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RETRACTED: Exact solutions for the incompressible viscous magnetohydrodynamic fluid of a rotating-disk flow with Hall current
International Journal of Non-Linear Mechanics 46 (2011) 1042–1048
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Exact solutions for the incompressible viscous magnetohydrodynamic fluid of a rotating-disk flow with Hall current M. Turkyilmazoglu Mathematics Department, University of Hacettepe, 06532-Beytepe, Ankara, Turkey
a r t i c l e i n f o
a b s t r a c t
Article history: Received 28 May 2009 Accepted 6 April 2011 Available online 15 April 2011
The focus of the present study is to obtain exact solutions for the flow of a viscous hydromagnetic fluid due to the rotation of an infinite disk in the presence of an axial uniform steady magnetic field with the inclusion of Hall current effect. In place of the traditional von Karman’s axisymmetric evolution of the flow, the rotational non-axisymmetric stationary conducting flow is taken into consideration here, whose governing equations allow an exact solution to develop bounded everywhere in the normal direction to the wall. The three-dimensional equations of motion are treated analytically yielding derivation of exact solutions, which differ from those of corresponding to the classical von Karman’s conducting flow. Making use of this solution, analytical formulas for the angular velocity components, for the current density field as well as for the wall shear stresses are extracted. The critical peripheral locations at which extrema of the local skin friction occur are also determined. It is proved from the analytical results that for the specific flow the properly defined thicknesses decay as the magnetic field strength increases in magnitude, approaching their hydrodynamic value in the limit of large Hall numbers. Interaction of the resolved flow field with the surrounding temperature is further analyzed via the energy equation. The temperature field is shown to accord with the dissipation function. According to the Fourier’s heat law, a constant heat transfer from the disk to the fluid occurs, though it increases by the presence of magnetic field, the increase is slowed down by the Hall effect eventually reaching its hydrodynamic limit. & 2011 Elsevier Ltd. All rights reserved.
Keywords: Exact solutions Magnetohydrodynamic flow Rotating disk Hall effect Heat transfer
1. Introduction A substantial interest in rotating magnetohydrodynamic viscous fluid flow motion has been witnessed in the past few decades, due to the reason that it provides practical applications to many engineering areas, for instance, pumping and levitation of liquid metals. von Karman’s swirling hydrodynamic viscous flow [1] is a well-documented classical problem in fluid mechanics, which has several technical and industrial applications. The original problem raised by von Karman, which is the most studied by researchers in the literature, is the viscous flow motion induced by an infinite rotating disk where the fluid far from the disk is at rest. Then the problem is generalized to include the case where the fluid itself is rotating as a solid body far above the disk. Another generalization is to consider the viscous flow between two infinite coaxial rotating disks. All these problems and also stability issues are attacked, theoretically, numerically and experimentally, by many researchers amongst many others, such as [2–13]. However, all of
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these results are either numerical or analytical-numerical. Moreover, [14] established the existence of an infinite set of solutions to the flow of two infinite parallel plates rotating with the same angular velocity about an axis. Extension to porous plates was given in [15,16], refer also to [17]. Additionally, heat transfer problem over a rotating disk was also studied, see for instance [18,19]. The cases when an exact solution for the Navier–Stokes equations can be obtained are of particular importance in investigations to describe fluid motion of the viscous fluid flows. However, since the Navier–Stokes equations are non-linear in character, there is no known general method to solve the equations in full, nor the superposition principle for non-linear partial differential equations does work. There are some wellknown exact unidirectional or parallel shear flows, a few sample cases contain the steady Poiseuille and Couette flow. In addition to this, there are exact cylindrically symmetric solutions with closed plane streamlines, see for instance [20,21]. Further exact solutions that we already know possess certain feature of the fluid motion, such as rectilinear motion, motions of the duct flows, axisymmetric flows and stagnation flows on plate with slip, etc., see for instance, the works of [22–24].
M. Turkyilmazoglu / International Journal of Non-Linear Mechanics 46 (2011) 1042–1048
The purpose of the current work is to present details of some new three-dimensional solutions of the Navier–Stokes equations governing the steady state stationary viscous flow of an incompressible Newtonian electrically conducting fluid associated with a single rotating disk. The disk is supposed to be non-electrically conducting under the impression of an external magnetic field of constant strength applied normal to the disk. The self-similar axisymmetric solutions in the presence of a transverse magnetic field were previously investigated numerically solving the underlying equations in [25–27] and recently in [28,29]. Additionally, analytic solutions have been recently presented in [30,31] without considering the Hall current effect. However, in the existence of a strong magnetic field Hall current is known to have a marked effect on the magnetic field, see [32–34]. Unlike the case of the classical von Karman’s axisymmetric similarity solution, the present motivation lies in searching for the velocity and pressure fields, affected by the existence of the magnetic field considering the Hall effect. It is shown in this study that under certain well-defined conditions the Navier–Stokes equations reduce to a system of ordinary differential equations whose translation to a complex format enables the generation of solutions in a closed form depending on magnetic field and Hall parameters. The analytical solution is in fact found by imposing on the infinite rotating disk no-slip condition for the velocity field (after setting no normal velocity at all) together with an assumption that the field is bounded with respect to the normal axis. The resulting equations are then used to obtain exact expressions for the angular velocity and current density components, for the displacement thicknesses as well as for the existing wall shear stresses. We further analyze the subject of heat transfer from the disk to the fluid by solving the energy equation, which, for the flow under consideration, is shown to be in balance with the dissipation function and the Joule heating term due to current density of the magnetic field. The effects of uniform magnetic field and Hall current on the development of the flow velocity, vorticity, displacement thicknesses, skin frictions and also heat transfer are fully discussed. The following strategy is pursued in the rest of the paper. In Section 2 the full governing equations representing the magnetohydrodynamic flow motion are outlined in cylindrical coordinates. Section 3 contains the analytical results, including the steady state flow and the physically relevant variables, obtained under self-consistent assumptions. Heat conducting case including the adiabatic and isothermal wall conditions is analyzed in Section 4, in which, an expression for the heat transfer is derived using the Fourier’s heat law. Finally our conclusions follow in Section 5.
