Cent. Eur. J. Phys. • 9(1) • 2011 • 167-175 DOI: 10.2478/s11534-010-0057-1

Central European Journal of Physics

Analytical solutions for the unsteady MHD rotating flow over a rotating sphere near the equator Research Article

Erik Sweet, Kuppalapalle Vajravelu, Robert A. Van Gorder ∗ Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA

Received 29 March 2010; accepted 13 June 2010

Abstract:

In this paper we investigate the three-dimensional magnetohydrodynamic (MHD) rotating flow of a viscous fluid over a rotating sphere near the equator. The Navier-Stokes equations in spherical polar coordinates are reduced to a coupled system of nonlinear partial differential equations. Self-similar solutions are obtained for the steady state system, resulting from a coupled system of nonlinear ordinary differential equations. Analytical solutions are obtained and are used to study the effects of the magnetic field and the suction/injection parameter on the flow characteristics. The analytical solutions agree well with the numerical solutions of Chamkha et al. [31]. Moreover, the obtained analytical solutions for the steady state are used to obtain the unsteady state results. Furthermore, for various values of the temporal variable, we obtain analytical solutions for the flow field and present through figures.

PACS (2008): 47.10.-g, 47.11.-j, 47.50.-d, 47.65.-d, 47.85.-g Keywords:

rotating flow • magneto-hydrodynamic flow • Navier-Stokes equations • nonlinear system • analytical solution © Versita Sp. z o.o.

1.

Introduction

Rotating flows over stationary or rotating bodies are known to have applications in several diverse areas, such as meteorology, geophysical and cosmical fluid dynamics, gaseous and nuclear reactors, and so on. The hydrodynamic flow of rotating electrically conducting viscous fluids in the presence of a magnetic field are encountered in several important problems in geophysics and astrophysics and can provide explanations for the observed maintenance and secular variations of the geomagnetic

field [1]. Such flow problems are also useful in the solar physics governing sun spot development, the solar cycle and the structure of rotating magnetic stars [2]. Banks [3] has theoretically and experimentally studied the rotating flow over a stationary sphere, whereas Singh [4] investigated this flow analytically for small values of the Reynolds number. At very large values of the Reynolds number the rotating flow near the pole of the rotating sphere reduces to the problem of a rotating flow over a rotating infinite disk. For the case of an axially symmetric flow, the Navier-Stokes equations can be reduced to a set of five first-order ordinary differential equations with the boundary conditions specified at two different locations.

∗

E-mail: [email protected]

As the rotating flow of a viscous fluid over a rotating 167

Analytical solutions for the unsteady MHD rotating flow over a rotating sphere near the equator

sphere has many applications, numerical solutions have been obtained by several investigators [5–10]. Furthermore, authors in references [11–16] have studied this problem theoretically in different settings. Banks [3] obtained the similarity solution near the equator of a stationary sphere in a rotating fluid. For the case of a sphere rotating in an ambient fluid, no similarity solution was found in the vicinity of the equator, and the nature of the flow in this region has been discussed by Stewartson [17], Banks [18, 19], Singh [20], and Dennis et al. [21]. Ingham [22] has studied the rotating flow in the vicinity of the equator of a rotating sphere numerically and found that no unique solution exists.

In the above studies, the flow was assumed to be axisymmetric, while the non-axisymmetric flow in rotating fluids has been considered in [23–26]. Recently, the spin-up and spin-down aspects over a rotating disk in a vertical plane in the presence of a magnetic field and buoyancy force have been investigated by Slaouti et al. [27]. In this problem the flow becomes non-axisymmetric due to the presence of the buoyancy force. Some further studies on the steady state rotating flow due to the rotating disk in an ambient fluid are given in [28–30]. The unsteady MHD flow of a rotating fluid on a rotating body is important in the temporal evolution of rotating magnetic stars, and was considered by Chamkha et al. [31]. The authors considered the case when there is an initial steady state which is perturbed by a sudden increase in the angular velocity of the sphere, causing unsteadiness in the flow field. The coupled nonlinear parabolic partial differential equations governing the boundary-layer flow of a rotating unbounded fluid in the vicinity of the equator of a rotating sphere were solved numerically by using a finite-difference scheme. Analytical solutions were then obtained for large values of the suction parameter.

