate, A., ‘Finite-difference version quasi-linearization applied to boundary layer equations’, AIAA J. 12 (1974) 558–560. Varga, R.S., Matrix Iterative Analysis, Springer, New York, 2000, pp. 220. Eringen, A.C. and Maugin, S.A., Electrodynamics in Continua, Vol. 2, Springer, Berlin 1990.