2. Formulation of the problem We consider the three-dimensional flow of an incompressible, electrically conducting viscous fluid on an infinite disk which rotates about its axis with a constant angular velocity O. An external uniform magnetic field B ¼ B0 , with the flux density B0 constant, is applied to the system along the direction perpendicular to the disk. The Navier–Stokes equations are non-dimensionalized with respect to a length scale L ¼ re , velocity scale Uc ¼ LO, time scale L=Uc , magnetic scale B0 and pressure scale rUc2 , where r is the fluid density. This leads to a Reynolds number Re ¼ Uc L=n, n being the kinematic viscosity of the fluid. Thus, relative to nondimensional cylindrical polar coordinates ðr, y,zÞ which rotate with the disk, the full time-dependent, unsteady magnetohydrodynamic equations governing the viscous fluid flow are as follows: @u @u v @u v2 @u þu þ þw @t @r r @y r @z
¼
1043
@p 1 2 @v u m þ r2 u 2 2 ½ubðvrÞ, 2 @r Re r @y r 1þb
@v @v v @v uv @v þu þ þw þ @t @r r @y r @z 1 @p 1 2 @u v m þ r2 vþ 2 2 ½ðvrÞ þ bu, ¼ 2 r @y Re r @y r 1þb @w @w v @w @w @p 1 2 þu þ ¼ þ ½r w, þw @t @r r @y @z @z Re
ð2:1Þ
@u 1 @v @w u þ þ ¼ 0, þ @r r @y @z r @T @T v @T @T J2 2 þu þ ¼ k½r T þ mF þ : cp r þw @t @r r @y @z s
ð2:2Þ
The Laplacian operator in cylindrical coordinates is defined as 2 @ 1 @2 @2 1@ : ð2:3Þ r2 ¼ þ 2 2þ 2þ 2 r @r r @y @r @z The parameters appearing in Eqs. (2.1)–(2.2) are, respectively, m ¼ sB20 =ro is the magnetic interaction parameter, cp is the specific heat, k is the thermal conductivity (assumed constant here), m is the viscosity coefficient of the fluid and s is the electrical conductivity of the fluid. Moreover, function F on the right-hand side of Eq. (2.2) represents the effects due to flow dissipation and J 2 =s denotes Joule heating, with J ¼ ðJr ,Jy ,Jz Þ being the current density. Furthermore, it is assumed that the motion of the conducting fluid has a negligible effect on the imposed magnetic field, but the effect of the magnetic field on the fluid motion is strongly felt, shown by the Lorentz force terms J B, at the end of the Eq. (2.1). It is also assumed that the electron–atom collision frequency is relatively large, so that the Hall current effect cannot be neglected revealed in the generalized Ohm’s law by J ¼ s½Eþ u BbðJxBÞ, where E is the electric field which results from charge separation and b is the Hall factor, that, whenever is positive, the magnetic flux points upwards and the electrons of the conducting fluid gyrate in the same sense as the rotating disk, with an opposite situation for negative b, see [34]. In this analysis the fluid is assumed to lie in the z Z0 semiinfinite space. Boundary conditions accompanying the Eqs. (2.1)–(2.2) are such that the fluid adheres to the wall at z¼0 with a prescribed (isothermal) or symmetric (adiabatic) wall temperature, and the quantities are bounded at far distances from the wall, with the temperature being uniformly distributed there. In the classical von Karman flow under the influence of a uniformly applied magnetic field normal to the wall, the rotational symmetry assumption is used which removes the y-dependence of the variables in Eqs. (2.1)–(2.2). However, in the present case we allow the y-dependence, enabling motion of the non-axisymmetric magnetofluid flow to develop, but instead assume w¼0 throughout. It should be remarked that the latter noaxial flow assumption is automatically satisfied by the third of the momentum equations (2.1), provided that the pressure p is not a function of z. The above constraints render the Eqs. (2.1)–(2.2) to a simplified version @u @u v @u v2 þu þ @t @r r @y r @p 1 2 @v u m þ r2 u 2 2 ½ubðvrÞ, ¼ 2 @r Re r @y r 1þb
M. Turkyilmazoglu / International Journal of Non-Linear Mechanics 46 (2011) 1042–1048
@v @v v @v uv þu þ þ @t @r r @y r 1 @p 1 2 @u v m þ r2 vþ 2 2 ½ðvrÞ þ bu, ¼ 2 r @y Re r @y r 1þ b @u 1 @v u þ þ ¼ 0, @r r @y r @T @T v @T J2 2 þu þ cp r ¼ k½r T þ mF þ : @t @r r @y s
G ¼ f ðZÞsinðyaÞ þ gðZÞcosðyaÞ, ð2:4Þ
ð2:5Þ
3. Steady-state mean flow We here restrict ourselves to the stationary hydromagnetic mean flow relative to the rotating disk. Since the energy equation in (2.2) is decoupled from the Eq. (2.1), this section focuses only on the solution of velocities and pressure. Within this framework, it is particularly clear that u ¼ 0,
v ¼ r,
p¼
r2 þ p0 , 2
ð3:1Þ
where p0 is a constant, constitute a solution for (2.4), which is known as the rigid-body rotation. pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Next, via a coordinate transformation Z ¼ ðRe =2Þz, the solution (3.1) is perturbed and we assume a new solution of the form u ¼ r0 Fðy, ZÞ,
v ¼ r þ r0 Gðy, ZÞ,
p¼
r2 rr 0 cosðyaÞ þ p1 , 2
GZZ
2m 1þ b
2
G2 1þ
0.4
2
F ¼ 2sinðyaÞ:
0.3
0.2 1 0.1
0 ð3:3Þ
-0.1
Introducing a new function of the form H¼F þiG transforms the pair of Eq. (3.3) into the subsequent single complex differential equation with real variable " !# m mb HZZ 2 þi 1 þ H ¼ 2ðcosðyaÞisinðyaÞÞ, 2 2 1þb 1þb
0
1
3
4
5
6
7
1.2 β=−1 β=1 β=−3 β=3
m=0
whose solution bounded with respect to Z can be immediately expressed as
0.9 1
After decomposing real and imaginary parts of the solution, F and G from (3.5) are found to be
-g
ð3:5Þ
where C1 is a complex integration constant depending on y. The latter is determined by the no-slip velocity condition on the wall. The constant l in (3.5) corresponding to magnetofluid case satisfies qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 l ¼ ðL1 þ iL2 Þ, L1 ¼ a þ a2 þb2 , L2 ¼ , L 1 ! m mb ,1þ ða,bÞ ¼ : 2 2 1þb 1þb
F ¼ f ðZÞcosðyaÞ þgðZÞsinðyaÞ,
2
η
ð3:4Þ
H ¼ C1 elZ iðcosðyaÞisinðyaÞÞ,
β=−1 β=1 β=−3 β=3
m=0
!
mb 1þb
where the form of the functions f and g in (3.6) is given as f ¼ eL1 Z sinðL2 ZÞ and g ¼ 1 þeL1 Z cosðL2 ZÞ. It can be seen that when m ¼ 0 we have the non-conducting fluid case, and when the magnetic with the absence of the Hall factor pffiffiffiffiffiffiffifield is large together pffiffiffiffiffiffiffi L1 2m and L2 1= 2m. Accounting for the asymptotic behaviors of f and g in the large Z limit in Eq. (3.6), it can be immediately deduced that the velocities far away from the disk turn out to be u ¼ r0 sinðyaÞ and v ¼ r þr0 cosðyaÞ, which differ from the no-slip velocities. The path along which the velocities vanish exactly through the space can be easily determined by setting zero the velocities in Eq. (3.2) with the consideration of (3.6). In the particular case when a ¼ y, the implication is that F ¼f and G ¼g, whose graphs are displayed in Fig. 1. These graphs clearly indicate the development of a boundary layer like structure near the surface of the disk, whose thickness varies with the magnetic strength parameter and Hall parameter (m¼1 for all b). As in the classical von Karman’s axisymmetric flow, see [33], an increase in the magnetic interaction number m provides a considerable reduction in the velocity components f and g for the problem under consideration, essentially due to the damping influence of the magnetic field. A dissimilar contribution, to the
ð3:2Þ
such that, as opposed to the well-known Karman solution, periodic solutions with respect to y of F and G are sought, subjected to a pressure field given by the third of the Eq. (3.2). Moreover, parameters r0 and a correspond to polar representation of a fixed point on the disk surface and p1 is a constant. Substituting equations (3.2) into (2.4), we see that the continuity equation is automatically satisfied, while the momentum equations, with the help of (3.1), give rise to ! 2m mb FZZ F þ 2 1 þ G ¼ 2cosðyaÞ, 2 2 1þb 1þb
ð3:6Þ
f
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0.6
0.3
0
0
1
2
3
4
5
6
7
η Fig. 1. The effects of magnetic strength parameter m and Hall parameter b on the development of velocity components (a) f and (b) g, respectively.