2. Three-dimensional flow over a rotating sphere In [31], the authors used spherical polar coordinates (r, θ, φ) with the origin at the center of the sphere of radius a and θ = 0 the axis of rotation. The motion is assumed to be axisymmetric (independent of the azimuthal angle φ). The magnetic field B is assumed to be applied in the r – direction and that only the applied magnetic field contributes to the Lorentz force. The components of the Lorentz force in the θ and φ directions are −Mv and B2 ) is the mag−M(w −r sin θ) respectively, where M = (σ (ρΩf ) netic parameter that depends on the strength of the magnetic field B, the electrical conductivity of the fluid σ , the fluid density ρ and the angular velocity of the distant fluid Ωf . The continuity and the Navier-Stokes equations governing the unsteady rotating flow over a rotating sphere are 1 ∂ 1 ∂ 2
(v sin θ) = 0, r u + (1) r 2 ∂r r sin θ ∂θ   2 ∂u ∂u v ∂u v + w2 +u + − ∂t ∂r r ∂θ r   (2) ∂p 2u 2v cot θ 2 ∂v =− + Re−1 ∇2 u − 2 − 2 − , ∂r r r ∂θ r2 ∂v v ∂v uv − w 2 cot θ ∂v +u + + ∂t ∂r r ∂θ r   2 ∂u v 1 ∂p −1 2 + Re ∇v+ 2 − − Mv, =− r ∂θ r ∂θ r 2 sin2 θ (3) ∂w ∂w v ∂w uw + vw cot θ +u + + ∂t ∂r r ∂θ r   (4) w (w − M − r sin θ) , = Re−1 ∇2 w − r 2 sin2 θ where ∇2 =

In the present paper, we extend the results of Chamkha et al. [31] by obtaining analytical solutions for arbitrary values of the physical parameters. These analytical solutions are then used to study the effects of the magnetic field and the suction/injection at the surface of the sphere on the steady flow. We provide an analysis of the error in the approximate analytical solutions employed. Our results agree very well with the numerical solutions of Chamkha et al. [31] and suggest interesting behaviors of the selfsimilar solutions which warrant further study. In order to study the transient flow leading to a steady state solution, we construct an analytical approximation to the transient flow by perturbation about the steady state analytical solutions. The profiles obtained for the transient flow are presented through figures and discussed. 168

1 ∂ r 2 ∂r

 r2

∂ ∂r

 +

1 ∂ r 2 sin θ ∂θ

 sin θ

∂ ∂θ

 .

In the above equations, u, v and w represent the dimensionless velocity components in the r, θ and φ directions respectively. The dimensionless velocity components are obtained by dividing the dimensional velocity components by aΩf where Ωf is the angular velocity of the fluid far away from the surface of the sphere. The boundary conditions are the no-slip conditions on the surface and the free stream conditions far away from the surface. Hence u(1, θ, t) = u0 , v(1, θ, t) = 0, w(1, θ, t) = λ(1 + ε), (5a) u(∞, θ, t) = 0, v(∞, θ, t) = 0, w(∞, θ, t) = 1.