M. Turkyilmazoglu / International Journal of Non-Linear Mechanics 46 (2011) 1042–1048
Z 0
1
f dZ ¼
L2 , L2
where L2 ¼ L21 þ L22 . It is no surprising to observe from these exact formulas that the thicknesses decay (see also Fig. 1) as the magnetic effect is considerably large. In addition to this, when the magnetic interaction number is kept fixed, allowing the absolute growth of the Hall parameter b will cause the thicknesses to recover their hydromagnetic correspondence 1/2. Having found the form of the velocities, the components of the stress tensor, which are non-zero, are only the radial tr and tangential ty shear stresses on the wall, and these can be calculated as rffiffiffiffiffi @u Re tr ¼ ¼ r0 ½L2 cosðyaÞL1 sinðyaÞ @z z ¼ 0 2 and
ty ¼
β=−1 β=1 β=−3 β=3
Jr1
-0.2
-0.5 m=0 -0.8 1 -1.1
0
1
@v ¼ r0 @z z ¼ 0
rffiffiffiffiffi Re ½L1 cosðyaÞ þ L2 sinðyaÞ: 2
When a is set to zero, the findings point out the fact that the minimum and maximum skin frictions offered against the flow within the presence of a transversely applied magnetic field take place, respectively at the locations, y ¼ tan1 ðL1 =L2 Þ and y ¼ tan1 ðL1 =L2 Þ þ p (the branch of tan1 is over ðp=2,3p=2Þ) for the radial stress and at the locations y ¼ tan1 ðL1 =L2 Þ and y ¼ tan1 ðL1 =L2 Þ þ p (the branch of tan1 is over ðp=2, p=2Þ) for
2
3
4
5
6
7
η 0.6
β=−1 β=1 β=−3
0.3
β=3
θ
0 1 -0.3 m=0
-0.6
0
1
L
0
dr ¼
0.1
J 1
axisymmetric rotating-disk flow, of the Hall parameter b is apparent in Figs. 1(a) and (b). As concluded in [33], the effect of Hall current on the steady axisymmetric swirling flow depends upon the sign and the absolute magnitude of the Hall parameter. However, as seen from the figure, for all the negative values of Hall parameter b, the effects of Hall current decrease magnetic damping on the flow velocities, as a consequence of which f and g increases. The increase is seen to be more effective in case of negative Hall number than of positive one. Actually, some small positive values of b acts in favor of increasing the magnetic damping as opposed to the negative values. For b growing in negative value, the magnetic field has a propelling effect on the velocity components, which exceed their hydrodynamic values. But, for large positive values of b, they keep increasing while being still less than their hydrodynamic values. It should be noticed that the large limit of b corresponds to the hydrodynamic limit by decreasing the magnetic force. To conclude, the typical impacts of the magnetic field together with the Hall current is better understood from the form of the exact velocity equations (3.6), which need to be resolved numerically for other type of fluid flow phenomena, see [25,35]. The current density components Jr ¼ r0 Jr 1 and Jy ¼ r0 Jy 1 for 2 the hydromagnetic flow, for which Jr 1 ¼ ðbF þ GÞ=ð1 þ b Þ and 2 Jy 1 ¼ ðF þ bGÞ=ð1 þ b Þ are demonstrated in Fig. 2. The effect of increasing the absolute value of Hall parameter is to increase the current Jr1 in the magnetic field, whereas the effect of increasing the Hall parameter is to increase the current Jy 1 followed by an initial decrease. Just an opposite behavior is observed in the current Jy 1 while the Hall parameter decreases, a decrease in the current is followed after an initial increase. The current densities depicted in Figs. 2(a) and (b) and formed under the influence of the Hall effect will drive the hydromagnetic flow motion accounted for in this study. R1 The thicknesses defined by dy ¼ 0 ð1 þgÞ dZ as the displaceR1 ment thickness in the tangential direction and by dr ¼ 0 f dZ as the displacement thickness in the radial direction are evaluated as Z 1 L1 dy ¼ ð1 þ gÞ dZ ¼ 2 ,
1045
2
3
4
5
6
7
η Fig. 2. The effects of magnetic strength parameter m and Hall parameter b on the development of scaled current densities (a) Jr 1 and (b) Jy 1, respectively.
the azimuthal stress, over one period of rotation. Moreover, an equal amount of resistance happens at the locations y ¼ cot1 ððL1 L2 Þ=ðL1 þ L2 ÞÞ and y ¼ cot1 ððL1 L2 Þ=ðL1 þ L2 ÞÞ þ p (the branch of cot1 is over ð0, pÞ). The corresponding locations can be easily determined from the above mentioned formulas for any prescribed magnetic field and Hall parameters m and b. In a similar manner, for non-zero a, the locations corresponding to the greatest and smallest stresses can be obtained. The vorticity components ðor , oy , oz Þ existing within the magnetofluid flow can also be computed exactly with the help of Eq. (3.6), which are respectively rffiffiffiffiffi @v R or ¼ ¼ r0 e GZ , @z 2
oy ¼
@u ¼ r0 @z
oz ¼ 2:
rffiffiffiffiffi Re FZ , 2 ð3:7Þ
Fig. 3 demonstrates the effects of normally applied magnetic field and of the Hall current on the radial and circumferential dimensionless angular velocity components for several magnetic field and Hall parameters, for the special case a ¼ y. A gradual decrease
1046
M. Turkyilmazoglu / International Journal of Non-Linear Mechanics 46 (2011) 1042–1048 2
which reduces only to J 2 =s ¼ rm=ð1þ b Þ2 ½ðubðvrÞÞ2 þ ðvr þ buÞ2 . Moreover, by virtue of the form of velocities in (3.2), and also defining the Prandatl number Pr ¼ cp m=k, the energy equation (2.5) can be simplified to
1.7 β=−1 β=1 β=−3 β=3
1.4 m=1
TZZ ¼ r02
1.1 ωr
k
½ðFZ2 þ G2Z Þ þ 2mðF 2 þ G2 þE23 Þ:
ð4:1Þ
Via the replacement Y ¼ ðTT1 Þ=ðTw T1 Þ (for which Tw is the prescribed wall boundary and T1 is the constant temperature far above the disk), defining the Eckert number Ec ¼ r02 =cp ðTw T1 Þ and taking into account the form of the solutions F and G in (3.6), it is possible to rewrite Eq. (4.1) in the form
0 0.8 0.5
YZZ ¼ PrEcL2 e2L1 Z :
0.2 -0.1
m
0
1
2
3
4
5
η 1.1
It should be remarked here that the structure of the Joule heating term (the second term in Eq. (4.1)) does not allow a formation of a uniform temperature at infinity, thus, it has been dropped from Eq. (4.2). Otherwise, keeping it would be only possible if further terms from the convection would have been preserved, which then results in a dependence of the temperature on the other coordinate axes giving rise to a complication of the problem. 4.1. Adiabatic wall
β=−1 β=1 β=−3 β=3
0.7
ð4:2Þ
It this case, Eq. (4.2) is accompanied by the boundary conditions
ωθ
YZ ð0Þ ¼ 0, Yð1Þ ¼ 0: Integrating Eq. (4.2), the solution Y is found to be Y ¼ PrEcðL2 =4L21 Þe2L1 Z , with the constriction Gi ¼ L2 ¼ 0 due to symmetry boundary condition. Hence the temperature distribution is constant everywhere given by Y ¼ 0.
0.3 m=1 -0.1
4.2. Isothermal wall
0 In this case, Eq. (4.2) is subject to the boundary conditions
-0.5
Yð0Þ ¼ 1, Yð1Þ ¼ 0, 0
1
2
3
4
5
η Fig. 3. The effects of normally applied magnetic field and Hall current on the (a) radial or and on the (b) circumferential oy angular velocity components are demonstrated.
can be captured from the figure, but a large gradient becomes unavoidable for or near the wall, specifically when the magnetic field is appreciably effective. Moreover, decreasing Hall number reduces or while increasing oy . The vorticity field acted upon by the Hall effect as depicted in Figs. 3(a) and (b) will drive the viscous hydrodynamic forces within the flow. It should be emphasized that analogous solutions for the Newtonian magnetofluid flow induced by two parallel infinite rotating disks can also be obtained in a similar manner, though not implemented here. Moreover, the solutions obtained in this work are not similarity solutions as given in [1].
4. Heat conducting case Due to the difference in temperature between the wall and the ambient fluid heat transfer takes place. Analysis carried out in the previous section enables us to look for temperature solution of Eq. (2.5) depending only on the normal direction z. As a result, for the flow under consideration the temperature will balance with the dissipation function which reduces only to F ¼ ð@u=@zÞ2 þ ð@v=@zÞ2 , and with the dissipation function due to Joule heating
it is easy to verify that the solution satisfying these conditions is Y ¼ e2L1 Z : Fig. 4 reveals the distribution of temperature profiles Y within the presence of the normal magnetic field and Hall current. Fig. 4(a) is for the near-wall behavior that is significant due to the contribution to the heat transfer. Whereas the effect of magnetic field is seen to decrease the temperature field in Fig. 4, the absolute increase in Hall parameter leads to a noticeable increase in the temperature through the domain of interest. At this stage the heat transfer q from the disk to the fluid can be computed in accordance with the Fourier’s heat law rffiffiffiffiffi @T Re dY q ¼ k ¼ k T Þ , ðT w 1 @z z ¼ 0 2 dZ Z¼0
which results in a dimensionless Nusselt number Nu to measure the heat transfer rate dY ¼ 2L1 : Nu ¼ dZ Z ¼ 0 An immediate conclusion from this formula is that a constant heat transfer takes place on the wall without any dependence upon the azimuthal direction as for the case of axisymmetric von Karman flow [1], but of course varying with the magnetic field and Hall current parameters m and b. In line with Figs. 4 and 5 shows an increase in the rate of heat transfer Nu, as the magnetic strength parameter increases. A further increase in the magnitude of the absolute value of Hall parameter results in a decrease in the heat transfer rate (a faster
M. Turkyilmazoglu / International Journal of Non-Linear Mechanics 46 (2011) 1042–1048
decrease can be observed for negative Hall numbers) since the flow will turn into its hydrodynamic state in the infinite Hall parameter limit. Finally, similar to the flow thicknesses as given in Section 3, the R1 thermal layer thickness defined by dt ¼ 0 Y dZ is evaluated as Z 1 1 dt ¼ Y dZ ¼ , 2L1 0
1 β=−1 β=1 β=−3 β=3
θ
1047
which proves why the thermal layer in connection with the temperature displayed in Fig. 3 gets thinned for stronger magnetic fields and recovers the non-conducting thermal layer thickness value 1/2 for the Hall parameter tending to infinity.
0.9
0 5. Concluding remarks
m=1 0.8
0.02
0
0.04
0.06
0.08
0.1
η 1 β=−1 β=1 β=−3 β=3
0.8
θ
0.6
0.4
0.2
0
m=1
0
1
0
2
3
η Fig. 4. The effects of normally applied magnetic field and Hall current on the development of dimensionless temperature field Y are demonstrated.
10
8
β=0
6 Nu
1 −1
4
3 2
−3
0 0
2
4
6
8
10
m Fig. 5. The variation of Nusselt number Nu against magnetic field strength m is demonstrated for a variety of Hall parameters b.
An exact solution of the motion of incompressible viscous magnetohydrodynamic fluid flow over a non-conducting rotating disk has been obtained which differs from that of corresponding to the classical von Karman’s flow, in that, the physical quantities are allowed to develop non-axisymmetrically, within the no normal flow restriction. The solution is influenced by a fixed point on the disk, whose polar representation is ðr0 , aÞ. The particular case r0 ¼0 is associated with the rigid-body rotation. The non-zero choice of r0 has enabled us to achieve solutions bounded far away from the disk. Solutions in the presence of a uniform magnetic field impressed in the normal direction to the surface of the disk point out that a boundary layer structure develops near the wall, whose far-field behavior is distinct from the near-wall solution. The analytic form of the solution further explains the well-known damping impact of the vertical magnetic field on the motion of swirling flows or similar. The inclusion of the Hall effect has revealed that except for some small positive values of Hall parameter, its contribution is in the way to overwhelm the suppressing effect of magnetic field, which is more pronounced in the case of negative values of the Hall parameter. It has been further shown that a large negative Hall number creates a propelling effect on the flow velocities before returning to their hydrodynamic values in the limit of infinite Hall parameter. The effects of the magnetic field and Hall current on the angular velocity and current density components have also been made clear from the obtained velocity field. Moreover, closed form expressions for the non-vanishing shear stresses on the disk have been worked out, which clearly demonstrate that stresses at the surface of the disk applied against the flow varies with the peripheral angle. In addition to this, the critical locations corresponding to the extremums of the skin frictions have also been determined. Furthermore, the properly defined displacement thicknesses in the radial and tangential directions as well as the thermal layer thickness have been evaluated for the flow under consideration. The assumptions made on the velocity and pressure dictate a balance on the temperature and the dissipation function in the energy equation. Thus, the temperature field within the magnetic field pertinent to the obtained form of solutions has been analytically derived. In addition to this, a heat transfer expression has been obtained using the Fourier’s law via a dimensionless heat transfer rate parameter. This relation reveals that the rate of heat transfer increases for all the magnetic field parameters, though the increase is slowed down by the Hall current effect, approaching its hydrodynamic value in the limit of large Hall numbers. The present solutions are in no doubt closely relevant to the fully laminar flow of an incompressible viscous flow motion of the Newtonian magnetofluid contained between two parallel plates
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M. Turkyilmazoglu / International Journal of Non-Linear Mechanics 46 (2011) 1042–1048
rotating about the same axis of rotation. The particular problem may also cover the flow cases, such as, Couette flow, radial flow and spiral flow. References [1] T.V. Ka´rma´n, Uber laminare und turbulente reibung, Z. Angew. Math. Mech. 1 (1921) 233–252. [2] W.G. Cochran, The flow due to a rotating disk, Proc. Cambridge Philos. Soc. 30 (1934) 365–375. [3] U.T. Boedewadt, Die drehstroemung uber festem grund, Z. Angew. Math. Mech. 20 (1940) 241–253. [4] N. Gregory, J.T. Stuart, W.S. Walker, On the stability of three dimensional boundary layers with applications to the flow due to a rotating-disk, Philos. Trans. R. Soc. London Ser. A 248 (1955) 155–199. [5] G.N. Lance, M.H. Rogers, The rotationally symmetric flow of a viscous fluid in the presence of an infinite rotating disk, J. Fluid Mech. 7 (1960) 617–631. [6] R. Benton Edward, On the flow due to a rotating disk, J. Fluid Mech. 24 (1966) 781–800. [7] B.I. Federov, G.Z. Plavnik, I.V. Prokhorov, L.G. Zhukhovitskii, Transitional flow conditions on a rotating-disk, J. Eng. Phys. 31 (1976) 1448–1453. [8] Y. Kohama, Study on boundary layer transition of a rotating-disk, Acta Mech. 50 (1984) 193–199. [9] P. Hall, An asymptotic investigation of the stationary modes of instability of the boundary layer on a rotating-disk, Proc. R. Soc. London Ser. A 406 (1986) 93–106. [10] D. Dijkstra, P.J. Zandbergen, Von Karman swirling flows, Ann. Rev. Fluid Mech. 19 (1987) 465–491. [11] S.L.G. Jarre, M.P. Chauve, Experimental study of rotating-disk instability. I. Natural flow, Phys. Fluids 8 (1996) 496–508. [12] M. Turkyilmazoglu, Linear absolute and convective instabilities of some twoand three dimensional flows, Ph.D. Thesis, University of Manchester, 1998. [13] H.A. Jasmine, J.S.B. Gajjar, Convective and absolute instability in the incompressible boundary layer on a rotating disk in the presence of a uniform magnetic field, J. Eng. Math. 52 (2005) 337–353. [14] R. Berker, An exact solution of the Navier–Stokes equation the vortex with curvilinear axis, Int. J. Eng. Sci. 20 (1982) 217–230.