(5b)

Erik Sweet, Kuppalapalle Vajravelu, Robert A. Van Gorder

Here Re = a νΩf is the Reynolds number, ν is the kinematic viscosity, λ = ΩΩb is the ratio of the angular velocity of the f sphere to the angular velocity of the distant fluid, t = Ωf t ∗ ∗ is the dimensionless time, r = ra is the dimensionless 2 radial distance, M = σρΩB = Ha is the magnetic parameter, Re 2

f

Ha= σ Bµ a is the Hartmann number, µ is the viscosity, a is the radius of the sphere, ε is a dimensionless constant and uo is the velocity at the surface of the sphere along the r – direction. uo < 0 corresponds to (radial) suction and uo > 0 corresponds to (radial) injection at the surface of the sphere. To simplify (2)-(4) we transform the radial coordinate as 2 2

η=

√

Re(r − 1),

and redefine the velocity components and pressure as

 π π θ− Fτ + θ − HFη 2 2 π 2 F − (cot θ)G 2 (8) + θ− 2    π π = − sin θ cos θ − θ − Fηη − θ − MF , 2 2   π Gτ +HGη −cot θ θ − F G = Gηη −M(G −sin θ). (9) 2 Equating like powers of (θ − π2 ) in (6) gives that either Hη + F = 0 or F = 0. Ignoring the trivial solution we take F = −Hη = −H 0 ; here prime denotes differentiation with respect to η. Substitution of this relation into (8) gives 

 π π  ∂H 0  HH 00 − θ− − θ− 2 ∂τ 2  
2 π + θ− H 0 − (cot θ)G 2 2  π π  000  H + θ− MH 0 . = − sin θ cos θ − θ − 2 2 0

Equating like powers of (θ − π2 ) gives that − ∂H − HH 00 + ∂τ 000 0 2 0 2 (H ) = −H + MH or −(cot θ)G = − sin θ cos θ. From the second equation with θ = π2 we see that G 2 −1 = 0 and thus the above equation can be re-written as

1 u(r, θ, t) = √ U(η, θ, τ), Re v(r, θ, t) = V (η, θ, τ), w(r, θ, t) = W (η, θ, τ),

H 000 − HH 00 + H 0

where t = τ and p = P. Under these transformations, the boundary layer equations become (see Chamkha et al. [31]) Uη + Vθ + V cot θ = 0, P=


2

− MH 0 + G 2 − 1 −

∂H 0 = 0. (10) ∂τ

Again using the relation from (6) in (9) and equating like + HG 0 = G 00 − M(G − 1) or powers of (θ − π2 ) gives ∂G ∂τ −(cot θ)H 0 G = 0. Ignoring the second trivial case we get

1 sin2 θ, 2

G 00 − HG 0 − M(G − 1) −

Vτ + UVη + V Vθ − W cot θ = − sin θ cos θ + Vηη − MV ,

∂G = 0. ∂τ

(11)

2

Wτ + UWη + V Wθ + V W cot θ = Wηη − M (W − sin θ) .

The boundary conditions (5) become √ H(0, τ) = A = u0 Re,

Since we are interested in the flow near the equator (θ ≈ π ) we assume that 2

H 0 (0, τ) = 0, G(0, τ) = λ(1 + ε),

(12)

0

H (∞, τ) = 0,

U(η, θ, τ) = H(η, τ),

G(∞, τ) = 1.

π V (η, θ, τ) = θ − F (η, τ) 2 

The assumption that state equations

and W (η, θ, τ) = G(η, τ).

∂G ∂τ

=

H 000 − HH 00 + H 0


2

∂H 0 ∂τ

= ε = 0 gives the steady-

− MH 0 + G 2 − 1 = 0,

(13)

Substitution into the above relations gives 

Hη + F + θ −

π

G 00 − HG 0 − M(G − 1) = 0, (cot θ)F = 0,

(6)

2 P=

1 sin2 θ, 2

(7)

(14)

subject to H(0) = A, H 0 (0) = 0, G(0) = λ, H 0 (∞) = 0, G(∞) = 1. (15) 169

Analytical solutions for the unsteady MHD rotating flow over a rotating sphere near the equator