[15] K.R. Rajagopal, A class of exact solutions to the Navier–Stokes equations, Int. J. Eng. Sci. 22 (1984) 451–455. [16] K.R. Rajagopal, Swirling flows of viscoelastic fluids, Int. J. Eng. Sci. 30 (1988) 143–149. [17] R.S. Sherman, Viscous Flow, McGraw-Hill, Inc., 1990. [18] E.M. Sparrow, J.L. Gregg, Heat transfer from a rotating disk to fluids of any Prandtl number, J. Heat Transfer 81 (1959) 249–251. [19] N. Riley, The heat transfer from a rotating-disk, Q. J. Mech. Appl. Math. 17 (1964) 331–349. [20] G.K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, 1967. [21] A.D. Polyanin, Exact solutions to the Navier–Stokes equations with generalized separation of variables, Dokl. Phys. 46 (2001) 726–731. [22] K.S. Sai, B.N. Rao, Magnetohydrodynamic flow in a rectangular duct with suction and injection, Acta Mech. 140 (2000) 57. [23] N.W. Vlachakis, S. Tsangaris, D. Kondaxakis, Int. J. Eng. Sci. 44 (2006) 1498. [24] K. Hanif, S. Asghar, T. Hayat, Meccanica 42 (2007) 141. [25] E.M. Sparrow, R.D. Cess, Magnetohydrodynamic flow and heat transfer about a rotating disk, J. Appl. Mech. 29 (1962) 181–187. [26] J.A.D. Ackroyd, On the steady flow produced by a rotating disc with either surface suction of injection, J. Eng. Phys. 12 (1978) 207–220. [27] W.I. Thacker, S.K. Kumar, L.T. Watson, Magnetohydrodynamic flow and heat transfer about a rotating disk with suction and injection at the disk surface, Comput. Fluids 16 (1988) 183–193. [28] A.M. Siddiqui, T. Hayat, S. Asghar, T. Haroon, Acta Mech. 151 (2001) 127. [29] P.D. Ariel, On computation of MHD flow near a rotating-disk, Z. Angew. Math. Mech. 82 (2001) 235–246. [30] M. Turkyilmazoglu, Exact solutions for the incompressible viscous fluid of a porous rotating disk flow, Int. J. Nonlinear Mech. 44 (2009) 352–357. [31] M. Turkyilmazoglu, Exact solutions corresponding to the viscous incompressible and conducting fluid flow due to a rotating disk, Z. Angew. Math. Mech. 89 (2009) 490–503. [32] K. Cramer, S. Pai, Magnetofluid Dynamics for Engineers and Applied Physicists, McGraw-Hill, Inc., 1973. [33] A.L. Aboul-Hassan, H.A. Attia, The flow due to a rotating disk with Hall effect, J. Phys. Lett. 228 (1997) 286–290. [34] H.A. Attia, Effect of Hall current on the unsteady MHD flow due to a rotating disk with uniform suction or injection, Appl. Math. Modelling 25 (2001) 1089–1098. [35] H. Schlichting, Boundary-layer Theory, McGraw-Hill, Inc., 1979.
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International Journal of Non-Linear Mechanics journal homepage: www.elsevier.com/locate/nlm
Exact solutions for the incompressible viscous magnetohydrodynamic fluid of a rotating-disk flow with Hall current M. Turkyilmazoglu Mathematics Department, University of Hacettepe, 06532-Beytepe, Ankara, Turkey
a r t i c l e i n f o
a b s t r a c t
Article history: Received 28 May 2009 Accepted 6 April 2011 Available online 15 April 2011
The focus of the present study is to obtain exact solutions for the flow of a viscous hydromagnetic fluid due to the rotation of an infinite disk in the presence of an axial uniform steady magnetic field with the inclusion of Hall current effect. In place of the traditional von Karman’s axisymmetric evolution of the flow, the rotational non-axisymmetric stationary conducting flow is taken into consideration here, whose governing equations allow an exact solution to develop bounded everywhere in the normal direction to the wall. The three-dimensional equations of motion are treated analytically yielding derivation of exact solutions, which differ from those of corresponding to the classical von Karman’s conducting flow. Making use of this solution, analytical formulas for the angular velocity components, for the current density field as well as for the wall shear stresses are extracted. The critical peripheral locations at which extrema of the local skin friction occur are also determined. It is proved from the analytical results that for the specific flow the properly defined thicknesses decay as the magnetic field strength increases in magnitude, approaching their hydrodynamic value in the limit of large Hall numbers. Interaction of the resolved flow field with the surrounding temperature is further analyzed via the energy equation. The temperature field is shown to accord with the dissipation function. According to the Fourier’s heat law, a constant heat transfer from the disk to the fluid occurs, though it increases by the presence of magnetic field, the increase is slowed down by the Hall effect eventually reaching its hydrodynamic limit. & 2011 Elsevier Ltd. All rights reserved.
Keywords: Exact solutions Magnetohydrodynamic flow Rotating disk Hall effect Heat transfer
1. Introduction A substantial interest in rotating magnetohydrodynamic viscous fluid flow motion has been witnessed in the past few decades, due to the reason that it provides practical applications to many engineering areas, for instance, pumping and levitation of liquid metals. von Karman’s swirling hydrodynamic viscous flow [1] is a well-documented classical problem in fluid mechanics, which has several technical and industrial applications. The original problem raised by von Karman, which is the most studied by researchers in the literature, is the viscous flow motion induced by an infinite rotating disk where the fluid far from the disk is at rest. Then the problem is generalized to include the case where the fluid itself is rotating as a solid body far above the disk. Another generalization is to consider the viscous flow between two infinite coaxial rotating disks. All these problems and also stability issues are attacked, theoretically, numerically and experimentally, by many researchers amongst many others, such as [2–13]. However, all of
E-mail address: [email protected] 0020-7462/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijnonlinmec.2011.04.023
these results are either numerical or analytical-numerical. Moreover, [14] established the existence of an infinite set of solutions to the flow of two infinite parallel plates rotating with the same angular velocity about an axis. Extension to porous plates was given in [15,16], refer also to [17]. Additionally, heat transfer problem over a rotating disk was also studied, see for instance [18,19]. The cases when an exact solution for the Navier–Stokes equations can be obtained are of particular importance in investigations to describe fluid motion of the viscous fluid flows. However, since the Navier–Stokes equations are non-linear in character, there is no known general method to solve the equations in full, nor the superposition principle for non-linear partial differential equations does work. There are some wellknown exact unidirectional or parallel shear flows, a few sample cases contain the steady Poiseuille and Couette flow. In addition to this, there are exact cylindrically symmetric solutions with closed plane streamlines, see for instance [20,21]. Further exact solutions that we already know possess certain feature of the fluid motion, such as rectilinear motion, motions of the duct flows, axisymmetric flows and stagnation flows on plate with slip, etc., see for instance, the works of [22–24].
M. Turkyilmazoglu / International Journal of Non-Linear Mechanics 46 (2011) 1042–1048
The purpose of the current work is to present details of some new three-dimensional solutions of the Navier–Stokes equations governing the steady state stationary viscous flow of an incompressible Newtonian electrically conducting fluid associated with a single rotating disk. The disk is supposed to be non-electrically conducting under the impression of an external magnetic field of constant strength applied normal to the disk. The self-similar axisymmetric solutions in the presence of a transverse magnetic field were previously investigated numerically solving the underlying equations in [25–27] and recently in [28,29]. Additionally, analytic solutions have been recently presented in [30,31] without considering the Hall current effect. However, in the existence of a strong magnetic field Hall current is known to have a marked effect on the magnetic field, see [32–34]. Unlike the case of the classical von Karman’s axisymmetric similarity solution, the present motivation lies in searching for the velocity and pressure fields, affected by the existence of the magnetic field considering the Hall effect. It is shown in this study that under certain well-defined conditions the Navier–Stokes equations reduce to a system of ordinary differential equations whose translation to a complex format enables the generation of solutions in a closed form depending on magnetic field and Hall parameters. The analytical solution is in fact found by imposing on the infinite rotating disk no-slip condition for the velocity field (after setting no normal velocity at all) together with an assumption that the field is bounded with respect to the normal axis. The resulting equations are then used to obtain exact expressions for the angular velocity and current density components, for the displacement thicknesses as well as for the existing wall shear stresses. We further analyze the subject of heat transfer from the disk to the fluid by solving the energy equation, which, for the flow under consideration, is shown to be in balance with the dissipation function and the Joule heating term due to current density of the magnetic field. The effects of uniform magnetic field and Hall current on the development of the flow velocity, vorticity, displacement thicknesses, skin frictions and also heat transfer are fully discussed. The following strategy is pursued in the rest of the paper. In Section 2 the full governing equations representing the magnetohydrodynamic flow motion are outlined in cylindrical coordinates. Section 3 contains the analytical results, including the steady state flow and the physically relevant variables, obtained under self-consistent assumptions. Heat conducting case including the adiabatic and isothermal wall conditions is analyzed in Section 4, in which, an expression for the heat transfer is derived using the Fourier’s heat law. Finally our conclusions follow in Section 5.
2. Formulation of the problem We consider the three-dimensional flow of an incompressible, electrically conducting viscous fluid on an infinite disk which rotates about its axis with a constant angular velocity O. An external uniform magnetic field B ¼ B0 , with the flux density B0 constant, is applied to the system along the direction perpendicular to the disk. The Navier–Stokes equations are non-dimensionalized with respect to a length scale L ¼ re , velocity scale Uc ¼ LO, time scale L=Uc , magnetic scale B0 and pressure scale rUc2 , where r is the fluid density. This leads to a Reynolds number Re ¼ Uc L=n, n being the kinematic viscosity of the fluid. Thus, relative to nondimensional cylindrical polar coordinates ðr, y,zÞ which rotate with the disk, the full time-dependent, unsteady magnetohydrodynamic equations governing the viscous fluid flow are as follows: @u @u v @u v2 @u þu þ þw @t @r r @y r @z
¼
1043
@p 1 2 @v u m þ r2 u 2 2 ½ubðvrÞ, 2 @r Re r @y r 1þb
@v @v v @v uv @v þu þ þw þ @t @r r @y r @z 1 @p 1 2 @u v m þ r2 vþ 2 2 ½ðvrÞ þ bu, ¼ 2 r @y Re r @y r 1þb @w @w v @w @w @p 1 2 þu þ ¼ þ ½r w, þw @t @r r @y @z @z Re
ð2:1Þ
@u 1 @v @w u þ þ ¼ 0, þ @r r @y @z r @T @T v @T @T J2 2 þu þ ¼ k½r T þ mF þ : cp r þw @t @r r @y @z s
ð2:2Þ
The Laplacian operator in cylindrical coordinates is defined as 2 @ 1 @2 @2 1@ : ð2:3Þ r2 ¼ þ 2 2þ 2þ 2 r @r r @y @r @z The parameters appearing in Eqs. (2.1)–(2.2) are, respectively, m ¼ sB20 =ro is the magnetic interaction parameter, cp is the specific heat, k is the thermal conductivity (assumed constant here), m is the viscosity coefficient of the fluid and s is the electrical conductivity of the fluid. Moreover, function F on the right-hand side of Eq. (2.2) represents the effects due to flow dissipation and J 2 =s denotes Joule heating, with J ¼ ðJr ,Jy ,Jz Þ being the current density. Furthermore, it is assumed that the motion of the conducting fluid has a negligible effect on the imposed magnetic field, but the effect of the magnetic field on the fluid motion is strongly felt, shown by the Lorentz force terms J B, at the end of the Eq. (2.1). It is also assumed that the electron–atom collision frequency is relatively large, so that the Hall current effect cannot be neglected revealed in the generalized Ohm’s law by J ¼ s½Eþ u BbðJxBÞ, where E is the electric field which results from charge separation and b is the Hall factor, that, whenever is positive, the magnetic flux points upwards and the electrons of the conducting fluid gyrate in the same sense as the rotating disk, with an opposite situation for negative b, see [34]. In this analysis the fluid is assumed to lie in the z Z0 semiinfinite space. Boundary conditions accompanying the Eqs. (2.1)–(2.2) are such that the fluid adheres to the wall at z¼0 with a prescribed (isothermal) or symmetric (adiabatic) wall temperature, and the quantities are bounded at far distances from the wall, with the temperature being uniformly distributed there. In the classical von Karman flow under the influence of a uniformly applied magnetic field normal to the wall, the rotational symmetry assumption is used which removes the y-dependence of the variables in Eqs. (2.1)–(2.2). However, in the present case we allow the y-dependence, enabling motion of the non-axisymmetric magnetofluid flow to develop, but instead assume w¼0 throughout. It should be remarked that the latter noaxial flow assumption is automatically satisfied by the third of the momentum equations (2.1), provided that the pressure p is not a function of z. The above constraints render the Eqs. (2.1)–(2.2) to a simplified version @u @u v @u v2 þu þ @t @r r @y r @p 1 2 @v u m þ r2 u 2 2 ½ubðvrÞ, ¼ 2 @r Re r @y r 1þb
M. Turkyilmazoglu / International Journal of Non-Linear Mechanics 46 (2011) 1042–1048
@v @v v @v uv þu þ þ @t @r r @y r 1 @p 1 2 @u v m þ r2 vþ 2 2 ½ðvrÞ þ bu, ¼ 2 r @y Re r @y r 1þ b @u 1 @v u þ þ ¼ 0, @r r @y r @T @T v @T J2 2 þu þ cp r ¼ k½r T þ mF þ : @t @r r @y s
G ¼ f ðZÞsinðyaÞ þ gðZÞcosðyaÞ, ð2:4Þ
ð2:5Þ
3. Steady-state mean flow We here restrict ourselves to the stationary hydromagnetic mean flow relative to the rotating disk. Since the energy equation in (2.2) is decoupled from the Eq. (2.1), this section focuses only on the solution of velocities and pressure. Within this framework, it is particularly clear that u ¼ 0,
v ¼ r,
p¼
r2 þ p0 , 2
ð3:1Þ
where p0 is a constant, constitute a solution for (2.4), which is known as the rigid-body rotation. pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Next, via a coordinate transformation Z ¼ ðRe =2Þz, the solution (3.1) is perturbed and we assume a new solution of the form u ¼ r0 Fðy, ZÞ,
v ¼ r þ r0 Gðy, ZÞ,
p¼
r2 rr 0 cosðyaÞ þ p1 , 2
GZZ
2m 1þ b
2
G2 1þ
0.4
2
F ¼ 2sinðyaÞ:
0.3
0.2 1 0.1
0 ð3:3Þ
-0.1
Introducing a new function of the form H¼F þiG transforms the pair of Eq. (3.3) into the subsequent single complex differential equation with real variable " !# m mb HZZ 2 þi 1 þ H ¼ 2ðcosðyaÞisinðyaÞÞ, 2 2 1þb 1þb
0
1
3
4
5
6
7
1.2 β=−1 β=1 β=−3 β=3
m=0
whose solution bounded with respect to Z can be immediately expressed as
0.9 1
After decomposing real and imaginary parts of the solution, F and G from (3.5) are found to be
-g
ð3:5Þ
where C1 is a complex integration constant depending on y. The latter is determined by the no-slip velocity condition on the wall. The constant l in (3.5) corresponding to magnetofluid case satisfies qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 l ¼ ðL1 þ iL2 Þ, L1 ¼ a þ a2 þb2 , L2 ¼ , L 1 ! m mb ,1þ ða,bÞ ¼ : 2 2 1þb 1þb
F ¼ f ðZÞcosðyaÞ þgðZÞsinðyaÞ,
2
η
ð3:4Þ
H ¼ C1 elZ iðcosðyaÞisinðyaÞÞ,
β=−1 β=1 β=−3 β=3
m=0
!