3. Analytical steady flow

solutions

for

the

We shall now discuss analytical solutions to the coupled system (13)-(15), governing the three-dimensional steady flow of a magnetohydrodynamic (MHD) fluid near the equator of a rotating sphere. To obtain such analytical solutions, we employ the Homotopy Analysis Method (see [32–38] for an outline of the method). The Homotopy Analysis Method may be used to solve nonlinear differential equations N[p] = 0 subject to boundary conditions B(p, p0 , p00 , . . .) = 0 for η ∈ ∂Ω, where Ω is the solution domain and B is a boundary operator. A soluP∞ n tion of the form P(η, q) = n=0 q pn (η) is assumed in which p0 (η), the initial approximation, is chosen to satisfy B(p, p0 , p00 , . . .) = 0 and the remaining pj (η), for all j > 1, are held subject to homogeneous boundary conditions. Once p0 (η) is selected, the generalized homotopy, H(η, q, ~), is constructed as H(η, q, ~) = (1 − q)L[P(η, q) − p0 (η)] − q~N[P(η, q)]. The method gives us the freedom to select an auxiliary linear operator, and such a linear operator L is chosen in such a way that L[p0 (η)] = 0. The ith term of the solution P(η, q) is found by solving the ith order deformation of H(η, q, ~). The ith order deformation is found by taking the ith derivative of H(η, q, ~) = 0 with respect to q, dividing by i! and setting q = 0. The final solution is then p(η) = P(η, 1). Since this solution is dependent on the convergence control parameter ~ (i.e., p(η) = p(η, ~)), and thus ~ must be chosen such that p(η) is convergent. In principle, one may select such optimal values for the convergence control parameter, allowing a maximal region of convergence for the obtained analytical solutions. For (13), we take the initial approximation of H(η) to be h0 (η) = A, while for (14) we take the initial approximation of G(η) to be g0 (η) = 1 + (λ − 1)e−η , in agreement with the boundary data. We assume the analytical solutions to be of the form H(η, q) =

∞ X

qn hn (η) and G(η, q) =

n=0

∞ X

qn gn (η), (16)

Since we are dealing with a coupled system we get two generalized homotopies, we first take H1 (η, q, ~) = (1 − q)L1 [H(η, q) − h0 (η)] −q~N1 [H(η, q), G(η, q)], where N1 [H(η, q), G(η, q)] =

2 ∂ ∂ H(η, q) + (G(η, q))2 − 1 − M H(η, q), ∂η ∂η (18) is the nonlinear operator corresponding to (13) and then take H2 (η, q, ~) = (1 − q)L2 [G(η, q) − g0 (η)] (19) −q~N2 [H(η, q), G(η, q)], 

where N2 [H(η, q), G(η, q)] =

d d3 − dη3 dη

∂2 G(η, q) ∂η2

∂ −H(η, q) G(η, q) − M (G(η, q) − 1) , ∂η

(20)

is the nonlinear operator corresponding to (14). In order to solve the coupled system (13)-(15), we take an iterative approach. To obtain the solutions we first solve the ith order deformation of (17) for the ith term in the solution H(η, q) and then plug it into the ith order deformation of (19) to find the ith term in the solution G(η, q). The ith order deformation equation corresponding to (17) is given by 0 h000 i − hi = ~

i−1 X

gk gi−1−k − (1 + ~M)h0i−1 + ~

k=0

i−1 X

h0k h0i−1−k

k=0

−~

i−1 X

hk h00i−1−k + (1 + ~)h000 i−1 + χi ~,

k=0

for all i > 0, where ( χi =

L1 =

∂3 ∂2 H(η, q) − H(η, q) 2 H(η, q) ∂η3 ∂η

+

n=0

While the auxiliary linear operators corresponding to nonlinear equations (13) and (14) are taken to be

(17)

0, i = 1, 1, i > 1.