mb 1þb
where the form of the functions f and g in (3.6) is given as f ¼ eL1 Z sinðL2 ZÞ and g ¼ 1 þeL1 Z cosðL2 ZÞ. It can be seen that when m ¼ 0 we have the non-conducting fluid case, and when the magnetic with the absence of the Hall factor pffiffiffiffiffiffiffifield is large together pffiffiffiffiffiffiffi L1 2m and L2 1= 2m. Accounting for the asymptotic behaviors of f and g in the large Z limit in Eq. (3.6), it can be immediately deduced that the velocities far away from the disk turn out to be u ¼ r0 sinðyaÞ and v ¼ r þr0 cosðyaÞ, which differ from the no-slip velocities. The path along which the velocities vanish exactly through the space can be easily determined by setting zero the velocities in Eq. (3.2) with the consideration of (3.6). In the particular case when a ¼ y, the implication is that F ¼f and G ¼g, whose graphs are displayed in Fig. 1. These graphs clearly indicate the development of a boundary layer like structure near the surface of the disk, whose thickness varies with the magnetic strength parameter and Hall parameter (m¼1 for all b). As in the classical von Karman’s axisymmetric flow, see [33], an increase in the magnetic interaction number m provides a considerable reduction in the velocity components f and g for the problem under consideration, essentially due to the damping influence of the magnetic field. A dissimilar contribution, to the
ð3:2Þ
such that, as opposed to the well-known Karman solution, periodic solutions with respect to y of F and G are sought, subjected to a pressure field given by the third of the Eq. (3.2). Moreover, parameters r0 and a correspond to polar representation of a fixed point on the disk surface and p1 is a constant. Substituting equations (3.2) into (2.4), we see that the continuity equation is automatically satisfied, while the momentum equations, with the help of (3.1), give rise to ! 2m mb FZZ F þ 2 1 þ G ¼ 2cosðyaÞ, 2 2 1þb 1þb
ð3:6Þ
f
1044
0.6
0.3
0
0
1
2
3
4
5
6
7
η Fig. 1. The effects of magnetic strength parameter m and Hall parameter b on the development of velocity components (a) f and (b) g, respectively.
M. Turkyilmazoglu / International Journal of Non-Linear Mechanics 46 (2011) 1042–1048
Z 0
1
f dZ ¼
L2 , L2
where L2 ¼ L21 þ L22 . It is no surprising to observe from these exact formulas that the thicknesses decay (see also Fig. 1) as the magnetic effect is considerably large. In addition to this, when the magnetic interaction number is kept fixed, allowing the absolute growth of the Hall parameter b will cause the thicknesses to recover their hydromagnetic correspondence 1/2. Having found the form of the velocities, the components of the stress tensor, which are non-zero, are only the radial tr and tangential ty shear stresses on the wall, and these can be calculated as rffiffiffiffiffi @u Re tr ¼ ¼ r0 ½L2 cosðyaÞL1 sinðyaÞ @z z ¼ 0 2 and
ty ¼
β=−1 β=1 β=−3 β=3
Jr1
-0.2
-0.5 m=0 -0.8 1 -1.1
0
1
@v ¼ r0 @z z ¼ 0
rffiffiffiffiffi Re ½L1 cosðyaÞ þ L2 sinðyaÞ: 2
When a is set to zero, the findings point out the fact that the minimum and maximum skin frictions offered against the flow within the presence of a transversely applied magnetic field take place, respectively at the locations, y ¼ tan1 ðL1 =L2 Þ and y ¼ tan1 ðL1 =L2 Þ þ p (the branch of tan1 is over ðp=2,3p=2Þ) for the radial stress and at the locations y ¼ tan1 ðL1 =L2 Þ and y ¼ tan1 ðL1 =L2 Þ þ p (the branch of tan1 is over ðp=2, p=2Þ) for
2
3
4
5
6
7
η 0.6
β=−1 β=1 β=−3
0.3
β=3
θ
0 1 -0.3 m=0
-0.6
0
1
L
0
dr ¼
0.1
J 1
axisymmetric rotating-disk flow, of the Hall parameter b is apparent in Figs. 1(a) and (b). As concluded in [33], the effect of Hall current on the steady axisymmetric swirling flow depends upon the sign and the absolute magnitude of the Hall parameter. However, as seen from the figure, for all the negative values of Hall parameter b, the effects of Hall current decrease magnetic damping on the flow velocities, as a consequence of which f and g increases. The increase is seen to be more effective in case of negative Hall number than of positive one. Actually, some small positive values of b acts in favor of increasing the magnetic damping as opposed to the negative values. For b growing in negative value, the magnetic field has a propelling effect on the velocity components, which exceed their hydrodynamic values. But, for large positive values of b, they keep increasing while being still less than their hydrodynamic values. It should be noticed that the large limit of b corresponds to the hydrodynamic limit by decreasing the magnetic force. To conclude, the typical impacts of the magnetic field together with the Hall current is better understood from the form of the exact velocity equations (3.6), which need to be resolved numerically for other type of fluid flow phenomena, see [25,35]. The current density components Jr ¼ r0 Jr 1 and Jy ¼ r0 Jy 1 for 2 the hydromagnetic flow, for which Jr 1 ¼ ðbF þ GÞ=ð1 þ b Þ and 2 Jy 1 ¼ ðF þ bGÞ=ð1 þ b Þ are demonstrated in Fig. 2. The effect of increasing the absolute value of Hall parameter is to increase the current Jr1 in the magnetic field, whereas the effect of increasing the Hall parameter is to increase the current Jy 1 followed by an initial decrease. Just an opposite behavior is observed in the current Jy 1 while the Hall parameter decreases, a decrease in the current is followed after an initial increase. The current densities depicted in Figs. 2(a) and (b) and formed under the influence of the Hall effect will drive the hydromagnetic flow motion accounted for in this study. R1 The thicknesses defined by dy ¼ 0 ð1 þgÞ dZ as the displaceR1 ment thickness in the tangential direction and by dr ¼ 0 f dZ as the displacement thickness in the radial direction are evaluated as Z 1 L1 dy ¼ ð1 þ gÞ dZ ¼ 2 ,
1045
2
3
4
5
6
7
η Fig. 2. The effects of magnetic strength parameter m and Hall parameter b on the development of scaled current densities (a) Jr 1 and (b) Jy 1, respectively.
the azimuthal stress, over one period of rotation. Moreover, an equal amount of resistance happens at the locations y ¼ cot1 ððL1 L2 Þ=ðL1 þ L2 ÞÞ and y ¼ cot1 ððL1 L2 Þ=ðL1 þ L2 ÞÞ þ p (the branch of cot1 is over ð0, pÞ). The corresponding locations can be easily determined from the above mentioned formulas for any prescribed magnetic field and Hall parameters m and b. In a similar manner, for non-zero a, the locations corresponding to the greatest and smallest stresses can be obtained. The vorticity components ðor , oy , oz Þ existing within the magnetofluid flow can also be computed exactly with the help of Eq. (3.6), which are respectively rffiffiffiffiffi @v R or ¼ ¼ r0 e GZ , @z 2
oy ¼
@u ¼ r0 @z
oz ¼ 2:
rffiffiffiffiffi Re FZ , 2 ð3:7Þ
Fig. 3 demonstrates the effects of normally applied magnetic field and of the Hall current on the radial and circumferential dimensionless angular velocity components for several magnetic field and Hall parameters, for the special case a ¼ y. A gradual decrease
1046
M. Turkyilmazoglu / International Journal of Non-Linear Mechanics 46 (2011) 1042–1048 2
which reduces only to J 2 =s ¼ rm=ð1þ b Þ2 ½ðubðvrÞÞ2 þ ðvr þ buÞ2 . Moreover, by virtue of the form of velocities in (3.2), and also defining the Prandatl number Pr ¼ cp m=k, the energy equation (2.5) can be simplified to
1.7 β=−1 β=1 β=−3 β=3
1.4 m=1
TZZ ¼ r02
1.1 ωr
k
½ðFZ2 þ G2Z Þ þ 2mðF 2 þ G2 þE23 Þ:
ð4:1Þ
Via the replacement Y ¼ ðTT1 Þ=ðTw T1 Þ (for which Tw is the prescribed wall boundary and T1 is the constant temperature far above the disk), defining the Eckert number Ec ¼ r02 =cp ðTw T1 Þ and taking into account the form of the solutions F and G in (3.6), it is possible to rewrite Eq. (4.1) in the form
0 0.8 0.5
YZZ ¼ PrEcL2 e2L1 Z :
0.2 -0.1
m
0
1
2
3
4
5
η 1.1
It should be remarked here that the structure of the Joule heating term (the second term in Eq. (4.1)) does not allow a formation of a uniform temperature at infinity, thus, it has been dropped from Eq. (4.2). Otherwise, keeping it would be only possible if further terms from the convection would have been preserved, which then results in a dependence of the temperature on the other coordinate axes giving rise to a complication of the problem. 4.1. Adiabatic wall
β=−1 β=1 β=−3 β=3
0.7
ð4:2Þ
It this case, Eq. (4.2) is accompanied by the boundary conditions
ωθ
YZ ð0Þ ¼ 0, Yð1Þ ¼ 0: Integrating Eq. (4.2), the solution Y is found to be Y ¼ PrEcðL2 =4L21 Þe2L1 Z , with the constriction Gi ¼ L2 ¼ 0 due to symmetry boundary condition. Hence the temperature distribution is constant everywhere given by Y ¼ 0.
0.3 m=1 -0.1
4.2. Isothermal wall
0 In this case, Eq. (4.2) is subject to the boundary conditions
-0.5
Yð0Þ ¼ 1, Yð1Þ ¼ 0, 0
1
2
3
4
5
η Fig. 3. The effects of normally applied magnetic field and Hall current on the (a) radial or and on the (b) circumferential oy angular velocity components are demonstrated.
can be captured from the figure, but a large gradient becomes unavoidable for or near the wall, specifically when the magnetic field is appreciably effective. Moreover, decreasing Hall number reduces or while increasing oy . The vorticity field acted upon by the Hall effect as depicted in Figs. 3(a) and (b) will drive the viscous hydrodynamic forces within the flow. It should be emphasized that analogous solutions for the Newtonian magnetofluid flow induced by two parallel infinite rotating disks can also be obtained in a similar manner, though not implemented here. Moreover, the solutions obtained in this work are not similarity solutions as given in [1].
4. Heat conducting case Due to the difference in temperature between the wall and the ambient fluid heat transfer takes place. Analysis carried out in the previous section enables us to look for temperature solution of Eq. (2.5) depending only on the normal direction z. As a result, for the flow under consideration the temperature will balance with the dissipation function which reduces only to F ¼ ð@u=@zÞ2 þ ð@v=@zÞ2 , and with the dissipation function due to Joule heating
it is easy to verify that the solution satisfying these conditions is Y ¼ e2L1 Z : Fig. 4 reveals the distribution of temperature profiles Y within the presence of the normal magnetic field and Hall current. Fig. 4(a) is for the near-wall behavior that is significant due to the contribution to the heat transfer. Whereas the effect of magnetic field is seen to decrease the temperature field in Fig. 4, the absolute increase in Hall parameter leads to a noticeable increase in the temperature through the domain of interest. At this stage the heat transfer q from the disk to the fluid can be computed in accordance with the Fourier’s heat law rffiffiffiffiffi @T Re dY q ¼ k ¼ k T Þ , ðT w 1 @z z ¼ 0 2 dZ Z¼0
which results in a dimensionless Nusselt number Nu to measure the heat transfer rate dY ¼ 2L1 : Nu ¼ dZ Z ¼ 0 An immediate conclusion from this formula is that a constant heat transfer takes place on the wall without any dependence upon the azimuthal direction as for the case of axisymmetric von Karman flow [1], but of course varying with the magnetic field and Hall current parameters m and b. In line with Figs. 4 and 5 shows an increase in the rate of heat transfer Nu, as the magnetic strength parameter increases. A further increase in the magnitude of the absolute value of Hall parameter results in a decrease in the heat transfer rate (a faster
M. Turkyilmazoglu / International Journal of Non-Linear Mechanics 46 (2011) 1042–1048
decrease can be observed for negative Hall numbers) since the flow will turn into its hydrodynamic state in the infinite Hall parameter limit. Finally, similar to the flow thicknesses as given in Section 3, the R1 thermal layer thickness defined by dt ¼ 0 Y dZ is evaluated as Z 1 1 dt ¼ Y dZ ¼ , 2L1 0
1 β=−1 β=1 β=−3 β=3
θ
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which proves why the thermal layer in connection with the temperature displayed in Fig. 3 gets thinned for stronger magnetic fields and recovers the non-conducting thermal layer thickness value 1/2 for the Hall parameter tending to infinity.
0.9
0 5. Concluding remarks
m=1 0.8
0.02
0
0.04
0.06
0.08
0.1
η 1 β=−1 β=1 β=−3 β=3
0.8
θ
0.6
0.4
0.2
0
m=1
0
1
0
2
3
η Fig. 4. The effects of normally applied magnetic field and Hall current on the development of dimensionless temperature field Y are demonstrated.
10
8
β=0
6 Nu
1 −1
4
3 2
−3
0 0
2
4
6
8
10
m Fig. 5. The variation of Nusselt number Nu against magnetic field strength m is demonstrated for a variety of Hall parameters b.
An exact solution of the motion of incompressible viscous magnetohydrodynamic fluid flow over a non-conducting rotating disk has been obtained which differs from that of corresponding to the classical von Karman’s flow, in that, the physical quantities are allowed to develop non-axisymmetrically, within the no normal flow restriction. The solution is influenced by a fixed point on the disk, whose polar representation is ðr0 , aÞ. The particular case r0 ¼0 is associated with the rigid-body rotation. The non-zero choice of r0 has enabled us to achieve solutions bounded far away from the disk. Solutions in the presence of a uniform magnetic field impressed in the normal direction to the surface of the disk point out that a boundary layer structure develops near the wall, whose far-field behavior is distinct from the near-wall solution. The analytic form of the solution further explains the well-known damping impact of the vertical magnetic field on the motion of swirling flows or similar. The inclusion of the Hall effect has revealed that except for some small positive values of Hall parameter, its contribution is in the way to overwhelm the suppressing effect of magnetic field, which is more pronounced in the case of negative values of the Hall parameter. It has been further shown that a large negative Hall number creates a propelling effect on the flow velocities before returning to their hydrodynamic values in the limit of infinite Hall parameter. The effects of the magnetic field and Hall current on the angular velocity and current density components have also been made clear from the obtained velocity field. Moreover, closed form expressions for the non-vanishing shear stresses on the disk have been worked out, which clearly demonstrate that stresses at the surface of the disk applied against the flow varies with the peripheral angle. In addition to this, the critical locations corresponding to the extremums of the skin frictions have also been determined. Furthermore, the properly defined displacement thicknesses in the radial and tangential directions as well as the thermal layer thickness have been evaluated for the flow under consideration. The assumptions made on the velocity and pressure dictate a balance on the temperature and the dissipation function in the energy equation. Thus, the temperature field within the magnetic field pertinent to the obtained form of solutions has been analytically derived. In addition to this, a heat transfer expression has been obtained using the Fourier’s law via a dimensionless heat transfer rate parameter. This relation reveals that the rate of heat transfer increases for all the magnetic field parameters, though the increase is slowed down by the Hall current effect, approaching its hydrodynamic value in the limit of large Hall numbers. The present solutions are in no doubt closely relevant to the fully laminar flow of an incompressible viscous flow motion of the Newtonian magnetofluid contained between two parallel plates
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