Similarly, the ith order deformation of (19) is given by g00i +g0i = g0i−1 +(1+~)g00i−1 −~

i−1 X

hk g0i−1−k −~Mgi−1 +χi ~M,

k=0

and L2 =

170

d2 d + . dη2 dη

for all i > 0. Letting q = 1 in (16), we obtain solutions H(η, ~) and G(η, ~) depending on the convergence control

Erik Sweet, Kuppalapalle Vajravelu, Robert A. Van Gorder

parameter ~. To find the values of ~ that provide convergent solutions we solve 

 ∂ ∂ = 0, HK (η, ~) + GK (η, ~) ∂η ∂η η=0

for ~, where HK (η, ~) and GK (η, ~) denote the K th order HAM approximations HK (η, ~) = h0 (η) +

K X

hn (η, ~)

n=1

and GK (η, ~) = g0 (η) +

K X

gn (η, ~).

n=1

Table 1 provides the optimal values of ~ chosen for the solution under a given choice of parameters, along with the number of terms in the solution and the maximal associated error over the domain. For brevity, we include the first two terms of the solutions, which are explicitly given by: h0 (η) = A, h1 (η) =

1 2 2 1 5 ~ − ~e−2η − ~e−η − ~ηe−η − ~λ + ~λe−η 6 6 3 3 3 1 1 1 +~ληe−η − ~λ2 − ~λ2 e−2η + ~λ2 e−η , 6 6 3 g0 (η) = 1 + (λ − 1)e−η ,

g1 (η) = ~ηe−η + A~ηe−η − M~ηe−η − ~ληe−η − A~ληe−η

Figure 1.

(a) Steady state profiles of the radial velocity H (η) for A = 1 2 , M = 1 and for different values of λ; (b) Steady state profiles of the negative meridional velocity H 0 (η) for A = 1 2 , M = 1 and for different values of λ.

+M~ληe−η . A M

λ Number of Terms

1 1 − 14 2 0 1 − 41 − 21 1 − 14 -1 1 − 41 1 1 − 12 2 1 1 0 2 1 1 21 2 1 1 43 2 1 1 0 2 1 3 0 2 2 1 2 0 2 1 5 0 2 2

Table 1.

30 30 30 30 28 28 28 28 28 28 30 30

~

Maximum Error

-0.241 -0.380 -0.443 -0.425 -0.258 -0.271 -0.292 -0.305 -0.271 -0.389 -0.350 -0.226

10−4 10−7 10−6 10−6 10−4 10−4 10−6 10−6 10−4 10−7 10−6 10−5

Values of ~ and maximum error in the approximate solutions.

4. Analytical solution method for the unsteady flow Let (G∗ , H∗ ) denote the solutions for the equations governing the steady flow, (13)-(15), as were obtained in section 3. In order to obtain analytical solutions to the unsteady e H) e to the flow problem (10)-(12), we assume a solution (G, unsteady flow problem which takes the form e (η, τ) = G∗ (η) + e−τ G1 (η) + e−2τ G2 (η) + · · · , G

(21)

e (η, τ) = H∗ (η) + e−τ H1 (η) + e−2τ H2 (η) + · · · . H

(22)

Substituting (21) and (22) into (10)-(11), we obtain the boundary value problems H∗000 − H∗ H∗00 + H∗0

In Figures 1-3 we present the velocity profiles of the problem (13)-(15) for several sets of values of the physical parameters A, M and λ.


2

− MH∗0 + G∗2 − 1 = 0,

G∗00 − H∗ G∗0 − M(G∗ − 1) = 0,

(23a)

(23b) 171

Analytical solutions for the unsteady MHD rotating flow over a rotating sphere near the equator

H∗ (0) = A,

H∗0 (0) = 0, H∗0 (∞)

= 0,

G∗ (0) = λ, (23c)

G∗ (∞) = 1,

H1000 − H∗ H100 − H∗00 H1 + 2H∗0 H10 − MH10 + 2G∗ G1 +H10 = 0, G100 + H∗ G10 + G∗0 H1 − MG1 + H10 = 0, H1 (0) = 0,

H10 (0) = 0, H10 (∞)

= 0,

(24b)

G1 (0) = 0, (24c)

G1 (∞) = 0,

H2000 − H∗00 H2 − H∗ H200 − H1 H100 + 2H∗0 H20 + H10


2

−MH20 + 2G∗ G2 + G12 + 2H20 = 0, G200 + H∗ G20 + G∗0 H2 + H1 G10 − MG2 + 2H20 = 0, H2 (0) = 0,

H20 (0) = 0, H20 (∞)

= 0,

(24a)

(25a)

(25b)

G2 (0) = 0, G2 (∞) = 0,

(25c)

and so on. These equations may be solved successively. Notice that (23) is identically satisfied, as it is simply (13)-(15) for which (G∗ , H∗ ) is a solution. One then solves (24), which is coupled yet linear in the unknown functions (G1 , H1 ). Subsequently, one solves (25), which is coupled yet linear in the unknown functions (G2 , H2 ). One may continue in this manner, to obtain an approximate analytical solution of desired accuracy. Placing the obtained solutions back into the expressions (21)-(22), the unsteady flow profiles are obtained. For instance, in order to obtain H1 and G1 we used the Homotopy Analysis Method under the assumption that the initial approximations are h10 (η) = 2e−η − e−2η − 1 and g10 (η) = 0, respectively. The auxiliary linear operators are as described in section 3, while the operators N1 and N2 are taken to be those given in (24). In solving the coupled boundary value problem, it was found that taking ~1 = −0.65 provided the optimal solution. Figure 4 shows the 8 term first order solutions of the unsteady flow problem for A = 21 , λ = 34 and M = 1. Due to the linearity of the terms in the perturbation solution, we can obtain the same order of accuracy in the solutions for (Gi , Hi ) (i > 1) as is given in the approximation to the steady flow, (G∗ , H∗ ). e (η, τ) = G∗ (η) and lim H e (η, τ) = H∗ (η). Note that lim G τ→∞

τ→∞

Hence, as τ → ∞, the solutions to the transient flow problem reduce to those solutions obtained for the steady flow. These analytical results agree with the numerical results of Chamkha [31]. We observe this behavior of the unsteady flow in Figure 4. 172

Figure 2.

(a) Steady state profiles of the negative meridional velocity H 0 (η) for λ = − 41 , M = 1 and for different values of A; (b) Steady state profiles of the rotational velocity G (η) for λ = − 14 , M = 1 and for different values of A.

If the solutions are to be continuous at τ = 0 with wellbehaved initial data, then the initial data for the boundary value problem (10)-(12) must satisfy ι1 (η) = G∗ (η) + G1 (η) + G2 (η) + · · · ,

(26)

ι2 (η) = H∗ (η) + H1 (η) + H2 (η) + · · · ,

(27)

e and ι2 (η) is the iniwhere ι1 (η) is the initial data for G e tial data for H. In other words, the terms in the series expansion in e−τ must satisfy the constraints G1 (η) + G2 (η) + · · · = ι1 (η) − G∗ (η) ,

(28)

H1 (η) + H2 (η) + · · · = ι2 (η) − H∗ (η) .

(29)

Furthermore, for small τ, the boundary conditions (12) imply that, for continuous solutions with well-behaved initial e must satisfy data, the initial data ι1 (η) for G ι1 (0) = λ(1 + ε) and lim ι1 (η) = 1, η→∞

(30)

Erik Sweet, Kuppalapalle Vajravelu, Robert A. Van Gorder

e must satisfy while the initial data ι2 (η) for H √ ι2 (0) = A = u0 Re,

ι02 (0) = 0 and lim ι02 (η) = 0. η→∞

(31)

Figure 4.

Figure 3.

(a) Steady state profiles of the radial velocity H(η) for λ = 0, A = 12 and for different values of M; (b) Steady state profiles of the negative meridional velocity H 0 (η) for λ = 0, A = 21 and for different values of M; (c) Steady state profiles of the rotational velocity G(η) for λ = 0, A = 12 and for different values of M.

5.

(a) Profiles of the radial velocity H(η, τ) for various values of τ: For A = 12 , λ = 43 and M = 1; (b) Profiles of the negative meridional velocity H 0 (η, τ) for various values of τ: For A = 12 , λ = 34 and M = 1; (c) Profiles of the rotational velocity G(η, τ) for various values of τ: For A = 12 , λ = 43 and M = 1.

Results and discussion

In the present paper, we have extended the results of Chamkha et al. [31] by obtaining analytical solutions for arbitrary values of the physical parameters. These analytical solutions are then used to study the effects of the 173

Analytical solutions for the unsteady MHD rotating flow over a rotating sphere near the equator

magnetic field and the suction/injection at the surface of the sphere on the steady flow. We provide an analysis of the error in the approximate analytical solutions employed. Our results agree very well with the numerical solutions of Chamkha et al. [31] and suggest interesting behaviors of the self-similar solutions which warrant further study. A similar problem, the steady viscous flows above a rotating plate, have been solved successfully by the homotopy analysis method; see [39]. Recently, the homotopy analysis has been applied to several transient flow problems; see for instance [40–42]. In the present paper, in order to study the transient flow leading to a steady state solution, we construct an analytical approximation to the transient flow by perturbation about the steady state analytical solutions. The profiles obtained for the transient flow are presented through figures and discussed. Furthermore, we discuss properties of the initial data required in order to allow for continuous solutions as τ → 0. The system of coupled nonlinear differential equations (13)-(15) governing the three-dimensional magnetohydrodynamic (MHD) fluid flow (near the equator) of a rotating sphere has been studied via an application of the Homotopy Analysis Method. The benefit of using the Homotopy Analysis Method is that it allows us the construction of analytical solutions to the similarity functions G and H governing the flow. We present the velocity profiles for several sets of the parameters λ (the ratio of the angular velocity of the sphere to that of the ambient fluid flow), M (the magnetic parameter) and A (the radial suction (A < 0)/injection (A > 0) from the surface of the sphere). We remark that the unsteady flow profiles demonstrate a dependence on the model parameters; so for brevity we include plots for the steady flow when discussing the qualitative behaviors of the obtained analytical solutions. Figure 1 shows that as the ratio of the angular velocity of the sphere to the angular velocity of the distant fluid decreases, the radial velocity decreases while the meridional velocity increases. This behavior is seen even in allowing the sphere and fluid to rotate in opposite directions which can be seen for negative values of λ. This is interesting for applications in which the rotation of the sphere can be varied, as such variation can influence the ambient fluid flow profiles (as might be expected). Increasing the radial velocity and decreasing the meridional velocity by increasing the ratio of the angular velocity of the sphere to the angular velocity of the distant fluid can be useful in cooling, as it permits heat to radiate away from the sphere more rapidly allowing for the sphere and ambient fluid to come more quickly into thermal equilibrium (and vice versa, for the case in which the heat of the sphere 174

is desired to be preserved). Figure 2a shows that the radial velocity decreases with decreasing injection. Figure 2b shows the quite opposite behavior in the rotational velocity. As the injection rate increases, the rotational velocity increases. Again, such can be useful in the cooling of molten materials in thermal baths, or in the cooling of spherical bearings. Figure 3 shows that an increase in the value of the parameter M causes an increase in the radial and rotational velocities and a decrease in the meridional velocity. An increase in the parameter M can be attributed to an increase in the electrical conductivity of the fluid or on increase of the magnetic field. An increase in M could also result from a decrease in the fluid density or a decrease in the angular velocity of the fluid away from the equator. In the case of the unsteady flow, we find that solutions to the unsteady flow problem reduce to those solutions obtained for the steady flow, in the limit τ. Such behavior of the solutions is agreement with the numerical results of Chamkha [31]. We observe this behavior of the unsteady flow in Figure 4.

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